model order reduction and controller design...

Model Order Reduction and Controller DesignTechniques

Dr. S. Janardhanan

Contents

1 Introduction to Large Scale Systems 11.1 What are Large Scale Systems ? . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hierarchial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Decentralized Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Large Scale System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Large Scale System Model Order Reduction and Control - Modal AnalysisApproach 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Davison Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Reduced Order Model Using Davison Technique . . . . . . . . . . . . 82.2.2 Alternative Method to Obtain Reduced Order Model through Davison

Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Improved Davison Technique . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 Suboptimal Control Using Davison Model . . . . . . . . . . . . . . . 142.2.5 Control Law Reduction Approach Using Davison Model . . . . . . . . 14

2.3 Chidambara Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Reduced Order Model Using Chidambara Technique . . . . . . . . . . 152.3.2 Suboptimal Control Using Chidambara Model . . . . . . . . . . . . . 162.3.3 Control Law Reduction Approach Using Chidambara Model . . . . . 17

2.4 Marshall Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Reduced Order model by Marshall Technique . . . . . . . . . . . . . 17

2.5 Choice of Reduced Model Order . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Model Order Selection Criterion by Mahapatra . . . . . . . . . . . . 182.5.2 Another Criterion for Order Selection and Mode Selection . . . . . . 20

3 Model Order Reduction and Control - Aggregation Methods 233.1 Aggregation of Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Properties of Aggregated System Matrix . . . . . . . . . . . . . . . . 243.1.2 Error in Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Determination of Aggregation Matrix . . . . . . . . . . . . . . . . . . . . . . 263.3 Modal Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Reduced Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Stability of Feedback System . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Aggregation by Continued Fraction . . . . . . . . . . . . . . . . . . . . . . . 32

4 Large Scale System Model Order Reduction - Frequency Domain BasedMethods 374.1 Moment Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Pade Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Pade Approximation Method for SISO Systems . . . . . . . . . . . . 404.2.2 Modal-Pade Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.3 Pade Approximation for Multivariable Systems in Frequency Domain 434.2.4 Stable Pade for Multivariable Systems . . . . . . . . . . . . . . . . . 454.2.5 Reduction of non-asymptotically stable systems . . . . . . . . . . . . 464.2.6 Time-Domain Pade Approximation for Multivariable Systems . . . . 474.2.7 Time-Domain Modal-Pade Method . . . . . . . . . . . . . . . . . . . 51

4.3 Routh Approximation Techniques . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Routh Approximation Method Using α− β Parameters . . . . . . . . 534.3.2 Routh Approximation Technique Using γ − δ Parameters . . . . . . . 544.3.3 Aggregated Model of Routh Approximants . . . . . . . . . . . . . . . 564.3.4 Optimal Order of Routh Approximant . . . . . . . . . . . . . . . . . 59

4.4 Continued Fraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.1 The Three Cauer Forms . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.2 A Generalized Routh Algorithm . . . . . . . . . . . . . . . . . . . . . 624.4.3 Simplified Models Using Continued Fraction Expansion Forms . . . . 64

5 Large Scale System Model and Controller Order Reduction - Norm BasedMethods 675.1 Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1.1 Norms of Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . 675.1.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 685.1.3 Grammian Matrices and Hankel Singular Values . . . . . . . . . . . . 705.1.4 Matrix Inversion Formulae . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Model Reduction by Balanced Truncation . . . . . . . . . . . . . . . . . . . 735.2.1 Balanced Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.2 Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.3 Steady State Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.4 Reduction of Unstable Systems by Balanced Truncation . . . . . . . . 765.2.5 Properties of Truncated Systems . . . . . . . . . . . . . . . . . . . . 765.2.6 Frequency-Weighted Balanced Model Reduction . . . . . . . . . . . . 81

5.3 Model Reduction by Impulse/Step Error Minimization . . . . . . . . . . . . 825.3.1 Impulse Error Minimization . . . . . . . . . . . . . . . . . . . . . . . 835.3.2 Step Error Minimization . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Optimal Model Order Reduction Using Wilson’s Technique . . . . . . . . . . 905.4.1 Impulse Error / White Noise Error Minimization . . . . . . . . . . . 90

6 Pole Placement Techniques 956.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2 What Poles to Choose ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.2 Interpretation of Responses from Pole-Zero Locations . . . . . . . . . 966.3 Pole Assignment in Single Input Systems . . . . . . . . . . . . . . . . . . . . 97

6.3.1 State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3.2 Optimal State Feedback ( Brief Introduction to LQR ) . . . . . . . . 996.3.3 Static Output Feedback in Single Input Systems . . . . . . . . . . . . 1026.3.4 Dynamic Output Feedback ( SISO Case ) . . . . . . . . . . . . . . . . 103

6.4 Pole Assignment and Placement in Multi-Input Systems . . . . . . . . . . . 1046.4.1 Concepts of Multivariable Systems . . . . . . . . . . . . . . . . . . . 1046.4.2 State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.4.3 Design of Optimal Control Systems with Prescribed Eigenvalues . . . 1086.4.4 Static Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 1166.4.5 Two Time-Scale Decomposition and State Feedback Design . . . . . . 123

7 Fast Output Sampling (FOS) 1277.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Controller Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3 Closed Loop Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.4 Techniques for Determining Fast Output Sampling Controller Gain . . . . . 130

7.4.1 Two Time-Scale Approach for Conditioning of State Feedback Gain F 1307.4.2 Singular Value Decomposition of Measurement Matrix C . . . . . . . 1327.4.3 Approach for Multi-Plant Systems . . . . . . . . . . . . . . . . . . . 132

7.5 An LMI Formulation of the design problem . . . . . . . . . . . . . . . . . . . 1337.6 A Modified Approach for Fast Output Sampling Feedback . . . . . . . . . . 133

8 Periodic Output Feedback (POF) 1358.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358.2 Periodic Output Feedback Controller Deduction . . . . . . . . . . . . . . . . 1358.3 Multimodel Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9 Robust Control of Systems with Parametric Uncertainty 1399.1 Concepts Related to Uncertain Systems . . . . . . . . . . . . . . . . . . . . . 139

9.1.1 Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399.1.2 Hermite-Bieler Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1409.1.3 Kharitonov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1429.1.4 Gerschgorin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1459.1.5 Simultaneous Stabilization of Interval Plant Family Based on Kharitonov

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.2 Bhattacharyya’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1479.3 Jayakumar’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.3.1 Routh Table Based Kharitonov Algorithm in Controller Design . . . 1549.3.2 An Alternative Proof for Existence of a Simultaneously Stabilizing

State Feedback for Interval Systems . . . . . . . . . . . . . . . . . . . 1569.4 Smagina & Brewer’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.4.1 The Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 1579.4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

9.4.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.5 State Feedback for Uncertain Systems Based on Gerschgorin Theorem . . . . 163

A Numerical Problems in Large Scale Systems 171A.1 Davison, Chidambara and Marshall Techniques . . . . . . . . . . . . . . . . 171A.2 Routh and Pade Approximations . . . . . . . . . . . . . . . . . . . . . . . . 172A.3 State and Output Feedback Design . . . . . . . . . . . . . . . . . . . . . . . 173A.4 Periodic Output Feedback and Fast Output Sampling . . . . . . . . . . . . . 174A.5 Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Chapter 1

Introduction to Large Scale Systems

1.1 What are Large Scale Systems ?

A great number of problems are brought about by the present day technology and societal andenvironmental processes which are highly complex and ’large in dimension and stochastic bynature’. The notion of ’large scale’ is highly subjective one in that one may ask : ’How largeis large?’. There has been no accepted definition for what constitutes a large scale system.Many viewpoints have been presented on this issue. One viewpoint has been that a systemis considered large scale if it can be decoupled or partitioned into a number of interconnectedsubsystems or small scale systems for either computational or practical reasons. Anotherviewpoint is that a system is large scale when its dimensions are so large that conventionaltechniques of modeling, analysis, control, design and computation fail to give reasonablesolutions with reasonable computational efforts. In other words a system is large when itrequires more than one controller.

Since the early 1950s, when classical control theory was being established, engineers havedevised several procedures, both within the classical and modern control contexts, whichanalyze or design a given system. These procedures can be summarized as follows.

1. Modeling procedures which consist of differential equations, input-output transfer func-tions and state space formulations.

2. Behavioral procedures of systems such as controllability, observability and stabilitytests and application of such criteria as Routh - Hurwitz, Nyquist, Lyapunov’s secondmethod etc.,.

3. Control procedures such as series compensation, pole placement, optimal control etc.,.

The underlying assumption for all such control and system procedures has been cen-trality i.e., all the calculations based upon system information and the informationitself are localized at a given center, very often a geographical position.

A notable characteristic of most large scale systems is that centrality fails to hold updue to either the lack of centralized computing capability or centralized information. Need-less to say, many real problems considered are large scale by nature and not by choice.

1

2 Large Scale Systems

The important points regarding large scale systems are that their hierarchial (multilevel)and decentralized structures depict systems dealing with society, business, management, theeconomy, the environment, energy, data networks, power networks, space structures, trans-portation, aerospace, water resources, ecology and flexible manufacturing networks to namea few. These systems are often separated geographically, and their treatment requires con-sideration of not only economic costs as is common in centralized systems but also suchimportant issues as reliability of communication links, value of information etc.,. It is forthe decentralized and hierarchial control properties and potential applications that manyresearchers throughout the world have devoted a great deal of effort to large scale systemsin recent years.

1.2 Hierarchial Structures

One of the earlier attempts in dealing with large scale systems was to decompose a givensystem into a number of subsystems for computational efficiency and design simplification.The idea of decomposition was first treated theoretically in mathematical programming byDantzig and Wolfe [1] by treating large linear programming problems possessing specialstructures. The coefficient matrices of such large linear programs often have relatively fewnonzero elements, i.e., they are sparse matrices. There are two basic approaches for dealingwith such problems, ’coupled’ and ’decoupled’. The coupled approach keeps the problem’sstructure intact and takes advantage of the structure to perform efficient computations. The’compact basis triangularization’ and ’generalized upper bounding’ are two such efficientmethods. The decoupled approach divides the original system into a number of subsystemsinvolving certain values of parameters. Each subsystem is solved independently for a fixedvalue of so-called decoupling parameter, whose value is subsequently adjusted by a coordina-tor in an appropriate fashion so that the subsystems resolve their problems and the solutionto the original system is obtained. Recently the decoupled approach has been termed as’multilevel’ or ’hierarchial’ approach. Consider a two level system shown in Fig .(1.1). Atthe first level, N subsystems of the original large scale system are shown. At the second levela coordinator receives the local solutions of the N subsystems, Si, i = 1, 2, · · · , N and thenproduces a new set of interaction parameters ai, i = 1, 2, · · · , N . The goal of the coordinatoris to arrange the activities of the subsystems to provide a feasible solution to the overallsystem.

1.3 Decentralized Control

Most large scale systems are characterized by a great multiplicity of measured outputs andinputs. For example, an electric power system has several control substations, each beingresponsible for the operation of a part of the overall system. This situation arising in acontrol system design is often referred to as decentralization. The designer for such systemsdetermines a structure for control which assigns system inputs to a given set of local con-trollers(stations), which observe only local system outputs. In other words, this approach,called decentralized control, attempts to avoid difficulties in data gathering storage require-ments, computer program debuggings and geographic separation of system components.

1.3 Decentralized Control 3

Figure 1.1: Hierarchial Control


Figure 1.2: Two Controller Decentralized Large Scale System

Fig. (??) shows a two controller decentralized system. The basic characteristic of anydecentralized system is that the transfer of information from one group of sensors or actuatorsto others is quite restricted. For example, in the system of Fig. (??), only the output y1 andthe external input u1 are used to find the control v1 and likewise the control v2 is obtainedthrough only the output y2 and external input u2. The determination of control signals v1

and v2 based on the output signals y1 and y2 respectively is nothing but two independentoutput feedback problems which can be used for stabilization or pole placement purposes. Itis therefore clear that the decentralized control scheme is of feedback form, indicating thatthis method is very useful for large scale linear systems. This is a clear distinction from thehierarchial control scheme, which was mainly intended to be an open loop structure.

In the previous part the concept of a large scale system and two basic hierarchial anddecentralized control structures were briefly introduced. Although there is no universaldefinition of a large scale system, it is commonly accepted that such systems possess thefollowing characteristics

1. Large scale systems are often controlled by more than one controller or decision makerinvolving ’decentralized’ computations.

2. The controllers have different but correlated ’information’ available to them, possiblyat different times.

3. Large scale systems can also be controlled by local controllers at one level whose controlactions are being coordinated at another level in a hierarchial(multilevel) structure.

4. Large scale systems are usually represented by imprecise aggregate models.

1.4 Large Scale System Modeling 5

5. Controllers may operate in a group as a ’team’ or in a conflicting manner with single-or multiple-objective or even conflicting-objective functions.

6. Large scale systems may be satisfactorily optimized by means of suboptimal or nearoptimum controls, sometimes termed as a satisfactory strategy.

An attempt is made here to consider primarily modeling and control of large scale systems.Most of the discussions are focussed on large scale linear, continuous time, stationary anddeterministic systems.

1.4 Large Scale System Modeling

Scientists and engineers are often confronted with the analysis, design and synthesis of real-life problems. The first step in such studies is the development of a ’mathematical model’which can be a substitute for the real problem.

In any modeling task, two often conflicting factors prevail - simplicity and accuracy. Onone hand, if a system model is oversimplified, presumably for computational effectiveness,incorrect conclusions may be drawn from it in representing an actual system. On the otherhand, a highly detailed model would lend to a great deal of unnecessary complicationsand should a feasible solution be attainable, the extent of resulting details may becomeso vast that further investigations on the system behavior would become impossible withquestionable practical values. Clearly a mechanism by which a compromise can be madebetween a complex, more accurate model and a simple, less accurate model is needed. Sucha mechanism is not a simple undertaking. The key to a valid modeling philosophy is to setforth the following outline.

1. The purpose of the model must be clearly defined, no single model can be appropriatefor all purposes.

2. The system’s boundary separating the system and the outside world must be defined.

3. A structural relationship among different system components which would best repre-sent desired or observed effects must be defined.

4. Based on the physical structure of the model, a set of system variables of interest mustbe defined. If a quantity of important significance cannot be labelled, step (3) mustbe modified accordingly.

5. Mathematical descriptions of each system component, sometimes called elementalequations, should be written down.

6. After the mathematical description of each system component is complete, they arerelated through a set of physical laws of conservation (or continuity) and compatibility,such as Newton’s, Kirchoff’s or D’ Alembert’s.

7. Elemental, continuity and compatibility equations should be manipulated and themathematical format of the model should be finalized.


8. The last step to a successful modeling is the analysis of the model and its comparisonwith real situations.

Chapter 2

Large Scale System Model OrderReduction and Control - ModalAnalysis Approach

2.1 Introduction

It is usually possible to describe the dynamics of physical systems by a number of simulta-neous linear differential equations with constant coefficients.

x = Ax + bu

But for many processes ( like chemical plants and nuclear reactors) the order of the matrixA may be quite large. It would be difficult to work with these complex systems in theiroriginal form. In such cases, it is common to study the process by approximating it to asimpler model. For instance, the response of an airplane is quite commonly approximated bya second order transfer function. These mathematical models correspond to approximatinga system by its dominant pole-zeros in the complex plane. They generally require empiricaldetermination of the system parameters. Many different methods have been developed toaccomplish the purpose by estimating the ’dominant’ part of the large system and finding asimpler ( or reduced order) system representation that has its behavior akin to the originalsystem.

2.2 Davison Technique

A structured approach to the model reduction problem was given by E. J. Davison in [2].The method suggests that a large (n× n) system can be reduced to a simpler (l × l) model(l ¿ n) by considering the effects of the l most dominant ( dominant in the sense of beingclosest to instability) eigenvalues alone. The principle of the method is to neglect eigenvaluesof the original system that are farthest from the origin and retain only dominant eigenvaluesand hence dominant time constants of the original system in the reduced order model.

The procedure to obtain the reduced order model can be described thus

7


2.2.1 Reduced Order Model Using Davison Technique

Suppose the original system is represented as

X = AX + Bu, where A = n× n matrix (2.1)

and the new mathematical model is given by

Y = A∗Y + B∗u, where A∗ = l × l matrix (2.2)

where, l < n, and Bu and B∗u are respectively their forcing functions.It can be shown that if x1, x2, · · · , xn are the normalized eigenvectors corresponding to

the eigenvalues λ1, λ2, · · · , λn of the matrix A with Re(λ1) ≥ Re(λ2) ≥ · · · ≥ Re(λn), thenthe transformation

Z = P−1X (2.3)

P =[

v1 v2 · · · vn

]

vi =[

x1,i x2,i · · · xn,i

]T

would transform the system into

Z = AzZ + Bzu

where,

Az = P−1AP,

Bz = P−1B

and Az would be either in the diagonal or the Jordan canonical form. Truncation ofnon-dominant eigenvalues is simpler in this case.

In this case, the state response of the system for an input Bu = Bu, can be shown to be

X =

t∫

0

PeAz(t−τ)P−1Budτ (2.4)

P−1 = [φij], i = 1..n, j = 1..n

which would give

X =n∑

i=1

−1 + eλit

λi

x1,i

x2,i...xn,i

(φi1b1 + φi2b2 + · · ·+ φinbn) (2.5)

If l states that need to be considered for the reduced order model are represented asxc1, xc2, · · · , xcl then the lth order reduced model can be derived as

2.2 Davison Technique 9

A∗ = A∗0 + A∗

1P1P−10 (2.6)

B∗ = P0[P−1B] (2.7)

Y =

xc1

xc2

xcl

(2.8)

where,

A∗0 =

ac1,c1 ac1,c2 · · · ac1,cl

ac2,c1 ac2,c2 · · · ac2,cl...

......

acl,c1 acl,c2 · · · acl,cl

(2.9)

A∗1 =

ac1,1 · · · acl,1

ac1,2 · · · acl,2...

...ac1,c1−1 · · · acl,c1−1

ac1,c1+1 · · · acl,c1+1...

...ac1,c2−1 · · · acl,c2−1

ac1,c2+1 · · · acl,c2+1...

...ac1,cl−1 · · · acl,cl−1

ac1,cl+1 · · · acl.cl+1...

...ac1,n · · · acl,n

T

(2.10)

P0 =

xc1,1 xc1,2 · · · xc1,l

xc2,1 xc2,2 · · · xc2,l...

......

xcl,1 xcl,2 · · · xcl,l

(2.11)

P1 =

x1,1 x1,2 x1,l

x2,1 x2,2 x2,l

xc1−1,1 xc1−1,2 xc1−1,l

xc1+1,1 xc1+1,2 xc1+1,l

xcl−1,1 xcl−1,2 xcl−1,l

xcl+1,1 xcl+1,2 xcl+1,l

xn,1 xn,2 xn,l

(2.12)


[P−1B] = first l rows of P−1B (2.13)

Derivation :

The principle involved in reducing the matrix is to neglect higher-order time constants ofthe system. The following equation is obtained if the first alone are retained from Eqn..(2.4),.i.e., considering only the first l terms of Eqn. (2.5), and taking into account only thel states considered (i.e., equating the other states to zero), the following is obtained.

Y =l∑

i=1ξi

xc1,i

xc2,i...xcl,i

(2.14)

ξi =−1 + eλi

λi

(φi1b1 + φi2b2 + · · ·+ φinbn)

and therefore

ξ1

ξ2...ξl

= P−1

0 Y (2.15)

then

x1

x2...xc1−1

xc1+1...xcl−1

xcl+1...xn

= P1

ξ1

ξ2...ξl

(2.16)

Substituting Eqn. (2.15) in Eqn. (2.16)


x1

x2...xc1−1

xc1+1...xcl−1

xcl+1...xn

= P1P−10 Y (2.17)

Considering the equations for xc1, xc2, · · · , xcl alone from Eqn. (2.1), the following equa-tion can be obtained

Y = A∗0Y + A∗

1

x1

x2...xc1−1

xc1+1...xcl−1

xcl+1...xn

+ P0PlBzu (2.18)

Pl =[

Il 0]

Substituting Eqn. (2.17) in Eqn. (2.18),

Y = A∗0Y + A∗

1P1P−1o + P0[P

−1B]u (2.19)

2.2.2 Alternative Method to Obtain Reduced Order Model throughDavison Technique

The Davison Model can also be computed thusInitially, the system states are rearranged in such a manner that the eigenvectors corre-

sponding to the states to be retained from Eqn. (2.1) are placed first. If P is representedas

P =

[P11 P12

P21 P22

](2.20)

then, if the state vector X is partitioned into dominant and non-dominant parts as X1

(consisting of the states to be retained) and X2 (consisting of the states to be discarded),


and the state and input matrices are also partitioned in appropriate fashion, then X can berepresented as

X =

[X1

X2

]=

[A11 A12

A21 A22

] [X1

X2

]+

[B1

B2

]u (2.21)

X =

[X1

X2

]=

[P11 P12

P21 P22

] [Z1

Z2

], (2.22)

where, Z1 and Z2 are the states of the decoupled system representation. Thus,

X1 = P11Z1 + P12Z2, (2.23)

X2 = P21Z1 + P22Z2. (2.24)

If the decoupled representation of the system is

Z =

[Z1

Z2

]=

[Λ1 00 Λ2

] [Z1

Z2

]+

[Γ1

Γ2

]u (2.25)

where,

[Λ1 00 Λ2

]= P−1AP, (2.26)

[Γ1

Γ2

]= P−1B. (2.27)

The modes in Z2 are non-dominant and therefore can be ignored (according to Davison[2]). Thus, setting Z2 to zero, and substituting in Eqns. (2.23 and 2.24) the following areobtained.

X1 = P11Z1 (2.28)

X2 = P21Z1 (2.29)

X2 = P21P−111 X1 (2.30)

and from Eqn. (2.21)

X1 = A11X1 + A12X2 + B1u (2.31)

Substituting Eqn. ( 2.29) in Eqn. (2.31),

X1 = A11X1 + A12P21Z1 + B1u

X1 =(A11 + A12P21P

−111

)X1 + B1u (2.32)

Eqn. (2.32) gives the reduced order model for the system computed through the alternatemethod. Both models in Eqn. (2.2) and Eqn. (2.32) represent the same system dynamics.


2.2.3 Improved Davison Technique

It was noted that the above said methods were able to reproduce the transient behaviorof the large scale system well. But, as pointed out in [3–5] and rectified in [6], there is anerror in the steady state value of the system response. An improvement in the method wasproposed by Davison in [6]. In the improved Davison Technique, the state equation in Eqn.(2.2) is replaced by

Y = DA∗D−1Y + DB∗u (2.33)

where,

D =

d1

d2

. . .

dl

(2.34)

dj =[A−1B]

j∗[A∗−1B∗]j

, if [A∗−1B∗]j 6= 0

= 1 if [A∗−1B∗]j = 0

j = 1, 2, · · · , l (2.35)

where [A∗−1B∗]j is the jth element of the l vector A∗−1B∗ and [A−1B]j∗ is the element of

the n vector A−1B which corresponds to the jth state retained in the simplified system. Thenew simplified system is equivalent to the following system.

X∗ = A∗X∗ + B∗u (2.36)

Y = DX∗ (2.37)

and so it can be seen that the response of the new system will have correct steady-state values for a step-function input (provided that [A∗−1B∗]j 6= 0), and will still maintain

satisfactory dynamic behavior.. It should be noted that if [A∗−1B∗]j = 0 for some j, thenthe steady-state values of the variable yj may be in error. Variables to be retained in thereduced order model should, therefore, always be chosen so that [A∗−1B∗]j 6= 0.

For the case of multi-input systems, the corresponding model would be

Xi = A∗X∗i + B∗

i ui, i = 1, 2, · · · , r

Y =r∑

i=1DiX

∗i (2.38)

where,

B∗ =[

B∗1 B∗

2 · · · B∗r

](2.39)

and Di, i = 1, 2, · · · , r is determined from Eqns. (2.34 - 2.35), using B∗i in place of B

where

B =[

B1 B2 · · · Br

](2.40)


2.2.4 Suboptimal Control Using Davison Model

For the large scale system represented in Eqn. .(2.1), an optimal controller may be deducedby minimizing the cost function

J =

∞∫

0

(XT QX + uT Ru

)dt (2.41)

where, Q and R are the weights of the states and inputs of the system respectively.Partitioning Q and using Eqn. (2.30),

XT QX = XT1 Q11X1 + 2XT

1 Q12X2 + XT2 Q22X2 (2.42)

= XT1

(Q11 + 2Q12P21P

−111 +

(P−1

11

)TP T

21Q22P21P−111

)X1 (2.43)

Thus,

J = JM =

∞∫

0

(XT

1 QMX1 + uT Ru)dt (2.44)

QM =(Q11 + 2Q12P21P

−111 +

(P−1

11

)TP T

21Q22P21P−111

)(2.45)

If the reduced order system is represented as

X1 = FX1 + Gu (2.46)

then, the suboptimal controller is

u = −R−1GT Ω (2.47)

where, Ω is the solution of the Riccati equation,

ΩF + F T Ω− ΩGR−1GT Ω + QM = 0 (2.48)

2.2.5 Control Law Reduction Approach Using Davison Model

A large scale system (as in Eqn. (2.1)) with a performance criterion described by Eqn. (2.41)can have its optimal control law described as

u∗ = Kx (2.49)

=[

K1 K2

] [X1

X2

](2.50)

Using the Davison model in Eqn. (2.30),

u∗ =(K1 + K2P21P

−111

)X1 (2.51)

2.3 Chidambara Technique 15

2.3 Chidambara Technique

Noting that the basic model of Davison in [2] does not give accurate steady-state response,Chidambara, in his correspondence with Davison ( [3–5]) had suggested an approach formodel order reduction. In this model only the transient response of the left out states areignored (i.e. X2), the steady-state contribution of these states are taken into account inorder to nullify the steady-state error seen in the basic Davison technique.

2.3.1 Reduced Order Model Using Chidambara Technique

For the Chidambara model, the system is represented as in Eqn. (2.21). The transformationP is computed as earlier and represented as in Eqn. (2.20).

Consider the differential equation for Z2 as in Eqn. (2.25).

Z2 = Λ2Z2 + Γ2u (2.52)

Performing Laplace transform on Eqn. (2.52),

(sI − Λ2)Z2(s) = Γ2U(s) (2.53)

Since only the steady-state contribution of Z2 is to be considered, setting s = 0,

−Λ2Z2(s) = Γ2U(s)

Z2 = −Λ−12 Γ2u (2.54)

Now substituting the value of Z2 in the relation in Eqn. (2.24),

X2 = P21Z1 − P22Λ−12 Γ2u (2.55)

Solving for Z1 in Eqn. (2.23) and substituting it in Eqn. (2.55),

X2 = P21P−111 X1 +

(P21P

−111 P12 − P22

)Λ−1

2 Γ2u (2.56)

= LX1 + Hu (2.57)

Substituting Eqn. (2.57) in Eqn. (2.31), the reduced order model of the large scalesystem is obtained using Chidambara technique as

X1 =(A11 + A12P21P

−111

)X1 +

(A12

(P21P

−111 P12 − P22

)Λ−1

2 Γ2 + B1

)u (2.58)

= FX1 + Gu (2.59)


2.3.2 Suboptimal Control Using Chidambara Model

The method of deducing a suboptimal control law for the reduced order Chidambara modelof a system was proposed by Rao and Lamba in [7]. The modified performance criterion ofthe reduced order system reflects the performance criterion desired from the original system.

If the cost function is defined as in Eqn. (2.41) and the original system as in Eqn. (2.21),then as in the case of the Davison model, the value of XT QX can be deduced using Eqn.(2.57) as,

XT QX = XT1 Q11X1 + 2XT

1 Q12(LX1 + Hu) + (XT1 LT + uT HT )Q22(LX1 + Hu)

The cost function can therefore be represented as

JM =

∞∫

0

(XT

1 (Q11 + 2Q12L + LT Q22L)X1 + 2XT1 (Q12H + HT Q22L)u + uT (R + HT Q22H)u

)dt

(2.60)Defining

Q1 = Q11 + 2Q12L + LT Q22L, (2.61)

R1 = R + HT Q22H, (2.62)

S1 = Q12H + HT Q22L, (2.63)

and

u = u + R−11 ST X1, (2.64)

the simplified model represented in Eqn. (2.59) is equivalent to

X1 =(F −GR−1

1 ST1

)X1 + Gu (2.65)

and the performance criterion in Eqn. (2.60) is equivalent to

JM =

∞∫

0

(XT

1 (Q1 − S1R−11 ST

1 )X1 + uT R1u)dt (2.66)

If the matrices R1 and QM =(Q1 − S1R

−11 ST

1

)are positive definite and positive semi-

definite, respectively, then an optimal solution of the problem represented be Eqns. (2.65and 2.66) is given as

u∗ = −R−11 GT ΩX1 (2.67)

where Ω is the solution of the Riccati equation

Ω(F −GR−1

1 ST1

)+

(F −GR−1

1 ST1

)TΩ− ΩGR−1

1 GT Ω + Q1 − S1R−11 ST

1 = 0 (2.68)

Thus

2.4 Marshall Technique 17

u∗ = u∗ −R−11 ST X1

= −R−1(GT Ω + ST

)X1 (2.69)

This optimal controller to the simplified system could also serve as a suboptimal controllerto the original system.

2.3.3 Control Law Reduction Approach Using Chidambara Model

If a control law u has been designed to control the original system then it can be applied onthe reduced order model thus.

Assuming the derived control law is of the form

u = KX

=[

K1 K2

] [X1

X2

]

= K1X1 + K2X2

Using the relation between X2 and X1 described in Eqn. (2.57), the control law can berepresented as

u = (K1 + K2L) X1 + K2Hu (2.70)

⇒ (I −K2H)u = (K1 + K2L)X1

If (I −K2H) is invertible then the control law can be represented for the reduced ordersystem as

u = (I −K2H)−1(K1 + K2L)X1 (2.71)

2.4 Marshall Technique

S. A. Marshall had proposed an alternate way to compute the reduced order model in [8].This technique is quite similar to the Chidambara technique since it too takes into accountthe steady-state values of the X2 states. The difference exists in the manner in which thereduced order state equation is obtained.

2.4.1 Reduced Order model by Marshall Technique

If a large scale system is represented as in Eqn. (2.21) and the transformation that producesa decoupled system be as in Eqn. (2.22), then its reduced order model may be deduced thus.

If the transformation relating the X states to the Z states be given as Q then,

[Z1

Z2

]= Z = P−1X = QX =

[Q11 Q12

Q21 Q22

] [X1

X2

](2.72)


Observing Eqn. (2.25), it can be seen that the submatrix Λ1 is associated with largertime constants of the system, whereas the response of any element in Z2 settles very fast.Thus, it may be approximated as an instantaneous step change. This is the essence of thetechnique.

Mathematically, this approximation is equivalent to putting

Z2 = 0 (2.73)

Substituting Eqn. (2.73) and P−1 = Q in Eqn. (2.25),

Z1 = Λ1Z1 + Γ1u (2.74)

0 = Λ2Z2 + Γ2u

and using Eqn. (2.72) in Eqn. (2.74),

Z2 = Q21X1 + Q22X2 = −Λ−12 Γ2u

X2 = −Q−122 Q21X1 −Q−1

22 Λ−12 Γ2u (2.75)

Substituting Eqn. (2.75) in Eqn. (2.31) and using the relationships between Pij and Qij,one obtains the reduced order model be Marshall technique as

X1 = P11Λ1P−111 X1 +

(B1 − A12Q

−122 Λ−1

2 Γ2

)u (2.76)

2.5 Choice of Reduced Model Order

The methods presented above give a reduced order model that is an approximation of theoriginal system. However, it is not yet clear how small the approximate model can be andyet accurately represent the process. It is seen that the smallness of the approximate modelcan be decided in terms of the largest eigenvalue neglected, the size of the original plant,and the reduced plant. This criterion is judged by comparing the time responses of variouslow order systems.

2.5.1 Model Order Selection Criterion by Mahapatra

A criterion for selecting the model order for a reduced order Davison model was proposed byMahapatra [9]. For a large scale system model in Eqn. (2.1), if the transformation matrixP is partitioned as

P =

[P11 P12

P21 P22

]=

[P1 P2

](2.77)

Mahapatra had made the assumption that the eigenvalues are real, negative and distinct.The initial conditions and the inputs are considered as zero and unit step respectively. Thenthe solution of the system would be

2.5 Choice of Reduced Model Order 19

X(t) = P1

[eΛ1t − I

]Λ−1

1 Γ1 + P2

[eΛ2t − I

]Λ−1

2 Γ2 (2.78)

The approximate solution provided by the reduced order model is

X(t) = P1

[eΛ1t − I

]Λ−1

1 Γ1

Hence, the error involved in ignoring the higher order modes λl+1, · · · , λn in state equationEqn. (2.78) is given by

E(t) = P2

[eΛ2t − I

]Λ−1

2 Γ2 (2.79)

‖E(t)‖ ≤ ‖P2‖ .∥∥eΛ2t − I

∥∥ .∥∥Λ−1

2

∥∥ . ‖Γ2‖ (2.80)

‖exp (Λ2t)− I‖ <√

n− l;∥∥Λ−1

2

∥∥ =

[n∑ (

1

λi

)2

i=l+1

]1/2

∼= 1

|λl+1| (2.81)

‖P2‖ < ‖P‖ ; ‖Γ2‖ < ‖Γ‖ =∥∥P−1B

∥∥ (2.82)

From Eqns. (2.80 - 2.82),

‖E(t)‖ ≤ ‖P‖ . ‖Γ2‖ .

√n− l

|λl+1| , 0 < t < ∞ (2.83)

For the given system in Eqn. (2.1), let

‖P‖ . ‖Γ2‖ = K (2.84)

Therefore,

‖E(t)‖ ≤ K

√n− l

|λl+1| (2.85)

Eqn. (2.85) shows that for a given system, the error in states due to neglecting the higher

modes depends upon

√n− l

|λl+1| . Therefore, the error can be made small when

√n− l

|λl+1| −→ 0

Mahapatra had extended his approach for Marshall’s model [10] and obtained the rela-tionship

‖E(t)‖ ≤ K

√n− l + 1

|λl+1| (2.86)

Using the above equations, it is suggested that a suitable value of l can be selected suchthat the error is within the tolerable limits. This criterion has following drawbacks.

1. It is based on the assumption that the eigenvalues selected are dominant.


2. The assumption which are made earlier are very crude.

3.

√n− l

|λl+1| is monotonically decreasing with increasing l and hence yields the trivial

result l = n as the best choice of l.

Rao et al have suggested an alternative approach to overcome the third drawback,

‖E(t)‖ ≤ KUl

where,

Ul =

√n− l

|λl+1|Vl =

Ul−1

Ul

Upper bound on error ‖E(t)‖ is proportional to Ul and hence Ul gives an idea about theactual error while Vl represents a measure of improvement achieved by increasing the modelorder from l − 1 to l.

However all these criteria are based on the assumption that the eigenvalues retained aredominant.

2.5.2 Another Criterion for Order Selection and Mode Selection

Let a SISO system be given by

X = AX + Bu (2.87)

y = CX

In this case scalar y is the output for evaluation and one is interested in constructing anlth order system whose output is close to that of the system for a given input.

