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Robust control of MIMO systems

Olivier Sename

Grenoble INP / GIPSA-lab

September 2017

Olivier Sename (Grenoble INP / GIPSA-lab) Robust control September 2017 1 / 162

1. Some definitionsTheH∞ norm definitionStability issues

2. Introduction to Linear Matrix InequalitiesBackground in OptimisationLMI in controlSome useful lemmas

3. What is theH∞ performance?H∞ norm as a measure of the system gain ?How to compute theH∞ norm?What isH∞ control?

4. WhyH∞ control is adapted to control engineering?Performance analysis and specification usingthe sensitivity functions: the SISO casePerformance analysis and specification usingthe sensitivity functions: the MIMO caseThe mixed sensitivityH∞ control designSome performance limitations

5. How to solve an H∞ control problem?The Static State feedback caseThe Dynamic Output feedback caseThe Riccati approach

The LMI approach forH∞ control design6. Uncertainty modelling and robustness analysis

IntroductionRepresentation of uncertaintiesDefinition of Robustness analysisRobustness analysis: the unstructured caseanalysis: the structured caseRobust control design

7. What else inH∞ approach?H2 and multi-objective problemsH∞ observer design

8. Introduction to LPV systems and controlWhat is a Linear Parameter Varying systems?Classes (models) of LPV systemsHow to approximate a nonlinear system by anLPV one ?Stability of LPV systemsControl & Observation

The Static State feedback caseThe Dynamic Output feedback caseLPV observer designSummary of LPV approach interests

O. Sename [GIPSA-lab] 2/162

About the course

Organization

• 20H lessons in class• 10H Matlab sessions (E3) and 16H (MISCIT)

Homework

• see given document +• course complements : tools for norm definitions, and Introduction to Linear Matrix Inequalities• Exercices• Matlab project

Evaluation

• One final exam• Homework• Ability and report on one Matlab project


Reference booksTo be studied during the course

• S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: analysis and design, JohnWiley and Sons, 2005.www.nt.ntnu.no/users/skoge/book, chap 1 to 3 available

• K. Zhou, Essentials of Robust Control, Prentice Hall, New Jersey, 1998.www.ece.lsu.edu/kemin, book slides available

• J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback control theory, MacmillanPublishing Company, New York, 1992.https://sites.google.com/site/brucefranciscontact/Home/publications, ,book available

• Carsten Scherer’s courseshttp://www.dcsc.tudelft.nl/ cscherer/. , Lecture slides available (MSc Course"Robust Control", MSc Course "Linear Matrix Inequalities in Control")

• + all the MATLAB demo, examples and documentation on the ’Robust Control toolbox’(mathworks.com/products/robust)

Other references (some in french)

• G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, New Jersey, 2001.csd.newcastle.edu.au

• G. Duc et S. Font, Commande Hinf et -analyse: des outils pour la robustesse, Hermès, France, 1999.

• D. Alazard, C. Cumer, P. Apkarian, M. Gauvrit, et G. Ferreres, Robustesse et commande optimale,Cépadues Editions, 1999.


0

Robust control in 1 slide ?


• Sensitivity function S(s) = 11+L(s)

• Complementary Sensitivity function :

y =G(s)K(s)

1 +G(s)K(s)r

=L(s)

1 + L(s)r = T (s).r

0





y =G(s)K(s)

1 +G(s)K(s)r

=L(s)

1 + L(s)r = T (s).r

Why S is the key function in control ?

S allows to characterize many things:• S = 1− T −→ S = r−y

r.

For performance analysis: (S(ω = 0) =steady-state error, bandwidth )

• if output disturbance dy then, ydy

= S.S to be minimized !

• distance from -1 to Nyquist plot =infω | −1− L(jω) |=[supω| 1

1+L(jω)|]−1

• Robustness w.r.t model uncertainties

Robust control: Find K s.t S satisfies allrequirements

0





y =G(s)K(s)

1 +G(s)K(s)r

=L(s)

1 + L(s)r = T (s).r

Example: an uncertain mass-spring-damper systemcontrolled by a proportional gain.

Robust stability condition : | S(jω) |<| 1W (jω)

|,∀ω

1 Some definitions












1 Some definitions

Definition of LTI systems

Definition (LTI dynamical system)

Given matrices A ∈ Rn×n, B ∈ Rn×nw , C ∈ Rnz×n and D ∈ Rnz×nw , a Linear Time Invariant(LTI) dynamical system (ΣLTI ) can be described as:

ΣLTI :

x(t) = Ax(t) +Bw(t)z(t) = Cx(t) +Dw(t)

(1)

where x(t) is the state which takes values in a state space X ∈ Rn, w(t) is the input taking valuesin the input space W ∈ Rnw and z(t) is the output that belongs to the output space Z ∈ Rnz .

The LTI system locally describes the real system under consideration and the linearizationprocedure allows to treat a linear problem instead of a nonlinear one. For this class of problem,many mathematical and control theory tools can be applied like closed loop stability, controllability,observability, performance, robust analysis, etc. for both SISO and MIMO systems. However, themain restriction is that LTI models only describe the system locally, then, compared to nonlinearmodels, they lack of information and, as a consequence, are incomplete and may not provideglobal stabilization.


1 Some definitions

Signal norms

Reader is also invited to refer to the famous book of Zhou et al., 1996, where all the followingdefinitions and additional information are given.All the following definitions are given assuming signals x(t) ∈ C, then they will involve theconjugate (denoted as x∗(t)). When signals are real (i.e. x(t) ∈ R), x∗(t) = xT (t).

Definition (Norm and Normed vector space)

• Let V be a finite dimension space. Then ∀ p ≥ 1, the application ||.||p is a norm, defined as,

||v||p =(∑

i

|vi|p)1/p (2)

• Let V be a vector space over C (or R) and let ‖.‖ be a norm defined on V . Then V is anormed space.


1 Some definitions

L? signal norms

Definition (L1, L2, L∞ norms)

• The 1-Norm of a function x(t) is given by,

‖x(t)‖1 =

∫ +∞

0|x(t)|dt (3)

• The 2-Norm (that introduces the energy norm) is given by,

‖x(t)‖2 =√∫+∞

0 x∗(t)x(t)dt

=√

12π

∫+∞−∞ X∗(jω)X(jω)dω

(4)

The second equality is obtained by using the Parseval identity.• The∞-Norm is given by,

‖x(t)‖∞ = supt|x(t)| (5)

‖X‖∞ = supRe(s)≥0

‖X(s)‖ = supω‖X(jω)‖ (6)

if the signals that admit the Laplace transform, analytic in Re(s) ≥ 0 (i.e. ∈ H∞).


1 Some definitions

L∞ and H∞ spaces

Definition (L∞ space)

L∞ is the space of piecewise continuous bounded functions. It is a Banach space ofmatrix-valued (or scalar-valued) functions on C and consists of all complex bounded matrixfunctions f(jω), ∀ω ∈ R, such that,

supω∈R

σ[f(jω)] <∞ (7)

Definition (H∞ and RH∞ spaces)

H∞ is a (closed) subspace in L∞ with matrix functions f(jω), ∀ω ∈ R, analytic in Re(s) > 0(open right-half plane). The real rational subspace of H∞ which consists of all proper and realrational stable transfer matrices, is denoted by RH∞.

Example

In control theorys+1

(s+10)(s+6)∈ RH∞

s+1(s−10)(s+6)

6∈ RH∞s+1

(s+10)∈ RH∞

(8)


1 Some definitions TheH∞ norm definition

H∞ norm

Definition (H∞ norm)

The H∞ norm of a proper LTI system defined by the state space representation (A,B,C,D) frominput w(t) to output z(t) and which belongs to RH∞, is the induced energy-to-energy gain(induced L2 norm) defined as,

‖G(jω)‖∞ = supω∈R σ (G(jω))

= supw(s)∈H2

‖z(s)‖2‖w(s)‖2

= maxw(t)∈L2

‖z‖2‖w‖2

(9)

Physical interpretations of the H∞ norm

• This norm represents the maximal gain of the frequency response of the system. It is alsocalled the worst case attenuation level in the sense that it measures the maximumamplification that the system can deliver on the whole frequency set.

• For SISO (resp. MIMO) systems, it represents the maximal peak value on the Bodemagnitude (resp. singular value) plot of G(jω), in other words, it is the largest gain if thesystem is fed by harmonic input signal.

• Unlike H2 , the H∞ norm cannot be computed analytically. Only numerical solutions canbe obtained (e.g. Bisection algorithm, or LMI resolution).


1 Some definitions Stability issues

Well-posedness


Consider

G = −s− 1

s+ 2, K = 1

Therefore the control input is non proper:

u =s+ 2

3(r − n− dy) +

s− 1

3di

DEF: A closed-loop system is well-posed if all the transfer functions are proper

⇔ I +K(∞)G(∞) is invertible

In the example 1 + 1× (−1) = 0 Note that if G is strictly proper, this always holds.


Internal stability

DEF: A system is internally stable if all the transfer functions of the closed-loop system are stable

Figure: Control scheme

(yu

)=

((I +GK)−1GK (I +GK)−1G

K(I +GK)−1 −K(I +GK)−1G

)(rdi

)For instance :

G =1

s− 1, K =

s− 1

s+ 1,

(yu

)=

(1s+2

s+1(s−1)(s+2)

s−1s+2

− 1s+2

)(rdi

)There is one RHP pole (1), which means that this system is not internally stable. This is due hereto the pole/zero cancellation (forbidden!!).



Input-Output Stability

Definition (BIBO stability)

A system G (x = Ax+Bu; y = Cx) is BIBO stable if a bounded input u(.) (‖u‖∞ <∞) maps abounded output y(.) (‖y‖∞ <∞).

Now, the quantification (for BIBO stable systems) of the signal amplification (gain) is evaluated as:

γpeak = sup0<‖u‖∞<∞

‖y‖∞‖u‖∞

and is referred to as the PEAK TO PEAK Gain.

Definition (L2 stability)

A system G (x = Ax+Bu; y = Cx) is L2 stable if ‖u‖2 <∞ implies ‖y‖2 <∞.

Now, the quantification of the signal amplification (gain) is evaluated as:

γenergy = sup0<‖u‖2<∞

‖y‖2‖u‖2

and is referred to as the ENERGY Gain, and is such that:

γenergy = supω‖G(jω)‖ := ‖G‖∞

For a linear system, these stability definitions are equivalent (but not the quantification criteria).



Small Gain theorem

Consider the so called M −∆ loop.

Figure: M −∆ form

Theorem

Suppose M(s) in RH∞ and γ a positive scalar. Then the system is well-posed and internallystable for all ∆(s) in RH∞ such that ‖∆‖∞ ≤ 1/γ if and only if

‖M‖∞ < γ


2 Introduction to LMIs












2 Introduction to LMIs Background in Optimisation

Brief on optimisation

Definition (Convex function)

A function f : Rm → R is convex if and only if for all x, y ∈ Rm and λ ∈ [0 1],

f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y) (10)

Equivalently, f is convex if and only if its epigraph,

epi(f) = (x, λ)|f(x) ≤ λ (11)

is convex.

Definition ((Strict) LMI constraint)

A Linear Matrix Inequality constraint on a vector x ∈ Rm is defined as,

F (x) = F0 +m∑i=1

Fixi 0( 0) (12)

where F0 = FT0 and Fi = FTi ∈ Rn×n are given, and symbol F 0( 0) means that F issymmetric and positive semi-definite ( 0) or positive definite ( 0), i.e. ∀u|uTFu(>) ≥ 0.



Convex to LMIs

Example

Lyapunov equation. A very famous LMI constraint is the Lyapunov inequality of an autonomoussystem x = Ax. Then the stability LMI associated is given by,

xTPx > 0xT (ATP + PA)x < 0

(13)

which is equivalent to,

F (P ) =

[−P 00 ATP + PA

]≺ 0 (14)

where P = PT is the decision variable. Then, the inequality F (P ) ≺ 0 is linear in P .

LMI constraints F (x) 0 are convex in x, i.e. the set x|F (x) 0 is convex. Then LMI basedoptimization falls in the convex optimization. This property is fundamental because it guaranteesthat the global (or optimal) solution x∗ of the the minimization problem under LMI constraints canbe found efficiently, in a polynomial time (by optimization algorithms like e.g. Ellipsoid, InteriorPoint methods).



LMI problem

Two kind of problems can be handled

Feasibility: The question whether or not there exist elements x ∈ X such that F (x) < 0 iscalled a feasibility problem. The LMI F (x) < 0 is called feasible if such x exists,otherwise it is said to be infeasible.

Optimization: Let an objective function f : S → R where S = x|F (x) < 0. The problem todetermine

Vopt = infx∈Sf(x)

is called an optimization problem with an LMI constraint. This problem involvesthe determination of Vopt, the calculation of an almost optimal solution x (i.e., forarbitrary ε > 0 the calculation of an x ∈ S such that Vopt ≤ f(x) ≤ Vopt + ε, orthe calculation of a optimal solutions xopt (elements xopt ∈ S such thatVopt = f(xopt)).



Semi-Definite Programming (SDP) Problem

LMI programming is a generalization of the Linear Programming (LP) to cone positivesemi-definite matrices, which is defined as the set of all symmetric positive semi-definite matricesof particular dimension.

