Robust stability of uncertain systems with input delay and output feedback controller

(1)

Robust Stability of Uncertain Systems with

Input Delay andl Output Feedback Controller

I-Kong Fong and C h i - h e y Jou

Departiiient of Electrical Engineering

N atioiial Tai wail Uiiiversi ty Taipei, Taiwan 10617, Republic of China Abstract--The robust stability of systems with para-

metric uncertaiiities, input delays, arid output feed- back controllers is aiialyzed iii this paper. First, the stability condition of Iioiiiiiial systems is expresscd in terms of the eigenvalues of a Hamiltonian ma- trix. Tlien, for the iioiiiiiially stable uiicertaiii system- s, a new rrietliod based on structured siiigular value and linear fractional traiisfoririatioii techniques is pro- posed, so that the Lyapunov stability tlicory can be

used to find a set of uncertain paraiiieters within wliich the systems reiiiaiii stable. Finally, it is reniarked that the results caii be extended to the uiicertaiii systeiiis with multiple time delays.

I . I N T R O D U C T I O N

T h e robust stability prohleiii of uiicertaiii systems with time delays is usually coniplicated. Recently, severa1 analysis rnethods are proposed for such systeiiis w i t h various types of unceitairities [ I , 2, 3 , 4, 5, 6 , 71. However, t o consider uncertain time-delay systems with out put feedback controller, these inetliods caii riot be applied tlirect- ly, because when there are uiicertaiiities in the input and o u t p u t matrices of tlie open-loop systeiii, rritilt iplicative terms of uiicertaiiities will appear in the resultiiig closed- loop system. Especially wlieii there are linearly clepeiideiit parametric iincertaiiities a n d delays in the s t a t e as well as input dynamics, t h e stability analysis often involves ex- pressions with high order multiplicative terms of the uncertain parameters. T h u s straightforward applications of these methods will usually lead to co~iservative results.

In this paper, tlie al,ove-inerit,ioiiccI coli-iplicated problem is studied. First, with the Lyap~iiiov riiethotl, h e stability coiiditiori of iiorniiial systeii-is is expressed in terins of the eigenvalue locations of a Hainiltoiiiaii iriatr,ix. 'Tlien, for the no rnin all y s t a b le ti11 cc r t a i 11 systems sat i:;fy i n g t lie condition, a iiew ii-iethocl based on the slructurctl singular value [8] and linear fractional traiisforiiiation (LF'I') techniques is proposed, so that tlie Lyapuiiov stability theory can h e used to find a set of uiicertaiii parameters withiii which the systems reiiiain stable. Finally, it is shown t h a t the results can be extended l o the uiicertain systerris with mu 1 tip le ti me del ay s ~

Coiisitler the uiicertaiii ziiigle tlelity systeiii (C,) described by t li e follow iii g f U 11 c t, io i i a l c1 i l h e i i t i a l eq ii a t i on

and coritrolletl by ont,put feedback

0-7803-2428-5195 $4.00 0 1995 IEEE

+

(Bo

+

E

k i B i ) U ( t - IL) i = l m y(t) = (CO

+

E

k i C i ) x ( t ) .(t) = h ( t )

v

t

E [-h,O]

4 2 )

= L Y ( t ) , i=l

where the s t a t e vector z E

R", t h e input vector

U E

RP,

the output vector y

E

R'

, t h e o u t p u t feedback matrix

L

R p x r ,

t,he delay time 11

>

0, the initial condition

41

is a continuous vector function, a n d the i t h uncertain parameter ki E

R

for i = 1 , . .

. ,

m. It is assumed t h a t maxi lkil

<

y,

aiid all A i ,

Bi,

Ci,

Di are known. Note t h a t although it seems t h a t each

ki

appears in all system matrices, there is no restriction in this formulation of uncertainties, since we caii siniply set t h e corresponding A i ,

E l i ,

Ci

,

or Di to zero rnatrices if ki does not affect the par- ticular system matrices. Under these basic assumptions, tlie closed-loop systern is

i ( t )

= (A0

+

,

:

E

k i A i ) x ( t )

+

(Eo

+

kiEi

+

Cyj=,

k i k j E i j ) z ( t - h ) , where Eo = A

Do+BoLCo, Ei

!?

D;+BoLCi+B;LCo, and Eij = A BiLCj.

