Feedback decoupling and stabilization for linear systems with multiple exogenous variables

Feedback decoupling and stabilization for linear systems with

multiple exogenous variables

Citation for published version (APA):

Woude, van der, J. W. (1987). Feedback decoupling and stabilization for linear systems with multiple exogenous

variables. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR274016

DOI:

10.6100/IR274016

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Published: 01/01/1987

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FEEDBACK DECOUPLING AND STABILIZATION

FOR LINEAR SYSTEMS

WITH MULTIPLE EXOGENOUS VARIABLES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE

TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG

VAN DE RECTOR MAGNIFICUS, PROF. DR. F.N. HOOGE, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DECANEN IN HET OPENBAAR TE VERDEDIGE;N OP

DINSDAG 1 DECEMBER 1987 TE 16.00 UUR

DOOR

JACOB WILLEM VAN DER WOUDE

GEBOREN TE SOESTDIJK

door de promotoren

Prof.dr.ir. M.L.J. Hautus

en

Prof.dr.ir.

J.e.

Willems

CONTENTS

PBEPACE

CHArTER 1. INTRODUCTION

1.1. Basic concepts and notation 1.2. Stability issues

1+3~ Invarianca issues

5 11 1. 4. Combined stabi 1 i ty and inV8d8nc.e issues 17

1.5. Dynamic feedback 20

CHAPTER 2. DIS1:URJ3ANCE DJlCOUPLINC AND OUTPUT STABlLiZATION BY

MEASUREMENT FEEDBACK WITH INTERNAL STABILITY 27

2.1. Problem formulation 30

2.2. Di~t"rbance decoupling and output stabilization by

st(lt~-feedbac]< with int<"mal slabili ty 34

2.3. Disturbance dccoul'h:d and "tabili~ed estimation with

internal stability 40

2.4. Main res,,)c 42

2.5. Special cases 49

GHAPTER 3. (ALMOST) NONINTERACTING CONTROL ~4

3.1. Fiest principles 57

3.2. (Almost) noninte'-(l.cting' control by state feedbllck; probl"m

formulation~ 66

3.3. Noninte~acting control by .tate feedback with input/output and

interna1 st,~bility 69

3.4. Almost noninteracting control by state feedback 74

CHAPTER 4, (ALMOST) NONINTERACTlNG CONTROL BY MEASUBEMENT PEEDBACK 77

4.1. introduction 78

4.2. On So common solution to a pale of linear m;;.t"ix eq""ti()ns 80 4.3. Nonin terac ting can trol by measurement fe~dback: suttie ien t

1,.5. Spec. t<.1 l (.,'=l~fI.!=i

4.6. (AlroD~t) noninturBcting control and (almuat) diagoo~l t.n:tnsfe.r pr~::;erviltio[l by measurement fe:'E:::dback

CllAfTr;R :;, (A~.M()S1') 'tRiANGULAR DECOUPLTNG 5.1. Problem form"l"tion.

5.2. Triangular decouplins by measurement feedback with interhal stabi lity

5.3. AlolOSt td3118\lLar decoGupling by measurement feedback

REFERENCES SUBJECt INDEx SYMBUL iNDH SN1ENV A TTT N G CURRiCULUM ViTAE 106 109 112 114 116 120 128 130 133 134

PREFACE

In the present monograph we study the exi£ten~e and design of automatic Controtler5 ~or dynamic~l ~y~tem~ fo. which the interaction with the out-side world plays an important role.

A (simplified) exa\T1ple of ~1,lch a ~y~tem .·8 " bleI1Ol.ng process. In a bleI1dil1g process one mixes two or more gasses (or liquids) with diff",rent concentrations of some subst~nce in order to obtain a blend with" pI"e-scribed concentration of that substance. In such a proLes", the concen-trations in the incoming gaEses are quantities tholt Jre brought into the process from the outside, whereas the concentration In tbe blend mDy be relevant in further proeessing Outside the pro<;e~s. Tn vi.ew of such l~ter processing we aSsume that it is OeS£1;"ea to holVC the (:onc~lltration in the blend con~t,m!;. Then, if by some means the concentrations i.n the incoming gasses are known and are constant, <Je <;(>n c31culate the ilmounts of flow of the different incOming gasse5 necessary for a proper bl<tnding. If, in addition, flow control valves ~re present in th .. pipes that take care of the transport.~t~on of the inc:omillg gasses, we "an actu.'] ly <lccomplish that th" proper amounts of the flo~ of the different gasses meet in the bh'nding point.

Howe:ve.r,. if for tlIly t'eaSon thE conce:nt'~'itions. in the incoming ga.sse.s change, also the i:tiuOunts of flow of the oj,Uerent gasses needed ror a proper blending lMy LhaIl.~e. In tUJ;"n, this may imply that also the v"lves have to be adjusted. In order to find out how we have to aojust these valves, we haVE to know what the COllCetltrations (or the ~.h"nges of the concentrations) in the incOmirlg ga~5E:.!;i aye+ The:refore, it is necessary to have equipment that C~n measure tbe concentrations in the incoming g,,gs~g and l'erh~ps also in the bl.:tld itself.

As announced above, in the present work we are inl;.ere;s.ted in automatic.:: controllers. For our blending proce~~, e·vch ~C\tomatic conlrollers will be mechanisms (software Or haI"dware) that based on si¥nals of the meS5urement

valve 'HljustlllcJ1lts, tlwt can be sent to the valve engines. Such control

dev1.c:.ec are feedhac.k mec.hanisms bec.ause they feed information abouL the: prucess back to the controls of the procesS.

The a\ltornatic controller that we .. ould like to have for our blending pro,es~ h~-~ C" be_ sIKh th~t in the <:.ont-.;-ol1eO proc,,~~ th" chang"~ in th~ concentration of the substance in the blend are ze1::o, no matter what the chclnge:s in thE:: cuncentration of the 5ubstdnce i!"l. th0 iIlCOI'I'l.illg gasses are.

If 'We c.an c.on~tTI.lct .~l.lch a controlle.r ~ we say that we have been ab le to

a~.hieve de(_",lpUng (between th~ ch;jl.nge$ tn cOn,,,,nC'.Hion of t;he inc<>ming g"s~es .3nd the outgoing blended g~$) by means of (mMsurement) feedback.

We stres" the f<1ct tll"t the above e)<;"Ulple is just a very simplified pr:es-entation of reality. Nevertheh_ss, the eX;jI.mp1e should make C1e;jl.T tl);jI.t the 'yst;e[!l~ ",e ,~~e inte~ested in may have two types of ~nput ;jI.nd two types of Ol,ltPl,lt ~

The fiqt_ typ" of input, <:alled control input, represents the ~ctions that

one caD. ulldert~ke to regulate the SyHem. The second type of input, called

C'XOgCI"lOU.s input, standr:; for (unknown) influencec. ~ntering the syst.ern from

ol,ltside. The fil.'"st type of output, called mellsurement output, >:eflects the state of P(lTt of the variables involved ill the system. The second type of oueput, c(l11ed exogenous output, represents the outcome of the system

r€lE::vant to the outRide world.

in contr,,"t to the above ""ample, all system" in this monograph will be ["CmuldLCd mathcmaliccilly only ~lld will have a fi,,-ite-di.\)ensional J)nea"

time-invariant state space representation.

If? foy a Biven ~ystern~ we have an undesired transf~r frOm the ~xogenuus \11Ptlt (~) to the exogenous output(s), this featu,e m"y be reflec:t",o in the m~.J.::;ur0nu~nl uulpul. th~n, noticing this;jo we. L<lU t'(y to improve cll1d

com-penl->ate t.hii-> undesired behavior by applying dn appropriate c.Ol1trol input.

Sint:~ Wi:! are interested in automatic. control~ 'We asSume that lhr2s~ c.ontrol input.:= .:in.: gr.:!n.:-r=.ted by .:J. dynamical :=ystem that is drivt:I1 by the:

measure-ment outputs. We. c,.ll this t.yp" of conlrolh:r dynamic (measurement)

feed-hac:k ,~nd we .!1.c:sume thAt these feedb .. '=I.ck :=;ystem.c: also ha.ve a finite-dimensional linear time-invariant state ~pace repr~sentation.

The cont,ol prob,eros that we ~tudy for a siven system will be formulat~d in te,ms of f~nd~ng a dynamic feedbac~ such that the 'controlled' Or IclD~ed leap' gystem satisfi~5 certain~ in advance giv~n propertie~. We say that a given control problem is solvable if we can actually find and construct a dynamic feedback that meets the requirement~ of the problem.

iii

In order to tackle the control problems in an ele.gant way, We shall adopt the so-called 'geometric approach' towards linear control theory. For a

fundamental treatment of thi. approach, we refer to Wonham (1979). The genera 1 izations and modifications of the theory presented in this refer-ence that we shall need, can be found in Schumach~r (1981) and Willems

(1981), (1982), r~spectively.

The main f!!aturES of the control problems successfully solved by this 'g~om~tric approach' involve decoupling (see e~arople abave), alroost deeoupling, internal st3bili~3tion and input/output stabilization.

