H-infinity control : an exploratory study

H-infinity control : an exploratory study

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van den Boom, A. J. J. (1988). H-infinity control : an exploratory study. (EUT report. E, Fac. of Electrical Engineering; Vol. 88-E-211). Eindhoven University of Technology.

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Hoo

-Control:

An Exploratory Study

by

A.J.J. van den Boom

EUT Report 88-E-211 ISBN 90-6144-211-7

ISSN 0167- 9706

Eindhoven University of Technology Research Reports

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering

Eindhoven The Netherlands

Hoo-CONTROL:

An exploratory study

by

A.J.J. van den Boom

EUT Report 88-E-211

ISBN 90-6144-211-7

Eindhoven

This report is submitted in partial fulfillment of the requirements for the degree pf electrical engineer (M.Sc.) at the Eindhoven University of Technology.

The work Was carried out from September 198? untill August 1988 in charge of ~ofessor Dr.Ir. P. Eykhoff under super-vision of Dr.Ir. A.A.H. Damen.

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Boom, A.J.J. van den

H~-control: an exploratory study / by A.J.J. vanden Boom.

-Eindhoven: Eindhoven University of Technology, Faculty of

Electrical Engineering. - Fig. - (EUT report, ISSN 0167-9708, 88-E-211)

Met lit. opg., reg. ISBN 90-6144-211-7

5I50 656.2 UDC 519.71 NUGI832

SUMMARY

The ~ control theory is a new technique , pioneered by Zames in

1981, for optimizing controllers of linear multivariable

feedback systems. A reason for using Hoo~optimization is that the

approach is more effective than linear quadratic control with

respect to plant variations and disturbances, and therefore leads to robustly stabilizing controllers.

Many control problems can be rephrased as follows: Find a real

rational proper controller that stabilizes the plant and

minimizes the Hoo-norm of some transfer function. Examples of such functions are the sensitivity. the complementary sensitivity, the

control sensitivity, the signal tracking error, or a mixed

criterion.

In another class of control problems we requIre

compensator stabilizes the plant under condItion

that that

the the

weighted model-error (additive or multiplIcative) has a Hoo-bound

the robust stability problem).

A theory to find the optimal controller IS presented. In WhICh we

need some mathematIcal tools as coprime factorization, spectral

factorization and optimal Hankel approxImation. This will result

in a design procedure. We discuss the implementation via

algorithms, using state-space description.

One problem in the proposed design procedure is the appearance of jw-axis zeros or poles and strictly proper plants. Two techniques

are described to calculate an approximant for the ~-optimal

contro 11 er .

Finally we present a case-study: we design three different

controllers for a laboratory process (ball-balancing system) ,i.e.

minimizing the signal tracking error, maximizing the robustness

and minimizing the mixed sensitivity (a combination of both). We

show the different steps in the design procedure and interprete

and compare the properties of each controller.

Boom, A.J.J. van den

Hw-CONTROL: An exploratory study.

Faculty of Electric~l Engineering, Eindhoven University of Technology,

19BB.

EUT Report BB-E-2ll

Address of the author:

ir. A.J.J. van den Boom,

Measurement and Control Group, Faculty of Electrical Engineering, Eindhoven University of Technology,

P.O. Box 513, 5600 MB EINDHOVEN,

CONTENTS

Summary Contents

List of used symbols Preface

O.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1. 1.1 1.2 1.3 1.4 1.5 1.6 2. 2.1 2.2 2.3 2.4 2.5 3. 3.1 3.2 3.3 4. 4.1 4.2 5. 5.1 5.2 6. 6.1 6.2 7. 7.1 7.2 7.3 7.4 8. Mathematics

Algebraic and Topological preliminaries Function spaces

Transfer functions

Operations on linear systems

Controllability and Observability

Hankel-matrix,Hankel-operator and Hankel-norm Properties of the Hoo-norm

Introduction

Some examples of control problems The Standard problem

The Model-matching problem The Nehari problem

Design procedure Example

Coprime factorization Introduction

Stabilizing controllers

Closed-loop Transfer matrices State-space realizations

Linear fractional transformations Inner matrices

Introduction

with Hoo-criteria

Transformation to the extended Nehari problem The minimal realization

The i-block Nehari problem Introduction

State-space realization

Reduction of the extended Nehari problem Reduction with spectral factorization

State-space realizations for spectral factorizations Zero's and poles on the jw-axls

Bilinear conformal mapping in the s-plane Cancelling zeros with mixed sensitivity

weighting functions Example: The ball balancing system

Introduction

Minimizing the signal tracking error Maximizing the robustness

Minimizing the mixed sensitivity Conclusions and Remarks

Appendix References 1 2 3 4 4 6 8 9 12 14 16 17 17 21 28 29 30 34 39 39 39 43 44 50 53 53 56 60 73 73 75 78 78 84 87 87 89 90 90

95

102 109 121

LIST OF USED SYMBOLS

R C L. (-IX>, IX» L. (j R, Crnxn ) fu (j R, Crnxn )

Lm

(j R, Cmxn )

lim

( j R, Crnxn ) prefix R prefix B

II' II

11'11'

II' II '"

superscript J. A'

A*

A-l

A+ 4>*

field of real numbers field of complex numbers time-domain Lebesgue space

frequency-domain Hilbert Lebesgue space frequency-domain Hilbert Hardy space frequency-domain Banach Lebesgue space frequency-domain Banach Hardy space real rational

the unit-ba 11

norm on Crnxn largest singular value

norm on L. (-"',"') norm on L",(jR,Crnxn ) orthogonal complement transpose of matrix A

complex-conjugate transpose of matrix A inverse of A

generalised inverse of A ( i.e. pseudo-inverse) adjoint of operator 4>

unit step function unit impulse

singular value

largest singular value eigenvalues

largest eigenvalue

Preface

A new approach in the control-theory is the Hoo-optimization.

Pioneered by Zames [1J many researchers developed useful

techniques to find an optimal Hoo-controller .

This report is aimed to presenting an introduction of the

techniques. that uses state-space representation. and concluding

with a case-study.

There will be readers. who are not familiar with the mathematical

tools. used in the literature about Hoo-control. Therefore we

give in chapter 0 a short introduction in the elementary concepts of the algebra and functional analysis. used in this literature.

Chapter 1 gives a framework of the design-procedure for finding

the optimal controller. Chapter 2 untill 6 gives a detailed

elaboration of all the different steps in this design-procedure.

The case-study. the design of three different controllers for a

laboratory process. is presented in chapter 7.

The author likes to thank A.Damen. J.Ludlage and M.Klompstra for

their help and their support to establish this report.

o

MATHEMATICS

In this chapter we will briefly introduce some mathematical tools that are used during the design of Hm-controllers.

0.1 Algebraic and Topological preliminaries This section is

Vidyasagar[5]. A Desoer [29] .

an abstract of appendices A to C few definitions are from Kreyszig[28]

from and

0.1.1 Rings.Fields and Ideals

Definition; A Ring is a nonempty set R together with operations +(addition) and (multiplication) such following axioms are satisfied;

(R1) (R,+) is a commutative group. This means that

a + b ER , for all a,b ER

a + ( b + C ) = ( a + b ) + c, for all a,b,c ER

a + b = b + a, for all a,b ER

There exists an element OER such that a + 0

=

0 + a = a, for all aER

two binary that the

For every element aER, there exists a corresponding element -aER such that a + (-a)-

o.