A

Let z t ( u ) = x ( t + a ) for -h

5

(T

5

0. For the riorriinal sys-

tem i ( t ) = A o z ( t )

+

Eoz(t - h ) , we find t h a t a Lyapunov function

V ( t ,

z t ) = ~ ~ ( t ) P z ( t ) + S : - ~ zT(a)z(a)da, where

P is soine positive defiiiite m a t r i x , can be used t o show the stability under some conditions. In fact, we have

Tireorem 1: If A0 is a stable matrix a n d for some a

>

0 the IIaiiiiltoiiian iriatrix

lias 110 eigenvalues on the jw-axis, then the norninal systeiii is asymptotically stable.

1351

Proof. From the assuiriptions aiid Leniina 4 in [9], it is

liiiowii t h a t t h e afgelJraic Riccati e q u a t i o n ( A R E )

(2)

has a positive definite solution P for some a

>

0. Consider the Lyapunov functioiial candidate

V ( t , z t ) = z T ( t ) p z ( t )

+

i:h

zT(g)z(a)du.

we want to ensure t h a t V ( t , zt) is negative definite for all .zt and iiiax, I C z /

<

y, or equivalently, t h a t S ( k ) is pos- itive definite for all rnax, Ik,I

<

y, we only need t o ensure the nonsingularity of S ( t ) for all max, Ikil

<

y. T h e key points are t h a t the eigenvalues of S ( k ) are continuous functions of k , and S ( 0 ) = Qo is positive definite. As long

as S ( k ) does not become singular t o have zero eigenvalues when C varies, the eigenvalues of S ( k ) will remain positive. T o find the largest value of y such t h a t

S(R)

is nonsingular for all max;

lt,l <

y, we first manage to express

~ ( k )

in the form of a n LFT. If this can be done, then we have

S ( k )

5'22

+

Sz1(4 - AS11)-'AS12, (4) ( 2 ) T h e n t h e Lyapunov derivative

V ( t ,

xt) is z T ( t ) ( A ; f p + PA' + ~ n ) x ( t ) + 2 ~ ( t ) ~ p E o z ( t

-

z T ( t - h ) z ( t - h ) -In

O

1 ,

A T P

+

PA0

+

P E o E T P

+

In 0

4 t )

[

-&P

;

] [

z ( t - h )

]

Since P is t h e solution of A R E ( l ) , we have V ( t , z i )

5

-allz(t)112. T h u s , we conclude this theorem by using the Lyapunov stability t>heorein [ l o , p. 1051.

To proceed, we assume the nominal system satisfies the assumptions of Theorem 1. T h u s the nominal system is asymptotically stable, a n d the A R E (1) has a positive definite solution P . We then t u r n t o our second objective: the robust stability of the uncertain system. More specifi- cally, we want to obtain the upper bound y and guarantee t h e stability of (E,) for inax, 1b,1

<

y.

Obviously, we can try t o use the sarne Lyapuriov function V ( t , x t ) t o establish the robust stability. T h e proof requires one t o show t h a t t h e Lyapunov derivative V ( t , z t ) is negative definite for all z t and inax, IC,/

<

y. By some tedious iiianipulations, it is not difficult to show t h a t

V ( t , z t ) is equal t o m x T ( t ) [ P A o

+

A T P

+

I,,

+

k , ( P A ,

+

A T P ) ] z ( 2 ) 2 = 1

+

2 x T ( t ) P H ( k ) ; c ( t

-

h ) - z T ( t - 1 L ) Z ( t - IL)

- S ( t )

0

. . .

k ; k j E ; j . Clearly, S ( k ) has the form

[

0 - I ,

] [

-Hi;.')TP

;

] [

.z(;!)IL)

]

where

k

=

[

ICl

t,,

I T ,

~ ( b ) = CUI,

+

P E ~ E T P

-

xzl

k i ( P A i + A T P ) - P N ( k ) H ( b ) * P , and H ( k ) =

Eo+

CE=,

kiEi

+

m ni

QO

- CkiQ; - k ; k j Q ; j - k i r k j k < Q ; j ~ - C j k j k ~ b C Q j j C E ~ (3) i , j , c = 1 i , j

,c

, E = 1

where

Q;,

Q i j , & ; j c , Q ; j c c are all kiiown coilstant syin- metric matrices, a n d Qo = is nonsingular. Thus. if

where the Sij's are constant matrices which depend only on the &-matrices in ( 3 ) , det S22

#

0,

A = diag{ClI,,

,

. .