I~ Chapter 1 of this monograph we introduce all the SyStem theoretic con-""pts needed for the developrrlellt of later chapten. Furthermore, in Chapter 1 we introduce moSt of the not~tion.

In Chapter 2 we study our £b:st actual control problem. l'he basic motiva-tion fOr the study of the problem li.es 1n that fact th .• t it in~1110es the extension towards measurement feedba>;k of control problems considered in llautus (1980) and Trentelrnan (1986).

In Chapters 3 and 4 we con~ider the <i'xtensio[1 towards measurement feedback o{ pl:"oblelns of 'nonintcrac ting control' and 'almost noni.nteraC ting control' a. formulated in Wi llem. (1980). Aho in Chapter 3, we extend and give alternative frequency domain oriented proof. of some of the results in the latter reference.

Finally, in the spirit of Chapters .3 and 4, and £nsph:ed by Wonhaffi (1979, Section 9.8), in ChaptH 5 we fOl."11luhte and study a nulrlber of new problems c:oncI<rning 'triangular decoupling' and 'almost triangular deco"p 1 ing' by measurement feedha~k.

INTRODuCTION

In chis first chapte~ we shall inCroduce all the systems theoretic con-cepts that will

be

important in the dG!veloJ?!llent of later chapter~ where we study actual control problems. Furthermore, in the present chapter .. e shall introdl)ce most o£ the notation that we use throughout thi~ wOrk.

Th" systems theoretic concepts Co be introduced all have their origin Ul th", geOmetric approach towards linear control theory (cf, Wonham (1979)). Origitl.cllly~ thobo concepts were d~fined in the time domain u$ing ~t;.,qt.e space descriptions of linear syst~ms. In this monograph we shall introduce the concepts in a uniform way by meanS of frequency domain descriptions, also starting from state space representations of linear Systems. The con-nection between our definitions and the original description 1n the time domain, and vadous properties of the con""pts will be given in the form of propositions and theorems.

The linear System that we shall use for the introduction of th~ various CDncepts, together wiCh its dual, will be desc~ibed in Section 1,1, Also in Section 1 t 1 ~ we revte.w some elementary facts from linear .a.1BebX"~ ~n,d matrix theory. In Section 1.2 we describe what we lIlean by stability and stabilizability of linear systems. In Section 1.3 we diBcUBS the notion of

inv~rianec and the ability to achieve invariance for linear systems. Section 1.4 will consist of a ~ombinstion of sOme of the concepts intro-duced in Sections 1.2 and 1.3. Finally, in Section 1.5 w~ give an intro-duction to the notion of dynamic feedback (cf. Schumacher (1981)). This notion will play an important roh in this monograph.

2

1.1. Basic concepts and notation

In thi s section we start with giving the description of the type of system tho t; p 1 ays •• n importanL r"h: throughout this monograph. These syst"m~ cO[lsist of a c.omOin .• tion of il. linear inhomogeneous first order ordinaJ:"Y dif(erenti~l equation and a llI"u::ar algebraic equation!

(1.1 il.) ~(t) Ax( t) + Bu( c) t " 0 , (O:(t) - ddt x(t»

(1. 1

bl

yet) ~ ex(t)

Here~ the variables Xt U and yare 1:"ea]-va.lued V€C.tol's and A, B J'Od Care

real conHant matricc" of suitable dimensions. The indopendent variable t will b~ inturptclcd us time.

As t1l1nou1)ced, systems of typ" (1.1) plil.y an important rol" ,n th" present work. On the one hllnd, this type of sys tern wi 11 be used in the pre>;eIlt chopt<'.r for the intr-octuetion of some Ey~temS theo~ecic concept. needed i,l the developlllent of Illter- Ghiipte.rs. On th", other hand, sySiems of type (1.1)

~an be considered to be the fundament of all the systems thac we shil11 stl)dy in tho"" lat"r chapter>;. In fact, the,,,, the systems always COIlsist of il cliEfcrential equation, nbtllined from (l.la) by adding extn term~ to

the ri"hL-ha"d >;ici<: uf (1.1<1), the algeoriiic equation (1.1b) and <1 numbe, of additional equations of type (LIb). Consequently, the vari<1bles x, \) ilIVJ y wi II be p~esent il;l ~_very system that we shdll study. Throughout this monograph we cal I the variable" tho:: stil.te, the Viirl.i101.e 11 the (control) inp\lt and the vaciahle y th" (measurement) output of the sy.tEm. Furthennore, WE::! always a:!::i~ume that :x is .a "{"eal n-vector. u i~ il real ,II-vector and y '" a r"al p-vcctor. Correspondingly, A is a real t'I x n-matrix, II is a real n x m-matrix ilnd C is iI real p ~ n-matrix.

In the present IJork we dc:nut.c the field of real numb"r~ by IR and the field uf "omplcx numbe):s by (:. F\lrthermore, we denote the re,.1 l i.near space of

~-v~c totS wLth real compOllel1t8 by IRa and the r~al linear spac"- of aX

b-mdt["i-c:c:.: wi.th t'C.:..'l.l

cl),tri~s

by J]fLxb,

He.n~.e,

formulated in this notationjo

we d,"umC thruughout thou x(c) E lR't. u(t) E IRffi and y(t) E J)l.p tor all

For obvious reasons the equac£on (1,1,,) will be called th" i;tat~ ~volution

"quat ion and the equation (1,1b) will be called the (measurement) output equation.

Throughout the present work we assume that any control input u: [0, ... ) + IRm is a piecewise continuous function. Given such" control input and the state at time t .. 0, x(O)

=

x

O' the solution of (1,10) is g~ven by the well-known variation of constants formula

t

(1 .2) _x ( ) _{t "e} tA _{xo +}

J

e(t-T)A Bu(")d·· . _{, ,}

o

Not .. that the linear system descdbed by (1.1) is complet~ly oetenni.n",d l:>y the matrices A, Band C.

By letting T denote matril< transpositio[l we can also associate anothl" Syst~nl of typ.e (1.1) to the matrices A, Il and C. This system is call",d the dual system of (1.1) and is described by the following equations:

(1.3a) ~. (t)

(1.3b)

We want to point ovt that o~r notion of d~al system i6 slightly different

from the \151).:>1 nocion (If du.:>l system (cf, K.:>lm3n (1969), Brockett (1970»).

We. conc1udli' ~he P:Ii'SIi'[lt 5ect~on l:>y reviewing some basic facts frOm matrix theory and linear alge.bra. First of 311, we shall use the symbol 0 for ,Hl.ything tha t is zero (as a number, vector, matrix, Jine.:>r subspace, etc.).

I f M E: ma'l:> is a given matrix. then we denace its im.:>ge by im M and its nullspacc by ker M, i.e.

im M {w EO IRa

I

::3 v E: IRb: w .. Mv} and ker M = {v E lRb

I

Mv = O} FurtheJ:more, we say that the matrix M is inj ec tiv~ (or: hao f\11l. co llJIlln rank) if ker M ~ 0, surjective (or: h~6 fun low 1:ank) if im M " ma and regular (or: non:lingular or inve.rtibl~) i f the matrix M is lLljectivc as well as surjective. 'rhe latter implies that a regular matrix is square and hilS full rank. We. denote the inverse of .:> regular matriK M by M-1, i.e. 11M-I = M-1 M .. I, where 1 denote.s the i.dentity matrix of appropriat"

4

~iil\~~115i.ons. All t'1~ ']ine~r !5'Ub5pace.~ t'1~'=J.t. we work with will be re:.~l vector

spaces, where sometime,s we use st<:mdard complexitic~ti.ons (cL Arnold ('973» .

tJ tJ • , b .

I f v, ,~nu v 2 ,He two llnear subspac:es 1n ill , then we dellote theu SUlU by

tI, "

tl2 and their int"r,,,cti()n by VI

n

liZ'

vf

will denoU: th" orthogonal c:omplf:mf:ol of V, in JRb with respect to the. stand.ud eue.lid""n ioner

prod-llct in IRb. if V, "is H

line:~r

sl"tb.sp.:'lce in lRb1 .)nd V

2

~s

a Ji.ne..;iX" 8UOSP0Ce

"2 ~ b +b

,n lR , then lI, 6:l V₂ denotes the li,~car subspa~(: in ill 1 2 defined as

1£ [oj E ll(b"b is " given .S'IUM·e. mat6x then oeM) denotes the set of b com-plex .eros of the. c.hl'ly,.c.teristic polynomial of M. FlL~ther",ore, if V is a lil''J.c<l:t' SUbb:p.:.H .

