multivariable control of a submarine using the lqg/ltr method. · 2016. 6. 12. ·...

Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1985

Multivariable control of a submarine using the

LQG/LTR method.

Mette, James Allen Jr.

http://hdl.handle.net/10945/21457

DUDLEY KNO.

.

NAVAL POSTGRADUA1MONTEREY, CALIFORNI

DEPARTMENT OF OCEAN ENGINEERINGMASSACHUSETTS INSTITUTE OF TECHNOLOGY* CAMBRIDGE, MASSACHUSETTS 02139

MULTIVARIABLE CONTROL OF A SUBMARINE

USING THE L3G/LTR METHOD

BY

JAMES ALLEN METTE, JR.

OESM(ME)

COURSE XIIIAMAY: 1985

MULTIVARIABLE CONTROL OF A SUBMARINEUSING THE LOG/LTR METHOD

byJAMES ALLEN ME77E, JR.

BSME, Purdue University (1978)

SUBMITTED TO THE DEPARTMENTS OF OCEAN ENGINEERINGAND MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREES OF

OCEAN ENGINEERAND

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

10 MAY 1985

©James Allen Mette, Jr., 1985

Permission is hereby granted to the Charles Stark Draper Laboratory,to the Massachusetts Institute of Technology and to the United StatesGovernment and its agencies to reproduce and to distribute copies ofthis thesis document in whole or in part.

MULTIVARIABLE CONTROL OF A SUBMARINEUSING THE LQG/LTR METHOD

by

James Allen Mette, Jr.

Submitted to the Departments of Ocean Engineering and MechanicalEngineering in Partial Fulfillment of the Requirements for theDegrees of Ocean Engineer and Master of Science in MechanicalEngineering.

ABSTRACT

A multivariable control system for a deeply submerged submarinewith active roll control is designed using the Linear QuadraticGaussian with Loop Transfer Recovery (LQG/LTR) method. A linearmodel of the submarine is developed for a 1 degree rudder deflectionat a speed of 15 knots. The linear model is then scaled for unitsand input/output weightings and augmented with integral control in

all four input channels. Using the properties of the linear model, a

Model Based Compensator (MBC) is designed by shaping the singularvalues of a Kalman Filter to meet desired performance criteria andthen recovering the singular value shapes using the Kwaakernaakrecovery process. During extensive testing at speeds from 15 to 30

knots, the compensator performed well enough so that gain schedulingwas not employed. Next, the compensator is compared to one designedfor a 30 knot model. Finally, an Anti-Reset Windup (ARW) strategy isemployed to counter the effects of control surface saturation.

THESIS SUPERVISORS: Dr. Michael Athans, Professor of Systems Scienceand Engineering

Dr. Lena Valavani, Assistant Professor ofAeronautics and Astronautics

ACKNOWLEDGEMENTS

I would like express my thanks to my thesis supervisors, MichaelAthans and Lena Valavani, for their support, guidance and patience.I would also like to thank Mr. Petros Kapasouris for his ideas andthe interaction that helped in many ways.

The Charles S. Draper Laboratory deserves special recognitionfor providing the submarine simulation facilities. A special thanksto Mr. William Bonnice for his continual programming and technicalsupport.

Last, but not least, I want to express my thanks to Sue, Lizzieand Jimmy for the support they have given not only during my stay atMIT but throughout my naval career. A man could not ask for a moreloving or supportive family.

This research was performed at the MIT Laboratory for Informationand Decision Systems with partial support provided by the Office ofNaval Research under contract 0NR/N00014-82-K-0582(NR606-003)

.

TABLE OF CONTENTS

ABSTRACT 2

ACKNOWLEDGEMENTS 3

TABLE OF CONTENTS 4

LI5T OF FIGURES 6

LIST OF TABLES 7

CHAPTER l: INTRODUCTION AND SUMMARY

1 .

1

Background and Discussion of Prior Wori< 3

1 .

2

Contributions of the Thesis 10

1.3 Outline of the Thesis 10

CHAPTER 2: MODELING OF THE SUBMARINE

2.1 Introduction 12

2.2 Description of the Nonlinear Submarine Model 122.3 Description of the Linear Submarine Model 16

2.4 Selection of the Output Variables 19

2 .

5

Summary 27

CHAPTER 3: LINEAR MODEL ANALYSIS

3 .

1

Introduction 23

3 .

2

Reduction of the Model 283.3 Scaling 293 .

4

Eigenstructure 31

3.5 Performance Specifications 41

3 .

6

Summary 47

CHAPTER 4: LINEAR COMPENSATOR DESIGN

4 .

1

Introduction 43

4.2 The LQG/LTR Design Methodology 434.3 Design of the LQG/LTR Compensator 554.4 Preliminary Testing of the Compensator 654.5 Summary 81

CHAPTER 5: ADDITIONAL ANALYSIS OF THE COMPENSATOR

5.1 Introduction 82

5.2 Additional Analysis of the S15R1 Compensator 825.3 Comparison of the 15 and 30 Knot Compensators 91

5.4 Summary 118

CHAPTER 6: IMPLEMENTATION OF ANTI-RESET WINDUP FEED3ACX

6.1 Introduction 1196.2 ARW Feedback Loop 119

6.3 Simulations for a System With ARW Feedback.. 1236 .

4

Summary 136

CHAPTER 7: SUMMARY AND DIRECTIONS FOR FURTHER RESEARCH

7 .

1

Summary 137

7.2 Conclusions and Directions for Further Research 138

REFERENCES 141

APPENDIX A: MATRICES OF THE S15R1 LINEAR MODEL

Al Original Plant Matrices Prior to Reduction and Scaling.. 144

A2 Matrices Scaled for Units 145A3 Matrices Scaied for Units and Weights 149

APPENDIX B: PROPERTIES OF THE 15 KNOT PLANT AND COMPENSATOR

Bl Poles , Zeros and Eigenvectors of S15R1 152

B2 Observability and Controllability Matrices 153

B3 Model Based Compensator Matrices 154

B4 Poles and Zeros of the Open and Closed Loop Systems 156

APPENDIX C: MATRICES OF THE 30 KNOT PLANT AND COMPENSATOR

CI State Space Matrices of the 530R1 Linear Model 159C2 Matrices of the 30 Knot Comoensatcr 162

:!3P:

2.1 Body-Fixed Coordinate System for a Submarine 12

2.2 Comparison of Nonlinear and Linear Dynamics for Model 515R1 20

2.3 Comparison of Noniinear and Linear Response for 202

Perturbation Above Nominal Point. 23

3.1 Block Diagram Representation After Scaling 31

3 .

2

3reakdown of 515R1 Modes by State 333 .

3

Effect of Control Surfaces by Mode 36

3.4 Singular Value Decomposition of S15R1 39

3 .

5

Feedback Loop Structure 423.6 Singular Value Requirements for G(s)K(s) 42

3.7 Plant with Integrators in Input Channels 433.8 Singular Values of Original Plant Gp(s) 44

3.9 Singular Values of Augmented Plant G(s) 44

3.10 MIMO Feedback Loop with Multiplicative Error 45

4.

1

Feedback Loop Structure with G (s) 49

4.2 Structure of the Model Based Compensator in a

Feedback Configuration 50

4.3a Singular Values of G_f ]_(s) 534.3b Singular Values of Gj^f(s) 53

4.3c Singular Values of G(s)K(s) 584.4 Block Diagram Representation of the Closed Loop System 59

4.5a Singular Values of K(s) 61

4.5b Singular Values of K q ( s

)

61

4 .

6

Singular Values of the Closed Loop 624.7 Singular Values of Kr <s) 624.8 Singular Value Decomposition of K_r(s) 634.9 Symmetry Comparison for +/- 1 Degree/Second Yawrate 66

4.10 Symmetry Comparison for +/- 5 Degree Roil Angle 72

4.11 Simulation for 1 Foot/Second Depthrate 77

4.12 Simulation for 4.5 Feet/Second and -10 Degree Pitch Angle 79

5.1 Comparison of Singular Values of 515R1 and S30R1 Linear Models.. 845.2 Robustness Comparison of G(s)K(s) and the Error E(s) 855.3 Comparison of 15 Knot Compensator Performance for

Simultaneous Turn and Dive During Speed Change 875.4 Comparison of 15 Knot Compensator Performance for

Simultaneous Roil and Dive During Speed Change 925.5 Comparison of 15 Knot and 30 Knot Compensators

for -1 Degree/Second Yawrate at 15 Knots 975.6 Comparison of 15 Knot and 30 Knot Compensators

for -1 Degree/Second Yawrate at 30 Knots 1025.7 Comparison of 15 Knot and 30 Knot Compensators for

Simultaneous Turn and Dive (30 to 15 Knots) 1075.8 Comparison of 15 Knot and 30 Knot Compensators for

Simultaneous Turn and Dive (15 to 30 Knots) 113

6.1 Structure of the Ciosea Loop System with ARW 120

6.2 3cde Plot, of Simple Lag for Varying o< 121

6.3a Singular Values of GCs)K(s) for o< = 0.1 122

6.3b Singular Vaiues of GCs)K(s) for o< = 0.05 1226.4 Singular Vaiues of the Closed Loop for oc = 0.05 124

6.5 Comparison of Systems Without and With ARW (o<= 0.05)for 2 Feet/Second Depthrate 126

6.6 Comparison of Systems Without and With ARW (o<= 0.05)

for 4 Feet/Second DeDthrate 131

LIST OF TABLES

2.1 State and Control Variables 15

2.2 20* Perturbations Applied to Model 515R1 19

3.1 Control Surface Rate and Deflection Limits 30

CHAPTER 1

INTRODUCTION AND SUMMARY

1.1 Background and Discussion of Prior Work

The introduction of nuclear propulsion and hydrodynaraically efficient

hullforms on submarines have resulted in platforms capable of very high

submerged speed for prolonged periods of time. At these high speeds, ship

control is a complex problem due to the dynamic forces and moments acting

on the submarine. During a maneuvering situation, the cross-flow and

nonsymmetric flow over the hull may create a situation in which the

submarine can not be safely operated using our present method of ship

control. This requires that speed and depth restrictions be imposed on the

employment of the submarine.

Presently, submarine control systems are designed by decoupling the

vertical and horizontal planes of motion, looking at the individual

input/output transfer functions and "shaping" the response as desired.

This requires the use of an oversimplified model of a very complex dynamic

system. The recent development of the Linear-Quadratic-Gaussian with Loop

Transfer Recovery (LQG/LTR) methodology has provided a powerful tool for

the design of multivariable control systems. However, as LQG/LTR is

relatively new, research is required to investigate the inherent

limitations of the methodology and develop practical applications for its

use.

Within the last few years, several examples of the practical use of

LQG/LTR have been researched. Multiple Input Multiple Output (MIMO)

8

controllers for gas turbine engines have been developed CI] C2] [33

.

Several examples of submarine control systems have also been developed.

The submarine controller design by Harris C43 dealt with use of an

inverted-Y sternplane configuration. Of particular interest though are the

designs of Milliken C5] , Dreher C63 and Lively [7] . These designs all used

the standard cruciform stern configuration presently in use on U.S.

submarines. Milliken' s controller was sufficient for depth and course

keeping. Acceptable performance during depth and/or heading changes was

obtained only for small changes over long periods of time (i.e. small depth

and heading rates) . Dreher designed two rate controllers. One controlled

depthrate and heading rate directly and the other attempted to control

these rates by using pitch angle and yawrate. In both cases, acceptable

performance was obtained only for small rates.

The design by Lively provided the best example of using the

Linear-Quadratic-Gaussian with Loop Transfer Recovery (LQG/LTR) design

methodology for submarine control. In an attempt to capture the

maneuvering dynamics of the submarine, the linear model for this design

used a diving turn for a nominal point. By using linear compensators

designed at speeds of 5, 10, 15, 20 and 25 knots, a nonlinear compensator

was employed using gain scheduling. This provided for control of the

submarine over the speed range of 5 to 25 knots during speed changes as

well as at discrete intermediate speeds.

Additionally, Martin C8] has recently shown the merits of active roll

control over the conventional cruciform configuration for a submarine at 30

knots. By comparing LSG/LTR controllers with and without differential

sternplane deflection capability, better maneuvering characteristics were

realized especially in the area of depthkeeping during turns.

1.2 Contributions of the Thesis

The purpose of this thesis is twofold; to develop a practical example

of the use of LQG/LTR and to help open an avenue for advancement in the

area of submarine ship control systems. It is intended that this thesis be

considered a feasibility design showing the promise of using active roll

control through differential deflection of the sternplanes and not a final

detailed design ready for implementation aboard a submarine. A project of

that magnitude is obviously beyond the scope of this thesis due to

financial and time considerations.

In comparison to the work of Lively, this thesis shows that using

active roll control through differential deflection of the sternplanes

eliminates the need for a gain scheduled nonlinear compensator to provide

adequate control for speeds of 15 to 30 knots. Further research may even

permit this speed range to be expanded.

To show the eixects of crossover frequency on the response and

stability of the closed loop system, a comparison is made to the

compensator designed by Martin [8] . Following this comparison, an

Anti-Reset Windup (ARW) feedback loop is employed to counter the effects of

integrator windup during control surface saturation.

1.3 Outline of the Thesis

The nonlinear and linear models of the submarine are discussed in

Chapter 2. A brief description of the nonlinear equations of motion used

for submarine simulation at the Charles S. Draper Laboratory (CSDL) is

presented. This is followed by a discussion of the linearized model

employed in the design of the LQG/LTR compensator.

10

Chapter 3 contains an analysis of the linear model. The analysis

begins with a reduction in the order of the linear model. Scaling is then

applied to provide the designer with a more understandable system of units

and to weight the inputs and outputs. Next, the pole/zero structure and

eigenvectors of the open loop submarine model are presented. The chapter

ends with a discussion of performance requirements and specifications. In

this section, the concept of singular values is introduced.

Chapter 4 begins with a discussion of the Model Based Compensator

(MBC) concept and the LQG/LTR methodology. The design of the compensator

for the linear model is then presented. The chapter ends with a

description of the compensator and results of some of the maneuvering

simulations.

Chapter 5 begins with a discussion of additional maneuvering

simulations for the 15 knot compensator, concentrating on more complex

maneuvers. A direct comparison between the compensator designed in this

thesis and that designed by Martin C83 is also presented in Chapter 5.

A brief discussion on the use of Anti-Reset Windup feedback to reduce

the effects of integrator windup in the presence of saturating actuators is

contained in Chapter 6. This is followed by implementation of an

Anti -Reset Windup feedback loop in the 15 knot compensator and presentation

of some results which demonstrate its effectiveness.

Chapter 7 contains a summary of results, conclusions and some

recommendations for future research.

11

CHAPTER 2

MODELING AND ANALYSIS OF THE SUBMARINE

2.1 Introduction

The design of any control system begins with a mathematical

representation of the real world nonlinear system which we want to control.

The model used for this thesis was the analytic version of the SUBMODEL

submarine simulation computer programs resident on the computer systems at

the Charles Stark Draper Laboratory (CSDL)

.

The chapter begins with a brief description of the nonlinear model and

relevant computer programs. This is followed by a description of the 15

knot linear model using differential deflection of the sternplanes to

provide the necessary degree of freedom to actively control the roll angle

of the submarine.