Definition (SDP problem)

A SDP problem is defined as,

min cT xunder constraint F (x) 0

(15)

where F (x) is an affine symmetric matrix function of x ∈ Rm (e.g. LMI) and c ∈ Rm is a givenreal vector, that defines the problem objective.

SDP problems are theoretically tractable and practically:• They have a polynomial complexity, i.e. there exists an algorithm able to find the global

minimum (for a given a priori fixed precision) in a time polynomial in the size of the problem(given by m, the number of variables and n, the size of the LMI).

• SDP can be practically and efficiently solved for LMIs of size up to 100× 100 and m ≤ 1000see ElGhaoui, 97. Note that today, due to extensive developments in this area, it may be evenlarger.


2 Introduction to LMIs LMI in control

The state feedback design problem

Stabilisation

Let us consider a controllable system x = Ax+Bu. The problem is to find a state feedbacku(t) = −Kx(t) s.t the closed-loop system is stable.

Using the Lyapunov theorem, this amounts at finding P = PT > 0 s.t:

(A−BK)TP + P(A−BK) < 0

⇔ ATP + PA−KTBTP − PBK < 0

which is obviously not linear...

Solution; use of change of variables

First, left and right multiplication by P−1 leads to

P−1AT +AP−1 − P−1KTBT −BKP−1 < 0

⇔ QAT +AQ + YTBT +BY < 0

with Q = P−1 and Y = −KP−1.

The problem to be solved is therefore formulated as an LMI ! and without any conservatism !



The Bounded Real Lemma

The L2-norm of the output z of a system ΣLTI is uniformly bounded by γ times the L2-norm ofthe input w (initial condition x(0) = 0).A dynamical system G = (A,B,C,D) is internally stable and with an ||G||∞ < γ if and only ifthere exists a positive definite symmetric matrix P (i.e P = PT > 0 s.t AT P + P A P B CT

BT P −γ I DT

C D −γ I

< 0, P > 0. (16)

The Bounded Real Lemma (BRL), can also be written as follows (see Scherer)I 0A B0 IC D

T

0 P 0 0P 0 0 00 0 −γ2I 00 0 0 I

I 0A B0 IC D

≺ 0 (17)

Note that the BRL is an LMI if the only unknown (decision variables) are P and γ (or γ2).



Quadratic stability

This concept is very useful for the stability analysis of uncertain systems.Let us consider an uncertain system

x = A(δ)x

where δ is an parameter vector that belongs to an uncertainty set ∆.

Theorem

The considered system is said to be quadratically stable for all uncertainties δ ∈ ∆ if there exists a(single) "Lyapunov function" P = PT > 0 s.t

A(δ)TP + PA(δ) < 0, for all δ ∈ ∆ (18)

This is a sufficient condition for ROBUST Stability which is obtained when A(δ) is stable for allδ ∈ ∆.


2 Introduction to LMIs Some useful lemmas

Some useful lemmas

• Schur lemma: allows to a convert a quadratic constraint (ellipsoidal constraint) into an LMIconstraint.

Lemma

Let Q = QT and R = RT be affine matrices of compatible size, then the condition[Q SST R

] 0

is equivalent toR 0

Q− SR−1ST 0

• Kalman-Yakubovich-Popov lemma: convert frequency inequalities into Linear MatrixInequalities (used in robust control for dissipative systems)

• Projection Lemma: allows to eliminate variable by a change of basis (projection in the kernelbasis). It is involved in one of the H∞ solutions, see (Doyle,Scherer).

• Completion Lemma: allows to simplify the number of variables when a matrix and its inverseenter in a LMI.

• Finsler’s lemma: This Lemma allows the elimination of matrix variables.


2 Introduction to LMIs Some useful lemmas

Interests of LMIs

LMIs allow to formulate complex optimization problems into "Linear" ones, allowing the use ofconvex optimization tools.Usually it requires the use of different transformations, changes of variables ... in order to linearizethe considered problmems: Congruence, Schur complement, projection lemma, Eliminationlemma, S-procedure, Finsler’s lemme ...

Examples of handled criteria

• stability• H∞, H2, H2/H∞ performances• robustness analysis: Small gain theorem, Polytopic uncertianties, LFT representations...• Robust control and/or observer design• pole placement• stability, stabilization with input constraints• Passivity constraints• Time-delay systems


3 What is theH∞ performance?












3 What is theH∞ performance? H∞ norm

How to define the system gain?

SISO systems

z = Gd, the gain at a given frequency is simply

|z(ω)||d(ω)|

=|G(jω)d(ω)||d(ω)|

= |G(jω)|

The gain depends on the frequency, but since the system is linear it is independent of the inputmagnitude

How to generalize to MIMO systems?

we may select:‖z(ω)‖2‖d(ω)‖2

=‖G(jω)d(ω)‖2‖d(ω)‖2

= ‖G(jω)‖2?

Which seems to be "independent" of the input magnitude. But this is not a correct definition.Indeed the input direction is of great importance



The gain of a MIMO system as induced L2 norm?

Let consider

G =

[5 43 2

]How to define and evaluate its gain ?Consider five different inputs:

d1 =

(10

)d2 =

(01

)d3 =

(0.7070.707

)d4 =

(0.707−0.707

)d5 =

(0.6−0.8

)The input magnitudes are:

‖d1‖2 = ‖d2‖2 = ‖d3‖2 = ‖d4‖2 = ‖d5‖2 = 1

But the corresponding outputs are

z1 =

(53

)z2 =

(42

)z3 =

(6.36303.5350

)z4 =

(0.70700.7070

)z5 =

(−0.20.2

)and the ratio are ‖z‖2/‖d‖2‖z1‖2‖d1‖2

= 5.83‖z2‖2‖d2‖2

= 4.47‖z3‖2‖d3‖2

= 7.27‖z4‖2‖d4‖2

= 0.99‖z5‖2‖d5‖2

= 0.28

So the gain value differs function of the input vector direction.



How the Singular Value Decomposition can provide such a MIMO gaindefinition?


Below is represented ‖z‖2/‖d‖2 as afunction of d20/d10 (where d = [d10, d20]T )

We can see that, depending on theratio d20/d10, the gain variesbetween 0.27 and 7.34 .,where σ(G) = 7.34 and σ(G) = 0.27.

We then have these mathematicaldefinitions:

MAXIMUM SINGULAR VALUE

maxd 6=0

‖Gd‖2‖d‖2

= σ(G)

MINIMUM SINGULAR VALUE

mind 6=0

‖Gd‖2‖d‖2

= σ(G)


Characterization of the H∞ norm as induced L2 norm

Finally, in the case of a transfer matrix G(s) : (m inputs, p outputs) u vector of inputs, y vector ofoutputs.

σ(G(jω)) ≤‖z(ω)‖2‖d(ω)‖2

≤ σ(G(jω))


Example of A two-mass/spring/damper system2 inputs: F1 and F2 2 outputs: x1 and x2

Singular Values

Frequency (rad/sec)S

ingu

lar V

alue

s (d

B)

10-1

100

101

102

-100

-80

-60

-40

-20

0

20

40

Hinf norm 11.4664 = 21.18 dB

smallest singular value

largest singular value

G=ss(A,B,C,D): LTI system[ninf,fpeak] = hinfnorm(G): Compute H∞ norm and freqnorm(G,inf): Compute H∞ normnormhinf(G): Compute H∞ normsigma(G): plot max and min SV

3 What is theH∞ performance? H∞ norm computation












3 What is theH∞ performance? H∞ norm computation

How to compute the H∞ norm?

As said before, H∞ norm cannot be computed analytically. Only numerical solutions can beobtained (e.g. Bisection algorithm, or LMI resolution).

Method 1: Since ‖G(jω)‖∞ = supω∈R σ (G(jω)), the intuitive computation is to get thepeak on the Bode magnitude plot, which can be estimated using a thin grid offrequency points, ω1, . . . , ωN, and then:

‖G(jω)‖∞ ≈ max1≤k≤N

σG(jωk)

Method 2: Let the dynamical system G = (A,B,C,D) ∈ RH∞ :||G||∞ < γ if and only if σ (D) < γ and the Hamiltonian H has no eigenvalueson the imaginary axis, where

H =

(A+BR−1DTC BR−1BT

−CT (In +DR−1DT )C −(A+BR−1DTC)

)and R = γ2 −DTD

Use norm(sys,inf)or hinfnorm(sys,tol)in Matlab.Method 3 (Bounded Real Lemma): A dynamical system G = (A,B,C,D) is internally stable and

with an ||G||∞ < γ if and only is there exists a positive definite symmetric matrixP (i.e P = PT > 0 s.t AT P + P A P B CT

BT P −γ I DT

C D −γ I

< 0, P > 0. (19)


3 What is theH∞ performance? What isH∞ control?













Towards H∞ control: the General Control Configuration

This approach has been introduced by Doyle (1983). The formulation makes use of the generalcontrol configuration.

P is the generalized plant (contains the plant, the weights, the uncertainties if any) ; K is thecontroller. The closed-loop transfer matrix from w to z is given by:

Tzw(s) = Fl(P,K) = P11 + P12K(I − P22K)−1P21

where Fl(P,K) is referred to as a lower Linear Fractional Transformation.



Problem definition

The overall control objective is to minimize some norm of the transfer function from w to z , forexample, the H∞ norm.

Definition (H∞ optimal control problem)

H∞ control problem: Find a controller K(s) which based on the information in y, generates acontrol signal u which counteracts the influence of w on z, thereby minimizing the closed-loopnorm from w to z.

Definition (H∞ suboptimal control problem)

Given γ a pre-specified attenuation level, a H∞ sub-optimal control problem is to design astabilizing controller that ensures :

‖Tzw(s)‖∞ = maxω

σ(Tzw(jω)) ≤ γ

The optimal problem aims at finding γmin (done using hinfsyn in MATLAB).

Remarks

• It is worth noting that the H∞ control problem is a disturbance attenuation, formulated in theworst-case performance analysis.

• z is then often defined as an "error signal" (to be minimized)


4 WhyH∞ control is adapted to control engineering?












4 WhyH∞ control is adapted to control engineering?

Introduction

Objectives of any control systemshape the response of the system to a given reference and get (or keep) a stable system inclosed-loop, with desired performances, while minimising the effects of disturbances andmeasurement noises, and avoiding actuators saturation, this despite of modelling uncertainties,parameter changes or change of operating point.This is formulated as:

Nominal stability (NS): The system is stable with the nominal model (no model uncertainty)

Nominal Performance (NP): The system satisfies the performance specifications with the nominalmodel (no model uncertainty)

Robust stability (RS): The system is stable for all perturbed plants about the nominal model, up tothe worst-case model uncertainty (including the real plant)

Robust performance (RP): The system satisfies the performance specifications for all perturbedplants about the nominal model, up to the worst-case model uncertainty(including the real plant).


4 WhyH∞ control is adapted to control engineering? Sensitivity functions

The control structure - SISO case

Figure: Complete control scheme

In the SISO case, it leads to:

y = 11+G(s)K(s)

(GKr + dy −GKn+Gdi)

u = 11+K(s)G(s)

(Kr −Kdy −Kn−KGdi)

Loop transfer function L = G(s)K(s)

Sensitivity function S(s) = 11+L(s)

Complementary Sensitivity function T (s) =L(s)

1+L(s)

N.B. S is often referred to as the ’Output Sensitivity’.



Stability and robustness margins ...


Classical definitions:• Stability⇔ | L(jωπ) |< 1, where ωπ

is the phase crossover frequencydefined by φ(L(jωπ)) = −π.

• Gain Margin: indicates the additionalgain that would take the closed loopto the critical stability condition(GM (dB) = −[|L(jωπ |]dB)

• Phase margin: quantifies the purephase delay that should be added toachieve the same critical stabilitycondition(ΦM = 180o + arg[L(jωc)], where|L(jωc|) = 1(0dB))

• Delay margin: quantifies themaximal delay that should be addedin the loop to achieve the samecritical stability condition, PM/ωc


Robustness margins ...


It is important to consider the modulemargin that quantifies the minimaldistance between the curve and thecritical point (-1,0j): this is a robustnessmargin.

∆M = 1/MSMS = max

ω|S(jω)| = ‖S‖∞

Good value MS < 2 (6dB)

• The MODULE MARGIN is a robustnessmargin. Indeed, the influence of plantmodelling errors on the CL transfer function:

T =K(s)G(s)

1 +K(s)G(s)

is given by:

∆T

T=

1

1 +K(s)G(s)

∆G

G= S.

∆G

G

• A good module margin implies good gain andphase margins:

GM ≥MS

MS − 1andPM ≥

1

MS

For MS = 2, then GM > 2 and PM > 30

• Last:MT = max

ω|T (jω)|

A good value : MT < 1.5(3.5dB)


Input/Output performances

Defining two new ’sensitivity functions’:Plant Sensitivity: SG = S(s).G(s) (often referred to as the ’Input Sensitivity’, e.g in Matlab)Controller Sensitivity: KS = K(s).S(s)



Using Matlab

% Determination of the sensitivity fucntionsL=series(G,Khinf) % Loop transfer function L=GKS=inv(1+L); % S= 1/(1+L)poleS=pole(S)T= feedback(L,1)poleT=pole(T)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SG=S*G;poleSG=pole(SG)KS=Khinf*S;poleKS=pole(KS)%%%%w=logspace(-3,3,500); %% to be adjustedsubplot(2,2,1), sigma(S,w), title('Sensitivity function')subplot(2,2,2), sigma(T,w), title('Complementary sensitivity function')subplot(2,2,3), sigma(SG,w), title('Sensitivity*Plant')subplot(2,2,4), sigma(KS,w), title('Controller*Sensitivity')


Remark: for SISO systems, use bodemag instead of sigma but not for MIMO ones !