.:~

lTI lRb, then we Ray t.hat V is

~n

M-invarisnt :t;:l,ll)sp.'lc.e if MV <; V.

b

Jf V, and _V2 arc two li"""r sub~p""cs 1n lR such that V, S Vl' then th"re exists .:.1. (uuIlunique.) regula.r matrix S € lRb:u:b

th~t

can be

parti.tion~d

afi

S = [S, .SZ,S) s\'ch thH. im 5, =

V,

and im [SI,5₂) = 11

2, If, in (1.dditic>n,

VI

,~nd

V₂ He M-ilwQ1:lant 6ubspil.:es, theI, with rCGpe.::t to the b<."lsis in IRb

formed by t.11e c.olullos of S, the m.Jtrix M hcl~ the fullowiI\g fur'"m~

wher", M

I, , M22 ilI,d MB are "quare matrices. Clenly, oeM) = ll(M,,)

U

O(M

22)

U

O(M33) ",here U denotes the union ..,ith any cOn""on C' lemcnt" r~p"at"d (ef. Wonh .• m (, 979» •

We ddi"" o(MllI

l) := o(M,I) "nd rl(Mlv/v1) := I1(H2) . I t can be proved th.~tt th'i 5 doF..finition is indc-pCI1dcI\l of the. p:i.rticl.,Ilar choice of the matrix S

=

[0,

,5

₂

,S).

In the present 8~~tion w~ introduce the c.on~ept$ of stobility, stabiliza-bility and detectastabiliza-bility for the cl~ss of line<lr systems described by (1.1). To thi,; ~nd.

w"

consider" l.inear systel)) of type (1.1) and for a st<lrt we aSSume that B = 0, Furthermore, for the time beirt~, we do not pay any attention to the measuremellt output ~quation (LIb). Hence, we are oea ling

wi

th the linear Utltollomous system descdoed oy the homogeneous di£ ferenda! equat iOll:

(1,4) ~(t) = h ( t ) .

I'ro11l (1.2) we kn<'w that for a given initial state

Xo

the solution of (1.4) reads x(t) - etA "0' A5 ,on i.ntroduction to the general

d,'£init~(m

of sta-bility of SyStem (1,4) we now define:

l)efini tion 1.1:

The linear syscem (1.4) is called

asymptotically staNe

U f~ x(t) for every initial state

Xo

E !Rll•

The ~ollowing thcor~m is well known (cf. Arnold (1973)).

Theorem 1,2:

The

foHwirlg

etatementl1 are equivaZent:

(1) The linear system

(1.4) is

asymptotically staH(l.

(2) cr(Al £;

q:-,

wheY',"

q:-

;= {z E: ~

I

Re " <: o} ,

(3) 1'he mUoltdt

matrix

(sl-A)-l has

aU its

poZes in

q:-.

o

throughout this morto!;raph we denote the set of r .• t~onJl functionS with l,'eal co~ffici"TIt~ by JR(s). IRO(s) and lR .. (s) will d~I'lot" the 5et of proper r<ltional func:t~ons with real codficients and th~ s"t of 5tri<;tly p~oper

rational hmctions with rca1 coefficients, re~p~ctively. lR<l(s) , JR~(s) ,,,,d JR:(s) will denote the set of a-vectors with entri,,~ in lR(s), lRO(s) and lR ( ) _{+ S , rGsp~ct~ve} . 1 _{y,. W11 B} I . 1 max)) ( ) _{~ t lRo} axb ( ) _{:s an} d lR~'b () . 11 d _{-+ !:i W~ enot.e t} h _e

~et of a " b-matrices with entries in lR(~), lEO(s) and IR+(s), respecrive-ly.

6

i f

L<:N

1:i O-'t·)· with (i E;; TRb fDr all i:.:

~

is the power ljeries of a ratio,Ml vector

t.

E

m

(s), Le. F,(s) = L~N ~i s - ' , and i f V is ;j. lin""" subHpacc

j." lRb, then we say that

r,

E V if si E V for all i

~

N. 11.1 p;j.,tic;ular, wt,o s~y that ~ = 0 if (i • 0 [or all i ~ N.

By Thcorem 1.~, ;j.symptotic H"bility of the systelTI (1 .4) i~ related to the particular ~ub~ct

q;

of the complex plano;:. III the p,ese.nt work wI< "h"ll relate ~tabiliLy of the system (1.4) to mor" gCr.eral subsats of the com-pIe" plane, su-called 'sta1)iHty regions'. w~ shDll restdct O1Jr~elv~" to

'SYLIU\loet ric' st(lbj 1i ty regions. Thal is, we sha 11 eons ider the s~ l of stL.bUfty ree;i""s 0, defin"d by f):~ {rf; ~

I

A E

r ..

A E

r

",Id IRllr

f

~}.

llere, tl1e overbay denotes comple" conjugation. It is cl"ar that C E (')

'l'hroughollt this first ch.~pt"r we "s~LIIlle Lhat a st~bility region 4:

g E fl is given ('g' sL:J.ncis for 'good'),. and we c.all B. rdtio!'Ldl function, ve~tor or m3td~ ~t,'bl" if th" latter has all its poles i.n IC

g.

(Ill l<1tcr chL1pters we sh.~.I] "ften be d""ling with pairs of otability regiuns,

ICf,f

_R E fl with C_f ~ 4:s

('f'

~tands for 'fast' and's' st8nds for

tHlow'). Th~rr.:!~ we ~h[.tll call a n'ttlon31 functioTIt v~ctor or m..JtJ.'"'.x

f-st"ble. ,-,t"bl" if the l"tter l1(!S ,,11 its p()I"~ in

t

f, (8' rcspc,c-Lively.) Thus, given i[:g E: (.). in~l'ired by the Lhi"rd stateme~t of TheorelTI 1.2, we define,

Definitit!n 1.3:

The liM,n

8yst~"IlI

(1.4) is c;j.\led

stable

if th" rational matrix (51- A)-l

1!'5 stable.

Then we: halVe:

~:t.:.:C.;9.rem 1. 4 :

Tlw linear' sylc'tern (1.4) {:l

,nable

·if and only

if

o(A) :;; 11:

8.

A~ ~n alternative for De£illitioo 1.3 we could have defincd~

The linear sy,Lem (1.4) is Col! led st.'ble i.f for eve.ry initial state "0 E ru1.1 -1

therali0Iwl vector (sT - A)

Xo

is st3ble, or:

The linear system (1.4) is "alJed Rtable if for every initi"l state "0 E lRn thHc cxisL~ u sLuble rational v~ctor ~ E lR~(s) ~~.c;h that

Xo

= (~1-A)I;(5).

still exist :i.uit:i.al ~t,.te~

Xo

E

mn

51lch that th~ ration"l vector -1

(sI -A) )(0 i.s stable. Therefox" t11e followiug defird.t:jQn makes sense,

Definition 1.5:

It can be "hown that

Xg

(A) is the ?UIn of th~ gen"raU"e.;l e:iB~n.spae~s d A corresponding to the eigenvalues of A ,n ¢:g' Fuxthermore, :i.t

where p~ is a real polynomial ",ith all j ts zeros j,n tg "nd P

g is a o:e81 polynomial with all it~ :;::ero~ ~n

q:"

~:B := {z; €

q;

I "

rt.

q; }

thcn X CA) =

g g

= ker P B (A) (cf. Wo"hilm (1979». Likewioe. Xb (A), the ",urn of the gener81-ized eigcngp~c:es of A co:a"sponding to the eig env .• l1J.es o~ A in

t"t

g, is equal to key Pb(A).

W<l now r"turn to th" lin"ar sys tern .;lese>:" Ded by (1.1 (1) <1Ltd IVC drop the assumptioIl that B = O. As an introduction to the concept of stabilizabil-ity for SyH<lms described by (1.1 a) we define:

nefinitioIl 1.6:

The liIlMr system (l.la) i~ called asymptotically stabiU~able if [or every initial state

Xo

E lR

n there ex~5Cs

a

piecewise continuous cOIltrol input u: (0,00) "' mID S1J.ch that liI!\ x(t) ~ o.

t _

Then

we

have

(d.

Hautus (1970). H"utU5 (1980):

Theo~em 1.7:

The

following statements

a:t'e equi'vaZent:

(1) The Unear'system (1.1a) is aSllmp~otieatly "ta'biZiJiJ(J,bk. (2) The)'e exists a matrix FE: m1l\Xn sw,h I;hal; a(A+Bf)!': it-.

en

For every initial stat" X

_o

E

mn

thrd"N '!-xist mtional vectol'S

r,

E: :1R~(s)

and

w € m~(s), 'both

with onty po/,6:[; in

4:-,

8=h that liD = (sI~ AH(~) - BUl(o).

8

In vicw of the tillrd ·Hiit"m"nl of th~ dbove thcorenl we define (see also Ilcl"tCls (1980) for., ~Iightly diH"r"nt definition):

Definition 1.B:

_ II ,

1f for a giv"" initial st'lte Xo I;. lR 8o'ng with th" lin"a< system (l.la) th"n: e:<ist rational vectors E lRI\s) and I~ E: lR"\E) such lhal

Xo = ("1- A)f,(sl - ll,,(s), thon tllC l~tteT expr"s~i.(}n for Xo is calle.d a

(r,''')-'''q."y,~.]ntaho" u[ Xo ("0 is s,lid to have (\ (t.,w)-re.prese.ntatiDn), We ;;hall e,dl ~

U;

,1.,)-represeLltation 1'('rJ,"l.Qi' if both ~ ",nd c.! •• re o;trictly

proper ratiDnal veclors dnd we shall (all c.1 (s,w)-repr.es~ntat:i.cm atabl.£:' if hoth nnd W Rre stable rational v~ctoc~.