Let λ1, λ2, · · · , λn be distinct and negative real eigenvalues and of these λ1, · · · , λl areretained in the reduced model.

y = CMZ =n∑

i=1

hifi

λi

[eλit − 1

](2.88)

where,M is the modal matrixZ is the transformed matrix (X = MZ)hi are the elements of the row vector h = CMfi are the elements of the column vector f = M−1BOutput of Marshall’s reduced order model can be obtained as

2.5 Choice of Reduced Model Order 21

Z1 =(eΛ1t − I

)Λ−1

1 F1

Z2 = −Λ−12 F2

Using y = CX = CMZ, y, output of the reduced model is obtained as

y =l∑

i=1

hifi

λi

[eλit − 1

]−n∑

i=l+1

hifi

λi

(2.89)

An ISE (Integral Square Error) criterion as a measure of evaluation can be derived as

E =

∞∫

0

e2dt =n∑

i=l+1

n∑

j=l+1

(hihjfifj

λiλj (λi + λj)

)(2.90)

For a given lth order reduced model, the procedure is to select a combination of l eigen-values and determine the ISE. This is repeated for all possible combinations and the one,which gives the least ISE is selected.

Example 1 Let us consider the example of a third order system

X =

0 1 00 0 1−10 −17 −8

X +

001

u

y =[

1 0 0]X

SOLUTION: Modal matrix M is given by

M =

1 1 1−1 −2 −51 4 25

M−1 =1

12

30 21 3−20 −24 −42 3 1

f = M−1B =1

12

3−41

h = CM =[

1 1 1]

λ1 = −1, λ2 = −2, λ3 = −5


Order Eigenvalues Retained ISE−1 1.95

1 −2 32.7−5 10.5

−1,−2 0.02782 −1,−5 6.94

−2,−5 31.3

Thus the result.¥

Chapter 3

Model Order Reduction and Control -Aggregation Methods

3.1 Aggregation of Control Systems

One of the most important techniques for modeling of large scale systems is aggregation. Theconcept of aggregation was first introduced not in control systems, but in economics literatureto address the appropriateness of the analogy between microeconomic and macroeconomicrelationships.. Aggregation of control systems was proposed by Aoki [11]. This aggregationapproach is discussed in the following.

Consider a large scale continuous dynamic system described by the state space equations

X = AX + Bu (3.1)

y = CX

whereX ∈ Rn is the state vectoru ∈ Rm is the control input vectory ∈ Rp is the output vectorThe matrices A,B and C are constant with appropriate dimensions and the triplet

(A,B, C) is completely controllable and observable. We wish to replace the large modeldescription by a satisfactory aggregated model given by

Z = FZ + Gu (3.2)

w = Hz

whereZ ∈ Rr is the aggregated state vectorw ∈ Rp is the aggregated output vectorThe aggregated model description is considered satisfactory if for a given class of inputs

u, the aggregated outputs w are good approximations of the original outputs y of the largemodel. Intuitively, the aggregated model has an order r such that m ≤ r ≤ n.

23


The link between the linear dynamic models in Eqn. (3.1) and Eqn. (3.2) could beestablished by a linear transformation of the form

Z = LX (3.3)

where L is an r × n constant aggregation matrix of rank r. Using the Eqn. (3.3), theequivalence between the original and aggregated models is achieved provided that the con-ditions

FL = LA (3.4)

G = LB (3.5)

Z(0) = LX(0) (3.6)

are satisfied. Since the r × n aggregation matrix L is assumed to be of full rank, it willpossess a pseudo-inverse and therefore a least squares solution of F is

F = LALT(LLT

)−1(3.7)

It is emphasized that the aggregated system matrix F obtained as above is an approxi-mate solution of the earlier equation and depends on the aggregation matrix L.

3.1.1 Properties of Aggregated System Matrix

It is often desirable to choose the dynamic structure of the aggregated system to reflect asignificant portion of the dynamics of the original system in an appropriate sense. In whatfollows, we outline the most important properties of the aggregated system matrix F .

It can be shown that when an aggregation matrix L exists, then the eigenspectrum of Fis contained in that of A.

σ(F ) ⊂ σ(A)

In particular, the matrix F retains some of the characteristic values of A. To see this,let λ1, · · · , λn be the n distinct eigenvalues of A, and let u1, · · · , un be the associatedeigenvectors. Then it follows that

λiui = Aui

Lλiui = LAui

λi (Lui) = F (Lui)

indicating that if Lui 6= 0, then the vector Lui is an eigenvector of F with the sameeigenvalue λi. Now a necessary and sufficient condition for the Eqns. (3.4 - 3.6) to have aunique solution for F is given below.

A necessary and sufficient condition for Eqns. (3.4 - 3.6) to have a unique solution isthat

3.1 Aggregation of Control Systems 25

ρ(AT LT ) ⊆ ρ(LT )

If in addition, Rank(L) = Rank(LA) = r then

η(L) = η(LA)

where ρ(•) and η(•) are the notations for range and null spaces of a matrix.Let us define

(LA)T , α1|α2| · · · |αrαi ∈ Rn

F T = f1|f2| · · · |fr fi ∈ Rr

By transposing Eqn. (3.4), we have

LT fi = αi (3.8)

It is well known that Eqn. (3.8) will have a unique solution for fi if and only if

αi ∈ ρ(LT )

Rank(LT ) = r

By virtue of the rank assumption on L Eqn. (3.4) will have a unique solution for F ifand only if

ρ(AT LT ) ⊆ ρ(LT )

which proves the necessary condition.Furthermore, if Rank(AT LT ) = r, then if spans an r− dimensional subspace and every

row of L can be expressed as a linear combination of the rows of LA. Hence,

ρ(LT ) ⊆ ρ(AT LT )

and from this it follows that

ρ(LT ) = ρ(AT LT )

or equivalently

η(L) = η(LA)

Another property of the aggregated system matrix F is that any polynomial in A, p(A),has p(F ) as its aggregation.

Lp(A) = p(F )L


In particular, if p(A) = [sIn − A]−1 then p(F ) = [sIr − F ]−1 is its aggregation. Where,Ir is the rth order identity matrix. Next, let us derive the transfer function G(s) betweenu(s) and Z(s).

G(s) = L [sIn − A]−1 B

In order for the dynamic exactness to hold, we must also have

G(s) = [sIr − F ]−1 LB

That is, the transfer function matrix must be realizable by either Eqn. (3.1) and Z = LXor by Eqn. (3.2). Since, Z has a lower dimension than X, the state space description definedin Eqn. (3.1) and Z = LX is nonminimal. This situation occurs if and only if there are(n− r) pole-zero cancellations.

In effect, the class of aggregation matrices is restricted to those creating zeros in the input-

output relationshipZ(s)

u(s)that cancel (n − r) poles of the relationship. These cancelled

poles are precisely the eigenvalues of A that are not retained in F.

3.1.2 Error in Aggregation

Let the aggregation error e(t) be defined as

e(t) = Z(t)− LX(t)

e(t) = Z(t)− LX(t)

= FZ(t) + Gu(t)− L(AX(t) + Bu(t))

= FZ(t) + Gu(t)− LAX(t)− LBu(t)

= F (e(t) + LX(t)) + Gu(t)− LAX(t)− LBu(t)

= Fe(t) + (FL− LA)X(t) (3.9)

which reduces to

e(t) = Fe(t)

if FL = LA is satisfied. If Z(0) = LX(0) is satisfied as well, then e(0) = 0 andconsequently the case of perfect aggregation which implies that e(t) = 0 for all t ≥ 0. Onthe other hand, if Z(0) = LX(0) is not satisfied but the aggregation matrix F is chosen tobe asymptotically stable matrix, then e(t) → 0 as t → ∞. In this case, we say that perfectaggregation is obtained asymptotically.

3.2 Determination of Aggregation Matrix

From the preceding section, it is clear that the aggregation matrix L constitutes the setof primary design parameters in constructing the aggregated models. Denoting the n−dimensional row vectors of L by

LT

i

, i = 1, · · · , r, we can express L as

3.2 Determination of Aggregation Matrix 27

L =[

LT1 LT

2 · · · LTr

]T(3.10)

Then the ith element of Z, Zi is given by LTi X. This means that Zi is the weighted sum

of some components of X. Given the freedom to select the elements of L, such that L has atmost one entry in each column, then the n components of X can be grouped into at most rseparate clusters. In this way, the vectors

LT

i

, 1 ≤ i ≤ r are mutually orthogonal which

implies that L has a minimal rank. This procedure constitutes a method of determining theaggregation matrix. Despite its simplicity it is rather arbitrary and essentially depends onthe physical model set up. We note that this procedure involves projecting the state vectorX into an r− dimensional subspace.

An alternative method to compute the matrix L can be developed by considering thecontrollability matrices of the systems defined in Eqns. (3.1 and 3.2). Define

WA ,[

B AB · · · An−1B]

(3.11)

WF ,[

F FG · · · F n−1G]

(3.12)

then from Eqns. (3.4 - 3.6),

LWA = WF (3.13)

Thus, using the pseudo-inverse, L can be obtained by

L = WF W+A

= WF

(W T

A

[WAW T

A

]−1)

(3.14)

By controllability assumptions WA is of full rank n. Thus, by specifying F = diag λ1, · · · , λrand choosing G so as to make Eqn. (3.2) completely controllable, Rank(WF ) = r, and Lcan be computed via Eqn. (3.14).

Example 2 (From [12])In order to illustrate the aggregation procedure, let us consider thefollowing system

X =

−1 0 0.01 0.05 0.250 −4 0 0.45 0.1−0.088 0.2 −10 0 0.221 0 0.075 −4 0.050.11 0.2 0.999 0.44 −3

X +

1 0.50 10.5 0.92 0.751 1

u

The object is to get a third order aggregated model.

SOLUTION: The system eigenvalues are:−10.03,−0.952,−0.2996,−4.073,−3.95 . Themodes to be retained are the average of the first and fourth modes, the second mode andthe average of the third and fifth modes. This gives L of the form


L =

0.5 0 0 0.5 00 1 0 0 00 0 0.5 0 0.5

Accordingly, the aggregated model has

F =

−1.975 0 0.19250.45 −4 0.10.231 0.2 −5.9

G =

1.5 0.6260 10.75 0.95

Using X(0) =[

0.5 0 0.25 0 −0.5]T

, C =[

0.5 1 0.5 0.5 0.5], Z(0) is ob-

tained as Z(0) =[

0.25 0 −0.125]T

.The comparative responses of the original and aggregated models are found to match

well. (See [12], pp. 115-117)¥

3.3 Modal Aggregation

3.3.1 Reduced Order Model

Consider a continuous time, linear dynamic plant represented as

X = AX + Bu

Where X is n− vector, u is m− vector and the matrices A and B are of appropriatedimensions. For the following discussion it is assumed that A has distinct eigenvalues withnegative real parts. Although it is not explicitly shown, yet the results presented here arealso applicable for the case when A has repeated eigenvalues. Let a simplified model berepresented by

Z = FZ + Gu

where Z is r− vector and r < n. Let the simplified state vector Z and the original statevector X be related via an aggregation matrix C where by

Z = CX

Then the matrices F and G are given by

F = CACT (CCT )−1; G = CB (3.15)

Let

3.3 Modal Aggregation 29

X = MY

where M is the modal matrix of A with its columns arranged from left to right in theorder of decreasing magnitudes of the corresponding eigenvalues. Then we get

Y = ΛY + Γu

where

Λ = M−1AM ; Γ = M−1B (3.16)

Assuming that the first r dominant eigenvalues are to be retained in the simplifiedmethod. Let

W = TY

T = [Ir : 0]r×n (3.17)

and where W is an r− vector of the simplified model in modal form.

W = TΛT T W + TΓu

In order to convert the modal representation of the simplified form into a general form,we utilize a reduced dimensional variation of X = MY., namely

Z = M0W (3.18)

The transformation matrix M0 is obtained in the following manner. Let the first rcolumns of the matrix M be represented by

M =

u11 u12 · · · u1r

u21 u22 · · · u2r...

......

un1 un2 · · · unr

If on physical grounds certain specific state variables are to be retained in the simplifiedmodel, then the matrix M0 is written directly from the above matrix. If, for example, statevariables x1, x4, xn−1 are to be retained in the model, then

M0 =

u11 u12 u13

u41 u42 u43

u(n−1)1 u(n−1)2 u(n−1)3

Consequently,

Z = M0TΛT T M−10 Z + M0TM−1Bu (3.19)


The r− dimensional state vector Z is given by

Z = M0W = M0TM−1X (3.20)

Thus the aggregation matrix is given by

C = M0TM−1 (3.21)

Example 3 Consider a third order system

x =

0.5 0.5 00 1 00.833 −2.1167 −0.333

x +

112

u

It is desired to find a reduced order model by modal aggregation. (From [13])

SOLUTION: The eigenvalues of A are λA = 0.5, 1,−0.333 , which indicates thatthe system is unstable. The modal matrix M can be found out to be

M =

0.5774 0.7071 00.5774 0 0−0.5774 0.7071 1

M0 =

[0.5774 0.70710.5774 0

]; M−1

0 =

[0 1.73211.4142 −1.4142

]

T =

[1 0 00 1 0

]

The resulting aggregated model using modal aggregation is given by

Z =

[0.5 0.50 1

]Z +

[11

]u

which has retained two eigenvalues with positive real parts.It is noted that this aggregated model is also unstable. It is a point worth to note that

the aggregation matrix is determined by a defined procedure in modal aggregation, whereasit is on an ’ad hoc’ basis in the case of exact aggregation. Thus, it may be said that modalaggregation is a special case of exact aggregation. The argument is handled in detail withthe above example in ( [13], pp. 12-19).¥

3.3.2 Stability of Feedback System

Let the optimal control policy for the simplified model be

u = −KZ

If this control law is applied to the higher order plant, the resultant feedback system isgiven by

3.3 Modal Aggregation 31

u = −KZ = −KCX

X = AX + Bu

= (A−BKC) X

Now applying the control law u = −KCMW to the transformed system

W = ΛW + Γu

= (Λ− ΓKCM) W

The eigenvalues of (A−BKC) and (Λ− ΓKCM) are identical. Consider the matrix

(Λ− ΓKCM) =

[Λ1 00 Λ2

]− ΓKM0TM−1M

=

[Λ1 00 Λ2

]− ΓKM0

[Ir 0

]

=

[Λ1 00 Λ2

]−

[D1

D2

] [Ir 0

]

=

[Λ1 00 Λ2

]−

[D1 0D2 0

]

=

[Λ1 −D1 0−D2 Λ2

]

Eigenvalues of (Λ− ΓKCM) is the disjoint sum of the eigenvalues of (Λ1 −D1) and Λ2.

Now the eigenvalues of (F −GK)

(F −GK) = M0Λ1M−10 −M0TM−1BK

= M0Λ1M−10 −M0TΓK

= M0Λ1M−10 −M0TΓKM0M

−10

= M0Λ1M−10 −M0

[Ir 0

] [D1

D2

]M−1

0

= M0Λ1M−10 −M0D1M

−10

= M0 (Λ1 −D1) M−10

It is obvious that eigenvalues of (Λ1 −D1) and M0 (Λ1 −D1) M−10 are the same.

Thus we have shown that the eigenvalues of (A−BKC) are the sum of the eigenvaluesof the matrix (F −GK) and the nondominant eigenvalues of the matrix A. The resultantfeedback matrix (A−BKC) is asymptotically stable.


3.4 Aggregation by Continued Fraction

One of the more popular methods for large scale systems order reduction has been the’continued fraction’ technique first introduced by Chen and Shieh [14] and extended bymany others.

The original technique (which would be handled in later chapters) is based on a Taylorseries expansion of the systems closed loop transfer function about s = 0. Our objectivehere is to use the continued fraction technique to obtain a reduced order model for largesingle-input single-output linear time invariant systems which falls under the concept ofaggregation.

Consider

x = Ax + Bu (3.22)

y = Cnx

where, without loss of generality, the matrix A is assumed to be in companion form

A =

0 1 0 · · · 00 0 1 · · · 0...

......

. . ....

0 0 0 · · · 1−a11 −a12 −a13 · · · −a1n

B =[

0 0 · · · 1]T

, Cn =[

a21 a22 a2n

]

Chen and Shieh have shown that the above system can be transformed to an aggregatedform using a transformation matrix P, corresponding to its continued fraction expansion,i.e.,

q = Hq + Ku (3.23)

v = Cqq (3.24)

where the transformed vector q is

q = Px (3.25)

(Note that it is NOT x = Pq, as in the previous cases)

Matrix P is obtained through the modified Routh-Hurwitz array

3.4 Aggregation by Continued Fraction 33

a11 a12 a13 a14 a15 · · · a1n 1

a21 a22 a23 a24... · · · a2n 0

a31 a32 a33... · · · 1

a41 a42... · · · 0

a51...

...a2n−1,1 1a2n,1 01

whose first two rows are extracted from the nth row of A and elements of output vectorCn. The remaining rows are calculated from the common Routh-Hurwitz iterative formula.

aij =ai−1,1.ai−2,j+1 − ai−1,j+1.ai−2,1

ai−1,1

The matrix P is then extracted from the table as

P =

a31 a32 a33 · · · · · · 10 a51 a52 · · · · · · 10 0 a71 · · · · · · 1...

......

...0 0 0 · · · a2n−1,1 10 0 0 · · · 0 1

(3.26)

The system in Eqn. (3.22) can be transformed as

q = PAP−1q + PBu (3.27)

which indicates that matrices H and K are

H = PAP−1

K = PB

The continued fraction expansion simplification of the system from nth to lth order cor-responds to retaining the first l variables of q. Let the first l elements of q be called z, thenit is clear that

z = Rq (3.28)

R = [Il : 0]

Now,


z = Rq

= R(Hq + Ku)

= RHq + RKu

which, when compared to the derived aggregated system state equation

z = Fz + Gu

= FRq + Gu

leads to

FR = RH = RPAP−1

G = RK = RPB

Using Eqns. (3.25 and 3.28),

z = Rq = RPx = Cx

where, C is the l × n aggregation matrix.A relation involving the aggregated system output matrix Cl from the corresponding

equation (w = Clz) can be obtained by equating y and w, which yields

Cn = ClC

Now, using the pseudo-inverse of matrices R and C, the aggregated system matrices are

F = RHR+ = RPAP−1RT(RRT

)−1(3.29)

G = RPB (3.30)

Cl = CnC+ = CnCT (CCT )−1 (3.31)

with the aggregation matrix expressed by

C = RP (3.32)

The proposed method is very convenient for computational purposes and its effectivenessis examined by an example.

Example 4 Consider a fourth order system in companion form

x =

0 1 0 00 0 1 00 0 0 1−120 −180 −102 −18

x +

0001

u

y =[

120 90 24.8 1.4]x

It is desirable to find a second order aggregated model.

3.4 Aggregation by Continued Fraction 35

SOLUTION: The transformation matrix P based on the Routh-Hurwitz array, and Rare

P =

90 77.2 16.6 10 95.76 17.07 10 0 13.2 10 0 0 1

R =

[1 0 0 00 1 0 0

]

From this the aggregated system and aggregation matrices are

F =

[ −1.34 0.1351−1.34 −0.8048

]; G =

[11

]

Cl =[

13.35 −1.274]; C =

[90 77.2 16.6 10 85.76 17.07 1

]

The comparison between the unit step responses of the original and aggregated systemsin Fig. (3.1) shows remarkable degree of coincidence.¥


Figure 3.1: Comparison of Full and Aggregated System Responses

Chapter 4

Large Scale System Model OrderReduction - Frequency Domain BasedMethods

4.1 Moment Matching

This method is based on determining a set of time functions for the full model and matchingthem to a simple model by choosing a number of appropriate parameters without havingto obtain the full model’s time or frequency responses [15]. The technique is essentially amatch of time-moments of the full model’s impulse response to those of the reduced model.Consider an nth order transfer function of the large scale system

G(s) =a21 + a22s + · · ·+ a2,m+1s

m

1 + a12s + a13s2 + · · ·+ a1,n+1sn,m ≤ n (4.1)

It is desired to find a lower order transfer function for this system. It can be done in thefollowing manner.

G(s) =

∞∫

0

g(t)e−stdt =

∞∫

0

g(t)

1− st

1!+

(st)2

2!− · · ·

dt

=

∞∫

0

g(t)dt− s

∞∫

0

tg(t)dt + s2

∞∫

0

t2g(t)dt− · · ·= c0 + c1s + c2s

2 + · · · (4.2)

where

ci =(−1)n

i!

∞∫

0

tig(t)dt =(−1)n

i!Mi (4.3)

where Mi is the ith time-moment of impulse response g(t). Direct division of Eqn. (4.1)yields

37


G(s) = a21 − a31s + a41s2 − a51s

3 + · · · (4.4)

In Eqn. (4.4), a21 is the zeroth-term coefficient of the numerator in Eqn. (4.1), and theremaining coefficients are obtained from the following recursion:

ak,l = ak−1,1.a1,l+1 − ak−1,l+1.a1,1 (4.5)

for k = 3, 4, · · · , and l = 1, 2, · · · , Note that once a Routhian array is formed based onEqn. (4.5), the jth moment Mj can be obtained by multiplying aj+2,1 with j!. The expansioncoefficient cj is then obtained by cj = (−1)jaj+2,1.

Let the full model be given by Eqn. (4.1); then by using Eqns. (4.2 - 4.5),Eqn. (4.6) canbe derived as

c0

c1

c2...cm

cm+1

cm+2...cm+n

=

0 0 · · · 0 0 · · · 0−c0 0

−c1 −c0. . . 0

.... . .

......

......

−cm−1 −cm−2 · · · −c0 0 0 0 · · · 0−cm −cm−1 · · · −c1 −c0 · · · · · · · · · 0−cm+1 −cm · · · −c1 · · · 0...

......

−cm+n−1 −cm+n−2 · · · 0 · · · 0

×

a12

a13

a14...a1,n+1

00...0

+

a21

a22

a23...a2,m+1

00...0

(4.6)

The (n + m + 1)-dimensional vector relation in Eqn. (4.6) can be rewritten in partitionedform as

[C1

C2

]=

[C11 0C21 C22

] [A1

0

]+

[A2

0

](4.7)

where C11, C21, C22 are (m+1)×n, n×n, n×(m+1) matrices respectively, and Ci, A1, i =1, 2 are vectors of (m + 1)st and nth dimension defined by Eqn. (4.6). Using Eqn. (4.7) andsolving for A1 and A2, one gets

4.1 Moment Matching 39

A1 = C−121 C2

A2 = C1 − C11C−121 C2 = C1 − C11A1 (4.8)

The submatrix C21 is generally nonsingular, and its singularity means that the givenset of moments can be matched with a simpler model. The following example explains themoment matching method.

Example 5 Consider a third-order asymptotically stable system with a transfer function

G(s) =s2 + 13s + 40

s3 + 13s2 + 32s + 20=

2 + 0.65s + 0.05s2

1 + 1.6s + 0.65s2 + 0.05s3

SOLUTION : The closed-loop poles of the system are at −1,−2 and −10. It is thereforereasonable to seek a second-order reduced model. Following the above discussions, theRouthian array of G(s) becomes

1 1.6 0.65 0.052 0.65 0.052.55 1.25 0.12.83 1.55752.9705...

which indicates that the first few expansion coefficients cj, j = 0, 1, · · · are c0 = 2, c1 =−2.55, c2 = 2.83, c3 = −2.9705, etc. The denominator and numerator coefficients of thereduced order model are obtained from Eqn. (4.8), i.e.e

A1 = C−121 C2 =

[ −c1 −c0

−c2 −c1

]−1 [c2

c3

]

=

[2.55 −2−2.83 2.55

]−1 [2.83−2.9705

]

=

[1.51440.5171

]

A2 = C1 − C11A1 =

[c1

c0

]−

[0 0−c0 0

] [a12

a13

]

=

[2−2.55

]−

[0 0−2 0

] [1.51440.5171

]

=

[20.48

]

Therefore, the second order reduced model is

R2 =a21 + a22s

1 + a12s + a13s2=

2 + 0.48s

1 + 1.5144s + 0.5171s2


which provides a pair of dominant poles at s1,2 = −0.1,−1.925, which indicates that astable second order reduced model results.

This result is not always obtainable. Moment matching is known for resulting in unstablereduced order models for stable full models and vice versa.¥

4.2 Pade Approximation Methods

4.2.1 Pade Approximation Method for SISO Systems

This approach stems from the theory of Pade [16] and was later used for model reductionby Shamash [17]. Before a formal presentation of the method is done, consider the followingdefinition.

Definition 1 Consider a function

f(s) = c0 + c1s + c2s2 + · · · (4.9)

and a rational function Um(s)Vn(s)

, where Um(s) and Vn(s) are mth and nth order polynomials

in s respectively, and m ≤ n. The rational function Um(s)Vn(s)

is said to be a Pade approximant

of f(s) if and only if the first (m+n) terms of the power series expansions of f(s) and Um(s)Vn(s)

are identical.

For the function f(s) in Eqn. (4.9) to be approximated, let the following Pade approxi-mant be defined.

Un(s)

Vn(s)=

a0 + a1s + a2s2 + · · ·+ an−1s

n−1

b0 + b1s + b2s2 + · · ·+ bn−1sn−1 + sn(4.10)

For the first (m + n) terms of Eqn. (4.9) and Eqn. (4.10) to be equivalent, it becomesapparent that the following set of relations must hold:

a0 = b0c0 (4.11)

a1 = b0c1 + b1c0

a2 = b0c2 + b1c1 + b2c0

...

an−1 = b0cn−1 + b1cn−2 + · · ·+ bn−1c0

0 = b0cn + b1cn−1 + · · ·+ bnc0 (4.12)...

0 = b0c2n−1 + b1c2n−2 + · · ·+ bn−2cn + cn−1

Once the coefficients ci, i = 0, 1, 2, · · · are found out using Eqn. (4.5) and cj = (−1)jaj+2,1,for the full model,

4.2 Pade Approximation Methods 41

G(s) =d0 + d1s + · · ·+ dm−1s

m−1

e0 + e1s + · · ·+ em−1sm−1 + emsm(4.13)

Eqns. (4.11 and 4.12) can be written in vector form as

cn cn−1 · · · c1

cn+1 cn cn−1 · · · c2

cn+2 cn+1 cn · · · c3...

. . .

cn

c2n−1 c2n−2 · · · cn

b0

b1

b2...bn−2

bn−1

=

−c0

−c1

−c2...−cn−2

−cn−1

(4.14)

c0 0 · · · 0c1 c0 0 · · · 0c2 c1 c0 0 · · · 0...

.... . . 0

...0

cn−1 cn−2 · · · c1 c0

b0

b1

b2...bn−2

bn−1

=

a0

a1

a2...an−2

an−1

(4.15)

It must be noted that in the above reformulation of Eqns. (4.11 and 4.12),bn = 1.

Example 6 Consider the sixth order system

G(s) =9 + 23.25s + 30.2s2 + 22.25s3 + 9s4 + s5

15 + 69.5s + 119s2 + 100s3 + 45s5 + s6

We would like to apply the Pade approximation method to find a reduced order model.

SOLUTION: A power series expansion of G(s) would result in

G(s) = 0.60− 1.230s + 2.95566s2 − 6.45325s3 + 13.45741s4 − 25.54s5

+46.4s6 − 85s7 + 160s8 − 307s9

Using the coefficients ci, and forming the matrices in Eqns. (4.14 and 4.15), the resultsobtained for a second-, third- and fourth-order reduced model are given as below

k th Order Reduced Model, Rk(s) Closed-Loop Poles

R2(s) =−0.19543 + 0.2238s

−0.3257− 0.2947s + s2 0.7368077,−0.442077

R3(s) =−0.16− 0.389s + 0.095s2

−0.266− 1.194s− 0.98s2 + s3 1.75,−0.385± j0.064

R4(s) =−0.066− 0.108s− 0.0986s2 + 0.265s3

−0.11− 0.4s− 0.45s2 + 0.32s3 + s4 0.88,−0.355± j0.358,−0.5

It can be seen that all the reduced order models are unstable ( whereas the full modelis a stable one). In order to obtain a stable reduced order model, preassign the first pole at


s = s1 = −3, corresponding to one of the full model’s original poles. Under these conditions,since s1 is a pole of the reduced system, the last equation is replaced by

0 = b0 + b1s1 + b2s21 + · · ·+ bnsn

1

which, in this particular example becomes

0 = b0 − 3b1 + 9b2 − 27b3 + 81

Using this new equation and solving for a1 and bi, the fourth order reduced model obtainedwould be

R′4(s) =

1.6× 10−6 + 1.436s + 1.34s2 + 1.358s3

2.1× 10−6 + 2.39s + 7.1404s2 + 5.114s3 + s4

Since there is a pole-zero cancellation (10−6 ' 0) in the above transfer function, a thirdorder reduced model results

R′3(s) =

1.436 + 1.34s + 1.358s2

2.39 + 7.1404s + 5.114s2 + s3

which is stable with a pole at s = −3.¥

Remark 1 Assume that the reduced order model R(s) of order k is required which retainsthe pole at s = s1 say,

Then the Pade approximant uses the concept of Pade approximation about more than onepoint and the last equation is replaced by the equation

0 = b0 + b1s1 + b2s21 + · · ·+ bks

k1

4.2.2 Modal-Pade Method

It can be seen that the Pade approximation method does not always give a reliable approx-imation of a high order system. Hence, it is generally recommended to combine the Pademethod with other approximation methods so that there is partial moment matching in theapproximation along with retention of stability.

Now suppose that the reduced order model R(s) is required to retain k dominant poles ofthe higher order system. Further suppose that the n dominant poles are known, then R(s)can be written as

R(s) =a0 + a1s + a2s

2 + · · ·+ an−1sk−1

(s + s1) (s + s2) · · · (s + sk)(4.16)

where the bi (i = 0, 1, · · · , k − 1) may be computed in terms of s1, · · · , sk . Then if R(s)is to approximate G(s) in the Pade sense, about s = 0, then the ai may be determined usingEqn. (4.15).


Pade Approximation using Routh Stability Criterion

One way of overcoming the instability problem is by comparing the denominator of R(s)such that its alpha parameters are the same as the first k alpha parameters of G(s). Thefirst k alpha parameters of G(s) may be determined as follows [17]

Let

Q(s) =1

1 +e0 + e2s

2 + · · ·e1s + e3s

3 + · · ·=

1

1 + α11

s+ 1

α21

s+

. . .1

αn1

s

The αi are termed as the ’alpha parameters ’ of the full model. The denominator of thereduced order transfer function is obtained by retaining the first k alpha parameters. Thedenominator of R(s) is thus given by the denominator of the truncated continued fraction.

Qk(s) =1

1 + α11

s+ 1

α21

s+

. . .1

αk1

s

Having obtained the denominator of R(s), the bi one known and hence the ai may beobtained by solving the first k equations.

4.2.3 Pade Approximation for Multivariable Systems in FrequencyDomain

The frequency domain method for Pade approximation can be extended to multivariablesystems [18]. Let us first consider that a second order model has to be derived for an nth

order m input - m output system. The method is only applicable for equal number of inputsand outputs.

Let the higher order system be expanded in the following manner.

G(s) =∞∑i=0

Cisi = C0 + C1s + C2s

2 + · · · (4.17)

LetR(s) =

[L0 + L1s + Is2

]−1[M0 + M1s] (4.18)


Now

[L0 + L1s + Is2

]−1=

[L0

(I2 + L−1

0 L1s + L−10 s2

)]−1

=(I + L−1

0 L1s + L−10 s2

)−1L−1

0

= I − L−10 L1s− L−1

0 s2 + L−10 L1L

−10 L1s

2

+2L−10 L1L

−10 s3 − L−1

0 L1L−10 L1L

−10 L1s

3 + . . .

= L−10 − L−1

0 L1L−10 s− L−1

0 L−10 s2 + L−1

0 L1L−10 L1L

−10 s2

+2L−10 L1L

−10 L−1

0 s3 − L−10 L1L

−10 L1L

−10 L1L

−10 s3 + . . .

Truncating higher order terms, R(s) can be represented as

R(s) = (L−10 − L−1

0 L1L−10 s− L−1

0 L−10 s2 + L−1

0 L1L−10 L1L

−10 s2 (4.19)

+2L−10 L1L

−10 L−1

0 s3 − L−10 L1L

−10 L1L

−10 L1L

−10 s3) (M0 + M1s)

= L−10 M0 +

(L−1

0 M1 − L−10 L1L

−10 M0

)s

+((

L−10 L1

)2L−1

0 M0 − L−10 L1L

−10 M1 −

(L−1

0

)2M0

)s2 +

((L−1

0 L1

)3L−1

0 M1

)s3

Comparing Eqn. (4.17) and Eqn. (4.19),

C0 = L−10 M0

M0 = L0C0 (4.20)

C1 = L−10 M1 − L−1

0 L1L−10 M0

L0C1 = M1 − L1L−10 M0

L0C1 = M1 − L1C0

M1 = L0C1 + L1C0 (4.21)

Similarly, it can be seen that

0 = L0C2 + L1C1 + C0 (4.22)

0 = L0C3 + L1C2 + C1 (4.23)

Eqns. (4.22 and 4.23) can be written as

[L0 L1

] [C2 C1

C3 C2

]=

[ −C0

−C1

]

[L0 L1

]D =

[ −C0

−C1

](4.24)

For unique solutions to exist for the above Pade equations. D must be nonsingular.Once, L1 and L0 are solved, M0 and M1 can be obtained using Eqns. (4.20 and 4.21).


Now let us consider an rth order model

R(s) =[L0 + L1s + L2s

2 + · · ·+ Lr−1sr−1 + Isr

]−1 [M0 + M1s + · · ·+ Mr−1s

r−1]

(4.25)

If the above equation is expanded up to first 2r terms and compared with Eqn. (4.17),we obtain the matrix equations

[L0 L1 · · · Lr−1

]

Cr Cr+1 · · · C2r−1

Cr−1 Cr · · · C2r−2...

......

C1 C2 · · · Cr−2

=

[ −C0 −C1 · · · −Cr−1

](4.26)

[L0 L1 · · · Lr−1

]

C0 C1 · · · Cr−1

0 C0 · · · Cr−2

0 0...

......

. . .

0 0 C0

=[

M0 M1 · · · Mr−1

](4.27)

4.2.4 Stable Pade for Multivariable Systems

Let the higher order model

G(s) =D0 + D1s + D2s

2 + · · ·+ Dn−1sn−1

c0 + c1s + c2s2 + · · ·+ cn−1sn−1 + sn(4.28)

= C0 + C1s + C2s2 + · · ·

where,ci are scalars and Ci, Di are matrices.

Let an rth order model

R(s) =A0 + A1s + A2s

2 + · · ·+ Ar−1sr−1

b0 + b1s + b2s2 + · · ·+ br−1sr−1 + sr(4.29)

The denominator of R(s) is fixed either by Routh method or by dominant mode retention.The numerator of R(s) is obtained by using the following r matrix equations.

A0 = b0C0

A1 = b1C0 + b0C1

... (4.30)

Ar−1 = b0Cr−1 + b1Cr−2 + · · ·+ br−1C0


4.2.5 Reduction of non-asymptotically stable systems

Definition 2 A system is said to be non-asymptotically stable if ’not every’ trajectory inthe unforced system follows the relation

‖x(t)‖ ≤ ae−bt ‖x(0)‖a, b ≥ 0,∀x ∈ <n

The methods presented above are based on the fact that the high-order system is asymp-totically stable. For the case of non-asymptotically stable systems the methods may failsince some of the alpha parameters of G(s) will be negative. Non-asymptotic stability isdue to the presence of poles with zero real parts. It is obviously important that the reducedmodel be non-asymptotically stable if the higher order system is non-asymptotically stable.Hence, the poles with zero real parts are retained in the reduced order model. Thus, letG(s), with n poles and m zeros and p non-asymptotic poles, be of the form

G(s) =1

ep(s)

(Um(s)

Vn−p(s)

)

=1

ep(s)

(Pm−n+p(s) +

Un−p−1(s)

Vn−p(s)

)

=Pm−n+p(s)

ep(s)+

1

ep(s)

Un−p−1(s)

Vn−p(s)(4.31)

where,

ep(s) = pth order polynomial in s consisting of the p non-asymptotic poles

Um(s) = mth order numerator of the full model.