Performance analysis and specification using the sensitivity functions:the SISO case- Dynamical behavior

As mentioned in Skogestad & Postlethwaite’s book:The concept of bandwidth is very important in understanding the benefits and trade-offs involvedwhen applying feedback control. Above we considered peaks of closed-loop transfer functions,which are related to the quality of the response. However, for performance we must also considerthe speed of the response, and this leads to considering the bandwidth frequency of the system.In general, a large bandwidth corresponds to a faster rise time, since high frequency signals aremore easily passed on to the outputs. A high bandwidth also indicates a system which is sensitiveto noise and to parameter variations. Conversely, if the bandwidth is small, the time response willgenerally be slow, and the system will usually be more robust.

Definition

Loosely speaking, bandwidth may be defined as the frequency range [ω1, ω2] over which control iseffective. In most cases we require tight control at steady-state so ω1 = 0, and we then simply callω2 the bandwidth. The word "effective" may be interpreted in different ways : globally it meansbenefit in terms of performance.



Performance analysis and specification using the sensitivity functions:the SISO case- Bandwidth definitions

Definition (ωS )

The (closed-loop) bandwidth, ωS , is the frequency where |S(jω)| crosses −3dB (1/sqrt2) frombelow.

Remark: |S| < 0.707, frequency zone, where e/r = −S is reasonably small

Definition (ωT )

The bandwidth (in term of T ), ωT , is the frequency where |T (jω)| crosses −3dB (1/sqrt2) fromabove.

Definition (ωc)

The bandwidth (crossover frequency), ωc, is the frequency where |L(jω)| crosses 1 (0dB), for thefirst time, from above.



Some remarks

Remark

Usually ωS < ωc < ωT

Remark

In most cases, the two definitions in terms of S and T yield similar values for the bandwidth. Inother cases, the situation is generally as follows. Up to the frequency ωS , |S| is less than 0.7, andcontrol is effective in terms of improving performance. In the frequency range [ωS , ωT ] control stillaffects the response, but does not improve performance. Finally, at frequencies higher than ωT ,we have S ' 1 and control has no significant effect on the response.

Remark

Usually ωS < ωc < ωT

Finally the following relation is very useful to evaluate the rise time:

tr ≈2.3

ωT



Performance analysis: answer to ....

Analysis of S:• Steady state error in tracking and output disturbance rejection : S(ω = 0) = 0 ?• Maximum peak criterion (Module Margin): ‖S‖∞ < 2 ?• bandwidth of S

Analysis of T :• Steady state error in tracking : T (ω = 0) = 1 ?.• Attenuation of measurement noise: |T (jω)| small when ω →∞ ?• Maximum of T , ‖T‖∞ < 1.5 ?• ωT bandwidth of T + rise time evaluation tr

Analysis of KS:• Input saturation: |u(t)| < |umax| ? (where |umax| < ‖KS‖∞ |rmax|).• Attenuation of measurement noise: |KS(jω)| small when ω →∞ ?

Analysis of SG:• Steady state error in input disturbance rejection : SG(ω = 0) = 0 ?• Attenuation of input disturbance effet: |SG(jω)| small in the frequency range of interest ? .



From analysis to specification ... templates

Objective : good performance specifications are important to ensure better control systemMean : give some templates on the sensitivity functionsFor simplicity, presentation for SISO systems first.Sketch of the method:

1 Robustness and performances in regulation can be specified by imposing frequentialtemplates on the sensitivity functions.

2 If the sensitivity functions stay within these templates, the control objectives are met.3 These templates can be used for analysis and/or design. In the latter they are considered as

weights on the sensitivity functions4 The shapes of typical templates on the sensitivity functions are given in the following slides

Mathematically, these specifications may be captured by an upper bound, on the magnitude of asensitivity function, given by another transfer function, as for S:

|S(jω)| ≤1

|We(jω)|, ∀ω ⇔ ‖WeS‖∞ ≤ 1

where We(s) is a WEIGHT selected by the designer.



Template on the sensitivity function S - Weighted sensitivity

Typical specifications in terms of S include:1 Minimum bandwidth frequency ωS2 Maximum tracking error at selected frequencies.3 System type, or the maximum steady-state tracking error ε04 Shape of S over selected frequency ranges.5 Maximum peak magnitude of S, ||S||∞ < MS .

The peak specification prevents amplification of noise at high frequencies, and alsointroduces a margin of robustness; typically we select MS = 2.

How to select the template function

It should:• be close to the control objectives• avoid too much under -or over- estimation• be simple enough to be used later in the control design step



Template on the sensitivity function S


S(s) =1

1 +K(s)G(s)

1

We(s)=

s+ ωbε

s/MS + ωb

Generally ε ' 0 is considered, MS < 2(6dB) or (3dB - cautious) to ensuresufficient module margin.ωb influences the CL bandwidth : ωb ↑• faster rejection of the disturbance• faster CL tracking response• better robustness w.r.t. parametric

uncertainties

Template on the Sensitivity function S

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

10−4

10−2

100

102

104

106−60

−50

−40

−30

−20

−10

0

10Template on the sensitivity function S

MS = 2 (6dB)

ε = 1e − 3

ωb s.t. |1/We| = 0dB


Template on the function KS


KS(s) =K(s)

1 +K(s)G(s)

1

Wu(s)=

ε1s+ ωbc

s+ ωbc/Mu

Mu chosen according to LF behavior ofthe process (actuator constraints:saturations)ωbc influences the CL bandwidth : ωbc ↓• better limitation of measurement

noises• roll-off starting from ωbc to reduce

modeling errors effects

Template on the Controller*Sensitivity KS

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

100

105

−60

−50

−40

−30

−20

−10

0

10

M u = 2

ω bc fo r |1 / W u| = 0 d B

ε c = 1 e -3


Template on the function T


T (s) =K(s)G(s)

1 +K(s)G(s)

1

WT (s)=

εT s+ ωbt

s+ ωbt/MT

Generally εT ' 0 is considered,MT < 1.5 (3dB) to limit the overshoot.ωbt influences the bandwidth hence thetransient behavior of the disturbancerejection properties: ωbt ↓• better noise effects rejection• better filtering of HF modelling errors

10−4

10−2

100

102

104

106−60

−50

−40

−30

−20

−10

0

10Template on the Complementary sensitivity function T

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B) MT =1.5

εT =1e-3

ωbT s.t. |1/WT | = 0 dB


Template on the function SG


SG(s) =G(s)

1 +K(s)G(s)

1

WSG(s)=

s+ ωSGεSG

s/MSG + ωSG

MSG allows to limit the overshoot in theresponse to input disturbances.Generally εSG ' 0 is considered,ωSG influences the CL bandwidth :ωSG ↑ =⇒ faster rejection of thedisturbance.

10−4

10−2

100

102

104

106−40

−30

−20

−10

0

10

20 Template on the Sensitivity*Plant SG

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B) MSG = 10

εSG=1e-2

ωsg s.t. |1/WSG| = 0 dB


Performance analysis and specification using the sensitivity functions:the MIMO case

Figure: Complete control scheme

The output & the control input satisfy the following equations :

(Ip +G(s)K(s))y(s) = (GKr + dy −GKn+Gdi)(Im +K(s)G(s))u(s) = (Kr −Kdy −Kn−KGdi)

BUT : K(s)G(s) 6= G(s)K(s) !!



Sensitivity functions- The MIMO case

Definitions

Output and Output complementary sensitivity functions:

Sy = (Ip +GK)−1, Ty = (Ip +GK)−1GK, Sy + Ty = Ip

Input and Input complementary sensitivity functions:

Su = (Im +KG)−1, Tu = KG(Im +KG)−1, Su + Tu = Im

Properties

Ty = GK(Ip +GK)−1

Tu = (Im +KG)−1KGSuK = KSy



MIMO Input/Output performances

Defining two new ’sensitivity functions’:Plant Sensitivity: SyG = Sy(s).G(s)Controller Sensitivity: KSy = K(s).Sy(s)



Performance analysis and specification using the sensitivity functions:the MIMO case- Some classical analysis criteria (1)


• The transfer function KSy(s) should beupper bounded so that u(t) does notreach the physical constraints, even fora large reference r(t)

• The effect of the measurement noisen(t) on the plant input u(t) can bemade « small » by making thesensitivity function KSu(s) small (inHigh Frequencies)

• The effect of the input disturbance di(t)on the plant input u(t) + di(t) (actuator)can be made « small » by making thesensitivity function Su(s) small (takecare to not trying to minimize Tu whichis not possible)


Performance analysis and specification using the sensitivity functions:the MIMO case- Some classical analysis criteria (2)


• The plant output y(t) can track thereference r(t) by making thecomplementary sensitivity functionTy(s) equal to 1. (servo pb)

• The effect of the output disturbancedy(t) (resp. input disturbance di(t) ) onthe plant output y(t) can be made «small » by making the sensitivityfunction Sy(s) (resp. SyG(s) ) « small »

• The effect of the measurement noisen(t) on the plant output y(t) can bemade « small » by making thecomplementary sensitivity functionTy(s) « small »


Trade-offs

ButS? + T? = I∗

Some trade-offs are to be looked for...These trade-offs can be reached if one aims :• to reject the disturbance effects in low frequencies• to minimize the noise effects in high frequencies

It remains to require:• Sy and SyG to be small in low frequencies to reduce the load (output and input) disturbance

effects on the controlled output• Ty and KSy to be small in high frequencies to reduce the effects of measurement noises on

the controlled output and on the control input (actuator efforts)



Synthesis

Provide a clear and detailed frequency-domain performance analysis using the sensitivityfunctions in order to explain the trade-off performance/robustness/actuator constraints.

Qualitative analysis

Use of Sy , Ty , KSy , SyG to:• predict the behavior of the output w.r.t different external inputs (reference, disturbance, noise)• predict the behavior of the control input w.r.t different external inputs (reference, disturbance,

noise)

Quantitative analysis

Use of Sy , Ty , KSy , SyG to:• Stability analysis and margins.• Compute the steady-state errors in tracking, output and input disturbance attenuations.• Give the maximum of the input/output gains to analyze the transient behaviors of the output

and control input (incl. saturation).• Give all the bandwidths of the sensitivity functions• Evaluate the rise time in tracking• Evaluate the robustness margins



A first insight into the performance specifications for the MIMO case

The direct extension of the performances objectives to MIMO systems could be formulated asfollows:

1 Disturbance attenuation/closed-loop performances:

σ(Sy(jω)) <1

|W1(jω)|

with |W1(jω)| > 1 for ω < ωb2 Actuator constraints:

σ(KSy(jω)) <1

|W2(jω)|

with |W2(jω)| > 1 for ω > ωh3 Robustness to multiplicative uncertainties:

σ(Ty(jω)) <1

|W3(jω)|

with |W3(jω)| > 1 for ω > ωt

However these objectives do not consider the specific MIMO structure of the system, i.e. theinput-output relationship between actuators and sensors.It is then better to define the objectives accordingly with the system inputs and outputs.



Towards MIMO systems

Let us consider a system with 2 inputs and 1 output and define:

G =(G1 G2

), K =

(K1

K2

)Therefore

GK = G1K1 +G2K2, KG =

(K1G1 K1G2

K2G1 K2G2

)and the sensitivity functions are:

Sy =1

1 +G1K1 +G2K2, KSy =

(K1

1+G1K1+G2K2K2

1+G1K1+G2K2

)

While a single template We is convenient for Sy it is straightforward that the following diagonaltemplate should be used for KSy :

Wu(s) =

(W 1u(s) 00 W 2

u(s)

)where W 1

u and W 2u are chosen in order to account for each actuator specificity (constraint).



The MIMO general case

Let us consider G with m inputs and p outputs.• In the MIMO case the simplest way is to defined the templates as diagonal transfer matrices,

i.e. using (MSi, ωbi , εi)

• In that case, a weighting function should be dedicated for each input, and for each output.• These weighting functions may of course be different if the specifications on each actuator

(e.g. saturation), and on each sensor (e.g. noise), are different.

In addition, during the performance analysis step, take care to plot, in addition to the MIMOsensitivity functions, the individual ones related to each input/output to check if the individualspecification is met. Hence, in the simplest case:

1 If the specifications are identical then it is sufficient to plot:• σ(Sy(jω)) and 1

|We(jω)| , for all ω• σ(KSy(jω)) and 1

|Wu(jω)| , for all ω

2 If the specifications are different, one should plot• σ(Sy(i, :)) and 1

|Wie|

, for all ω, i = 1, . . . , p

• σ(KSy(k, :)) and 1

|Wku |

, for all ω, k = 1, . . . ,m.

i.e. p plots for all output behaviors and m plots for the input ones.3 In a very general case, plot σ(Sy) with σ(1/We)



More on weighting functions

When tighter (harder) objectives are to be met .....the templates can be defined more accurately by transfer functions of order greater than 1, as

We(s) =

(s/MS + ωb

s+ ωbε

)k,

if a roll-off of −20× k dB per decade is required.Take care to the choice of the parameters (MS , ωb, ε) to avoid incoherent objectives!