(In l~ter ChJpt~1"5 when l[:[,(:s E: (:l ,.r" two ~t"bil.ity regions, we shdll call

il (~,~)-r.pr •• "ntaLiun f-.lablQ, s-stablc if both rational Vectors € and w are f-stahle, s-stable, ~e~pcctivcly.)

We (Il~{m tl'~t ~ r~8ul~r (~~w)-r~pr~H~nlatiorl uf Xo can be considered 8S rh~. 1.:'1'1"0" ~":In"fm-m <Of (I. la) wilh iniliul stelte x(Q) =

Xo

ono

u: [0,,:.:.) ,). IR('(I A pic:ct::;wis\::': <:':oIlLinu(Jus t:onlrol input, provid~d ('.t;':!rtain (".on-dLtions "'"" ""tiofi~d. indeed, if both functiol1s )( ~!1d 1) ~s~oci .• t:~d with

(I. I,,)

a,."

slIch that ltl~i," L"pliiU: lranS[OnnS exist Mld ~r" strlc'-ly prop€:~r l:-ational v~ct()rs~ then thE:! ~xpr~b":l:::iiOIl xl) = (sr- A),:-,(s)

b~ ';0I18~d",.",(i to be. the T ... 'pl,,~.e trandorm of (1. 13) wilh x(O)

FunctiollS tiltH h~ve <l strictly prOpel; ,,,tiO\'JI T .. "pl,,<:e transform ar" called .~".,h; /:I~:"I .. {O"'; (eL Trelltelmdn (1986). I f h is a Bohl function th~n it

CCln he !::;bown tkLt there ~K..i..!::;t r'r:::dl mdtric:c;8. f, G ,Jod H of suit~~bl~

rlimo::n-si"ns

8\I~h

thClt h(t) = Feet H. 1f in this representation o[ h the Jimension ,of G io; t.'k"n [Ilinim"l, then '''' d.fin<: o(h) ;= a(G).

Inspi.rol.::!d hy th~ thi.rd atat~m.::nt of 1'hcllrcm 1.7 we now clef:l.~,t.."! th~ concept

nf st:li)iliz.Jbilit.y nf ~hr. line." "y"tem (lola) as follows. To this end, let ~g E

e

be a given stability region.

ncfj\~i tion 1.9:

- - _ . , _ . _ - _ ...

Th" lineilr SystClll (1. L1) is called stabU.i"iClbk i f ev"ry iuili.<ll slate

Theoreffi 1.10:

The fO

Uawing

statemen ts are equiva I.en t:

(1) The tineal'system (lola)

i$

stabUi;;able.

(2) Thin'e flxists a matdx ," EO mffi'n suah that a(A+ BF) .. ~g'

(3) Fcp flveT'J/ initial. state "0 E lRn there exictc (J ['"U""]1J1:~e ocmtinu.ou<'

aontpol input u, [0,"') -, lRffi sWJh that o·(x) SO

«:g

(tfleanitlg i.Jlat x: [0,=) +

mffi

is a Baht luneticn and o(x) SO I[:g)'

(4) Im (Al-A) + im B =

m!\

lop all,

E

C"C .

I;

(5) Thepe e:c'ist stable pational matrices X E 1R~xn(5) and U E 1R:xn(s) such

that

1 = (81 - A)X(s) - BU(s).

Proof:

Se~ Hautus (1970), Hautu5 (1980).

•

Note that the stabilizability of (l.la) only depend~ on th~ m~tr£ces A and B. Therefore, instead of the stating that the linear system (I,la) is stabilizable we shall often say that the pair (A,B) is stabilizable. I f th~ linear system is not SC~bilizablc, th(!r" may still "xist initial

n

states "0 E lR that dQ h~ve a stable regular (I;,m)-representation. There-fore, we define (ct. Hautus (1980)):

Definition 1.11:

\tab (A,B) := {X

_{o (}

lRn

I

KO has a stable re~ul"r «(:,w)-representation}. Xstab(A,B) is c.alled the $t{)bili'JabZe suk:pace associated wi.t.h the liacOlr system (1.1a) (d. Hautus (1980». I t can be ahown (d, SchUffiachtor (1981)) that

"'here we

have denoted

<A

I

iffi

B>

:~ im B

+

A ,m B

+ " . +

A

u

-

1

im

B

. I

Ai im B • l~O

Here, the last eq~ality is based On the Cayl~y-Hamilton theorem (of. Gantmacher (1959»). <A

I

,m

B~ i~ knowp as the contpoZZab~$ subspaoe {cf.

10

WOnllHlTt (1979)) clnd is ~'1u,d to th~ smalle:H A-iuvnrLlut s\.1b~pacc that con-l.jJin~ im 13.

UnLi 1 !lUW W~ hav~~ ign('H-cci th(:;: m~.!::I.~ur-!::!rnE::!nt output ~qua.tion (J .1b) and WI.::: ildV8 he en [~(ln(~~l~nerl with the gtate evoJlltion equaticm (1.1a) only? where, in first 1n!-ltnllcc~.;t We! havc-='! (\~:f:l.J\11E(l t.h(l:t :R = (J~ In the rem ... dnd~.r ot th-E:!

rr(l;~t~r~t ~,(~(tjol) w(, ~~b~\ll ..:.\g~l:i.l) dSSum(' thD.t B "'" 0, but tlO~ IN(: do .i.flcli .. nj(: ce'1I"'tL<1Il (I.lb). J.1"·n(~, in

ttl"

'-"IlL(I\n<i0r of this sec.tioll we (1\"(' (I"(IHllg

wir.h tl'H~ line;"ll" sy~tC'lI), J(:~s(rib('J by:

(1.4) (1.1 b)

In Seni,)!l 1.1 w(·j,,'v0 introJu(('u thl' lil~uclr' Systolll (1.3), the dUill of ~y~l'elll (1.1). it i.s clear LhaL lhc LinedL' S.y::;LE2m dual to the lin!:::::c.r. ~y::::Lem (1.4), (1.1b) is dc"crlbeJ by;

(1. 'lit) A T x'(t) + C 'j' lI'(t)

Tht:l, i(1ttl;'"r" 1,$ () lix~(';,,~r .:;y.stC·m of th(~ ki.L)J w.i.th respc(:t to ~h:i.ch Wl' bave'

J.1.)trOU1.1«(\J Lh..:.' Il~)ti~m i.)[ ~l'lbiliz.:lbilily t:-2!rlil;.;( lh.i..:::i ~(.'(:li'JrL Now w(:

cJe[i.I1e~

Ilet in it _{, ...} i. on 1. 12 : " . ""

T);" I in",,,' "y"LClil (1.4), CI.IJ) (or ,he, I":lfr (C,A)) is cclllcJ dinedab1.e if tilt" li",,,,,' "y(;t('1II (1.3.1) L.: H,'bil.(~,\bl('.

Dy dUillizatiiJIl of Theo['l~m 1.10 we obluin:

1'hC0r't:m 1.13;

(1) Ti", 7.illew' :JY')/.(:'IIJ (1.4), (1.10)

·i.:;

ddt?e/.able.

(2) Th,,)"',-.: fo':I:{'II .. " rl rnuL-r""i.:t; .T E: IR1\XP ",w:h 1"1Iu/. o(A + -Ie) c: _.

t .

_g 0) kef (AT - A)

n

kcr C

= ()

Irn'

(:/.1. ~ E t"~g'

(t,) 'l'llerl:' e:ri:!I.. :;l.nhle l'ut-imwi. mal..d.e,",;~ l E: J.R~·"n(") and Y E lR:xp(o) ,),,1.' h i .. ha I, I = 7. (s) (~ T - A) - Y (s)

c .

Ddini tion 1.14:

X

de t (A ,C) ;.. ""lled th .. Ul1rkteatab le subspaae a •• oc i"ted wi th the liMar system (1.4), (LIb). We have Xdct(A,C:) = Xb(A)

n

<ker cIA~ (cf. Scl1\)m~cher (1981» where we have oenoted

<ker

ciA>

ker

C

n

key

An ...

n

ker

CA

n-1

n

i;:O

On(e more, the la5t equality is ba~ed on the Cllyley-Hamiltoll theorem. <ker C I A~ i~ known "5 the unobser'Vabl.e c·ubcpaa" L1ssociat~d with th~ linca,

syste.m (1.l.), (1.1b) "no i~ "ql'''] to the largest A-invariant sllbsp~ce con-tained in k~r C.

1.3. Invariance issues

In this section we (:ondder 1 ine~r systems of type (1.1) in conn"ction with line,':lr subspaces that ..:lre invJ.riclI~t Or tha.t can be made invari.'int+ For this purpose, w;, again start with tohe linear autOrlomous system described by (1.4) and we def\ne:

Definition 1.1S:

A lil'ear subspace I! in ful1 is called dynann:aally invariant (with respect to the line~r syst(!m (1.4» if x(t) E II for [l.Jl t ~ 0 al1d [or every initial s e[l.te

Xo

E V.