Vn−p(s) = Denominator of full model after exclusion of non-asymptotic poles

Pm−n+p(s) = Quotient of the polynomial division Um(s)Vn−p(s)

Un−p−1(s) = Reminder of the polynomial division Um(s)Vn−p(s)

Now, assigning

G1(s) =Un−p−1(s)

Vn−p(s)(4.32)

and obtaining the reduced order model R1(s) of order r1 < n− p, by any of the previousmethods, the reduced order model of the original system can be obtained as

R(s) =Pm−n+p(s)

ep(s)+

1

ep(s)R1(s) (4.33)

This reduced model would be of the order r = p+r1. Thus reiterating the condition thatthe reduced order model should be of an order greater the number of non-asymptotic poles inthe system. If this condition fails to apply then the system is not reducible.¥


4.2.6 Time-Domain Pade Approximation for Multivariable Sys-tems

Time Domain Pade for SISO Systems

In this method, Pade equations are derived by assuming a state-space representation for thereduced model. Let us consider a nth order system while also assuming that an rth reducedmodel is to be derived. Pade equations are obtained by equating the first 2r time momentsof the model with that of its system. Let us assume that the rth order model is representedin phase canonical form by

Z = FZ + Gu (4.34)

y = LZ

where

F =

0 1 0 00 0 1 0

0. . .

......

...... 1

−b0 −b1 −b2 · · · −br−1

, G =

000...1

(4.35)

L =[

a0 a1 a2 ar−1

]

Further, let c0, c1, · · · c2r−1 be the first 2r time moments of the system and let ai, bi be theunknown parameters of the model. After equating the first 2r time moments of the modelwith that of the system, the following equations are obtained.

c0 = −LF−1G

c1 = −LF−2G... (4.36)

c2r−1 = −LF−2rG

On simplification, the equations in Eqn. (4.36) take the form of Eqns. (4.11 and 4.12).The parameters of the reduced model can be evaluated by a solution of the Pade equations.

Time Domain Pade Approximation for MIMO Systems

In this part, Pade equations for multivariable systems are derived [19]. It will be shown thatin the time domain, only a partial Pade approximation is possible for multivariable systems.Let us first assume that an 8th order Pade approximant for an nth order 2-input 2-outputsystem is to be derived. Also, let C0, C1, · · · , C7 be the first eight moment matrices of the nth

order system and that the higher order system and its reduced order models are completelycontrollable.


Let the controllable canonical form of the 8th order model be given by

Z = FZ + GU (4.37)

Y = LZ

where

Z ∈ R8, U ∈ R2, Y ∈ R2

F =

0 I 0 00 0 I 00 0 0 I−B0 −B1 −B2 −B3

, G =

000I

(4.38)

L =[

A0 A1 A2 A3

]

Let us assume F is nonsingular. Bi and Ai are the unknown matrices each of dimension(2× 2). Also, 0 and I are the null and identity matrices each of dimension (2× 2). Equatingthe first eight time moment matrices of the system and the model, the following equationsare obtained.

C0 = −LF−1G

C1 = −LF−2G... (4.39)

C7 = −LF−8G

The first of the above equations can be written as

C0 = − [A0 A1 A2 A3

]

−B−10 B1 −B−1

0 B2 −B−10 B3 −B−1

0

I 0 0 00 I 0 00 0 I 0

000I

= A0B−10

A0 = C0B0

As F is nonsingular, B0 is nonsingular and B−10 exists. The second equation of the above

set can be written as

C1 = −LF−2G

− [A0 A1 A2 A3

]

(fBB1)−(B−1

0 B2

)(fBB2)−

(B−1

0 B3

)(fBB3)−B−1

0 fB

−B−10 B1 −B−1

0 B2 −B−10 B3 −B−1

0

I 0 0 00 I 0 0


fB = B−10 B1B

−10

000I

= C1

C1 = −A0B−10 B1B

−10 + A1B

−10

A1 = C0B1 + C1B0

Similarly, the other six equations are obtained with reasonable effort to yield the followingset of Pade equations.

A0 = C0B0

A1 = C0B1 + C1B0

A2 = C0B2 + C1B1 + C2B0

A3 = C0B3 + C1B2 + C2B1 + C3B0 (4.40)

and

0 = C0 + C1B3 + C2B2 + C3B1 + C4B0

0 = C1 + C2B3 + C3B2 + C4B1 + C5B0

0 = C2 + C3B3 + C4B2 + C5B1 + C6B0

0 = C3 + C4B3 + C5B2 + C6B1 + C7B0 (4.41)

The 32 unknown parameters in Ai and Bi can be uniquely obtained from a solution ofEqns. (4.40 and 4.41). Thus, it is seen that, in the time domain, an eighth-order Padeapproximant can be derived by matching eight time moment matrices only. Applying induc-tion, it can be shown that, for an rth order reduced model of the nth order 2-input 2-outputsystem, only r time moment matrices can be matched to obtain the Pade approximant. Inthe following, it will be shown that, for an nth order m-input p-output system, an rth orderPade approximant will match fewer time moments than in the previous case.

For that, let us consider the rth order model in controllable canonical form and representedby Eqn. (4.37) with

F =

0 I 0 0 · · · 00 0 I 0 · · · 00 0 0 I · · · 0...

......

.... . .

I−B0 −B1 −B2 −B3 · · · −Bk−1


G =

000...I

(4.42)

L =[

A0 A1 A2 · · · Ak−1

](4.43)

k = r/m

Bi and Ai are the unknown matrices of dimension (m×m) and (p×m) respectively. 0and I are null and identity matrices each of dimension (m × m). It is assumed that bothr/m and r/p are integers. If the first (r/m) + (r/p) time moment matrices of the reducedmodel are equated with that of tis system, the following equations are obtained.

C0 = −LF−1G

C1 = −LF−2G... (4.44)

C(r/m)+(r/p)−1 = −LF−((r/m)+(r/p))G

where C0, C1, · · · , C(r/m)+(r/p)−1 are the first ((r/m) + (r/p)) time moment matrices ofthe nth order system.

Then by using the results in Eqns. (4.40 and 4.41) and by employing the principle ofinduction, Eqn. (4.44) can be simplified to the form

A0 = C0B0

A1 = C0B1 + C1B0

A2 = C0B2 + C1B1 + C2B0

A3 = C0B3 + C1B2 + C2B1 + C3B0

...

A(r/m)−1 = C0B(r/m)−1 + C1B(r/m)−2 (4.45)

+C2B(r/m)−3 + · · ·+ C(r/m)−1B0

0 = C0 + C1B(r/m)−1 + C2B(r/m)−2

+C3B(r/m)−3 + · · ·+ C(r/m)B0

0 = C1 + C2B(r/m)−1 + C3B(r/m)−2

+C4B(r/m)−3 + · · ·+ C(r/m)+1B0

...

0 = C(r/p)−1 + C(r/p)B(r/m)−1 + C(r/p)+1B(r/m)−2

+C(r/p)+2B(r/m)−3 + · · ·+ C(r/m)+(r/p)−1B0 (4.46)

The first set of equations contains r/m linear matrix algebraic equations ( or pr linearsimultaneous equations). The second set contains r/p linear matrix algebraic equations


( or rm linear simultaneous equations). Each Ai and Bi contains pm and m2 unknown,respectively. So the total unknowns in the matrices are (rm + pr). Thus, the total numberof scalar equations in Eqn. (4.45 and 4.46) can be used to solve for Ai and Bi. Thus,it is proved that, in the time domain, an rth order m-input p-output model can match(r/m) + (r/p) time moment matrices with that of its system. For m = p, the number ofmoment matrices matching will be 2r/m.

Further, Eqn. (4.46) can be written as

CT1 CT

2 CT3 · · · CT

r/m

CT2 CT

3 CT4 · · · CT

(r/m)+1

CT3 CT

4 CT5 · · · CT

(r/m)+2...

......

CT(r/p) CT

(r/p)+1 CT(r/p)+2 · · · CT

(r/m)+(r/p)−1

BT(r/m)−1

BT(r/m)−2

BT(r/m)−3

...BT

0

=

−CT0

−CT1

−CT2

...−C(r/p)+(r/m)−1

≡ DV = C

V = D−1C (4.47)

Hence, for the existence of a unique solution for Bi and Ai in the time-domain Padeapproximant, the matrix D must be nonsingular.

4.2.7 Time-Domain Modal-Pade Method

As the stability is not guaranteed in an ordinary Pade approximation, a time-domain coun-terpart of the stable Pade approximant [20] which retains the dominant or desired poles ofthe higher order system is proposed. It is assumed that the system is completely controllableand observable. In case this is not so, a controllable and observable part of the system isfirst derived and a model is derived and a model order reduction is obtained for this partonly [21]. The slow or the dominant modes are retained in the reduced order model. Fora SISO system, the output matrix of the model is selected by matching r time momentswith that of its system where r is the order of the model. The model so obtained bearsboth the characteristics, viz, retention, of dominant eigenvalues and matching of the first rtime moments. But, for the MIMO system, it can be shown that an rth order modal-Padeapproximant matches less than r time moments. For this purpose, let us consider the statespace description of a multivariable system given by

X = AX + BU (4.48)

Y = EX

where X ∈ Rn, U ∈ Rm, Y ∈ Rp. Also, let the system be completely controllable andobservable with controllability index n. A modal transformation X = MW yields

[W1

W2

]=

[Λ1 00 Λ2

] [W1

W2

]+

[B1

B2

]U (4.49)

Y =[

E1 E2

] [W1

W2

](4.50)


where W =[

W1 W2

]T. Let Λ1 contain the r dominant eigenvalues of the system. The

state equation of the rth order model can be obtained by a truncation of the representationin Eqn. (4.49) as follows W1 =

[Ir 0

]W, thereby

W1 = Λ1W1 + B1U (4.51)

As the system is completely controllable, the state equation of the model in Eqn. (4.51)can be transformed into controllable canonical form and can be represented by Eqn. (4.37)with F, G given in Eqn. (4.42). The output matrix of the reduced model is assumed to beunknown and is represented by L in Eqn. (4.43). In this situation, Bi matrices are knownwhile Ai matrices contain unknown elements. These unknowns can be obtained from thesolution of Eqn. (4.45). Thus, it can be seen that, for a multivariable system in the timedomain, a stable Pade approximant matches r/m time moment matrices with that of itssystem.

Exact Aggregation for Modal-Pade Procedure

An exact aggregation matrix also exists for the modal-Pade procedure. The derivation ofthis is given in the following.

The transformation matrix M transforms the system in Eqn. (4.48) into the modal formin Eqn. (4.49). So

W = M−1X

The relation between the state vectors W1 and W is given by

W1 =[

Ir 0]W

Let the transformation matrix T transform the representation into controllable canonicalform in Eqn. (4.42). So

Z = TW1

Now

Z = TW1 = T[

Ir 0]W = T

[Ir 0

]M−1X

≡ CX (4.52)

where C is the desired aggregation matrix and is given as

C = T[

Ir 0]M−1

4.3 Routh Approximation Techniques 53

4.3 Routh Approximation Techniques

4.3.1 Routh Approximation Method Using α− β Parameters

.Consider the nth order transfer function

Un(s)

Vn(s)=

b1 + b2s + b3s2 + · · ·+ bns

n−1

a0 + a1s + a2s2 + · · ·+ an−1sn−1 + sn(4.53)

It has been shown by Hutton [23] and Friedland [22] that G(s) = 1sG

(1s

)can be repre-

sented in a canonical fashion as

G(s) = β1f1(s) + β2f1(s)f2(s) + · · ·+ βnf1(s)f2(s) · · · fn(s) (4.54)

=n∑

i=1βi

i∏j=1

fj(s)

where βi, i = 1, 2, · · · , n, and fk(s), k = 2, 3, · · · , n are determined by the following con-tinued fraction

fk(s) =1

αks +1

αk+1s+1

. . .

αn−1s +1

αns

(4.55)

f1(s) =1

1 + α1s(4.56)

Eqns. (4.54 - 4.56) are called alpha-beta expansions of G(s) . The kth order model isobtained by using the following algorithm. The αi and βi are computed through their alphaand beta Routh tables.(Refer [13, 22])

Routh-Hurwitz Approximation Algorithm

Step 1: Determine the reciprocal of the full model G(s).

Step 2: Construct the α− β tables corresponding to G(s).

Step 3: For a kth order reduced model use recursive formulae in Eqn. (4.64) to find Rk(s) =Pk(s)/Q(s).

Step 4: Reverse the coefficients of Pk(s) and Q(s) back to find Rk(s) = Pk(s)/Qk(s).

The Routh-Hurwitz method can be used to obtain reduced order models for stable fullsystems. By this method, the less dominant poles of the full model are retained.


4.3.2 Routh Approximation Technique Using γ − δ Parameters

It can be clearly seen that the α−β method involves two transformations and one reductionof model. Krishnamurthi and Sheshadri [24] proposed a simple straightforward technique toachieve the same end.

In this method, termed as γ − δ method ( as in [25]). The method represents G(s) in adifferent manner as

G(s) =1

s

n∑i=1

δi

i∏j=1

Wj(s) (4.57)

where the δi for i = 1, 2, · · · , n are constants and Wi(s), i = 1, 2, · · · , n are defined by thecontinued fraction expansions

Wi(s) =1

γi

s+

1

γi+1

s+

1. . .

γn−1

s+

1

γn

s

(4.58)

W1(s) =1

1 +γ1

s+ W2(s)

The values of the γ and δ parameters can be obtained using the gamma and delta Routhtables ( or inverse Routh tables). The n parameters γk, k = 1, 2, · · · , n, of this expressioncan be found in the following fashion

a00 = a0 a0

2 = a2 a04 = a4 a0

6 = a6 · · ·a1

0 = a1 a12 = a3 a1

4 = a5 · · ·γ1 = a0

0/a10 a2

0 = a02 − γ1a

12 a2

2 = a04 − γ1a

14 a2

4 = a06 − γ1

1a26 · · ·

γ2 = a10/a

20 a3

0 = a12 − γ2a

22 a3

2 = a14 − γ2a

24 · · ·

γ3 = a20/a

30 a4

0 = a22 − γ3a

23 a4

2 = a24 − γ3a

34 · · ·

γ4 = a30/a

40 a5

0 = a32 − γ4a

24 · · ·

γ5 = a40/a

50 · · · · · ·

γ6 = a50/a

60 · · ·

...

Table 4.1: Gamma Table

The δi parameters can be similarly obtained using the coefficients of the numeratorbj, j = 1, 2, · · ·n, as

The recursive formula to compute the entries of gamma and delta tables can be obtainedfrom


b10 = b0 b1

2 = b2 b14 = b4 · · ·

b20 = b1 b2

2 = b3 b24 = b5 · · ·

δ1 = b10/a

10 b3

0 = b12 − δ1a

22 b3

2 = b14 − δ1a

24 · · ·

δ2 = b20/a

20 b4

0 = b22 − δ2a

32 b4

2 = b24 − δ2a

34 · · ·

δ3 = b30/a

30 b5

0 = b32 − δ3a

42 b5

2 = b34 − δ3a

44 · · ·

δ4 = b40/a

40 b6

0 = b42 − δ4a

52 · · ·

δ5 = b50/a

50 · · ·

δ6 = b60/a

60 · · ·

...

Table 4.2: Delta Table

ai+10 = ai−1

2 − γiai2

ai+12 = ai−1

4 − γiai4

... (4.59)

ai+1n−i−2 = ai−1

n−i − γiain−i, i = 1, 2, · · ·n− 1

For the case when n− i is odd, the last equation is replaced by

ai+1n−i−1 = ai−1

n−i+1

and

γi = ai−10 /ai

0 i = 1, 2, · · · , n

bi+2j−2 = bi

j − δiaij

j =

2, 4, · · · , n− i, for n− i even2, 4, · · · , n− i− 1, for n− i odd

i = 1, 2, · · · , n− 2

(4.60)

and

δi = bi0/a

i0 (4.61)

The kth Routh reduced model using the alpha-beta expansion, Rk(s) for the full modelG(s) is found by truncating the expansion in Eqn. (4.54) and rearranging the retained termsas a rational transfer function. Truncating the continued fraction in Eqn. (4.55) after thekth term and denoting it by gj,k(s), the reduced model transfer function Rk(s) is similar toEqn. (4.54).

Rk(s) =k∑

i=1γi

i∏j=1

gj,k(s) (4.62)


gj,k(s) =1

γi

s+

1

γi+1

s+

1. . .

γk−1

s+

1

γk

s

(4.63)

Denote the numerator and denominator of Rk(s) by Pk(s) and Qk(s), respectively, definedbelow

Pk(s) = δksk−1 + s2Pk−2(s) + γkPk−1(s) (4.64)

Qk(s) = s2Qk−1(s) + γkQk−2(s)

for k = 1, 2, · · · and

P−1(s) = P0(s) = 0

Q−1(s) = 1/s

Q0(s) = 1

Routh Approximation for Unstable Systems

For an unstable transfer function G(s), a shift of the imaginary axis would provide a modifiedasymptotically stable transfer function

G(s) = G (s + a)

where the real, positive parameter a is chosen to be a > R λm , where λm is the closed-loop pole with the highest positive real part. The next step would be to find Rk(s) as usualand finally shift back the imaginary axis to its original position providing

Rk(s) = Rk(s− a)

the kth Routh approximant of the unstable system.

4.3.3 Aggregated Model of Routh Approximants

In the previous section, it has been shown that the Routh approximation technique has anattractive feature of derivation of all simplified models simultaneously via a single set ofcomputations, the stability of the generated lower order models is always guaranteed. Itis therefore worthwhile to explore the possibility of obtaining time-domain representationanalogous to the Routh approximants.

The development of aggregated models of the Routh approximant is based on the formu-lation of G(s) in phase canonical form.


x = Ax + bu (4.65)

y = Cx

It has been shown in [26] that the linear relation

v = Px (4.66)

exists which transforms Eqn. (4.65) to its α− β expansion in the phase canonical form

v = Rv + mu (4.67)

y = Ev

where,

R =

−γ1 0 −γ3 0 · · · · · · −γn

0 0 γ3 0 · · · · · · γn

−γ1 −γ2 −γ3 0 · · · · · · −γn

0 0 0 0 γ5 γn...

......

... · · · ...−γ1 −γ2 −γ3 −γ4 −γ5 · · · −γn

if n is odd

m =[

1 0 1 0 · · · 1]T

(4.68)

R =

0 γ2 0 γ4 · · · · · · γn

−γ1 −γ2 0 −γ4 · · · · · · −γn

0 0 0 γ4 · · · · · · γn

−γ1 −γ2 −γ3 −γ4 −γn...

......

......

−γ1 −γ2 −γ3 −γ4 −γ5 · · · −γn

if n is even

m =[

0 1 0 1 · · · 1]T

with, in both cases

E =[

β1 β2 β3 βn

](4.69)

It is obvious from Eqns. (4.65-4.67) that

R = PAP−1

m = Pb (4.70)

E = CP−1

In terms of the direct Routh approximation method (DRAM) [24], the linear transfor-mation P is given as


P =

a10 0 a1

2 0 · · · a1n−2 0 1

0 a20 0 a2

2 · · · · · · a2n−2 0

0 0 a30 0 · · · · · · · · · 1

.... . .

......

...0 0 0 0 0 1

if n is odd

(4.71)

P =

a10 0 a1

2 0 · · · a1n−2 0

0 a20 0 a2

2 · · · 0 10 0 a3

0 0 · · · · · · 0

0 0 0 a40

. . ....

...... an−1

0 00 0 0 0 0 1

if n is even

Since the matrix P is upper-triangular and sparse, the computation of its inverse wouldbe comparatively easy. The derivation of the rth order model which approximates the nth

order model in Eqns. (4.68 - 4.69) is carried by simply retaining the blocks of order r fromthe matrices R, m,E and discarding the remaining (n− r) blocks to yield

z = Fz + gu (4.72)

y = Hz

where

F =

−γ1 0 −γ3 0 · · · · · · −γr

0 0 α3 0 · · · · · · γr

−γ1 −γ2 −γ3 0 · · · · · · −γr

0 0 0 0 γ5 γr...

......

... · · · ...−γ1 −γ2 −γ3 −γ4 −γ5 · · · −γr

if n is odd

g =[

1 0 1 0 · · · 1]T

(4.73)

F =

0 γ2 0 γ4 · · · · · · γn

−γ1 −γ2 0 −γ4 · · · · · · −γn

0 0 0 γ4 · · · · · · γn

−γ1 −γ2 −γ3 −γ4 −γn...

......

......

−γ1 −γ2 −γ3 −γ4 −γ5 · · · −γn

if n is even

g =[

0 1 0 1 · · · 1]T


In both cases,

H =[

δ1 δ2 · · · δr

](4.74)

Define

Cr =[

Ir 0r×(n−r)

](4.75)

then it is readily verified that the system in Eqn. (4.72) is an aggregated model of thesystem in Eqn. (4.65) provided that

z = CrvF = CrRCT

r

g = CrmH = ECT

r

(4.76)

Furthermore, from Eqns. (4.66 and 4.76) the transformation matrix L, transforming thevector x into z, is given by

L = CrP (4.77)

In summary, the dynamic model in Eqn. (4.72) is an approximant of the original modelin Eqn. (4.65) using L as an aggregation matrix.

Remark 2 It is to be noted that though the Routh technique assures simultaneous derivationof reduced models of all orders, and retains the ’stability’ property during reduction, thereduced order models do not retain the eigenvalue subset of the original system, nor are theaggregated models of Routh approximation exact. The second and third equalities of Eqn.(4.76) are not preserved during the Routh-based aggregation.

4.3.4 Optimal Order of Routh Approximant

The impulse energy of the full model and its approximants defined as

‖G‖ =

∞∫

0

G2(t)dt (4.78)

are related as

‖Rk+1‖ =k+1∑

i=1

(δ2i

2γi

)= ‖Rk‖+

δ2k+1

2γk+1

(4.79)

and generally

0 ≤ ‖R1‖ ≤ ‖R2‖ ≤ · · · ≤ ‖Rn‖ = ‖G‖

As k → n, the eigenvalues of the approximant tend towards the actual eigenvalues of thefull model.(k → n =⇒ λRk

→ λG).


Thus, as the model order increases, the closeness of fit also increases. But, using thislogic would indicate that the full model is the ideal ’reduced’ model ( which, though true,loses the purpose of model order reduction). Hence, the maximum change in impulse energywith increase of order of model by one is calculated.

µk = ‖Rk‖ − ‖Rk−1‖ =δ2k

2γk

, k = 1, 2, · · · , n (4.80)

Now, the optimal reduced order model is calculated as

Rk(s), k = i|µi = max(µ1, µ2, · · · , µn) (4.81)

Remark 3 The first k terms of the power series expansion of G(s) and Rk(s) coincide, thatis

Rk(s)−G(s) =∞∑

j=1θjs

k+j

Evaluation of the above equation in accordance with the concept of Pade-type approximatereadily shows that the Routh approximant is only a ’partial’ Pade approximant and matchesthe initial r time moments.

4.4 Continued Fraction Method

Another model reduction method is the first Cauer continued fraction expansion proposedinitially Chen and Shieh [14].

4.4.1 The Three Cauer Forms

Consider a SISO closed-loop transfer function

G(s) =a21 + a22s + a23s

2 + · · ·+ a2nsn−1

a11 + a12s + a13s2 + · · ·+ a1,n+1sn(4.82)

In principle, G(s) can be expanded into several continued fraction forms; however, thereare three basic forms of particular relevance to systems engineering. These are called theCauer forms :

The First Cauer Form

G(s) =1

h1s+1

h2+1

h3s+1. . .

(4.83)

corresponds to a combination of multiple feedback loops comprising differentiator blocksand feed-forward paths having proportional blocks. Thus, the MIMO first Cauer form is

4.4 Continued Fraction Method 61

G(s) =

[H1s +

[H2 +

[H3s + [· · ·]−1]−1

]−1]−1

(4.84)

which has 2n matrices Hi and it represents a Maclaurin’s-series expansion about s = ∞.

The Second Cauer Form

G(s) =1

k1 +1

k2

s+

1

k3+1. . .

(4.85)

can be represented by a combination of multiple feedback loops having proportionalblocks and multiple feed-forward paths, including integral blocks. In case of MIMO systems,the second Cauer form is

G(s) =

K1 +

K2

1

s+

[K3 +

[K4

1

s+ [· · ·]−1

]−1]−1

−1

−1

(4.86)

which has 2n matrices Ki and represents a Maclaurin’s-series expansion about s = 0.

The Third Cauer Form

G(s) =1

d1 + f1s +1

d2

s+f2+

1

d3 + f3s+1

d4

s+f4+

1. . .

(4.87)

which corresponds to a combination of multiple feedback loops with blocks each con-taining proportional plus differential and feed-forward paths comprising proportional plusintegral blocks. The third Cauer form can be represented for MIMO systems as

G(s) =

D1 + F1s +

D2

1

s+ F2 +

[D3 + F3s +

[D4

1

s+ F4 + [· · ·]−1

]−1]−1

−1

−1

(4.88)

which has n matrices Di and Fi and is equivalent to a Maclaurin’s-series expansion aboutboth s = 0 and s = ∞.

The quotients hi in Eqn. (4.83) can be obtained by long synthetic division, or, alter-natively, by using Routh’s algorithm. The same algorithm could be used to determine the


quotients ki in Eqn. (4.85) or by arranging the polynomials in ascending order and thenperforming long synthetic division. The generalized Routh algorithm [28] can then be usedto compute the quotients di and fi in Eqn. (4.87).

It should be noted that the quotients in the MIMO cases, Hi, Ki, Di and Fi are constantm×m quotient matrices. Observe that any algorithm that computes the quotients Di andFi is capable of computing the Hi quotients (by setting all the Di’s equal to zero) and ofcomputing the Ki quotients by suppressing the quotients Fi throughout the implementation.

Next, we present an algorithmic procedure to compute the quotients Di and Fi of Eqn.(4.88) for transfer function matrices.

4.4.2 A Generalized Routh Algorithm

The computation of the matrix quotients in the third (mixed) matrix Cauer form is carriedout using an algorithm procedure based on a matrix Routh array [29]. Extension of thescalar transfer function in Eqn. (4.82) to the multivariable case is given by :

G(s) =[A2nsn−1 + A2,n−1s

n−2 + · · ·+ A22s + A21

]

× [A1,n+1s

n + A1,nsn−1 + · · ·+ A12s + A11

]−1(4.89)

where A2j, j = 1, · · · , n, are constant m ×m matrices and A1i = aiIm, i = 1, · · · , n + 1,where each ai is a coefficient of

∆(s) =n+1∑i=1

aisi−1

the common-denominator polynomial. Using double subscript notation, the matrixRouth array can be expressed as :

A11 A12 A13 A14 A15 · · ·A21 A22 A23 A24 · · ·A31 A32 A33 · · ·A41 A42 · · ·...

(4.90)

where the elements of the first and second rows of Eqn. (4.90) are the matrix coefficientsof the transfer function matrix in Eqn. (4.89). The elements on any subsequent row can beevaluated by the following generalized matrix Routh algorithm:

For j = 3, 4, · · · , n + 1

Ajk = Aj−2,k+1 −Dj−2Aj−1,k+1 − Fj−2Aj−1, k (4.91)

where for p = 1, 2, · · ·

Dp = Ap1 [Ap+1,1]−1 (4.92)

Fp = Ap,(n−p+2)

[Ap+1,(n−p+1)

]−1


Assuming the indicated inverses exist, the matrix Routh array is completed as follows

A11 A12 · · · A1,n A1,n+1

D1 = A11A

−121 F1 = A1,n+1A

−12,n

A21 A22 · · · · · · A2,n

D2 = A21A

−131 F2 = A2,nA−1

3,n−1

A31 A32 · · · · · · A3,n−1

D3 = A31A

−141 F3 = A3,n−1A

−14,n−2

A41 A42 · · · · · · A4,n−2

......

......

...An−1,1 An−1,2 · · · An−1,3

Dn−1 = An−1,1A

−1n,1 Fn−1 = An−1,3A

−1n,2

An,1 An,2 · · · −→

Dn = An,1A

−1n+1,1 Fn = An,2A

−1n+1,1

An+1,1 · · · −→

(4.93)It should be noted that the above procedure is directly amenable to machine computation.

To illustrate the calculations, let us consider a simple example :

Example 7

G1(s) =1

s2 + 2

[2s− 3 2s− 2 s− 2

]

In terms of Eqn. (4.89), G1(s) takes the form

G1(s) = [A21 + A22s][A11 + A12s + A13s

2]−1

with

A21 =

[ −3 2−2 −2

]A22 =

[2 01 1

]

A11 =

[2 00 2

], A12 =

[0 00 0

], A13 =

[1 00 1

]


Applying the matrix Routh array procedure, we obtain[2 00 2

] [0 00 0

] [1 00 1

]

D1 =

[ −0.4 −0.40.4 0.6

]F1 =

[0.5 0−0.5 1.0

]

[ −3 2−2 −2

] [2 01 1

]

D2 =

[ −1.15 0.36−0.67 −0.67

]F2 =

[0.727 0.1210.333 0.333

]

←−

[2.7 −0.60.3 3.6

]−→

Going back to the generalized matrix Routh array, it should be clear that the array is atriangular one. In the special case, where Fi = 0, which corresponds to the second Cauerform, the resulting array is a regular zig-zag pattern.

The following subsections discuss about derivation of simplified models using the matrixcontinued fraction expansion forms.

4.4.3 Simplified Models Using Continued Fraction Expansion Forms

The principle philosophy underlying the derivation of simplified models using continuedfraction expansion is that the representation resembles multiple feedback loops and feed-forward paths with blocks corresponding to the quotients. As the quotients descend lowerand lower in position, or equivalently the blocks develop to more and more inner loops,they have less and less significance as far as the overall system performance is considered.Therefore, truncating the continued fraction after some terms is equivalent to ignoring theinner, less important loops. In this respect, three candidate simplified models are readilyavailable through the use of the three Cauer forms. Because of the common features amongthe matrix continued fraction expansion forms, we would be concentrating on only one ofthem.

Considering the second matrix Cauer form, the initial step is to manipulate the transferfunction matrix G(s) = C [sI − A]−1 B to yield the general form given by Eqn. (4.89). Then,using the matrix Routh algorithm, with Fi = 0 and Di = Ki for all i, the quotient matricesKi are computed. A simplified model of order r is derived by truncating the matrix continuedfraction in Eqn. (4.86) after 2r matrices. This gives the simplified transfer function matrixin the form :

Gr(s) =

K1 +

K2

1

s+

K3 +

K4

1

s+

[· · ·

[K2r

1

s

]−1

· · ·]−1

−1

−1

−1

−1

(4.94)


which requires only the computation of K1, · · · , K2r.Given the quotient matrices Ki of Eqn. (4.86), a state-space formulation of G(s) in the

form of Eqn. (4.65) exists with the matrices given by :

A =

K1K2 K1K4 K1K6 · · · K1K2n

K1K2 (K1 + K3) K4 (K1 + K3) K6 · · · (K1 + K3) K2n

K1K2 (K1 + K3) K4 (K1 + K3 + K5) K6 · · · (K1 + K3 + K5) K2n...

......

...K1K2 (K1 + K3) K4 (K1 + K3 + K5) K6 (K1 + K3 + · · ·+ K2n−1) K2n

(4.95)

B =

Im

Im

...Im

C =[

K2 K4 · · · K2n

]

Note that the order of the system matrix is nm × nm. A simplified model can beobtained by partitioning. For example, the 2mth order model can be derived from the upperleft corners if Eqn. (4.95) is partitioned as shown.

The above formulation can also be obtained via aggregation for a SISO system in thefollowing manner. For a SISO system represented in phase canonical form,

x =

0 1 0 · · · 00 0 1 · · · 0...

......

...0 0 0 · · · 1−a11 −a12 −a13 · · · −a1,n+1

x +

00...01

u

y =[

a21 a22 a23 · · · a2,n

]x

A =

the aggregation matrix L =[

I2m 0]P , which results in the reduced state space model

described in Eqn. (4.95), has

P =

a31 a32 a33 · · · a3n

0 a51 a52 · · · a5,n−1

0 0...

...0 0 0 · · · a2n+1,1

0 0 · · · 0 1

It should be noted here that in this case too the aggregation is not exact as the eigenspectrumretained in the reduced order model is not a subset of the eigenspectrum of the original fullmodel.

The most important properties of continued fraction expansion are


1. It converges faster than other series expansions.

2. It contains most of the essential characteristics of the original model in the first fewterms.

3. It does not require any knowledge of the model eigenspectrum.

Since the denominator coefficients of the simplified model depend on both the numeratorand denominator coefficients of the original model, stability of the simplified model cannotbe guaranteed, even if the original model is stable.

Next let us look at the derivation of simplified transfer functions models that give goodapproximations to both the initial transient and the steady state responses. The third Cauerform , or any equivalent expansion, provides the basis of such models. This is due to thefact that it considers Maclaurin’s expansions about origin and infinity. Thus, a simplifiedmodel matrix Gr(s) could be obtained by truncating the continued fraction expansion inEqn. (4.88) after r terms. This means that r quotient matrices Di and Fi have to bedetermined using the matrix Routh array algorithm. A quite similar expansion of the thirdCauer form is given by

G(s) =

[D1 + s

[D2 +

[D3 + s [D4 + · · ·]−1]−1

]−1]−1

(4.96)

which considers the expansion of G(s) into a matrix Cauer type continued fraction abouts = 0 and s = ∞ alternately. Note that there are 2n Di constant (m×m) matrices. Thus,this method puts equal emphasis on its approximations to the initial transients and the steadystate responses. In principle, there is no restriction to the relative number of truncated termsabout each side of the series expansion. For example, increasing the number of terms in theseries expansion about s = 0 will have the effect of yielding more accurate approximationsto the steady state response, and vice versa. This class of simplified models are often termed’biased simplified models’.

In conclusion, it is interesting to note that the three matrix Cauer forms [29] could bereduced to the form in Eqn. (4.89).

Remark 4 In the case of the second Cauer form, the first 2r terms of the power seriesexpansion about s = 0 ( or the first 2r time moments) for of Gr(s) agree with those of G(s).In case of the first Cauer form, the coefficients of expansion about s = ∞ agree with oneanother ( Markov parameters). The third Cauer form is able to match both parameters.

Chapter 5

Large Scale System Model andController Order Reduction - NormBased Methods

5.1 Introductions

5.1.1 Norms of Vectors and Matrices

The norm is a measure of the magnitude of the vector or matrix. It is a measure that hasthe following important properties..

• For two elements a and b, and a scalars α and β

‖a‖+ ‖b‖ ≥ ‖a + b‖α ‖a‖ = ‖αa‖

‖(α + β) a‖ = α ‖a‖+ β ‖a‖

• In relation to the zero vector

‖0‖ = 0

‖x‖ = 0 ⇒ x = 0

It may be defined in the following manner.

Vector Norms

The p norm For an n-dimensional vector x, the p norm is defined as

‖x‖p = p

√√√√n∑

i=1

xpi (5.1)

67


The 2 - norm Thus, by the aforementioned definition, the 2 - norm of a vector is

‖x‖2 =

√√√√n∑

i=1

x2i = ‖x‖ (5.2)

The term “norm” of a vector is usually used synonymously to the 2 -norm of the vector.This norm also gives a ‘measure’ of the ’length’ of the vector in the Rn space.

The ∞− norm (Infinity Norm) The infinity norm denoted as ‖•‖∞ is simply the max-imum value of a vector-component in any of its coordinate directions.