100

101

102

103

104

105

106

−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

Mu=2, wbc=100, epsi1=0.01

Wu and Wu square without parameter modification

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

100

101

102

103

104

105

106

−50

−40

−30

−20

−10

0

10

20

Original weight (specifications) Mu=2, ε1 = 0.01, ωb c=100 for Wu

Mu =√

2, ε1 =√

0.01,ωb c=100 for 1/W 2

u case 1

Mu =√

2, ε1 =√

0.01,ωb c=200 for 1/W 2

u case 2

Mu =√

2, ε1 =√

0.01,ωb c=400 for 1/W 2

u case 3

$1/W_u$ and $1/Wu_^2$ with parameter modification

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)



Final objectives

In terms of control synthesis, all these specifications can be tackled in the following problem: findK(s) s.t. ∥∥∥∥∥∥∥∥

WeSyWuKSyWTTy

WSGSyG

∥∥∥∥∥∥∥∥∞

≤ 1 ⇒ ‖WeSy‖∞ ≤ 1 ‖WSGSyG‖∞ ≤ 1‖WuKSy‖∞ ≤ 1 ‖WTTy‖∞ ≤ 1

Often, the simpler following one (referred to as the mixed sensitivity problem) is studied:

Find K s.t.

∥∥∥∥ WeSyWuKSy

∥∥∥∥∞≤ 1

since the latter allows to consider the closed-loop output performance as well as the actuatorconstraints.


4 WhyH∞ control is adapted to control engineering? The mixed sensitivityH∞ control design













How to consider performance specification in H∞ control ?

Illustration on the H∞ SISO problem: ‖Tew(s)‖∞ =

∥∥∥∥ WeSWuKS

∥∥∥∥∞≤ γ

In that case the closed-loop system Tew(s) must have 1 input and 2 outputs. Since S = r−yr

andKS = u

r, the control scheme needs only one external input r.

Control objectives:

y = Gu = GK(r − y)⇒ tracking error : ε = Sru = K(r − y) = K(r −Gu)⇒ actuator force : u = KSr

To cope with that control objectives the following control scheme is considered:

Objective w.r.t sensitivity functions: ‖WeS‖∞ ≤ 1, ‖WuKS‖∞ ≤ 1.Idea: define 2 new virtual controlled outputs:

e1 = WeSre2 = WuKSr



The mixed sensitivity H∞ control design - Problem definition

The performance specifications on the tracking error & on the actuator, given as some weights onthe controlled output, then leads to the new control scheme:

The associated general control configuration is :



The mixed sensitivity H∞ control design - Problem definition

The corresponding H∞ suboptimal control problem is therefore to find a controller K(s) such that

‖Tew(s)‖∞ =

∥∥∥∥ WeSWuKS

∥∥∥∥∞≤ γ with

Tew(s) = Fl(P,K) = P11 + P12K(I − P22K)−1P21

=

[We

0

]+

[−WeGWu

]K(I +GK)−1I

=

[WeSWuKS

]in Matlab



What about disturbance attenuation ?

To account for input disturbance rejection, the control scheme must include di:

When Wd = 1 this corresponds to the closed-loop system:

Tew =

[WeSy WeSyGWuKSy WuTu

]The new H∞ control problem therefore includes the input disturbance rejection objective, thanksto SyG that should satisfy the same template as S, i.e an high-pass filter!Remarks: Note that WuTu is an additional constraint that may lead to an increase of theattenuation level γ since it is not part of the objectives. Hopefully Tu is low pass, and Wu as well.The input weight has to be on u not u+ di which would lead to an unsolvable problem.



Improve the disturbance attenuation

The previous problem, allows to ensure the input disturbance rejection, but does not provide anyadditional d-o-f to improve it (without impacting the tracking performance). In order to ’decouple’both performance objectives, the idea is to add a disturbance model that indeed changes thedisturbance rejection properties.Let then consider : di(t) = Wd.d. In that case the closed-loop system is This corresponds to theclosed-loop system.

Tew =

[WeSy WeSyGWd

WuKSy WuTuWd

]and the template expected for SyG is now 1

Wd.We.

First interest: improve the disturbance weight as Wd = 100... but this has a price (see Fig. belowfor an example)

10-2 10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

Sensitivity*Plant SG

Frequency (rad/sec)

with Wd=1

with Wd=100

10-2 10-1 100 101 102 103-50

-40

-30

-20

-10

0

10

20

30

40

Controller*Sensitivity KS

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

with Wd=100

with Wd=1



More generally...

To include multiple objectives in a SINGLE H∞ control problem, there are 2 ways:1 add some external inputs (reference, noise, disturbance, uncertainties ...)2 add new controlled outputs

Of course both ways increase the dimension of the problem to be solved....thus the complexity aswell. Moreover additional constraints appear that are not part of the objectives ....General rule: first think simple !!



Some extensions: the 2-DOF case

In some cases it is intresting to decouple the transient response in tracking fropm the stabilizationloop (as in RST controllers). This is the case of 2 dof control structure.


Pay attention when building P since:• External inputs: r, di, dy and n• Control Input: u• Controlled ouputs z1 and z2• Measurements: r and y + n

The controller solution will the be such as

u =[Kr Ky

] [ r−y − n

]

4 WhyH∞ control is adapted to control engineering? Some performance limitations













Introduction

Framework

Main extracts of this part: see Goodwin et al 2001. "Performance limitations in control are not onlyinherently interesting, but also have a major impact on real world problems."Objective : take into account the limitations inherent to the system or due to actuators constraints,before designing the controller..... Understanding what is not possible is as important asunderstanding what is possible!

Example of structural constraints

S + T = 1, ∀ω

We then cannot have, for any frequency ω0, |S(jω0)| < 1 and |T (jω0)| < 1. This implies that,disturbance and noise rejection cannot be achieved in the same frequency range.

Bode’s Sensitivity Integral for open-loop stable systems

It is known that, for an open loop stable plant:∫ ∞0

log|S(jω)|dω = 0

Then the frequency range where |S(jω)| < 1 is balanced by the frenquencies where |S(jω)| > 1



Bode sensitivity

Nice interpretation of the balance between reduction and magnification of the sensitivity.

For open-loop unstable systems we have a stronger constraint:∫ ∞0

log|det(S(jω))|dω = π

Np∑i=1

Re(pi),

where pi design the Np RHP poles. Therefore, in the presence of RHP poles, the control effortnecessary to stabilize the system is paid in terms of amplification of the sensitivity magnitude.



The interesting case of systems with RHP zeros

Theorem

Let G(s) a MIMO plant with one RHP zero at s = z, and Wp(s) be a scalar weight. Then,closed-loop stability is ensured only if:

‖Wp(s)S(s) ‖≥ |Wp(s = z)|

To illustrate the use of that theorem, if Wp is chosen as:

Wp(s) =

(s/MS + ωb

s+ ωbε

),

and , if the controller meets the requirements, then

‖Wp(s)S(s) ‖∞≤ 1

Therefore a necessary condition is:

|z/MS + ωb

z + ωbε|≤ 1

To conclude, if z is real, and if the performance specifications are such that : MS = 2 and ε = 0,then a necessary condition to meet the performance requirements is :

ωb ≤z

2


5 How to solve an H∞ control problem?












5 How to solve an H∞ control problem? The Static State feedback case

A first case: the state feedback control problem

Let consider the system:

x(t) = Ax(t) +B1w(t) +B2u(t) (20)

z(t) = Cx(t) +D11w(t) +D12u(t)

The objective is to find a state feedback control law u = −Kx s.t:

‖Tzw(s)‖∞ ≤ γ

The method consists in applying the Bounded Real Lemma to the closed-loop system, and thentry to obtain some convex solutions (LMI formulation).This is achieved if and only is there exists a positive definite symmetric matrix P (i.e P = PT > 0 s.t (A−B2K)T P + P (A−B2K) P B1 (C −D12K)T

? −γ I DT11? ? −γ I

< 0, P > 0. (21)


5 How to solve an H∞ control problem? The Static State feedback case

Solution of the state feedback control problem

Use of change of variables

First, left and right multiplication by diag(P−1, In, In), and use Q = P−1 and Y = −KP−1. Itleads to AQ +B2Y + QAT + YTBT2 B1 QCT + YTDT12

? −γ I DT11? ? −γ I

< 0, Q > 0. (22)

The state feedback controller is then:K = −YQ−1


5 How to solve an H∞ control problem? The Dynamic Output feedback case













The Dynamic Output feedback case

It will be shown how to formulate such a control problem using "classical" control tools. Theprocedure will be 2-steps:

Get P : Build the General Control Configuration scheme s.t. the closed-loop systemmatrix does correspond to the tackled H∞ problem (for instance the mixedsensitivity problem). Use of Matlab, sysic tool.A state space representation of P , the generalized plant, is needed.

Compute K: Use an optimisation algorithm that finds the controller K solution of theconsidered problem.The calculation of the controller, solution of theH∞ control problem , can then bedone using the Riccati approach or the LMI approach of the H∞ control problem[Zhou et al.(1996)Zhou, Doyle, and Glover] [Skogestad and Postlethwaite(1996)].

Notations:

P

x = Ax+B1w +B2uz = C1x+D11w +D12uy = C2x+D21w +D22u

⇒ P =

A B1 B2

C1 D11 D12

C2 D21 D22

withx ∈ Rn: plant state variables ∪ state variables of weightsw ∈ Rnw : external inputs u ∈ Rnu control inputsz ∈ Rnz : controlled outputs y ∈ Rny measured outputs (inputs of the controller)



Problem formulation

Let K(s) be a dynamic output feedback LTI controller defined as

K(s) :

xK(t) = AK xK(t) +BK y(t),u(t) = CK xK(t) +DK y(t).

where xK ∈ Rn, and AK , BK , CK and DK are matrices of appropriate dimensions.Remark. The controller will be considered here of the same order (same number of state variables)n than the generalized plant, which here, in the H∞ framework, the order of the optimal controller.With P (s) and K(s), the closed-loop system N(s) is:

N(s) :

xcl(t) = ACL xcl(t) + BCL w(t),z(t) = CCL xcl(t) +DCL w(t),

(23)

where xTcl(t) =[xT (t) xTK(t)

]and

ACL =

(A+B2 DK C2 B2 CK

BK C2 AK

),

BCL =

(B1 +B2 DK D21

BK D21

),

CCL =(C1 +D12 DK C2, D12 CK

),

DCL = B1 +B2 DK D21.

The aim is of course to find matrices AK , BK , CK and DK s.t. the H∞ norm of the closed-loopsystem (23) is as small as possible, i.e. γopt = min γ s.t. ||N(s)||∞ < γ.


5 How to solve an H∞ control problem? The Riccati approach













Asuumptions for the Riccati method

A1: (A,B2) stabilizable and (C2, A) detectable: necessary for the existence of stabilizingcontrollers

A2: rank(D12) = nu and rank(D21) = ny : Sufficient to ensure the controllers are proper, hencerealizable

A3: ∀ω ∈ R, rank(

A− jωIn B2

C1 D12

)= n+ nu

A4: ∀ω ∈ R, rank(

A− jωIn B1

C2 D21

)= n+ ny Both ensure that the optimal controller does

not try to cancel poles or zeros on the imaginary axis which would result in CL instability

A5:D11 = 0, D22 = 0, D12

T [ C1 D12 ] =[

0 Inu

],

[B1

D21

]D21

T =

[0Iny

]not necessary but simplify the solution (does correspond to the given theorem next but can beeasily relaxed)



The problem solvability

The first step is to check whether a solution does exist of not, to the optimal control problem.

Theorem (1)

Under the assumptions A1 to A5, there exists a dynamic output feedback controlleru(t) = K(.) y(t) such that the closed-loop system is internally stable and the H∞ norm of theclosed-loop system from the exogenous inputs w(t) to the controlled outputs z(t) is less than γ, ifand only if

i the Hamiltonian H =

(A γ−2 B1 BT1 −B2 BT2

−CT1 C1 −AT)

has no eigenvalues on the

imaginary axis.

ii there exists X∞ ≥ 0 t.q. AT X∞+X∞ A+X∞ (γ−2 B1 BT1 −B2 BT2 ) X∞+CT1 C1 = 0,

iii the Hamiltonian J =

(AT γ−2 CT1 C1 − CT2 C2

−B1 BT1 −A

)has no eigenvalues on the

imaginary axis.

iv there exists Y∞ ≥ 0 t.q. A Y∞ + Y∞ AT + Y∞ (γ−2 CT1 C1 − CT2 C2) Y∞ +B1 BT1 = 0,

v the spectral radius ρ(X∞ Y∞) ≤ γ2.



Controller reconstruction

Theorem (2)

If the necessary and sufficient conditions of the Theorem 1 are satisfied, then the so-called centralcontroller is given by the state space representation

Ksub(s) =

[A∞ −Z∞L∞F∞ 0

]with

A∞ = A+ γ−2B1BT1 X∞ +B2F∞ + Z∞L∞C2

F∞ = −BT2 X∞, L∞ = −Y∞CT2Z∞ =

(I − γ−2Y∞X∞

)−1

The Controller structure is indeed an observer-based state feedback control law, with

u2(t) = −BT2 X∞ x(t),

where x(t) is the observer state vector

˙x(t) = A x(t) +B1 w(t) +B2 u(t) + Z∞ L∞(C2 x(t)− y(t)

). (24)

and w(t) is defined as

w(t) = γ−2 BT1 X∞ x(t).