(R2) (R,') is a semigroup. This means that

a·b ER for all a,b ER

a' (b·c)

=

(a·b)·c, for all a,b,c ER.

(R3) Multiplication is distributive over addition. This means that

a' (b + c) = a· b + a' c,

(a + b)·c

=

a·c + b·c , for all a,b,c,ER.

A ring is said to be commutative if a·b - b·a ,for all a,b ER , and is said to have an identity if there exists an element 1ER such that l·a

=

a·1

=

a, for all aER.

Suppose R is a commutative ring with an identity. An element xER is called a unit of R if there exists a yER such that x·y = y·x

-1. It can be easily shown that such a y is unique; y is called the inverse of x and is denoted by x- 1 .

Definition; A field is a commutative ring F with an identity, satisfying two additional assumptions;

(Fl) F contains at least two elements. (F2) Every nonzero element of F is a unit.

A subset S of a ring R is called a subring if it is a ring in its own right.

A subset I in a ring R is said to be left ideal i f I is a subgroup of the additive group of R, and aEI , xER imply that x·aEI . I is a right ideal if I is a subgroup of the additive group of Rand aER, xEI imply that a·xEI . I is an ideal if it

is both a left ideal and a right ideal. 4

The set of all elements of the form x-aERo i.e. the set of all left multiples of a. is called the left principal ideal generated

by a. Similarly. the right principal ideal generated by a is the

set of all elements a-x where xER.

In the following definitions and facts "ring" means a commutative ring with an identity.

A ring R is said to be a principal ideal ring if every ideal in R is principal.

If x and yare elements of a ring R with x~O • we say y is called

a multiple of x . if there is an element zER such that y=x-z;

this is denoted by xl y . If x and yare elements of a ring R such

that not both are zero. a greatest common divisor (g.c.d) of x.y

is any element dER such that (GCD1) dlx and dly

(GCD2) it clx . Cly then cld

Fact: Suppose R is

elements x.yER. not can be expressed in

d = Pox + q-y

a principal both of which the form

for appropriate elements p.qER.

ideal ring. Then every pair of

are zero. has a g.c.d. d which

Two elements x.y are called relatively prime or simply coprime if every g.c.d of X.y is a unit. so there are appropriate elements p

and q such that pox + q-y - 1 (This is the diophantine or

Bezout equation) .

Note that because the commutativity also holds x-p + y-q = 1

which is contrary to a non-commutative ring with an identity

Two elements x.y are called right-coprime if there are

appropriate elements p and q such that Pox + q-y

=

1

Two elements x.y are called left-coprime if there are

appropriate elements p and q such that x-p + y-q = 1

0.1.2 Topological spaces

Definition: Let S be a set. A collection T of subsets of S is a topology i f

(TOP1) Both Sand 0 (the empty set) belong to T.

(TOP2) A finite intersection of sets in T again belongs to T. (TOP3) An arbitrary union of sets in T again belongs to T.

Suppose R is a ring and T is a topology on R. The pair (R.T) is a

topological space if the functions(x.y) -) x-y and (x.Y) -) x-y

are continuous functions from RxR into R. when RxR is given the

product topology.

Suppose A is an space over R (the

complex numbers). Then ( A,II'II) is a

define a norm

11'11

on A such t.hat

real numbers) or C (the

normed space if one can

(NS 1 )

II

all 1 0 f or a II aEA, II a II

=

0 <

=

> a

=

0 (NS2) lIaall - lal'lIall for all aER (or C) and all aEA

(NS3) lIa+bll i lIall + IIbll for all a,b E A

real numbers) inner product

or C (the space if

Suppose A is an space over R (the

comp lex numbers). Then ( A,

<, "

»

is a one can define a inner product on A such

(I PSI) < a, a> 1 0 f or a 11 a, b EA,

<

a, a>

that

=

0

<a>

a - 0 (IPS2) <aa,b> = a'<a,b> for all aER (or C) and all a,b E A (I PS3) < a + b, c> a < a, c > + < b, c > f or a 11 a, b, c E A

*

(IPS4) <a,b) = <b,a) for all a,b E A.

Fact: A inner product space is always a normed space where the

inner product induces the norm: < a, a) :

=

II

a II'

A space A is said to be complete if every convergent sequence in

A has a limit which itself is an element of this space. We deal with the following special spaces:

Banach spaces are complete normed spaces.

Hilbert spaces are complete inner product spaces,

0.2 Function The theory Doyle[2J and Spaces

in

the sections Francis [4J .

0.2 0.5 is mainly taken from

R is real axis, jR is imaginary axis, C is complex plane.

a[

A J is the largest singular value of matrix A. Definition:

For each pE[I,m) we define the Hardy-space Hp as

+ mxn Hp := ( f: C -) C I sup 0)0 m

[ I :

f (o+jw) : p dw J lip < m } -m

For each pE[I,m) we define the Lebesgue-space

Lp

as

m

Lp . = ( f: jR -) Cmxn

I

0 :

f (jw) : p dw J lip

<

m )

Continuous time domain :

L. (-m,co) Lebesgue space of squared integrable vector-valued function on R, with inner product

co

<

_{f.g ) =}

J [

f (t) • get) * ] dt

-co

L. [0, co)

subspace of L.-functions zero for almost all

Lo (-m,o] subspace of L.-functions zero for almost all

Continuous frequency domain

t<o t )0

Hilbert space of matrix-valued function on jR,

with inner product

co

< F,G )

=

~ ~coJ

trace [ F(jw)*'G(jw) ] dw

subspace of L.-functions F(s) analytic in Re s)O and satisfying

co

sup

J

trace [ F(o+jw)*'F(o+jw) ] dw < co

0)0

-H. (jR,~n)l.: subspace of L.-functions F(s) analytic in Re s<O and satisfying

co

sup

J

trace [ F(o+jw)*'F(o+jw) ] dw < co

0<0 -co

Banach space of (essentially) bounded matrix-valued functions on jR , with norm

subspace of Lco-functions F(s) analytic and bounded in Re s)O

Hco(jR,~n)-: subspace of Lco-functions F(s) analytic and bounded in Re s(O

The prefix R denotes real-rational ( RL., RLco ' RH., RHco ),

The pref ix B denotes the uni t ba 11 ( BL., BLco, BH., BHco ) .

The fourier transform is a Hilbert space isomorphism ( i t is a

linear surjection and it preserves inner products) L. (-co, co) '" L. (j R, Cmxn )

L. [0, co) "H. (j R , Cmxn ) L. (-co, 0] ;;; H. (j R, Cmxn).L

The norms of these spaces are all denoted by

11'11.

7

Ever¥ FEL. has a unique decomposition F = FI + F2 where FIEH. and

F2EHt. This decomposition is an orthogonal projection and

II

F II~ = II FI II~ + II F2 II~

A decomposition G = _{G1 + G2' where GEL}oo ' GIEHoo and G2EHoo- is not

necessarily unique and

If we consider a transfer-matrix F E RHoo(jR,Cmxn) and an

input-signal x E L. (jR,Cn ) , i t is easy to see that

II F

11m

ess sup

xEL. (jR,Cn ) _IIxll·=l ess sup II F· x II'

so

11.1100

is the induced operator-norm in a mapping L. -) L.

