Sufficient conditions for BIBO robust stabilization : given by the gap metric

Sufficient conditions for BIBO robust stabilization : given by

the gap metric

Citation for published version (APA):

Zhu, S. Q., Hautus, M. L. J., & Praagman, C. (1987). Sufficient conditions for BIBO robust stabilization : given by the gap metric. (Memorandum COSOR; Vol. 8726). Technische Universiteit Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Faculty of Mathematics and Computing Science

Memorandum COSOR 87-26 Sufficient conditions for bibo robust stabilization: given by

the gap metric by

S.Q.Zhu, M.L.J.Hautus, C.Praagman

Eindhoven, Netherlands October 1987

SUFFICIENT CONDITIONS FOR BIBO ROBUST STABILIZATION ·

GIVEN BY THE GAP METRIC

S.Q.Zhu M.L.~T.Hautus '='.Praae:man Facul ty of Ma thema tics and computing science Eindhoven uni verSl ty of Technology

The Netherlands

Abstract

A relation between coprlme fractions and the gap metric is presented. USlng this result we provlde some sufficient condi Hons for BIBO robust stabll1za tion for a very wlde class of systems. These conditions allow the plant and compensator to be disturbed simutaneously.

K.e.Y words: Robust stabllization; Gap metric; Coprime fraction.

1. Introduction

In a very real sense, almost all control system design problems are concerned wlth robust stabll1zation. One uses a mathematical model to deslgn a controller that produces a stable feedback system when eHher the model or the physlcal system is 1n the loop. Mathematical design procedures often produce a hl.gh order controller, whlle the englneers prefer a low order one which can be easily manlpula ted. On the whole, l.n control system design 1 t 1s necessary to consl.der the robust stabil1za tion problem in whl.ch both plant and controller are subjected to uncertain tles.

In order to investtga te the system uncertainties involved in robust stabil1 ty problem 1n a general sense, vldyasagar[l1) and

Zames et al.(13) proposed the graph topology and the gap topology respectively. Zhu[15] reformulated the two topologies in a very general setting.

It is known[6,7.11.12,13,15] that both the sraph topoloSY and the gap topology are the weakest topolog~es in which feed-back stability is a robust property. More precisely. any plant

P 2 can be stabilized by a controller which stabilizes plant Pi if the plant _{P 2 is in a} ne~ghborhood of Pi in the graph topology (or in the gap topology). So, these topolo-gies are good measures of plants in the robust stability problem. We list some propert~es of the two topologies from [6.7,11.12,13.15J in the following

1) The gap topology can be defined for a more general class of systems than the graph topOlogy. If one confines them to the same plant set. they coincide.

2) Restricted to stable plants, the graph topology ( or the sap topology ) is identical to the norm topology.

3) Both the gap topology and the graph topology can be metrized.

4} The set of stable plants viewed as a subset of all plants ( including stable plants and unstable plants ) is open ln the graph topology ( or the gap topology).

5) A plant sequence _{(P n ) con verges to Po in the} graph topology ( or in the gap topology 1f and only 1f any controller C which stabilize Po also stabilizes _{P n 1f n} large enough and the closed loop transfer ma trlx sequence H(Pn.Cn)(see Figure 2.1) converges to _{H(PO'C o )'}

When one wants to apply these topologies to practical prob-lems, the metriC descriptions of the topolog~es are needed. Vldyasagar [11,12) deSigned a graph metriC for lumped linear time invariant (LTI) systems, and by uSing this metriC he offered a sufficient condition for robust stabill ty. Callier et al. [3] extended this metriC to single-input-single-output (SISO) distri-buted LIT systems, and Zhu [ill] presented a graph metriC for a class of mul tiple-lnpu t-m ul tiple-ou tpu t (MIMO) distributed LTI systems. Generally, i t is difflCUlty to extend the definition of the graph metrlc to the distrlbuted LTI systems because of the the spectral factorization problem is involved. Praagman (10)

o'f'fered another graph metrlc which has a slmple 'form and can be easily computed in the SISO case.