‖x‖∞ = maxi=1,2,···,n

|xi| (5.3)

Matrix Norms

In case of a square matrix, the norm is defined as

‖A‖ = maxx6=0

∥∥xT Ax∥∥

‖x‖ (5.4)

This operation cannot be performed in case of non-square matrices. Hence, a more generaldefinition of the matrix norm can be given as follows

‖A‖ = max(λ

(AT A

))= max

(λ

(AAT

))(5.5)

5.1.2 Singular Value Decomposition

Singular Value Decomposition(SVD) is a mathematical technique by which any (m × n)matrix A of dimension (m× n), can be represented as a product of three matrices

A = UΣV ∗ (5.6)

where,

• U is a unitary matrix of dimension (m × m). The columns of U are composed ofeigenvectors ui, of the (m × m) matrix (AAT ), arranged in an order such that thecorresponding eigenvalues λi are in descending order.

• Σ is an (m × n) diagonal matrix with σij = 0, i 6= j and the diagonal elements σi

arranged as σi ≥ σi+1. The σi are termed as the ’singular values’ of the matrix A. Theygive an estimate about the ’gain’ of the matrix A. σi can be determined as

σi =√

λi

where,

λi are the eigenvalues of AAT .

5.1 Introductions 69

• V is a unitary matrix of dimension (n×n). The columns of V are composed of eigenvec-tors vi, of the (n×n) matrix (AT A), arranged in an order such that the correspondingeigenvalues λi are in descending order.

If r = min(m,n) then the first r columns of V the right singular vectors and the first rcolumns of U the left singular vectors.

Properties of Singular Values and SVD

From the components of the SVD, we can determine many properties of the original system.

• It can be seen that the decomposition A = UΣV ∗ = UΣV T for real matrices, can bewritten as

Avi = σiui

if σi = 0, then Avi = 0 and vj is in the null space of A, whereas of σi 6= 0, then uj is inthe range space of A.

• In the general case of N dimensions, the length (or norm) of a vector x is defined by

‖x‖ =√

(x21 + x2

2 + · · ·+ x2N) =

√(xT x)

When a vector x is multiplied by a matrix A, the length of the resulting vector Axchanges according to the matrix A. If A is orthogonal, the length is preserved. Oth-

erwise, the quantity‖Ax‖‖x‖ measures how much A stretches x. Thus, calculating the

norm of a matrix intuitively means finding the maximum stretch factor. If the SVD ofa matrix is given, this computation is simplified.

The Euclidean norm of a matrix, sometimes referred to as the L2 norm, is defined asfollows.

Let x be an N dimensional vector, and A be an m× n matrix, then

‖A‖E = max‖x‖=1

‖Ax‖

An alternative norm for A is the Frobenius norm, which is the Euclidean norm of avector constructed by stacking the columns of A in one mn vector. The Frobeniusnorm is then

‖A‖F =

(m∑

i=1

n∑j=1

|aij|2) 1

2

Using SVD, these norms can be computed as


‖A‖E = σ1

‖A‖F =

min(m,n)∑i=1

σ2i

12

5.1.3 Grammian Matrices and Hankel Singular Values

Let us consider the system of form

x = Ax + Bu

y = Cx

to be a minimal state-space realization. Although the system may be unstable, wesuppose that λi 6= λj for (i 6= j) and for any eigenvalue λ of A. In this hypothesis, thereachability Grammian Wr ( sometimes referred to as the controllability Grammian Wc) andthe observability Grammian Wo can be defined as the solutions of the equations:

AWr + WrAT = −BBT (5.7)

AT Wo + WoA = −CT C (5.8)

Eqns. (5.7 and 5.8) are called the Lyapunov equations of the system.If the eigenvalues of A are assumed to be strictly in the left half-plane then we can define

the controllability-reachability Grammian and the observability Grammian, respectively, as:

Wr =

∞∫

0

exp(At)BBT exp(AT t)dt (5.9)

Wo =

∞∫

0

exp(AT t)CT C exp(At)dt (5.10)

The state space representation is controllable if and only if Wr is positive semi-definiteand is observable if and only if Wo is positive semi-definite.

Proposition 1 For two algebraically equivalent systems with state vectors x and x the fol-lowing relations exist:

Wr = TWrTT (5.11)

Wo =(T T

)−1WoT

−1 (5.12)

where T is the n× n non-singular matrix such that x = Tx.

5.1 Introductions 71

The proof has been provided in [30].¥

Proposition 2 The eigenvalues λi of the product matrix Wr · Wo are invariant quantitiesunder any state coordinate transformation.

Proof : Let us consider, in fact, the matrix product Wr · Wo. Taking into account Eqns.(5.11-5.12) we have:

Wr · Wo = TWrTT (T T )−1WoT

−1

= TWr ·WoT−1¥

Definition 3 If a system is asymptotically stable, then the Hankel singular values of M(s)are defined as

σv(M(s)) = (λi (Wr ·Wo))12 , (i = 1, 2, · · · , n)

Generally they are ordered in a decreasing manner for i = 1 to n.

Definition 4 The Markov parameters of a system are defined as

Hk = CAk−1B

The Markov parameters are also given as the values of the system impulse response andits ith derivative computed at t = 0.

Definition 5 The Hankel matrix is defined as the doubly infinite matrix whose (i, j)th blockis Hi+j−1. Matrix H can be expressed as

H = MoMc

being

Mo =[

CT AT CT · · · (AT

)kCT · · ·

]T

Mc =[

B AB · · · AkB · · · ]

5.1.4 Matrix Inversion Formulae

Let A be a square matrix partitioned as follows

A :=

[A11 A12

A21 A22

]

where A11 and A22 are also square matrices. Now suppose A11 is nonsingular, then A hasthe following decomposition.


[A11 A12

A21 A22

]=

[I 0

A21A−111 I

] [A11 00 ∆

] [I A−1

11 A12

0 I

]

with ∆ := A22 − A21A−111 A12, and A is nonsingular if and only if ∆ is nonsingular.

Dually, if A22 is nonsingular then[

A11 A12

A21 A22

]=

[I A12A

−122

0 I

] [∆ 00 A22

] [I 0

A−122 A21 I

]

with ∆ := A11 − A12A−122 A21, and A is nonsingular if and only if ∆ is nonsingular. The

matrix ∆(∆) is called the Schur complement of A11(A22) in A.Moreover, if A is nonsingular, then

[A11 A12

A21 A22

]−1

=

[A−1

11 + A−111 A12∆

−1A21A−111 −A−1

11 A12∆−1

−∆−1A21A−111 ∆−1

]

and[

A11 A12

A21 A22

]−1

=

[∆−1 −∆−1A12A

−122

−A−122 A21∆

−1 A−122 + A−1

22 A21∆−1A12A

−122

]

The above matrix inversion formulae are particularly simple if A is block triangular:

[A11 0A21 A22

]−1

=

[A−1

11 0−A−1

22 A21A−111 A−1

22

]

[A11 A12

0 A22

]−1

=

[A−1

11 −A−111 A12A

−122

0 A−122

]

The following identity is also very useful. Suppose A11 and A22 are both nonsingularmatrices, then

(A11 − A12A

−122 A21

)−1= A−1

11 + A−111 A12

(A22 − A21A

−111 A12

)−1A21A

−111

As a consequence of the matrix decomposition formulae mentioned above, we can cal-culate the determinant of a matrix by using its submatrices. Suppose A11 is nonsingular,then

det(A) = det(A11) · det(A22 − A21A−111 A12)

On the other hand, if A22 is nonsingular, then

det(A) = det(A22) · det(A11 − A12A−122 A21)

In particular, for any B ∈ Cm×n and C ∈ Cn×m, we have

det

[Im B−C In

]= det (In + CB) = det(Im + BC)

and for x, y ∈ Cn

det (In + xy∗) = 1 + y∗x

5.2 Model Reduction by Balanced Truncation 73

5.2 Model Reduction by Balanced Truncation

5.2.1 Balanced Realization

In [31, 32], the idea of balanced realization of a stable transfer function is derived from theconcept of principal component analysis in statistical methods: that is, the controllabilityand observability grammians are supposed to be decomposed into principal components forevaluating the contribution of each mode.

The key property of a balanced realization is that the state coordinate basis is selectedsuch that the controllability and observability grammians are both equal to some diagonalmatrix

∑, normally with the diagonal entries of

∑in descending order. The state space

representation is then called balanced realization. The magnitudes of the diagonal entriesreflect the contributions of different entries of the state vector to system responses. Thestate vector entries that contribute the least are associated with the smallest σi. Thus, wecan use truncation to eliminate the unimportant state variables from the realization. Thefollowing two Lyapunov equations give the relation to the system matrices A, B, C for abalanced realization:

AΣ + ΣAT + BBT = 0 (5.13)

ΣA + AT Σ + CT C = 0 (5.14)

Hence Σ = diag [σi] and σ1 ≥ σ2 ≥ · · · ≥ σn. The σi are termed the Hankel singularvalues of the original system.

Let us consider how a balanced realization may be obtained [33].Let P and Q be respectively the controllability and observability grammians associated

with an arbitrary minimal realization A, B, C of a stable transfer function, respectively.Since P and Q are symmetric, there exist orthogonal transformations Uc and Uo such that

P = UcScUTc (5.15)

Q = UoSoUTo (5.16)

where Sc, So are diagonal matrices. The matrix

H = S1/2o UT

o UcS1/2c (5.17)

is constructed and a singular value decomposition is obtained from it:

H = UHSHV TH (5.18)

Using these matrices, the balancing transformation is given by

T = UoS−1/2o UHS

1/2H (5.19)

The balanced realization is[

A BC D

]=

[T−1AT T−1BTC D

](5.20)


Through simple manipulations it can be confirmed that

T T QT = T−1P(T T

)−1= SH

5.2.2 Balanced Truncation

Consider a stable system G ∈ RH∞ and suppose G =

[A BC D

]is a balanced realization.

Denoting the balanced Grammians by Σ, we have

AΣ + ΣA∗ + BB∗ = 0 (5.21)

ΣA + A∗Σ + C∗C = 0 (5.22)

Now partition the balanced Grammian as

Σ =

[Σ1 00 Σ2

]

and partition the system accordingly.

G =

A11 A12

A21 A22

B1

B2

C1 C2 D

Then Eqns. (5.21 and 5.22) can be written in terms of their partitioned matrices as

A11Σ1 + Σ1A∗11 + B1B

∗1 = 0 (5.23)

Σ1A11 + A∗11Σ1 + C∗

1 C1 = 0 (5.24)

A21Σ1 + Σ2A∗12 + B2B

∗1 = 0 (5.25)

Σ2A21 + A∗12Σ1 + C∗

2 C1 = 0 (5.26)

A22Σ2 + Σ2A∗22 + B2B

∗2 = 0 (5.27)

Σ2A22 + A∗22Σ2 + C∗

2 C2 = 0 (5.28)

By virtue of the method adopted to construct Σ, the most energetic modes of the systemare in Σ1 and the less energetic ones are in Σ2. Thus, the system with Σ1 as its balancedgrammian would be a good approximation of the original system.

Thus the procedure to obtain a reduced order model would be

• Obtain the balanced realization of the system.

• Choose an appropriate order r, of the reduced order model. Partition the systemmatrices accordingly.

• The reduced order model obtained as Gr =

[A11 B1

C1 D

]


5.2.3 Steady State Matching

Steady State Error

The algorithm discussed above has a basic disadvantage.. Though the responses of thereduced system get closer and closer to the original system, there is no certainty that theywould match at steady state. This is because of the upper bound on the approximation erroris dependent on the ignored Hankel singular values.

∥∥∥C(jωI − A

)−1B − C1

(jωI − A11

)−1B1

∥∥∥∞

≤ 2 (σr+1 + · · ·+ σn)

= 2Tr (Σ2)

This steady state error occurs because the ignored states even if not contributing muchto the dynamics of the system, do contribute to its steady state. Hence, they should beconsidered while deriving the reduced order model without steady state error. This steadystate error can be eliminated by modifying the reduced order model using the concept ofsingular perturbations.

Steady State Error Elimination

Consider the balanced realization of the full model as

[xz

]=

[A11 A12

A21 A22

] [xz

]+

[B1

B2

]u (5.29)

y =[

C1 C2

] [xz

]+ Du

Assuming that the states z are fast and stable, they would settle quickly and hence itcan safely be assumed that z = 0. Hence,

x = A11x + A12z + B1u (5.30)

0 = A21x + A22z + B2u (5.31)

Thus, z can be computed to be

z = −A−122 A21x− A−1

22 B2u (5.32)

Now, using Eqn. (5.32) in Eqn. (5.29), the modified reduced order system

Gr =

[A11 − A12A

−122 A21 B1 − A12A

−122 B2

C1 − C2A−122 A21 D − C2A

−122 B2

](5.33)


5.2.4 Reduction of Unstable Systems by Balanced Truncation

Unstable systems cannot be directly reduced using balanced truncation. Thus, the followingmethod proposed in [35] needs to be adopted.

The original system is transformed through schur transformations into

[x1

x2

]=

[A− Ac

0 A+

] [x1

x2

]+

[B−B+

]U (5.34)

y =[

C− C+

] [x1

x2

]+ DU

where, the A− has all its eigenvalues stable and A+ has all its eigenvalues unstable.This is system is then converted to two systems S1 and S2 with

S2 ⇒ x2 = [A+] x2 + [B+] U (5.35)

S1 ⇒x1 = [A−] x1 +

[B− Ac

] [Ux2

]

y = [C−] x1 +[

D C+

] [Ux2

] (5.36)

Now, using the above described methods, a reduced order model can be obtained for thestable system S1. Let the reduced system obtained from S1 represented as S′1

S′1 ⇒z = [Ar] z +

[BrU Brx2

] [Ux2

]

y = [Cr] z +[

DrU Drx2

] [Ux2

] (5.37)

The reduced order model for the original system can then be formulated as

[zx2

]=

[Ar Brx2

0 A+

] [zx2

]+

[BrU

B+

]U (5.38)

y =[

Cr Drx2

] [zx2

]+ [DrU ] U

Thus, even unstable systems can be reduced using balanced truncation.

5.2.5 Properties of Truncated Systems

Lemma 1 Suppose X is the solution of the Lyapunov equation

A∗X + XA + Q = 0 (5.39)

then

1. Re (λi (A)) ≤ 0 if X > 0 and Q ≥ 0


2. A is stable if X > 0 and Q > 0

3. A is stable if X ≥ 0, Q ≥ 0 and (Q,A) is detectable.

Proof : Let λ be an eigenvalue of A and v 6= 0 be a corresponding eigenvector, thenAv = λv. Premultiply Eqn. (5.39) by v∗ and postmultiply it by v to get

2Re (λ (v∗Xv)) + v∗Qv = 0

Now, if X > 0 and v∗Xv > 0, and it is clear that Re(λ) ≤ 0 if Q ≥ 0 and Re(λ) < 0if Q > 0. Hence, (1) and (2) hold. To see (3), we assume Re(λ) ≥ 0. Then we must havev∗Qv = 0, i.e., Qv = 0. This implies that λ is an unstable and unobservable mode, whichcontradicts the assumption that (Q,A) is detectable.¥

Lemma 2 Consider the Sylvester equation

AX + XB = C (5.40)

where A ∈ Fn×n, B ∈ Fm×m and C ∈ Fn×m are given matrices. There exists a uniquesolution X ∈ Fn×m if and only if λi(A) + λj(B) 6= 0,∀i = 1, 2, · · · , n, j = 1, 2, · · · ,m.

Proof : Eqn. (5.40) can be written as a linear matrix equation by using the Kroneckerproduct:

(BT ⊕ A

)vec(X) = vec(C)

Now that equation has a unique solution if and only if BT ⊕ A is nonsingular. Sincethe eigenvalues of BT ⊕ A have the form λi(A) + λj(B

T ) = λi(A) + λj(B), the conclusionfollows.¥

Theorem 1 Assume that Σ1 and Σ2 have no diagonal entries in common. Then both sub-systems (Aii, Bi, Ci), i = 1, 2 are asymptotically stable.

Proof : It is sufficient to show that A11 is asymptotically stable. The proof for thestability of A22 is similar.

Since Σ is a balanced realization, by the properties of SVD, Σ1 can be assumed to bepositive definite without loss of generality. Then it is obvious that λi(A11) ≤ 0 by the Lemma1. Assume that A11 is not asymptotically stable, then there exists an eigenvalue at jω forsome ω. Let V be a basis matrix for ker(A11 − jωI). Then

(A11 − jωI

)V = 0 (5.41)

which gives

V ∗ (A∗

11 + jωI)

= 0

Adding and subtracting jωΣ1, Eqns. (5.23 and 5.24) can be rewritten as


(A11 − jωI

)Σ1 + Σ1

(A∗

11 + jωI)

+ B1B∗1 = 0 (5.42)

Σ1

(A11 − jωI

)+

(A∗

11 + jωI)Σ1 + C∗

1 C1 = 0 (5.43)

Multiplication of Eqn. (5.43) from right by V and from left by V ∗ gives V ∗C∗1C1V = 0,

which is equivalent to

C1V = 0.

Multiplication of Eqn. (5.43) from the right by V now gives(A∗

11 + jωI)Σ1V = 0

Analogously, first multiply Eqn. (5.42) from the right by Σ1V and from the left by V ∗Σ1

to obtain

B∗1Σ1V = 0

Then multiply Eqn. (5.42) from the right by Σ1V to get(A11 − jωI

)Σ2

1V = 0

It follows that the columns of Σ21V are in ker

(A11 − jωI

). Therefore, there exists a

matrix Σ1 such that

Σ21V = V Σ2

1

Since Σ21 is the restriction of Σ2

1 to the space spanned by V, it follows that it is possibleto choose V such that Σ2

1 is diagonal. It is then also possible to choose Σ1 diagonal and suchthat the diagonal entries of Σ1 are a subset of the diagonal entries of Σ1.

Multiply Eqn. (5.25) from the right by Σ1V and Eqn. (5.26) by V to get

A21Σ21V + Σ2A

∗21Σ1V = 0

Σ2A21V + A∗12Σ1V = 0

which gives(A21V

)Σ2

1 = Σ22

(A21V

)

This is a Sylvester equation (refer [34]) in A21V. Because Σ21 and Σ2

2 have no diagonalentries in common it follows from Lemma 2 that

A21V = 0 (5.44)

is the unique solution. Now Eqn. (5.44 and 5.41) imply that[

A11 A12

A21 A22

] [V0

]= jω

[V0

]

which means that A-matrix of the original system has an eigenvalue at jω. This con-tradicts the fact that the original system is asymptotically stable. Therefore, A11 must beasymptotically stable.¥


Theorem 2 Suppose G(s) ∈ RH∞ and

G(s) =

A11 A12

A21 A22

B1

B2

C1 C2 D

is a balanced realization with Grammian Σ =diag(Σ1, Σ2).

Σ1 = diag(σ1Is1 , σ2Is2 , · · · , σrIsr)

Σ2 = diag(σr+1Isr+1 , σr+2Isr+2 , · · · , σNIsN)

and

σ1 > σ2 > · · · > σr > σr+1 > · · · > σN

where σi has multiplicity si, i = 1, 2, · · · , N and s1 +s2 + · · ·+sN = n. Then the truncatedsystem

Gr(s) =

[A11 B1

C1 D

]

is balanced and asymptotically stable. Furthermore

‖G(s)−Gr(s)‖∞ ≤ 2 (σr+1 + σr+2 + · · ·+ σN)

and the bound is achieved if r = N − 1, i.e.,

‖G(s)−GN−1(s)‖∞ = 2σN

Proof : The stability of Gr follows from Theorem 1. We shall now give a direct proofof the error bound for the case si = 1 for all i. Hence, we assume si = 1 and N = n.An alternative proof will be given later where the singular values σi are not assumed to bedistinct.

Let

φ(s) : =(sI − A11

)−1

ψ(s) : = sI − A22 − A21φ(s)A12

B(s) : = A21φ(s)B1 + B2

C(s) : = C1φ(s)A12 + C2

then using the partitioned matrix results in section 5.1.4,

G(s)−Gr(s) = C(sI − A)−1B − C1φ(s)B1

=[

C1 C2

] [sI − A11 −A12

−A21 s− A22

]−1 [B1

B2

]− C1φ(s)B1

C(s)ψ−1(s)B(s)


computing this quantity on the imaginary axis to get

σ [G(jω)−Gr(jω)] = λ1/2max

[ψ−1(jω)B(jω)B∗(jω)ψ∗−1(jω)C∗(jω)C(jω)

](5.45)

Expressions for B(jω)B∗(jω) and C∗(jω)C(jω) are obtained by using the partitionedform of the internally balanced Grammian equations ( Eqns. (5.23-5.28).)

An expression for B(jω)B∗(jω) is obtained by using the definition of B(s), substitutingfor B1B

∗1 , B1B

∗2 and B2B

∗2 from the partitioned form of the grammian Eqns. (5.23-5.25), we

get

B(jω)B∗(jω) = ψ(jω)Σ2 + Σ2ψ∗(jω).

The expression for C∗(jω)C(jω) is obtained analogously and is given by

C∗(jω)C(jω) = Σ2ψ(jω) + ψ∗(jω)Σ2.

These expressions for B(jω)B∗(jω) and C∗(jω)C(jω) are then substituted in Eqn. (5.45)to obtain

σ [G(jω)−Gr(jω)] = λ1/2max

[Σ2 + ψ−1(jω)Σ2ψ

∗(jω)] [

Σ2 + ψ∗−1(jω)Σ2ψ(jω)]

Now consider one-step order reduction, i.e., r = n− 1, then Σ2 = σn and

σ [G(jω)−Gr(jω)] = σnλ1/2max

[Σ2 + Θ−1(jω)

][Σ2 + Θ(jω)]

(5.46)

where Θ := ψ∗−1(jω)ψ(jω) = Θ∗−1 is an ’all pass’ scalar function. (This is the only placewe need the assumption of si = 1). Hence |Θ(jω)| = 1.

Using the triangle inequality we get

σ [G(jω)−Gr(jω)] ≤ σn [1 + |Θ(jω)|] = 2σn (5.47)

This completes the bound for r = n− 1.The remainder of the proof is achieved by using the order reduction by one step results and

by noting that Gk(s) =

[A11 B1

C1 D

]obtained by the ’k-th’ order partitioning is internally

balanced with balanced grammian given by

Σ1 = diag (σ1Is1 , σ2Is2 , · · · , σkIsk)

Let Ek(s) = Gk+1(s)−Gk(s) for k = 1, 2, · · · , N − 1 and let GN(s) = G(s). Then

σ [Ek(jω)] ≤ 2σk+1

since Gk(s) is a reduced order model obtained from the internally balanced realization ofGk+1(s) and the bound for one step reduction, Eqn. (5.47) holds.

Noting that


G(s)−Gr(s) =N−1∑

k=rEk(s)

by definition of Ek(s), we have

σ [G(jω)−Gr(jω)] ≤N−1∑

k=rσ [Ek(s)] ≤ 2

N−1∑k=r

σk+1

This is the desired upper bound.To see that the bound is actually achieved when r = N −1, we note that Θ(0) = I. Then

the right hand side of Eqn. (5.46) is 2σN at ω = 0.¥

5.2.6 Frequency-Weighted Balanced Model Reduction

Given the original full order model G ∈ RH∞, the input weighting matrix Wi ∈ RH∞ andthe output weighting matrix Wo ∈ RH∞, our objective is to find a lower order model Gr

such that

‖Wo (G−Gr) Wi‖∞is made as small as possible. Assume that G,Wi and Wo have the following state space

realizations.

G =

[A BC 0

],Wi =

[Ai Bi

Ci Di

],Wo =

[Ao Bo

Co Do

]

with A ∈ Rn×n. Note that there is no loss of generality in assuming D = G(∞) = 0 sinceotherwise it can be eliminated by replacing Gr with D + Gr.

Now the state space realization for the weighted transfer matrix is given by

WoGWi =

A 0 BCi

BoC Ao 00 0 Ai

BDi

0Bi

DoC Co 0 0

=:

[A B

C 0

]

Let P and Q be the solutions of the following Lyapunov equations

AP + P A∗ + BB∗ = 0 (5.48)

QA + A∗Q + C∗C = 0 (5.49)

Then the input weighted Grammian P and output weighted Grammian Q are defined by

P : =[

In 0]P

[In

0

]

Q : =[

In 0]Q

[In

0

]


It can be shown easily that P and Q can satisfy the following lower order equations

[A BCi

0 Ai

] [P P12

P ∗12 P22

]+

[P P12

P ∗12 P22

] [A BCi

0 Ai

]+

[BDi

Bi

] [BDi

Bi

]∗= 0

[Q Q12

Q∗12 Q22

] [A 0BoC Ao

]+

[A 0BoC Ao

]∗ [Q Q12

Q∗12 Q22

]+

[C∗D∗

o

C∗o

] [C∗D∗

o

C∗o

]∗= 0

The computation can be further reduced if Wi = I or Wo = I. In the case of Wi = I, Pcan be obtained from

PA∗ + AP + BB∗ = 0 (5.50)

while in the case of Wo = I, Q can be obtained from

QA + A∗Q + C∗C = 0 (5.51)

Now let T be a nonsingular matrix such that

TPT ∗ =(T−1

)∗QT−1 =

[Σ1

Σ2

]

(i.e., balanced) with

Σ1 = diag(σ1Is1 , σ2Is2 , · · · , σrIsr)

Σ2 = diag(σr+1Isr+1 , σr+2Isr+2 , · · · , σnIsn)

and partition the system accordingly as

[TAT−1 TBT−1C 0

]=

A11 A12

A21 A22

B1

B2

C1 C2 0

Then the reduced order model Gr is obtained as

Gr =

[A11 B1

C1 0

]

Remark 5 Unfortunately, there is generally no known a priori error bound for the approx-imation error and the reduced model Gr is not guaranteed to be stable either.

5.3 Model Reduction by Impulse/Step Error Minimiza-

tion

In this approach, an error function, which is constructed from the time responses of the sys-tem and the reduced order models is converted into frequency domain and the minimizationis carried out in this domain. The denominator is selected by retaining the dominant poles

5.3 Model Reduction by Impulse/Step Error Minimization 83

or by the Routh model. The numerator of the reduced order model is obtained by minimiz-ing the step/impulse response error along with a steady state constraint. The step/impulseerror is directly expressed in terms of the model parameters by evaluating certain integralswhich need the inversion of rth and (n + r)th order matrices. The impulse/step response er-ror function minimization problem is converted to a problem of solving simultaneous linearequations. These equations are solved to obtain the model parameters uniquely.

5.3.1 Impulse Error Minimization

Let the transfer function of an nth order SISO system be given as

H(s) =a0 + a1s + · · ·+ an−1s

n−1

b0 + b1s + · · ·+ bn−1sn−1 + bnsn(5.52)

and the rth order (r < n) reduced order model of the system (5.52) with unknown coef-ficients given by

Hr(s) =c0 + c1s + · · ·+ cr−1s

r−1

d0 + d1s + · · ·+ dr−1sr−1 + drsr(5.53)

The denominator coefficients of Hr(s) are obtained by dominant pole retention or Routhapproximation methods. The numerator is obtained by minimizing the impulse responseerror while satisfying the steady state constraint.

To match the steady state values of the system and the model,

c0

d0

=a0

b0

Let h(t) and hr(t) be the impulse response of the system and the reduced model, thenthe impulse response error is defined as

e , ‖h(t)− hr(t)‖2 =

∞∫

0

[h(t)− hr(t)]2 dt

=

∞∫

0

h2(t)dt+

∞∫

0

h2r(t)dt− 2

∞∫

0

h(t)hr(t)dt (5.54)

or

e =1

2πj

j∞∫

−j∞H(s)H(−s)ds +

1

2πj

j∞∫

−j∞Hr(s)Hr(−s)ds− 2

2πj

j∞∫

−j∞H(s)Hr(−s)ds

(5.55)The integrals in Eqn. (5.55) can be evaluated by a process given in [36] in terms of the

coefficients ai, bi of H(s) and ci, di of Hr(s). A table providing the values of the definiteintegrals in terms of the coefficients has also been provided in [37]. Only the coefficients


ci, i = 1, 2, · · · , r − 1 are unknown at this stage. The integral 12πj

j∞∫−j∞ H(s)Hr(−s)ds can

also be expressed in terms of ci, by extending the approach in [36] and is discussed in [38].Thus,

e = K + Ar−1 − [En−1 + Dr−1]

where the components Ar−1, En−1, Dr−1 are computed by solving the following equations.The value of Ar−1 can be found from the solution of

BA = C (5.56)

where,

B =

d0 0 0d2 −d1 d0 0

d4 −d3...

......

0 0 (−1)r−1dr−1

A =

A0

A1...Ar−2

Ar−1

,C =

C0

C2...C2r−4

C2r−2

where

2Cm =m∑

k=0(−1)k ckcm−k for 0 ≤ m ≤ r − 1

=r−1∑

k=m−r+1(−1)k ckcm−k for r ≤ m ≤ 2r − 2

Here, the vectors A and C are unknown. By inverting D, an expression for Ar−1 can bedeveloped i.e.,

Ar−1 =[Last Row of B−1

][C] (5.57)

Similarly the third term for the expression of ’e’ (En−1 +Pr−1) can be obtained by solving

d0 0 0 · · · 0 b0 0 · · · 0−d1 d0 0 · · · 0 b1 −b0 · · · 0d2 −d1 d0 · · · 0 b2 −b1 · · · 0−d3 d2 −d1 · · · 0 b3 −b2 · · · 0...

......

......

......

0 0 · · · 0 (−1)r dr 0 0 · · · bn

E0

E1...En−1

D0...Dr−1

=

P0

P1

P2...

Pn+r−2

Pn+r−1

(5.58)


where

Pm =m∑

k=0(−1)m−k akcm−k for 0 ≤ m ≤ r − 1

=m∑

k=m−r+1(−1)m−k akcm−k for r ≤ m ≤ n− 1

=n−1∑

k=m−r+1(−1)m−k akcm−k for n ≤ m ≤ n + r − 1

or symbolically NF = P where N is a (n + r)× (n + r) known square matrix, F and Pare (n + r)× 1 unknown vectors. Again by inverting N, the [En−1 + Pr−1] can be evaluatedas

[En−1 + Dr−1] =([

nth row of N−1]+

[(n + r)th row of N−1

])[P ] (5.59)

Thus, the expression for ’e’ becomes

e = K +[Last Row of B−1

][C] +

([nth row of N−1

]+

[(n + r)th row of N−1

])[P ] (5.60)

The minimization of ’e’ with respect to ci, will yield (r − 1) linear equations which canbe solved to obtain a unique solution and thus ci, i = 1, 2, · · · , r− 1 can be obtained. Differ-entiating to find the minimizers,

∂e

∂ci

= 0, i = 1, 2, · · · , r − 1

=∂Ar−1

∂ci

− ∂ [En−1 + Pr−1]

∂ci

= 0

Now,

Ar−1 =[

α0 α1 · · · αr−1

]

C0

C2...C2r−2

where,[

α0 α1 · · · αr−1

]is the last row of B−1 or

Ar−1 =[

α0 α1 · · · αr−1

]

0.5c20

c0c2 − 0.5c21

...−cr−3cr−1 + 0.5cr−2

−0.5c2r−1

(5.61)

= αQ (5.62)


The last term in the column vector in Eqn. (5.61) is c2r−1 for r being odd. Differentiating,

we get

∂Ar−1

∂cj

=

(r−1)/2∑i=0

αj+ic2i for even j (5.63)

= −(r−1)/2∑

i=1αj+ic2i−1 for even j

From Eqn. (5.63), the following may be inferred:

1. for r - even

(a) for i-even, all terms from q0 to qi−2 are zeros, from qi to qr+i−2 are cj, j =0, 2, · · · , r − 2, and from qr+i to q2r−2 are zeros.

(b) for i-odd, all terms from q0 to qi−1 are zeros, from qi+1 to qr+i−1 are -cj, j =1, 3, · · · , r − 1, and from qr+i+1 to q2r−2 are zeros.

2. for r - odd

(a) for i-even, all terms from q0 to qi−2 are zeros, from qi to qr+i−3 are cj, j =0, 2, · · · , r − 1, and from qr+i−1 to q2r−2 are zeros.

(b) for i-odd, all terms from q0 to qi−1 are zeros, from qi+1 to qr+i−2 are -cj, j =1, 3, · · · , r − 2, and from qr+i to q2r−2 are zeros.

Again,

[En−1 + Dr−1] =[

β0 β1 β2 · · · βn+r−1

]

P0

P1

P2...Pn+r−1

where,βi = pi + ri, i = 0, 1, · · · , n + r − 1 (5.64)

pi and ri are the elements of the nth and the (n + r)th rows of N−1. Thus,

[En−1 + Dr−1] =[

β0 β1 β2 · · · βn+r−1

]

a0c0

a1c0 − a0c1

a2c0 − a1c1 + a0c2...an−1cr−2 − an−2cr−1

−an−1cr−1

0

(5.65)


Thus,

∂ [En−1 + Dr−1]

∂cj

= (−1)jn−1∑

i=0βi+jai, j = 1, · · · , r − 1 (5.66)

Using Eqns. (5.63 and 5.66), the set of differential equations

∂e

∂ci

=∂Ar−1

∂ci

− ∂ [En−1 + Pr−1]

∂ci

= 0

can be expressed thus, in matrix form as,If ’r’ is even

α1 0 α2 · · · 0 αr/2

0 α2 0 · · · αr/2 0α2 0 α3 · · · 0 α r

2+1

......

......

...0 αr/2 0 · · · αr−2 0αr/2 0 α r

2+1 · · · 0 αr−1

c1

c2

c3......cr−2

cr−1

=

β1 β2 · · · βn

β2 β3 · · · βn+1

β3 β4 · · · βn+2...

......

βr−2 βr−1 · · · βn+r−2

βr−1 βr · · · βn+r−1

a0

a1

a2...

an−2

an−1

−

0α1c0

0α2c0...αr/2c0

0

(5.67)

If ’r’ is odd

α1 0 α2 · · · α r−12

0

0 α2 0 · · · 0 α r+12

α2 0 α3 · · · α r+12

0...

......

......

α r−12

0 α r+12

· · · αr−2 0

0 α r+12

0 · · · 0 αr−1

c1

c2

c3......cr−2

cr−1

=

β1 β2 · · · βn

β2 β3 · · · βn+1

β3 β4 · · · βn+2...

......

βr−2 βr−1 · · · βn+r−2

βr−1 βr · · · βn+r−1

a0

a1

a2...

an−2

an−1

−

0α1c0

0α2c0...0α r−1

2

(5.68)


or symbolically

MC = β (5.69)

where M is a known ((r − 1)× (r − 1)) known matrix and β is an ((r − 1)× 1) knownvector. Thus. C can be uniquely obtained from the above set of equations. For existence ofa unique solution, the matrix M must be non-singular.