. , k m l T m } ,

r1

+

r2

+

. . .

+

rm = p , (5) and T , ' S are positive integers. Under t h e circumstances,

we caii use the following theorem [ I l l .

Theorem 2: T h e L F T of S ( k ) in (4) is well-defined (i.e., ( I F - AS,,)-' exists) a n d S ( k ) keeps nonsingularity for all inax;

I

b;

I

<

y if a n d only if

3:

<

1 / m a x { P ( s l l ) , P ( ~ l l

-

S12S,-,1S21)}, where ,U(.) is the structured singular value with respect t o

{A : k, E

R

for i = 1, . . .

,

nc}.

Clearly, the y obtained in Theorem 2 serves as a n upper bound on max; Ik;l t o keep t h e asymptotic stability of the uncertain delay system

(Es).

T h u s , our development will be complete when we describe how t o systematical- lv express S ( k ) in the form of a n L F T in the following. Ti'lie method we adopted is basically from t h e works of [ l a , 131, where it is shown how t o convert a matrix with mixed uristrcctured and bounded rational type structured uncertainties to the form of a n L F T . However, note our objective here is to use the method t o convert a perturbed matrix to the form of a n LFT, b u t not t o adjust some uncertain system into the standard configuration suitable for p a n a l y s i s , i.e., a feedback configuration with t h e nominal systern in the forward p a t h a n d uncertainty in t h e return p a t h . Hence there is a fundamental difference between the present work and the previous ones. In view of the facl tliat we only have structured uncertainties in our perturbed inatrix S ( k ) , we only need t o describe the part of coiiversioii techniques developed in [13] for t h a t kind of uncer t ain t ies.

For o u r purpose, let F ( M , A ) denote t h e

LFT

M~~

+

~ ~ ~A M ~ ~ ) - ~ A . M ~ ~ , ( 1

-where M I i , M12, M 2 1 , and M22 are constant matrices of suitable dimensions, Ad is t h e joint coefficient matrix

,

and A has t h e sarne diagonal form as t>lie uncertaiiit,y matrix A in (5)) except the multiplicity of each k; may be tlifl'erelit. From ( 3 ) , it is easy t o see t h a t the matrix S ( k ) has elements t h a t are multivariable poly- noinials of k ; ' s . Therefore with the techniques proposed in

I

[

A421

I

h f 2 2

MI 1

I

M 1 2

(3)

[13], ~~ every element of S ( k ) can be expressed in the forrn of a n LFT F ( M i j l Aij(k)), where M i j =

[#2]

is a

Mi;

constant joint coefficient matrix, and A i j ( k ) is a (diagonal matrix with elements of k repeated on the diagonal. After transforming the elements of S ( k ) t o LFTs individually, we have

S ( k ) = [F(ILlij, Aij(k))]

= _{M 2 2}

+

_{G 2 1 ( I d -}

_{&(k)M1l)-la(k)n/rl,,}

where

A ( k ) = ~ i a g { A i l ( k ) ,

...

. ,

A i n ( k ) ,

.

. . ,,Ann(!)}, and I d is the identity matrix with the sarne diinension as t h a t of

A(k). In general, the size of A(k) will be unnecessarily large for S ( k ) t o be expressed as an LFT, and size reduction of

A ( k )

is needed. To proceed,

A ( k )

can be trans- formed first by interchanging rows and columns t o become a diagonal matrix with elernents of k appearing in order, which is exactly the same as the matrix A . Note that this rearrangement of the diagonal elements of A(k) does not affect the size of the uncertainty matrix, but makes the resulting L F T easier t o handle. After this rearrangement, it can be seen t h a t the size reduction of the resulting uncertainty matrix is equivalent t o the dirnension redluction of a multi-dimensional state-space realization, andl the procedure discussed in [13] can be used t o reduce the size of the uncertainty matrix. Though the proposed reduction procedure does not guarantee t o give a minimal sized uncertainty matrix, examples show that the size reduction is often effectively achieved. No matter the mmirnal di- mension is reached or n o t , a t this stage S ( k ) has been expressed in the forin of (4). Thus our task is completed.

111. EXTENSIONS

T h e above procedure caii be extended t o handle systeiiis with multiple non-commensurate time delays, e.g., system (E,) described by m m i = l 9 m m m Y(t) =

(CO

+

h G ) z ( t )

.(t)

=

41(1)

v

t E [ - ~ m a z , O ] u ( t ) =

Mt),

i = l

where h l ( t ) , 1 = 1 , . . .