If we consider an input-signal x E

fu

(jR,Cn ), we see that II F

11m =

ess sup

xEH. (j R, Cn)

II F·x II' II x II'

=

ess sup II F·x II' IIxll·=l

so

11.1100

is also the induced operator-norm in a mapping H. -) H •.

0.3 Transfer functions

Consider the linear time-invariant ordinary differential equation described by

.

_{x = Ax + Bu}

y ~ Cx + Du

where x(t) ERn is the state, u(t) ERm is the input, and y(t)ERP

is the output. The A,B,C and D are appropriately dimensioned real matrices.

Associated with this differential equation is the convolution

equation yet) g (t) = (g~ u) ( t ) At

=

C· e .B·l+(t) + D·S(t)

and upon taking Laplace transforms, the resulting transfer

function is:

yes) = G(s). U(s)

G(s) C·(sI-A)-l.B+D

To expedite notation

calculations involving transfer functions the

[*C

B

D

]

- C'(sI-A)-l'B + D

will be adopted.

Suppose G(s) is a real-rational transfer matrix which is proper, i.e. analytic in s=oo. Then there exists a state-space model (A,B,C,D) such that

The quadruple (A,B,C,D) is called a realization of G.

Definition: A transfer matrix G in RHa, is inner i f G*G = I

A transfer matrix G in RHa, is coinner if GG* I A sfuare transfer matrix G in RHa, is outer i f

G- E RHa,

If a tall matrix G is an inner then there exists a matrix GL,

called the complementary inner factor ,such that [G GL] is square and inner.

If a wide matrix G is a coinner then there exists a matrix GL,

called the complementary coinner factor ,such that [

~L]

is square and inner.

In electrical engineering terminology, an inner function is stable and all-pass with unit magnitude and an outer function is stable and minimum phase.

0.4 Operations on linear systems 1. Cascade

[*]

G2 Cl Dl

[*]

C2 D2

[*],[*]

Cl Dl C2 D2

[

A1 B1,C2 B1' D2

]

[

A2 0 _{A2 B2 Bl' C2} C1 _D1,C2 D1' D2 Dl,C2 9 0 _B2

]

A1 B1' D2 C1 Dl' D2

2. Paralle I

[*1

[*1

Gl G2 Cl Dl C2 D2 Gl+G2

[*1+[*]

C₁ Dl _C2 D2

-

[

~,

0 _Bl

1

A2 B2 Cl C2 Dl+D2 3. Change of variables x -) x = T-x y - ) _{Y R'Y}

.

u -) u - P-u

.

_.

[*1 --) [*1

=

[ 6

~

1· [*]. [

r

1 ₀

]

p-l 4. State feedback u - - ) u + F·x [ A+B·F C+D'F 5. Output injection '.

x

= Ax + Bu --) x = Ax + Bu + Hy 6. Transpose G --) G' 10

7. Conjugate G --) G* 8. Inversion G --) G+ i f D+ is a

[*]

right - - ) (left) inverse of D . G+

= [

A-B·D+·C -B.D+ D+·C D+

Then G+ is a right (left) inverse of G (not necessarily minimal).

9. bi linear map

G(s) --) G(s) G(s+a/1+bs)

]

[*

C B] D [ (A+al)' (I+ba)-l (l-ba). (I+bA)-l'B ]

--)

: + ,

-C. (I +ba) -1 D - b. C. (I +bA) -1. B 10. factorization a) Suppose G = _{G1' G"2}1 where

[

~~

]

[

:,

B D1

_]

C2 D2 then

[

A-B'D21 ,C2 -B.D"21

]

G

=

D1'D21,CZ-C1 _D1'D"21 b) Suppose

G

G

y

1.

G

₂ where

G1 G2

1

= [ :

I :: :: ]

then

G

= Df1.C D

y

1.D2 11

0.5 Controllability and Observability Consider the system

.

x = Ax + Bu

The system or the pair t1>0 and final state such that the solution Theorem:

x(O)=O (0.5.1)

(A, B) is controllable i f , for xl, there exists a (continuous) of (0.5.1) satisfies x(t 1 ) = xl

The following are equivalent: 1) (A,B) is controllable.

2) The matrix [B AB A'B ... J has independent rows.

each time input u(*)

3) The matrix [ A-~I,B J has independent rows for all ~ in C. 4) The eigenvalues of A+BF can be freely assigned by suitable

choice of F.

The matrix C = [B AB A'B ... J matrix.

is called the controllability

The matrix A is said to be asymptotically stable if all its eigenvalues satisfy Re ~

<

O. The matrix A is said to be anti-stable if all its eigenvalues satisfy Re ~ > O. The system is said to be stabilizable if there exists an F such that A+BF is stable.

Theorem:

[

The following are equivalent: 1) (A,B) is stabi lizable.

2) The matrix [ A-~I B J has independent rows for all Re ~ 10.

We will now consider the dual notions of observability and detectability with the system

•

x

=

Ax x(O)

=

0

y = Cx

The system or the pair (C,A) is observable if , for every t1>0 , the function y(t), tE[O,t1J, uniquely determines the initial state x(O)

Theorem:

The following are equivalent: 1) (C,A) is observable.

21 Th.

m.'<'x [

~~'l

h •• 'ad.,.ad.a' .o'umn.

3) The matrix [

AC~I]

has independent columns for all

~EC.

4) The eigenvalues of A+HC can be freely assigned by suitable

choice of H.

5) (A',C') is controllable. 12

The system. or the pair (C.A) is detectable if A+HC is stable for some H .

Theorem:

The following are equivalent: 1) (C.A) is detectable.

2) The matrix [

A~~I]

has independent columns for all re

~

20 . 3) (A'. C') is stabi I izable.

When we use a state-space description of a dynamic system we also need the controllability and observability grammian :

For a continuous stable system we define the contollability

grammian

P :-

J

exp(t·A)·B·B'·exp(t·A')·dt

o

and the observability grammian : Q .=

J

exp(t·A')·C',C·exp(t·A)·dt

o

For a discrete system we define the controllability grammian P .

=

~ Ak. B. B '. (A' ) k = C I. C

k=O

and the observability grammian

Q.= ~ (A,)k,C'·C·Ak

k=O

=

0 . 0 I

For a stable system (A.B.C) we can calculate (and for an

anti-stable system (A.B.C) we can define) the grammians by solving the Lyapunov equations: for the continuous case:

A·P

+

P·A'

+ B·B' = 0

A ' • Q + Q. A + C'· C 0

and for the discrete case

A·

p.

A'

P

+ B· B' 0

A ' • Q. A - Q + C'· C 0

Notice that for asymptotically stable systems the grammians are semi-positive definite, and that also hold:

P is positive definite if and only if controllable,

(A,B) is Q is positive definite if and only if (C,A) is observable.

For anti-stable systems the grammians are semi-negative definite,

and i t ho Ids:

P is negative definite i f and only i f (A,B) is controllable.