The gap metrlc can be defined 'for distrlbuted LTI systems as well as lumped LTI systems. In thlS paper, we are gOlng to gl ve a su'f'ficient condition for robust stabllity using the gap metrlC. ThiS is a parallel work to the sufficient result given by Vidya-sagar[12] in the graph metric. Our result depends upon a relation obtained in this paper between the copr1me fraction and the gap metric. The concept of stablli ty concerned in this paper is

bounded-input-bounded-output (BIBO) stabil1 ty. For lumped LTI system. BIBO stabill ty 1S ident.lcal to the internal stability (or exponential stability), whereas for d.lstributed LTI system this property is lost. The equivalence of BIBO stablli ty and internal stability 'for a very wide class o'f in'finite dlmensional systems has been offered by CUrtain [5].

This paper is organlzed ln the followlng way: In section 2, we will introduce the framework as well as the defin1tion o'f the gap metric. A relation between the gap metric and copr1me 'frac-tlons are presented in section 3. Finally. our main results, su'f'fic1ent conditions for robust stability, are gl ven in section 4.

~ Framework

In this section, we present the framework which was bUllt in [15], and the de'finition of the gap metriC.

Let H be an integral domain and F. which contains H, be a subset o'f the quotient f.leld of H. Assume that X is a H1lbert space. Our framework is based on the 'follOWing

Basic Assum ption Each element P E F is a linear operator mapping X to X and this operator is bounded i'f'f PE H.

We consider F as the universe of the plants and H as the set o'f the stable plants. The 'follOWing examples are given in order to demonstrate that the basic assumptlon 1s reasonable and in-cl uding many import an t cases.

Example 1: Let H be the set of all proper ra tional 'functions without poles in the closed right half plan (RHP) and F be the set o'f all rational functions. The input and output space is chosen to be H2 (C+), the Hardy space. In this case, the basic assumption 1S satls'fied.

Example 2: TaKe H to be A_(O), the algebra of the transfer functions studied by Calher and Desoer [1,2], taKe F to

be B(O), and taKe X to be H2(C+>' Then the baslc

assump-tion holds.

As usual, let M(H) and M(F) denote the set of ma trl.ces with

entries in H and F respectively. If necessary we wrJ.te

H(.)nxm to indica te the dimensions.

According to the baSIC assumption, each element pEH is a

bounded opera tor mapPIng X to X. By [15, lemma 2.3), one can

easl.ly mapping

show that each elemen t fEF IS a closed opera tor

element PEM(H)nxm X to 1s a bounded X. Consequently. operator maPPIng Xm each

to Xn and each element

PEM(F)nxm is

_a

closed operator maPPl.ng

xm

to

X n

We say tha t PEM(F)nxm has

t10n (r.c.f.) over the set of bounded

HEB(X m , Xn) and DEB(X m )

D is invertible;

a r1ght coprlme

opera tors, if there

such tha t

frac-eXl.st

1 )

2) There eXIst X and Y 1n the set of bounded

such that

operators,

XN + YD :: I

3) p : ND- 1 .

The left coprIme fractlon (I.c.f.) can be def1ned 1n the same

way.

In thl.s paper our result only holds for a subset R(F) of H(F) rather than M(F) itself, where R(F) consists of all elements in M(F) which has both r1gh t and left coprime fractions· over the set of bounded opera tors.

Let us consider the standard feedbacK system in Figure 2.1,

where P is the plant and C is the compensator. the closed loop

transfer matrix 1S

-P (I+CP )-1

H(P,C) ::

It 1s assumed that the system is well posed, so that the indica ted inverse exists.

The feedback system lS to be stable 1ff

H(P,C)EM(H).

How we are 1n a posi t.1.on to define the gap metric. We know tha teach element P in H(F) is a closed opera tor mapping Xm to x n , Denote the graph of P by G(P). Then G(P) is a closed subspace in xmxXn. Let IT(P) denote the orthogonal prOjection on the graph G(P). Then gap metric can be defined as

The topology genera ted by the gap metric loS called the gap

topology.