,

q , are continuous functions satisfying 0

5

h l ( t )

5

hi

<

cc and & ( t )

5

dl

<

1, h,,, = maxl h i , and d,,, = maxl d l . T h e closed-loop system is i ( t )

=

(A0

+

k i A i ) z ( t )

+

kiFilz(t

-

hr(t))

+

(Eo

+

kiE;

+

Cyj=,

k i k j E i j ) z ( t - h ) , where

Eo,

Ei,

and Eij are defined in Section 11. Again, we assume the nominal system with the output feedback gain L

is asyrnptotically stable, but due t o the assumption t h a t there exists a positive definite matrix P satisfying the following ARE

A T P

+

PA0

+

PEoEFP

+

( a

+

2)In =

0,

(Y

>

0. (6)

This i s equivalent t o assuming t h a t the ARE in (1) has a positive definite solution for sorne a

>

1. In other words, we make a little stronger assumption t o cope with the additional delay terms i n

(Em).

For the uncertain system

(Em),

consider the Lyapunov functional candidate V ( t , z t ) =

(7) Then Lyapunov derivative V ( t , z t ) can be written as

m z T ( t ) [ P A o

+

A T P

+

21,

+

ki(PA;

+

A T P ) ] z ( t ) i = l

+

2 z T ( t ) P H ( k ) z ( t - h ) - z T ( t - h ) z ( t - h )

+

2 z T ( t ) P k i F i , Z ( t - h , ( t ) ) - - C(1 - / l l ( t ) ) Z T ( t

-

h ( t ) ) z ( t

-

h l ( t ) )

5

z T ( t ) T Q T T z ( t ) , q m ,=1 i = l 1 ‘ l = 1 where z ( t ) =

[ z T ( t )

z T ( t

-

h ) z T ( t - h l ( t ) ) . . .

z T ( t

-

h,(t))lT, T is a n upper triangular rnatrix with unities a- long its diagonal, Q

=

- d i a g { S m ( k ) , I,, - I q , } , and

S,(k) = &In -

Czl

k i ( P A ;

+

A T P )

-

P H ( k ) H ( k ) V P E o E T P . Thus the remaining task is t o ensure t h a t

S m ( k ) is positive defiiiite, and the rest of the discussions follow the sarne line of development as in Section 11, except now the L F T of S m ( k ) is in general different from t h a t of S ( k ) .

- 2 ( 1 - d m a z )

E’,

yjzl

k i k j P(FilFj7;

+

FjlF$)P

+

Moreover, it is also possible t o use the same approach

on the even inore general system. For exam le, the system ( B o

+

xy:l

k ; B , ) u ( t - 11) and/or output part changed t o

(Cm)

with input part changed t o (Bo

+

xi=,

x

k i B i ) u ( t )

+

(4)

(CO

+

k i C i ) z ( t )

+

(CO

+

Czl

k i c i ) z ( t - h ) can be handle similarly. T h e only difference is that the corresponding Lyapunov derivative will be more complicated.

IV. AN

E X A M P L E

Consider a n uncertain system (‘E,q) \ ., with the following d a t a : AO

=

[

1;

-:

1,

Bo

=

[

1 ,

COT

=

[

” f ] ,

DO =

[

::!5

:],

A i =

[

1: i ] ,

Bi

=

[

i ] ,

CT =

[:

1 ,

D1

=

[:

: ] ,

A z , B z , C I , and

D7

are zero matrices of suitable dimensions, and the out- p i t feedback gain

L =

-0.5. By using Theorem 1, it is found t h a t

ARE

(1) has a positive definite solution

=

[

52.08

2.640

]

2.640 16.38

a t (Y = 125. Hence the nominal system is asymptotically

stable. From these information, S ( k ) =

be obtained, where s12 = 5 1 2 , and S I I , s12 and 5 2 2 are,

respectively,

[

:::

:;;

]

can 125

+

78.91k1

-

1353k2

-

2.178kf - 748.6k;

+

6.969klkz

+

3.484kfk2

-

72.23klk;

-

1.742kfk;, 189.5k1

-

47.13k2 - 13.51k: - 2GO.211.:

+

43.23k1k2

+

21.62k:k2 - 236.6klki

-

1 0 . 8 1 k ~ k ~ , and 125

-

45.31k1

+

130.6k2 - 83.82kB - 90.41k;

+

268.2klk.2

+

134.1k;kz - 155.7klk;

-

67.05kfk;. Following the procedure described in Section I1 and using the p-analysis package [14], it is possible t o get a bound of 0.08 on maxi Ik;l t o keep the nonsingularity of S ( k ) , and hence the asymptotic stability of the above system.