5.3.2 Step Error Minimization

The model reduction approach using step response error minimization has been handled byLamba et al in [39]. Let C(t) and Cr(t) be the step response of the system and the reducedorder model, the step response error is given by

e = ‖C(t)− Cr(t)‖= ‖g(t)− gr(t)‖ (5.70)

where g(t) = C(∞)− C(t) and gr(t) = Cr(∞)− Cr(t) are the transient part of the stepresponse. To match steady state values of the system and the model.,

Cr(∞) =c0

d0

= C(∞) =a0

b0

Then, G(s) = L−1 [g(t)] and Gr(s) = L−1 [gr(t)] can be expressed as

G(s) =C(∞)

s− H(s)

s=

a0

b0s− a0 + a1s + · · ·+ an−1s

n−1

(b0 + b1s + · · ·+ bnsn) s

=a0 + a1s + · · ·+ an−1s

n−1

b0 + b1s + · · ·+ bnsn

and

Gr(s) =c0 + c1s + c2s

2 + · · ·+ cr−1sr−1

d0 + d1s + d2s2 + · · ·+ drsr

where

ai =a0

b0

bi+1 − ai+1, i = 0, 1, · · · , n− 2, an−1 =a0

b0

bn (5.71)

and

ci =c0

d0

di+1 − ci+1, i = 0, 1, · · · , r − 2, cr−1 =c0

d0

dr (5.72)

Then,


e = ‖g(t)− gr(t)‖2 = ‖g(t)‖2 + ‖gr(t)‖2 − 2 〈g(t), gr(t)〉

=1

2πj

j∞∫

−j∞G(s)G(−s)ds+

j∞∫

−j∞Gr(s)Gr(−s)ds+

j∞∫

−j∞G(s)Gr(−s)ds

The integrals can be evaluated as for the impulse error case, then

e = K + A− [En−1 + Dr−1

]

here cr−1 is completely known from Eqn. (5.72). Thus, the minimization of e with respectto ci, i = 0, 1, · · · , r − 2 by

∂e

∂ci

= 0, i = 0, 1, 2, · · · , r − 2

yields (r− 1) linear equations, which can be obtained with reasonable effort by applyinga procedure similar to that for the impulse error minimization. These equations can berepresented in matrix form as

α0 0 α1 · · · αr/2−1

0 α1 0 · · · 0α1 0 α2 · · · αr/2...

......

...0 αr/2−1 · · · αr−3 0αr/2−1 0 αr/2 · · · αr−2

c0

c1

c2...cr−3

cr−2

=

β0 β1 · · · βn−1

β1 β2 · · · βn

β2 β3 · · · βn+1...

......

βr−3 βr−2 · · · βn+r−4

βr−2 βr−1 · · · βn+r−3

a0

a1

a2...an−2

an−1

−

0αr/2cr−1

0...αr−2cr−1

0

(5.73)

for r- even.For r- odd, the matrix equation is

α0 0 α1 · · · α r−32

0

0 α1 0 · · · 0 α r−12

α1 0 α2 · · · α r−12

0...

......

...α r−3

20 α r−1

2· · · αr−3

0 α r−12

0 · · · αr−2

c0

c1

c2...cr−3

cr−2

=

β0 β1 · · · βn−1

β1 β2 · · · βn

β2 β3 · · · βn+1...

......

βr−3 βr−2 · · · βn+r−4

βr−2 βr−1 · · · βn+r−3

a0

a1

a2...an−2

an−1


−

α r−12

cr−1

0α r+1

2cr−1

...0αr−2cr−1

(5.74)

or symbolically

RC = T (5.75)

where,R is a known ((r − 1)× (r − 1)) matrix and T is a known ((r − 1)× 1) vector. Thus

C can be uniquely found. To find the coefficients ci, i = 1, · · · , r − 1 the following matrixequation can be used.

c1

c2

c3...cr−2

cr−1

=c0

d0

d1

d2

d3...dr−2

dr−1

−

c0

c1

c2...cr−3

cr−2

. (5.76)

Thus the reduced order model can be obtained.

5.4 Optimal Model Order Reduction Using Wilson’s

Technique

The problem of deducing a reduced order model of a large system by error minimization wasconsidered by Wilson first in [?,?]. The full and reduced order models were represented instate space form and the error in their impulse response or white noise response is reduced.

5.4.1 Impulse Error / White Noise Error Minimization

The technique can be used for the reduction of multivariable systems represented in thefollowing structure.

x = Ax + Bu (5.77)

y = Hx

where, x ∈ Rn, u ∈ Rm, y ∈ Rp.The problem is to find an rth order reduced state-space representation with p ≤ r ≤ n of

the form

5.4 Optimal Model Order Reduction Using Wilson’s Technique 91

xr = Arxr + Bru (5.78)

yr = Hrxr

To synthesize the reduced order model given by Eqn. (5.78), a functional of the reductionerror will be minimized which is given as

J =

∞∫

0

eT (t)e(t)dt

where, e(t) = y(t)− yr(t) = Hx(t)−Hrxr(t)

eT (t)e(t) = [Hx(t)−Hrxr(t)]T [Hx(t)−Hrxr(t)]

=[

xT xTr

] [HT H −HT Hr

−HTr H HT

r Hr

] [xxr

]

= ZT MZ (5.79)

where,

Z =

[xxr

](5.80)

M =

[HT H −HT Hr

−HTr H HT

r Hr

]

Now,

J =

∞∫

0

ZT (t)MZ(t) = ZT (0)PZ(0) = Tr(PZ(0)ZT (0)

)(5.81)

It was shown by Wilson [?] that the minimization of J with respect to the parameters ofAr, Br, Hr involves the solution of the matrix equations

F T P + PF + M = 0 (5.82)

FR + RF T + S = 0 (5.83)

where,

S = Z(0)ZT (0) =

[BNBT BNBT

r

BrNBT BrNBTr

]⇒ J = Tr(PS)

N = p× p positive definite, symmetric noise matrix.

F =

[A 00 Ar

]

R =

∞∫

0

Z(t)ZT (t)dt ⇒ J = Tr(MR) = Tr(RM)


It is a well known fact that R would the solution of the linear matrix equation (5.83).Now, consider the problem of finding the derivatives of the cost function above with

respect to a parameter β appearing in the elements of F and M.

∂J

∂β= Tr

(∂P

∂βS

)+ Tr

(P

∂S

∂β

)(5.84)

Considering the evaluation of J, by using white noise input would give J =Tr(MR) =Tr(RM)Substituting for S in Eqn. (5.84)

∂J

∂β= −Tr

(∂P

∂βFR

)− Tr

(∂P

∂βRF T

)+ TrP

∂S

∂β

= −2Tr

(∂P

∂βFR

)+ Tr

(P

∂S

∂β

)(5.85)

Differentiating Eqn. (5.82), we have

∂P

∂βF + P

∂F

∂β+ F

∂P

∂β+

∂F T

∂βP +

∂M

∂β= 0 (5.86)

Postmultiplying by R and taking trace

0 = Tr

(∂P

∂βFR

)+ Tr

(P

∂R

∂βR

)+ Tr

(F

∂P

∂βR

)

+Tr

(∂F T

∂βPR

)+ Tr

(∂M

∂βR

)

−2Tr

(∂P

∂βFR

)= 2Tr

(∂F

∂βPR

)+ Tr

(∂M

∂βR

)(5.87)

Substituting Eqn. (5.87) in Eqn. (5.85)

∂J

∂β= 2Tr

(∂F

∂βPR

)+ Tr

(∂M

∂βR

)+ Tr

(P

∂S

∂β

)(5.88)

Since P and R are symmetrical matrices, let

P =

[P11 P12

P T12 P22

], R =

[R11 R12

RT12 R22

]

It has been shown with considerable elaboration in Wilson’s paper that using Eqn. (5.88)for the derivative of the cost function with respect to the parameters in F ,M and S, thefollowing conditions on Ar, Br and Hr can be found

1. Equating∂J

∂br= 0, where br is an element of Br,

Br = −P−122 P T

12B (5.89)

5.4 Optimal Model Order Reduction Using Wilson’s Technique 93

2. Equating∂J

∂hr= 0, where hr is an element of Hr

Hr = HR12R−122 (5.90)

3. Equating∂J

∂ar= 0, where ar is an element of Ar

Ar = −P−122 P T

12AR12R−122 (5.91)

Defining Θ1 = −P−122 P T

12 and Θ2 = R12R−122 , we get

xr = Θ1AΘ2xr + Θ1Bu (5.92)

y = HΘ2xr

The solution equations, Eqns. (5.89 - 5.91) contain four unknown matrices. The followingrelation is available to have four relations for the four unknown matrices.

RT12P12 + R22P22 = 0

However, no explicit solution for these matrices is apparently possible. It is possible tothink in terms of the two composite unknown matrices Θ1 and Θ2 which define the reducedmodel equation. Two equations in terms of Θ1 and Θ2 can also be obtained by substitutingrelations in Eqn. (5.92) in Eqns. (5.82 and 5.83).

This yields a set of nonlinear equations in Θ1 and Θ2 with unique solutions. Select-ing Ar and Br by Routh approximations, the unknown Hr can be evaluated of the abovedeliberations in the following manner.

From Eqn. (5.83), we get,

[A 00 Ar

] [R11 R12

RT12 R22

]+

[R11 R12

RT12 R22

] [A 00 Ar

]+

[BNBT BNBT

r

BrNBT BrNBTr

]= 0

which implies

AR12 + R12ATr + BNBT

r = 0 (5.93)

ArR22 + R22ATr + BrNBT

r = 0 (5.94)

Eqns. (5.93 and 5.94) are Sylvester equations in R12 and R22. Solving them would lead tothe unique solution of Hr using Eqn. (5.90). A more detailed discussion is provided in [42].

Chapter 6

Pole Placement Techniques

6.1 Introduction

The design of control system by pole assignment has two main features - the choice of polelocations to meet the design specification and the design of the controller to achieve thesedesired pole locations. In dealing with pole assignment it should be remembered that it dealsbasically with the design of system transient response, a very specific quantity. Since thisis but one aspect of system design in some ways pole assignment is restricted, specializeddesign technique in case of SISO systems. On the other hand, in multivariable system designit is a very direct way of bringing out and exploiting the extra degrees of freedom inherent inmulti-input systems and it is likely that a complete solution to the pole assignment problemwill provide a vehicle for a much broader design than purely transient response.

6.2 What Poles to Choose ?

6.2.1 General

The transfer function of a linear time-invariant model of a single-input single-output systemcan be represented in pole-zero configuration as

G(s) =k (s− z1) (s− z2) · · · (s− zm)

(s− p1)(s− p2) · · · (s− pn), n ≥ m

=k1

(s− p1)+

k2

(s− p2)+ · · ·+ kn

(s− pn)(6.1)

It is uniquely defined by its poles s = p1, p2, · · · , pn, and its zeros s = z1, z2, · · · , zm andthe gain multiplier k. The unit impulse response

g(t) = k1ep1t + k2e

p2t + · · ·+ knepnt

is closely linked to the pole-zero configuration. It is this link which is main attractionsof the technique, it allows the time response to be interpreted from the pole-zero locationsand conversely, it specifies where the poles and zeros should assigned to achieve a desiredtime response and other system specifications.

95


6.2.2 Interpretation of Responses from Pole-Zero Locations

Single real pole

G(s) =1

s + a, g(t) = e−at

As the pole is moved to the left, the response speeds up; time constant is1

a, the response

is substantially complete in five time constants.

Two poles

In the normalized form

G(s) =ω2

n

s2 + 2ςωns + ω2n

, 0 ≤ ς < 1

has a complex-conjugate pair lying on semicircle of radius ωn with real parts= ςωn andimaginary parts=ωn

√(1− ς2) = damped oscillation frequency of response. Time to first

peak of step response isπ

ςωn. Thus moving out on the radius from the origin speeds up the

response with its form unchanged, moving away from negative-real axis reduces damping.ς = 1, double pole at s = −ωn, critically damped response. 1 < ς ≤ 2, two real poles, onemoving to s = −∞ and the other to s = 0 as ς is increased.

Addition of a real pole to complex-conjugate pair

G(s) =ω2

n

(s2 + 2ςωns + ω2n) (s + a)

When the pole is well to the left, the residue at the real pole would be small and thecontribution to response over in short time, so effect is negligible. As ’a’ decreases, the effectis stabilizing; response is aperiodic when a ≤ ςωn, and becoming sluggish.

Addition of a real zero to complex-conjugate pair

G(s) =

ω2n

(1 +

s

a

)

(s2 + 2ςωns + ω2n)

The addition of a zero adds derivative of the original system response scaled by1

a,

having a destabilizing effect as ’a’ is reduced.

6.3 Pole Assignment in Single Input Systems 97

Poles well to the left

If well to the left, the effect is negligible as in 6.2.2 above, and as a rule of thumb polescan be neglected if the real parts are at least six times as far from the imaginary axis asdominant poles.

Dipoles

Poles and zeros close together are called as dipoles. They effectively cancel and pole hasnegligible residue. In the same way a cluster of P poles and Z zeros can be replaced by(P − Z) poles at the centre of gravity.

Broad specification of pole-zero location

As part of the control system specification, the step response rise-time, overshoot and com-pletion time will be specified. A broad boundary can be placed on the pole zero locationscan be placed as a result. For example, for completion in T sec all poles and zeros must lie

to the left of s = − 5

T; the precise response is of course determined by actual location of

poles within this boundary.

Note : Refer [43] for further explanations in this area.

6.3 Pole Assignment in Single Input Systems

There are now several methods for designing state and output feedback controllers for poleplacement in single-input systems. Attention is mainly restricted here to one particularmethod.

6.3.1 State Feedback

The basis of pole assignment stemmed from the state variable representation of linear sys-tems. This representation and results of optimal control theory led to the concept of feedingback the system states rather than a single input as in the case of classical control. Theconcept automatically introduces more degrees of freedom in the state-feedback coefficients,for freer pole assignment than in the classical case. It also allows direct choice of coefficientsrather than the iterations of the classical method.

State Feedback for Pole Assignment

Consider a completely controllable and completely observable system single-input system.

x = Ax + bu (6.2)

y = Cx


where x is a n-column state vector, u is the scalar-control input and y is a p-columnoutput vector. The aim is to find the relation between the states and the input (u = Fx)so that the system responds in a desired manner (i.e., the closed loop pole assignment, poleplacement region, performance criteria conformation etc.,).

As discussed above, the state feedback method is used to design a controller that wouldgenerate a control input as a linear combination of the system states.

u =n∑

i=1kixi = kx (6.3)

where k is a n-column gain vector. If the poles of the closed loop system need to be placedat specific locations, then the state feedback gain that would achieve it can be computed inthe following manner.

1. Transform the system representation (A, B) into its controllable canonical form (Ac, Bc)using a transformation, say Tz = x.

Then the transformation matrix would be T = PM (From [44])

where

P =[

B AB · · · An−1B]

M =

a1 a2 · · · an−1 1a2 a3 · · · 1 0...

......

...an−1 1 · · · 0 01 0 · · · 0 0

2. Compute the open loop characteristic polynomial and the desired closed loop charac-teristic polynomial of the system. Let open loop characteristic polynomial be

sn + an−1sn−1 + an−2s

n−2 + · · ·+ a1s + a0 = 0 (6.4)

and that desired from the closed loop system be

sn + an−1sn−1 + an−2s

n−2 + · · ·+ a1s + a0 = 0 (6.5)

Now, if a state feedback F =[

f1 f2 · · · fn

]is applied to the system in the phase

variable canonical form, it is well known that the characteristic polynomial in Eqn.(6.4) would change to

sn + (an−1 + fn−1) sn−1 + (an−2 + fn−2) sn−2 + · · ·+ (a1 + f1) s + (a0 + f0) = 0 (6.6)

3. Equating the coefficients of the powers of s in Eqns. (6.5 and 6.6), the values offi, i = 1, 2, · · · , n and hence F can be found.


4. The state feedback gain F ′, that gives the desired closed loop characteristic equationwhen applied to the (A, B) system representation can be calculated as

F ′ = FT−1 (6.7)

6.3.2 Optimal State Feedback ( Brief Introduction to LQR )

It is often the case that the closed loop system needs to have a minimal value of a ’cost’function, rather than having a set of pre-determined closed loop poles. The cost function (or, to be more precise, cost functional) would be of the form [45]

J(u) =

∞∫

0

(xT Qx + uT Ru

)dt (6.8)

The aim is to find an input function u∗(t) so that J is minimized.

u∗ = minu

∞∫

0

(xT Qx + uT Ru

)dt

(6.9)

Using the relationship between u and x from Eqn. (6.2), and incorporating it into Eqn.(6.9) with a constant (but unassigned) weighting factor λ,

u∗ = minu

∞∫

0

(xT Qx + uT Ru

)+ λ (x− Ax−Bu) dt

(6.10)

Now, using the Euler-Lagrange equation

L = min

∫H(p, p)dt

∣∣∣∣ ⇒ H(p, p) = (p∗, p∗)

⇒ ∂H

∂p

∣∣∣∣(p∗,p∗)

=d

dt

(∂H

∂p

)∣∣∣∣(p∗,p∗)

and the Pontryagin’s maximum principle, we would have

2Qx− AT λT = λ = 0

2Ru−BT λT = 0

Solving for u in terms of x using λ, we get the optimal state feedback as

u∗ = −R−1BT Px (6.11)

where P is the solution of the algebraic Riccati equation

AT P + PA− PBR−1BT P + Q = 0 (6.12)


Pole Placement to the left of a line real(s) = −α

Consider the system in (6.2). It is desired to design a state feedback that would minimizethe performance index specified in (6.9). The procedure discussed in the previous sectionperforms the task an provides a stabilizing compensator. But, there is no indication of thedegree of stability assured by the compensator. In other words, the distance at which thepoles would lie to the left of the imaginary axis in s plane. Anderson and Moore [46] hadderived a procedure by which the closed loop poles can be obtained with a desired degree ofstability.

Now, we would try to incorporate the constraint of the closed loop poles to be to theright of a line real(s) = −α, α > 0, by including the factor e2αt in the performance index,thus changing it to

J1 =1

2

∞∫

0

(xT Qx + uT Ru

)e2αtdt (6.13)

and minimize the index subject to the constraint (6.2).Let

x = xeαt, u = ueαt (6.14)

then Eqn. (6.2) is equivalent to

.

x= (A + Iα) x + Bu (6.15)

and the performance index in (6.13) is equivalent to

J1 =1

2

∞∫

0

(xT Qx + uT Ru

)dt (6.16)

Now using the Riccati equation

(AT + Iα

)P ′ + P ′ (A + Iα)− P ′BR−1BT P ′ + Q = 0 (6.17)

the optimal state feedback can be calculated as

u = −R−1BT P ′x

ueαt = −R−1BT P ′eαtx

u = −R−1BT P ′x (6.18)

Thus, the poles of the closed loop system can be ensured to be to the left of a linereal(s) = −α by finding the optimal controller for the auxiliary system

.

x= (A + Iα) x + Bu

Pole Placement within a Cone

The procedure of obtaining an optimal controller gain with the closed loop poles within acone in the complex s plane was discussed by Pal and Mahalanabis [47]. The technique thatis adapted to obtain the optimal controller gain is as follows.


Consider the transfer function form of the state equation in the space representation(6.2).

G(s) = (sI − A) BK (6.19)

=g(s)

F (s)(6.20)

where F (s) is the characteristic equation of the state matrix.

F (s) = det (sI − A)

Now, consider the return difference of the closed loop system,

T (s) = 1 + KT G(s)

Then, for frequency domain condition for optimality,

T (s)T (−s) = 1 + R−1GT (−s)QG(s) (6.21)

Note that T (s) can be expressed as

T (s) =h(s)

F (s)

where h(s) = det (sI − A + BK) .The problem is to choose a Q = ddT , where d is a n-column vector satisfying the criterion

(A, d) being observable. This will render the closed loop system stable [45,48]. This is donein the following manner.

Consider the polynomial

H(z) = h1(z)h2(z)

h1(z) = h(zej(π/2+θ)

)

h2(z) = h(zej(π/2−θ)

)

It can be verified that the above transformations translate the cone bounded by lines

tan−1 real(s)/Im(s) = −θ

in the s plane to the imaginary axis in the complex z plane. Thus checking for the Hurwitzstability of H(z) ensures that the closed loop poles of the system lie within the stable conewith an included angle of 2θ.

Example 8 Consider the system

x =

[0 1−1 0

]x +

[01

]u

then desired θ = 60.


SOLUTION: Let

h(s) = s2 + c1s + c0

H(z) = z4 + a3z3 + a2z

2 + a1z + a0

Checking the stability of H(z) using Hurwitz determinants, we obtain

a0 = c20 (6.22)

a1 = 2c0c1 cos (π/2− θ) =√

3c0c1 (6.23)

a2 = c21 + 2c0 cos(π − 2θ) = c2

1 + c0 (6.24)

a3 = 2c1 cos(π/2− θ) =√

3c1 (6.25)

0 ≤ a1a2a3 − a21 − a0a

23 (6.26)

The last inequality, obtained from the Hurwitz determinant being positive, can be rep-resented in terms of ci as

c21c0

(c21 − c0

) ≥ 0

Choosing c1 =√

3/2 and c0 = 3/2 would satisfy make H(z) Hurwitz.Now,

h(s) = s2 + 1.255s + 1.5

For R = 1, and Q = ddT =

[d2

1 d1d2

d1d2 d22

], we get the equations,

2− q22 = 3/2

1 + q11 = 9/4

from which Q and K are obtained to be

Q =

[1.25 0.7910.791 0.5

]

K =[

0.5 1.225]

K is the stabilizing state feedback gain that provides closed loop poles with damping ratiogreater than cos (60) ¥

6.3.3 Static Output Feedback in Single Input Systems

The state feedback technique is the simplest way of designing a control system, provided, thestate are available for measurement. But, generally, this is not the case. In most practicalsystem, all states are not measurable, though they are observable. Thus, it would be moregeneral to find a controller based on the system output.

A static output feedback is of little use in the single input case as it would be able tomatch at most the number of poles equal to the number of outputs of the system. Hence,it would require n-outputs to assign all the n-poles of the system. But, in such a scenario,the system states themselves would be expressible as linear combinations of the system


outputs. Hence, the output feedback would be a state feedback in disguise and thus of noreal significance.

Since in most cases, the number of outputs is less than the system order, static outputfeedback would not be a viable option for single input systems. Hence, the concept ofdynamic output feedback comes into picture.

6.3.4 Dynamic Output Feedback ( SISO Case )

In the dynamic output feedback method, the feedback function is a transfer function ratherthan a constant vector, as in the previous cases.

The Guillemin-Truxal Design Procedure

The Guillemin-Truxal method (Refer [49]) is based upon designing a compensator to yielda specified or desired closed loop system. The design method involves three steps, namely

1. Specifying the desired zeros, poles and numerator constant of the desired closed-loop

functionC(s)

R(s)

2. Solving for the required cascade compensator transfer function Gc(s) for the plantGp(s) using the formula

Gc(s) =C(s)

[R(s)− C(s)] Gp(s)(6.27)

3. Synthesizing a physically realizable network, preferably a passive unit.

This method would be able to match both the zeros and poles of the system using adynamic compensator. However, the order of the compensator would be large. If the systemhas n poles and m zeros, the compensator would be of order (m + n). Moreover, since theemphasis here is on pole placement, we would look into a dynamic output feedback methodthat would place the 2n-poles of the closed loop system. The procedure is thus :

Let the plant have a transfer function Gp(s) =Np(s)

Dp(s)and the dynamic compensator

Gc(s) =Nc(s)

Dc(s). The closed loop transfer function then would be

G(s) =Gp(s)

1 + Gp(s)Gc(s)

=Np(s)Dc(s)

Np(s)Nc(s) + Dp(s)Dc(s)=

C(s)

R(s)

For the compensator to match the 2n poles of the closed loop system, the denominator ofthe closed loop system should match R(s). Matching the coefficients of the two polynomials,


one would get 2n linear equations in 2n unknowns, which can be solved to obtain thenumerator and denominator coefficients of the compensator.

If the transfer functions are of the form

Gp(s) =

n−1∑i=0 ais

i

sn+n∑

i=0 bisi

Gc(s) =

n−1∑i=0 cis

i

sn+n−1∑

i=0 disi

R(s) = sn+2n−1∑

i=0gis

i

then the linear equations would be

b0 0 · · · 0 a0 0 · · · 0b1 b0 · · · 0 a1 a0 · · · 0...

.... . .

......

......

bn−1 bn−2 · · · b0 an−1 an−2 · · · a0

1 bn−1 · · · b1 0 an−1 · · · a1

0 1 · · · b2 0 0 · · · a2...

. . ....

......

. . ....

0 0 · · · 1 0 0 · · · 0

d0

d1...dn−1

c0

c1...cn−1

=

g0

g1...gn−1

gn

gn+1...g2n−1

−

00...0b0

b1...bn−1

(6.28)

MeX = g − b (6.29)

Solving for ci and di would lead to the desired dynamic compensator.Note : The matrix Me is defined as the eliminant matrix of the two polynomials Np(s)

and Dp(s) and it is non-singular if and only if that Np(s) and Dp(s) are relatively prime(i.e., there is no pole-zero cancellation in the system). A proof of this has been given in [50].

6.4 Pole Assignment and Placement in Multi-Input Sys-

tems

6.4.1 Concepts of Multivariable Systems

Generalized Frobinius Canonical Form

The notion of a ’controllable canonical form’ is not restricted to SISO systems [51,52]. It canbe extended to a more general multivariable case. In particular, consider any controllablesystem of the form

x = Ax + Bu

6.4 Pole Assignment and Placement in Multi-Input Systems 105

with, B being assumed to be of full rank m ≤ n. The assumption, which can be removedlater on, implies that all m available inputs are mutually independent, which is usuallythe case in practice. We now define C as the (n× n) matrix obtained by selecting fromleft to right as many (n) linearly independent columns of the controllability matrix D =[

B AB · · · An−1B]

as possible. Since the system was assumed to be controllable, itfollows that D has a full rank n and hence that n = n. Therefore, C has full rank n and|C| 6= 0. We now reconstruct the nonsingular (n× n) matrix L by simply reordering the ncolumns of C,beginning with a ’power ordering’ of those first (d1) columns of C which involveb1, the first column of B, and then employing those (d2) columns of C which involve b2 nextand so forth. In particular,

L =[

b1 Ab1 · · · Ad1−1b1 b2 Ab2 · · · Ad2−1b2 · · · Adm−1bm

](6.30)

Now, let us set

σk =k∑

1di, k = 1, 2, · · · ,m (6.31)

The transformation Q, which would transform the system to its multivariate controllablecanonical form can then be computed in the following manner.

Q =

q1

q1A...q1A

d1−1

q2

q2A...q2A

d2−1

...qmAdm−1

(6.32)

where, qi is the σthi row of L−1 for i = 1, 2, · · · ,m.

The multivariate controllable canonical form or the generalized Frobinius Canonical formwould be

.

x = Ax + Bu

A = QAQ−1 =

A11 A12 · · · A1m

A21 A22 · · · A2m...

......

Am1 Am2 · · · Amm


=

0 1 0 · · · 0 0 0 · · · 0 0 0 · · · 00 0 1 · · · 0 0 0 · · · 0 0 0 · · · 0

. . ....

...... · · · ...

......

1x x · · · x x x · · · x x x · · · x0 0 · · · 0 0 1 0 · · · 0 0 0 · · · 00 0 · · · 0 0 0 1 · · · 0 0 0 · · · 0...

......

. . . · · · ......

...1

x x · · · x x x · · · x x x · · · x...

.... . .

...0 0 · · · 0 0 0 · · · 0 0 1 0 · · · 00 0 · · · 0 0 0 · · · 0 0 0 1 · · · 0...

......

......

... · · · ......

.... . .

...1

x x · · · x x x · · · x x x · · · x

(6.33)

B = QB =

0 0 · · · 00 0 · · · 0...

......

1 x x · · · x0 0 · · · 00 0 · · · 0...

......

0 1 x · · · x...

0 0 · · · 00 0 · · · 0...

......

0 0 0 · · · 1

(6.34)

where the diagonal blocks Aii are each upper right identity companion matrices of di-mension di, while the off-diagonal blocks, Aij, i 6= j are each identically zero except for theirrespective final rows. We therefore note that all information regarding the equivalent statematrix A can be derived from knowledge of the m ordered controllability indices di and them ordered σi rows of A . The same can also be said about B, since we note that only thesesame ordered σi rows of B are nonzero. This particular structured form for the controllablepair (A, B) plays an important role in controller design.

Contollability Indices

The m integers di are defined as the controllability indices of the system with respect to theinput vectors bi respectively. The controllability index of the system is denoted by the Greek


letter µ = max (di) , i = 1, 2, · · · ,m. The controllability index of the system is a measure forthe extent of controllability of the system with each input.

Observability Indices and Multivariable Observable Canonical Form

Applying the procedure of multivariable controllable canonical form to the dual system

.x= AT x + CT u

the similarity transformation obtained Q gives rise to the observable canonical form

.

x = Q−T AQT x + Q−T Bu (6.35)

y = CQT x

The controllability indices of the dual system are equivalent to the observability indicesof the original system.

6.4.2 State Feedback

State feedback in multivariable systems is much more flexible than state feedback in singleinput systems. In the sense that pole placement as well as optimization of cost can besimultaneously performed.

Pole Assignment in Multi-Input Systems

The general procedure of assigning poles is as follows [53]

1. Transform the multi-input system (A,B,C) into its controllable canonical form usinga transformation z = Qx. Let the controllable canonical from be (A, B, C).

2. Select a (n× n) matrix Ad which has the same characteristic equation as the desiredcharacteristic equation.

3. Find the state feedback for the controllable canonical structure as

K =(BT B

)−1BT (Ad − A) (6.36)

4. The required state feedback for the original system can be found out as

K = KQ (6.37)

It can be seen that there would a different K for each choice of Ad. Thus, there would bea flexibility of choice of the state feedback gain. This choice, however, does not exist in caseof single input systems as the above procedure would result in the same K for all choices ofAd with the same characteristic equation.

This extra flexibility can be used to design optimal state feedback that also realizes thedesired closed loop poles.


Remark 6 A direct method of pole placement for multi-input systems has been discussedin [55]. The method converts the multi-input case into a related single input problem andsolves it using the methods of SISO state feedback computation. A different approach forpole assignment of multi-input systems has been discussed in [56]. It presents an efficientmethod for pole assignment, however, both the methods are not suitable to be used for optimalcontroller deduction.

6.4.3 Design of Optimal Control Systems with Prescribed Eigen-values

For a linear time-invariant dynamic multivariable system of the form

x = Ax + Bu (6.38)

y = Dx

where x, the state, is an n vector; u, the control is an r vector; and y the output, is an mvector.

Let us assume a linear feedback control law of the form:

u = Gx (6.39)

The feedback control matrix G may be derived through two distinct approaches. One isto choose G in order to minimize a quadratic performance index of the form:

J =1

2

∞∫

0

[xT Qx + uT Ru

]dt (6.40)

where Q is a positive-semidefinite matrix and P is a positive definite matrix.

The second approach is to choose G so that the closed-loop system:

x = (A + BG) x

achieves certain prescribed eigenvalues.

If the weighting matrices Q and P are given, then the eigenvalues of the closed loopsystem are also uniquely determined. However, these eigenvalues may not give the systemthe desired degree of stability.

On the other hand, using the second approach, we can find a feedback matrix, G, thatwill give the system the desired eigenvalues( This has been discussed earlier). This G matrixis, however, usually not unique, and to select one that is ’better’ than all the others is notclear.

We therefore need a method that combines the both , i.e., find a G that minimizes theperformance index as well as assigns the desired closed loop eigenvalues. A method to attainthis end was given by Solheim [54].


Optimal Feedback Control Systems

Consider the optimal control of the dynamic system in Eqn. (6.38) with the quadraticperformance index (6.40). The co-state is defined by:

p = −Qx− AT p (6.41)

The optimal control is given by the linear control law

u = −R−1BT p = −R−1BT Px = Gx (6.42)

where R is the solution of the degenerate Riccati equation:

P = −PA− AT P + PBR−1BT P −Q = 0 (6.43)

Combining Eqns. (6.38,6.41 and 6.42) gives the canonical system:

[xp

]=

[A −BR−1BT

−Q −AT

] [xp

]= F

[xp

](6.44)

This system has n eigenvalues with negative real parts and n with positive real parts,and the eigenvalues are located symmetrically about the imaginary axis. The eigenvalues ofthe optimal feedback system x = (A + BG) x are identical to those eigenvalues of F withnegative real parts. Therefore, it is possible to study the eigenvalues of F instead of those of(A + BG) . This has a great advantage that the eigenvalue dependence upon Q and R maybe studied without solving the Riccati equation.

The particular problem to be considered here is to determine a weighting matrix, Q, thatgives the feedback system a set of prescribed eigenvalues. The method to be presented isbased on the decoupled system.

Systems with Real, Distinct Eigenvalues Using the modal matrix M, the system inEqn. (6.38) can be diagonalized into

z = Λz + Γu (6.45)

Λ = M−1AM

Γ = M−1B

x = Mz (6.46)

We may express the performance criterion (6.40) in terms of the new state z as:

J =1

2

∞∫

0

[zT MT QMz + uT Ru

]dt

=1

2

∞∫

0

[zT Qz + uT Ru

]dt (6.47)

Q = MT QM


The co-state of the system (6.45) with the performance criterion (6.47) is defined by:

.

p= −Qz − Λp (6.48)

The optimal control law is then given by:

u = −R−1BT M−T p (6.49)

Combining (6.45,6.48 and 6.49) yields the canonical system:

[z.

p

]=

[Λ −H

−Q −Λ

] [zp

]= F

[zp

](6.50)

where,

H = M−1BR−1BT M−T (6.51)

The eigenvalues of the canonical system F are identical to the eigenvalues of the canonicalform F and can be obtained from the characteristic equation:

∣∣∣sI − F∣∣∣ = 0 (6.52)

Using the transformation:

[I 0

−Q (sI − Λ) I

] [sI − F

]=

[s− Λ H

0 sI + Λ−−Q (sI − Λ)−1 H

](6.53)

we get:

∣∣∣sI − F∣∣∣ = |sI − Λ|

∣∣∣sI + Λ−−Q (sI − Λ)−1 H∣∣∣ (6.54)

Suppose now the weighting factor Q has only one non-zero element, namely qjj. Thismeans that only mode zj is being considered in the performance criterion.

The second determinant on the right-hand side of Eqn. (6.54) then becomes

∣∣∣sI + Λ−−Q (sI − Λ)−1 H∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

s + λ1 0 · · · 00 s + λ2 · · · 0...

......

......

−qjj

s− λjhj1 · · · s + λj − qjj

s− λjhjj · · · −qjj

s− λjhjn

......

......

...0 · · · · · · 0 s + λn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(6.55)

By combining Eqns. (6.54 and 6.55) the characteristic equation in Eqn. (6.52) may bewritten:


∣∣∣sI − F∣∣∣ =

n∏i=1

(s− λi)

[(s + λj − qjj

s− λj

hjj

) n∏i=1,i 6=j

(s + λi)

]

= [(s + λj) (s− λj)− qjjhjj]n∏

i=1,i 6=j((s + λi) (s− λi)) = 0 (6.56)

The eigenvalues of the canonical system F are thus

si = ±λi, i 6= j, i = 1, 2, · · · , nsj = ±

√(λ2

j + qjjhjj

)

(6.57)

If sj of the optimal feedback system is given, we can find qjj from the above expressionas:

qjj =s2

j − λ2j

hjj

(6.58)

The only element of the H-matrix(H = −M−1BR−1BT M−T

)which is needed is thus

hjj.With Q known, we can obtain the optimal feedback gain, by solving the Riccati equation:

−PΛ− ΛP + PM−1BR−1BT M−T P − Q = 0 (6.59)

u = −R−1BT M−T PM−1x (6.60)

In the optimal feedback system, x = (A + BG) x, we have now shifted one eigenvalueof the open loop system to the specified position. We may now start with a new system(A1 = A + BG) and shift the next eigenvalue. The result is a recursive procedure that iseasy to implement. The sequence in which the eigenvalues are shifted may be arbitrary, buta different sequence will give a different Q matrix.

Before we summarize the procedure, we will mention that the system we start with mayalready be an optimal system, where the feedback gain, G0, is the result of an optimizationusing a weighted matrix Q0. If some of the eigenvalues in this optimal system are located tooclose to the imaginary axis, we may use the present method to shift them to more desiredlocations.

The Algorithm Let us assume that we shall shift k eigenvalues, where k ≤ n. We getthe following recursive procedure.