2.2 Description of the Nonlinear Submarine Model

Motion of the submarine is described in six degrees of freedom, three

translational and three rotational. Three force equations (X, Y and 2)

define the translational motions of surge (axial), sway (lateral) and heave

(normal). The rotational motions of roll, pitch and yaw are described by

three moment equations (K, M and N)

.

The complex hydrodynamic forces and moments which act on a submerged

submarine are best defined in a right-hand orthogonal coordinate system

fixed in the body. Figure 2.1 shows this coordinate system (and sign

12

STARBOARD VIEW

REFERENCE

TOP

Kj

HORIZONTALREFERENCE

'*

STERN VIEW

Figure 2.1 Body-Fixed Coordinate System for a Submarine

13

convention) with the origin at the center of gravity (CG) . The equations

of motion are more easily derived in and control of the submarine requires

knowledge of the submarine's motion (i.e. course, speed, depth and

orientation) with respect to an inertial or earth-fixed coordinate system.

The earth-fixed system is also a right-handed orthogonal system, but is

fixed at some geographic point on the surface of the water. The

orientation and motion of the submarine in these two reference frames are

related by Euler angles C9] , each representing a rotation about a

body-fixed axis. By convention, the rotations are applied in the following

order.

1. V (rotation about the z-axis)

2. 8 (rotation about the y-axis)

3. (rotation about the x-axis)

From the discussion above, it is evident that we are talking about

nine state variables, six representing the translational and rotational

velocities in the body-fixed reference frame and three representing the

orientation of the body-fixed axes with respect to earth reference. The

submarine's depth represents a tenth state variable. Control of the

submarine is accomplished by deflecting the rudder, stern planes, fairwater

planes and/or changing propeller speed.

The SUBMODEL simulation facilities at CSDL are based on the "2510

Equations" CIO] , updated with the cross-flow terms of the "Revised

Equations" [11] . Several additional features have been added to provide a

constant RPS constraint on the propeller and a differential deflection

option for the sternplanes. The constant RPS constraint reflects the

current operating policy used on submarines. The use of differential

14

sternplane deflection provides an avenue for active roil controi that is

not otherwise possible. Table 2.1 contains a summary of state and conr.ro!

variables.

State Variables

u = Xi, axial velocity (along x axis) - ft/sec

v = X2» lateral velocity (along y axis) - ft/sec

w = X3, heave velocity (along z axi3) - ft/sec

p = X4, roll rate (angular velocity about x axis) - rad/sec

q = X5, pitch rate (angular velocity about y axis) - rad/sec

r = Xfe, yaw rate (angular velocity about z axis) - rad/sec

= X7, roll (rotation about x axis) - radians

8 = X3, pitch (rotation about y axis) - radians

l|/ = Xg, yaw (rotation about z axis) - radians

z = X10, depth - feet

Control Variables

db = uj_, fairwater plane deflection - radians

dr = U2> rudder deflection - radians

dsi = U3, port stern plane deflection - radians

ds2 = U4, starboard stern plane deflection - radians

Table 2.1 State and Control Variables

It should be noted that these equations of motion do not account for

actuator dynamics. This will be addressed further in the discussion of

performance requirements in Section 3.5. Intuitively, one would expect the

dynamics of the submarine to be independent of the depth (z) and the

15

heading angle (IJJ) . This will prove true as we wiii see in the next.

section.

The SUBMODEL program provides the following analytic capabilities:

1. Integration of the nonlinear equations of motion

2. Determination of a local equilibrium point in

the nonlinear equations

3. Calculation of the linearized dynamics about

the equilibrium point

4. Integration of the linearized equations of motion

During integration of the equations of motion, the control surface

deflections may be set to initial values, varied as a function of time or

calculated using full state feedback or the dynamic compensation derived

using the LQG/LTR method. The analytical results of the integration may be

output graphically, in tabular form or both. A full description of the

submarine model may be found in [12] through [15]

.

2.3 Description of the Linear Submarine Model

The compensator design procedure is based on a Linear Time Invariant

(LTD model of the nonlinear system. The nonlinear equations described in

Section 2.2 can be generally put in the form

x(t) = f[x(t),u(t)] (2.1)

y_(t) = g_[x(t)] (2.2)

where

x(t) is the 3tate vector

u(t) i3 the control inDut vector

16

y_(t) is the output vector

These equations can then be linearized about aoie nominal point

(Xo,!^) by using a Taylor Series expansion. Neglecting the higher order

terms of the expansion, the linear dynamics may be expressed in the sLare

space form

x(t) = A x(t) + B u(t) (2.3)

y_(t) = C x(t) (2.4)

where

A = (al/dx.>|5Lo»lLo

C = (a£/dx.)|5Lo

The nominal point used here corresponds to a local equilibrium point.

This point is found by integrating the nonlinear equations of motion for a

specific set of initial conditions. At their steady-state value, each

state variable is then perturbed so that its local minimum is found.

Since the linear dynamics change with different nominal points, the

question arises as to what is the "correct" nominal or operating point

about which to linearize. Selection of a "benign" nominal point such as

that by Milliken C5] is probably satisfactory for an "autopilot" type of

controller. This controller however, will be used for maneuvering as well

as course and depth keeping. For this reason, a nominal point which

"captures" the maneuvering dynamics of the submarine is required. To

achieve this, the submarine must be oriented such that it experiences

crossflow over the hull much like it would encounter while simultaneously

turning and diving.

17

The nominal point used in this thesis was determined using an initial

forward velocity (surge) of 15 knots with a rudder deflection of 1 degree.

This model is named S15R1 to reflect the speed and rudder deflection. This

nominal point was selected so the linear model would have dynamics similar

to that used by Martin C83 and provide a basis for comparison. The nominal

point provides sufficient cross-coupling between the horizontal and

vertical planes of motion to capture the desired maneuvering

characteristics but is not too "radical". As a result, acceptable

performance was obtained for straight and level as well as maneuvering

trajectories. Figure 2.2 shows the excellent agreement between the time

response of the linear and nonlinear models. The A and B matrices and

values of the nominal states may be found in Appendix Al. Inspection of

the last two rows of these matrices indicate that our intuition about the

effect of z and ty on the other states was correct. This will be of vaiue

when we consider reducing the order of the system.

To determine the range of accuracy of the linear model, it can be

perturbed from its nominal point by a set of initial conditions, integrated

and then compared to the nonlinear system for the same set of initial

conditions. Figure 2.3 shows the results of the initial condition

perturbation of 20* above the nominal point. The perturbations are

summarized in Table 2.2.

18

u = 5.08 ft/aec

v = 0.150 ft/sec

w = 0.395E-02 ft/sec

p = -.115E-04 rad/sec

q = 0.352E-04 rad/sec

r = -.125E-02 rad/sec

= -.325 deg

8 = -.105 deg

Table 2.2 20% Perturbations Applied to S15R1

2.4 Selection of the Output Variables

Several factors must be considered in the selection of the output

variables. First and foremost is the intent of the controller and the

inherent capabilities of the system to be controlled. For example, an

attempt to control roll angle without using differential sternplane

deflection cannot possibly succeed.

Next, a constraint of the methodology is that the dimension of the

output vector must equal the dimension of the control input vector C16 ]

.

Since there are four control inputs (db, dr, dsl, ds2), there must be four

system outputs.

Finally, the effect of each control input "felt" at the output must be

considered. Two states may have influence on the results the designer is

trying to affect. The state which requires the least control surface

deflection to achieve the desired results is the better one to use since it

will require less energy, result in iower compensator gains and reduce the

chance of saturating the inputs.

19

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A submarine is presently maneuvered by deflecting the control surfaces

until a particular turning rate, depth rate or attitude is achieved.

Knowing the handling characteristics of the submarine, the operators

maintain the rates until a predetermined point when the surfaces are

deflected to counteract the rates which have developed. If done correctly

„

the submarine will "meet" the desired course and/or depth with little or no

overshoot. For this reason, the yaw rate ty and depth rare z will be used

as output variables.

The attitude of the submarine in pitch and roll strongly affects its

performance. As the pitch angle becomes excessive, it becomes increasingly

difficult to maintain depth or control depth rate because the fairwater

planes saturate. As roll angle increases, the forces generated by the

control surfaces act out of the plane they were meant to. To visualize

this problem imagine the submarine rolling 900. At this point, the rudder

will act as a sternplane and the sternplanes as rudders. Additionally, the

force and moment generated by crossflow over the sail is quite significant

and contributes to roll and pitch. These illustrations show the importance

of roll and pitch 8.

For this thesis then, the output vector y_(t) is given by

y_(t) = C0(t) 8(t) lj/(t) z(t)]T (2.5)

where

l//(t) = -u*sin8 v*cos8»sin0 w*cos8»cos0 (2.6)

z(t) = (r»cos0 * q*sin8)/cos8 (2.7)

The linearized equations for $ and z are easily derived from the state

space equation (2.3). The elements of the C matrix which represent ty and z

are simply the last 2 rows of the A matrix.

26

2.5 Summary

This chapter has briefly described the origin of the nonlinear model

and the linearization process. The state space form of the linear model

and the reasoning behind selection of the output variables were aiso

presented. Chapter 3 will cover model reduction, scaling, the

eigenstructure of the linear model and performance specifications.

27

CHAPTER 3

LINEAR MODEL ANALYSIS

3.1 Introduction

This chapter begins with a reduction of the tenth-order model and

discussion of the scaling. Next, the eigenstructure and pole/zero

structure of the reduced model will be presented. Finally, system

performance in the area of steady state error and crossover frequency will

be discussed. In the section on performance, the concept of singular

values and their use will be introduced. The reader already familiar with

the concept and its application to MIMO control system design may wish to

skip to the last paragraph of that section for a summary of the performance

specifications applicable to this design.

3.2 Reduction of the Model

As noted in Section 2.3, the depth and heading angle (yaw) have no

effect on the dynamic response of the other eight states. This may be

verified by inspection of the last two columns of the A matrix in Appendix

Al. Had the assumption of a deeply submerged submarine not been made, one

would expect depth and heading to have a significant effect on the states

due to near-surface suction forces and direction of the seas.

Additionally, inspection of the last four rows of the B matrix reveals that

the control surfaces have no direct effect on (J/, 9, or z.

28

Since this controller does not use W or z as outputs and the other

states are not functions of these variables, they may be deleted from the

model. This is accomplished by deleting the last two columns of the A

matrix and the last two rows of the A and B matrices. As we will see in

ChaDter 4, this will also reduce the order of the comoensator state vector.

3.3 Scaling

The use of scaling has been employed to change the state and control

variables of the linear system into units which are more easily understood

and to provide weighting on the inputs and outputs of the system. Kappos

C2] and Boettcher [17] have addressed scaling and it3 effects. Obviously,

the singular values of the system will be changed but the effect on system

robustness is not fully understood.

To provide the designer with a more easily understood system of units,

rotational or angular variables are expressed in units of degrees and

degrees per second vice radians and radians per second. Transiational

variables remain in units of feet and feet per second.

The unsealed state space system as expressed in equations (2.3) and

(2.4) is scaled by defining new state, control and output vectors where

x' = Sx x

u' = Sn u

y_' = Sy y_ (3.1)

The scaling matrices S_x, Sq and Sy are diagonal matrices whose elements

provide the desired transformation of units. Consequently, the system

matrices (scaled for units) become

29

A' = S_x A S.x-1

B' = S.x B Su"l

C' = Sy C S.x-1 (3.2)

Details of the unit transformation scaling matrices and scaled system

natrices may be found in Appendix A2.

Mow that the system has been scaled for units, we must consider

scaling the outputs and inputs to reflect their relative importance. These

weights were selected to coincide with those for a compensator being

concurrently developed by Martin C8] . Weights were selected for the

outputs such that yaw rate and depth rate were more important than roll and

pitch and are given by

Sy =

.1

.1

1

1

Input weights were selected by comparing the maximum deflection of each

control surface. Rate and deflection limits are listed in Table 3.1.

Control Rate Limit Deflection Limit

db 7 deg/sec 20 degrees

dr 4 deg/sec ±_ 30 degrees

dsl»dS2 7 deg/sec + 25 degrees

Table 3.1 Control Surface Rate and Deflection Limits

For this thesis, the control surface dynamics (deflection rates) will be

treated as high frequency modeling error. Considering the deflection

limits above, the selected input scaling is given by

30

0.S670.3

0.8

Figure 3.1 shows the block diagram representation of the system after

applying the scaling for units and input/output weights.

Sx — A

Figure 3.1 Block Diagram Representation After Scaling

The effect of scaling on the plant transfer function matrix is given by

G'(3) = Sy Sy G(s) Sjq"! Su"1 (3.3)

where G(s) is the plant transfer function matrix prior to any scaling. The

input and output scaling as well as the unit transformation is identical to

that used by Martin C8] so that a better comparison of the two compensators

may be made (presented in Chapter 5). A summary of the plant

matrices after scaling can be found in Appendix A3.

3.4 Eiqenatructure

To examine the modes of the state space system defined by (2.3) and

(2.4), the linear transformation (3.4) is applied to the state vector where

x(t) = P z(t) (3.4)

31

the matrix P is the constant and nonsmgular matrix of eigenvectors Pj.j

through P_ni as defined in (3.5). The new state space (3.6) representation

P =

I I

EliJ

P2i| 23 £niI

Z(t)

V_(t)

P-l A P z(t) P-1 B u(t)

C T z(t)

(3.5)

(3.6)

consists of n decoupled equations describing the modal response of the

system. The matrix P~l A P is a diagonal matrix whose eieraent3 are the

eigenvalues of the state space. Each eigenvector describes the motion of

its associated submarine mode along the coordinate axes of the

8-dimensional A matrix. Since the dynamic response of the submarine is a

linear combination of these modal responses, useful information may be

obtained by analyzing the contribution of each state to a particular mode.

Figure 3.2 shows the state breakdown of each normalized eigenvector

and it3 associated eigenvalue in bar chart form. The vertical scale of 0.0

to 1.0 has been selected to reflect the percent contribution of each state

to the overall magnitude of the eigenvector. Due to the difference in

variables and units (translational and rotational), cross-coupling between

terms and the complex nature of the dynamics involved, a clear cut physical

explanation of all the modes is not always possible. Several modes do

however exhibit clear physical meaning leading to the following

interpretation.

(1) Modes 1 and 2 are a complex conjugate pair representing

the oscillatory nature of pitch.

32

«J4-1

cn

>

<Q

0)

TJO

• pnijuSouj •pn||u6oui

-1 1 r« N «d d d

t 1 1 1 rn * •* «i ••

d d d d d

I

\

-

a. o

i

&

I"

(Xin•-i

CO

o

c3o-a

o0)

uCO

(N

aen

•pn||uSeui • pn||u6oiu

33

-»J

<0

•uen

>-Q

<0

uT3O

•pn||u6oui

&£

men

vio

c3OT3

"eo

(1)

UCO

<U

u3

• pr\||u6oui

34

(2) Mode 3 is heavily dependent on forward 3peed and is

therefore most likely related to drag.

(3) Modes 5 and 6 are a complex conjugate pair representing

the oscillatory nature of roll.