Q is negative definite if and only if (C,A) is observable. Finally we define:

A

system is called balanced if it holds

an )

A

system is called minimal if and only if PQ is not singular. For an asymptotically stable and minimal system this balanced realisation is unique, up to a transformation S, satisfying

SL

=

LS and S'S

=

I

If ai~aj for i~j then S is a sign-matrix.

0.6 Hankel-matrix, Hankel-operator and Hankel-norm

The theory in this section is mainly taken from Glover[6] and van der Linden[7]. In this section we consider asymptotically stable systems.

For a discrete system, the transfermatrix can be written as H(z) C· (zI-A)-l. B + D

(Xl

where Hk k=l, 2, ...

Then the Hankel-matrix is defined as

[

HI H2 H3

"J

H2 H3 . . . . r{H(z)}:= ~3 ~ • . • .

It is easy to see that r{H(z)} = O·G

The singular values of r{H(z)} are equal to the roots of the eigenvalues of p.Q, where

P

and

Q

are the grammians of a minimal realization (A,B,C) of H(z), so

For an asymptotically stable continuous system the

Hankel-operator

rg: H2L 00> H2 is defined as (rg.u) (t)

:~

J

G(t+r)·u(-r)·dr,

o

where G(t) ~ C·etA.B , o<t<ro

We now define the Hilbert-adjoint operator (T'):

Let T:Hl -> H2 be a bounded linear operator, where Hl and H2 are

Hilbert spaces. Then the Hilbert-adjoint operator T of T is the

operator T.:H2

-1

Hi' such that for all xEHl and yEH2 <Tx,y> - <x,T y> .

The singular values of an operator are defined as

a· ₁ (T) - /.l~(TT*) ₁

We now can see that also for continuous time it holds: ai( rg ) = /.l~(rgrg*) = /.l~(P.Q)

In the discrete time case we define the Hankel-norm as follows: II H(z)

IIH

= a( r(H(z)} )

and in the continuous time case: II G(s)

IIH

= a( rg

Define

u(t) E L, (-ro, 01 , so u (t) 0 for UO. and

v (t) _[

~

_{rg.U) (t)} for t<O _{for tlO} so

v (t) E L, (-ro,oo) then

II vet) II'

II

G(s) IIH sup

uEL, (-00, 0 1

_II

u(t)

_II'

0.7 Properties of the Hro-norm

This section is based on Doyle[2] . Francis[4] and Glover[6]. Properties of dilation: and and Theorem: and

II[~

1100

>

a iff.

II

X(a' I-A*A)-~

₁₁₀₀

>

1

II[~

1/ 00

a iff.

II

X(a' I-A*A)-~

1100 -

1

II

[~/loo

<

a if f .

II

X(a'I-A*A)-~

1100

<

1 Theorem ( special case of the Nehari-theorem ):

[

For REHro- and XEHoo it holds that

II

RI-s)

IIH

= min

II

R - X

1100

XEHro

We will prove these theorems in chapter 4 and 5 .

1.0 INTRODUCTION

The Hoo control theory is a new technique for optimizing

controllers of linear multivariable feedback systems.

Zames [lJ developed a theory that uses a minimization of the

Hoo-norm for systems in a general setting of Banach algebras. He

studied the effects of feedback on uncertainty. that occurs

either in the form of an additive disturbance at the output of a

plant or an multiplicative/additive perturbation in the plant.

representing "plant uncertainty".

For some control problems there is no good spectral information

avaible of the disturbance. Then it is desirable to limit the

maximum value of the disturbance frequency response (

Hoo-optimization). rather than to use an LQG-design that assumes good disturbance models.

When we study plant uncertainty we desire a robust design. In

that case Hoo-optimization will be better than

LQG-optimization.

This chapter will give a short overview on a design procedure for

Hoo optimal control. First we will consider some examples of

control problems using Hoo-criteria.

1.1 Some examples of control problems with Hoo-criteria

e

C P

u +

Fig. 1.1 Plant P with controller C Sensitivity:

Fig. 1.1 shows a plant P with a controller C. We have a

disturbance and/or modelling error n .acting on the real

plant-output y . with n=V·n and bounded energy IInll'

<

00 •

so nEH, .We

can consider this disturbance n as the worst-case signal. If we

want to minimize its influence on the output y • we will have to minimize the plants weighted sensitivity

W_{y '} S. V W_{y '} (1 - p. C) -1. V

so our problem is find inf

Cst [

SuP nEH, with Cst is a stabilizing controller.

II y II'

II

n

II'

1

inf

II

Wy·S·V

1100

Cst

Reference: Zames [lJ. Jonckheere

&

Juang [16J and Grimble

&

Biss [26J

Complementary Sensitivity:

Now we suppose that Vn is just measurement noise. So the real

system output is represented by z. If we want to reduce the

effects of measurement noise on z via the feedback, which is

needed e.g. for stabilizing the plant, we have to minimize the

plants weighted complementary sensitivity

Wz • T· V

=

_{Wz • (I -S)· V}

=

_{Wz •} p. C· (1-P. C) -1. V = _{Wz •} p. C. S. V so our problem is : find inf [ sup

Cst nEH.

with Cst is a stabilizing controller.

II

z

II'

II

n

II'

1

= inf

II

Wz •

T·

V

1100

Cst

Reference: Foo

&

Postlathwaite[12J, [13], Jonckheere

&

Juang [16] and Grimble

&

Biss [26J

Control Sensitivity:

Whatever n represents, we now want to limit the controller output

x, e.g. in order to avoid saturation of the actuator. We then

have to minimize the plants weighted control sensitivity W

_{x •}

R· V = W

_{x •}

C· (1-P· C) -1. V = W

_x•

C· S· V

so our problem is : find inf [ sup

Cst nEH.

II

X

II'

1

-II

n

II'

with Cst is a stabilizing controller.

Reference: Glover [14J and Grimble

&

Biss [26J Robustness:

~ Additive bounded model error:

inf

II

Wx • R· V

1100

Cst

We consider the stabilization of a closed-loop system, with a

modelled rational transfer-function P(s), an absolute model error

Q(s), satisfying

II

Q

1100

< 6

and a controller C(S). The real

transfer-function of the system is (P+Q) . The configuration is shown in Fig. 1.2

J Q I I I u +r, _J P I + + I I + I _C

l.-I

I

Fig. 1.2 Plant (P+Q) and controller C Consider the unperturbed sensitivity matrix

S

= (

I - P.C )-1 .

18

The perturbed sensitivity will become Sp

-

I (P+Q). C J -1

--

[ I Q·C - p·c J- 1

-I - Q. C. {I -po C}-l ). ( _I

-

p·C ) J -1

-

[ I Q·R ). S-l )-1 S I

-

Q·R )-1

If we assume that P and (P+Q) have the same number of poles in

the right half plane and further that the unperturbed sensitivity

matrix S is bounded. a necessary and sufficient condition for

robust stability is:

Doyle and Stein (21) and Chen and Desoer (34) showed that a

necessary and sufficient condition is

II

Q. R

II '"

<

1 •

this means a necessary and sufficient condition is

II

R

II",

=

II

C·S

II",

~ 6- 1

Consequently if we want to maximize the allowed radius 6 of a

ball for Q (ie. allow for the largest perturbations and still get

a stable system). we have to minimize

II

robustness in this context coincides

sensitivity.