From [~l, one knows that if P E R(F). then

fi(P) :

[:}

( D*D+N*N ) -1 [ D* I Nit ]

=

I

-r::

J

. . N

where (H,D) and (H.D) are any r igh t and left coprime fraction pair of P respectively. and D* means the dual of D. ( according to the basic assumption, D ( or H etc. ) is a bounded operator, so the dual eXlst.)

~ GaP metric ~ coprlme fractions

The main purpose of this section 1s to dig out the relation between the gap metric and the coprime fractions. This relation plays an important role in our maln result in the next section.

How we start with the following lemma.

Lemma U. Assume that PER(F)mxn,

and If one regards P as an operator mapping Xm to Xn and denotes the graph of P by G(P). then

:{ (DZ, Nz) ( 3 . 1 )

iff (N,D) is an r.c.f. palr of P.

Proof The sufficient part can be found in (12), here we Just prove the necessary part.

Assume (D_1, N₁₎ is an r.c.f. sufficient part, one knows that

Let X,Y be the operators such that

+

₌

Define

U;: XN + YD

pair of P and by the

I

By (lI) and (3.1), one knows that for every x in Xm there 1s a unique y 1n Xm such that

=

y

and vice versa. EqUivalently the opera tor U: Xm_- Xm

Uy

=

x

1s bijectl ve. Consequently.

u-

1 _{as well as U is a bounded}

operator. As a result,

Hence (H,D) is an r.c.f. paJ.r of P.

The next lemma J.S an al terna t i ve version of a result in Xrasnosel'skli Lemma TI(P 1) maps et al. [9,p206]. b1jectively (1=1,2). onto G(P 1) iff

Using lemma 3.1 and lemma 3.2, we can prove Theorem be an

{

::}

L1

r.c.f. Let paJ.r

f

::J

< 1 Proof (suffJ.ciency) By PiER(F)nxm (i=1,2)' of Pl' Define 3.2, lemma therefore we know ( It )

From lemma 3.1, orie knows that (D 2 , H2 ) 1s an r.c.f. pair of _{P E ,}

Then

and

maps

(necessi ty) From the given condition and lemma 3.1 one can

easily check that maps onto

biJecti vely. Furthermore, according to lemma 3.2, one has

RemarK: The sufficient part of this result is also obtained by Vldyasagar[11).

here.

Bu t the proof is different from the one given

Now we turn our attentlon to the left coprime fraction and we wish to get the similar result as Theorem 3.1.

For a given plant PER(F)nxm, let be

any l.c.f. pair of P, i.e.

N

1) D is in vertlble;

2) there exist bounded operators X and Y such that

N N NX + DY = I 3) P

=

D-

1

N.

Define Tp :

=

N* ( -D* -

1 )

N*

N*

.

.

Then ( N ,-D) 1S _{an r.c.f. palr of Tp , i.e.}

1)

-D*

is inVertlble;

2 ) X*N* + Y*D* = I

RemarK: Tp is uniquely determined by P and independent

of the chOlce of an I.c.f. pair of P.

One can readily prove the folloWing lemma. Lemma DEB(X n ) . . N and Suppose NEB(X n , t h a t PER(F)nxm, X m). Then (D,N) is an I.c.f. pair of P iff (N*,-D*) is an r.c.f. pair of Tp

Lemma LJj- Let PER(F)mxn.

and NEB(Xm,X n ). Then

N N

G (P) : Ker [N I -D )

1££

(D,N)

1s an I.c.f. pair of P.

Proof One can easHy checK the sufficient part. To prove

"

"

the necessity, we taKe one of the I.c.f. sufficiency. we Know that

palr (D,N). By the G(P) : Ker [N. -D) " " Hence N N A ~ Ker [N,-D] = Ker [N.-D) And Ker

[N:

I

-D

r~' = Ker [N. -D] " " . L l.e.

Because the risht hand side of the above equality 1S G (Tp) • and by lemma :;.1 we set that

(N*. -D*)

is an r.c.f. pair of Tp. furthermore. by lemma :;.:;

(D.N)

is an l.c.f.

pair of P.

details.