(4) The submarine is open loop 3table since all poles are

in the left half plane (LHP).

The eigenvalues, eigenvectors (modal matrix) and transmission zeros for the

linear model are presented in Appendix Bl.

System stabilizablility (unstable modes are controllable) and

detectability (unstable modes are observable) can also be addressed by

simple inspection of the T" 1 B and C T matrices in Appendix B2. The

elements of these matrices represent the link between the original state

space system and the decoupled system and a zero element in one of these

matrices would indicate a that the link is not present. The breakdown of

the controllability matrix in Figure 3.3 shows the relative impact of the

control inputs on each mode. Similarly, the observability matrix

represents the relative contribution of each mode to the system output

variables. When used in conjunction with the eigenvector breakdown,

additional physical information can be gained about the system dynamics.

Inspection of Figure 3.3 reveals that Modes 1 and 2 which represent

pitch of the submarine are mostly affected by the fairwater planes and

sternplanes. The dominance of Mode 3 by rudder deflections explains the

reason for the large dependence of Mode 3 on forward velocity. All control

surface deflections cause a loss in forward speed due to added drag. The

rudder, being the largest control surface, has significant effect even for

small deflections. Mode 4 which has no real dominant state is also most

35

a3

o<2

0.9 -

o.a -

0.7 -

o.s -

0.5 -

0.4 -

\\

\Z7\ <*B IXNl dRMODEE2 dsi

S 7

S3 dS2

Fiaure 3.3 Effect of Control Surfacss by 3ode

36

affected by rudder deflection. Since roil angle is a major contributor to

this mode followed cioseiy by forward speed, this probably represents the

speed loss and roil angle induced by the crossflow over the sail of a

turning submarine. Modes 5 and 6 which represent the roll characteristics

of the submarine are dominated by the sternplanes with a smaller

contribution from the rudder. Obviously, the differential deflection of

the sternplanes will affect roll. The effect of the rudder on roll has

already been mentioned. Mode 7 which is dominated by pitch with lesser

contributions from pitch-related states is mostly controlled by the

fairwater planes. No clear interpretation of the rudder deflections which

control Mode 8 is possible.

To provide an indication of the coupling between control inputs and

output variables, a singular value decomposition of the plant transfer

function G(s) is presented. Details of the theory behind singular value

decomposition may be found in Athans CIS] and Lehtomaki [18] . The plant

transfer function matrix ia given by

G(s) = C (si - A)'l B = U E V.H

where

U is the matrix of left singular vectors

E is a diagonal matrix of the singular values

\/H is the matrix of right singular vectors

and at s = by

G(0) = C (-A)-l B

37

Therefore, since

y_(t) = G(0) u<t) = U E VH u(t)

we can define

y/ (t) = E u ' (t)

where

y_' <t> = U _1 y_<t)

u' (t) = V.H u(t)

Since E is a diagonal matrix, a direct coaparison of the left and right

singular vectors for each singular value will yield the desired information

about input/output coupling. The left and right singular vectors for each

singular value are presented in Figure 3.4. For Oil, we see that

deflection of the 3ternplanes results in the coupled response in pitch

angle and depthrate. Intuitively, one would think that sternplanes would

affect pitch more that depthrate. One explanation is that these variables

are strongly coupled but have different units. As is obvious from o"22>

deflection of the rudder affects primarily z but does exhibit some effect

on roll angle. This is a reflection of the effect of rudder movement and

should not be interpreted as a feasible way to control roll angle. The

effect of differential deflection of the sternplanes is exhibited by O33.

As shown by 0*44, deflection of the fairwater planes affects primarily pitch

and depthrate to a somewhat lesser extent. Again, the difference in

variables and coupling between pitch and depthrate is the probable reason

for this behavior.

38

a m6 6

i r« - o6 6

OO

CM

IT)

en

H

— oc

a:in

en

o

09

oQeooa>

QCD

•HO>

-i r

3CT

C-HCO

CO

09

3

39

%

5

o

in

r

-

1

1 1 1 1 1 1 1 r

666666666

O

5

5

<N

'OvvvVNN \\\\ v\\\\\ W\\ V '

OS,

in

en

(0

oQeou<Da

3

>uo-^3ac

en

cn

0)

u3

40

3.5 Performance Specifications

The performance requirements used in this thesis represent the

choices of the designer and may not necessarily meet any established

criteria set by legal authority. The specifications here will deal with

steady-state error, disturbance rejection and stability robustness. First,

the concept of singular values will be briefly discussed.

A logical extension of the Bode plot can be made from the SISO case to

the MIMO system by consideration of the singular values of the system. The

singular values of a complex matrix M are defined by

where

oVW) = / *i CM.HM]'

(3.4)

o~i is the i tn singular value of M

M.H is the complex conjugate transpose of M

Aj_ is the ith eigenvalue of CM^M]

Referring to (3.4), a matrix M is considered "large" when its minimum

singular value is large and "small" when its maximum singular value is

small. Using these definitions, we can now address command following,

disturbance rejection, reduction in sensitivity to modeling errors and

response to sensor noise.

Our ultimate goal is to design a robust, dynamic compensator, K(s) to

control the Linear Time Invariant (LTD plant, Gp(s). Figure 3.5 3hows the

feedback loop structure with unity negative feedback, command input r(s),

plant input u(a), system output y_<3), output disturbance vector d(s) and

measurement noise n(s) . From Figure 3.5, the following frequency domain

*U

d

^O

Figure 3.5 Feedback Loop Structure

relation can be easily derived

y_(s) s £1 + G(3)K(a)]-lG(3)K(3) r(s)

CI G(s)K(s)]-l d(s)

« CI G(s)K(3)]-i n<3) (3.5)

From (3.5), we see that to have "good" command following and

disturbance rejection, the Loop Transfer Matrix G(s)K(s) and therefore

o"a i n [G(3)K(s) } must be "large" in the frequency range of the reference

commands. To reduce the sensitivity to modeling errors, o,

max CG(s)K(3)

}

must be "small" in the frequency range of the modeling errors. Similarly,

for "good" noi3e rejection, 0"max {G(3)K(s) } must be "small" in the frequency

range of the noise. Figure 3.8 shows the Bode-like plot interpretation of

L09 crtjwl

7777771777//low frequency//performance //BARRIER m „ /

a_-JG <N»> <(iv»H

•ma.ISIi-'Sf-'l

logjwj

777777777'

UNMO0EL2DOYNAMICS/SENSOR '

NOISE

Figure 3.6 Singular Value Requirements for G(3)K(3)

h2

these singular value requirements. More detail on thi3 subject nay be

found in Athans [16]

.

With respect to command following, it is desired to have zero steady

state error to step inputs. This coincides with the singular value

requirement discussed above. As such, integrators were placed in all four

input channels. Figure 3.7 shows the new plant configuration where the

Figure 3.7 Plant With Integrators in Input Channels

state space representation of the augmentation is defined by

G-(a) = (I/-) (3.6)

and the resultant plant transfer function matrix is

G(s) = Gp(3)G_a(s) (3.7)

The augmented plant is now a 12-dimensional system, however the 4

integrator states are included as part of the plant only during the L2G/LTR

compensator design. During implementation, the integrators become part of

the compensator. The singular values of the original plant Gp(s) and the

augmented plant G(s) are shown in Figures 3.8 and 3.9 respectively. Note

that augmenting the plant has increased the singular values at .001

radians/second by approximately 60 db. The matrices of the augmented plant

nay be found in Appendix A3.

43

o3

o

o

90 -

ao -

70 -

so -

50 -

AO -

30 -

20 -

10 -

n — ^-10 -

-20 - ^-30 -

-AO -

-SO -

-60 -

-70 -

-80 -

-90 -

-100 - 1— " T" " -"-"-""

.001 •01 .1 1 10omega (radians/second)

100

Figure 3.8 Singular Values of Original Plant Gp (s)

100

cm

O

o

.001 .1 1

omega (radians/second)100

'igure 3.9 Singular Valuea of Augmented Plant G(a)

kk

The next area which needs to be addressed is crossover frequency. As

crossover frequency is increased, the system responds faster and one would

expect better performance. At slower speeds however, the control surfaces

have less effect for a given deflection since the forces and moments

generated are proportional to speed squared. Since this compensator will

be compared to the one being concurrently developed by Martin C8] , a

comparable crossover frequency is desired. To avoid frequent control

surface saturation and in some cases limit cycling due to the decrease in

effectiveness, a compromise must be reached. Additionally, actuator

dynamics which exist at approximately 2 to 3 radians/second C5] C6] , have

not been included in this model and must be treated as modeling error. For

this thesis, a crossover frequency of 0.1 radian per second was found to be

suitable.

Finally, a brief discussion of robustness is in order. According to

Lehtomaki C183 , the error E(3) between the real system G(s) and the linear

model G(s> can provide a measure of stability robustness. These errors are

characterized as additive, subtractive, multiplicative or division in

nature. In particular, the multiplicative and additive errors can be used

to provide a measure of the relative and absolute errors, respectively. A

block diagram representation of multiplicative error as defined by (3.8) is

shown in Figure 3.10. Additive error is defined by (3.9). Using these

r(s) * ">.

£(s)

+ <r

K(s) G(s)Ji

vfe)

Figure 3.10 MIMQ Feedback Loop With Multiplicative Error

^5

G(s) s CI E(s)] G(s) = L(s)G(s)

E(s) = CG(s) - G(s)] G(s)-1 (3.8)

G(s) = E(s) G(s)

E(s) = Z(.a) - G(s) (3.9)

relationships, it can be shown that if the singular value inequalities

(3.10) and (3.11) hold, we are guaranteed a closed loop 3table 3V3tem .

o"*ax<k(s) - 13 < o-min {I. CG(s)K(s)] "-} (3.10)

o-max (L-l(s) -I) < <r«infl G(s)K(s)} (3.11)

The modeling errors may be difficult to quantify but looking at it from

another aspect, if we calculate the right hand aide of the inequalities,

then the maximum tolerable error is known. These relationships provide

conservative stability bounds so that a closed loop system may still be

stable if the bounds are crossed. System stability would then have to be

determined through extensive testing and evaluation. Additional

information on stability robustness may be found in C19] through C223

.

Performance specifications may therefore be quickly summarized aa

(1) The singular value requirements in Figure 3.6 must

be met

(2) Integrators in all input channels will insure zero

steady-state error to step inputs

(3) Maximum crossover frequency is 0.1 radian/3econd

46

3.6 Summary

In Chapter 3, the order of the iinear model has been reduced by 2

states and scaled for units and input/output weightings. The poie/zero

structure and eigenvectors have been presented. In the section on modal

analysis, the relative contribution of the states to each mode was

considered and the issues of stabilizability and detectability were

addressed. The coupling between inputs and outputs was also presented.

Finally, the performance specifications in terms of singular value

requirements and crossover frequency were detailed. In Chapter 4, the

LQG/LTR methodology and a summary of the compensator design are presented,

^7

CHAPTER 4

LINEAR COMPENSATOR DESIGN

4. i Introduction

This chapter covers the design of linear compensator for the S15R1

model discussed in Chapters 2 and 3. It begins with a general discussion

of the Model Based Compensator (MBC) concept and the Linear Quadratic

Gaussian with Loop Transfer Recovery (LQG/LTR) methodology tailored for

performance and robustness at the plant output. This is followed by a

summary of the linear compensator design for the S15R1 model. Finally, the

results of several simulations with the compensator integrated in the

nonlinear analytic model at CSDL are presented. The compensator is further

analyzed and critiqued in Chapter 5 where it is compared to the S30R1

compensator designed by Martin C8j .

4.2 The LQG/LTR Desicn Methodology

Figure 4.1 again shows the feedback loop structure with unity negative

feedback. Note that the original plant Gp(s) has been replaced with the

augmented plant G(s) as defined by (3.7). The compensator K(s) must

provide closed loop performance commensurate with the criteria noted in

Chapter 3. This will be accomplished through the design of a Model Based

Compensator (M8C) [163 using the Linear Quadratic Gaussian with Loop

Transfer Recovery (LQG/LTR) methodology [23] through [25]

.

48

uyuT

G(s)

+ 'F

^O

Figure 4.1 Feedback Loop Structure with G(s)

The LQG/LTR methodology guarantees the designer a MIMO compensator

K(3) that is closed loop stable while providing the necessary degrees of

freedom for "loop shaping" (shaping of the singular values of G(s)K(s)) to

achieve desired performance while maintaining stability robustness. Figure

4.2 shows the structure of the feedback loop using the MBC. The state

vector of the MBC is z(t)ERn (i.e. the same dimension as x(t), the plant

state vector) , hence we ultimately wind up with an open loop system

G(s)K(s) which is 2n-dimensional. The reduction of the model described in

Section 3.2 has therefore reduced the resultant closed loop system by a

total of 4 states.

The MBC derives its name from the fact that the A, 3 and C matrices

defined in the state-space representation of the plant appear in

the dynamics of the compensator. The open-loop plant dynamics are

x(t) = A x(t) + B u(t)

y_(t) = C x(t)

(4. 1)

(4.2)

where x(t)ERn, u(t)ER"» and y_(t)ER». Note that u(t) and y_(t) are the same

dimension as constrained by the methodology. The frequency domain

representation for the plant transfer function G(s) is given by

^9

ss

>.l

ol

xt

<!

col

ai

z<-Ia.

ol xl

.- ai Ol>l

XJn XI

++

t-i OlJS i cal31 +ml *l j-. <l+ Ol ^^ u*

|a>i

N|w» (A

I-t *i Ola 1 ii

•mi >l 31 31 ^\

CQl31

ail wt1

+ 31 <l1

•^ ** —i

xl XI to u*

<l Ol Ol olu u 1 u

mm*** ^i «• ^.•XI >>l >-l Ol

<0

3a

coo-Xo«0

j3T3OJ

<u

u.

to

c

uo*J<B

<D

cai

a£Oo-oai

(0

(0

CO

ai

ao=ai

ai

3Uo3

4-1

cn

CM

41

U3cn

50

G(s) = C (si - A)"i 3 (4.3)

Dynamics of the MBC in the time domain are

z(t) = CA - B G - H C] z(t) - H e(t)

u(t) = -G z(t)

(4.4)

(4.5)

and the coraoensator transfer function K(s) 13

K(s) =G(sI-A+BG+H C)~- K (4.6)

where the Filter Gain Matrix H and Control Gam Matrix G are design

parameters for the compensator.

As derived in C16] , by defining a state estimation vector (4.7), the

dynamics of the closed loop system can be represented by (4.3).

x(t)

w(t)

w(t) = x(t) - z(t)

(A - B G) -3 G

(A - H Z)

x(t)+

w(t) -H

r(t)

(4.7)

(4.8)

By examining the closed loop A matrix above, it is evident that the roots

of (4.9) yield the 2n eigenvalues of the closed loop system and for closed

loop stability, (4.10) and (4.11) must hold.

detCsI - A + B G ] • dettal - A H C3 =

ReCXiCA - B G ] } < ; i=l,2, . . . ,n

ReOiCA - HC11 < ; i = l,2, . . .,n

(4.9)

(4.10)

(4.11)

It now becomes clear that the compensator design decomposes into

finding G and H 3uch that (4.10) and (4.11) hold. The Linear Quadratic

Gaussian (LOG) methodology provides the necessary tools to make

51

"intelligent" choices for the compensator gain matrices thus the L3G

controller is a special type of Model Based Compensator. A review of

equations (4.4) and (4.5) reveai the LOG controller to be a cascaded

combination of a Kalman-Bucy Filter and a full-state feedback Linear

Quadratic Regulator.