Reference: Glover (14)

~ Multiplicative bounded modelerror :

R

II", .

Thus maximal

witn optimal control

If we consider the stabilization of a closed-loop system

Pp = (I+A)· P with a re lative mode 1 error 4 satisfying

II

A

II",

<

11

a necessary and sufficient condition for robust stabilIzation

will be:

II

T

II",

=

II

p·C·S

This can be seen easily by substituting Q

=

A·p

.

From the

previous derivation we get

II

QR

II",

=

II

lI.PR

II",

=

II

AT

II",

<

1

this means a necessary and sufficient condition is

We see that minimizing the complementary

minimizing the power transfer function R

robustness of the controled system. Reference: Foo & Postlathwaite(12), (13)

19

sensitivity T or

Mixed Sensitivity:

If we want to design a controller that not only mInImIzes the sensitivity, but also gives a sensitivity-function that is itself insensitive to perturbations, we can introduce a new criterion. This criterion is a combination of the sensitivity function with the complementary sensitivity function or the control sensitivity function.

We call this the mixed sensitivity: M1 _ [ Wy ' S· V ]

Wz·T·V or M2 = [ Wx·R·V Wy· S· V ]

where Wy , Wz ' Wx and V are the weighting matrices. We see that the criterion M1 guarentees:

II

Wy ' S· V

II",

<

II

M1

II",

II

Wz·T.V

II",

<

II

M1

II",

and the criterion M2

II

Wy.S.V

_II",

<

_II

M2

II",

II

Wx·R·V

II",

<

_II

M2

_II",

Reference: Foo

&

Postlewaithe [12], [13] , Jonckheere

&

Juang [16] and Doyle [ 30] .

A

Signal Tracking Problem

x

r x

P y

C2

Fig. 1.3 Plant P with controllers C1 and C2

Consider the system in Fig. 1.3 representing a plant P with two controllers C1 and C2' The output y is now to track a reference signal r, which is not a known fixed signal, but may be modelled as belonging to a class r : { r=W·u for some uEL., lIull.i1

l.

To ensure the existence of an optimal proper contr01ler, we will also include the weighted control-signal Wx·x in the cost-function, where Vu and Wx are weighting functions. So now we have two criteria:

Signal Tracking criterion: M1 - (y-r)/u Robustness criterion : M2 Wx·x/u We combine these two criteria to

II

M1

II _II

[ ( I -PC 2) - lPC1- I ]. VU

II

M

II",

= M2 '" Wx' (I-PC2) -1. C1• Vu

Reference: Francis [4] .

1.2 Standard problem

Many control problems

the H -criterion will

intro~uce the standard

can be reduced to a simular problem, where

have the same form. Therefore we will

problem. Suppose we have a invariant system, rational matrix) vectorvalued. multi-input multi-output with a transfermatrix G(s)

The input- and output

w(s) is the exogeneous input. u(s) is the control input.

z(s) is the output to be controlled. yes) is the measured output.

causal ( G

linear

time-is a

real-signal w,u,z,y, are

We want to control the system with a controller K(s) that has the

properties that it stabilizes system G(s) and minimizes the

H -

₀₀

norm of the transferfunction M(s) from w(s) to z(s)

w z

G

u y

I _K I.

I I

Fig. 1.4 The Standard Problem

[ G" (,:

G12(s)

1

G(s) can be partitioned as

G21 (s) G22(s)

so the algebraic equations become:

z (s) G11 (s) , w(s) + _{G12 (s)} u(s)

y (s) G21 (s) , w(s) + G22(s) u(s)

u (s) K(s) yes)

z(s) - M(s) w( s) ~ M(K(s) ) w( s)

From now on we will drop the Laplace variable .. (s)

the cases where confusion may arIse.

, except in

The transfermatrix from w to z is now a linear-fractional

transformation of K :

z ~

M,

w

~

~ [ G11 + G12'K' ( I - G₂₂'K )-l'G21 J'w

Our goal is to minimize the

H

-norm of tranfermatrix M over all

stabilizing controllers Kst ' ~o we have the criterion:

a inf [ sup Kst wEL.

II

z

II'

II

w

II'

1

~

inf Kst

II

M

1100

So finally we have the next definition:

Standard problem : find a real-rational minimize the Hoo-norm of the transfermatrix

the constraint that

K

stabilize G:

proper controller K to

M

from w to z under

a= inf

II

Gl1 + G12' K· ( I - G22' K ) -1. G21 1100 Kst

Remark: Sometimes we are not interested in the minimum of

II

M 1100,

but are we satisfied with an stabilizing controller

K ,

such that

holds:

II

_{M(Kst ) 1100}

<

6 . where 6

>

a

It will appear that the routines to find the optimal controller

will tend to be numerically unstable when we approach

K

_{opt '}

Let G have a state-space realization given by:

[

:,

B1 B2

]

[*]

G _{or Gij} =

Dl1 D12

C2 D21 D22

Then there exists a stabilizing controller for the closed-loop

system in Fig.1.4 i f and only if (A.B2) is stabilizable and

(C2· A) is detectable see Francis et a I. [3] )

Remark: We assume G12 to be "tall" and G21 to be "wide". I f this

is not the case we deal with too many degrees of freedom in

controller

K

and we have to define a reduced standard problem.

If Gq is "wide" we choose a G₁₂ - Gp' U . G22 = G22' U , such

that G12 is square and stabilizable. If

K

is the set of optimal

controller for this reduced problem, then

K = V·K

.for all

V

satisfying V·U =1 is the set of optimal controllers K. We can do

the same for G21: choose

G

₂₁

=

U.G

_f

l

and G22

=

U.G22 . such that

G21 is square and detectable. then K = K·V .for all

V

satisfying

a·V

= I is the set of optimal controllers K.

Remark: The design procedure as proposed in this report will fail

when we deal with strictly proper plants or jw-axis poles or

zeros i.e. all entries _{of G12 and G21 should show direct feed}

through or equivalently D12 and D21 are of full rank. Techniques

to approximate an optimal controller otherwise will be shown in

chapter 6.

We will now show that the examples of section 1.1 can indeed be brought into the figuration of the standard problem.

sensitivity:

If Fig. 1.5 we see the same system as in Fig. 1.1 in where the

system G is given by the dotted box

• • . . • • • . . . • • . • • . • • . . . • . . . . • . • • G n :

J

_V

l.tr.

J

W

I

: y

,

: '1 1

t .

'1 y 1 : : : :

I

_P

I

:

x

: e 1

....

. .

.

.

.

.

.

. . .

.

. .

.

. . .

.

. . .

.

J

C I. 1

r

Fig. 1.5 standard problem for the sensitivity Now we can write the following:

This brings us to the next criterion

a = inf

II

Wy'V + Wy ' PC· (I-PC) -1. V

11m

=

Cst

inf

/I

Wy ' Cst

(! -PC). (! -PC) -1 +PC· (! -PC) -1

J.

V

II

inf

/I

_{Wy ' (I-PC) -1. V}

/I",

Cst

inf

/I

Wy.S.V

/I",

Cst

complementary sensitivity:

In Fig. 1.6 the dotted box gives System G

n

x

. • • • • • . . . • • . . • . . • • . . . • G

...