Remark. w(t) is an estimation of the worst case disturbance. Z∞ L∞ is the filter gain for the OEproblem of estimating x(t) in the presence of the worst case disturbanceO. Sename [GIPSA-lab] 87/162

5 How to solve an H∞ control problem? The LMI approach forH∞ control design













The LMI approach for H∞ control design- Solvability

In this case only A1 is necessary. The solution is base on the use of the Bounded Real Lemma,and some relaxations that leads to an LMI problem to be solved [Scherer(1990)].when we refer to the H∞ control problem, we mean: Find a controller K for system P such that,given γ∞,

||Fl(P,K)||∞ < γ∞ (25)

The minimum of this norm is denoted as γ∗∞ and is called the optimal H∞ gain. Hence, it comes,

γ∗∞ = min(AK ,BK ,CK ,DK)s.t.σACL⊂C−

‖Tzw(s)‖∞ (26)

As presented in the previous sections, this condition is fulfilled thanks to the BRL. As a matter offact, the system is internally stable and meets the quadratic H∞ performances iff. ∃ P = PT 0such that, ATCLP + PACL PBCL CTCL

BTCLP −γinfty2I DTCLCCL DCL −I

< 0 (27)

where ACL, BCL, CCL, DCL are given in (23). Since this inequality is not an LMI and nottractable for SDP solver, relaxations have to be performed (indeed it is a BMI), as proposed in[Scherer et al.(1997)Scherer, Gahinet, and Chilali].



The LMI approach for H∞ control design- Problem solution

Theorem (LTI/H∞ solution [Scherer et al.(1997)Scherer, Gahinet, and Chilali])

A dynamical output feedback controller of the form K(s) =

[AK BKCK DK

]that solves the H∞

control problem, is obtained by solving the following LMIs in (X, Y, A, B, C and D), whileminimizing γ∞,

M11 (∗)T (∗)T (∗)TM21 M22 (∗)T (∗)TM31 M32 M33 (∗)TM41 M42 M43 M44

≺ 0

[X InIn Y

] 0

(28)

where,M11 = AX + XAT +B2C + C

TBT2 M21 = A +AT + CT2 D

TBT2

M22 = YA+ATY + BC2 + CT2 BT

M31 = BT1 +DT21DTBT2

M32 = BT1 Y +DT21BT

M33 = −γ∞Inu

M41 = C1X +D12C M42 = C1 +D12DC2

M43 = D11 +D12DD21 M44 = −γ∞Iny

(29)



Controller reconstruction

Once A, B, C, D, X and Y have been obtained, the reconstruction procedure consists in findingnon singular matrices M and N s.t. M NT = I −X Y and the controller K is obtained as follows

DK = DCK = (C−DcC2X)M−T

BK = N−1(B− YB2Dc)

AK = N−1(A− YAX− YB2DcC2X−NBcC2X− YB2CcMT )M−T

(30)

where M and N are defined such that MNT = In −XY (that can be solved through a singularvalue decomposition plus a Cholesky factorization).Remark. Note that other relaxation methods can be used to solve this problem, as suggested by[Gahinet(1994)].


6 Robust analysis












6 Robust analysis Introduction

Introduction

• A control system is robust if it is insensitive to differences between the actual system and themodel of the system which was used to design the controller

• How to take into account the difference between the actual system and the model ?• A solution: using a model set BUT : very large problem and not exact yet

A method: these differences are referred as model uncertainty.The approach

1 determine the uncertainty set: mathematical representation2 check Robust Stability3 check Robust Performance

Lots of forms can be derived according to both our knowledge of the physical mechanism thatcause the uncertainties and our ability to represent these mechanisms in a way that facilitatesconvenient manipulation.Several origins :• Approximate knowledge and variations of some parameters• Measurement imperfections (due to sensor)• At high frequencies, even the structure and the model order is unknown (100• Choice of simpler models for control synthesis• Controller implementation

Two classes: parametric uncertainties / neglected or unmodelled dynamics


6 Robust analysis Representation of uncertainties













Example 1: uncertainties

Let consider the example from (Sokestag & Postlewaite, 1996).

G(s) =k

1 + τse−sh, 2 ≤ k, h, τ ≤ 3

Let us choose the nominal parameters as, k = h = τ = 2.5 and G the according nominal model.We can define the ’relative’ uncertainty, which is actually referred as a MULTIPLICATIVEUNCERTAINTY, as


G(s) = G(s)(I +Wm(s)∆(s))

with Wm(s) = 3.5s+0.25s+1

and ‖∆‖∞ ≤ 1


Example 2: unmodelled dynamcis


10-8 10-6 10-4 10-2 100 102 104-250

-200

-150

-100

-50

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/s)

(Greal-Gnom)/Gnom

Wm

Let us consider the system:

G(s) = G0(s)1

1 + τs, τ ≤ τmax

This can be modelled as:

G(s) = G0(s)(I +Wm(s)∆(s))

with Wm(s) = τmaxjω1+τmaxjω

and ‖∆‖∞ ≤ 1

This can be represented as

with

N(s) =

[N11(s) N12(s)N21(s) N22(s)

]=

(0 I

G0Wm(s) G0(s)

)


Example 3: parametric uncertainties

Consider the first order system:

G(s) =1

s+ a, a0 − b < a < a0 + b

Define now:a = a0 + δ.b with |δ| < 1

Then it leads:1

s+ a=

1

s+ a0 + δ.b=

1

s+ a0(1 +

δ.b

s+ a0)−1

This can then be represented as a Multiplicative Inverse Uncertainty:


withz = y∆ = 1

s+a0(w − bu∆)


Example 3 (cont.) same example with state space formulation

Let us first the transfer function G(s) = 1s+a

as

G :

x = (−a0 − δ.b)x+ wz = x

(31)

In order to use an LFT, let us define the uncertain input:

u∆ = δx,

Then the previous system can be rewritten in the following LFR:


where ∆ and y∆ are given as:

∆ =[δ], y∆ = (x)

and N given by the state space representation:

N :

x = −a0x− bu∆ + wy∆ = xz = x

(32)


Example 4: parametric uncertainties in state space equations

Let us consider the following uncertain system:

G :

x1 = (−2 + δ1)x1 + (−3 + δ2)x2

x2 = (−1 + δ3)x2 + uy = x1

(33)

In order to use an LFT, let us define the uncertain inputs:

u∆1= δ1x1, u∆2

= δ2x2, u∆3= δ3x2

Then the previous system can be rewritten in the following LFR:


where ∆ and y∆ are given as:

∆ =

δ1 0 00 δ2 00 0 δ3

, y∆ =

x1

x2

x2

and N given by the state space representation:

N :

x1

x2

==

−2x1 − 3x2 + u∆1+ u∆2

−x2 + u+ u∆3

y = x1

(34)


Towards LFR (LFT)

The previous computations are in fact the first step towards an unified representation of theuncertainties: the Linear Fractional Representation (LFR).Indeed the previous schemes can be rewritten in the following general representation as:

Figure: N∆ structure

This LFR gives then the transfer matrix from w to z, and is referred to as the upper LinearFractional Transformation (LFT) :

Fu(N,∆) = N22 +N21∆(I −N11∆)−1N12

This LFT exists and is well-posed if (I −N11∆)−1 is invertible



LFT definition

In this representation N is known and ∆(s) collects all the uncertainties taken into account for thestability analysis of the uncertain closed-loop system.∆(s) shall have the following structure:

∆(s) = diag ∆1(s), · · · ,∆q(s), δ1Ir1 , · · · , δrIrr , ε1Ic1 , · · · , εcIcc

with ∆i(s) ∈ RHki×ki∞ , δi ∈ R and εi ∈ C.Remark: ∆(s) includes• q full block transfer matrices,• r real diagonal blocks referred to as ’repeated scalars’ (indeed each block includes a real

parameter δi repeated ri times),• c complex scalars εi repeated ci times.

Constraints: The uncertainties must be normalized, i.e such that:

‖∆‖∞ ≤ 1, |δi| ≤ 1, |εi| ≤ 1



Uncertainty types

We have seen in the previous examples the two important classes of uncertainties, namely:• UNSTRUCTURED UNCERTAINTIES: we ignore the structure of ∆, considered as a full

complex perturbation matrix, such that ‖∆‖∞ ≤ 1.We then look at the maximal admissible norm for ∆, to get Robust Stability and Performance.This will give a global sufficient condition on the robustness of the control scheme.This may lead to conservative results since all uncertainties are collected into a single matrixignoring the specific role of each uncertain parameter/block.

• STRUCTURED UNCERTAINTIES: we take into account the structure of ∆, (always such that‖∆‖∞ ≤ 1).The robust analysis will then be carried out for each uncertain parameter/block.This needs to introduce a new tool: the Structured Singular Value. We then can obtain morefine results but using more complex tools.

The analysis is provided in what follows for both cases.In Matlabthis analysis is provided in the tools robuststab and robustperf.


6 Robust analysis Definition of Robustness analysis













Robustness analysis: problem formulation

Since the analysis will be carried you for a closed-loop system, N should be defined as theconnection of the plant and the controller. Therefore, in the framework of the H∞ control, thefollowing extended General Control Configuration is considered:

Figure: P −K −∆ structure

and N is such thatN = Fl(P,K)



Robust analysis: problem definition

In the global P −K −∆ General Control Configuration, the transfer matrix from w to z (i.e theclosed-loop uncertain system) is given by:

z = Fu(N,∆)w,

with Fu(N,∆) = N22 +N21∆(I −N11∆)−1N12.and the objectives are then formulated as follows:

Nominal stability (NS): N is internally stable

Nominal Performance (NP): ‖N22‖∞ < 1 and NS

Robust stability (RS): Fu(N,∆) is stable ∀∆, ‖∆‖∞ < 1 and NS

Robust performance (RP): ‖Fu(N,∆)‖∞ < 1 ∀∆, ‖∆‖∞ < 1 and NS



Summary of the methodology

Uncertainty definition

Parameters, neglected dynamics ...

Unstructured case

Model type (additive, multiplicative ...)

Small Gain theorem

Sensitivity function template

RS (|S?| < 1/|W?|?) RP (∼ NP +RS < 1?)

Structured case

Structured Small Gain theorem

RS (µ∆r (N11) < 1?) RP (µ∆(N) < 1?)


6 Robust analysis Robustness analysis: the unstructured case













Towards Robust stability analysis

Robust Stability= with a given controller K, we determine wether the system remains stable for allplants in the uncertainty set.According to the definition of the previous upper LFT, when N is stable, the instability may onlycome from (I −N11∆). Then it is equivalent to study the M −∆ structure, given as:

Figure: M −∆ structure

This leads to the definition of the Small Gain Theorem

Theorem (Small Gain Theorem)

Suppose M ∈ RH∞. Then the closed-loop system in Fig. 7 is well-posed and internally stablefor all ∆ ∈ RH∞ such that :

‖∆‖∞ ≤ δ(resp. < δ) if and only if ‖M(s)‖∞ < 1/δ(resp. ‖M(s)‖∞ ≤ 1)



Definition of the uncertainty types

Figure: 6 uncertainty representations



Robust stability analysis: additive case

Objective: applying the Small Gain Theorem to these unstructured uncertainty representations.


Let us consider the following simple controlscheme as:

Figure: Control scheme with the uncertain system

The objective is to obtain:

Additive case:G(s) = G(s) +WA(s)∆A(s).Computing the N −∆ form gives

N(s) =

(−WAKSy WAKSy

Sy Ty

)Output Multiplicative uncertainties:G(s) = (I +WO(s)∆O(s))G(s).Then it leads

N(s) =

(−WOTy WOTySy Ty

)


General results

Theorem (Small Gain Theorem)

Consider the different uncertainty types, and assume that NS is achieved, i.e M ∈ RH∞ for eachtype. Then the closed-loop system is robustly stable, i.e. internally stable for all ∆k ∈ RH∞ (fork =A, 0, I, iO, iI) such that :

Additive : ‖WAKSy‖∞ ≤ 1Additive Inverse: ‖WiASy‖∞ ≤ 1Output Multiplicative: ‖WOTy‖∞ ≤ 1Input Multiplicative: ‖WITu‖∞ ≤ 1Output Inverse Multiplicative: ‖WiOSy‖∞ ≤ 1Input Inverse Multiplicative: ‖WiISu‖∞ ≤ 1

This gives some robustness templates for the sensitivity functions. However this may beconservative.



Illustration on the SISO case

Here Robust Stability is analyzed through the Nyquist plot. For illustration, let us consider the caseof Multiplicative uncertainties (Input and Output case are identical for SISO systems), i.e

G = G(I +Wm∆m)

Then the loop transfer function is given as:

L = GK = GK(I +Wm∆m) = L+WmL∆m;


According to the Nyquist theorem, RS isachieved the the closed-loop system isstable for any L should not encircle, i.e Lshould not encircle -1 for all uncertainties.According to the figure, a sufficient conditionis then:

|WmL| < |1 + L| , ∀ω⇔∣∣∣WmL

1+L

∣∣∣ < 1, ∀ω⇔ |WmT | < 1 ∀ω


A first insight in Robust Performance

Objective: applying the Small Gain Theorem to these unstructured uncertainty representations.