Selection of the Filter Gain Matrix H is made through the solution of

a Kalman-Bucy Filter (KBF) problem based on equation (4.4). The Filter

Gain Matrix is defined by

H = (1/^j) E C_T (4.12)

where E = E* >_ is the unique solution of the Filter Algebraic Riccati

Equation (FARE)

= AE*E_aJ + LLJ- (l/^i) E CT C E (4.13)

subject to the constraints

» CA,C_] detectable

* CA,L] stabilizable

• p >

The Control Gain Matrix G is defined by (4.14) as determined through

the solution of the "cheap" LQR (Linear Quadratic Regulator) control

problem

u(t) = -G z(t) = -R-l B_T K (4.14)

where K = Xj >_ is the unique solution of the Control Algebraic Riccati

Equation (CARE)

= -K A - Aj K - q C? C K 3 R/l B~ K (4.15)

52

subject, to the constraints

* CA,B] stabilizabie

« CA,C] detectable

8.s I

» q >

The compensator gains could new be determined if the design parameters

q, u and L were known. To determine the "correct" values for these design

parameters, we now turn our attention to Loop Transfer Recovery (LTR)

.

Refering to Figure 4.2, performance and stability robustness can be

measured at the plant input or plant output by breaking the loop at the

corresponding point. For the design of the submarine controller, physical

significance at the plant output has been retained (point (2) on Figure 4.2)

and therefore the latter approach will be used. The reader is referred to

[23] and [24] for an explanation of the methodology with respect to

performance and robustness at the plant input.

The design of the controller can now be performed in the following

fashion.

(1) Design a Kalman Filter with appropriate singular values

using the KBF equations (4.12) and (4.13)

(2) Recover the singular value shapes using the LQR equations

(4.14) and (4.15)

The following definitions of the filter open loop and Kalman Filter

transfer functions are useful in the development of the design procedure

Gf l(s) = C (si - A)*l L (4.16)

G^f(s) = C (sl_ - A)"l H (4.17)

53

Note that (4.17) represents the ioop transfer function when the LQG

loop in Figure 4.2 is broken at point (l) . The relationship between these

transfer functions is given by the following Return Difference Identity

(4.18) also known as the Kalman Frequency Domain Equality (FDE) for the

filter.

CI+Gjcf Cs)] CI+Gfcf <s)]H = I d/u)Gfol (s)GHfol ( S ) (4.18)

By looking at the singular values of both sides of the FDE (4.19), the

approximation (4.20) holds for 0"i(G_^f(s)} >> 1, i.e. at low frequency.

0-j.CI Gj<f(s)} = /l (l/^)o-i2(Gf i(s))1

(4.19)

0-i{G_i<f(s)} = 0"i((l/y7i)Gfo i(s)} (4.20)

This means the singular values of the KBF can be "shaped" to meet the

desired performance, crossover and robustness requirements by selecting

1. L for the desired loop shape

2. u for the desired crossover frequency

Using the value of }i and the matrix L from above, H is calculated

using the KBF equations and the following properties are guaranteed.

* Closed-loop stability (for the filter)

» -6db < Gain Margin < oo

* -60° < Phase Margin < 60°

Now that the Filter Gain Matrix has been determined, the zeros of the

plant tranfer function G(s) are checked. If the plant is minimum phase, we

are guaranteed recovery of the "nice" shape and properties of the Kalman

Filter using the Kwakernaak "sensitivity recovery" C26] via the LSR

5^

equations. At this time, the effects of non-minimua phase (NM?) zeros are

not. clearly understood, but suffice it to say that some degracaticn in the

recovery process and resulting cicsed loop performance is certain if one or

more NMP zeros are within the bandwidth of the controller. If the plant is

non-minimum phase, all that can presently be done is to proceed and use

extensive testing and evaluation to detect instabilities or other problems.

The LQR equations are now solved (as q

—

go) to determine the Control

Gain Matrix G. Using the G and H matrices, the LQG compensator matrix KCs)

is determined using (4.6). The singular values of G(s)K(s) are then

compared to the singular values of Gj<f(s). If the Cm (G(s)K(s) } are

sufficiently different from o"i (G^f (s) 1 , q must be increased and the LGR

equations solved for a new Control Gain Matrix. Recovery of the Kalman

Filter loop shape is considered satisfactory when there is good agreement

for at least 1 decade past crossover. At that point, the roiloff of the

singular values is at least -40 db/decade (two pole roiloff) . The Loop

Transfer Matrix G(s)K(s) is then checked for robustness as discussed in

Section 3.5.

4.3 Design of the LQG/LTR Compensator for the S15R1 Model

As discussed in the previous section, the design of the compensator is

performed as follows.

(1) Design a Kalman Filter loop with appropriate singular

value shapes

(2) Attempt to recover these singular value shapes using

the Kwaakernaak recovery method

55

The reader is once again reminded that during the LQG/L7R design process,

the plant matrices which are used are that for the augmented scaiec plant

G(s) . Any augmentation is separated from the plant model and included in

the compensator prior to implementation for testing.

Design of the Kalraan Filter loop hinges on selection of the parameter

)i (magnitude of the measurement noise intensity) and the matrix L so that

the singular values of Gf l<s) as defined by (4.16), have a "nice" shape

like that shown in Figure 3.5 and meet the performance specifications set

in Section 3.5. To enhance chances of a good recovery of these shapes in

the LQR loop, it is desired to have the minimum and maximum singular vaiues

natch. This is accomplished by selecting L such that the singular values

match at high and low frequencies.

The augmented plant transfer function G(s) = QB (.a)Gp(a) is defined by

G(s) = C (si - A)"l B (4.21)

whereA =

0.

Bp Cp

C = " Cp]

and therefore

[si - A]si

-Bp (si. - Ap) (4.22)

Looking at (4.22) at low frequency, (si. - Ap) —*> -Ap and a good

approximation for (si. - A) "I is

(si - A)"l £<i/sn

-(l/ 3 )Ap-l|p -Ap" 1 (4.23)

provided Ap~l exists. Therefore by (4.16), Gfol ( s) is approximately

56

Gf i(s) ^ CpJ

(1/S)I

-(l/ s )Ap~lBp -Ap-1

(l/ s )CpAp-lBpLi - CpAp _1 L2

kl

k2

(4.24)

Note that the matrix L has been partitioned into L± and L2 associated with

the low frequency and high frequency behavior respectively. From (4.24),

low frequency matching of singular values occurs if

Li [CoAo-lBol-1 (4.25)

Similarly, at high frequency, (sl_ - Ap) — sl_ and

(si - A)"l £3

(l/a)IS'±.

(l/ 32)Bp (i/s)!

(1/S )I

-(l/s2)Bp (1 / S >I

(!/s2)CpBpLi (l/ 3 )CpL2

(4.26)

Gf l(s) & Cp 1k2

(4.27)

As s —* 00, the second term in (4.27) dominates and the singular values are

matched if

L2 = CpT<CpCpT)-l (4.28)

The L matrix (listed in Appendix B3) has now been selected for low and high

frequency matching, however the behavior of the singular values at middle

frequencies has not been considered. Unacceptable differences at these

frequencies are dealt with by appropriate scaling of the plant inputs and

outputs such as that described in Section 3.3.

Once an appropriate L matrix has been determined, the singular values

are shifted up or down by varying the scalar parameter p until the desired

crossover frequency is obtained. The value of p. = 132 produced the desired

57

ioo

.001 .1 1 10

oaeqa (radians/second)

Figure 4.3a Singular Values of G-foi (a)

100

ICO

.001 .1 1 10

omega (radians/second)

Figure 4.3b Singular Values of Gir-f(s)

100

.001 .1 1


Figure 4.3c Singular Values of G(s)K(s)

100

58

maximum crossover frequency of 0.1 radians/second. Figure 4.3a shows the

plot of o"i { (l//JJ)Gf i (s) ) for these parameters.

Now, using the p. and L such that meet the desired specifications, the

FARE is solved and the Filter Gain Matrix H determined from (4.12). Next,

the singular values of the Kalman Filter o"i(Gj<f(s)} are calculated as shown

in Figure 4.3b and compared to the performance specifications. Thi3 is the

"loop shape" of G(s)K(s) which results from the Kwaakernaak recovery.

The next step in the LOG/LTR process is solution of the CARE (the MIMO

transmission zeros have already been checked in Section 3.4) and

calculation of the Control Gain Matrix using (4.14). Using the H and G

matrices, the compensator K(s) is calculated using (4.6) and the singular

values of the Loop Transfer Matrix G(s)K(s) are compared to the singular

values of the Kalman Filter loop. As stated in Section 4.2, satisfactory

recovery is said to occur when o~i (G(s)K(s) ) match o"i£Gj<f(s)) for at least

one decade past crossover. Figure 4.3c shows the recovered singular value

loop shape for q = 105. The crossover bandwidth for this design was from

approximately 0.05 to 0.1 radians/second. The G and H matrices are listed

in Appendix B3.

As stated in Section 3.5, the integrators are actually part of the

final compensator matrix Ka(s). The final compensator must also include

the effects of scaling on the inputs and outputs. Figure 4.4 shows the

block diagram representation of the resultant closed loop system. The

Figure 4.4 Block Diagram Representation of Closed Loop Sv3tam

59

singular values for the LOG compensator K(s) and the augmented and scaled

compensator |£a(s) are shown in Figures 4.5a and 4.5b respectively. From

these singular vaiue plots, the lead-lag characteristics, particularly the

effects of the integrators at low frequency, are evident. The spread in

singular values at frequencies below crossover reflects the different

amplification requirements for certain directions so that the singular

values of the recovered Loop Transfer Matrix G(s)K(s) match.

The singular values of the closed loop plant as defined by (4.3) are

shown in Figure 4.6. They reflect the desired chararacteristic of db

gain until crossover or break frequency and at least 2 pole rolloff

(-40 db/decade) after crossover. The poles and zeros of the open loop and

closed loop systems are found in Appendix B4.

By examining the transfer function matrix for the loop broken at the

plant input, information can be obtained on how the control inputs vary

with respect to reference commands. The transfer function relating r(s) to

u(s) is given by

u(s) = CI + K(s)G(s))-l K(s) r(s) = Kr(s)r(s) (4.29)

Figure 4.7 shows the singular values for the frequency range of interest.

The spread in singular values indicates that certain directions have higher

gains than others. The singular value decomposition of (4.29) shown in

Figure 4.3 allows us to examine the coupling of reference commands to

control inputs. The singular value decomposition was accomplished in the

manner discussed in Section 3.4 on plant eigenstructure. From Figure 4.8,

we see that for <T\\, the controller responds to a depthrate command with

deflection of the fairwater planes which makes physical sense. For 0*22 » it

is evident that if a yawrate command causes deflection of the rudder and

6o

100

3

C

o

o

.001 .1 1 10


Figure 4.5a Singular Values of K(a)

100

100

o"3

c

a

o

.001 .1 1

omega (radians/second)100

Figure 4.5b Singular Values of Ka (3)

61

>uO3

c

O

O

100 -r-

90 -

SO -

70 -

SO -

50 -

40 -

30 -

20 -

10 -

O --lO -

-20 -

-30 -

-40 -

-50 -

-SO -

-70 -

-30 -

-90 -

-100 -

.001

—

i

r-

.01 .1 1


100

Figure 4.6 Singular Values of the Closed Loop

TOO

o3

a

©

.001 .01 .1 1


100

Figure 4.7 Singular Values of Kr <3)

62

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the sternpianes. The rudder obviously deflects to induce the yawrate.

Movement of the sternpianes can be attributed to differential deflection to

counter the initial roll angle caused by the rudder deflection and the

steady state roll angle induced by crossflow over the sail. Furthermore,

the 0*33 information indicates that roll commands will cause the rudder and

sternpianes to respond. Differential sternplane deflection is obviously

the only feasible way to control roll so the question arises as to why the

rudder deflects so much. This is most likely due to the coupling between

the rudder and sternpianes. Finally, 044 3hows that a pitch command

results in deflection of the sternpianes. In this case we would expect

deflection in the same general direction with possible slight differences

due to induced roll angles.

4.4 Preliminary Testing of the Compensator

As a preiude to further analysis, preliminary testing of the

compensator was performed to gain some insight on possible limitations of

the design. Simulations were performed for single command inputs and

multiple command inputs using step and square commands.

First, a aeries of simulations were made to examine the effects of

symmetry on the response of the submarine. Figure 4.9 shows the results of

the 15 knot nonlinear simulations for ^1 degree/second yawrate step inputs

applied at t = 5 seconds and removed at t = 75 seconds. As the submarine

enters into the turn, it initially rolls slightly outward and then due to

the large force generated by crossflow over the sail, snap rolls inward.

To counter this effect, the sternpianes deflect differentially to generate

a righting moment which drives the roll angle to zero. To counter the

effect of the initial rudder deflection on pitch and depthrate, there is

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70

slight movement of the fairwater planes. The slight differences which do

exist are most likely due to propeller torque with a smaller contribution

probably due to the -1 degree/second command (turn to port.) being in the

direction of the nominal point whereas the +1 degree/second (turn to

starboard) is in the opposite direction.

Another maneuver which demonstrates motion about a plane of symmetry

is displayed in Figure 4.10. At t = 5 seconds, a ^_5 roll angle step input

is applied. The sternpianes deflect differentially to a steady-state valu

of approximately +7.5 degrees to achieve the desired roll angle. The

rudder and fairwater piane3 deflect only slightly in reaction to the

initial transients and then go to zero.

Various other maneuvers were simulated for depthrate with and without

pitch to ensure control surface deflections for motion in the vertical

plane were practical. The results of the simulation for a 1 foot/second

depthrate are presented in Figure 4.11. The singular value decomposition

in Figure 4.8 predicts that the fairwater planes should deflect in response

to this reference command. Thi3 does indeed occur a3 shown in Figure 4.11

and is accompanied by a sraail deflection of the sternpianes which produce a

moment to counter the slight trimming moment induced by the fairwater

planes.

Figure 4.12 shows the results of the simulation for a 4.5 feet/second

at -10 degree pitch. The initial decrease is attributed to the additional

drag generated by movement of the control surfaces. A3 the control surface

movements decrease, forward speed is regained. The sraail transient

disturbances in the states are a result of of control surface deflection

and to a lesser degree, mild cross-coupling. Referring again to the

singular value decomposition in Figure 4.8 (d*n and 044), one would predict

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deflection of the fairwater planes in response to the ciepthrate command,

aternpianes in response to the ordered pitch angle and minor deflection of

the rudder. Figure 4.12 confirms this to be the case.

Subsequently, simulations which flexed the compensated system in the

vertical and horizontal planes of motion simultaneously were made at speeds

from 15 to 30 knot3 as a prelude to gain scheduling. These simulations and

justification for not employing gain scheduling are discussed in Chapter 5.