C

Fig. 1.6 standard problem for the complementary sensitivity

z

e

We get the following equation:

This gives us the next criterion:

a inf Cst

II

Wz·PC. (I-PC)-l.V

II

inf

II

Wz·T.V

11m

control sensitivity: 00

-In Fig. 1.7 the dotted box gives system G

• . . . • . • . • . . . • . . G n : : J 1+

rl

W 1 : 1 V

r"""1

+

x 1 : : : :

I

~

I

I

: x : :

,

r

...

I

C

L

1

r

Fig 1.7 standard problem for the control sensitivity

We can write the following:

and we get the criterion

a

=

inf

II

_{Wx • C· (I-PC) -1. V}

II",

=

Cst inf

II

_{Wx • R· V}

II",

Cst 24 X ~ e

Mixed sensitivity Ml :

In Fig. 1.8 the dotted box gives system G

. . . • . . . G n

L

+ ".

: y

:---1

_..I

Wy :

1

V

I

->j

1

: I' • : -: : Z : J W

I

: ·1

z

1 : :

I

_P

I

: x : e T

.

,

...

1

_{C L}

1

r

Fig. 1.8 standard problem for the Mixed sensitivity M1

We can write the following:

~

]

So the criterion becomes a

=

inf Cst inf Cst

II

Wy' I-PC)-l. V W

_{z '}

PC· (I-PC) -1. V

II

Wy' S· V

Wz·T·V

inf

II

Ml

II",

Cst Mixed sensitivity M2 :

In Fig. 1.9 the dotted box gives system G

. . . • . . . G n : V

I-tr

: ,I

'"'

W 1 : -I 1

f+

·1 y 1 : : : :

I

_P

I

J W 1 :

""'I

x 1 : x : :

,

_...

J

I.

L

C

J'

Fig. 1.9 standard problem for the Mixed sensitivity M2

y

-X

We derive:

J. [

~

]

So the criterion becomes:

a

=

inf

"

Wy ' ( I -PC) -1. V

IL

Cst Wx '

c.

(I-PC) -1. V inf

"

Wy.S.V 1100 Cst W . R· V x inf

_II

_{M2 1100} Cst

A Signal tracking problem:

Fig. 1.10 gives the standard problem for the signal tracking

problem of : We derive . . . G u

_{: J}

_I

-

:

.1

Vu

I

_t

.~~ wl:

:

xI.

: : X

.I

P

I

: "I

I

• • • • • • • • • • • • • • • I • • • • • • • • • •

+

+

_{I L} 1\ I

C1

r

I c2 I, I

r

Fig 1.10 standard problem for the signal tracking problem

~

]

26 y-r

-x

y r

If we now choose

and

then we can write the criterion as follows a

=

i nf

II

M

II

CD

=

(Cl,C2 lst

inf

II

_Gl1 + G12K.(!-G22Kl-l.G21IlCD Kst

1.3 The model-matching problem

The standard problem can be transformed into a model-matching

problem. as will be shown in chapter 2 . Therefore we will first

define the model-matching problem:

Model-matching problem: Find a controller Q

model. represented by the transfer-function cascade T12 'Q'T21' with TIl' T12 and T2l E

w

E RHoo such that the Tll is matched by the RHoo .

+

z

Fig. 1.9 The model matching problem The transfermatrix from w to z is now affine in Q

z

=

M·

w =

Our goal is to minimize the H -norm of stabilizing controllers Q. mFor Tll' it is easy to see that the set of all

is RHm. So our criterion will be

tranfermatrix M over all Tl2 and T2l are all in RHm

stabilizing controllers Q a= inf QERHoo [ SUP wEL,

II

z

II'

II

w

II'

1

=

i nf

II

M

II

m QERHoo

So finally we have the next definition:

Model-matching problem find a stable

controller Q to minimize the Hm-norm of the

w

to z :

28

real-rational proper transfermatrix M from

1.4 The Nehari problem

The model-matching problem will be transformed into the

Nehari-problem, as will be shown in chapter 3. Therefore we will first

define this Nehari-problem:

Nehari-problem : Find a transfer-matrix Q E RHoo such that the

Hoo-norm of

L-Q

where

L E RLoo

is minimized, so

a

= inf

0

L -

Q

Om

QERHoo

We now introduce an extended Nehari problem, where Q may partly

consist of zero blocks. This extended Nehari-problem can have 3

forms: NPI :

[

NP2 or NP3 :

[

a a a a inf

0

QERHoo inf

II

QERHoo = inf

0

QERHoo = i nf

II

QERHoo L Q

₁₁₀₀

with L E RLoo L with L E RLoo L [Q 0 J

_Om

with L E RLoo L

₁₁₀₀

with L E RLoo

To avoid confusion we will write for the extended Nehari problem: a

=

inf

0

L - Q_e

Om

QERHoo

where Q_{e may have zero blocks.}

NPI is called the I-block Nehari problem. As will be shown in

1.5 Design procedure

The aim of this section is

illustrates the different steps

to present a procedure which

in the design of a Hoo optimal

contro 11 er .

We start with the original problem:

1.[ Find a controller K that is stabilizing the criterion

0

M Om

and that minimizes

According to the first part of this chapter it is possible to

rewrite this original problem to the standard problem

2.[ Find a controller K that is stabilizing and that minimizes the criterion

0

Gll + G_{12 '} K, ( I - G22' K ) -1, G21 Om

The next step will be to transform this standard problem into a

model-matching problem

3. Find a matrix Q E

RHoo

that minimizes

the criterion

0

Til - Tl2'Q'T21 Om

with Til ' T12 and T21 E

RHoo

Q E

RHoo

will parametrize all K's ,that stabilize G This model-matching problems will be transformed into the extended Nehari-problem :

4.[

Find a matrix Q

E

RHoo

that the crlterion

0

L - _{Qe Om}

with

L E

RLoo

minimizes

This extended Nehari problem can be reduced to a 1-block Nehari

problem ( in general this will be an iterative procedure)

5.[ Find a matrix

X E

RHoo

that minimizes

the criterion

0

R -

X

Om wi th R E RH;;;

This I-block Nehari-problem can be solved and we will find our

minimum

inf

0

M Om -

0

R(-s) OH Kst

This is the Hankel-norm of matrix R .

We will calculate the optimal Q and using this Q we will

calculate the optimal, possibly approximated controller K.

Now a short elaboration of the various steps will be given Step 1: Write original problem as the standard problem

In this step we define the transfer matrices G11 ,G12 G21 and

G22' Further we have to fix the structure of the controller K. Very important in this step is the choice of weighting matrices

, see Grimble

&

Biss [26J Step 2: Coprime factorization

In this step we will use the coprime factorization. Purpose of

this factorization is to transform the standard problem to the

model-matching problem.

For each proper real-rational matrix G22 there exist non uniquely eight RHm-matrices satisfying the equations

MY

N X

] = I

If we have done this coprime

set of all real-rational

system G by:

factorization we can parametrize the

proper matrices

K

that stabilize the

K

= (

Y - M"Q )" (

X -

N"Q )-1

where Q E RHm .