1. Initialize: Q = Q0, G = G0, i = 0.

2. Ai = A + BG

3. Compute Λi,Mi and Hi = M−1i BR−1BT M−T

i .

4. i = i + 1


5. The eigenvalue λj is to be shifted to sj. Compute:

(qjj)i =s2

j − λ2j

(hjj)i−1

.

6. With Qi =(qjj)i

compute the optimal feedback gain Gi.

7. Gi = GiM−1i−1, G = G + Gi

8. Qi = M−Ti−1QiM

−1i−1, Q = Q + Qi

9. If the number of eigenvalues shifted are less than k, then change j and go back to 2.

Example 9 Consider the system:

A =

[ −2 01 −1

], B =

[1 00 1

], R =

[1 00 5

], λ1 = −2, λ2 = −1 (6.61)

The eigenvalues of the optimal closed-loop system are specified as s1 = −8, s2 = −5.

Solution :

M0 =

[1 0−1 1

],M−1

0 =

[1 01 1

], H0 =

[1 11 1.2

].

Using Eqn. (6.58),

(q11)1 =s21 − λ2

1

(h11)0

= 60

or

Q1 =

[60 00 0

]

giving the feedback gain:

G1 =

[ −6 00 0

],

Q1 = M−T0 QM−1

0 =

[60 00 0

]

G1 = G1M−10 =

[ −6 00 0

]

We have now shifted one eigenvalue. We now proceed to shift the next one:

A1 = A + BG =

[ −8 01 −1

],

M1 =

[7 0−1 1

],M−1

1 =

[17

017

1

], H1 =

1

49

[1 11 54

5

],


(q22)2 =s22 − λ2

2

(h22)1

= 109,

Q2 =

[0 00 109

], G2 =

[0 −2.60 −3.63

],

Q2 = M−T1 Q2M

−11 =

[2.2 15.615.6 109

],

G2 = G2M−11 =

[ −0.37 −2.6−0.52 −3.63

]

The final result is:

G = G1 + G2 =

[ −6.37 −2.6−0.52 −3.63

]

Q = Q1 + Q2 =

[62.2 15.615.6 109

]

But, as mentioned above there exist a number of Q matrices that will give the prescribedeigenvalues. Different Q matrices may be obtained by changing the order in which we shiftthe eigenvalues, and also by changing the numbering . If in this example we first shift λ2 tos2, and then λ1 to s1, we get:

Q =

[22.86 48.648.6 306

], G =

[ −3.62 −6.2−1.24 −6.38

]

Further Q matrices may be obtained by changing the numbering so that s1 = −5, s2 =−8.¥

Systems with Complex Eigenvalues Consider a system with two complex eigenvalues:

Λ =

−α + jβ 0 0 · · · 00 −α− jβ 0 · · · 00 0 λ3 · · · 0...

......

. . ....

0 0 0 · · · λn

(6.62)

The eigenvector matrix M will also be complex. It is often advantageous to work withreal transformation matrices, so instead of using M directly, we introduce an auxiliary trans-formation:

Λ′ = L−1ΛL (6.63)

where

L =

12− j

20 · · · 0

12

j2

0 · · · 00 0 1 · · · 0...

......

. . ....

0 0 0 · · · 1

(6.64)


and

Λ′ =

−α β 0 · · · 0−β −α 0 · · · 00 0 λ3 · · · 0...

......

. . ....

0 0 0 · · · λn

(6.65)

The overall transformation becomes:

Λ′ = L−1M−1AML = T−1AT (6.66)

where the transformation matrix T = ML is now real.We now consider shifting the two complex eigenvalues. We choose a weighting matrix:

Q =

q11 0 0 · · · 00 q22 0 · · · 00 0 0 · · · 0...

......

. . ....

0 0 0 · · · 0

(6.67)

where q11 = q22.Thus, we can arrive at the characteristic equation for that part of the canonical system

F that concerns the two eigenvalues:

s4 − [2(α2 − β2

)+ q11 (h11 + h22)

]s2 + q2

11

(h11h22 − h2

12

)

+q11 (h11 + h22)(α2 + β2

)+

(α2 + β2

)2= 0 (6.68)

The H-matrix defined in Eqn. (6.50) becomes in this case:

H = T−1BR−1BT T−T (6.69)

The two complex eigenvalues are now shifted to either two new complex positions orpositions on the real axis. In this case, we do not have complete freedom of choice. If wechoose to shift to a complex pair, say

s1 = −γ + jδ, s2 = −γ − jδ

With these two eigenvalues, the characteristic equation corresponding to Eqn. (6.68)becomes:

s4 − 2(γ2 − δ2

)s2 +

(γ2 + δ2

)2= 0 (6.70)

Equating the coefficients for the s2 terms in Eqn. (6.68) and Eqn. (6.70) yields:

q11 =2 (γ2 − α2)− 2 (δ2 − β2)

h11 + h22

(6.71)

As mentioned above, γ and δ cannot both be chosen arbitrarily, but they must satisfy agiven constraint. Equating s0 coefficients in Eqn. (6.68) and Eqn. (6.70):

(γ2 + δ2

)2=

(α2 + β2

)2+ q11 (h11 + h22)

(α2 + β2

)+ q2

11

(h11h22 − h2

12

)(6.72)


where q11 may be eliminated through Eqn. (6.71).Having chosen for example γ, if δ, determined from Eqn. (6.72) becomes imaginary, it

indicates that for the chosen γ we cannot obtain complex eigenvalues.Let us next consider the case where the new eigenvalues are to be real:

s1 = γ1, s2 = γ2

The characteristic equation corresponding to Eqn. (6.68) becomes:

s4 − (γ2

1 + γ22

)s2 + γ2

1γ22 = 0 (6.73)

Equating the coefficients of the s2 terms in Eqn. (6.68) and Eqn. (6.73) yields:

q11 =(γ2

1 + γ22)− 2 (α2 − β2)

h11 + h22

(6.74)

The constraint relating to γ1 and γ2 is obtained by equating the coefficients of the s0

terms in Eqn. (6.68) and Eqn. (6.73) as:

γ21γ

22 =

(α2 + β2

)2+ q11 (h11 + h22)

(α2 + β2

)+ q2

11

(h11h22 − h2

12

)(6.75)

If for certain γ1, γ2 becomes imaginary, this indicates that for this value of γ1 it is notpossible to obtain two real eigenvalues.

It is worth while to note that as soon as we shifted the two original complex eigenvalues topositions on the real axis, then each of these new real eigenvalues may be shifted arbitrarilyto positions on the real axis following the procedure given above.

The Algorithm The recursive procedure for shifting complex eigenvalues is:

1. Initialize: Q = Q0, G = G0, i = 0.

2. Ai = A + BG

3. Compute Λi,Mi , Li, Ti and Hi = T−1i BR−1BT T−T

i .

4. i = i + 1

5. The two complex conjugate eigenvalues λj and λj+1 are to be shifted to sj and sj+1.If sj and sj+1 are complex conjugate, use Eqns. (6.71 and 6.72) to determine (qjj)i . Ifsj and sj+1 are real, use Eqns. (6.74 and 6.75) to determine (qjj)i .

6. With:

Qi =

0 0. . .

(qjj)i

(qj+1,j+1)i. . .

0 0

where (qjj)i = (qj+1,j+1)i . Next compute the optimal feedback gain, Gi.


7. Gi = −R−1BT T−Ti−1P , Gi = GiT

−1i−1, G = G + Gi

8. Qi = T−Ti−1QiT

−1i−1, Q = Q + Qi

9. If there are more complex eigenvalues to be shifted, then change j to a relevant numberand go back to 2.

An example for this sort of a system has been solved in [54].

Systems that Cannot be Diagonalized If a system matrix A cannot be diagonalized, (a necessary condition for this is that A has multiple eigenvalues), the procedure given abovemay still be used.

The system is transformed into the Jordan canonical form through the transformation:

J = U−1AU (6.76)

where U is a transformation matrix which is not the eigenvector matrix M.To use the procedure given above we must replace the transformation matrix M with

U and we also have to start at the bottom of each Jordan block. This is illustrated by thefollowing example.

Given:

J =

λ 1 00 λ 10 0 λ

Then J contains in this case only one Jordan block and this means we have to start at thebottom with (q33)1 . The first iteration gives the new system:

J1 =

λ 1 00 λ 00 0 s3

Next we have to determine (q22)2 , which gives:

J2 =

λ 0 00 s2 00 0 s3

This system now has distinct eigenvalues, and we may now use the procedure given above.

6.4.4 Static Output Feedback

The state feedback technique is the simplest way of designing a control system, provided, thestate are available for measurement. But, generally, this is not the case. In most practicalsystem, all states are not measurable, though they are observable. Thus, it would be moregeneral to find a controller based on the system output.

For the system in Eqn. (6.2), the static output feedback would be of the form


u = Ky (6.77)

Now, the closed loop system would be of the form

x = (A + BKC) x

For a single input n-state, p-output system it can be seen that the existence of a K thatwould deliver the desired closed loop poles is not always possible if p < n (n simultaneousequations in p unknowns to be satisfied to match the characteristic equations). However,when n = p, there is a possibility of a unique solution to the problem. There would be nochance of having multiple solutions as in that case the outputs would be linearly dependent.

Basic Output Feedback Pole Assignment Algorithm

The procedure to solve for K is similar to that of the pole placement in state feedback case.

1. Let

K =[

k1 k2 · · · kn

]

2. Compare the coefficients of the characteristic equations of (A + BKC) and the desiredcharacteristic equations, leading to n simultaneous linear equations in ki

3. Solve for ki and obtain the static output feedback gain K

Alternative Approach to Constant Output Feedback (Munro & Vardulakis)

For the system (6.2), with m inputs, l outputs and n states, the aim is to design a constantm× l output feedback gain Ky such that the closed-loop eigenvalues of the system

x = (A + BKyC) x (6.78)

confirm to specified eigenvalues. The procedure to find such a Ky, ( if it exists) has beengiven in [57,58].

The procedure involves finding a m×n state feedback matrix Kx such that the eigenvaluesof the closed-loop system

x = (A + BKx) x (6.79)

confirm to the desired eigenvalues. This can be easily accomplished using the proceduresdescribed in the previous section.

The output feedback problem can therefore be viewed as that of determining a matrixKy such that

BKyC = BKx (6.80)

KyC = Kx (6.81)

where B and C are given, and Kx is any one member of the set of feedback matrices whichachieve the desired pole placement using state feedback.


Using the notation given by Pringle and Rayner [59], a q × p matrix Cg1 is said to be ag1− inverse of the p× q matrix C if

CCg1C = C (6.82)

The matrix equation KyC = Kx is consistent if and only if

KxCg1C = Kx (6.83)

If the consistency condition given in Eqn. (6.83) is satisfied then the general solution forKy is given by

Ky = KxCg1 + Z (Il − CCg1) (6.84)

where Z is an arbitrary m× l matrix. Since Z is arbitrary, setting it to be the null matrixgives Cy as

Ky = KxCg1 (6.85)

Thus, the following theorem can be stated:

Theorem 3 A necessary and sufficient condition for all poles of a system described by Eqn.(6.2) to be arbitrarily assigned using constant output feedback is that at least one of the setof state feedback matrices Kx, which achieves the same pole placement, and one of the g1

inverses of C satisfy the consistency relationship KxCg1C = Kx.

Procedure of finding Cg1 Given a p×q matrix C with rank r ≤ min p, q and p > q thenit is shown by Pringle and Rayner [59] that if the matrix C is augmented on the right by aunit matrix of order p.

C =[

C Ip

](6.86)

and if the augmented matrix C is reduced using elementary row operations to the form

C =

[Ir C12 C13

0 0 C23

](6.87)

=[

E F]

(6.88)

then the first q rows of F will be g1− inverse of C., i.e. Cg1 = Fq

If the p × q matrix C has p < q, then the algorithm given above can be carried out onCT , and the required g1− inverse is given by F T

q .

Pole Shifting Using Output Feedback (Seraji)

The condition for existence of a constant output feedback gain given in Eqn. (6.81), can beused to check for the existence of Ky in the following procedure suggested by Seraji [60].

Since the l × n matrix C is of full rank l, it contains l linearly independent columns.Forming an l × l matrix C1 from these l columns and an n × l matrix Il from the sameselection of l columns of the n× n identity matrix I, Eqn. (6.81) can be written as

Ky

[C1 C2

]= Kx

[I1 I2

](6.89)


where C2 and I2 are the l × (n− l) and n × (n− l) matrices formed from the remaining(n− l) columns of C and I respectively. Then, Eqn. (6.89) can be written as two equations

KyC1 = KxI1 (6.90)

and

KyC2 = KxI2 (6.91)

Eqn. (6.90) can be solved for Ky as

Ky = KxI1C−11 (6.92)

On substitution this solution in Eqn. (6.91), we obtain the consistency condition for Eqn.(6.81) as

KxI1C−11 C2 = KxI2

or

Kx

[I2 − I1C

−11 C2

]= 0 (6.93)

Eqn. (6.93) gives the necessary and sufficient condition on the matrix C and Kx for theexistence of a matrix Ky satisfying Eqn. (6.81). If this condition is satisfied, Eqn. (6.92)can be used to calculate the required output feedback matrix Ky.

Partial Pole Placement

An algorithm for ’almost’ arbitrary eigenvalue assignment using constant gain output feed-back for systems with m inputs, p outputs and n states has been provided by Misra andPatel [62]. The algorithm for m = 1 and m > 1 are separate and are presented as follows.

Algorithm for Eigenvalue Assignment in Single Input Multi Output Systems EVA-1

Step I : (Initialization):

• Set kT = kT0 = 0, b1 = b, A1 = A,Q = In, i = 1, l = number of eigenvalues to be

assigned (l ≤ p).

Step II : (Real Eigenvalues):

1. Set ki = 0, Ti = Ip; If λi is complex, goto Step III;

2. If i = l = n, goto Step II-9; else determine an orthogonal matrix Pi such that aTi Pi =

±‖ai‖2 e2n where aT

i = [0, 0, · · · , an,n−1, an,n − λi] is the last row of Ai − λiIn, ‖ai‖2 =

(a2i ai)

1/2, and en is a vector of length n defined as [0, 0, · · · , 0, 1]T .

3. Set Ai = P Ti AiPi, bi = P T

i bi, and Ci = CiPi.


4. Reduce Ai to an upper Hessenberg matrix(UHF) [?] Ai by means of plane rotationsPi,j = 1, 2, · · · , n− i− 1, i.e.,

Ai = P Ti,n−i−1 · · ·P T

i,1AiPi,1 · · ·Pi,n−i−1

and let

bi+1 = P Ti,n−i−1 · · ·P T

i,1bi

Ci = CiPi,1 · · ·Pi,n−i−1

5. Find an orthogonal matrix Ti such that Ci+1 = TiCi, is in the lower row echelon form.

If the ith column of Ci is a zero vector (λi is a transmission zero of the system), go toStep II-8, else, continue.

6. Determine a feedback vector kTi such that the (i + 1, i)th element of Ai − bi+1k

Ti Ci+1

becomes zero. A suitable choice of the vector kTi ∈ Rp is

[0, · · · , 0, kp−i+1, 0, · · · , 0

]

where the only nonzero element of kTi is the (p− i + 1)th element.

7. Set T = TiT, kT = kT + kTi T T , Ai+1 = Ai − bi+1k

Ti Ci+1

If i = l STOP; else, set i = i + 1 and goto Step II.

8. Set Ai+1 = Ai; (bi+1 and Ci+1 have already been defined).

If i = l, STOP; else, set i = i + 1 and goto Step II

9. If the last column vector of Cn is a zero vector (λn is a transmission zero of the system),then STOP; else, determine a feedback vector kT

n such that the (n, n)th element of(An − bnk

Tn Cn

)is equal to λn. The vector kT

n will have only the first element k1 as

nonzero and Tn will be a n× n identity matrix.

Comment: The nonzero element in kTn is determined as

kl =an,n − λn

bn,nc1,n

(6.94)

where an,n denotes the (n, n)th element of An, bn,n denotes the nth element of bn,

and c1,n denotes the (1, n)th element of Cn

10. Set kT = kT + kTn and STOP.


Step III : (Complex-Conjugate Pairs of Eigenvalues):

1. If i = n− 1, goto Step III-8; else determine an orthogonal matrix Pi such that

aTi Pi = ±‖ai‖2 eT

n where aTi = [0, 0, · · · , an,n−2, an,n−1, an,n]

is the last row of (Ai − λiIn) (Ai − λ∗i In) , λ∗i being the complex-conjugate of λi. Theelements an,n−2, an,n−1, an,n are given by

an,n−2 = an−1,n−2an,n−1

an,n−1 = an,n−1 [an−1,n−1 + an,n − (λi + λ∗i )]

an,n = (an,n)2 + an,n−1an−1,n − an,n (λi + λ∗i ) + λiλ∗i

where ai,j denotes the (i, j)th element of A.

2. Set Ai = P Ti AiPi, bi = P T

i bi, Ci = CiPi and ki = 0

3. Apply plane rotations Pi,j, j = 1, 2, · · · , 3 (n− i− 1) in order to make A(i)22 as close to

upper Hessenberg as possible, i.e.,

Ai = P Ti,3(n−i−1) · · ·P T

i,1AiPi,1 · · ·Pi,3(n−i−1)

and let

bi+2 = P Ti,3(n−i−1) · · ·P T

i,1bi

Ci = CiPi,1 · · ·Pi,3(n−i−1)

4. Find an orthogonal matrix Ti such that Ci+2 = TiCi, is in the lower row echelon form.

If the ith and (i + 1)th columns of Ci are zero vectors (λi and λ∗i are complex-conjugatetransmission zeros of the system), goto Step III-7; else, continue

5. Determine a feedback vector kTi such that the (i + 2, i)th and (i + 2, i + 1)th elements

of Ai − bi+2kTi Ci+2 are eliminated. The vector kT

i ∈ Rp is given by

kTi =

[0, · · · , 0, kp−i, kp−i+1, 0, · · · , 0

]

where

kp−i =ai+2,i

bi+2cp−i,i

kp−i+1 =ai+2,i+1

bi+2cp−i+1,i+1

Comment: The feedback described above results in a 2 × 2 matrix in the ith and(i + 1)th rows and columns of the closed loop matrix Ai − bi+2k

Ti Ci+2 with eigen-

values (λi, λ∗i )


6. Set T = TiT, kT = kT + kTi T T , Ai+2 = Ai − bi+2k

Ti Ci+2

If i = l, STOP; else, set i = i + 2 and goto Step II.

7. Set Ai+2 = Ai; (bi+2, ci+2 have already been defined).

If i = l, STOP; else, set i = i + 2 and goto Step II.

8. If the last two columns of Cn−1 are zero vectors (λn−1 and λ∗n−1 are transmission zeros

of the system), then STOP; else, determine a feedback vector kTn such that the 2 × 2

matrix in the last two rows and columns of An−1 − bn−1kTn−1Cn−1 have the desired

complex-conjugate pair of eigenvalues at λi and λ∗i .

Comment: The vector kTn will be a vector of length n with only the last two elements

being nonzero. The last two rows and columns of An−1 − bn−1kTn−1Cn−1 are given

by [an−1,n−1 an−1,n

an,n−1 an,n

]−

[bn−1

0

] [k1 k2

] [0 c1,n

c2,n−1 c2,n

](6.95)

The elements k1 and k2 are given by

k1 =1

(an,n−1bn−1c1,n).[λn−1λ

∗n−1 + an−1,nan,n−1 + a2

n,n − an,n

(λn−1λ

∗n−1

)]

− c2,n

bn−1c1,nc2,n−1

[an−1,n−1 + an,n −

(λn−1λ

∗n−1

)](6.96)

and

k2 =1

bn−1c2,n−1

[an−1,n−1 + an,n −

(λn−1λ

∗n−1

)](6.97)

The effect of applying the feedback kTn−1 is to change the first row of the 2 × 2

matrix above so that by appropriate choices of the two nonzero elements of thefeedback vector, we can ensure that the 2× 2 matrix in (6.95) has eigenvalues atλi and λ∗i .

9. Set kT = kT + kTn−1 and STOP.

As shown in Step II-9 and Step III-8, if i = l = n for a real eigenvalue or i = l−1 = n−1for a complex-conjugate pair of eigenvalues, then the eigenvalues are assigned directly, sincewe cannot form implicit shifts in these cases.

Algorithm for Multi-Input,Multi-Output Systems with (m+p > n) EVA-2 Forsimplicity of algorithm, it is assumed that the eigenvalues are arranged in such a mannerthat the first p− 1 eigenvalues are real.


Step I : (Assign the First p− 1 Eigenvalues):

1. Set l = p − 1, and Λ = λ1, λ2, · · · , λl , the set of eigenvalues to be assigned in thisstep..

2. Obtain a controllable pair (A, b) , where b = Bd1.

Comment: The vector d1 can be generated randomly or as a linear combinationof the rows of B such that the above controllability condition is satisfied. If thematrix A is not cyclic, then a randomly generated output feedback Kr should beapplied to make the resulting state matrix cyclic.

3. Reduce (A, b, C) to its UHF and apply Algorithm EVA-1 to get the system (Ap−1, Bp−1, Cp−1)and output feedback matrix K1 = d1k

T1 where kT

1 is the output feedback vector requiredto assign the desired (p− 1) eigenvalues for the single-input system (A, b, C) .

Step II : (Assign the Remaining Eigenvalues):

1. Form the dual system, i.e., set F = (Ap−1)T , G = (Cp−1)

T , H = (Bp−1)T , and partition

F,G, H as follows

F ,[

F11 0F21 F22

], G ,

[G1

G2

], H =

[H1 H2

]

2. Determine d2 ∈ Rp such that G1d2 = 0 and G2d2 6= 0. If (F22, G2) is a controllable pairgoto Step II-3, else change Λ and goto Step I.

3. Set A = F11, b = G2d2, C = H2, n = n− p + 1, and p = m.

4. Set l = n (the number of eigenvalues to be assigned) and reduce the single-input,multi-output system (A, b, C) m to a UHF and apply Algorithm EVA-1 to assign thel eigenvalues and get an output feedback vector kT

2

5. Set the output feedback matrix K = Kr + K1 + d2kT2

6.4.5 Two Time-Scale Decomposition and State Feedback Design

Realistic models of large scale systems involve interacting dynamic phenomena of widelydifferent speeds. Typical examples are found in the field of power engineering and elec-tromechanical systems. In a power system model, voltage and frequency transients rangefrom intervals of seconds, corresponding to generator voltage regulator, speed governor ac-tion and shaft energy storage, to several minutes, corresponding to load voltage regulatoraction, prime mover fuel transfer times and thermal energy storage. This emphasizes thefact that the existence of multi-modes and multi-time-scales manifests itself in a broad classof physical models. Simulation studies of such models usually require expensive integrationroutines due to the stiffness of the models. It is therefore considered desirable to seek anapproach which overcomes the stiffness difficulties.


Block-Diagonalization

In this section, we consider the problem of block-diagonalizing large, continuous time-invariant systems whose eigenspectra (the sets of eigenvalues) are formed by clusters oflarge eigenvalues and clusters of small eigenvalues.

Consider an n-dimensional system

x = Ax + Bu (6.98)

where x ∈ Rn, u ∈ Rm and A and B are matrices of appropriate dimensions. The system issaid to possess two-time scale property if has eigenvalues distinctly clustered and the ratioof their magnitudes differing greatly from unity.

The basic idea of using the time-scale approach in generating lower-order models is todecouple the dominant (slow) modes from the non-dominant (fast) modes. This is per-formed through use of two-stage linear transformations. But, before proceeding into thedecomposition of the large scale system, its two-time scale property should be ascertained.

Let us represent the state x as

x =

[x1

x2

],

where the sizes of x1, x2 , (n1 and n2) correspond respectively to the sizes of the fast andslow modes existing in the system. Then Eqn. (6.98) can be written as

[x1

x2

]=

[A11 A12

A21 A22

] [x1

x2

]+

[B1

B2

]. (6.99)

The two-time-scaledness of the system can now be checked by evaluating the norm condition

∥∥A−122

∥∥ <1

3(‖A0‖+ ‖A12‖ . ‖L0‖)−1 (6.100)

where

A0 = A11 − A12L0 (6.101)

L0 = A−122 A21

However, Eqn. (6.100) gives only the sufficiency condition, it is not necessary for a two-time scaled system to satisfy Eqn. (6.100). Once the system is confirmed to have two-timescale property, the first of the two linear transformations is performed as follows. The firststage is to apply the change of variables

[x1

z2

]= T1

[x1

x2

](6.102)

T1 =

[I1 0L I2

]

to system (6.98) and choose the (n2 × n1) matrix L such that

LA11 + A21 − LA12L− A22L = 0 (6.103)


Thus,[

x1

z2

]=

[I1 0L I2

] [A11 A12

A21 A22

] [I1 0−L I2

] [x1

z2

]+

[I1 0L I2

] [B1

B2

]u

=

[A11 − A12L A12

LA11 + A21 − LA12L− A22L A22 + LA12

] [x1

z2

]+

[B1

LB1 + B2

]u

=

[Fs A12

0 Ff

] [x1

z2

]+

[B1

G2

]u (6.104)

The numerical value of L is computed using an iterative algorithm

Lk+1 = A−122 (LkA11 + A21 − LkA12Lk) (6.105)

with the initial value of L taken to be as L0 from Eqn. (6.103).Now the second linear transformation is applied as

[z1

z2

]= T2

[x1

z2

](6.106)

T2 =

[I1 M0 I2

]

thus transforming the system as[

z1

z2

]=

[I1 M0 I2

] [Fs A12

0 Ff

] [I1 −M0 I2

] [z1

z2

]+

[I1 M0 I2

] [B1

G2

]u(6.107)

=

[Fs A12 + MFf − FsM0 Ff

] [z1

z2

]+

[G1

G2

]u

G1 = B1 + MG2

Now, if M is so chosen that A12 + MFf − FsM = 0, them the system would attain ablock diagonal form. The value of M to achieve this is computed by an iterative algorithm.

0 = A12 + MFf − FsM

0 = A12 + M (A22 + LA12)− (A11 − A12L) M

M = − (A12 + MLA12 − A11M + A12LM) A−122

Thus, the iterative formula would be

Mk+1 = − (A12 + MkLA12 − A11Mk + A12LMk) A−122 (6.108)

M0 = A12A−122

Design of State Feedback Controller for Two-Time Scaled Systems

In order to design the state feedback for the two-time scaled system, a two step procedureis used. The input u is represented as u = u1 + u2. The input u1 is computed as

u1 =[

K1 0] [

z1

z2

](6.109)


Substituting the value of u1 from Eqn. (6.109) in Eqn. (6.107) yields

[z1

z2

]=

[Fs + G1K1 0

G2K1 Ff

] [z1

z2

]+

[G1

G2

]u2 (6.110)

Now again using a transformation

[z1

g2

]= T3

[z1

z2

](6.111)

T3 =

[I1 0N I2

]

Applying this transformation to the system in Eqn. (6.110), we obtain the transformedsystem

[z1

g2

]=

[Fs + G1K1 0

N (Fs + G1K1)− FfN + G2K1 Ff

] [z1

g2

]+

[G1

G2 + NG1

]u2

Now obtaining the value of N so that the term N (Fs + G1K1)−FfN +G2K1 becomes zero,the system would then be diagonal. This is done by using the iterative algorithm

Nk+1 = F−1f (Nk (Fs + G1K1) + G2K1) (6.112)

N0 = F−1f (G2K1)

Now apply the second input u2 as

u2 =[

0 K2

] [z1

g2

](6.113)

After the application of u2, the closed loop system is of the form

[z1

g2

]=

[Fs + G1K1 G1K2

0 Ff + (G2 + NG1) K2

] [z1

g2

].

The control input u = u1 + u2 can be expressed in terms of the original state vector x inthe following manner.

u = u1 + u2

=[

K1 0] [

z1

z2

]+

[0 K2

] [z1

g2

]

=[

K1 0]T2T1

[x1

x2

]+

[0 K2

]T3T2T1

[x1

x2

]

=[

K1 + K2N + ((K1 + K2N) M + K2) L (K1 + K2N) M + K2

] [x1

x2

](6.114)

Chapter 7

Fast Output Sampling (FOS)

7.1 Introduction

The problem of simultaneous stabilization has received considerable attention. Given afamily of plants in state space representation (Φi, Γi) , i = 1, · · · ,M , find a linear statefeedback gain F such that (Φi + ΓiF ) is stable for i = 1, · · · , M , or determine that no suchF exists. But the method is of use only in the case where whole state information is available.

One way of approaching this problem with incomplete state information is to use observerbased control laws, i.e. dynamic compensators. The problem here is that the state feedbackand state estimation cannot be separated in face of the uncertainty represented by a familyof systems. Assuming that a simultaneously stabilizing F has been found, it is possible tosearch for a simultaneously stabilizing full order observer gain, but this search is dependenton the F previously obtained. If no stabilizing observer for this state feedback exists, nothingcan be said because there may exist stabilizing observers for different feedback gains.

With the Fast output sampling approach proposed by Werner and Furuta in [64], it isgenerically possible to simultaneously realize a given state feedback gain for a family oflinear, observable models. For fast output sampling gain L to realize the effect of statefeedback gain F , find the L such that (Φi + ΓiLC) is stable for i = 1, · · · ,M , If there exista set of F ’s, there should also exist a common L for given family of plants. One of theproblems with this approach is that large feedback gains tend to render the system verynoise -sensitive. The design problem can be posed as a multiobjective optimization problemin an LMI formulation [?] [66]

The fast output sampling controller obtained by the above methods requires only constantgains and hence is easier to implement online. This approach can be used for tracking purposealso as described in [?] and [?].

7.2 Controller Deduction

Consider a plant described by linear model is of the form

x = Ax + Bu

127


y = Cx (7.1)

. Here (A,B) and (A,C) are assumed to be to controllable and observable respectively.Choose an effective sampling time τ during which control signal u is held constant. Thesampling interval is divided into N subintervals of length ∆ = τ/N, and the output mea-surements are taken at time instant t = l∆,where l = 0, 1, ...,. The control signal u(t), whichis applied during the interval kτ < t < (k + 1)τ , is then constructed as linear combinationof the last N output observations. Here N ≥ v the observability index.

Definition 6 (Observability Index) Given an observable pair (A,C) ∈ Rn×n×Rq×n, andrank(C) = q, the observability index of the system with respect to any particular row ci of Cis the minimum value of νi such that the row ciA

νi is dependent on the rows before it in thefollowing series

c1, c2, · · · , cq, c1A, c2A, · · · , cqA, · · · , c1Aνi , · · · , ciA

νi , · · ·The observability index of the entire system is defined as ν = max (νi) .

The following sampled data control is applied to the system.

u(t) = [L0 L1 L2......LN−1]

y(kτ − τ)y(kτ − τ + ∆)...y(kτ −∆)

= Lyk (7.2)

For kτ < t ≤ (k + 1)τ, , where the matrix blocks Lj represent output feedback gains andthe notation L and yk have been introduced for the convenience. Note that 1/τ is the rate atwhich the loop is closed, whereas output samples are taken at the N -times faster rate 1/∆.

Let (Φτ , Γτ , C) denotes the system (A,B, C) sampled at the rate 1/τ,i.e. Φτ = eAτ , Γτ =τ∫0

eAsdsB and (Φ, Γ, C) the same system sampled at the rate 1/∆ . Consider the discrete

time systems having at t = kτ the input uk = u(kτ),states xk = x(kτ) and the outputyk.Then we have

xk+1 = Φτxk + Γτuk

yk+1 = C0xk + D0uk (7.3)

Where

C0 =

CCΦ..CΦN−1

, D0 =

0CΓ..

C∑N−2

j=0 ΦjΓ

(7.4)

7.3 Closed Loop Stability 129

Next assume that a state feedback gain has been designed such that (Φτ + ΓτF ) has noeigenvalues at the origin. For this state feedback one can define the fictitious measurementmatrix

C(F, n) = (C0 + D0F )(Φτ + ΓτF )−1 (7.5)

which satisfies the fictitious measurement equation

yk = Cxk

Let

L = [L0 L1.......LN−1],then the feedback law Eqn. (7.2) can be interpreted as staticoutput feedback

uk = Lyk

For L to realize the effect of F ,in the system described by Eqns. (7.3) and (7.4) with themeasurement matrix C. it must satisfy

LC = F (7.6)

7.3 Closed Loop Stability

At time t = 0, the control signal u(t) = u0 for 0 < t ≤ τ cannot be computed from Eqn.(7.2). since output measurements are not available for t < 0. If initial states x0 is known onecan take u0 = Fx0. If x0 is unknown and estimated with error ∆x0, the value of u0 will differby ∆u0 = F∆x0 from the control signal which would be applied if the initial states wereknown. For k 1 1, uk can be computed from Eqn. (7.2), but if ∆u0 6= 0 ,the assumptionuk = Fxk and therefore yk = Cxk does not hold, and the effect of initial error ∆u0 wouldpropagate through the closed loop response of the system. One can verify that closed loopdynamics is governed by

[xk+1

∆uk+1

]=

[Φτ + ΓτF Γτ

0 LD0 − FΓτ

] [xk

∆uk

](7.7)

where ∆uk = uk − Fxk

Let Ψ = LD0 − FΓτ . Thus we have the eigenvalues of the closed loop system undera fast output sampling control law in Eqn. (7.2) as those of Φτ + ΓτF together with thoseof LD0 − FΓτ .

The above design was carried out for a simple model, but it can also be used to finda fast output sampling gain which simultaneously assigns prescribed closed -loop poles formultiple models.


7.4 Techniques for Determining Fast Output Sampling

Controller Gain

In most of the cases of designing a Fast Output Sampling Controller, there is no singlesolution for the Eqn.(7.6). In most cases, there are multiple solutions and in a few cases(of multi-plant systems) there may not even be a single exact solution. To solve, the Eqn.(7.6),the solution needs to be an optimal one rather than insisting on an exact one.

Even in cases where an exact solution is possible, the solution may not be a desirableone. The value of ’L’ may be badly scaled. These problems would arise in systems wherethe eigenvalues vary over several orders of magnitude. To rectify the problem, the followingmethods may be adopted.

7.4.1 Two Time-Scale Approach for Conditioning of State Feed-back Gain F

Introduction

When a system has its eigenvalues many orders of magnitude apart, with the unstable partbeing slow and the stable part being fast,the system is termed as two time-scale systems.The interaction of these slow and fast modes makes the system stiff.

.This method can be adopted to get a better conditioned L for such systems. The StateFeedback Gain F , that stabilizes the τ system needs to be conditioned for this.

Two time Scale System Description

Let the two time-scaled system be represented as

x(k + 1) = Φ1x(k) + Γ1u(k) (7.8)

y(k) = Cx(k)

with n states, n1 of which are close to unity ( slow modes) and n2 eigenvalues are aroundthe origin (fast modes). We assume the system is asymptotically stable, completely control-lable and completely observable. Note that re-indexing of states using permutational matrixwhose columns are elementary vectors and re-scaling the resultant model using appropriatediagonal matrix is necessary to isolate fast and slow states.

State Feedback Conditioning for Two time Scale Systems.