Let us consider the following simple controlscheme as:

Figure: Control scheme

Case of Output Multiplicative uncertainties:G(s) = (I +WO(s)∆O(s))G(s).Computing the N −∆ form gives

N(s) =

[N11(s) N12(s)N21(s) N22(s)

]=

(−WOTy WOTy−WeSy WeSy

)

We wish to get:The objectives are then formulated asfollows:

NS: N is internally stable

NP: ‖WeSy‖∞ < 1 and NS

RS: ‖WOTy‖∞ < 1 and NS

RP: ‖Fu(N,∆)‖∞ < 1 ∀∆, ‖∆‖∞ < 1,Sufficient condition: NS andσ(WOTy) + σ(WeSy) < 1, ∀ω


Illustration on the SISO case

Here Robust Performance is analyzed through the Nyquist plot. For illustration, let us consider thecase of Multiplicative uncertainties (Input and Output case are identical for SISO systems), i.e

G = G(I +Wm∆m)

Then the loop transfer function is given as:

L = GK = GK(I +Wm∆m) = L+WmL∆m;


First NP is achieved when:|WeS| < 1 ∀ω, ⇔ |We| <|1 + L| , ∀ω.Therefore RP is achieved if∣∣∣WeS

∣∣∣ < 1, ∀S,∀ω

⇔ |We| <∣∣∣1 + L

∣∣∣ , ∀L, ∀ωSince

∣∣∣1 + L∣∣∣ ≥ |1 + L| − |WmL∆m|, a

sufficient condition is actually:

|We|+ |WmL| < |1 + L| , ∀ω⇔ |WeS|+ |WmT | < 1, ∀ω

6 Robust analysis analysis: the structured case













The structured case

∆ = diag∆1, · · · ,∆q , δ1Ir1 , · · · , δrIrr , ε1Ic1 , · · · , εcIcc ∈ Ck×k (35)

with ∆i ∈ Cki×ki , δi ∈ R, εi ∈ C,where ∆i(s), i = 1, . . . , q, represent full block complex uncertainties, δi(s), i = 1, . . . , r, realparametric uncertainties, and εi(s), i = 1, . . . , c, complex parametric uncertainties.Taking into account the uncertainties leads to the following General Control Configuration,

∆(s)

K(s)

y

e

z∆r

ω

v

P (s)

v∆r

Figure: General control configuration with uncertainties

where ∆ ∈ ∆.



The structured singular value

Let consider the M −∆ structure with structured uncertainties. We look for the smalleststructured ∆ which makes (I −M∆) singular.We need to introduce µ, the structured singular value, defined as:

Definition (µ)

µ∆(M) :=1

minσ(∆) : ∆ ∈ ∆, det(I −∆M) = 0, for M ∈ Cn×n

Theorem (The structured Small Gain Theorem)

Let M(s) be a MIMO LTI stable system and ∆(s) a LTI uncertain stable matrix, (i.e. ∈ RH∞).The M −∆ structure is stable for all ∆(s)∆ ∈ ∆ with σ(∆) < 1 if and only if:

∀ω ∈ R µ∆ (M(jω)) ≤ 1, with M(s) := N11(s)

More generally both following statements are equivalent• For µ ∈ R, M(s) and ∆(s) belong to RH∞, and

∀ω ∈ R, µ∆ (M(jω)) ≤ µ

• the M −∆ structure is stable for any uncertainty ∆(s) of the form (35) such that :

||∆(s)| |∞ < 1/µ



Build the whole control scheme



Introduction of a fictive block

Usually only real parametric uncertainties (given in ∆r) are considered for RS analysis. RPanalysis also needs a fictive full block complex uncertainty, as below,

∆f

∆(s)

∆r

N(s)ω e

v∆rz∆r

N(s)

Figure: N∆

where N(s) =

[N11(s) N12(s)N21(s) N22(s)

], and the closed-loop transfer matrix is:

Tew(s) = N22(s) +N21(s)∆(s)(I −N11(s))−1N12(s) (36)



Robust analysis theorem

For RS, we shall determine how large ∆ (in the sense of H∞) can be without destabilizing thefeedback system. From (36), the feedback system becomes unstable if det(I −N11(s) = 0 forsome s ∈ C,<(s) ≥ 0. The result is then the following.

Theorem ([Skogestad and Postlethwaite(1996)])

Assume that the nominal system N22 and the perturbations ∆ are stable. Then the feedbacksystem is stable for all allowed perturbations ∆ such that ||∆(s)| |∞ < 1/β if and only if∀ω ∈ R, µ∆ (N11(jω)) ≤ β.

Assuming nominal stability, RS and RP analysis for structured uncertainties are therefore suchthat:

NP ⇔ σ(N22) = µ∆f(N22) ≤ 1, ∀ω

RS ⇔ µ∆r (N11) < 1, ∀ω

RP ⇔ µ∆(N) < 1, ∀ω, ∆ =

[∆f 00 ∆r

]Finally, let us remark that the structured singular value cannot be explicitly determined, so that themethod consists in calculating an upper bound and a lower bound, as closed as possible to µ.



Summary

The steps to be followed in the RS/RP analysis for structured uncertainties are then:• Definition of the real uncertainties ∆r and of the weighting functions• Evaluation of µ(N22)∆f

, µ(N11)∆rand µ(N)∆

• Computation of the admissible intervals for each parameter

Remark: The Robust Performance analysis is quite conservative and requires a tight definition ofthe weighting functions that do represent the performance objectives to be satisfied by theuncertain closed-loop system. Therefore it is necessary to distinguish the weighting functionsused for the nominal design from the ones used for RP analysis.


6 Robust analysis Robust design












6 Robust analysis Robust design

Brief overview

In order to design a robust control, i.e. a controller for which the synthesis actually accounts foruncertainties, some of the methods are:• Unstructured uncertainties: Consider an uncertainty weight (unstructured form), and

include the Small Gain Condition through a new controlled output. For example, robustnessface to Ouptut Multiplicative Uncertainties can be considered into the design procedureadding the controlled output ey = WOy, which, when tracking performance is expected, leadsto the condition ‖WOTy ‖∞≤ 1.

• Structured uncertainties: the design of a robust controller in the presence of suchuncertainties is the µ− synthesis. It is handled through an interactive procedure, referred toas the DK iteration. This procedure is much more involved than a "simple" H∞ controldesign and often leads to an increase of the order of the controller (which is already the sumof the order of the plant and of the weighting functions).

• Use other mathematical representation of parametric uncertainties,[Scherer and Wieland(2004)], as for instance the polytopic model. In that case the set ofuncertain parameters is assumed to be a polytope (i.e. a convex) set. The stability issue inthat framework is referred to as the ’Quadratic stability’ i.e find a single Lyapunov function forthe uncertainty set. While in the general case this is an unbounded problem, in the polytopiccase (or in the affine case), the stability is to be analyzed only at the vertices of the polytope,which is a finite dimensional problem.This approach can then be applied to find a single controller, valid over the potyopic set. Notethat this approach gives rise to the LPV design for polytopic systems, as described next.


7 Further studies












7 Further studies H2 and multi-objective problems

H2 design

The H∞ norm considered above gives the system gain when input and output are measuredusing the L2 norm. Rather than bounding the output energy, it may be desirable to keep the peakamplitude of the controlled output below a certain level, e.g. to avoid actuator saturations.Now, when we refer to the H2 control problem, we mean: Find a controller C for system M suchthat, given γ∞,

||Fl(M,C)||2 < γ2 (37)

The H2 problem can be expressed as follow[ATK +KA KBBTK −I

]< 0

[K CTC Z

]> 0 , Trace(Z) < γ2

where ACL, BCL, CCL, DCL are given in (23).


7 Further studies H2 and multi-objective problems

H∞/H2 problem formulation

Useful to deal with different objectives functions of the external signal types (noise, disturbance..).


z∞w∞

u y

P (s)

K(s)

z2w2

Objectives:T∞ =

∥∥∥ z∞w∞ ∥∥∥∞ < γ∞

T2 =∥∥∥ z2w2

∥∥∥2< γ2

The resulting LMI based problem formulation consistsin solving the following problem subject toK = KT 0 (note that to obtain LMIs, the samechange of variable as introduced in the H∞ andH2 problems can be applied).

ATCLK +KACL KB∞ CT∞BT∞K −γ2

∞I DT∞1

C∞ D∞1 −I

< 0[AT

CLK +KACL KB2

BT2 K −I

]< 0

[K CT2C2 Z

]> 0 , Trace(Z) < γ2

Even after the change of basis, it is impossible (non convex problem) to minimize simultaneouslythe H∞ and H2 criteria. As a consequence, the problem is usually reformulated as one of theproblems below:• A linear combination of γ∞ and γ2, e.g.:

γmix = αγ∞ + (1− α)γ2 , where α ∈ [0 1] (38)

• Minimize γ∞ (resp. γ2) while fixing γ2 (resp. γ∞).

7 Further studies H∞ observer design

H∞ observer definition


x(t) = Ax(t) + Ew(t) +Bu(t) (39)

y(t) = Cx(t) + Fw(t)

The objective is

˙x(t) = Ax(t) +Bu(t) + L(y(t)− Cx(t))x0to be defined

(40)

where x(t) ∈ Rn is the estimated state of x(t) and L is the n× p constant observer gain matrix tobe designed.The estimated error, e(t) := x(t)− x(t), satisfies:

e(t) = (A− LC)e(t)+(E − LF )w(t) (41)

Problem definition

System (75) is said to be a H∞ observer for the above system if:

limt→∞

e(t)→ 0 for w(t) ≡ 0 (42)

‖Tew(s)‖∞ ≤ γ under e(t = 0) = 0 (43)



H∞ observer design

The method consists (as for state feedback design) to apply the BRL to the error equation and usesome chage of variables to get some LMIs.This is achieved if and only is there exists a positive definite symmetric matrix P (i.e P = PT > 0 s.t (A− LC)T P + P (A− LC) P (E − LF ) In

? −γ I 0? ? −γ I

< 0, P > 0. (44)


Let define Y = −KPL. It leads to AP + YC + PAT + CTYT PE + Y F In? −γ I 0? ? −γ I

< 0, Q > 0. (45)

The observer gain is then:L = −P−1Y



Other interests of the H∞ approach

Control

Using LMis the previous methods can be designed to take into account• Pole placement constraints: useful to avoid fast dynamics and high frequencies in the

controller (to facilitate digital implementation).• Input and state constraints: some results allow to included together with H∞ performance,

the saturation constraints on the input (to provide an anti-windup scheme) (and stateconstraints)

• Passivity performance: used to enforce dissipative properties of the closed loop (thisproperty is widely used in e.g. electrical systems, robotic applications). This property ensuresthat the introduced energy is dissipated into the system. This approach is linked with thepassivity theory.

Observer design, Fault Diagnosis and Fault Tolerant Control

• Design of H∞ observer and robust observers.• Design H∞ observesr for Fault Detection and Isolation (FDI) (sometimes using a bank of

observers) and for Fault Estimation as well.• Reconfiguration of (state or dynamic output) feedback control: The controller changes

according to detected faults

Last be not least: all what has been seen in the course does exist for discrete-time systems.O. Sename [GIPSA-lab] 129/162

8 LPV systems












8 LPV systems What is a Linear Parameter Varying systems?

LPV systems

Definition of an Linear Parameter Varying system

Σ(ρ) :

xzy

=

A(ρ) B1(ρ) B2(ρ)C1(ρ) D11(ρ) D12(ρ)C2(ρ) D21(ρ) D22(ρ)

xwu

x(t) ∈ Rn, ...., ρ = (ρ1(t), ρ2(t), . . . , ρN (t)) ∈ Ω, is a vector of time-varying parameters (Ωconvex set), assumed to be known ∀t



LPV systems


Σ(ρ) :

xzy

=


xwu


System (ρ)

ExogeneousInputs

Controlledoutputs

Measuredoutputs

z

ControlInputs

w

u y

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



LPV systems


Σ(ρ) :

xzy

=


xwu


The frozen Bode plots forc = 1 and k ∈ [1, 3]

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



LPV systems (2)

Let the LPV system be:

Σ(ρ) :

xzy

=


xwu

x(t) ∈ Rn, ...., ρ = (ρ1(t), ρ2(t), . . . , ρN (t)) ∈ Uρ, is a vector of time-varying parameters (Uρconvex set)• ρ(.) varies in the set of continously differentiable parameter curves ρ : [0,∞)→ RN .

It is assumed to be known or measurable.• The parameters ρ are always assumed to be bounded:

ρ ∈ Uρ ⊂ RN and Uρ compact (46)

defined by the minimal ρi, and maximal ρi values of ρi(t)

ρi(t) ∈ [ρi, ρi], ∀i

• The system matrices A(.) .... are continuous on Uρ



LPV systems (3): about the parameters

• Parameters are exogenous if they are external variables. The system is in that case nonstationary.See the previous damped mass-sping system.

• Parameters are endogenous if they are function of the state variables, ρ = ρ(x(t), t), and, inthat case, the LPV system is referred to as a quasi-LPV system.This case is encountered when approximating Nonlinear systems.For instance:

x(t) = x2(t) = ρ(t)x(t)

with ρ(t) = x(t).• It is sometimes required that the derivative of the parameters are bounded, i.e:

ρ ∈ Uρ ⊂ RN and Uρ compact (47)

defined by the minimal νi, and maximal νi values of ρi(t)

ρi(t) ∈ [νi, νi], ∀i

This corresponds to the case of slow varying parameters• Other representations can be considered if ρ is piecewise-constant, or varies in a finite set of

elements (ρ(t) ∈ 0, 1 for switching systems)

Next, several classes of LPV models are presented, and some ways to go from one class toanother are given.