V. C O N C L U S I O N

In this paper we develop a n LFT approach to deal with the robust stability analysis problern of delay systems. A

condition on the stability of the nominal system is proposed in terms of t h e eigenvalue locations of a Hamiltonian matrix. Then a systematic procedure is described for the computations of the bound of the uncertain parameters to keep the stability. Note t h a t because the Lyapunov theory is used, the stability is guaranteed even for time-varying uncertainties. As the main objective of this paper is to iii- troduce a new method for transforming the robust stability problem into a inanageable f o r m , the riuiiierical corri- putation issues related to the structured singular values are not addressed. These issues, inclucling how to find an acceptable bound with simple computation requirements, will be discussed in the future papers.

A C I< N 0 W L E D G hl E N T

This research is supported by the National Science Council of the Republic of China under Grant NSC S3- 0404-E002-004.

REFERENCES

[l]

S.

Wang, B.

S.

Chen, and

T.

P. Lin, “Robust stability of uncertain time-delay s y s t e ~ ~ s , ” Int. J. Control,

vol. 46, pp. 963-976, 1987.

[‘’I

J .

H .

Chou,

I. R.

Horng, a n d

B.

S.

Chen, “Dynam- ical feedback compensator for uncertain time-delay systems containing saturating actuator,” Int. J . Con- trol, ~ 0 1 . 4 9 , pp. 961-968, 1989.

[3] E. Cheres, Z . J . Palmor, and S. G u t m a n , “Quanti- tative measures of robustness for systems including delayed perturbations,” IEEE Trans. Automat. Con- trol, vol. 34, pp. 1203-1204, 1989.

[4] A. Hmamed, “Further results o n the robust stability of uncertain time-delay systems,” Int. J. Systems SCL., vol. 22, pp. 605-614, 1991.

[5]

S.

Phoojaruenchanachai and K. Furuta, “Memory- less stabilization of uncertain linear systems including time-varying state delays,” IEEE Trans. Automat.

Confrol, vol. 37, pp. 1022-1026, 1992.

[6]

T. J . Su and C. G . Huang, “Robust stability of delay dependence for linear uncertain systems,” IEEE Trans. Automat. Control, vol. 37, pp. 1656-1659,

171 W. Yu, K. M. Sobel, a n d E. Y. Shaprio, “A time domain approach t o the robustness of time delay systems,” Proc. of the 31st IEEE Con5 on Decision and

Control, Tucson, Arizona, pp. 3726-3727, 1992. [SI J . C. Doyle, “Analysis of feedback systems with struc-

tured uncertainties,” Proc. IEE, p t . D, vol. 129, pp.

[9]

J . C. Doyle, K. Glover, P. P. Khargoneckar, and B. A. Francis, “State-space solutions to standard H2 and

H,

control problern,” IEEE Trans. Autonaat. Con- trol, vol. 34, pp. 831-846, 1989.

[lo]

J . Hale, Theory of Functional Differential Equations,

New York, IVY: Springer-Verlag, 1977.

[ l l ] A . I<. Packard, What’s New With ,U: Structured Un-

certainty in Multivariable Control. P h . D. disserta- t’ion, University of California, Berkeley, 1988. [12]

M .

Steinbuch, J . C. Terlouw,

0.

H . Bosgra, and

S.

G.

Si-nit, “Uncertainty Modeling and Structured Singu- lar Value Computation Applied t o an Electromechan- ical Systeirr,” Proc. IEE, p t . D , vol. 139, pp. 301-307, 1992.

[13] P. Lambrechts, J . Terlouw, S. Bennani, a n d M . Stein- buch, “Pararrietric uncertainty modeling using LFT-

s,” Proc. American Contr. Conf.., San Francisco, Cal- ifornia, pp. 267-272, 1993.

[14] G . J . Balas, J . C . Doyle, K. Glover, A . Packard, and

R. Sniit.h, p.-Analysis and Synthesis Toolbox - User’s G u i d e , Natick, MA: Musyn Inc. and T h e MathWorks 242-250, 1982.

IIlC.> 1991.