The Transformation Matrix

With re-indexing and/or re-scaling of states, the eigenvalues can be arranged as

|λ1| ≥ |λ2| ≥ · · · ≥ |λn1| > |λn1+1| ≥ · · · ≥ |λn|

To accomplish this, the system described in Eqn. (7.8) can be represented as

7.4 Techniques for Determining Fast Output Sampling Controller Gain 131

x(k + 1) =

[A11 A12

A21 A22

] [x1(k)x2(k)

]+

[B1

B2

]u(k) (7.9)

y(k) =[

C1 C2

] [x1(k)x2(k)

]

From this representation a transformation matrix T [69] can be derived which woulddecouple the fast and slow states using explicitly invertible linear transformation, where,

T =

[I1 −ML −MQ I2

](7.10)

where, Ii is a ni × ni identity matrix, for i = 1, 2 .M is n1 × n2 and Q is n2 × n1. With Qand M satisfying the relations

A21 + QA12 − A22Q−QA12Q = 0 (7.11)

(A11 − A12Q)M −M(A22 + QA12) + A12 = 0 (7.12)

then it follows that the transformation[

xs(k)xf (k)

]= T

[x1(k)x2(k)

]

reduces the system in Eqn. (7.9) to block diagonal form

[xs(k + 1)xf (k + 1)

]=

[As 00 Af

] [xs(k)xf (k)

]+

[Bs

Bf

]u(k)

y(k) =[

Cs Cf

] [xs(k)xf (k)

]

where

[As 00 Af

]= TAT−1,

[Bs

Bf

]= TB (7.13)

[Cs Cf

]= CT−1

Derivation of Conditioned State Feedback

To derive a well-conditioned F for the system,stabilizing state feedback is derived sepa-rately for the slow and fast subsystems, so that the systems (As + BsFs) and (Af + BfFf )are both stable. But, since the fast dynamics are stable by assumption, the value of Ff canbe taken to be zero.

The conditioned state feedback to be applied on the original system in Eqn. (7.8) isderived as

F =[

Fs Ff

]T (7.14)


7.4.2 Singular Value Decomposition of Measurement Matrix C

It may happen that the measurement matrix C is ill-conditioned with some of its elementsseveral orders of magnitude lower than others. In such cases, the FOS Controller Gain wouldbe of a large magnitude and would be unrealizable. Hence, to circumvent this problem, themeasurement matrix should be conditioned. The conditioning is done using Singular ValueDecomposition.

The measurement matrix is decomposed as

C = UΣVT (7.15)

The matrix of singular values Σ, is then analyzed.. Singular values very close to theorigin are discarded.

If σmin is the minimum value of singular value that is considered as significant and ifσ1 ≥ σ2 ≥ . . . ≥ σq ≥ σmin ≥ σq+1 ≥ . . . ≥ σr., then the measurement is approximated to

CA = UAΣAVTA (7.16)

Where,

ΣA = diag(σ1, σ2, ..., σq)

UA =[

U1 U2 · · · Uq

]

VA =[

V1 V2 · · · Vq

](7.17)

With the new measurement matrix calculated from Eqn. (7.16), the FOS Controller Gainis calculated using Eqn. (7.6).

7.4.3 Approach for Multi-Plant Systems

To compute the fast output sampling controller for a multi-model system, the initial part ofdesign up to the calculation of the measurement matrix

Ci(F, n) = (C0,i + D0,iFi)(Φτ,i + Γτ,iFi)−1 (7.18)

is done independently for each plant. The controller gains are found by solving theequation

L˜

C= F (7.19)

where,

C =[

C1 C2 . . CM

](7.20)

F =[

F1 F2 . . FM

]

7.5 An LMI Formulation of the design problem 133

7.5 An LMI Formulation of the design problem

With this type of controller, the unknown states are estimated implicitly, using the mea-sured output samples and assuming that control is generated by state feedback. If externaldisturbance causes an estimation error, then decay of this error will be determined by theeigenvalues of the matrix Ψ which depend on L and whose dimension equals the number ofcontrol input . For stability, these eigenvalues have to be inside the unit disc, and for fastdecay they should be as close to origin as possible. This problem must be taken into accountwhile designing L.

The second problem is that the gain matrix L may have elements with large magni-tude. Because these values are only weights in linear combination of output samples, largemagnitudes do not necessarily imply large control signal. In practice, they may amplifymeasurements noise, and it is desirable to keep these values low. This objective can beexpressed by an upper bound ρ1 on the norm of the gain matrix ‖ L ‖< ρ1.When trying todeal with these problem, it turns out that it is better not to insist on an exact solution to thedesign equation one can allow a small deviation and use an approximation LC ≈ F , whichhardly effects the desired closed -loop dynamics, but may have considerable effect on thetwo problems described above. Instead of looking for an exact solution to the inequalities,the following inequalities are solved

‖ L ‖< ρ1

‖ Ψ(L) ‖< ρ2

‖ L˜

C −F ‖≤ ρ3

Here three objectives have been expressed by the upper bounds on matrix norms , andeach should be as small as possible. If ρ3 = 0 then L is exact solution.

Using the schur complement, it is straight forward to bring these conditions in the formof LMI (Linear Matrix inequalities) [70].

[ −(ρ1)2I L

LT − I

]< 0 (7.21)

[ −(ρ2)2I Ψ(L)

ΨT (L) − I

]< 0 (7.22)

−(ρ3)

2I L˜

C −F

(L˜

C −F )T − I

< 0 (7.23)

7.6 A Modified Approach for Fast Output Sampling

Feedback

The fast output sampling feedback technique described above has a restriction that noneof the closed loop poles of the system (Φτ + ΓτF ) be at the origin. This was to ensure the


invertibility of the measurement matrix C. However, this is no longer a necessary condition.A modified approach of design of an exact FOS controller has been developed in recenttimes [72, 73]. This controller does not employ the concept LMI and hence is used to solvecontroller design problems for single plants only. Further it uses the previous input u(k− 1)to determine the control input. The control design methodology is as follows.

Consider the lifted system in (7.3), since the system is assumed to be observable, the liftedoutput matrix C0 is of rank n. If the value of N is chosen as greater than the observabilityindex of the system, then for a p output system, we would necessarily have Np ≥ n and C0

would be of dimension Np× n. Thus,

yk+1 = C0x(k) + D0u(k)

CT0 yk+1 = CT

0 C0x(k) + CT0 D0u(k) (7.24)

Here, CT0 C0 is a n× n matrix of rank n and hence invertible. Therefore, the state vector can

be determined as

x(k) =(CT

0 C0

)−1CT

0 (yk+1 −D0u(k)) (7.25)

x(k + 1) = Φτx(k) + Γτu(k)

= Φτ

(CT

0 C0

)−1CT

0 yk+1 +(Γτ − Φτ

(CT

0 C0

)−1CT

0 D0

)u(k)

Thus,

x(k) = Φτ

(CT

0 C0

)−1CT

0 yk +(Γτ − Φτ

(CT

0 C0

)−1CT

0 D0

)u(k − 1) (7.26)

Now, if the state feedback control input is designed as u(k) = Fx(k), it can be convertedinto an output feedback based control by simply substituting for x(k) from Eqn. (7.26) toobtain

u(k) = FΦτ

(CT

0 C0

)−1CT

0 yk + F(Γτ − Φτ

(CT

0 C0

)−1CT

0 D0

)u(k − 1) (7.27)

The advantage of the improved version of the Fast output sampling controller (or multi-rate output feedback based controller) as proposed in [72,73] is that Eqn. (7.26) can be usedto realize any state based controller, not just the static gain state feedback u (k) = Fx(k).

Chapter 8

Periodic Output Feedback (POF)

8.1 Review

The problem of pole assignment by piecewise constant output feedback was studied byChammas and Leondes [71] for linear time-invariant systems with infrequent observation.They showed that, by use of periodically time-varying piecewise constant output feedbackgain, the poles of the discrete-time control system could be assigned arbitrarily (within thenatural restriction that they be located symmetrically with respect to real axis) [64,74].

8.2 Periodic Output Feedback Controller Deduction

Consider a discrete time invariant system with sampling interval τ sec

x (k + 1) = Φτx (k) + Γτu (k) ,

y (k) = Cx (k) , (8.1)

where x ∈ Rn, u ∈ Rm, y ∈ Rp and Φτ , Γτ and C are constant matrices of appropriatedimensions.

The following control law is applied to this system. The output is measured at the timeinstant t = kτ, k = 0, 1, · · · . We will consider constant hold function because they are moresuitable for implementation. The output sampling interval is divided into N subintervals oflength ∆ = τ/N, and the hold function is assumed constant on these subintervals. Thus thecontrol law becomes

u (t) = Kly (kτ) ,

kτ + l∆ ≤ t ≤ kτ + (l + 1) ∆, Kl+N = Kl (8.2)

for l = 0, 1, .....N − 1.Note that a sequence of N gain matrices K0,K1,....., KN−1 when substituted in Eqn.(8.2)

generates a time-varying, piecewise constant output feedback gain K(t) for 0 ≤ t ≤ τ .

135


Consider the following system obtained by sampling the system in Eqn.(8.1)at samplinginterval ∆ = τ/N and which is denoted by (Φ, Γ, C) :

x (k + 1) = Φx (k) + Γu (k) ,

y (k) = Cx (k) . (8.3)

A useful property of the control law in Eqn.(8.3) is given by the following lemma. But,before venturing into the lemma it is necessary to understand the following definition.

Definition 7 (Controllability Index) Given an observable pair (A,B) ∈ Rn×n × Rn×m,and rank(B) = m, the controllability index of the system with respect to any particular columnbi of B is the minimum value of νi such that the column AνiB is dependent on the columnsbefore it in the following series

b1, b2, · · · , bm, Ab1, Ab2, · · · , Abm, · · · , Aνib1, · · · , Aνibi, · · ·

The controllability index of the entire system is defined as ν = max (νi) .

The following sampled data control is applied to the system.

Lemma 3 Assume (Φτ , C), is observable and (Φ, Γ) is controllable with controllability indexν such that N ≥ ν,then it is possible to choose a gain sequence Kι such that the closed loopsystem, sampled over τ,takes desired self-conjugate set of eigenvalues [71].

Proof: Define

K =[

K0 K1 . . . KN−1

]T,

u (kτ) = Ky (kτ) =

u (kτ)u (kτ +4)..u (kτ + τ −4)

,

then a state space representation for the system sampled over τ is

x (kτ + τ) = ΦNx (kτ) + Γu,

y (k) = Cx (k) ,

whereΓ = [ΦN−1Γ, · · · ., Γ].

Applying periodic output feedback in Eqn.(8.3), i.e., Ky (kτ) is substituted for u (kτ) ,the closed loop system becomes

8.3 Multimodel Synthesis 137

x (kτ + τ) =(ΦN + ΓKC

)x (kτ) . (8.4)

The problem has now taken the form of static output feedback problem. Eqn.(8.4) suggestthat an output injection matrix G be found such that

ρ((ΦN + GC

)< 1, (8.5)

where ρ (•) denotes the spectral radius. By observability one can choose an outputinjection gain G to achieve any desired self-conjugate set of eigenvalues for the closed loopmatrix

(ΦN + GC

), and from N ≥ ν it follows that one can find a periodic output feedback

gain which realizes the output injection gain G by solving

ΓK =G. (8.6)

for K.¥The controller obtained from the above equation will give desired behavior, but might

require excessive control action. To reduce this effect we relax the condition that K exactlysatisfy the above linear equation and include a constraint on the gain K. Thus we arrive atthe following in equations

‖K‖ < ρ1

‖ΓK−G‖ < ρ2 (8.7)

[ −ρ21I K

KT −I

]< 0

[ −ρ22I (ΓK−G)

(ΓK−G)T −I

]< 0 (8.8)

In this form the LMI Tool Box Matlab can be used for synthesis [?].

8.3 Multimodel Synthesis

For multimodel representation of a plant, it is necessary to design controller which willrobustly stabilize the multimodel system. Multimodel representation of plants can arise inseveral ways. When a nonlinear system has to be stabilized at different operating points,linear models are sought to be obtained at those operating points. Even for parametricuncertain linear systems, different linear models can be obtained for extreme points of theparameters. The models are used for stabilization of the uncertain system.

Let us consider a family of plant S = Ai,Bi,Ci ,defined by

.x = Aix + Biu

y = Cix i = 1, 2, · · · ,M


By sampling at the rate of 1/4 we get a family of discrete systems ΦiΓiCi .

Assume that

(Φ

˜Ni , Ci

)are observable. Then we can find output injection gains Gi such

that

(Φ

˜Ni + GCi

)has required set of poles. Now consider the augmented system defined

below.

vΦ=

Φ1 0 . 00 Φ2 . .. . . .0 . . ΦM

,

vΓ=

Γ1

Γ2

.ΓM

,

vG=

GG.G

The linear equation

[ΦN−1Γ . . . Γ

]

K0

.

.

.

KN−1

= G (8.9)

has a solution if(Φ, Γ

)is controllable with controllability index ν, and N > ν. This

periodic output feedback gain realizes the designed G for all plants of the family. It has beenshown in the [64] the controllability of individual plants generically implies the controllabilityof the augmented system.

The controller obtained from the above equation will give desired behavior, but mightrequire excessive control action. To reduce this effect of gain we relax the condition thatthe Eqn. (8.9) satisfies exactly and include a constraint on the gain. Thus we consider thefollowing inequations

‖K‖ < ρ1

‖ΓiK−Gi‖ < ρ2i, i = 1....M (8.10)

This can be formulated in the framework of Linear Matrix Inequalities as follows

[ −ρ21I K

KT −I

]< 0

[ −ρ22iI (ΓiK−Gi)

(ΓiK−Gi)T −I

]< 0 (8.11)

In this form the LMI Tool Box Matlab [?] can be used for synthesis.The robust periodic output feedback controller obtained by the above method requires

only constant gains and hence is easier to implement.

Chapter 9

Robust Control of Systems withParametric Uncertainty

9.1 Concepts Related to Uncertain Systems

9.1.1 Interval Arithmetic

The concept of interval analysis is to deal with uncertainty by representing it to lie withincertain bounds and then dealing with it using interval arithmetic. The concept of intervalanalysis was first discussed by Teruo Sunaga [76] and later made popular by Ramon Moore[77,78]. The applicability of interval analysis has been discussed in [79].

In interval arithmetic, the numerics are expressed as lying between a lower and an upperbound .

A = [a, a] ≡ a|a ∈]a, a[

Basic Arithmetic Operations

If A = [a, a], B = [b, b] are two intervals then the basic interval arithmetic can be put as

• A + B = [a + b, a + b]

• A−B = [a− b, a− b]

• A.B = [min(ab, ab, ab, ab), max(ab, ab, ab, ab)]

• A/B = A. (1/B)

where, 1/B = 1/b|b ∈ Bprovided 0 /∈ B

The concept of interval analysis and interval arithmetic is useful in the study of stabilityand robust control of systems with parametric uncertainty within known bounds. The un-certain parameter is replaced by an interval entity and the robust stability of the system isanalyzed by various methods.

139


9.1.2 Hermite-Bieler Theorem

Axiom 1 Consider a polynomial of degree

P (s) = p0 + p1s + · · ·+ pnsn

P (s) is said to be Hurwitz if and only if all its roots lie in the open left half of the complexplane. For a Hurwitz polynomial with real coefficients we have the following two elementaryproperties.

1. If a polynomial P (s) is Hurwitz then all its coefficients are non zero and have the samesign, either all positive or all negative.

2. If a polynomial P (s) is Hurwitz and of degree n, then arg(P (jω)) is a continuous andstrictly increasing function of ω on (−∞,∞) . Moreover the net increase in phase from−∞ to ∞ is

arg(P (+j∞))− arg(P (−j∞)) = nπ (9.1)

The proofs of the above have been discussed in [80].

The Interlacing Property

Let us define the even and odd parts of P (s) as

P even(s) : = p0 + p2s2 + · · ·

P odd(s) : = p1s + p3s3 + · · · (9.2)

We also define P e(ω) and P o(ω) as follows:

P e(ω) = P even(jω) = p0 − p2ω2 + · · ·

P o(ω) =P odd(jω)

jω= p1 − p3ω

2 + · · · (9.3)

P e(ω) and P o(ω) are both polynomials in ω2 and as an immediate consequence their rootsets will always be symmetric about the origin of the complex plane.

Suppose now that the degree of the polynomial P (s) is even, that is n = 2m,m > 0. Inthat case we have

P e(ω) = p0 − p2ω2 + · · ·+ (−1)m p2mω2m

P o(ω) = p1 − p3ω2 + · · ·+ (−1)m−1 p2m−1ω

2m−2

Definition 8 We say P (s) satisfies the interlacing property if and only if

9.1 Concepts Related to Uncertain Systems 141

1. p2m and p2m−1 have the same sign.

2. All the roots of P e(ω) and P o(ω) are real and the m positive roots of P e(ω) togetherwith the m− 1 positive roots of P o(ω) interlace in the following manner:

0 < ωe,1 < ωo,1 < ωe,2 < · · · < ωo,m−1 < ωe,m

If on the contrary the degree of P (s) is odd then n = 2m + 1, m ≥ 0, and

P e(ω) = p0 − p2ω2 + · · ·+ (−1)m p2mω2m

P o(ω) = p1 − p3ω2 + · · ·+ (−1)m p2m+1ω

2m

and the definition of the interlacing property for this case is naturally modified to

1. p2m and p2m+1 have the same sign.

2. All the roots of P e(ω) and P o(ω) are real and the m positive roots of P e(ω) togetherwith the m positive roots of P o(ω) interlace in the following manner:

0 < ωe,1 < ωo,1 < ωe,2 < · · · < ωo,m−1 < ωe,m < ωo,m

Theorem 4 (Hermite Bieler Theorem) A real polynomial P (s) is Hurwitz if and onlyif it satisfies the interlacing property.

Proof: The proof of this theorem is rather involved and has been discussed in detailin [80].¥

Lemma 4 Let

P1(s) = P even1 (s) + P odd(s) (9.4)

P2(s) = P even2 (s) + P odd(s)

denote two stable polynomials of the same degree with the same odd part P odd(s) anddiffering even parts P even

1 (s) and P even2 (s) satisfying,

P e1 (ω) ≤ P e

2 (ω) ∀ω ∈ [0,∞] (9.5)

Then,

P (s) = P even(s) + P odd(s),

is stable for every P even(s) satisfying

P e1 (ω) ≤ P e(ω) ≤ P e

2 (ω) ∀ω ∈ [0,∞] (9.6)


Proof: Since P1(s) and P2(s) are stable, P e1 (ω) and P e

2 (ω) both satisfy the interlacingproperty with P o(s). In particular, P e

1 (ω) and P e2 (ω) are not only of the same degree, but

the sign of their highest order coefficient is also the same since it is in fact the same as thatof the highest coefficient of P o(ω). Given this it is easy to see that P e(ω) cannot satisfyEqn. (9.6) unless it also has this same degree and the same sign for its highest coefficient.Then, the condition in Eqn. (9.6) forces the roots of P e(ω) to interlace with those of P o(ω).Therefore, P even(s) + P odd(s) is stable.¥

The dual of this lemma can be stated as

Lemma 5 Each P (s) = P even(s) + P odd(s) satisfying

P o1 (ω) ≤ P o(ω) ≤ P o

2 (ω) ∀ω ∈ [0,∞]

is stable if both the polynomials

P1(s) = P even(s) + P odd1 (s) (9.7)

P2(s) = P even(s) + P odd2 (s)

are stable.

9.1.3 Kharitonov Theorem

Consider the family F of real interval polynomials,

P (s) :=p(s) = qnsn + · · ·+ q0 : qi ∈

[q−i , q+

i

], i = 0, 1, · · · , n

The Kharitonov’s theorem provides a surprisingly simple necessary and sufficient condi-tion for the Hurwitz stability of the entire family.

Theorem 5 (Kharitonov’s Theorem) Every polynomial in the family F is Hurwitz ifand only if the following four extreme polynomials are Hurwitz [81]:

K1(s) = q−0 + q−1 s + q+2 s2 + q+

3 s3 + q−4 s4 + · · ·K2(s) = q−0 + q+

1 s + q+2 s2 + q−3 s3 + q−4 s4 + · · · (9.8)

K3(s) = q+0 + q−1 s + q−2 s2 + q+

3 s3 + q+4 s4 + · · ·

K4(s) = q+0 + q+

1 s + q−2 s2 + q−3 s3 + q+4 s4 + · · ·

Proof: The proof given allows for the interpretation of Kharitonov’s theorem as a gen-eralization of the interlacing property of Hurwitz polynomials.

Let us introduce the hyper-rectangle or box B of coefficients of the perturbed polynomials

B =q|q ∈ Rn+1, q−i ≤ qi ≤ q+

i , i = 0, 1, · · · , n(9.9)

The four Kharitonov’s polynomials are built from two different even parts Kevenmax (s) and

Koddmin(s) and two different odd parts Kodd

max(s) and Koddmin(s) defined below:


Kevenmax (s) = q+

0 + q−2 s2 + q+4 s4 + q−6 s6 + · · ·

Kevenmin (s) = q−0 + q+

2 s2 + q−4 s4 + q+6 s6 + · · · (9.10)

and

Koddmax(s) = q+

1 s + q−3 s3 + q+5 s5 + q−7 s7 + · · ·

Koddmin(s) = q−1 s + q+

3 s3 + q−5 s5 + q+7 s7 + · · · (9.11)

The motivation of the subscripts max and min is as follows. Let q(s) be an arbitrarypolynomial with its coefficients lying in the box B and let qeven(s) be its even part. Then

Kemax(ω) = q+

0 − q−2 ω2 + q+4 ω4 − q−6 ω6 + · · ·

qe(ω) = q0 − q2ω2 + q4ω

4 − q6ω6 + · · · (9.12)

Kemin(s) = q−0 − q+

2 ω2 + q−4 ω4 − q+6 ω6 + · · ·

so that

Kemax(ω)− qe(ω) =

(q+0 − q0

)+

(q2 − q−2

)ω2 +

(q+4 − q4

)ω4 + · · ·

and

qe(ω)−Kemin(ω) =

(q0 − q−0

)+

(q+2 − q2

)ω2 +

(q4 − q−4

)ω4 + · · ·

Therefore,

Kemin(ω) ≤ qe(ω) ≤ Ke

max(ω), ω ∈ [0,∞] (9.13)

Similarly, if qodd(s) denotes the odd part of q(s), it can be verified that

Komin(ω) ≤ qo(ω) ≤ Ko

max(ω), ω ∈ [0,∞] (9.14)

To proceed, note that the Kharitonov polynomials in Eqn. (9.8) can be rewritten as

K1(s) = Kevenmin (s) + Kodd

min(s) (9.15)

K2(s) = Kevenmin (s) + Kodd

max(s) (9.16)

K3(s) = Kevenmax (s) + Kodd

min(s)

K4(s) = Kevenmax (s) + Kodd

max(s)

If all the polynomials with the coefficients in the box B are stable, it is clear that theKharitonov polynomials in Eqn. (9.8) must also be stable since their coefficients lie inB. For the converse, assume that the Kharitonov polynomials are stable, and let q(s) =


qeven(s)+ qodd(s) be an arbitrary polynomial with coefficients in the box B with its even partqeven(s) and its odd part qodd(s).

Since K1(s) and K2(s) are stable and Eqn. (9.14) holds, we conclude from Lemma 5applied to K1(s) and K2(s) in Eqn. (9.15), that

Kevenmin (s) + qodd(s) is stable.

Similarly, from Lemma 5 applied to K3(s) and K4(s) in Eqn. (9.15), we conclude that

Kevenmax (s) + qodd(s) is stable.

Now, since Eqn. (9.13) holds, we can apply Lemma 4 to the two stable polynomialsKeven

max (s) + qodd(s) and Kevenmin (s) + qodd(s) and we conclude that

qeven(s) + qodd(s) = q(s) is stable

This completes the proof. ¥

Kharitonov’s theorem for Complex polynomials

Consider the family of complex interval polynomials F∗ :

q(s) = (α0 + jβ0) + (α1 + jβ1) s + · · ·+ (αn + jβn) sn (9.17)

where

αi ∈ [α−i , α+i ], βi ∈ [β−i , β+

i ], i = 0, 1, · · · , n (9.18)

Kharitonov extended his result for real polynomials to the above complex interval familyby introducing two sets of complex polynomial as follows:

K+1 (s) : =

(α−0 + jβ−0

)+

(α−1 + jβ+

1

)s +

(α+

2 + jβ+2

)s2 +

(α+

3 + jβ−3)s3 (9.19)

+(α−4 + jβ−4

)s4 +

(α−5 + jβ+

5

)s5 + · · ·

K+2 (s) : =

(α−0 + jβ+

0

)+

(α+

1 + jβ+1

)s +

(α+

2 + jβ−2)s2 +

(α−3 + jβ−3

)s3 (9.20)

+(α−4 + jβ+

4

)s4 +

(α+

5 + jβ+5

)s5 + · · ·

K+3 (s) : =

(α+

0 + jβ−0)

+(α−1 + jβ−1

)s +

(α−2 + jβ+

2

)s2 +

(α+

3 + jβ+3

)s3 (9.21)

+(α+

4 + jβ−4)s4 +

(α−5 + jβ−5

)s5 + · · ·

K+4 (s) : =

(α+

0 + jβ+0

)+

(α+

1 + jβ−1)s +

(α−2 + jβ+

2

)s2 +

(α−3 + jβ+

3

)s3 (9.22)

+(α+

4 + jβ+4

)s4 +

(α+

5 + jβ−5)s5 + · · ·

and


K−1 (s) : =

(α−0 + jβ−0

)+

(α+

1 + jβ−1)s +

(α+

2 + jβ+2

)s2 +

(α−3 + jβ+

3

)s3 (9.23)

+(α−4 + jβ−4

)s4 +

(α+

5 + jβ−5)s5 + · · ·

K−2 (s) : =

(α−0 + jβ+

0

)+

(α−1 + jβ−1

)s +

(α+

2 + jβ−2)s2 +

(α+

3 + jβ+3

)s3 (9.24)

+(α−4 + jβ+

4

)s4 +

(α−5 + jβ−5

)s5 + · · ·

K−3 (s) : =

(α+

0 + jβ−0)

+(α+

1 + jβ+1

)s +

(α−2 + jβ+

2

)s2 +

(α−3 + jβ−3

)s3 (9.25)

+(α+

4 + jβ−4)s4 +

(α+

5 + jβ+5

)s5 + · · ·

K−4 (s) : =

(α+

0 + jβ+0

)+

(α−1 + jβ+

1

)s +

(α−2 + jβ−2

)s2 +

(α+

3 + jβ−3)s3 (9.26)

+(α+

4 + jβ+4

)s4 +

(α−5 + jβ+

5

)s5 + · · ·

Theorem 6 (Complex Variable Kharitonov’s Theorem) The family of polynomials F∗

is Hurwitz if and only if the eight Kharitonov polynomials K+1 (s), K+

2 (s), K+3 (s), K+

4 (s), K−1 (s), K−

2 (s), K−3 (s), K−

4 (s)are all Hurwitz.

Proof : The proof of this theorem has been discussed in [80].¥

9.1.4 Gerschgorin Theorem

Let A = [aij] , i, j = 1, 2, · · · , n be a n× n matrix.Let

ri =n∑

j−1,j 6=i|aij| , i = 1, 2, · · · , n (9.27)

Let Zi denote the circle in the complex plane with centre aii and radius ri.

Zi = z ∈ C| |z − aii| ≤ ri (9.28)

Theorem 7 (Gershgorin Theorem) A as above. Let λ be an eigenvalue of A then λbelongs to one of the circles Zi. Moreover if m of the circles form a connected set S, disjointfrom the remaining n−m circles, then S contains exactly m of the eigenvalues of A, countedaccording to their multiplicity as roots of the characteristic polynomial of A.

Proof: [82]Let λ be an eigenvalue of A and x its corresponding eigenvector. Let k besuch that

|xk| = max1≤i≤n

|xi| = ‖x‖∞ (9.29)

then from Ax = λx, the kth component

n∑j=1

akjxj = λxk

(λ− akk) xk =n∑

j=1,j 6=kakjxj

|λ− akk| |xk| ≤n∑

j=1,j 6=k|akj| |xj| ≤ rk ‖x‖∞


Cancelling ‖x‖∞ proves the theorem. ¥

Remark 7 Since A and AT have the same eigenvalues and characteristic polynomial, theseresults are also valid if summation within the column rather than in the row is used definingthe radii in Eqn. (9.27)

9.1.5 Simultaneous Stabilization of Interval Plant Family Basedon Kharitonov Theorem

Consider the interval plant family F of the transfer function form

F =N(s)

D(s)(9.30)

where

D(s) = sn+n−1∑

i=0 aisi, ai = [a−i , a+

i ]

N(s) =n−1∑

i=0 bisi, bi = [b−i , b+

i ]

This system in Eqn. (9.30) can be represented in controllable canonical form as

X = AX + Bu,

y = CX (9.31)

where,

A =

0 1 · · · 0...

.... . .

...0 0 · · · 1−a0 −a1 · · · −an−1

, B =

0...01

,

C =[

b0 b1 · · · bn−1

]. (9.32)

A state feedback vector F is said to be stabilizing the above interval system if the inputu = FX makes the characteristic polynomial of the closed loop system

f(s) = det(sI − A− bF ) , α0 + α1s + · · ·+ αn−1sn−1 + sn

a hurwitz invariant polynomial; i.e., all the roots of the uncertain polynomial are in the strictleft half of the complex plane.

where F =[

f0 f1 · · · fn−1

]and αi = ai − fi, i = 0, 1, .., n− 1.

Lemma 6 (Simultaneously Stabilizing State Feedback for Interval Systems) An in-terval plant with D(s) as its characteristic polynomial is completely stabilized by the statefeedback gain F designed to stabilize the four Kharitonov polynomials of D(s).

9.2 Bhattacharyya’s Method 147

Proof : The four Kharitonov plants with have their characteristic equations as

K1(s) = a−0 + a−1 s + a+2 s2 + a+

3 s3 + a−4 s4 + · · ·+ sn

K2(s) = a−0 + a+1 s + a+

2 s2 + a−3 s3 + a−4 s4 + · · ·+ sn (9.33)

K3(s) = a+0 + a−1 s + a−2 s2 + a+

3 s3 + a+4 s4 + · · ·+ sn

K4(s) = a+0 + a+

1 s + a−2 s2 + a−3 s3 + a+4 s4 + · · ·+ sn

If these plants were represented in phase variable canonical form as shown in Eqn.(9.32), anda simultaneously stabilizing state feedback gain F is obtained. This state feedback whenapplied to the interval plant, would change its characteristic polynomial from D(s) to D(s),where

D(s) = d0 + d1s + d2s2 + · · ·+ dn−1s

n−1 + sn

where, di = ai − fi, i = 0, 1, · · · , n− 1

This would result in the new interval polynomial D(s), with its four Kharitonov polynomialsas

K1(s) = d−0 + d−1 s + d+2 s2 + d+

3 s3 + d−4 s4 + · · ·+ sn (9.34)

K2(s) = d−0 + d+1 s + d+

2 s2 + d−3 s3 + d−4 s4 + · · ·+ sn

K3(s) = d+0 + d−1 s + d−2 s2 + d+

3 s3 + d+4 s4 + · · ·+ sn

K4(s) = d+0 + d+

1 s + d−2 s2 + d−3 s3 + d+4 s4 + · · ·+ sn

all of which are Hurwitz (by design). Since the four Kharitonov polynomials of D(s) arestable , it is implied from the Kharitonov theorem, that the entire family of polynomials isstable. This implies that the state feedback F , designed to stabilize the four Kharitonovplants stabilizes the entire family of plants.

9.2 Bhattacharyya’s Method

Let us consider the following problem. Suppose that you are given a set of n nomi-nal parameters

a0

0, a01, a

02, · · · , a0

n−1

, together with a set of prescribed uncertainty ranges,

∆a0, ∆a1, ∆a2, · · · , ∆an−1, and that you consider the family F0 of monic polynomials

δ(s) = δ0 + δ1s + δ2s2 + · · ·+ δn−1s

n−1 + sn

where

δi ∈[a0

i −∆ai

2, a0

i +∆ai

2

], i = 0, 1, · · · , n− 1

Suppose now that you can use a vector of n free parameters K = (k0, k1, k2, · · · , kn−1) ,to transform the family F0 into a family Fk described by


δ(s) = (δ0 + k0) + (δ1 + k1) s + (δ2 + k2) s2 + · · ·+ (δn−1 + kn−1) sn−1 + sn

The problem of interest, then, is the following: Given ∆a0, ∆a1, ∆a2, · · · , ∆an−1 theperturbation ranges fixed a priori, find, if possible, a vector K so that the new family ofpolynomials Fk is entirely stable. This problem arises, for example, when one applies astate-feedback control to a single input single output system where the matrices A, B are incontrollable canonical form, and the coefficients of the characteristic polynomial of A aresubject to bounded perturbations.

Lemma 7 Let n be a positive integer and let P (s) be a stable polynomial of degree n− 1 :

P (s) = p0 + p1s + p2s2 + · · ·+ pn−1s

n−1

pi > 0, i = 0, 1, 2, · · · , n− 1

Then there exists an α > 0 such that:

Q(s) = P (s) + pnsn = p0 + p1s + p2s

2 + · · ·+ pn−1sn−1 + pns

n

is stable if and only if pn ∈ [0, α) .

Proof: To be absolutely rigorous there should be four different proofs depending onwhether n is of the form 4r or 4r + 1 or 4r + 2 or 4r + 3. We will give the proof of thislemma when n is of the form 4r and one can check that only slight changes are needed if nis of the form 4r + j, j = 1, 2, 3

If n = 4r, r > 0, we can write

P (s) = p0 + p1s + p2s2 + · · ·+ p4r−1s

4r−1

and the even and odd parts are given by :

Peven(s) = p0 + p2s2 + · · ·+ p4r−2s

4r−2

Podd = p1s + p3s3 + · · ·+ p4r−1s

4r−1

Let us also define

P e(ω) : = Peven(jω) = p0 − p2ω2 + p4ω

4 − · · · − p4r−2ω4r−2

P o(ω) : =Podd(jω)

jω= p1 − p3ω

2 + p5ω4 − · · · − p4r−1ω

4r−1

P (s) being stable, we know by the Hermite-Bieler theorem that P e(s) has precisely 2r−1positive roots ωe,1, ωe,2, · · · , ωe,2r−1, that P o(ω) has also 2r−1 positive roots ωo,1, ωo,2, · · · , ωo,2r−1

and that these roots interlace in the following manner:


0 < ωe,1 < ωo,1 < ωe,2 < ωo,2 < · · · < ωe,2r−1 < ωo,2r−1

It can also be checked thatP e(ωo,j) < 0 if and only if j is odd, and P e(ωo,j) > 0 if and only if j is even.That is

P e(ωo,1) < 0, P e(ωo,2) > 0, P e(ωo,3) < 0, · · · , P e(ωo,2r−2) > 0, P e(ωo,2r−1) < 0 (9.35)

Let us denote

α = minj odd

−P e(ωo,j)

(ωo,j)4r

(9.36)

By (9.35), we know that α is positive. We can now prove the following:

Q(s) = P (s) + p4rs4r is stable if and only if p4r ∈ [0, α)

Q(s) is certainly stable when p4r = 0. Let us now suppose that

0 < p4r < α (9.37)

Qo(ω) and Qe(ω) are given by

Qo(ω) = P o(ω) = p1 − p3ω2 + p5ω

4 − · · · − p4r−1ω4r−1 (9.38)

Qe(ω) = P e(ω) + p4rω4r = p0 − p2ω

2 + p4ω4 − · · · − p4r−2ω

4r−2 + p4rω4r

We are going to show that Qo(ω) and Qe(ω) satisfy the Hermite-Bieler theorem providedthat p4r remains within the bounds defined in (9.37).