Then we h[l.ve (cf. Hautus (1980»:

Theorem 1,16;

2'he foUowing statements are eq!~l.vat[~n./,:

(1) The Unear subspace II in mn -is dynamf(:aUy invariant.

(2) AI! !>; II.

(3) For every initiaZ fiLatf.; "0 E: I! theroe exists a rational veetO)"l

Observe. th.:1l [((rILl th0 S.r.?:('.cmd ~t.atE:::m~!lt of TheOTP-m 1 f 16 it. fullo'Ws that

nntiOH!-i o[ A-invi.H'i~lnC(", .1R i.ntrodw..:~d ill. Section 1 f 1, and dY!'l.i:l:.mic

inv;J"("i-~1\1('€~.,. ~:lH defint:d ubuvet nrc in fact thl.::: S.:.J.mC .. TheX'ef0t:'~, in thE:! !::iequt:l we

~h<.I'I·\ rcd('~1- t.oO dyn.a.mi<..:. iIl'V~Jt'lunce DS A-inv.9.ridnce.

If Vi:; I)ot <.111 A-{nv.'1.l~i:·lnt :::ub~p.a<..:c in JP.n tbere. m.'i)' ~lill ~2';ist \J"lltial

.,t"U's xI) E V fur which there exist., ~ "atiul\al "",(to, <; ( 1R:(H) Bueh thnt

Xo

= (,1- A)f,(,) and S ( V.

A55ume! tI1.-::t.t V = ko:::r H fur ~O!TlC" i...lppropriate mat.rix H. Th C' (I 'w(~ C~H'I cif!fine.:

Del'inition 1.17,

*

I (ker H;A) - {.x

o € ker II I thtF!:t"e. .:!xi~t= a rut.i.i)[l;Jl vector

Oh~H~I-V~~ th:l.t, iF

tlw" ((~)

Y

_L: O

'f m:(s)

su~h

that

Xo

= (~l~A)f,(s) .~nd

H(

=

0)

tht: rULillfHl1 Vt;.-.ctor t f lRn(/-;)

i - (i+ I) . ' . "'.

A xo'~ . I f t J.n D0e'l1 t'lon,

u; ,;u(h th'H

Xo

= (.'1- A)i;(:,;) HI;(") ~ 0 [l1Col) HAL Xo = (I

o.

So, Wi.:.' c·ould ju::;.l <.)::;. wI.-.J·1 l~<.lVi? dc~f:in.:!d~

l*(I",r H;A)

,=

{xU

c

mn I there EeXiRtS d ration<ll v<"nor

<. ( lli~(o) :,;uch tlwt xO= (sf-A)F,(,) illld

W;

~ 0),

1* (kcr H;A) _{Of"" all} :. 0) .

It fol.low:,; LhilL 1."(Kt·r lI;A) = <:k"" HIA>. Hellce, r*(k(,,· ll;A) is th" l"rse~t

l\-inv:-1ri.::Int HllhHP.:l.t:C l:unlulned iii k!:'r Hr

Ag.:.1111 ... W~l. r ... ~turn to Lho..: linC:i..tr system J~!;l(~rih~d by (1.1<1) .:.n(l W't~ doC:finc (d. Wonllcllll (1~79), (Hallt\!" (1980), Be\sile (11'd M.,rr" (1969)):

l),,[illi Li()11 1.18:

A I incHr Hub::;p..:.i<..:(,,\ V in

mn

i$ (,:Rll.l!d a eont·p()7..7.,,:~d ·t~nvay-"-{ant C<.H·~ (A~B)"'"

z:nl!al'ianl,) ;;llh,p'-,cCo (wi th Ie Sl'<"ct. t." the linear system (1. 1 .•

»)

i.f every

If we omit th.:. assumption th8t the (;;,w)-rel're"~nl"tio'l should be r:egular, We obtain (d. Wille~ls (1981), Trentelman (1986»):

Definition 1.19:

A line.u 5ubspac" V in mn is called an a7,modt contI'oPed {n:D(iy·i,mt (or: almost (A,Bj-'irn)ci1'iar(t) 5ubsp""" (with r~sp"ct to the linear system (1.la» if every initial state "0 E: V hafi a (f"III)-n,pr(:"cntation such that ~ E V.

The names attached to th" two type>; of 8ubs]Jaccs dofi[\ed above wi) 1 be~ome more clear 8£ter we hav", stated the following resl.llt~ (d. Wonham (1979), Hautu5 (1980), B~sile 8nd Marro (1969»;

Theorem 1.20:

The foUawing statements are equivalent:

(1) The linear' SUb8PW;:,' V {:; (/{mtlvU(ld l.nval"ianL.

(2) AV

~

V

+

im

R.

(3) 'f'hef'e ex'U;t8 a matr-ix ~. E

np'n

{JW]il U"d (A + Bl') V s; V.

(4) For' every initiaZ state Xo € V thrJ.f>e (lx{:;I.:; (1 pi$c'ewi;;e cOl1th!uouc control u: [0,00) -> JR111 ~uc:h

that

,,(t) E: V fop all. t ~ O.

from l'h"orem 1.20 it is dear that starting in a controlled inv~riant hl)b-space ODe can St8Y in that subspace by I)sing an appcopriate cOntrol input, Moreover, it follows that the I.:<tter cal~ be (lone by m~.:<n~ of. a f~,ed1)ack

~ontrol law.

Theorem 1.21:

Th,] foUcrwing statements are equivalent:

(1) The l in eaT' subspace V is an a i.mo8t con tx'o /. Zed {m'apian I; subspace. (2) FoT' aU ~ > 0 tlJere exists a matrix F E: IRffixn 8WIJ that

inf l!e,(A+BI'l NO - vII :: c for' all t

~

0 and for every "0 E V with

vEV

1111011 ;; 1.

(3) Fm' a

n

~ > 0 md tm' <>vePy

Xo

E V I;h,ue exists a p-i(?'Nwise can tinuous aontT'ol u: [0,=) + mm Buah that inf 1I)«t) - vII ~ ~ foY' aE t ~ O.

vE;V

proof:

14

l'r"m Tt"",t'<':1II 1.21 it fOllow" that starting in an almost C()"O:oll"d

invari-(lnt i-iUbI;i.P';'H,:'r.:: Ol)e C,:-!l1 ~t.,~y arbitralrily ("lo!;le to lh<lt Subsp,3ce hy using

spp["opriatc ("ot'l.tr01 input.H. Moreover, it follOWn LhC1t the l~t.t.e"( c.:J.n be: il'l~o achi~v~d by ;)Pl"l,,(OP1-j::]te. fe.~dbc:a(:k (:OI.1t"(cd 'IHW~.

t.\gi,dn, a:=;.strrn/:::: thal V .... ker H for ~omc.:. ilpprt'1prL.1te mat.rix H. Then, much l.n

tho spirit, M DefiniLion 1.17, "'''' "'in d~fLIlC (cf. H"uttlo (1980)):

Ocf.L[liti()Ll 1. 2<:

lI"(ker' II) (= V*(k~l- H;A,ll)) :~ (x

O E k"r H

I \)

lius a re&ular «"I",)-r('fWCS'~t1t~ti()n ouch Uwt lt~ = 0) .

N(")t:~:, that we :iu~l n::: well CO\lld h.:lve defi.ned:

1J*(k",- H) := {x

o

E' ron

I Xo

has d reguL:.r

(f"I.')-r~l'r"H"nt"L.LOn

Huch th<1t H~ = OJ .

J.nclcc'd, if ('11(' '-,<ltionid v(oe turs ~ C 1R~(") lind I"~ E m:($)

.n"

such thOlL fur n

Xo

r;

m

we have "0 = (s]-i\)~(~) - l\),,(s), then it follow,; Lhat s«") = xO+ S'(,) whcr~

VCs)

= /lsC~)+ 11)"(0) E

m:cs).

I f , in addition,

lit, ~ 0,

tI"'"

HxO

=

0 ""d Hr,' - o. H","c~,

Xo

E k~[' H.

V*(k~r H) (= V*(kc[' lI;Il,B» ;= {x

O

e

keI' II I x() l,as "

«(,),,)-:! ~l

['c:Pl'('Sc"t>Jtinn ,uch LkLL 11'; 0).

Wt:· -/'ikn now TloL rt:placc kl)r 11 hy LRI~. ~\h.i.::; ib c ... I..Ist;!d by thi::: [ncl that the (f:.~I!))-rl,;-'prl:'.Sflnt.;;:ti()n i~ ~ll1l nc.c:cssnrily n:.guliir-. Th0.refo:r:-~.:::~ WoC~ d.::.f:i.rH.: ,('p~lr"te1y (~f. S"humachc[' (1983));

V~(kc1"

H) (=

V~(kcr

H;J\,fJ») :=

ixo

~

IJ{" I xI) hOi,; " (1"",)-l'epresentilti on such that H~ = OJ.

*

*

Of cOI.P:se, V.1(ker H)

=

Vb(kcr Il)

n

kel" R. Furthennon" we have (eL llDut\'~

(1980), Schum~cheT (1983»):

Proposition 1.25:

(1* (ker H) ic the Zar'rwst (:IYI1LY>oUed invariant <Jtlb~paee -in ker Il.