Suppose PiER (F )nxm (1 ': 1. 2) •

N IV

~ll~Q~=ID ~ _{and (D 1 ,N 1 )}

1s an l.c.f. pa1r of Pi' Define

f:},

(n(P

2) )""

r:::}

Then

(-i)*,N

IE,) is an l.c. f. palr of P 2 iff

PrQQf Notlce

...

And by lemma 3.3, i t is equ1 valent to prove that (N,D) is an r.c.f. pair of iff

< 1

This is the result of theorem 3.1. So the conclusion is true.

~ Suff1ci=nt cQnditiQns :fQr ~ robust stability

NoW' we are ready to state our maln result. Let Po and Co in R(F)nxm be the nom1nal plant and controller respect! vely wlth a stable closed loop transfer matrix

H(PO'C O)' Take any r.c.f. pair of

N N

and I.c.f. pair (Do,N

_{o )}

of Co respectively. Denote AO=

f

:oJ

and

...

...

B _{O. :} [ D _{0 .N o ]} 10

Define

u

0 : It is m follows a l:>ounded B O·A 0 from [12] operator that which is stable biJecti vely iff onto Remark: Because we have assumed that _{H(PO'C O)} is stable, U

_o

can l:>e chosen as the identity.

Suppose that P,C ~n R(F) are the plant and controller consi-dered to l:>e disturl:>ed from Po and Co respectively.

For the saKe of convenience, denote

Theorem U If

o

(C I CO) +0 (P ,P 0) (

w -

1 (4. 1 )

then H(P,C) is stable.

Proof First, one can easily checK that the right hand side of (4.1) is smaller than 1, According to theorem 3.1 and theorem

3.2, we can define an r.c.f. palr (N,D) of P and an I.c.f. pair

(b,N)

of C respecti vely with

and

=TT(C).L

r:::}

Denote

and

-

....

B

=

[DIIN]

then

~ UAU II (B - B 0 ) II + II ( A - A 0)11 liB 011

Therefore, BA is invertible and the ~n verse ~s also a bounded

operator. Consequently Hep,e) is stable.

We can also glve a sufflcient condition by using only r.c.f. pairs of both plant and con troller.

As before, let _{Po' Co} in R(F)nxm and

H(PO'C o ) is stable. Assume that (Npo·DpO> and

(NCO·DCO) are any r.c.f. pa~rs of _Po and _Co

respecti. vely. Denote

r

NPO ) AO :; DpO

reo }

Bo :; NCO U_o

.

.

-

-

_{[ BO' AO )} 12

IV :max[ IIAdl • liB ell I

and.

m =

w IIU

_o

-1₁₁

It follows from [ 12] that _{H{PO'C O)} is stable iff

U

_o

is bijecti ve.

As above, suppose P, C ~n R(F)nxm to be the

dis-turbed plant and controller respectively. l:tl.CQ;t:CIP ~ If

<

m -

1 ( ~ . 2 )

then H(P.C) ~s stable.

PrQQf According to theorem 3.1, we can define an r.c.f. pair (Np.Dp) of P and an r.c.f. pa~r _{(DC,N C)} of C

respect! vely wlth Dp DpO = n (-P) -Np _-NpO and Dc DCO = n{C) NC Nco Denote -Np A

=

Dp DC B : Nc 13

and U : [ B, A ] then IIU - Uoll :II[B-BO' A-l'.O]1I <liB -Boll+lIl'. -A

011

<IITT(C)-TT(Colll IIBoll+IITT(-P)-TT(-PO>l1 IIAoll

: 0 (C ,co ) liB 011+0 ( -P • -Po )IIAOII

: 6 (C .CO )UBolI+o (P .Po )IIAOII

< [0 (C ,COl+o (P .PO) Jw

Therefore. U is biJect1ve. Consequently. H(P.Cl is stable.

In the same way. we can also gl ve another suff1c1en t condl tion by uSing only l.e.f. pa1rs. For the technlques are the same we om1 t 1 t.

Yz

Figure 2.1 Feedback. System

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functions for distrl.buted l.i.near tlme-lnvarl.ant systems.

IEEE Trans. Circul. t Syst., CAS-25. pp. 651-663. 1978;

Correction: VoL CAS-26, pp. 360, 1978.

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[6J

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'7

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