( X -

Q"

N )

-1" ( Y

Using the properties of this factorization we can rewrite our

standard criterion as follows

G_{l l} + _G12"K"( I - _G22" K )-1"G21 G_{l l} + _{G12"M"Y"G21 - G12"M"Q"ji.i"G 21} Tll T12"Q"T21 We see matching choosing

that we transform the standard problem to the

model-problem by coprime factorization of matrix G22 and

T11 _G11 + G12"M"Y"G21

T12 G_{12 "} M

T21 ji.i"G21

Step 3: Rewrite to Nehari-problem

Purpose of step 2 is to rewrite the model-matching problem to the extended Nehari-problem: a = inf

II

Tll QERHm inf

II

L QERHm with L E RL",

In the coprime factorization, as introduced in step 2, we still have some degrees of freedom. When we first scale the process, we

can use these degrees of freedom to make the matrices T12 and T21 inner respectively coinner. In that case it holds that T12*oT12 -I and T21oT21* - I . This will be shown in chapter 3.

For T12 _{has the same size as G12 and} T21 has the same size as G21 we know that T12 is "tall" and T21 is "wide".

*

If T12 and T21 are square we can write the following: a

=

_{i nf} II Tll T 12'Q,T 21 II", QERHa, inf II T 12 ,Tll,T2 1

*

•

-

Q _II", QERHa, i nf II L Q _II", QERHa,

Remark: In this derivation we use the property II F II",

=

II U*'F II",

=

II F,V*

II",

with L E RL",

where U is square and inner and V is square and coinner .

*

If T12 is square following:

and T21 is non-square we can write the

a = _{inf II Tll} T₁₂,Q,T 21 II", QERHa, inf II Tl1 T12' Q 0 _{J ' [} T21

]

_II", QERHa, T21J..

*

T21

*

*

(Q , inf II T12· T ll' T 21J.. _J - , 0 J ₁₁₀₀ QERHa,

=

i nf II L [ Q 0 J ₁₁₀₀ with L E RLoo QERHa,

*

If T12 is non-square and T21 is square we can write the

following: a

=

inf II Tll - _{T 12 ,} Q, T21 1100 QERHa, inf II Tll - ( T12 : T 12J.. J, [

g ],

T21 ₁₁₀₀ QERHa,

[

*

]'T l1 'T 21

*

[ g ]

inf II T12 * _II", QERHa, T12J..

inf II L

[ g ]

₁₁₀₀ with L E RLoo

QERHa,

* I f _{T12 and T21 are both} non-square we can write the following: a ~ _inf

II

QERHa, Tll T 12'Q'T21

1100

Q 0 _T21 inf

II

Tll - [ QERHa, T12 T12.L l' [ 0 0

1, [

T21.L

1

1100

inf

II

QERHa, ~ i nf

II

QERHa,

*

[ T12 * l'T ll ' [ T21* _T12.L L with L E RLoo

We see that the model-matching problem is transformed into the

extended Nehari-problem.

Step 4 : Reduce to the 1-block Nehari problem

This step is probably the most difficult step

procedure. We will need an iterative process to

minimum a. This will be discussed in chapter 5.

1n the design

approximate the

Step 5 : Calculate optimal Q for the 1-block Nehari problem

In the I-block case Nehari's theorem gives a value for the

minimum-norm. Suppose L E RLoo and Q E RHoo . First we have to

find a decomposition L ~ Ls + La ' such that LsERHoo and

LaERHoo-(All decompositions give the same result). When we deal with a

square standard problem, and we use the algorithm as proposed in

step 3, L will already be in

RHm-,

and L~La and Ls~O . Now define R La E RHa,- and X ~ Q - Ls E RHoo . Then it holds:

a _{x~~k,,11 R - X}

₁₁₀₀

_~

_II

_{R(-s) IIH}

where II ' IIH denotes the Hankel-norm.

The solution for all optimal X is given by Glover [6J

chapter 4.

Step 6 : Calculate optimal K

, see

From the optimal X we calculate the optimal Q and from the

optimal Q we calculate the optimal controller K by

K ~ ( Y - M. Q ), ( X - N, Q )-1

where Q E RHa, .

In the following chapters we will discuss the

detail we will first give an example

different steps in the design procedure.

1.6 Example

different steps in

to illustrate the

Consider a feedback system as in Fig. 1.10 , with a plant

P -

s 2 and K is a controller to be designed.

s 18

u

K

+

v _y

+

Fig. 1.10 Plant P with controller K

Suppose we have the objective to

weighted disturbance signal v refer~ed

P. We assume v to be a function in the

attenuate the effect of a

to the output of the plant class

{ v v

=

V·n for some nEL2

_II

_n

₁₁₂

~ 1 }

In this example we choose V

=

s _s + 3

+ 30

[

II

Y

₁₁₂

1

so our problem is now find inf sup

=

inf

_II

S·V

II",

K n

_II

_n

₁₁₂

K

wi th K a stabilizing contro ller.

Step 1 : Write as standard problem We have our criterion M(K)

M = S·V V + P.K. ( I - P.K )-l·V Gll + _{G12• K· ( I - G2 2' K ) -1. G21} so we can write s + 3 s - 2

[

V P

1

s + 30 s 19 G (s)

-s + 3 V P s 2 s +

30

s 18 34

and we get the standard set-up as in Fig. 1.11

{j

G

I _K I.

I I'

Fig. 1.11 standard set-up Step 2 : coprime factorization

G22

=

N.Mrl Mil.N

MY

N X ] =

I

a possible coprime factorization is

N

s

-

2

M

N = M = s + 4 -30.25 s Y Y X = X s + 4 y s

-

18 s + 4 - 4.25 s + 4

another coprime factorization ( a very nice one, as will be shown later in 5tep 3 ) i5 5 - _{2 5} - 18 N ₊

₂

M ₊

₂

5 5 -13.78 s+50.57 5 - 2.78 Y X 5 +

2

5 + 30 5 +

2

2 30 18 5 + 30 5

-

5 + 5 -N +

18

+ 3 M +

18

+ 3 5 5 5 S

Y

-13.78

₁₈

5+50.57 ₃ X 5 - 2.78

₁₈

.

5 + 30 3 5 + 5 + 5 + 5 +

In ca5e we choo5e the la5t coprime factorization then S·V

v

+ p. K· ( I -Gll + _{G12 • K· ( I} P.K )-l. V - _{G22• K ) -1. G21}

N·Q·M·V

v

+

M·

5(.

V

5 - 18 5 +

18

N·Q·M·V

5 - 2.78 s + 2 5 - 2 s +

2

Q Q 5 - 18 5 +

18

Step 3 : Rewrite to (extended) Nehari-problem The coprime factorization

property that

introduced as above has the desired

s - 2 s - 18

is inner and T21 is coinner.

s + 2 s + 18

T12 and T21 are square. so we can write the following

Step We a ~ inf

II

T 11 QERHa, = inf QERHa, inf QERHa, inf QERHa, inf QERHa,

II

s - 18 s +

18

s + s s 2 2 -s L s - 2.78 s +

2

s -18

18

.

s + 2.78 2

4 : Reduce to I-block Nehari

see that L is in RHa,- ( we

factorization)

.

so we choose

R = L E RHoo- and X = Q E RHa,

s -s + s - 2 s +

2

2.78 2 Q Q 1100 problem • Q • s + 18 s 18 1100

used the 'special'

s - 18 s +

18

Q

coprime

Step 5 Calculate optimal Q for the I-block Nehari-problem

R 1 + -0.7813 _s

₂

This gives

II

R(-s)

IIH -

0.1953

00

00

With an algorithm of Francis[41.pp. 72 (step 2 untill 6) the optimal Q

we get

Qopt ~ 1.1953

When we fill in this optimal Q we get:

L Qopt

=

-0.1953 • s _s + 2

₂

The weighted sensitivity is equal to so S S'V = V = = = = = + p.