(1* (ker Il) is the ZaY'gest aUnost (!ollt!'olkd i111xu>{Qnt :J!<kJpaee ·in k(:r H. <l

In the sequ"l

w"

"hall not need the. subspace V* (ker H). The subspac,,"

<l

(I*(I:;er H) and V~(k"r H) will be used quite frequentJy in later chapters. Therefore, it wi 11 he necessary to giv" st:hemcs tlwt can be used for the calculation of these sub5p.~<:es. To th • . t 'md, we "t«Le the following (llgo-rithms where we assume. that the m_~trices A, B _.nd H are given

Ccf _

Willem" (1981), Wonham (1979)).

Algori chIllS 1. 2~:

COIlsider th.e following iter~tion processes:

(1. $)

_Vo

_{ker H, V} k+ 1

=

ker H

n

{x E mn

I

Ax € (Vk+iro B)} for sll k ~ 0 (1. 6) _"0 im B

,

_R k+1 im B + {" E

mn

I

x E A(Rk

n

k~r

H)}

for all k 0

By induction it C~n \:>e proverl tl1~~ 0 S Vk+l ~ V_k S IRn and 0 S R

k:;; "k+l £; S JRn {or- alJ k ~ O. So J.t h cle~r ~h~t. the limits V= ('nd R"" exist and are. determ),ned within" finite nl.lI\1her

U

n) of tterstions.

Then we have V*(ker H) =

V~

and V:(ke,. H) V +

R=.

*

Of "ou r" e, Va (kE! r H) = k e r H n (V ~ +

R,) .

We now consider th~ lincar System dcscrib0d by (1.4), (1. I\:» together with ~he corresponding dual system d,,~(;r ibed by (1.3,,).

We define (ef. Schunwcher (1979), Ila,ii e and Marro (1969), Willcms (1982»:

Definition 1.27: ---~--~

A linear subsp_~c_e S in IR i, called a eonditlonc,d lnpariant (or:

(C,A)-'it!vQyiQrJ~) subspace? (with respect to the. linenr system (1.4), (L1h» if th" lin"ar "ub,;p'"'''' S1. in illn is u controlled invuriunt

subsp~ce

(wit11 r~spect [0 th~ dual "ySt~m (1.3u»).

Ie

l)c' [i'\i.ti.o."_ .. ..!~:

A linl'nr SUUsp.'lce S in l1\L~ in c.:al1l'..:u ':"'([1 a7..1Tfll,<;!. (:on(lit-i.on.(~:d -(n,1JGF1:ant' (or:

a/.lrr,,;·!! (c,II)-hllJcu'/cml.) :::"h:rXW(1 (wilh r'c~p(:ct to tllelin""r ,ystE!m (1.1,), (1.Ib) if lhl' Linenr

sl'l>~r('C"

S1. in IR" io an

~l[[]ost

control'leci inv,.r;,.nt sub."p;I"O willI (e"peel lO thc' do,ll $yst",m (1.1il».

Tht:ur"(.'1U 1. ;~r9 ~ _._---,,-_ ... ,~".

(2) ~(s

n

key C) C

s.

(3) 'l'her'<I' (';1:,:,,1.;; a 1'".li..l'lIX· J E lRnxp "uuh tllol; (1\ + JC)S <;: S.

JI/I!'! /..'lnclZli r~/..tb.~:pO'(!r'

.s

I.:r (HI i',~7.mc):1!.. c}orldi-'t'{onf!d /rr()(+r·'·'>l'Y1'f.. I;~ih~~pnoe l.j' ant.."i

(lilly i1' 1'01' al!. '.-. ,.. 0 t·Ie,.'I·I.' I::".{"I .. ,' (.) I'IQ/:r'l:.: .1 E mll'P l'lAl:h I.hal.

Tl!i.. .... Ju~ll \.'i:.rRl(in~ of the fourth :=.l-ULcmcnt of 'Theo).-ern 1.20 and the third .!i.LaU.=.:me1.lL o[ Theorem 1.21 uo not h.rIVe. ,1'1 ~trCl.ighL[UI'wi.\rd i!"J.teX'p1:·~7.t':"!t.JQn in lil" l'(.'nLl'XL of tllc' I ine.~r syst"m (1.4), (l.lb) and Llt'c therefore left aut of Thcc:.rcl1\s I. 09 ~ndl. 'jO_

If, lU (.\ddit'J(l!~ to the"! m.'ltrices: A and C, .:.l uwtrix C ot ~llitable dirni.::!l.siO(l.S

is glv("n, t}),('11 by c:hJ"11i7.,~ti(')n W~ ~8n ubU.lin Lll(' fo11owing (ef. will~m~ .t:H'Ld

C(,mmc\u'IL (1981), SChUIMC[1",- (1981), WilLems (1982)):

VefiLllLlun I. 31 :

'k

C) (= S* (im (;;A ,C»)

(V*

(ker CT;AT,CT

»)1.

S Om

.-.s~(im c;) (= S*(im C;A,C) ) ~..:.

[v*

(kor GT;AT,CT))J. il

"

*('

Proposition 1.32:

S*(itnG) i$ the r;maUu;t

aondit.io)wd

-i.nvCll'iant

subspaae

C(Jrlt,1irl"ing imG. S*(im G) ie the $rfitlZr'lst ai.most c{)T/dilc7:orl,ld

ilwadant

~:ub8pc!l]e nm.M{ninu imG.

a

Algorithm 1.33:

C"n~id"r th" iteration pco~es~es (compare (1.5) and (1.6))

(1.7) far all k ~ 0 , (1.8) h r C n {x E

mn

I

Ax E (N k + i.m G») for all k O. n n Then 0 ~ Sk!:: Sk+l S; 1R and 0 f:; N_k+ 1 S;; Nk f:; lR for all k ~ O.

Hence, Sand N exi.st (lnd are determined within a finite number of itel"(l-tions.

N~W,

S*(7m

G)

~

S_,

Sb*(im G)

S

n Nand S*(im G)

=

im G +

'oN 00 00 c.1

+ (5",

n

N,).

1.4. Combined stabi lity and invariance issues

In the pre"ent section we c<msid"r th~ c<Jl!lbinaLion of some of the concepts

that we introduc.ed in the pre.viou~ two s~ction~. Tu thiS '.::r'l.d w~ '(:OI"l.sicicr

the line.ar system (1.1) and w':' as.umC thal (;g £ c,) is a giver"! stability region.

Definition 1.34,

A l h"\~3, Subsp3ce V 1.n mn ~ ~ <cal led .~ ~tabi ~i;;abdity 8ubl!paae (wi th respect to the linear system (l.la» it every initial state

Xo

E

V

has a stable regular (E,w)-repr~sentation with t

e

v.

From this definition it i~ clear that ~very stabilizability sub~pac~ i~ a controlled invariant subspace. Furthermore, it will turn out that the linear system obtained by restricting the linear system (l.la) controlled by a suitable state feedback to a stabilizobility subspace is 0 stabilig-ab Ie system. In f.~c t , we h.qve th" fOllowing

Cd.

H.qt)tllS (1980). Sch\ID1acher

TIH:!O ri:::m I • .l~:

'('hi'~ [1)1. [..ol,J'EH(j H (,(,I t"c,'rr)('N. l,(; (,i:1"O (,)ql.A.lvc:r.ttP'1 t: ,"

(2) (Al-A)V" i.m Il = V ... im Il ./"01' all. E ~..., ~ •

~

(3) 'I'iler'e e:t:i,,/.iC (I I"II!U.r,·r.:;r, F E mlllXLl mE!; (hat (A+ nr")v '" V 117I.d [) (Ie + HI-I V) .<:::

It

g •

(4) Fur' u\!cr11 initial d/:Cll-e

Xo

E V U",J:'[, "od.s/.r; a

'''.I'Ii-r·o/. u: [0,.') ., nin ''1wh that x(t) E 1/ /01'

ObSc'l'v0 th'IL, .i.[ V··

m",

ChC'11 the st~te!J\en<. thM. V is a sLahil.i.zab.i.lity

huh:-ip.:,;t..:r..: impli~o that lhl'...: pi-lit'

(A,rn

i.s st,;,J.td.li~(lb'I{~,. Fl.lrth-erm()re, if

4:g

= ~;~ l!t~n evr..:['y ~uIllrulled inVi..lrL';')llt ::;ubSp;JC'0 is ;:tls() .1 lit::lbiliil . .=:l.bility

i->Ub'<;'PilC~. if lhe liul'dr .sub::;pacc V i[1 lRIl is ~l. st~'biliz~bility Rut~p.=:l.(!~ .:J.nd lh€ pair (A,B) iii stHbilizBblc~ lhi.!.!1 lhcrc. exists a mo,trix f E lRlII).:CJ. such

tl'I,"lt (A+ HF)V >; V lind a(Ie+ Ill>") '" {:8 ( d . Wonham (1979)).