4.5 Summary

The first section of this chapter covered the LQG/LTR methodology for

robustness and performance at the piant output. Next, the design and

analysis of the linear compensator for the S15R1 model were presented.

Finally, results of some simulations at 15 knots were presented. These

simulations demonstrated the symmetry of yawrate and roll angle and

confirmed the predictions of the singular value decomposition for the

transfer function relating the control inputs to the reference commands.

Chapter 5 will cover some additional simulations which were performed for

speeds from 15 to 30 knots. A comparison to the compensator designed by

Martin C8] is also presented.

81

CHAPTER 5

ADDITIONAL ANALYSIS OF THE COMPENSATOR

5.1 Introduction

This chapter covers additional analysis performed on the compensator

designed for the S15R1 linear model at speeds from 15 to 30 knots. This is

followed by a comparison to the S30R1 compensator designed by Martin C8]

.

As a result of thi3 analysis, it was determined that while gain scheduling

would offer improved performance, it was not required to assure a closed

loop stable system and therefore would not be implemented.

5.2 Additional Analysis of S15R1 Compensator

It has been shown that using active roll control through differential

deflection of the sternplanes reduces the depth excursion problem

experienced by a high speed submarine in a turn. In this section, it

becomes obvious that active roll control also provides improved performance

for depth changing and coordinated maneuvers over a wide speed range.

To implement a gain scheduling scheme, linear compensators must be

designed for various discrete speeds in the speed range of interest. A

question arises as to how to select those discrete speeds. In this case,

quantifying the modelling error between the linear model and the actual

plant is difficult. We can use the robustness singular value inequalities

(3.10) and (3.11) in a similar manner by assuming the linear model at one

speed is the real system G(s), the linear model at another speed is the

82

model G(s) and using the relationships (3.8) and (3.9) as a gauge of the

errors between the models. This is not. strictly accurate representation of

the errors and the following comments should be kept in mind. Due to this

assumption, the error between the S15R1 model and the actual 30 knot system

is probably larger in some directions and smaller in others. The

inequalities provide a conservative estimate of stability robustness 30

that the system may be stable even if the relations (3.10) and (3.11) do

not hold. These two conditions tend to offset one another and

determination of system stability must be confirmed through extensive

simulation.

A comparison of the singular values of the S15R1 and S30R1 linear

models is shown in Figure 5.1. At low frequency, the singular values of

S15R1 are generally lower than those of S30R1 with the exception of the

minimum singular values which are identical up to approximately 0.05

radians/second. We see that U3e of integral augmentation has increased the

gain of all singular values by approximately 60 dB at 0.001 radians/second

and has shifted the maximum crossover frequency somewhat higher. The state

space matrices for S30R1 can be found in Appendix CI.

Preliminary simulations showed that the compensator designed for the

15 knot model adequately controlled the nonlinear model even at 30 knots.

Assuming the S30R1 model as representative of the real or nominal system

G(s) and the S15R1 model as the system model G(s), we can get a feel for

the error of the S15R1 model at 30 knot3. Figure 5.2 shows the comparison

of the singular values of CI (G(s)K(s) ) "1] and [I G(s)K(s)] with the

singular values of CL(s) - JJ and CL~l(s) - I_] and reveals that the

singular value inequalities (3.10) and (3.11) do not hold. As previously

stated, the relations provide a conservative estimate of stability and the

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85

direction of the instability (if it does exist.) must be determined through

testing. The results of some nonlinear simulations for the 15 knot

compensator are now presented to support this claim.

The maneuvers presented in Chapter 4 were rerun for various speed

between 15 and 30 knots with favorable results. Since these maneuvers were

not very stressful, two maneuvers which flexed the system were simulated.

The first maneuver consisted of a combined dive and turn performed at 15

knots, at 30 knots, as speed increased from 15 to 30 knots and finally as

speed decreased from 30 to 15 knots. The maneuver begins with a 4.5

feet/second depthrate, -10 degree pitch angle and -2 degree/second yawrate

initially applied at t = 5 seconds. At t = 80 seconds, the yawrate command

is zeroed, followed at t = 130 by the depthrate and pitch commands.

Results of these simulations were satisfactory. For the sake of brevity,

only the speed change maneuvers are presented. Figure 5.3 shows the

results with the 15 to 30 knot run on the left. Both simulations were

completed without encountering instability however, there were notable

differences in the yawrate command following and minor differences in depth

change. The roll angle for both simulations was adequately controlled

after the initial transients due to control surface movement died out.

Maximum pitch errors of approximately 3 degrees and 2 degrees respectively

were experienced. The bottom line of this maneuver though was to effect a

course and depth change through appropriate rate commands during a speed

change to stress the system. Considering the time the rate commands were

in effect, the depth should have increased to approximately 1060 feet and

the course change have been 150 degrees. The 15 to 30 knot simulation

displayed some overshoot in depth but both converged to within a few

percent of the intended depth as did the 30 to 15 knot simulation. Even

86

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though the yawrate command following for the 30 to 15 knot was not felt to

be adequate, both simulations converged on a 150 decree course change.

The second maneuver consisted of a dive with an ordered roll angle

during speed changes 3uch as that in the previous maneuver. At t = 5

seconds, a 20 degree roll angle, 4.5 feet/second depthrate and -10 degree

pitch angle were ordered. The 20 degree roil angle remains for the entire

simulation however, the depthrate and pitch angle orders are removed at

t = 130 seconds. Figure 5.4 shows the results with the 15 to 30 knot

simulation on the left again. The simulation for 15 to 30 knots showed

favorable results by maintaining the ordered 20 degree roll angle. The

depthrate and pitch angle responses exhibited some overshoot but were

converging on the commanded values as the fairwater planes came out of

saturation. Throughout the maneuver only slight disturbances to the

yawrate were present. The 30 to 15 knot simulation showed that the system

was stable, but due to the speed decrease, the roil angle could not be

maintained as is evident by sternpiane saturation at t = 95 seconds.

5.3 Comparison of 15 and 30 Knot Compensators

Since the 15 knot compensator performed adequately, the next logical

step was to determine how it's performance at various speeds compared to

the compensator designed by Martin for the 30 knot model. For details of

the design as well as a comparison to a system without active roil control

the reader is referred to C83 . The matrices for the 30 knot compensator

are found in Appendix C2.

First, several benign maneuvers were performed to observe how the

control inputs and outputs of the two systems responded. The first set of

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simulations shown in Figures 5.5 and 5.6 were for a -I degree/second

yawrate step input applied at t s 5 seconds at speeds of 15 and 30 knots

respectively. The response for the 15 knot compensator is on the left and

the 30 knot on the right. In general, both exhibited good command

following with the 30 knot compensator reaching the commanded yawrate

quicker. It is evident that this occurs because the control surfaces move

ituch faster than the 15 knot compensator. It should also be noted that

this causes major disturbances to the states and other outputs as compared

to the 15 knot compensator. Of particular interest is that the 15 knot

compensator showed more oscillation at 30 knots and the 30 knot compensator

showed more oscillation at 15 knots. This is due to the particular

simulation being far from the nominal point. The oscillations for the

system using the 30 knot compensator were generally of larger amplitude

than those of the system with the 15 knot compensator. This set of

simulations was followed by a similar set for a 1 foot/second depthrate

with similar results.

The two maneuvers presented in Section 5.2 above were then repeated

for the 30 knot compensator. The results of the simulations are again

shown with the 15 knot compensator on the left and the 30 knot compensator

on the right. The 30 to 15 knot simulation for the turn and dive is

presented in Figures 5.7. The results are generally the same a3 previously

stated. The 30 knot compensator responds quicker, is more oscillatory and

exhibits more overshoot. Of particular interest though is the saturation

of the fairwater planes at t = 80 seconds due to the sternplanes deflecting

quicker than those of the 15 knot compensator as the yawrate command is

removed. This results in growing errors in pitch and depthrate until these

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commands are removed. The results of the roil and dive maneuver provided

similar results and are therefore not presented.

As shown in Figure 5.8, very interesting results were obtained when

the simulation was repeated for 15 to 30 knots. The results are similar

until the yawrate command is removed. At that time, the system with the 30

knot compensator experiences a large 3nap roll, almost twice that of the 15

knot compensator. At t = 110 seconds, the fairwater planes of the 30 knot

compensator saturate which the system also handles but with a larger error

in pitch and depthrate. At t = 130 however, the pitch and depthrate

commands are zeroed and the system goes unstable in the vertical plane as

evidenced by the growing depthrates and pitch angles and repeated

saturation of the fairwater planes and the oncoming saturation of the

sternplanes.

As shown by Martin, the 30 knot compensator performs well for large

yawrates. It appears then that the problem arises only for motion in the

vertical plane. The reason or reasons why the system with the 30 knot

compensator goes unstable and the system with the 15 knot compensator does

not are unknown but the following is offered as a possible explanation.

(1) The higher crossover frequency for the 30 knot compensator

causes the control surfaces to move faster. These quick

movements result in quicker control surface saturations resulting

in large intergated errors.

(2) Since we are operating away from the nominal design point, the

model is no longer accurate for motion in the vertical plane.

(3) The drag terms associated with the 15 knot compensator

simulation are actually 4 times larger at 30 knot3 than at

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15 knots (drag is proportional to velocity squared) where the

compensator was designed. This has sufficiently slowed the

system response at 30 knots, particularly in the directions

related to motion in the vertical plane.

Determination of the exact cause is a subject which requires further

research. In any case, the 30 knot does exhibit better performance in mild

maneuvers so to implement a global control system for the submarine at

various speed and for variouys maneuvers, there may be 3ome merit to using

a quick controller for mild maneuvers and a slower controller for more

radical maneuvers.

As a result of this analysis, it was determined that implementing a

gain scheduling algorithm wouid not be feasible without either redesigning

the 30 knot compensator for a lower crossover frequency or limiting the

maneuvers at higher speeds. As shown by Lively C7] , implementing a gain

scheduling scheme is rather straightforward and in this case, it was felt

that nothing new would be learned from the exercise.

5.4 Summary

This chapter has covered some of the more stressful maneuvers which

the 15 knot compensator performed. The 30 knot compensator exhibited

better behavior for less stressful maneuvers and at lower speeds where the

quick control surface movements did not excite any instabilities like it

did at higher speeds. As a result of this comparison, gain scheduling was

not used as it was felt that more could be learned from investigating the

effect of control surface saturation. The use of an Antireset Windup (ARW)

feedback loop is investigated in Chapter 6.

118

CHAPTER 6

IMPLEMENTATION OF ANTI-RESET WINDUP FEEDBACK

6. 1 Introduction

In Chapter 5, it became apparent that as the submarine is ordered to

do more difficult maneuvers, control surface saturation enters into the

picture, While the 15 knot compensator remained stable throughout the

testing, the 30 knot compensator did not. As was evident in the

simulations presented in Chapter 5, control surface saturation of the

fairwater planes occurred frequently and in some cases for long periods of

time. To decrease the time required for a control surface to come out of

saturation, an Anti-Reset Windup (ARW) feedback loop is installed. The

chapter begins with a discussion of the ARW feedback loop. This is

followed by an analysis of the impact of the ARW loop on the open loop and

closed loop systems. Finally, several simulations which demonstrate the

use of ARW are presented.

6.2 ARW Feedback Loop

As a result of prolonged control surface saturation, the system

experiences a phenomena known as integrator windup or reset windup. The

delay in the control surface or surfaces coming out of saturation after the

saturating condition is no longer in effect is caused by the integration of

the error between the ordered deflection and the saturation level. If the

119

error is large and/or the period of saturation is long, the integrated

error may also become large and quite significant delays can be

encountered. To counter this problem, a nonlinear feedback loop is

employed whose effect is to "turn off" the integrator for the saturating

actuator.

The use of ARW strategy in a MIMG control system is not fully

developed and the method employed here is based on C27] and personal

interactions with it3 authors, Xapasouns and Athan3. Figure 5.1 shows the

closed loop system with the ARW feedback loop installed around the

:Q—Ka-Q -ffll

:I 7L

bfl- G»

Figure 6.1 Structure of the Closed Loop System With ARW

integrators. The nonlinearity prior to the plant represents the saturating

control surfaces of the submarine, with the saturation limits as listed In

Table 3.1. When the control surfaces are not in saturation, the feedback

loop is "deenergized" through the "dead-zone" as shown. As a control

surface saturates, the feedback loop for that control channel energizes.

As a result of this feedback, the integrator is replaced by a lag network

as shown by (6.1). The amount of feedback used is a function of the

uCs) = (I/ 3 )CI +«(i/ s )3" 1 K(s) e(s)

= K(a) CI/(a «)] e<a) (6.1)

120

variable oc in the loop but the nee effect of the feedback is to reduce the

gain at low frequency. Figure 6.2 shows the affect of the parameter °< on

the Bode plot of a lag network. A3 <X i3 decreased, the behavior of the

100 o

Figure 6.2 Bode Plot of Siiaole Lag for Varying o<

loop approaches that for the integrator. As <* is increased, the gain at

low frequency decreases. At present, there is no known straighforward

method for selecting °<, especially for the MIMO case. To gain some insight

on a method of selection, several areas were examined.

Figure 6.3 shows the effect the ARW method has on the singular values

of the loop transfer matrix G(s)K(s) where G(s) now contains the dynamics

of the ARW feedback loop instead of the integrators as presented in Section

3.5. At low frequencies, the singular values have spread with two 3howmg

slight increases, one a slight decrease and the one associated with the

saturated control surface a drastic decrease. When o< was decreased by a

factor of 1/2 from 0.1 to 0.05, the minimum singular value increased by a

factor of 2 or 6 dB. There was little or no effect on the transfer

function relating reference commands to control inputs (4.30) or its

singular value decomposition. Finally, the effect on the singular values

121

3

C

t3»

O

©

lOO(alpha = 0.1)

.001 .01 .1 1


igure 6.3a Singular Value3 of G(s)K(s) for <* = 0.1

100

100(alpha = 0.05)

O>l.

c

o

o

.1 1 10omega (radians/second)

'igure 6.3b Singular Values of G(3)K(s) for o< = 0.05

100

122

of the cicsed loop system for c* = 0.05 is shown in Figure 6.4. Again,

spreading of the singular values occurs at low frequency and one would

expect problems in the channel with the -10 dB minimum singular value. For

this design then, the only noticabie effect is on the singular values and

not the singular vectors. Therefore, selection of the parameter o< appears

to be based on how much the minimum singular value of the unaugmented

system needs to be increased at low frequency to assure acceptable command

performance. In the next section, several simulations which shew the merit

of using an ARW feedback loop are presented.

6.3 Simulations for System With ARW Feedback

Using the ARW feedback loop as described in Section 6.2, several

simulations were made for various values of c<. Because the minimum

singular value of G(s)K(s) exibits such a large dependence on <X, it was

thought that to have adequate performance and robustness, c< would have to

be selected such that the value of the minimum singular value was at least

20 dB at low frequency, consistent with the low frequency barrier in Figure

3.6. 7hi3 proved not to be the case as an c< of 0.05 seemed to provide

adequate command following. This may only be sufficient for the series of

tests performed by the author. Additional testing may indeed indicate that

command following is not satisfactory for some combination of command

inputs. But that is a subject for further research.