K·

(

Tll

s - _{18 s} s + 18 s

-

18 s s + 18 -0.1953 -0.1953 • I

-

P'K

T12

.

Q s s - 2.78 + 2 - 2.78 + 2 s - ₁₈ s + 18 s - 18 s +

19

) -1. V T21 s - ₂ s + 2 - _1.1953· s + 30 s + 3 s - 18 Q s + 18 s - 2 s

-

18 s + 2 s + 19

In plot 1.1 we see that the sensitivity for lower frequencies is

1.953 . for higher frequencies the value becomes 0.1953 . Step 6; Calculate the optimal K

The optimal controller K_opt will be;

K

= (

Y - M.Q ). ( X - N'Q )-1 =

Finally S = [ -13.78

.

s + 50.57

-1.1953· s + 2 s + 30 -s 2.78

.

s + 30

_-

_1.1953· s + 2 s + 30 -1.1953· ( s

+

2 ) ( s

+

21.53 -0.1953· ( s + 2 ) ( s + 30 ) 6.1203. ( s + 21. 53 ) ( s + 30 )

we check this by filling I

-

P·K

)-1 6.1203· ( s + 21.53 1

-s + 30 -5.1203· s + 18 s + 3 s - 18 s + 30 -0.1953 s - 18 s s +

19

s in ) ( (

J-

1 + 30 + 3 s - 18

.

s + 30 s + 2 s + 30 s

-

2 s + 30 s + 2 s + 30 s - 2 )

_J-

₁ s - 18)

101 Amplitude sensitivity ... -;, . ....•.... ... -,_. ,., ... ... -..•. ,. ·)··;··j·H·)f . -, ... ~ ... ; .. " .. ; .)..;.;,.;; ...•.. " ... ! .. .. ; ... ( .. (.~.. ... . ... ;. .. , ... ~ ... I .;.;.;.); ... :, .... ; .i.; . · .

:: ...

:.~ r~'

,', ... ; ..

+,! .... :.::.:: ... ~ ... ~.~ '!'!' · . iSI . ... .. ~.~ .. ~. .~.;... . ~.~ ... . 1.953

-:-

, 100 _{... ':-.} " ) ... ( ... ( .. , ... ; .... . ~. .. ... ... "-, '~'.' ,', ... ; .. " ' " "

... ..

••.•• < .; •• ;. ':' • -, ••• ~ •••• ~ .,! .. ~.!.!.:.!~ ... : .... ) .. ! .. ) • • .. .: .. ~ .~.~ .;. ... ~ .... ~ ... ! .. ;.:.:~.!~ ... ; , . . . . . . ~ " •••• ~ •••• ~ ••• ! .. ~.:.:~.:! ... ~. ··~··r·:· .;. ... ! .~ . . ~.~ ••••.•. _{~::,j.l.· .~}

...

_~

:::! ... ;

_.~.:.:~.;;-

... : ....

_~

..

! .! • • !.~! ·~·:·:T::··· ':':::" ...

-

... -~ . .:. ... ~ ... ~. {. (-) ... .

f

H::::::: :::: :: ..

0.1953 ... ... "., .... _ . A _, , ~

'"

~

.. .... .. . . . . . , . , · .

10.1 L--'---.L..L'-'-""'--'---'-'u... "--'-..i....i..lli. ... i..-...L.-i....i...i..W ... _ ... -i...i..i.. i.J.< ... Li... ___ '--.i....i..i.iiliJ··

10.3 _10.1 _10-1

loo

₁₀1 _{102 103}

frequeIII:y

->

Plot 1.1 Amplitude sensitivity Phase sensitivity -140,-..."r-rrTITIr-"""'TTrr'-r-T..-nmrr-rronTIrr---'''-'-TTTTrTr--'-'rTTTTTl -145 -150 -155 -160 -165 -170 -175 .. . ... . ... "

...

: . ; ;;:;

.. ...,." '" 'j"",,,r·:

0

·'l,,'··'·'·'i"'j I

.~ . . . :,: .•. :,,'1,. ~,~, ... :",.. . . . .. : ... ''''':'' ... ~, ..• "'".' .:,"':. ;"'".':"'" .. :'" ;.~ ;,; . . . i .... : .. t. • . .:. ',: ,. . . ; ; ;:: .. ...

::.

··i"i

':!

:m··· ..

·'Tnn:

.~.:.~~~

... ; ..

~.:

.. ;;;;; ··r

:·~T~T~!·

....

:.~~~...

···:·T

·rr:!·· .. ··· .... : ...

~.~.:.:!. ... .~-:.!:.:-... : ... ~.: .~~.:.~; · .. ... ' .... ~ ... ~.~.:.:.:-... ~ .. :··:··:··;.i:·! .. · ' ... ~ .. !. ; .. ! .... ; .... ·i·;i·; ... .. ... : ... ~.~,.;. .;.:.:.;, ... , ... , .. , ... , ... :,; :~ ... . .... ; . .;.. ,.:.~. ;..:,~, .... --.

r:nli

_... .... ... ... _ .... . .. :,. . .~.~. i .~.~

....

_~.

':"+'i'jj;

. ...

-ISO l.-l...i..l...ill1.~. -==,,:d::t:lil. ·JJ..,--..i...JcJ...iW· . ·w"...i....L.l..ill·w· .. _L....LlJ.l..w.. ·L· -=:t::::b:h·~···

10-3 _10-2 _10-1 _{100 10}1 ₁₀2 ₁₍₎₃

frequency --> Plot 1.2 Phase sensitivity

2.

COPRIME FACTORIZATION

2.1 Introduction

The RHoo-space is a non-commutative ring wIth identity. Suppose four matrices N,M.N,M E RHoo.

From section 0.1.1 page 4, we know:

Two matrices M and N are right-coprime iff . there exist two matrices

X

, Y E RHoo such that [ X

-y

J •

[

~

]

= I

Two matrices N and Mare left-coprime iff . there exist two matrices X , Y E RHoo such that

[ -if

M

J. [

~]

=

I

A right-coprime factorization (ref) is a factorization P

where Nand M are right-coprime RHoo-matrices.

A left-co~rime !.actorization (let) is a factorization P where Nand M are left-coprime RHoo-matrices.

A doubly-coprime factorization (dcf) is a factorization

P

=

N'M-1

=

M-1.N where

Lemma:

MY

N X

[

For all proper real-rational matrices P there exists a doubly-coprime factorization.

We will prove this lemma constructively in section 2.4 .

2.2 Stabilizing Controllers Consider the system in Fig. 2.1

Fig. 2.1: System with Controller

We introduce the following doubly-coprime factorizations:

P N.Mi1

=

Mi1.N

K U.V-1

V-

1

.U