In thE~ ~p1.rit lit' th,(:~ p1·E.~vioUH liE::.ctionH W(! d.r.,:finl..:~

UC'finl.tic\J\ 1.36:

V*(l«'r TI) (= V*(k"r H;A,B)) ;= {x

O ( koor It

I

zO° has a sttlble

g g

regllL"3.T U~,IIi)-r:<::pr.r.,::=enu.:.lli0n wi.th H( ::..: O}

'fhcll wc have ("f. HauLu" (1980»;

Propul:;.iliOII. 1.37:

V* (k<'r H) iii U,,·) i.atVu,;! DI.ab,U·I:;wl.nTUy dU/.r:;[HWI"" In k~!.' \i. II

"

!\.!!:i.C),. tht!:!. linl::::U:' o\lb~p,.li.,:c Vg(l~cr H) wj,'jl ht;! llsed f-t:-equelll:ly ill future

ch~lrters. Th<.:n:for.r.,:~ WI..: pt.r.,:sc"t th(' f('J,[']oi.Jing nlgorithm by whi.ch thjs

suh-Al gorilbll 1.38:

1.et F E ill"l' " h" t\ IlIJlrix sech th(lt. (A+ llF)V*(ker' H) ~V*(kor II), where V*(k"r H) ca" be ""lcululcu by means

or

t.h" it"riiti"TI process dccscdlJed by (1..0). For the compuLiJLiml of lbc' Iniltdx f

w,'

rej'"" to Wonham (1979),

V* (ker H)

=

(X (A + fl1')

n

V* (kcr 1I» + (Reo

n

V*

Ck~r

l:l» ,

8 g

where R~ i~ the limit of the iteration process d~scribed by (1.6). The validity of the above expr""ssion for

V*(I~"r

H) can be

prov~d

st<1rti,)g

*

g

fr<lm th~ <i:xpress.i.cm for (/gCker II) given i,l Wonham (1979), sect~Qn 5.S.

Note that, if ¢g

=

L then V;(ker fl)

=

V* (ker Ill, alld not~ th<1t V*(ker B)

n

im B ~ V*(ker H)

nRc

V*(ker H) .

00 - g

Furthermore. note that, if H = 0 (ker H = rn.n), then V lRn, implying

. ~

that AV_ _~ ~ (/_. and R_ _{~ ~} ~ _Li~O Al im B = <A

I

1m B>. Hence,

' \ (A) + <A

I

irn B>

=

X_stab (A,lll •

We consider now the linear system described by (1.4), (l.lb) togeth"r with ito d~al

(1.3a).

Oefin~ion 1.39):

A linear subspace S in JRn is ell 11 ed a detec cabUity subspace (wi th re-spect to the linear System (1.4), (1.1b) if the lineilr subspace S,L in

nt

is a stabilbability subspi1ce (with rcsp~ct to th.:, dUill system (1.3<1».

Theorem 1.40:

The following 8tatemen te aN equivalen t:

(1) S

·is

(1 det(l<:I,<ibd"ity .'lUbS[!(1(.'('.

(2) (A- H)-l S

n

ker G = S

n

ker G fOT' aU A E

~

....

~

g

(3) Til(!!'''' ex'l:sts a mat'r>j,x J E JRnxp 8ucll that (A + JC)S <:::: Sand o(A+JG I JRn/S) ~

t

g•

Note that, if S

=

0, thell the statement thut S h a dctCC::li1bility subspace implies thilt th" pair (e,A) i, det"c:tabl.e. If q:g =

q;

then .• ny conditioned invariant ~lJb5p"ce j,s •• Iso 8 detect,ability subspace. Finally, if the pair (C,A) i , det~ctable and S is a detectability subspace, thea there exists a me.tdx J E lRn~p such thilt (A+ JCjS ~ S und a(A + .le)

'=

~g'

20

DI~.filliti{Jn 1.41 ~

s*

Urn G) (= _c:;'~ (im C;A,C))

8 g

PropO~_~"t,isl~ 1 + 42 I

S~ (illl C) /0 tt/<'.' IJmal'/"{!'~;i.. deteotal)7~.?,/t .. y DUb~p(b .. ·!e (!orl.Lain"l:ng lIT! G.

g

A I go r_~_~.llln, ...

1.

.!!J. '

LcL J EmIl"» h" " matrix SClC[) t.h.H (A + .lC)Sw £; S"" th.",

(Xb(!\+JC) •

S )

n

(N",+S)

l,-,Iht.:r(~ S-oo ,111d Nco, ,~ri': the limits of the iler,,-)tion proci::!::i.!::ir.2S deSCTfbed by (1.7) <,lLla (1.8), "e~p","livcly.

1.5. Dynamic feedback

Ag.:.lit1~ W("": ('on-, .. ddE:~r th~ linC' .. u- system described by (~ .. 1).

In tbo:.' prE'sent. se.ct.i.on we 8h.111 be de,~ling with '!.lyn[lmi(,.1 f~e.db<H.:k .J.round Lhis "yst~m. This being in cuntrost with the prQvious sectione vhHr~ the ;:;;y~tl'll1 th(~()"('!i:t1c:', ('.onC".~pts intrudu(:l." . .;j were relF.lted l(J 'stati<":' f.:!e.dback in

lhe [()t~lL or J st .. ~t" f.edback u(L)

=

ex(C) or an output 'injection' Jy(t).

!\~ ~1\L\OUllCCd, il1 the pr""",,L ,,,,,::tioo we sh,ll!. h" ac,lling with lincdr dynamic [ccdb!lCk Ulld it wi I], turn out tl)Dt tl,i5 kind of fcedba~~ c:an be ~QnHid~rt.:d a gcncralizDtion of !-;Lat.:.i.C- output feo!:dbat:k o[ the foXTll

lI(t) = Ny(t). The ,)cijC'ctivl" 'dynamic' r"[l,,c[s the t3"t that Lhe feedback can b~ considored a dynamit! :::y:;.tctl) tn thp. 1l~1.lH.l .::;cnse w1th a f:itatc

evolu-t:.i<.>!l. ,.:quutlOI) Clnd ,:In output ~qui:.\tion.

DY!1.:.tmil..: [('ecJbn~l< I .'1'1 ~o ea L led ciyn':'Hld c (',ompo.::n::;.c.t Lion,. wi'l 'I play an important

called (dyn<1mic) compen~ators, th"t we shilll Ub"

en

tho sequel are de-scribed as follows: (1.9,,) (1.9b) .;; (t) 1,1 ( t) Kw(t) .. Ly(t) Mw ( t) + Ny ( t)

Here, wet) E JRk denoCes the state of the

c()mpe(.~d[()r

alld K E JRk'k, L eo IRk>p, M E mmxk and N E :1F:nxp•

Th~

integer k ::: 0 is

call~d

thE (JX'&'" of thE c()mp~nsat()r. In the case ch<lt k : 0 we itr", d"".ling with .o;tatic Dut-put fe"dba~k (ze~() orMr compensator) of the form u(t) : Ny(t), while for k ~ 0 dynamics will be involved in the compensator.

'rh" closed 1",,1' (or: controlled) linear system obtall'ed by the intercon-nection of the linear system (1.1) and the compensutor (1.9) j5 de~crib"d by:

(1.10)

X

(t) '" A x et) ,

e e e

where we have denoted

(1,11 .. ) "e(t)

:: [,,(tl]

wet) alld [ A+BNC BKM] . (1. 11 b) A

.-e LC

Note that "e(t) E:

m

n+lc, the state space of the closed loop system.

I t should be "lear that, when designing a compensator of t.ype (1.9) for the linear system (1.1), one always has certain control objectives in mind (for instance: stabilization, decoupl1ng, etc.). in many cases these control obj"ctiv,," require th" cl,,"ed loop "y"t"m (1.10) [u satisfy c~rtai[\ conditions (for inst~nc,,: the closed loop system should be "t~ble, in the state space of the closed loop syst.em there should exist linear subspaces with certDJ.n properties, etc.). However, when starting thE' design of .:my

~Qmpensator of type (1.9) Olle only has tho;, to-be-colltroll,}d system avail-Jbl~. Thf.::tl.:.:forl'.:!, it is impo["tant to €:::::~tabli!::j:h I'"~lation~ b~lwct:n sy!::'Lt:m

tli<c80 l"Cliltioll~ hJve t.o m"k" cle3T how the dcdred control obje<:tiv".~ h"VB to h~ tran,laled into lleceH",ry prop<'Tti"" of ~yoL"m (1.1). WI') le, on lhc aliter lWlld, thee" relalions hJV~. t.o e"plain which control

objective. can be ochieved when system (1.1) s~tisfles certuin properties.

It tllrn ... out that, for C'st[lblishing m.,ny of Lhese relation~, the following

lWO linear SllbspaceR

d~p~Ilding

on

~

linear

~Ub5p~CC

We in ffiD+k

ar~

cruci81

(cL Sdn.nllC\cller (1981); in fact 011 Illi.1terial prese.nted

H'

this s,,~.~ion can be [ouod in thi, referenc.).

pew )

oenotee the projecd",\ of W ont.o lRn alon;l

m

k an<l i(W )

den[)t"~

the

~ .: e

i"Lerl",ction o[ W with IRn. Of COUTSC', ieW ) c pew ) for all W c

re

n+k

e e - C e

-Now we can est"b li,h th" following reLHio\15 O".tw""n th" closed loop

sy~-tell, (1.10) and the 'to-he-conlrOlled' systeln (1.1). To thi~ <:nd, let ¢g E 0 he a given stabilily r0gioo and let AD be given by (1.11b).