First we know the roots of Qo(ω) = P o(ω). Then we have that Qe(0) = p0 > 0, and also

Qe(ωo,1) = P e(ωo,1) + p4r(ωo,1)4r

But, by (9.36) and (9.37), we have

Qe(ωo,1) < P e(ωo,1)− P e(ωo,1)

(ωo,1)4r (ωo,1)

4r

︸︷︷︸=0

Thus Qe(ωo,1) < 0. Then we have

Qe(ωo,2) = P e(ωo,2) + p4r (ωo,2)4r

But, by (9.35), we know that P e(ωo,2) > 0, and therefore we also have


Qe(ωo,2) > 0

Pursuing the same reasoning we could prove in exactly the same way that the followingqualities hold

Qe(0) > 0, Qe(ωo,1) < 0, Qe(ωo,2) > 0, · · · , Qe(ωo,2r−1) < 0 (9.39)

From this we already conclude that Qe(ω) has precisely 2r− 1 roots in the open interval(0, ωo,2r−1) , namely

ω′e,1, ω

′e,2, · · · , ω

′e,2r−1

and that these roots interlace with the roots of Qo(ω)

0 < ω′e,1 < ωo,1 < ω

′e,2 < ωo,2 < · · · < ω

′e,2r−1 < ωo,2r−1 (9.40)

Moreover, we see in (9.39) that

Qe(ωo,2r−1) < 0

and since p4r > 0, we obviously have

Qe(+∞) > 0

Therefore Qe(ω) has a final positive root ω′e,2r which satisfies

ωo,2r−1 < ω′e,2r (9.41)

From (9.40) and (9.41) we conclude that Qo(ω) and Qe(ω) satisfy the Hermite-Bielertheorem and therefore Q(s) is stable.

To complete the proof of this lemma, notice that Q(s) is obviously unstable if p4r < 0,since we assumed all pi to be positive. Moreover, it can be shown by using (9.36) that forp4r = α, the polynomial P (s) + αs4r has a pure imaginary root and therefore is unstable.Now, it is impossible that P (s) + αs4r to be stable for some p4r > α,because otherwise wecould use the Kharitonov’s theorem and say,

P (s) +α

2s4r and P (s) + p4rs

4r both stable ⇒ P (s) + αs4r stable

which would be a contradiction. This completes the proof of the theorem when n = 4r.¥For the sake of completeness, let us make precise that in general we have,

if n = 4r, α = minj odd

−P e(ωo,j)

(ωo,j)4r

if n = 4r + 1, α = minj even

−P o(ωo,j)

(ωo,j)4r+1


if n = 4r + 2, α = minj even

P e(ωo,j)

(ωo,j)4r+2

if n = 4r, α = minj odd

P o(ωo,j)

(ωo,j)4r+3

The details of the proof for the other cases are omitted. ¥We can now enunciate the following theorem to answer the question raised at the begin-

ning of this section.

Theorem 8 For any set of nominal parameters a0, a1, a2, · · · , an−1 , and for any set ofpositive numbers ∆a0, ∆a1, · · · , ∆an−1, it is possible to find a vector K such that the entirefamily Fk is stable.

Proof :The proof is constructive [83].Step 1: Take any stable polynomial R(s) of degree n−1. Let ρ(R(.)) be the radius of the

largest stability hypersphere around R(s).

ρ(R(.)) = rmax = max |r| such that r ∈ C, R(s− r) is HurwitzIt can be checked that for any positive real number λ, we have

ρ(λR(.)) = λρ(R(.))

Thus it is possible to find a positive real λ such that if P (s) = λR(s),

ρ(P (.)) >

√∆a2

0

4+

∆a21

4+ · · ·+ ∆a2

n−1

4(9.42)

If we denote

P (s) = p0 + p1s + p2s2 + · · ·+ pn−1s

n−1

we conclude from Eqn. (9.42) that the following four Kharitonov polynomials of degreen− 1 are stable.

P 1(s) =

(p0 − ∆a0

4

)+

(p1 − ∆a1

4

)s +

(p2 +

∆a2

4

)+ · · ·

P 2(s) =

(p0 − ∆a0

4

)+

(p1 +

∆a1

4

)s +

(p2 +

∆a2

4

)+ · · · (9.43)

P 3(s) =

(p0 +

∆a0

4

)+

(p1 − ∆a1

4

)s +

(p2 − ∆a2

4

)+ · · ·

P 4(s) =

(p0 +

∆a0

4

)+

(p1 +

∆a1

4

)s +

(p2 − ∆a2

4

)+ · · ·

Step 2: Now, applying Lemma 7, we know that we can find four positive numbersα1, α2, α3, α4, such that


P j(s) + pnsn, is stable for 0 ≤ pn < αj, j = 1, 2, 3, 4

Let us select a single positive number α such that the polynomials,

P j(s) + αsn (9.44)

are all stable. If α can be chosen to be equal to 1 ( that is if all four αj are greater than

1) then we do choose α = 1; otherwise we multiply everything by1

αwhich is greater than

1 and we know from (9.44) that the four polynomials

Kj(s) =1

αP j(s) + sn

are stable. But the four polynomials Kj(s) are nothing but the four Kharitonov polyno-mials associated with the family of polynomials

δ(s) = δ0 + δ1s + · · ·+ δn−1sn−1 + sn

where,

δi =

[1

αpi − 1

α

∆ai

2,1

αpi +

1

α

∆ai

2

], i = 0, 1, 2, · · · , n− 1

and therefore this family is entirely stable.

Step 3: It suffices now to choose the vector K such that

ki + a0i =

1

αpi, i = 0, 1, · · · , n− 1

The vector K so obtained would be the stabilizing state feedback for the interval plantrepresentation (A, b) in phase variable canonical form. The state feedback K would renderthe closed loop system

x = Ax + bu

u = −Kx

stable.¥

Remark 8 It is clear that in Step 1 one can determine the largest box around R(.) withsides proportional to ∆ai. The dimensions of such a box are also enlarged by a factor λ whenR(.) is replaced by λR(.). This change does not affect the steps that follow.


Example 10 Suppose that our nominal plant is

s6 − s5 + 2s4 − 3s3 + 2s2 + s + 1

and with the uncertainty ranges

∆a0 = 3, ∆a1 = 5, ∆a2 = 2, ∆a3 = 1, ∆a4 = 7, ∆a5 = 5

Our aim is find the stabilizing vector K.

Proof : Here we have(a0

0, a01, a

02, a

03, a

04, a

05

)= (1, 1, 2,−3, 2,−1)

Step 1: Consider the following stable polynomial of degree 5

R(s) = (s + 1)5 = 1 + 5s + 10s2 + 10s3 + 5s4 + s5

The calculation of ρ(R(.)) gives: ρ(R(.)) = 1.On the other hand, we have

√∆a2

0

4+

∆a21

4+ · · ·+ ∆a2

n−1

4= 5.31

Therefore, taking λ = 6, we have

P (s) = 6 + 30s + 60s2 + 60s3 + 30s4 + 6s5

has a radius ρ(P (.)) = 6 that is greater than 5.31.Thus the four polynomials P j(s) are given by

P 1(s) = 4.5 + 27.5s + 61s2 + 60.5s3 + 26.5s4 + 3.5s5,

P 2(s) = 4.5 + 32.5s + 61s2 + 59.5s3 + 26.5s4 + 8.5s5,

P 3(s) = 7.5 + 27.5s + 59s2 + 60.5s3 + 33.5s4 + 3.5s5,

P 3(s) = 7.5 + 32.5s + 59s2 + 59.5s3 + 33.5s4 + 8.5s5.

Step 2: The application of Lemma 7 gives the following values

α1 ' 1.360, α2 ' 2.667, α3 ' 1.784, α4 ' 3.821

and therefore we can choose α = 1, so that the four polynomials

K1(s) = 4.5 + 27.5s + 61s2 + 60.5s3 + 26.5s4 + 3.5s5 + s6,

K2(s) = 4.5 + 32.5s + 61s2 + 59.5s3 + 26.5s4 + 8.5s5 + s6,

K3(s) = 7.5 + 27.5s + 59s2 + 60.5s3 + 33.5s4 + 3.5s5 + s6,

K3(s) = 7.5 + 32.5s + 59s2 + 59.5s3 + 33.5s4 + 8.5s5 + s6.


are stableStep 3: We just have to take

K = (k0, k1, k2, k3, k4, k5) = (p0, p1, p2, p3, p4, p5)−(a0

0, a01, a

02, a

03, a

04, a

05

)

= (5, 29, 58, 63, 28, 7)

Thus, the state space representation

x =

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1[−2.5, 0.5] [−3.5, 1.5] [−3,−1] [2.5, 3.5] [−5.5, 1.5] [−1.5, 3.5]

x +

000001

u

u =[ −5 −29 −58 −63 −28 −7

]x

is stable.¥

9.3 Jayakumar’s Method

9.3.1 Routh Table Based Kharitonov Algorithm in Controller De-sign

Consider an interval plant family F , represented in the phase variable canonical form. It hasalready been shown that if there exists a state feedback K, such that it stabilizes all the fourKharitonov plants of the family (K1, K2, K3, K4) simultaneously, then K would stabilize theentire interval plant family.

Jayakumar [?] had proposed a method of deriving a simultaneously stabilizing statefeedback for interval plant families. The method involves preparing the closed loop routhtable for the four Kharitonov characteristic equations of the plant family and finding thecoefficients of K that would render all the four plants stable using nonlinear programmingtechniques.

The method can be summarized thus:

1. Represent the plant family F in the phase variable canonical form and obtain the fourKharitonov characteristic equations, K1(s), K2(s), K3(s), K4(s).

Ki(s) = ai0 + ai

1s + ai2s

2 + · · ·+ ain−1s

n−1 + sn

i = 1, 2, 3, 4

where

aij, j = 0, 1, 2, · · · , n− 1, i = 1, 2, 3, 4, are one or the other of the extremal points of the

intervals [a−j , a+j ] depending on the value of i.

9.3 Jayakumar’s Method 155

2. Let the simultaneously stabilizing state feedback be K =[

k0 k1 · · · kn−1

]. Then,

applying this feedback to the Kharitonov plants would transform their characteristicequations to

Ki′(s) =(ai

0 + k0

)+

(ai

1 + k1

)s + · · ·+ (

ain−1 + kn−1

)sn−1 + sn

i = 1, 2, 3, 4

3. Now prepare the Routh Table of the four polynomials. The first row of the table wouldbe nonlinear expressions in ki. Setting all the first row elements of the Routh Tablesof the four Kharitonov polynomials to be positive, one would get a set of nonlinearinequalities in ki.

4. These inequalities when solved using nonlinear programming, with the aim of mini-mizing ‖K‖2 .

Example 11 The problem is to compute the simultaneously stabilizing state feedback for thesystem

x =

0 1 00 0 1[−3,−2] [−2,−1] [−2, 1]

x +

001

u

SOLUTION: The characteristic polynomial of the interval system is

P (s) = s3 + [−1, 2]s2 + [1, 2]s + [2, 3]

The four Kharitonov polynomials can be computed to be

K1(s) = 2 + s + 2s2 + s3

K2(s) = 2 + 2s + 2s2 + s3

K3(s) = 3 + s− s2 + s3

K4(s) = 3 + 2s− s2 + s3

The Kharitonov polynomials of the closed loop system would be

K1′(s) = (2 + k0) + (1 + k1)s + (2 + k2) s2 + s3

K2′(s) = (2 + k0) + (2 + k1) s + (2 + k2) s2 + s3

K3′(s) = (3 + k0) + (1 + k1)s + (−1 + k2)s

2 + s3

K4′(s) = (3 + k0) + (2 + k1) s + (−1 + k2)s

2 + s3

This gives rise to eight nonlinear inequalities which when solved give the value of K as

K =[ −0.6 1 2.2

]

The interval plant family is stable for u = −Kx


9.3.2 An Alternative Proof for Existence of a Simultaneously Sta-bilizing State Feedback for Interval Systems

Let X = AX +bu be a linear time invariant structured uncertain SISO system. The meaningof structured uncertain system is that the system parameters vary on some bounded intervalon the real line. It is assumed that the system is in the controllable canonical form, P (s) bethe interval characteristic polynomial of the system.

The four Kharitonov polynomials of P (s) lead to four state matrices which are denotedby A1, A2, A3 and A4.

Proposition 3 The following statements are equivalent.

1. V (X) is the Lyapunov function of X = (A− bK) X.Where u = −KX and K is statefeedback gain. V (X) = XT PX and

V (X) = XT(P (Ai − bK) + (Ai − bK)T P

)X = −XT QX, i = 1, 2, 3, 4

By using appropriate choices of positive definite P and Q matrices for i = 1, 2, 3, 4 inthe above equation find state feedback gain K such that closed loop uncertain system isstable.

2. Find a state feedback gain K such that real part of eigenvalues of (Ai − bK) is lessthan zero for all i. (where i = 1, 2, 3, 4)

Proof:

2 ⇒ 1 Suppose the second statement is true, then there exist positive definite matrices P andQ such that the following equation is true.

P (Ai − bK) + (Ai − bK)T P = −Q (9.45)

It may be difficult to construct such a P and Q, but there exist such a P and Q tosatisfy Eqn. (9.45) and it is true for all i from 1 to 4.

1 ⇒ 2 Suppose the first statement is true. All the four extremal systems are stable. ByKharitonov theorem, it is clear that the second statement is true.

Hence concludes the proof. ¥

Proof of Existence of Simultaneously Stabilizing K

Let P1(s), P2(s), P3(s), P4(s) be the Kharitonov polynomials constructed from the intervalpolynomial P (s).

Proposition 4 Let Ri(s) = Pi(s) + Q(s) be an nth degree polynomial. For a given Pi(s) itis possible to select Q(s) such that Ri(s) is stable.

9.4 Smagina & Brewer’s Method 157

Proof : Assume i = 2 and also assume P2(s) has its roots inside a circle centered at theorigin and radius rmax. Transform P2(s) to R2(s) by using αrmax. Bound on α can be derivedas follows.

R2(s) = P2 (s + αrmax) = P2(s) + Q(s)

Add the same Q(s) to other polynomials P1(s), P3(s) and P4(s).

R1(s) = P1(s) + Q(s) = P2(s) + Q(s) + q1(s)

R3(s) = P3(s) + Q(s) = P2(s) + Q(s) + q3(s)

R4(s) = P4(s) + Q(s) = P2(s) + Q(s) + q4(s)

Where qj = cj1s

n−1 + cj2s

n−2 + · · ·+ cjn for j = 1, 3 and 4.

Maximum possible value of root of qj(s), j = 1, 3, 4 is cmax. Where

cmax = max

1+n

Σi=1

∣∣cji

∣∣

j=1,3,4

If cmax < αrmax− rmax then it would be impossible for any of these qi(s) to shift the rootsof R2(s) + qi(s) to the right half of the complex plane

From this it is clear that it is always possible to choose α such that Ri(s) is stable forall i = 1, 2, 3, 4. by using the following relation.

α >

cmax

rmax

+ 1

(9.46)

The above α value may be high, but it makes the interval polynomials P (s) stable byadding Q(s) with it.

Existence of stable closed loop state feedback gain K is nothing but existence of Q(s).Thus the existence of simultaneously stabilizing K for a given A1, A2, A3 and A4 is proved.

¥

9.4 Smagina & Brewer’s Method

Smagina and Brewer have presented a method of pole assignment by state feedback forinterval plants [85]. The closed characteristic polynomial of the interval system would besuch that they are included in a given interval polynomial.

9.4.1 The Problem Statement

Let us consider a linear multivariable control system in state space

x = [A] x + [B] u (9.47)

where x is the n-state vector and u is the input r− vector. The elements [aij] , [bik] , i, j =1, 2, · · · , n, k = 1, 2, · · · , r, are intervals with known upper and lower bounds. These matricesdescribe the sets of matrices A ∈ [A] , B ∈ [B] with real elements aij ∈ [aij] , bik ∈ [bik] .


For the robust state feedback control

u = Kx (9.48)

The problem is to find a real r × n matrix K satisfying the inclusions

det (sI − A−BK) ⊆ [D(s)] (9.49)

for every real matrix A ∈ [A] , B ∈ [B] where [D(s)] is an assigned interval asymptoticallystable polynomial.

[D(s)] = sn + [dn−1] sn−1 + · · ·+ [d1] s + [d0] (9.50)

that describes a set of asymptotically stable polynomials D(s) = sn + dn−1sn−1 + · · · +

d1s + d0 with real coefficients di ∈ [di] .

9.4.2 Main Results

Case r = 1 : If r = 1 then [B] = [b] is a column vector, k is a row vector. In such a case,consider the following criterion.

Controllability Criterion 1 (Sufficient) The pair ([A] , [b]) is said to be controllable forany A ∈ [A] , b ∈ [b] if an n× n interval controllability matrix.

[Y ] =([b] , [A] . [b] , · · · , [A]n−1 [b]

)(9.51)

satisfies the condition

0 /∈ det [Y ] (9.52)

where det [•] is the interval extension of the function det (•) .

If the pair ([A] , [b]) do not satisfy the above criterion then there is no guarantee of theexistence of a simultaneously stabilizing state feedback.

Introduce a n× n interval matrix

[P ] = [Y ] .

− [α1] − [α2] · · · − [αn−1] −1− [α2] − [α3] · · · −1 0...

......

...−1 0 · · · 0 0

(9.53)

and an interval row vector

[f ] = ([d0]ª [α0] , [d1]ª [α1] , · · · , [dn−1]ª [αn−1]) (9.54)

where,


[αi] , i = 0, 1, · · · , n− 1 are the coefficients of the interval characteristic polynomial of theinterval matrix [A] (φ(s) = sn + [αn−1] s

n−1 + · · ·+ [α1] s + [α0]) . In Eqn. (9.54), ª denotesa non-standard interval subtraction defined as

[a]ª [b] =[min(a− − b−, a+ − b+), max

(a− − b−, a+ − b+

)](9.55)

Let us define the width of an interval number as

w ([a]) =(a+ − a−

)(9.56)

Theorem 9 If the pair ([A] , [b]) is controllable and the widths of the polynomial coefficientssatisfy the inequalities w ([di]) > w ([αi]) , i = 0, 1, · · · , n − 1 then a n-row vector k of astabilizing state feedback might be calculated from the interval inclusion

K. [P ] ⊂ [f ] (9.57)

Proof : If the pair ([A] , [b]) is controllable then all pairs (A, b) with A ∈ [A] , b ∈ [b] arecontrollable. Then, a feedback row vector k can be calculated from the following matrixequation:

k.(b, Ab, · · · , An−1b

)

−α1 −α2 · · · −αn−1 −1−α2 −α3 · · · −1 0...

......

...−1 0 · · · 0 0

+(α0, α1, · · · , αn−1) = (d0, d1, · · · , dn−1)

(9.58)where α0, α1, · · · , αn−1 are the coefficients of the characteristic polynomial of A, d0, d1, · · · , dn−1

are the coefficients of the assigned asymptotically stable polynomial D(s).The characteristic polynomial coefficients αi(A), i = 0, 1, · · · , n− 1 can be considered as

multi-linear functions of the elements of matrix A. Denote the row vector from the left-handside of Eqn. (9.58) by f(k, A, b). Its coordinates are rational functions of the elements ofmatrices k, A, b. Selecting an interval extension F (k, [A] , [b]) of the function f(k, A, b), thenwe can represent the left-hand side of Eqn. (9.58) in the interval form as

F (k, [A] , [b]) = k. [P ] + ([α0] , [α1] , · · · , [αn−1]) (9.59)

where the n× n interval matrix [P ] is defined in Eqn. (9.53),[αi] , i = 0, 1, · · · , n− 1 areinterval extensions of the rational functions αi(A), A ∈ [A] .

If we can find a real row vector k satisfying the following inclusion:

k. [P ] + ([α0] , [α1] , · · · , [αn−1]) ⊂ ([d0] , [d1] , · · · , [dn−1]) (9.60)

then from Eqn. (9.58) we can calculate di, i = 0, 1, · · · , n − 1 for every A ∈ [A] , b ∈ [b]and, according to (9.60), they are to belong to the corresponding interval coefficients di ∈[di] , i = 0, 1, · · · , n − 1. Function F (k, [A] , [b]) is a sum of an unknown interval row vectork. [P ] and a known interval row vector ([α0] , [α1] , · · · , [αn−1]) . In order to transform (9.60)into the form (9.57) we would subtract the known interval row vector ([α0] , [α1] , · · · , [αn−1])


from both sides of the relation (9.60) leaving the unknown member k. [P ] in the left-handside. Since the regular interval subtraction of an interval number [a] from itself does notresult in a zero ([a]− [a] 6= 0) we cannot use the usual interval arithmetic operation to solveEqn. (9.60) for k. [P ] . The desired zero result can be achieved if we apply a nonstandardinterval subtraction ª as defined in Eqn. (9.55). Application of this arithmetic operation toEqn. (9.60) results in the formula in Eqn. (9.57).

The inequalities w [di] > w [αi] , i = 0, 1, · · · , n − 1 follow from Eqn. (9.54) and thedefinition of the regular interval width. Thus, the theorem has been proved. ¥

Theorem 10 Vector inclusion in (9.57) has a solution of the form

k = M [f ] P−1 (9.61)

if the pair ([A] , [b]) is controllable and for assigned asymptotically stable interval polyno-mial as in Eqn. (9.50), the following inclusion

M [f ] P−1. [P ] ⊂ [f ] (9.62)

takes place.

In Eqns. (9.61 and 9.62), the matrix P ∈ [P ] is a real nonsingular matrix, M [•] denotesa real matrix of interval element midpoints.

Corollary 1 The pair ([A] , [b]) controllability is a necessary condition for the modal P-regulator u = kx existence:

Remark 9 If a regular interval number [a] is not a degenerate (point) interval then it hasa positive width w [a] = w [a−, a+] = (a+ − a−) > 0. So, the elements of interval vector [f ]in Eqn. (9.54) have to have positive widths and therefore, they must satisfy the inequalitiesw [di] > w [αi] , i = 0, 1, 2, · · · , n−1. Thus, it is recommended to select an asymptotically stableor stable interval polynomial [D(s)] with interval coefficients [di] , i = 0, 1, 2, · · · , n − 1 thatshould be wide enough to guarantee the required inequalities.

In practice, a real matrix P may be chosen as P = M [P ] . Let us obtain the restrictionson the interval coefficient widths of the assigned stable polynomial [D(s)] to ensure problemsolvability.

Theorem 11 If the pair ([A] , [b]) controllable according to the Controllability criterion 1and P = M [P ] then the inclusion (9.57) has a solution of the form

k = M [f ] (M [P ])−1 (9.63)

provided that the widths w [dj] , j = 0, 1, 2, · · · , n − 1 of the interval polynomial [D(s)]coefficients satisfy the inequalities

n∑i=1

abs(ki)w [pij] + w [αj] < w [dj] (9.64)

where ki are the row vector k elements, w [pij] are the widths of the elements of the matrix[P ] in Eqn. (9.53).


Remark 10 For some control systems it is possible to calculate k that does not satisfy someof the inequalities in (9.64). In order to comply with the restrictions in (9.64), we can try torelax the intervals [dj] without losing the stability of the interval polynomial [D(s)] and thenrecalculate k.

Therefore, a stabilizing state-feedback control u = kx exists if the pair ([A] , [b]) is con-trollable and the coefficients of some stable interval polynomial [D(s)] satisfy the inequalitiesin (9.64).

Case r ≥ 2 : For multivariable interval control systems we use a concept of intervalmatrix rank.

For an l×p interval matrix [C] we consider all minors det [C]m×m of order m ≤ min(l, p).These minors are equal to some interval numbers.

A minor is referred to as singular if it contains a zero.

Definition 9 An interval matrix [C] has a rank [C] equal to the maximal order of its non-singular minors.

Based on the above definition for n× n matrix [A] and n× r matrix [B] , we introduce

Controllability Criterion 2 The pair ([A] , [B]) is controllable for any A ∈ [A] , B ∈ [B]if and only if the interval controllability matrix

[Y ] =([B] , [A] . [B] , · · · , [A]n−1 . [B]

)(9.65)

satisfies the equation

rank [Y ] = n (9.66)

9.4.3 The Algorithm

Consider the following algorithm for the design of a stabilizing state feedback.

Step 1: For the pair ([A] , [B]) analyze controllability criterion in Eqn. (9.66) to determine ifthe problem has a solution. Note that for the pair ([A] , [b]) conditions (9.51 and 9.52)can be checked out.

Step 2: Calculate the interval coefficients [αi] of the characteristic equation of [A] .

Step 3: Select interval coefficients [di] of the polynomial [D(s)]such that w [di] > w [αi] , i =0, 1, · · · , n − 1 and [D(s)] is asymptotically stable. If for certain i both conditionscannot be satisfied, then the problem has no solution.

Step 4: If r = 1, then go to Step 5, otherwise choose a real r-vector q such that the pair([A] , [B] .q) is controllable.


Step 5: Compute k = M [f ] P−1 or k = M [f ] (M [P ])−1 , if P = M [P ] . If inequalities in(9.64) hold then goto Step 6, otherwise return to Step 3 and increase the widths of theintervals [di] , preserving, if possible, the stability of the polynomial [D(s)] .

Step 6: If r = 1, then K = k. If r ≥ 2 then K = qk.

Example 12 Let us consider a stabilization problem for the helicopter longitudinal motionspeed model with n = 3, r = 2 and interval matrices [A] and [B]

[A] =

[a11] [a12] −9.8[a21] [a22] 00 1 0

, [B] =

[b11] 00 [b22]0 0

(9.67)

where

[a11] = [−0.031,−− 0.0128] , [a12] = [−3.4,−0.1] , [a21] = [−0.00077,−0.0007]

[a22] = [−0.32,−0.31]

[b11] = [−18,−15] , [b22] = [−3.3,−3]

and

[D(s)] = s3 + [3, 4] s2 + [2, 8] s + [0.5, 5.5]

SOLUTION : It can be shown that the pair ([A] , [B]) is controllable and for a selectedvector q = (0.8, 1.2)T , the pair ([A] , [b]) is controllable. Calculations of the characteristicequation of [A] yields

[φ(s)] = s3 + [0.3228, 0.351] s2 + [0.00135, 0.00985] s + [−0.00755,−0.00686]

The matrix [P ] is then found to be

[P ] =

[−43.3833,−30.5269] [−10.036, 4.5404] [12, 14.4][−0.0920, 0.0953] [−0.1161, 0.2556] [3.6, 3.96][−0.1161, 0.2556] [3.6, 3.96] 0

Since, det (M [P ]) 6= 0 and w [di] > w [αi] then

k = M [f ] (M [P ])−1 = (−0.0793, 1.1160, 1.2471).

The gain matrix K for the regulator has the following form.

K = qk =

[ −0.0634 0.8228 0.9977−0.0958 1.3392 1.4963

]

Thus solved. ¥

9.5 State Feedback for Uncertain Systems Based on Gerschgorin Theorem 163

9.5 State Feedback for Uncertain Systems Based on

Gerschgorin Theorem

Lemma 8 Making use of the Gerschgorin theorem, it can be verified that if a matrix A hasits elements such that

aii +n∑

j=1,j 6=i|aij| < 0 (9.68)

aii +n∑

j=1,j 6=i|aji| < 0

then the systemx = Ax

is stable since its eigenvalues would lie only in the strict left-half of the complex plane.

Using the above lemma, an algorithm can be derived for the stabilizing state feedbackfor parametric uncertain systems in state space.

Assume that the system is

x = [A] x + [B] u (9.69)

where [A] , [B] n× n and n× r interval matrices respectively representing the state andinput matrices of the uncertain system. Let the elements of [A] and [B] be denoted as[aij] and [bip] , i = 1, 2, · · · , n, p = 1, 2, · · · , r. respectively. Now, if a state feedback K =kij , i = 1, 2, · · · , r, j = 1, 2, · · · , n is applied to the system, the stability condition in (9.68)is translated to

sup ([aii]) +r∑

j=1sup ([bij] . [kji]) +

n∑j=1,j 6=i

∣∣∣∣∣[aij] +r∑

p=1[bip] [kpj]

∣∣∣∣∣ < 0 (9.70)

sup ([aii]) +n∑

j=1sup ([bij] . [kji]) +

n∑j=1,j 6=i

∣∣∣∣∣[aji] +r∑

p=1[bjp] [kpi]

∣∣∣∣∣ < 0 (9.71)

where, sup ([a]) = a+, the supremum of an interval number and |[a]| = max(|a−| , |a+|), the absolute value of an interval number. The inequalities in (9.70 and 9.71) produce 2nlinear inequalities in kij, which when solved under the minimizing constraint of min (‖K‖) ,would give the stabilizing state feedback for the interval uncertain system with its sensitivityfunction minimized.

Example 13 Consider the uncertain system

x =

[−1,−0.5] 0 00 [−0.5, 0] 00 0 [−0.24, 0.25]

x +

[−1.2,−1.1]11

u (9.72)

The problem is to design a stabilizing state feedback for the system.


SOLUTION : Let the stabilizing state feedback gain be K =[

k1 k2 k3

].

Then, applying the conditions in (9.70 and 9.71) would give the six inequalities,

−0.5 + sup([−1.2,−1.1] .k1) + |[−1.2,−1.1] .k2|+ |[−1.2,−1.1] .k3| < 0

k2 + |k1|+ |k3| < 0

0.25 + k3 + |k1|+ |k2| < 0

−0.5 + sup([−1.2,−1.1] .k1) + |2k1| < 0

k2 + |[−1.2,−1.1] .k2|+ |k2| < 0

0.25 + k3 + |[−1.2,−1.1] .k3|+ |k3| < 0

Using Linear Programming, the optimum value of K is found to be

K =[

0 −0.05 −0.301]

Thus, solved. ¥

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170

Appendix A

Numerical Problems in Large ScaleSystems

A.1 Davison, Chidambara and Marshall Techniques

1. For the third order system

X =

−2 −1 1−0.5 −1.5 −0.5

0.5 0.5 −2.5

X +

20.5

−0.5

u

find the second order Davison approach such that the second state is not retained.

2. For the above system, find the first order Chidambara model and also find the sub-optimal controller that would minimize the quadratic performance index for Q =

1 0 10 1 0

−2 1 1

, R = 10 using the same model.

3. Consider the following third order system

X =

−1 −2 2−2 −1 10 0 −2

X +

001

u

Derive the second order reduced model for the system using Davison’s technique.

4. Use Marshall’s Technique to solve the above problem.

5. Use Chidambara’s Technique to design a suboptimal control for the system for aquadratic performance index with Q = 10I3, R = 50.

6. Consider the fourth order system

X =

0 1 0 0−255 −279 −65 130

3 4 −3 −2−515 −563 −125 263

X +

0102

u

171

Obtain a third order model that is composed of the average of consecutive modes ofthe original system.

7. Use aggregation by continued fraction to find a second order approximation for thesystem

X =

−2 0 0 0−1 −4 3 0

0 3 −4 00 0 1 −6

X +

1111

u

A.2 Routh and Pade Approximations

1. Obtain the third order Pade approximation for the system

G(s) =s3 + 12s2 + 44s + 48

s4 + 16s3 + 86s2 + 176s + 105

such that one of the poles of the reduced model is -1.

2. Find the Second order Pade approximation for the third order system

G(s) =12s + 6

s3 + 6s2 + 11s + 6

3. Consider the third order system

G(s) =b0 + b1s + b2s

2

a0 + a1s + a2s2 + s3

What are the first and second order Routh approximations of the system using γ − δexpansion.

4. Using the Second Cauer Form, find the second order approximation of the system

G(s) =6s2 + 17s + 5

s3 + 3s2 − s− 3

5. Find the first order approximation of the second order 2× 2 system

P (s) =1

s2 + 5s + 6

[s + 1 s− 22s− 1 s + 1

]

using the multivariable extension of the third Cauer form.

6. Compute the second order time-domain Pade approximation for the system

X =

0 1 00 0 1−15 −23 −9

X +

001

u

y =[

8 6 1]x

172

7. Obtain the second order reduced order model using γ − δ parameters for the system

G (s) =s2 + 4s + 3

s3 + 5s2 + 2s− 8

8. Find the first order model for the second order system using γ − δ canonical form

x =

[0 1−2 −3

]x +

[01

]u

y =[

1 0]x

9. Find the second order reduced model, using routh approximants for the system

X =

0 1 0 00 0 1 00 0 0 1−120 −180 −102 −18

X +

0001

u

y =[

1200 900 248 14]X

10. Illustrate a technique to determine the optimal order for a reduced order model throughrouth approximants for the above system.

11. Find the second order approximation for the following system using the second cauerform

G (s) =1441.53s3 + 78319s2 + 525286.125s + 607693.25(

s7 + 112.04s6 + 3755.92s5 + 39736.73s4 + 363650.56s3 + 759894.19s2

+683656.25s + 617497.375

)

A.3 State and Output Feedback Design

1. For the second order system

X =

[ −1 24 −9

]X +

[11

]u

(a) Check for two-time scale property

(b) Find the state feedback gain, by decomposing the system, so that the closed loopeigenvalues of the system are −50,−50.

2. Find the robust state feedback gain F =[

f1 f2

]that would stabilize the interval

system so that the system closed loop system never has complex eigenvalues and arealways to the left of s = −1. Find F so that the performance index J = f 2

1 + f 22 is

minimized.

X =

[0 1a1 a2

]X +

[01

]u

a1 = [−1, 1], a2 = [−1, 1]

173

3. Design a static output feedback controller for the system

X =

0 1 00 0 10 0 −3

X +

001

u

y =

[0 1 01 1 0

]X

so that the closed loop eigenvalues are −2,−0.5± 0.5j

A.4 Periodic Output Feedback and Fast Output Sam-

pling

1. Two first order systems have the transfer functions

G1 =1

s + 1, G2 =

1

s + 3

respectively. At the input to these systems, a sample and hold circuit with samplingperiod of 1 sec has been fixed. Design a fast output sampling controller for this plant-setso that the closed loop eigenvalue is λ = −2 in continuous time

2. What are the conditions for the existence of periodic output feedback gain ?

3. How do you calculate the output injection gain for the system

X(k + 1) = ΦτX(k) + Γτu(k)

y(k) = CX(k)

(a) Define Controllability Index.

(b) In the case of Periodic Output Feedback, for restricting the value of the gain Kthe constraint imposed is [ −ρ2

1I KKT −I

]< 0

what does this mean. How will the satisfaction of this constraint assure that themagnitude of K is restricted.

4. Derive the expression for the lifted output in terms of state and input in the contextof fast output sampling feedback technique.

5. Derive the expression for the dynamics of the error input in the context of fast outputsampling technique.

6. For a three model case, each being controllable and observable and of 10th order, whatare the conditions for the existence of a robust periodic output feedback gain.

174

7. Compute the fast output sampling feedback control input for the system

X(k + 1) =

[0 1−4 −4

]X(k) +

[01

]u(k)

y =[

1 0]X(k)

so that the closed loop characteristic equation is Z2 = 0.5Z.

8. Given the representation of a system at an input sampling period of τ sec as

X (k + 1) =

[0.64 00 1

]X (k) +

[11

]u

y (k) =[

1 2]X (k)

Find the fast output sampling gain L such that the closed loop eigenvalues are placedexactly at λ = 0.6, 0.7 . The output sampling rate is ∆ = τ/2.

9. Assuming that the state feedback F is so designed that the closed loop system (Φτ + ΓτF ) isnon-singular, derive the formula of the fast output sampling controller gain for a system

x(k + 1) = Φτx(k) + Γτu (k)

y (k) = Cx(k) + Du(k)

Use and assign appropriate notations wherever necessary.

A.5 Uncertain Systems

1. Find the minimal set of conditions for the stability of the uncertain polynomial

P (s) =[a−4 , 1

]s4 + [4, 5] s3 +

[6, a+

2

]s2 + [3, 4] s +

[a−0 , 1

]

2. Using the above result, find the state feedback gain F, that would assure the closedloop stability of the system

X =

0 1 0 00 0 1 00 0 0 1[−1, b+

0

][−4,−3]

[b−2 ,−6

][−5,−4]

X +

0001

u

175

model order reduction and controller design...

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