Since saturation of the fairwater planes occurs most frequently, the

first two simulations deal with an ordered depthrate with zero commanded

pitch as this would easily cause saturation. A comparison of the closed

loop system response for a 2 feet/second commanded depthrate with and

123

100

a3o»

"5?

o»o

o

.001 .1 1


100

Figure 6.4 Singular Values of the Closed Loop for <* = 0.05

12^

without ARW feedback is shown in Figure 6.5. A "mild" saturation is

created by application of the reference command at t = 5 seconds. Noting

the change in scale on the graphs, we see that the responses remain

identical until t = 130 seconds when the reference command is zeroed. At

that time, sternplane and rudder deflections are the same but a slight

difference in the fairwater piane deflection occurs with the ARW simulation

displaying a slight overshoot. Review of the outputs show that the

oscillations have been effectively decreased and the output errors slightly

reduced in magnitude.

To demonstrate the effects of a prolonged "hard" saturation, another

simulation was made with a 4 feet/second commanded depthrate as shown in

Figure 6.6. There is a slight difference in the rudder and sternplane

deflections but an obvious improvement in fairwater plane response is

noted. At t = 130 seconds when the reference command is zeroed, the

fairwater planes for the system with ARW immediately deflect to reduce the

depthrate to zero while the fairwater planes for the 3y3tera without ARW do

not come out of saturation for another 110 seconds! A comparison of output

variables again shows that the ARW feedback loop has decreased oscillations

and more importantly, decreased the error by as much as a factor of 2 in

the case of depthrate and yawrate.

These simulations have 3hown that for a system which experiences

prolonged control surface saturation, the use of an ARW feedback loop will

decrease oscillations in the outputs and sometimes significantly decrease

the magnitude of the maximum error. Additionally, the use of the fairwater

planes as an effective dynamic control input is regained almost immediately

instead of waiting until the integrated error is nulled.

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6.4 Summary

This chapter has covered the use o£ an Anti Reset Windup feedback loop

to improve performance of the submarine in the presence of saturating

control surfaces. In particular, the effect on fairwater plane saturation

was demonstrated with notable improvements by reduction in the oscillation

and maximum error of the output variables and the time lag until dynamic

control of the saturating surface is regained.

136

CHAPTER 7

SUMMARY AND DIRECTIONS FOR FURTHER RESEARCH

7. 1 Summary

This thesis has presented the design of a multivariate control system

for a submarine using the Linear Quadratic Gaussian with Loop Transfer

Recovery (LQG/LTR) methodology. The submarine model was based on the NSRDC

2510 equations of motion modified to include the crossfiow terms of the

updated equations, differential control of the sternplanes and a constant

RPS constraint on, the propeller.

The 10th order linear model based on a nominal point of 15 knots

forward speed and 1 degree rudder deflection was reduced to an 8th order

system and then scaled for units and input/output weightings. Using

LQG/LTR, a Model Based Compensator was designed for a crossover frequency

of 0.1 radians/second for the plant augmented with integrators to reduce

steady state error to step inputs. The design was extensively tested at

speeds from 15 to 30 knots and then compared to the compensator design by

Martin C83 . That design is based on a 30 knot model of the same submarine

but has a crossover frequency of 0.5 radians/second.

Subsequently, an Anti-Reset Windup (ARW) feedback loop was

incorporated to reduce the effects of saturating control surfaces on the

dynamic response of the submarine. This is accomplished by replacing the

integrator in the input channel with a simple lag when that channel issues

a command above the saturation level of the control surface.

137

7.2 Conclusions and Directions for Further Research

The LQG/LTR multivariable control system design methodology has been

used to design a controller which considers rather than neglects the cross

coupling present in the dynamics of a submarine. This provides for better

control over a wider variety of maneuvers since the modei is vaiid for

larger perturbations than one which is based on decoupled vertical and

horizontal planes of motion. Active roil control through the differentia-

deflection of the sternplanes also helps by limiting perturbations in roll

angle during maneuvers, thereby limiting cross-coupling.

The use of modal analysis and singular vaiue decomposition have proven

to be an excellent means "of providing the designer with insight on the

states and control inputs which dominate the response of the submarine.

Additionally, the singular value decomposition of the transfer function

relating the reference commands to the control inputs Kr-(s), provides

important information on their coupling, i.e. what control inputs are

generated in the controller for a given reference command.

Comparisons of the 15 knot compensator designed in this thesi3 and the

30 knot compensator designed by Martin [8] indicate that a higher crossover

frequency is desirable for improved performance during mild maneuvers of

the submarine, but a lower crossover design is preferable for more radical

maneuvers. This was evident by the instability experienced during the turn

and dive maneuver by the system using the 30 knot compensator.

The effectiveness of an ARW feedback loop was demonstrated in a system

that experienced control surface saturation. It was determined that the

feedback parameter alpha must be selected such that the gain of G(s)K(s)

remains high enough to guarantee adequate command following. There was no

138

effect on the singular value decomposition of Kt-(s) and only a slight

effect on its singular values.

This thesis has provided an avenue for advancement in the area of

submarine control system design but prior to any actual system being built

and put to sea, much more research must be performed. This thesis and

Martin'3 have demonstrated the advantage of using differential deflection

of the sternpianes to provide active roll control for the submarine.

Additional research is required however, in the area of casualty control in

the event of a sternplane or other control surface failure. In Chapter 5,

it was shown that the 15 knot and 30 knot compensators could be operated at

speeds and operating points far away from nominal but thatthe severity of

the maneuver had to be restricted for the 30 knot compensator due to

instabilities. The reason for this unstable condition is mostly attributed

to the higher crossover frequency, but the effect of the higher nominal

speed must be investigated. The use of gain scheduling will obviously be

required for an actual controller design which effectively controls the

submarine at all speeds.

The singular value decomposition of the plant and the reference to

command input transfer function Kj-Cs), provided useful information about

the coupling of the system at low frequency, but the singular values and

singular vectors change as a function of frequency. Additional research in

this area will provide insight into how the coupling of the system changes

as a function of frequency.

The last area which requires further research is the use of Anti -Reset

Windup feedback. The improvement in the submarine's performance was

evident, but the method of selecting the feedback parameter o< needs

further development. Additionally, the effect of prolonged multiple

139

saturations while using as ARW feedbackloop and the effect, of the ARW loop

on closed loop stability must be addressed.

1^-0

REFERENCES

1. Kapasouris, P., "Gain-Scheduled Multivariable Control for the GE-21

Turbofan Engine Using the LGR and LQG/LTR Methodologies", Master'sThesis, MIT, Cambridge, MA, May 1984.

2. Kappos, P., "Robust Multivariable Control for the F100 Engine",

Master's Thesis, MIT, Cambridge, MA, September 1983.

3. Pfeil, W., "Multivariable Control for the GE T700 Engine Using theLQG/LTR Design Methodology", Master's Thesis, MIT, Cambridge. MA,

August 1984.

4. Harris, K., "Automatic Control of a Submersible", Masters' Thesis,

MIT, Cambridge, Ma, May 1984.

5. Milliken, L., "Multivariable Contron of an Underwater Vehicle",Engineer's Thesis, MIT, Cambridge, MA, May 1984.

6. Dreher, L., "Robust Rate Control System Designs for a

Submersible", Engineer's Thesis, MIT, Cambridge, MA, May 1984.

7. Lively, K., "Multivariable Control System Design for a Submarine",Engineer's Thesis, MIT, Cambridge, MA, May 1984.

8. Martin, R., "Multivariable Control System Design for a SubmarineUsing Active Roll Control", Engineer's Thesis, MIT, Cambridge, MA, May1985.

9. Abkowitz, M., Stability and Motion Control of Ocean Vehicles ,

MIT Press, Cambridge, MA, 1969.

10. Gertler, M. and Hagen, G., "Standard Equations of Motion forSubmarine Simulation", NSRDC Report 2510, Washington, DC, June 1967.

11. Feldman, J., "DTNSRDC Revised Standard Submarine Equations ofMotion", DTNSRDC/SPD-0393-09, Bethesda, MD, June 1979.

12. Bonnice, W. and Valavani, L., "Submarine Configuration andControl", Charles S. Draper Laboratory Memo SUB 1-1083, October 1983.

13. Bonnice, W. and Valavani, L., "Submarine Configuration andControl", Charles S. Draper Laboratory Memo SUB 2-1183, November 1983.

14. Bonnice, W., "Propulsion and Drag Models for the SubmodelSubmarine Simulation", Charles S. Draper Laboratory Memo SUB 3-984,September 1984.

15. Bonnice, W., CONSTRPS Fortran and Command List Computer Programs,Charles S. Draper Laboratory, May 1984.

lM

16. Athans, M., Class Notes for Course 6.232. MIT, Cambridge, MA,

Spring 1984.

17. Boettcher, K., "Analysis of Multivariable Control Systems withStructural Uncertainty", Doctoral Dissertation, MIT, Cambridge, MA,

1983.

18. Lehtomaki, N., "Practical Robustness Measures in MultivariableControl System Analysis", Doctoral Thesis, MIT, Cambridge, MA, May

1981.

19. Lehtomaki, N., Sandell, N. and Athans, M., "Robustness Results in

Linear-Quadratic-Gaussian Based Multivariable Control Designs",IEEE Transactions on Automatic Control , Vol.AC-26. No.l,

February 1981.

20. Safanov, M., "Robustness and Stability Aspects of StochasticMultivariable Feedback System Design", Doctoral Thesis, MIT,

Cambridge, MA, 1979.

21. Lehtomaki, N., et al, "Robustness Tests Utilizing the Structure ofModelling Error", LIDS Report P-1151, MIT, Cambridge, Ma, October1981.

22. Lehtomaki, N., et al, "Robustness and Modelling Error

Characterization", LIDS Report P-1311, MIT, Cambridge, MA, June 1983.

23. Doyle, J. and Stein, G., "Multivariable Feedback Design: Conceptsfor a Classical/Modern Synthesis", IEEE Transactions on AutomaticControl . Vol. AC-26, No.l, February 1981.

24. Stein, G., "LCG-Based Multivariable Design: Frequency DomainInterpretation", AGARD-LS-117, 1981.

25. Stein, G. and Athans, M., "The LQG/LTR Procedure for MultivariableFeedback Control Design", LIDS Report P-1384, MIT, Cambridge, MA, May1984.

26. Kwakernaak, H. and Si van, R. , Linear Optimal Control Systems,Wiiey, New York, NY, 1972.

27. Kapasouris, P., and Athans, M., "Multivariable Control Systemswith Saturating Actuators Antireset Windup Strategies", Draft of Paperfor American Control Conference, Boston, MA, June 1985.

1>2

APPENDIX A

MATRICES 0? THE S15R1 LINEAR MCOEL

143

APPENDIX Al

ORIGINAL PLANT MATRICES PRIOR TO SCALING AND 1QDEL REDUCTION

A MATRIX (10 x 10)

-1.9012E-02 -1.1096E-02 -4.2241E-04 -9.594tE-03 -1.6035E-02 1.5933E+00 O.OOOOE+00 2.9550E-04 O.OOOOE+OO O.OOOOE+00

5.3364E-04 -6.0771E-02 3.4469E-04 -6.46B2E-01 9.48B2E-02 -7.8164E+00 1.3L76E-01 -3.4021E-05 O.OOOOE+00 O.OOOOE+00

7.2953E-06 -7.3932E-05 -5.3862E-02 -B.0323E-01 6.1384E+00 9.7422E-03 O.OOOOE+OO 7.6175E-03 O.OOOOE+OO O.OOOOE+OO

1.1097E-04 -5.8726E-03 -7.1679E-04 -2.2072E-01 -1.2851E-01 -1.4504E-02 -1.6206E-01 4.1B45E-05 O.OOOOE+OO O.OOOOE+OO

-6.5774E-07 -2.2004E-06 6.7287E-04 -5.7594E-03 -2.0682E-01 1.4001E-04 O.OOOOE+00 -2.5124E-03 O.OOOOE+00 O.OOOOE+00

1.3843E-05 -1.0317E-03 3.7870E-06 -3.5813E-03 4.7420E-04 -1.9379E-01 2.6177E-04 -6.7592E-08 O.OOOOE+OO O.OOOOE+00

O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO l.OOOOE+OO 2.5811E-04 -9.1543E-03 -1.3327E-12 -6.2735E-03 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 9.9960E-01 2.8184E-02 6.2730E-03 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO -2.8186E-02 9.9965E-01 1.4553E-10 5.7451E-05 O.OOOOE+00 O.OOOOE+00

9.1576E-03 -2.B183E-02 9.9956E-01 O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO 7.9107E-01 -2.4B18E+01 O.OOOOE+OO O.OOOOE+OO

B MATRIX 14 x 10)

-4.2090E-04 -1.5065E-02 7.2244E-04 7.2244E-04

O.OOOOE+OO 5.9604E-01 -4.3727E-02 4.3727E-02

-3.7257E-01 -3.8218E-07 -2.5391E-01 -2.5391E-01

O.OOOOE+OO 1.0988E-02 5.3783E-02 -5.3783E-02

3.5786E-03 1.2605E-07 -6.1424E-03 -6.1424E-03

O.OOOOE+OO -1.5105E-02 -8.6874E-05 8.6874E-05

O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+00

O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00

O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO O.OOOOE+00

1*44

APPENDIX A2

MATRICES TO PERFORM UNIT TRANSFORMATIONS

Matrix used to premiltiply the A and B aatrices:

1.0O0OE+M O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 0.0OOOE+O0 O.OOOOE+OO 0.0OOOE+OO O.OOOOE+OO

O.OOOOE+OO l.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+O1 O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO

Matrix used to postaultiply the A latrix:

l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.74S2E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO- O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO

145

APPENDIX A2

Matrix used to postaultiply the B aatrix:

1.7452E-02 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00

0.0000E+00 1.74S2S-02 O.OOOOE+00 O.OOOOE+00

O.OOOOE+00 O.OOOOE+00 1.7452E-02 O.OOOOE+00

O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO 1.7452E-02

Matrix used to pree-ultiply the C aatrix:

5.7300E+01 O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO

Matrix used to postaultiply the C utrix:

l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO l.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO

146

a?pe:;d:x A2

PLANT MATRICES SCALED FOR UNIT TRANSFORMATION ONLY

A MATRIX (8 x 8)

-1.9012E-02

5.8364E-04

7.2953E-06

6.35B4E-03

-3.7689E-05

-7.9321E-04

O.OOOOE+OO

O.OOOOE+OO

-1.1096E-02

-8.0771E-02

-7.8932E-05

-3.3650E-01

-1.2608E-04

-5.9116E-02

O.OOOOE+OO

O.OOOOE+OO

•4.2241E-04

3.4469E-04

5.3862E-02

4.1072E-02

3.8555E-02

2.1699E-04

O.OOOOE+OO

O.OOOOE+OO

1.6744E-04

1.1288E-02

1.4018E-02

2.2072E-01

5.7594E-03

3.5813E-03

l.OOOOE+OO

O.OOOOE+OO

2.7984E-04

1.6559E-03

1.0713E-01

1.2B51E-01

•2.0682E-01

4.7420E-04

2.581 tE-04

9.9960E-01

2.7806E-02

1.3641E-01

1.7002E-04

•1.4504E-02

1.4001E-04

1.9379E-01

•9.1543E-03

2.8184E-02

O.OOOOE+OO

2.2994E-03

O.OOOOE+OO

1.6206E-01

O.OOOOE+OO

2.6177E-04

•1.3327E-12

6.2730E-03

5. 1570E-06

-5.9374E-07

1.3294E-04

4.1845E-05

-2.5124E-03

-6.7592E-08

-6.2735E-03

O.OOOOE+OO

B MATRIX (8 x 4)