Some comments

• LPV systems can model uncertain systems (ρ fixed but unknown) or parameter-varyingmodels (ρ(t))

LPV=linear or nonlinear?

• What is often referred to as gain-scheduling control, corresponds to Jacobian linearization ofthe nonlinear plant about a family of equilibrium points Shamma (90), Rugh & Shamma (2000)In terms of control design this means, lineraization around operating conditiosn, design (ateach operating points) of a LTI controller, and interpolation of the LTI controllers in betweenoperating conditions (often used in Aerospace and Automotive industries).Pros: Simplicity of design for a non linear systemCons: No a priori guarantee of stability nor robustness

• But: this differs from quasi-LPV representations where nonlinearities are hiddden in someparameter descriptions (as seen later in the course)

LPV=LTV

• Theoretical analysis of LPV system properties (stability, controllability, observability), oftenfalls into the framework of LTV systems or of nonlinear ones (for quasi-LPV representations),see (Blanchini).



Some references

Those not to be ignored

• Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth) + 2011 TCST Special Issue on"Applied LPV modelling and identification"

• Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker, Seiler, Grigoriadis ...)• Stability, stabilization (Scherer, Wu, Blanchini ...)• Geometric analysis (Bokor & Balas)• Survey paper: Hoffmann & Werner, 2015• Fault tolerant control: special issues by

• Balas, 2012: in International Journal of Adaptive Control and Signal Processing• Casavola, Rodrigues & Theilliol, 2015: in International Journal of Robust and Nonlinear Control

Some recent books

• R. Toth, Modeling and identification of linear parameter-varying systems, Springer 2010• J. Mohammadpour, C. Scherer, (Eds), Control of Linear Parameter Varying Systems with

Applications, Springer-Verlag New York, 2012• O. Sename, P. Gaspar, J. Bokor (Eds), Robust Control and Linear Parameter Varying

Approaches: Application to Vehicle Dynamics, Springer, 2013


8 LPV systems Classes (models) of LPV systems

Class 1: Affine parameter dependence

In this case, the system matrices are such that:

A(ρ) = A0 +A1ρ1 + ...+ANρN

This is the case of the damped mass-sping system:

p+ cp+ k(t)p = u

considering the state space representationx(t) = A(k(t))x(t) +Bu(t),

y(t) = Cx(t) +Du(t)(48)

with x(t) ∈ R2 and

A(k(t)) =

(0 1−k(t) −c

), B =

(01

), C =

(1 0

), D = 0

Denoting the varying parameter ρ(t) = k(t), we get:

A0 =

(0 10 −c

), A1 =

(0 0−1 0

)



Class 2: Polynomial parameter dependence

In this case, the system matrices are such that:

A(ρ) = A0 +A1ρ+A2ρ2 + ...+ASρ

S

Example 1

Let consider for instance the following example in (Briat, 2015):

x(t) = x3(t)

which can be written asx(t) = −ρ(t)2x(t),with ρ(t) = x(t)

Example 2

Another example is the sampling-dependent discrete-time system representation (Robert et al,2010):

Gd :

xk+1 = Ad(h) xk +Bd(h) ukyk = C(h) xk +D uk

(49)

where Ad(h) = eAh is approximated using Taylor expansions, such as:

Ad(h) ≈ I +N∑i=1

Ai

i!hi



Class 3: Polytopic models

A polytopic system is represented as

Σ(ρ) =Z∑k=1

αk(ρ)

[Ak BkCk Dk

], with

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

where[Ak BkCk Dk

]are LTI systems.

This representation is often used to rewrite an affine LPV system. Indeed, assuming that theparameters are bounded (ρi ∈

[ρi

ρi]), the vector of parameters evolves inside a polytope

represented by Z = 2N vertices ωi, as

ρ ∈ Coω1, . . . , ωZ (50)

It is then written as the convex combination:

ρ =Z∑i=1

αiωi, αi ≥ 0,Z∑i=1

αi = 1 (51)

where the vertices are defined by a vector ωi = [νi1, . . . , νiN ] where νij equals ρj or ρj .

The LTI system[Ak BkCk Dk

]here corresponds to the LPV system frozen at the vertex k.


8 LPV systems How to approximate a nonlinear system by an LPV one ?

Nonlinear vs Linear Differential Inclusion

See (Boyd et al, 1994).Let consider the nonlinear system

ΣNL :

x = f(x(t), w(t))z = g(x(t), w(t))

(52)

Suppose that, for each x, w and t, there is a matrix G(x,w, t) ∈ Ω s.t.:[f(x,w)g(x,w)

]= G(x,w, t)

[xw

](53)

where Ω ∈ R(nx+nz)×(nx+nu).As said in (Boyd et al, 1994):"Then of course every trajectory of the nonlinear system (52) is also a trajectory of the LDI definedby (53). If we can prove that every trajectory of the LDI defined by (53) has some property (e.g.,converges to zero), then a fortiori we have proved that every trajectory of the nonlinear system(52) has this property."



Example of a one-tank model

Usually the hydraulic equation is non linear and of the form

SdH

dt= Qe −Qs

where H is the tank height, S the tank surface, Qe the input flow, and Qs the output flow definedby Qs = kt

√H.

Definition the state space model

The system is represented by an Ordinary Differential Equation whose solution depends on H(t0)and Qe. Clearly H is the system state, Qe is the input, and the system can be represented as:

x(t) = f(x(t), u(t)), x(0) = x0 (54)

with x = H, f = − ktS

√x+ 1

Su

A LPV model

Following the LDI method, we define:

ρ(x) =

√x

x

which leads to the LPV model

x(t) = − ktS.ρ(x).x+ 1

Su, x(0) = x0 (55)



Example of a one-tank model (cont.)

Consider the variations qe, qs, h around the steady state as:Qe = Q0 + qe; Qs = Q0 + qs; H = H0 + h.This leads to the equation :

Sdh

dt= qe − kt(

√H0 + h−

√H0)

Denoting the state variable x = h, the control input u = qe, the output y = h, the tangentlinearization gives

x = Ax+Bu (56)

y = Cx (57)

with A = − kt2S√H0

, B = 1S

and C = 1.Now implement 3 types of models and compare with the nonlinear one:• The linear model around a fixed operating point• A LPV model obtained defining an adequate LDI• A LPV model defined in the LFR framework



LPV systems properties

Let consider the LPV syetsm

Σρ

x(t) = A(ρ(t))x(t) +B(ρ(t))u(t), x(0) = x0

y(t) = C(ρ(t))x(t) +D(ρ(t))u(t)(58)

What kind of properties we should pay attention to?

When ρ is fixed (constant) the previous system is LTI and• controllability, observability, stability, are uniquely defined• controllability⇔ reachabillity, observability⇔ reconstructibility• these properties are equivalent by a state change of basis.

But when ρ(t) is time varying .....

• these facts may not be true (asymptotic and exponential stabilty may differ)• need to study properies of Linear Time-Varying systems.• A generalization of the exp(At) is needed, defining the state transition matrix Φ(t, t0, ρ(t))

• For a change of basis T (t) with x(t) = T (t)xnew(t) then, x(t) = T (t)xnew(t) + T (t)xnew(t)


8 LPV systems Stability of LPV systems

Problem statement and facts

Recall

For LTI systems all notions of stability are equivalent: global/local, asymptotic/exponential,time-domain (Lyapunov)/frequency-domain (Bode, poles...).

Why stability analysis fo LPV systems is not an easy task?

Let consider x = A(ρ(t))x. Stability analysis is more involved (as for LTV systems) since:• there is a set of solutions for a given x0 (family of systems from ρ variations)• the system may be stable for frozen parameter values and unstable for varying parameters

(as for switching systems)• asymptotic and exponential stability are no more equivalent and cannot be characterized by

the eigenvalues of A(ρ(t)).• In term of design, we will often rely on the notion of quadratic stability (using quadratic

Lyapunov function V (x) = xTPx) which is stronger but easier to check for stability andsimpler to use for control and observer design, see (Wu, PhD 95)

Robust or LPV? (Blanchini,00 & 07)

• Robust analysis and control: dedicated to LTI systems subject to time-varying uncertainties• LPV (or gain-scheduling) analysis and control: dedicated to LTV systems or to linearizations

of non linear systems along the trajectory of ρO. Sename [GIPSA-lab] 143/162


Quadratic stability for time-varying parameters

Let us consider the LPV systemx = A(ρ(t))x

where ρ(t) is an time-varying parameter vector that belongs to an uncertainty set Ω.

Use of a single Lyapunov function

If there exists P = PT > 0 such that:

A(ρ(t))TP + PA(ρ(t)) < 0, ∀ρ(t) ∈ Ω

then the system is stable for arbitrarily fast time-varying uncertainties

Remarks

• Quadractic stability imples exponential stability (Wu, 95)• It is an infinite dimension problem (can be relaxed for polytopic uncertainties)• It could be conservative since stability is checked for any variation of the parameters !

Pay attentation in what follows: LPV system means TIME-VARYING parameters so a polytopicLPV system is not an uncertain polytopic system (in the latter case the coefficient αi of thepolytopic description are constant even if unknown)



L2 stability of LPV systems (Wu, 95)

Definition

Given a parametrically dependent stable LPV system Σρ = (A(ρ), B(ρ), C(ρ), D(ρ)) for zeroinitial conditions x0. The induced L2 norm is defined as:

||Σρ||i,2 = supρ(t)∈Ω

supw(t)6=0∈L2

‖y‖2‖u‖2

which is often referred to as (by abuse of langage) the H∞ gain ||Σρ||∞ of the LPV system.

Theorem

A sufficient condition for the L2 stability of system Σρ is the generalized BRL, using parameterdependent Lyapunov functions, i.e assuming |ρi| < νi, ∀i, if there exists P (ρ) > 0, ∀ρ s.t A(ρ)TP (ρ) + P (ρ)A(ρ) +

∑Ni=1 νi

∂P (ρ)

∂ρiP (ρ) B(ρ) C(ρ)T

B(ρ)T P (ρ) −γ I D(ρ)T

C(ρ) D(ρ) −γ I

< 0, ∀i. (59)

then ||Σρ||i,2 ≤ γ


8 LPV systems Control & Observation

Towards LPV control

The "gain scheduling" approach

System (ρ)Controller (ρ)

Adaptationstrategy

Measured or estimatedParameters

References ControlInputs

Outputs

Externalparameters

Some references

• Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth)• Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker ...)• Stability, stabilization (Scherer, Wu, Blanchini ...)• Geometric analysis (Bokor & Balas)


8 LPV systems The Static State feedback case

State feedback control design: pole placement (1)

Let consider the system Σ(ρ):

x(t) = A(ρ)x(t) +B(ρ)u(t) (60)

y(t) = C(ρ)x(t) +D(ρ)u(t)

The objective is to find a state feedback control law u = −F (ρ)x+G(ρ)r, where r is thereference signal s.t:• the closed-loop system is stable• the output y tracks the reference r (unit closed-loop gain y/r)

The closed-loop system is

x(t) = (A(ρ)−B(ρ)F (ρ))x(t) +B(ρ)G(ρ)r (61)

y(t) = (C(ρ)x(t)−D(ρ)F (ρ))x(t) +D(ρ)G(ρ)r

Then we must consider the following issues:• What controllability property shall we consider?• What parameter dependency should we define for (F (ρ), G(ρ))?• How to choose the poles (dynamics) of the closed-loop system?



State feedback control design: pole placement (2)

Here we can tackle several problems:

Robust SF control:• Design a nominal state feedback control (F,G) (for Σ(ρnominal))• Check quadractic stability and performances of the closed-loop system, with

(A(ρ)−B(ρ)F ) ...

LPV SF control with fixed performances:• Choose some desired poles (p1, p2, . . . , pn) for the CL system• Design F (ρ) such that eig(A(ρ)−B(ρ)F (ρ)) = (p1, p2, . . . , pn). Could be

analytic design or ’frozen-type’.• Check stability (quadractic?) of the CL system when ρ is time-varying

LPV SF control with varying (adaptive) performances:• Choose some desired poles (p1(ρ), p2(ρ), . . . , pn(ρ)) for the CL system• Design F (ρ) such that eig(A(ρ)−B(ρ)F (ρ)) = (p1(ρ), p2(ρ), . . . , pn(ρ)).

Could be analytic design or ’frozen-type’.• Check stability (quadractic?) of the CL system when ρ is time-varying• This latter case allows to schedule the performances according to the

parameter changes, so to handle the trade-off, function of the parametervariations, between closed-loop system transient dynamics and control costlimitations



The H∞ state feedback control problem


x(t) = A(ρ)x(t) +B1(ρ)w(t) +B2(ρ)u(t) (62)

y(t) = C(ρ)x(t) +D11(ρ)w(t) +D12(ρ)u(t)

The objective is to find a state feedback control law u = −K(ρ)x s.t:

‖Tyw(s)‖∞ ≤ γ

The method consists in applying the Bounded Real Lemma to the closed-loop system, and thentry to obtain some convex solutions (LMI formulation).Following the framework of quadratic stabilty, this is achieved if and only is there exists a positivedefinite symmetric matrix P (i.e P = PT > 0 s.t (A(ρ)− B2(ρ)K(ρ))T P + P (A(ρ)− B2(ρ)K(ρ)) P B1(ρ) (C(ρ)−D12(ρ)K(ρ))T

? −γ I DT11(ρ)

? ? −γ I

< 0.