Theorem 1.41,:

1) If (l(A,) ~ 1(:8' Uwn the P(P:.·" (A,H)

t:,;

,~taNUrwbk and iohrl pal.l' (C,A) -,~~~ de/~l~~c.~t..ahr..e.

~)

1]' We 7:<' WI A,,--£nIJllFianl:

~Ub8pa('e

'iTt mn+k, U"m p(W..lie (l (,o!li-r'o1."ied

'inl'at'~:ant 6ub8[xH.'e r;rl.d i(W,,) I:; a condit1:orwd {rwr;n,iarli; sub~pace in

l)':ll •

3) If "CAe)

~

'l:

g a.nd We iD an

J\e-irl"ra},,>.~nl;

s/1.bt;paae In. JRI1+k, then p(W.,,)

·u~ a ;)U1i.d/,{;;abn.i.I . .y ~'l<bDpa"e and i(W..l i.9 a rl,l{;BIJtabiUty 8u/:!spacB -in

jj{"

Proof:

2) h1llows [rom J) by ."p"cifyin~ I/:

g "' f.

3)

L~\:

x E

peW).

Then there oxists wE: JRk "Heh that x"

DetillE!

t (,) = ;= (81 - A

l x ,

[

F,(Sl]

-1

e. w(s) " e

From Theorems 1.4 and 1.16 it follows that " .. to We ;lCld G_e is stublc. since l~

( We

it is ~lear that ~ E p{W

_e).

Furthermor~, it follow" that X = (sl - A)t{5) -1l(NC~(5) + Mw(~», where ~, w "nd c.onsequently NC'.+ M,., are stable und strictly proper. Hence,

pew,)

is u stabilizabi 1 ity

5ub-sPd~e.

DUdlly, we CdCl prove that i(W.) is a d~t"ctability "ub"pac~.

Coroll-Hi 1.45:

) ( 1R('J.,+'k,

4 If a Ae) J;;; ¢:g and We ia an Ae-lnj)ay~:ant ~'ubapace 'in then the [~~i.r (A.B) i~ Eltabili,;aMe, the pah' (C,A) is detectable, peW,,) iD a

8 t.abil i;,abi li ty DubDpace and J.(W.,l 1:3 a det,w tahi.li ~y suhspace?

ThcoreTII 1.44 st~tes which prcperti~, will he gatisfieo <Jith re5p~~.t to system (1.1) if th~ closed loop 5y~t"m (1, 10) s~ti.sUes cenain ("ontn») objectives. HO<Jever, <J .. c.an ,,180 give results for thC' conversC'. 10 this end, we. c.ondder ~yst",m (1,1) and we l~t a stability region <tg E: (~ be

givE:::n.

'thEorem 1.46:

[,[It S, V btJ Zhlear' 8ubapaces 'in mn such that .$ ~ V, .$ iD a ddi?Cl;abUity subspace? and V t:1!i (2 $t(1biUR(~bHity subspace, and Zet the ea·ir (A ,Il) be, DtaMUzable and th8 pair (C,A) brfJ detectabk. then there e;.iste a Car/-pm/satOl' (1,9) and

a

/.'i:rwa:r fmbspace We

in

mn+k. th8 [ltai.!. CJpac,.

of

01.<-)

r>(l8uZtinrJ aloMd loop system (1.10), CiI.<£3h th21. a(A) ~ <l:

g, A~Wo.

s:

We (ind

sllloc:w

c:vmm

k_.

- e

-The proof of -Theorem 1.46 will be given lat"" in thj.$ .~(".cti.on jn cho:' form of a construc.tion. We proceed with two specLll cusos of Thcorol~ 1.46.

Corollary 1.47:

•

1)

.Ii

the pah> (II ,B) iD DtaM t-izabtG: and aU? lX2ir (C ,A) -[.,~ (i(</.(i(:I,(1bk, then therG: existe a compenDatm' (1.9) f!'w)h -that the ,,/.QS(ld loop syvtem (1.10) is stab7.e.

24

2) j,c)j, S I"", a ,'(m,h'l;iOrlf;r:/ {nv<H"iant Dubt'pa('<' md Id: V he a contI'oUrid

inlHlr'i(1n'!, {)uhnpac;,:: aw~h th(I'~ S C V, Then there I'JX'£8t'i <2 <:omp<?ru;atol'

, Il+k

(1.9) (I1!(i (I UIWal' Duhc;p(we We '/,n 11\ ,the state "'paee of the Y'o'l-[;u!.t'in~J a/'o",[,d ZOO£! :ly,.;{em (1.10), $W::'h I,hal, A",Wc !:: We arId

(S @ 0) !:: We

~

(V

~

lRk). :,1'00£ :

1) Follows from Thi::::orcIII 1.46 by $pec.ifying S ~ ₀ ,"1nd V j}(n.

2) Fu 11 Ow:; fron) Theorem 1.1,6 by specHying ¢:

=

¢:. g

A:::. onnounced, we sha L! pruvidc ",,\ proof of Ih€or~TU 1.46 by roe.'ins of d.

(:Ofl"-Btrnctio!1. We give tl"-d,s ~on6tructiofl. iLL full detail here, bec~I.J~e it will be u"~d fr~quently in leter chapter •.

Cons truet,i"?J._l~'

Conside'r the linea); sys\,em (1.1) alld let ~ EO be" given st~bility

g

reeion. Suppo~e th<lt S ~nd V are linear subsp,~ces in lRn such th<lt S,= V. Fllrth"rmurc, suppose tj)3t F E JRm~n, J E )Jlnxp and N E lRmxp .we matri""s ollch th'lt (1\+ BF)V ~ V, (A + JC),S S S, (i'>+ BNC)S'; V, o(A+ SF),;; {:g and a (A + JC)

'=

~

.

"

•

(nearly, tllis impli~" thill th~ P<l),r (A,B) is .tabiltzol)le, the pair (C,A) is dctcctob1e. S is d d"uoct'lbilic.y subsp.~ce and V is 0 stabilizability subspoce. )

Now, 1.<," k = nand h,t K E lR,lX<" L f j}(n.p dnd M E lRffi'o be matrices defined by K:~A+BF+.JC-llNC. L:=BN-,J 8nd M:=l·'-NC. Furthermore, let

We h" the

lin,,~r

subspace in rn2n defined by

(1.12)

Proof;

It i~ clear thnt with th~s(' specj,fic: choic:.:cs of thl;""- matT.LCe~ K, L .~md M WE. hHV~ tb.=-ll Ae = At!.' w'here.

(1,13)

A

_e [ A+BNC (BN-J)C B(F-NC) ] MBHJC-BNC

: = [

11· Now let S E lR2nx2n \>e defined by S

e e since cJ (A ) " _e er (A + Bin

U

d (A + JC) ,

_ [A+OBF

S

A

S-l _ " e e fiF-BNC] A+JC

~l,

Then

NeJ(!;:, let ){e Ii: We' Le. Xe

=

[~l

+

[~l

with s E S and v E V. Then A x

e "

=

Ae (["]

+

[v])

=

[(A+.JC) s]

+

[(MBNCh] _ [(A+JC)S] + [(A+BF)V] ,

o

v 0 (MBNC)s

(A+JC)s

(A+IlF)v

from which it is clc~r that \ / \ E We' Hence, AeW,,- ~ We'

FinJlly, by (1.12) it i . clear that (5 ~ 0)." W" S;; (V 'ij:i mn).

Speciol cas,," 1.49;

1) Let F E Il\mxn dnd J E: IRn<p b", matri"es such !;:h.H a (A + BF) S (:g <:tnci

a(A +

JC)

c: I\: , and

IH N

E: mm"p be an ar\>i!;:r"~y

mHrix.

- g

Then er(i': ) .<:::; I(: • where

A

15 given Py (1.13).

~ g ~

•

2) Let $, V be linear subspaces in IRn such that S ~ V, and Jet F E lRmx". J E JRnxp and N E IRffiXP be matrices such !;:h"t (A+ BF)V S;; (I,

(A + JC)S .<:::; S

,ina

(A + BNC)$ co (I, Then

A

W c W where

A

is given by

- e ~ - ~ e

(1.13) and We is givell by (1.12).

Proof of

Theorem

1.46:

From Section 1.4 the exist"n"e of matrices 17 eo lRrn>n and J E

nt"P

such that (A + BF)(I ~ V, (A+ .JC)S !:: S, a(A+ BF) ~ It and cJ(A+ .TC) C ¢: is clecH.

g ,;;Xp g

Fl:om tbe lIext hmma the existence follows of a matrix N E JR such that (A + BNC)S .<:::; (I. whereupon .'ppliunioI"l of construction 1,48 completes the

26

L~rnrn.:.J 1.50~

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