7.3455E-06

O.OOOOE+OO

6.5021E-03

O.OOOOE+OO

3.5786E-03

O.OOOOE+OO

O.OOOOE+OO

O.OOOOE+OO

2.6291E-04

1.0402E-02

6.6698E-09

1.0988E-02

1.2605E-07

•1.5105E-02

O.OOOOE+OO

O.OOOOE+OO

1.2608E-05

•7.6312E-04

4.4313E-03

5.3783E-02

•6.1424E-03

•8.6874E-05

O.OOOOE+OO

O.OOOOE+OO

1.2608E-05

7.6312E-04

-4.4313E-03

-5.3783E-02

-6.1424E-03

8.6874E-05

O.OOOOE+OO

O.OOOOE+OO

1^+7

APPENDIX A2

C MATRIX (4 x 8)

O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO l.OOOOE+00 O.OOOOE+00

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE+00

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO -2.3185E-02 9.9964E-01 1.4552E-10 5.7450E-05

9.1576E-03 -2.8183E-02 9.9956E-01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.3806E-02 -4.3313E-01

148

APPENDIX A3

PLANT MATRICES SCALED FOR UNIT TRANSFORMATION AND WEIGHTINGS

ON INPUTS AND OUTPUTS

A MATRIX (8 x 8)

-1.9012E-02 -1.1096E-02 -4.2241E-04 -1.6744E-04 -2.7984E-04 2.7806E-02 O.OOOOE+OO 5.1570E-06

5.8364E-04 -8.0771E-02 3.4469E-04 -1.1288E-02 1 . 6559E-03 -1.3641E-01 2.2994E-03 -5.9374E-07

7.2953E-06 -7.8932E-05 -5.3862E-02 -1.4018E-02 1.0713E-01 1.7002E-04 O.OOOOE+OO 1.3294E-04

6.3584E-03 -3.3650E-01 -4.1072E-02 -2.2072E-01 -1.2851E-01 -1.4504E-02 -1.6206E-01 4.1845E-05

-3.7689E-05 -1.2608E-04 3.8555E-02 -5.7594E-03 -2.0682E-01 1.4001E-04 O.OOOOE+OO -2.5124E-03

-7.9321E-04 -5.9116E-02 2.1&99E-04 -3.5813E-03 4.7420E-04 -1.9379E-01 2.6177E-04 -6.7592E-08

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO 2.5811E-04 -9.1543E-03 -1.3327E-12 -6.2735E-03

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 9.9960E-01 2.8184E-02 6.2730E-03 O.OOOOE+OO

B MATRIX (8 x 4)

-7.3455E-06 -3.9417E-04 1.5760E-05 1.5760E-05

O.OOOOE+OO 1.5595E-02 -9.5390E-04 9.5390E-04

-6.5021E-03 -9.9997E-09 -5.5391E-03 -5.5391E-03

O.OOOOE+OO 1.6473E-02 6.7228E-02 -6.7228E-02

3.5786E-03 1.8898E-07 -7.6780E-03 -7.6780E-03

O.OOOOE+OO -2.2646E-02 -1.0859E-04 1.0859E-04

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO


1^+9

APPENDIX A3

C MATRIX (4 x 8)

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 l.OOOOE-Oi O.OOOOE+OO

O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE-Ot

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -2.3185E-02 9.9964E-01 1.4552E-10 5.7450E-05

9.1576E-03 -2.8183E-02 9.9956E-01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.3806E-02 -4.3313E-01

150

APPENDIX 3

MATRICES OF THE 15 KNOT PLANT AND COMPENSATOR

151

APPENDIX 3:

POLES, ZEROS AND EIGENVECTORS

EIGENVALUES

2.0413E-02 -2.0413E-02 -2.0618E-02 -3.3461E-02 -1.0603E-01 -1.0603E-01 -2.2040E-01 -2.4761E-01

1.4066E-02 -1.4066E-02 O.OOOOE+OO 0.0000E+OO 3.8084E-01 -3.8084E-01 0.0OOOE+OO O.OOOOE+OO

TRANS?1ISSI0N ZEROS

3.1712E+08 3.3004E+06 5.8381E+03 -1.9007E-02 -1.2729E-01 -5.3321E+03 -1.2904E+07 -7.9657E+08

EI6ENVECT0RS (MODAL MATRIX)

5.4585E-03 5.4585E-03 9.7809E-01 5.2178E-01 -4.3995E-05 -4.3995E-05 -5.4014E-04 3.3478E-02

2.1218E-03 -2.1218E-03 O.OOOOE+OO O.OOOOE+OO 2.9056E-04 -2.9056E-O4 O.OOOOE+OO O.OOOOE+OO

2.0006E-03 2.0006E-03 6.9651E-02 3.4612E-01 -7.5258E-04 -7.5258E-04 2.7841E-03 -3.7199E-01

1.8330E-03 1.8330E-03 O.OOOOE+OO O.OOOOE+OO 1.2856E-02 -1.2856E-02 O.OOOOE+OO O.OOOOE+OO

-1.1406E-02 -1.1406E-02 1.4151E-02 6.0006E-02 7.784AE-03 7.7846E-03 1.3399E-01 -4.6995E-03


-5.2527E-03 -5.2527E-03 1.0682E-03 2.2280E-02 3.4260E-01 3.4260E-01 -2.2220E-02 -1.9978E-01

3.3729E-03 -3.3729E-03 O.OOOOE+OO O.OOOOE+OO -1.3309E-01 1.3309E-01 O.OOOOE+OO O.OOOOE+OO

6.0737E-03 6.0737E-03 4.7564E-03 1.5018E-02 1.8075E-03 1.8075E-03 -2.1241E-01 -1.7366E-02


-5.5494E-04 -5.5494E-04 -2.8418E-02 -1.3178E-01 -1.9431E-03 -1.9431E-03 4.6092E-03 -4.2513E-01

4.9147E-04 -4.9147E-04 O.OOOOE+OO O.OOOOE+OO 3.0331E-03 -3.0331E-03 O.OOOOE+OO O.OOOOE+OO

7.7598E-04 7.7598E-04 -1.1242E-01 -7.3935E-01 -5.5690E-01 -5.5690E-01 1.2856E-01 7.9363E-01


•7.4935E-01 -7.4935E-01 -1.5755E-01 -1.9905E-01 1.9061E-03 1.9061E-03 9.5910E-01 9.8394E-02

6.5659E-01 -6.5659E-01 O.OOOOE+OO O.OOOOE+OO 4.0419E-03 -4.0419E-03 O.OOOOE+OO O.OOOOE+OO

152

APPENDIX 3:

CONTROLLABILITY MATRIX

1.6552E-01 1.0504E-01 3.5334E-01 3.6637E-01

-5.7600E-01 -1.8233E-01 -2.9672E-01 -3.0245E-01

1.6552E-01 1.0504E-01 3.5334E-01 3.6637E-01

5.7600E-01 1.8233E-01 2.9672E-01 3.0245E-O1

-6.8062E-03 -3.9264E-01 -5.6056E-03 -4.4330E-03

2.0B19E-10 -1.5146E-09 -3.3261E-09 3.2745E-09

4.0734E-03 7.1184E-01 1.7878E-03 -2.5278E-03

3.1612E-10 2.7899E-09 6.1340E-09 -6.1256E-09

2.3352E-02 1.2539E-01 4.3242E-01 -3.9234E-01

3.4091E-03 2.8794E-01 2.9887E-01 -3.2909E-01

-2.3352E-02 1.2539E-01 4.3242E-01 -3.9234E-01

-3.4091E-03 -2.3794E-01 -2.98B7E-01 3.2909E-01

-5.2958E-01 -2.3894E-02 1.4670E-01 1.5495E-01

-2.0413E-11 3.4676E-10 7.9478E-10 6.7265E-10

-5.4850E-03 2.2961E-01 -5.6294E-03 9.4034E-03

-9.9081E-12 2.1548E-11 4.4051E-11 -2.7408E-11

OBSERVABILITY MATRIX

7.7598E-04 7.7598E-04 -1.1242E-01 -7.3935E-01 -5.5690E-01 -5.5690E-01 1.2856E-01 7.9363E-01


7.4935E-01 -7.4935E-01 -1.5755E-01 -1.9905E-01 1.9061E-Q3 1.9061E-03 9.5910E-01 9.8394E-02

6.5659E-01 -6.5659E-01 O.OOOOE+OO O.OOOOE+OO 4.0419E-O3 -4.0419E-03 O.OOOOE+OO O.OOOOE+OO

7.6899E-04 -7.6899E-04 -2.8551E-02 -1.3217E-01 -1.9932E-03 -1.9932E-03 1.0649E-02 -4.2448E-01

1.2112E-03 -1.2U2E-03 O.OOOOE+OO O.OOOOE+OO 2.8946E-03 -2.8946E-03 O.OOOOE+OO O.OOOOE+OO

3.1317E-01 3.1317E-01 8.7825E-02 1.3101E-01 -7.1196E-04 -7.1196E-04 -2.7979E-01 -2.5567E-02

-3.5528E-01 3.5528E-01 O.OOOOE+OO O.OOOOE+OO -1.3537E-03 1.3587E-03 O.OOOOE+OO O.OOOOE+OO

153

APPENDIX E3

MODEL BASED COMPENSATOR MATRICES

L MATRIX (12 x 4)

-5.2859E-01 -3.6891E+01 -1.9800E-02 -8.9967E+00

3.1664E-03 5.6612E-02 -3.0290E+00 1.0072E-02

1.2335E+01 2.0142E+00 -4.8602E+00 7.1824E-01

-1. 1585E+01 -7.66B1E-01 5.2128E+00 1.1306E-01

-1.2643E-03 3.9664E-02 -4.5390E-16 9.1576E-03

3.B909E-03 -1.2207E-01 1.3969E-15 -2.8183E-02

-1.3800E-01 4.3294E+00 -4.9544E-14 9.9956E-01

0.0000E+00 0.0000E+00 0.0000E+00 O.OOOOE+00

4.1013E-11 1.M91E-05 -2.B183E-02 2.8272E-14

-1.4546E-09 -5.7425E-04 9.9956E-01 -1.0027E-12

1.0000E+01 6.4753E-11 1.0100E-18 4.3422E-U

2.9472E-10 1.0000E+01 8.5B22E-14 2.9325E-09

154

APPENDIX B3

MODEL BASED COMPENSATOR MATRICES

CONTROL 6AIKI MATRIX - G (4 x 12)

1.9075E+00 1.7309E-02 3.3175E-01 3.3927E-01 -2.2603E+00 6.3586E+00

2.3374E+02 6.0251E-02 1.1518E+02 2.9814E+O0 -2.4295E+00 1.2755E+02

1.7309E-02 3.6408E+00 2.7069E-02 -5.3234E-02 8.8669E-02 9.1610E+00

•4.3139E+00 1.4955E+00 7.0839E+00 -2.8537E+02 1.26&7E+00 1.3071E+00

3.3175E-01 2.7069E-02 2.2008E+00 -4.1713E-01 -1.1099E+00 -3.4200E+00

1.3371E+02 2.5322E+01 -1.5522E+01 1.1943E+01 1.736BE+01 4.0942E+01

3.3927E-01 -5.3234E-02 -4.1713E-01 2. 1780E+00 -1.3532E+00 1.0440E+01

1.3407E+02 -2.5060E+01 -1.0521E+01 -3.B677E+00 -2.1052E+01 4.189BE+01

FILTER 6AIN HATRIX - H (12 x 4)

-5.9202E-02 -3.2113E+00 -3.B250E-02 -7.79B3E-01

3.2510E-02 8.4010E-03 -2.6152E-01 -3.6334E-03

1.1180E+00 1.8450E-01 -2.8643E-01 1.646BE-02

-1.0548E+00 -7.6073E-02 3.2537E-01 5.3966E-02

9.5416E-04 2.0790E-03 3.5B01E-02 7.4520E-04

1.9485E-03 -5.3268E-03 -B.6929E-02 -3.5640E-03

-1.0162E-02 3.76B9E-01 1.0342E-03 8.7100E-02

5.2396E-03 6.0975E-03 -1.4220E-02 4.6202E-04

-3.9137E-03 -3.B591E-05 -1.9355E-03 B.6603E-05

-6.6970E-03 3. 1A29E-04 6.5711E-02 1.3147E-03

9.2545E-01 1.3114E-02 -6.5B35E-02 -3.1072E-02

1.3114E-02 B.7089E-01 3.6730E-03 -1.2924E-03

155

APPENDIX B4

POLES AND ZEROS OF THE OPEN AND CLOSED LOOP SYS7EKS

OPEN LOOP POLE:

(1 x 24)

PE£L PART

7. 1012E-09 i.3i4QE-09 e.0203E-iO a.020SE-10 -1.P007E-02 -2.0413E-42

-2.0*13E-02 -2.0ai=E-v2 -3.3461E-02 -L.0603E-01 -:.0cv3E-vt -i.272?E-01

-2.2040E-01 -2.47elE-01 -3.sOS3EH)i -3.a083c-0i -7.0970E-0! -7.0970E—)1

-7. ?225E-01 -1.3002E-00 -i.3002rX>O -i.cOOOE-OO -i.?273E-H>0 -1.P273E-00

IIWSI.NARY PART

O.OOOOE-00 O.OGOOE-HJO 5.ai4S£-09 -5.al4fiE-39 Q.3000&MJ0 1.4CSdE-02

-i.406iE-v2 0.0000=^)0 Q.QOOOS-HX) 3.3084SHJI -3.3vS4E-)i O.OOOOE^OO

0.0OOOE-00 O.OOOOE-rOO B.0017E-01 -a.OOlTE—>I 1.323SE-00 -L323SE-00

tOOE+OQ -1.3200E-HJ0 0.QCOQE+O0 i.'?220E-00 -t.?22OE~0O

OPEN LflO? ZEROS

(i : 24)

9.2849E+02 9.224PE-02 I.2343E*02 1.2343E-02 i.3i37E*)C L.5099E+O0

•i.?0O7E-32 -i.?IaiE-$2 -2.04O4E-02 -2.0404E-O2 -5.7I43E-02 -l.MME-M-1.M19EHU -1.2725E-01 -2.0323E-O1 -2.2075E-01 -1.2357E-HJ2 -i.2357E*<S