(63)



Solution of the state feedback control problem


First, left and right multiplication by diag(P−1, In, In), and use Q = P−1 and Y(ρ) = −K(ρ)P−1.It leads to Q > 0 and A(ρ)Q + B2(ρ)Y(ρ) + QAT (ρ) + Y(ρ)TBT

2 (ρ) B1(ρ) QCT (ρ)− YT (ρ)DT12(ρ)

? −γ I DT11(ρ)

? ? −γ I

< 0. (64)

The state feedback controller is then:

K(ρ) = −Y(ρ)Q−1

How to solve (64)?

• It is indeed not an LMI in ρ due to B2(ρ)Y(ρ) and to D12(ρ)Y(ρ), and still infinite dimensional• For polytopic systems, a solution exists if either we choose Y time-invariant or if B2 and D12

are parameter independent• In the general case this requires to impose some parameter dependency on Y(ρ) (affine,

polynomial ...) and to solve the problems trying to linearize it (or using gridding techniques(Balasz, Packard, Seiler)).


8 LPV systems The Dynamic Output feedback case

The H∞/LPV control problem

Definition

Find a LPV controller C(ρ) s.t the closed-loop system is stable and for γ∞ > 0, sup ‖z‖2‖w‖2< γ∞,

• Unbounded set of LMIs (Linear Matrix Inequalities) to be solved (ρ ∈ Ω)• Some approaches: polytopic, LFT, gridding. See Arzelier [HDR, 2005], Bruzelius [Thesis,

2004], Apkarian et al. [TAC, 1995]...

A solution: The "polytopic" approach [C. Scherer et al. 1997]

• Problem solved off line for each vertex of a polytope (convex optimisation) (using here asingle Lyapunov function i.e. quadratic stabilization).

• On-line the controller is computed as the convex combination of local linear controllers

C(ρ) =

2N∑k=1

αk(ρ)

[Ac(ωk) Bc(ωk)Cc(ωk) Dc(ωk)

],

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)

• Easy implementation !!



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)

Problem on standard form

CL (ρ)

w: exagenous input u: control input

z: output to minimize y: measurement

Dynamical LPV generalized plant:

Σ(ρ) :

xzy

=


xwu

(65)

LPV controller structure:

S(ρ) :

[xc

u

]=

[Ac(ρ) Bc(ρ)Cc(ρ) Dc(ρ)

] [xc

y

](66)

LPV closed-loop system:

CL(ρ) :

[ξz

]=

[A(ρ) B(ρ)C(ρ) D(ρ)

] [ξw

](67)



LPV control design

H∞ criteria Apkarian et al. [TAC, 1995]

Stabilize system CL(ρ) (find K > 0) while minimizing γ∞. A(ρ)TK +KA(ρ) KB∞(ρ) C∞(ρ)T

B∞(ρ)TK −γ2∞I D∞(ρ)T

C∞(ρ) D∞(ρ) −I

< 0

Infinite set of LMIs to solve (ρ ∈ Ω) (Ω is convex)

LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004]

LFT, Gridding, Polytopic



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N∑k=1

αk(ρ)


]where,

αk(ρ) =

∏Nj=1 |ρj − Cc(ωk)j |∏N

j=1(ρj − ρj),

where Cc(ωk)j = ρj if (ωk)j = ρj

or ρj otherwise.

2N∑k=1

αk(ρ) = 1 , αk(ρ) > 0



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N∑k=1

αk(ρ)


]

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)



LPV/H∞ control synthesis

Proposition - feasibility (brief) Scherer et al. (1997)

Solve the following problem at each vertices of the parametrized points (illustration with 2parameters):

γ∗ = min γs.t. (69) |ρ1,ρ2s.t. (69) |ρ1,ρ2s.t. (69) |ρ1,ρ2s.t. (69) |ρ1,ρ2

(68)

AX +B2C(ρ1, ρ2) + (?)T (?)T (?)T (?)T

A(ρ1, ρ2) +AT YA+ B(ρ1, ρ2)C2 + (?)T (?)T (?)T

BT1 BT1 Y +DT21B(ρ1, ρ2)T −γI (?)T

C1X +D12C(ρ1, ρ2) C1 D11 −γI

≺ 0

[X II Y

] 0

(69)



LPV/H∞ control synthesis

Proposition - reconstruction (brief) Scherer et al. (1997)

Reconstruct the controllers as,solve (71) |ρ1,ρ2

(71) |ρ1,ρ2(71) |ρ1,ρ2(71) |ρ1,ρ2

(70)

Cc(ρ1, ρ2) = C(ρ1, ρ2)M−T

Bc(ρ1, ρ2) = N−1B(ρ1, ρ2)

Ac(ρ1, ρ2) = N−1(A(ρ1, ρ2)− Y AX −NBc(ρ1, ρ2)C2X

− Y B2Cc(ρ1, ρ2)MT)M−T

(71)

where M and N are defined such that MNT = I −XY which may be chosen by applying asingular value decomposition and a Cholesky factorization.


8 LPV systems LPV observer design

Definition LPV observers

Definition

Let consider the LPV system:

x(t) = A(ρ)x(t) +B(ρ)u(t)y(t) = C(ρ)x(t)

(72)

The following LPV state space representation

˙x(t) = A(ρ)x(t) +B(ρ)u(t) + L(ρ)(y(t)− y(t))y(t) = C(ρ)x(t)

x0to be defined(73)

is said to be an observer for (72) if

limt→∞

(x(t)− x(t))→ 0 ∀ρ(t) ∈ Ω

where x(t) ∈ Rn is the estimated state of x(t) and L(ρ) is the n× p observer gain matrix to bedesigned.



Some issues for LPV observer design

The estimated error, e(t) := x(t)− x(t), satisfies:

e(t) = (A− LC)(ρ)e(t) (74)

The two main problems to be handle are then• What observability property shall we consider?• What parameter dependency should we define for L(ρ)?

Quadractic detectability (Wu, 95)

A simple solution is to consider a single Lyapunov function in order to guarantte the quadraticdetectability, i.e:

(A(ρ)− L(ρ)C(ρ))T P + P (A(ρ)− L(ρ)C(ρ)) < 0

Some remarks:• The previous problem can be solved using a polytopic approach only if C(ρ) = C, a constant

matrix• If this is not solvable, one can try using Parameter dependent Lyapunov functions, but the

coupling between L(ρ) and P(ρ) will lead to soved non affine LMIs (a polynomial or a griddingapproach is then needed).



Some issues for LPV observer design (2)

On key issue in observer implementation concerns the knownledge of ρ(t). While previously theresult is valid if ρ(t) is perfectly known, such a following observer description must be used if ρ(t)is estimated:

˙x(t) = A(ρ)x(t) +B(ρ)u(t) + L(ρ)(y(t)− C(ρ)x(t)) (75)

Denoting ∆A = A(ρ)− = A(ρ), ∆B = B(ρ)−B(ρ), ∆C = C(ρ)− C(ρ), and∆L = L(ρ)− L(ρ), this leads for the estimation error equation:

e(t) = (A− LC)(ρ)e(t) + (∆A+ L(ρ).∆C)x+ ∆Bu(t) (76)

If C(ρ) = C and B(ρ) are constant matrices, then we get the uncertain estimated error system

e(t) = (A(ρ)− L(ρ)C)e(t) + ∆Ax(t) (77)

The stability analysis is indeed more involved due to the state vector x (see (Daafouz et al, 2010)for the discrete-time case). Either ∆Ax(t) should be considered as a disturbance, or a stateaugmentation approach is to be used (which has to be done in closed-loop control).

Observer-based control

For control design in the latter case, the following state feedback should be used:

u(t) = −F (ρ)x(t)


8 LPV systems Summary of LPV approach interests

Interest of the LPV approach

LPV is a key tool to the control of complex systems.

Some examples :

Modelling of complex systems (non linear)

• Use of a quasi-LPV representation to include non linearities in a linear state space model(even delays)

• Transformation of constraints (e.g. saturation) into an ’external’ parameter• Modelling of LTV, hybrid (e.g. switching control)

BUT :

A q-LPV system is not equivalent to the non linear one:• stability: ρ = ρ(x(t), t) is assumed to be bounded... so are the state trajectories• controllability: some non controllable modes of a non linear system may vanish according to

the LPV representation• observability: unobservability may occur for some specific parameter variations



Interest of the LPV approach

Some of works using LPV approaches - former PhD students

Gain-scheduled control

• Account for various operating conditions using a variable "equilibrium point": (Gauthier 2007)• Control with real-time performance adaptation using parameter dependent weighting

functions from endogenous or exogenous parameters (Poussot 2008, Do 2011)• Analysis and control of LPV Time-Delay Systems: delay-scheduled control Briat 2008• Control under computation constraints: H∞ variable sampling rate controller with sampling

dependent performances (Robert 2007, Roche 2011, Robert et al., IEEE TCST 2010))

Coordination of several actuators for MIMO systems

• An LPV structure for control allocation Poussot et al. (CEP 2011)• Selection of a specific parameter for the control activation (of each actuator) Poussot et al.

(VSD 2011), Doumiati et al (EJC 2013), Fergani et al (IEEE TVT 2015)

Incorporate fault-(diagnosis, accomodation, tolerant control) properties

• LPV fault-scheduling control: see Sename et al (Systol 2013, ICSTCC 2015).



Some PhD students on robust and/or LPV control

• Waleed Nwesaty, " LPV/H∞ control design of on-board energy management systems for electrical vehicles", PhD GIPSA-lab, Univertisté GrenobleAlpes, 2015.

• Soheib Fergani, "H∞ /LPV robust MIMO control of vehicle dynamics", PhD, GIPSA-lab, Université Grenoble Alpes, 2014.

• Caroline Ngo, "Surveillance du sysyème de post-traitement essence et contrôle de chaîne d’air compressé", PhD, GIPSA-lab / RENAULT, GrenobleINP, 2014.

• Tinghong Wang, "Robust control approach to battery health accommodation for Energy Management Systems in hybrid vehicles ", PhD, GIPSA-lab,Université Grenoble Alpes, 2013.

• Maria Rivas, "Modeling and Control of a Spark Ignited Engine for Euro 6 European Normative", PhD, GIPSA-lab / RENAULT, Grenoble INP, 2012.

• Ahn-Lam Do, "LPV Approach for Semi-active Suspension Control & Joint Improvement of Comfort and Security", PhD, GIPSA-lab, Grenoble INP,2011.

• David Hernandez, "Robust control of hybrid electro-chemical generators", PhD, GIPSA-lab / G2Elab, Grenoble INP, 2011.

• Emilie Roche, "Commande Linéaire à Paramètres Variants discrète à échantillonnage variable : application à un sous-marin autonome", PhD,GIPSA-lab, Grenoble INP, 2011.

• Sébastien Aubouet, "Semi-active SOBEN suspensions modelling and control", PhD, GIPSA-lab / SOBEN, INP Grenoble, 2010.

• Charles Poussot-Vassal, "Robust LPV Multivariable Global Chassis Control", PhD , GIPSA-lab, INP Grenoble, 2008.

• Corentin Briat, "Robust control and observation of LPV time-delay systems", PhD, GIPSA-lab, INP Grenoble, 2008.

• Christophe Gauthier, "Commande multivariable de la pression d’injection dans un moteur Diesel Common Rail", PhD, LAG / DELPHI, Grenoble INP,2007.

• David Robert, "Contribution à l’interaction commande/ordonnacement", PhD, LAG, Grenoble INP, 2007.

• Marc Houbedine, "Contribution pour l’amélioration de la robustesse et du bruit de phase des synthétiseurs de fréquences" , PhD, LAG / STMicrolectronics, Grenoble INP, 2006.

• Alessandro ZIN, "Sur la commande robuste de suspensions automobiles en vue du contrôle global de châssis", PhD, LAG / Grenoble INP, 2005.

• Julien Brely, " Régulation multivariable de filières de production de fibre de verre", PhD, LAG / ST Gobain Vetrotex, Grenoble INP, 2003.

• Giampaolo Filardi, "Robust Control design strategies applied to a DVD-video player", PhD, LAG / ST Microlectronics, Grenoble INP, 2003.

• Damien Sammier, "Modélisation et commmande de véhicules automobiles: application aux éléments suspension", PhD, LAG / Grenoble INP, 2001.

• Anas Fattouh, "Observabilité et commande des systèmes linéaires à retards", PhD, LAG / Grenoble INP, 2000.



P. Gahinet.

A linear matrix inequality approach toH∞ control.International Journal of Robust and Nonlinear Control, 4:421–448, 1994.

C. Scherer.

The riccati inequality and state-spaceH∞-optimal control.Ph.D. dissertation, University Wurzburg, Germany, 1990.

C. Scherer and S. Wieland.

LMI in control (lecture support, DELFT University).2004.

C. Scherer, P. Gahinet, and M. Chilali.

Multiobjective output-feedback control via LMI optimization.IEEE Transaction on Automatic Control, 42(7):896–911, july 1997.

S. Skogestad and I. Postlethwaite.

Multivariable Feedback Control. Analysis and Design.John Wiley and Sons, Chichester, 1996.

K. Zhou, J. Doyle, and K. Glover.

Robust and Optimal Control.New Jersey, 1996.


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