•i.44a3Er03 -i.*663c*03 -2.-0371Ei03 -4.?i9SE*03 -4.?!<?SE-03 -7.2S51E+Q4

ItlASI.NASY PART

1.7100E-03 -i.7I00E*03 1.2349E-02 -1.2349E+02 7.6mE*M -T.ii?lE-04

O.OOOOE-KW 0.3000S-00 1.4944E-02 -1.4044EH52 0.0O00E-KJ0 3. 74?0£-^H

-3.7490E-01 0.9000E-00 O.OOOOErOO O.OOOGE^OO 1.234PE-02 -i.234=E-02

2.3463S-H)3 -2.34a3E*03 O.OOOOE-00 i.a495£«)4 -l.a4?5E*04 0.0OOCE*)O

156

APPENDIX B4

POLES AKD ZEROS OF THE OPEN A KD CLOSED LOOP SYSTEMS

pi ncrr, t rmo s™ -5

(1 : 24)

REAL PART

L.S917E-02 -J.9007E-Q2 -2.Q299E-02 -2.0299E-02 -4.2237E-02 -4.2237E-02

-B.5722-02 -5.7075E-02 -9.3104E-02 -i.07A0E-0i -1.0740E-01 -I.2729E-01

-2.2005E-01 -2.e290E-01 -3.542SE-01 -3.342SE-01 -7.07I4E-Q1 -7.0714E-01

-7.1471E-0! -1.2584E+00 -1.2584E+0O -1.4I26E+0C -1.S944E+00 -1.B944E+00

0.0000E+O0

0.000GE+00

O.OOOOE^OO

O.OOOOE---00

Q.0000S+00

O.OOOOE+00

O.OOOOE+00

1.2795E+C0

IHAelMABY PART

I.4076E-02 -1.407&E-02 3.2491E-02 -3.2491E-02

O.OOOOE+00 3.62C2E-01 -3.8202-01 0.0000E+00

5.5033E-01_c r

033E-01 1.2751E+00 -1.2751E-C0

1.2795E+00 O.OOOOE+00 00 -1.BS98E+00

CLEEED LAPP ZEROS

(1 s 24)

REAL PART

1.9711E+03 9.7852E+01 9.7852E+01 -1.9007E-02 -1.9161E-02 -2.0403E-02

-2.0403E-02 -5.7143E-02 -l.MME-01 -1.0619E-01 -1.2729E-01 -2.0323E-01

-2.2C75E-01 -3.1904E+01 -3.1904E+01 -9.9733E+01 -9.9733E+01 -1.JC79E+02

-1.1079E+02 -1.799EE+03 -1.799SE+03 -1.2192+04 -1. 21921+04 -B.623SE+C4

MASHtftgY PART

-9.9187E+C2 9.8791E+01 -9.B791E+01 O.OOOOE+00 O.OOOOE+00 1.4O43E-02

-1.4O43E-02 0.0OO0E+00 3.7490E-01 -3.7490E-01 O.OOOOE+00 0.000OE+O0

O.OOOOE+00 2.1934£t03 -2.1984E+03 9.S791E+01 -9.6791E+01 2.7447E+04

-2.7447E+04 1.1325E+03 -1.1325E+03 2.3498E+04 -2.3498E+04 O.OOOOE+00

157

APPENDIX C

MATRICES OF THE 30 KNOT PLANT AND COMPENSATOR

158

APPENDIX CI

S

T

A7Z SPACE MATRICES OF THE S30R1 LINEAR MODEL

ORIGINAL MATRICES PRIOR TO SCALIN6

A BATRII

-3.3245E-02 -2.1911E-02 -2.7720E-03 -1.3964E-02 -2.9363E-01 3.1674E+00 0.0OOOE+OO 2.9326E-04 O.OOOOE+OO O.OOOOE+OO

1.1461E-03 -1.3919E-01 -1.933SE-03 -1.1464E+00 1.1276E-01 -1.5397E+01 1.3004E-01 -1.7564E-03 0.0000E+OO O.OOOOE+OO

2.4222E-05 4.4499E-04 -1.0631E-01 -1. 39846*00 1.2070E+01 3.0t94E-02 O.OOOOE+00 7.5597E-03 O.OOQOE+00 0.00OOE+OO

2.4614E-04 -I.1680E-02 -1.3226E-03 -4.344SE-01 -2.3879E-01 -7.1773E-03 -1.5995E-01 2.1603E-O3 O.OOOOE+OO O.OOOOE+00

-5.3732E-06 -1.3585E-05 1.3207E-O3 -1.1380E-02 -4.0755E-01 1.0074E-04 O.OOOOE+00 -2. 4934E-03 O.OOOOE+OO O.OOOOE+00

-2.7564E-03 -2.0277E-03 2.4063E-05 -8.1034E-03 3.4042E-03 -3.3180E-01 2.583AE-04 -3.4895E-06 O.OOOOE+00 O.OOOOE+00

O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO 1.0000E+00 1.3427E-02 -1.2348E-01 -2.0244E-10 -1.26A0E-02 O.OOOOE+OO O.OOOOE+00

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 9.9414E-01 1.08IOE-01 1.2467E-02 O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -1.0893E-01 I.0018E+00 1.4423E-09 1.5405E-03 O.OOOOE+00 O.OOOOE+00

1.2326E-01 -1.0728E-01 9.36S4E-01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.5702E+00 -4.3493E+01 O.OOOOE+00 O.OOOOE+OO

8 BATRII

-1.4315E-03 -5.3396E-02 2.3022E-03 2.3022E-03

O.OOOOE+00 2.3119E+00 -1.4950E-01 1.4950E-01

-1.4442E+00 -1.4813E-0A -9.3474E-01 -9.3476E-01

O.OOOOE+OO 4.2586E-02 2.0848E-01 -2.0848E-01

1.3872E-02 4.8862E-07 -2.382SE-02 -2.3823E-02

O.OOOOE+OO -5.3593E-02 -3.3476E-04 3.2676E-04

O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 O.OOOOE+00


O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00

O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 O.OOOOE+00

159

APPENDIX CI

STATE SPACE MATRICES OF THE S2QR1 LINEAR MODEL

REDUCED AND SCALED PLANT MATRICES WITH APPROPRIATE C MATRIX

A MATRIX

•3.3269E-02 -2.1964E-02 -2.7533E-03 -3.3173E-04 2.0734E-03 5.5394E-02 O.OOOOE+00 5. 12S5E-0&

1.1417E-03 -1.5939E-01 -3.3786E-05 -2.3578E-02 2. S353E-03 -2.6860E-0L 2.2745E-03 -2.5914E-05

-4.7476E-04 1.3910E-03 -9.4526E-02 -2.7949E-02 2.1163E-01 7.&140E-04 O.OOOOE+00 1.3221E-04

1.3945E-02 -6.6430E-01 -8.0931E-02 -4.3452E-01 -2.S262E-01 -2.1920E-O2 -1.6030E-01 1.3264E-03

7.1418E-05 -2.5929E-04 7.3U7E-02 -1.1406E-02 -4.0815E-01 -7.7327E-04 O.OOOOE+00 -2.4985E-03

-1.5782E-03 -1.1622E-01 3.4035E-04 -8.00IIE-03 2.2809E-03 -3.3201EH31 2.5893E-04 -2.9501E-06

O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 1.0000E+OO 1.1328EH32 -1.0538E-OI -4.93S2E-10 -1.2635E-02

0.0O00E+0O O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO 9.9427E-01 1.0689E-01 1.2494E-02 O.OOOOE+OO

3 HATRII

-I.2666E-03 -1.3279E-03 9.3625E-05 9.3625E-05

O.OOOOE+OO &.0491E-02 -3.6976E-03 3.6976E-03

-2.5204E-02 -3.3763E-08 -2.1483E-02 -2.1483E-02

O.OOOOE+OO 6.3847E-02 2.6060E-01 -2.6060E-01

1.3873E-02 7.3256E-07 -2.9781E-02 -2.9781E-02

O.OOOOE+00 -3.784AE-02 -4.2094E-04 4.2094E-04

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+00


160

APPENDIX CI

SPACE MATRICES 07 THE S30R1

C HATCH

O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE-Ol O.OOOOE+00

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE-Ol

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -1.0749E-01 9.9984E-01 4.4827E-09 1.3316E-03

1.0539E-01 -1.0629E-01 9.3873E-01 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 2.7U9E-02 -8.4843E-01

161

APPENDIX C2

MATRICES 07 THE 30 KN'OT CC.M.PSSSAT33

FILTH SAIN MATRIX

-1.02B2E+00 -1.3198E+01 -3.1019E-01 -2.1680E+00

1.I923E-01 7.9327E-02 -7.5141E-01 2.3994E-02

1.9031E+00 1.76ME+Q0 -3.6529E-01 1.5857E-01

-1.4904E+00 4.2619E-01 1.2014E+00 1.141SE-01

9.3005E-03 4.4870E-01 9.2493E-HJ2 5.4981E-42

6.3777E-03 -3.4783E-0I -2.43546-01 -3.3983E-42

2.4814E-02 4.1892E+O0 2.1197E-01 4.9849Eh)1

LS704E-01 2.4765E-01 -3.5220EHJI 4.9200E-02

-3.0473E-02 -4.2343E-03 -3.1907EHJ2 3.5337E-44

-4.2073E-02 2.1331E-02 3.3510EHJI -4.2349E-03

5.343IE+00 2.1792E-01 -3.3301E-01 -1.5210E-01

2.1792E-01 4.9887E+00 2.3426E-01 -4.34I4Eh)3

CONTROL SAIN MATRIX

1.3487E+00 L4722EHH -I.8437E-03 3.2330E-03 -2.3935E+00 2.1865E+00

-2.3081E+01 4.6679E-02 2.3366E+0I 2.0238E+00 -2.3789E-01 2.5951E+01

1.4722E-03 2.0983E+00 -1.1733E-02 -4.2&71E-02 -4.7100E-02 2.5212E+00

•7.4947E-01 1.90S3E-01 L7016E+00 -2.3I96E+01 3.3982E-02 3.4430E-01

1.3437E-03 -1.I733E-02 1.3132E+00 -3.5292E-01 -7.9515E-01 -1.3192E+00

•3.6155E+00 2.2982E+00 -1.9369E+00 1.4596E+00 1.4934E+00 4.3954E+00

3.2330E-03 -4.2471E-02 -3.5292E-01 1.2835E+00 -9.4105E-01 2.7612E+00

-3.7112E+00 -2.5731E+00 -5.3521E-01 6.3046E-01 -1.9771E+00 5.4309E+00

162

APPENDIX C2

MATRICES OF THE 30 KNOT COMPENSATOR

L HATRIX

-a.0839E-Ol -3.M79E+01 5.i433E-0l -4.3555E+00

1.5610E-02 B.7114E-02 -1.5284E+00 &. 4870E-03

3.3627E+00 3.5527E+00 -2. 4355E+00 4.4871E-01

-2.6540E+00 8.S502E-01 2.7545E+00 1.5325E-01

-2.3532E-02 8.9419E-01 S.1547E-14 1.0S39EHH

2.8826E-02 -9.0182E-01 -8.2242E-14 -1.0629E-01

-2.6814E-01 8.3888E+00 7.4502E-13 9.3S73E-01

O.OOOOEtOO O.OOOOE^OO O.OOOOE+00 O.OOOOE+00

4.9774E-09 1.4154E-03 -1.0629E-01 -4.4369E-13

-4.6300E-08 -1.3166EHJ2 9.3873E-01 4.1272E-12

i.OOOOE+Ol -1.4691E-09 -3.3444E-17 1.2488E-11

-1.3286E-10 l.OOOOE+Ol 6.9211E-12 -5.5841E-09

OPEN LOOP EIGENVALUES

9.5843E-09 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -1.4176E-02 -3.3412E-02

•4.4084E-01 -4.4084E-OI -4.5U4E-01 -5.0484EhJI -5.3965E-01 -3.8965E-01

-4.0661E-02 -4.2B86E-02 -7.1364E-02 -1.9689E-01 -1.96B9E-01 -2.5114E-01

9.5148E-01 -9.S148E-01 -1.01O4E+OO -1.3887E+00 -1.3887E+00 -1.4462E+00

O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00

4.7901E-01 -4.7901E-01 O.OOOOE+00 O.OOOOE+00 1.1366E+00 -i.i34aE+00

O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO 3.1300E-01 -3.1300E-01 O.OOOOE+OO

1.0148E+00 -1.0148E+00 O.OOOOE+00 1.3186E+00 -1.31S6E+00 O.OOOOE+00

163

APPENDIX C2

MATRICES OF THE 20 KNOT COMPENSATOR

OPEN LOOP TRANSMISSION ZEROS

1.0000E+30 1.0000E+30 1.0000E+30 1.0000E+30 1.1625E+08 1.0192E+08

•3.9767E-02 -3.9767E-02 -1.9869E-01 -2.0469E-01 -2.0469E-01 -2.5097E-01

1.2816E+05 1.4260E+04 1.2133E+00 1.2202E+00 -1.425AE-02 -3.8414E-02

2.3357E-01 -4.5944E-01 -1.4263E+04 -1.2816E+05 -8.4484E+07 -5.303BE+09

O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00

1.6S40E-03 -1.M40E-03 O.OOOOE+00 2.3343E-01 -2.3343E-01 O.OOOOE+00

O.OOOOE+00 O.OOOOE+00 1.0851E+04 -1.0851E+04 O.OOOOE+00 O.OOOOE+00

O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00

. CLOSED LOOP EIGENVALUES

-1.4220E-02 -3.3414E-02 -3.8479E-02 -4.1399E-02 -1.0317EHJI -1.0317E-01

5.1064E-01 -5.10ME-01 -5.15UE-01 -5.2355E-01 -5.2355E-01 -7.1513E-01

-2.3970E-01 -2.3970E-01 -2.5106E-01 -2.7292E-01 -2.7292E-01 -4.1228E-01

-7.4533E-01 -7.5232E-01 -7.5232E-01 -1.0388E+00 -1.1957E+00 -1.1957E+00

O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.0A59E-01 -1.0659E-01

7.04OIE-O2 -7.0401E-02 O.OOOOE+OO 9.2146E-01 -9.2144E-01 O.OOOOE+00

3.3649E-01 -3.3449E-01 O.OOOOE+OO 2.7737E-01 -2.7737E-01 O.OOOOE+OO

O.OOOOE+00 8.5953E-01 -3.5953E-01 O.OOOOE+00 1.1439E+00 -1.1639E+00

CLOSED LOOP TRANSHISSION ZEROS

7.5203E+10 I.1350E+04 8.i830E+03 6.7422E+03 2.5300E+02 2.5300E+02

-3.9768E-02 -3.9768E-02 -1.98&9E-01 -2.0469E-01 -2.0469E-01 -2.5097E-01

L5900E+00 U4499E+00 2.3446E-01 7.5507E-0I -1.4253E-02 -3.84I4E-02

2.3357E-01 -4.5944E-01 -5.0698E+02 -6.7412E+03 -8.4823E+03 -1.1349E+04

O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 4.3878E+02 -4.3878E+02

1.A527E-03 -1.M27E-03 O.OOOOE+00 2.3343E-01 -2.3343E-01 O.OOOOE+00

2.4470E+05 -2.4470E+05 1.5450E+04 -1.5450E+04 O.OOOOE+OO O.OOOOE+00

O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

164

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