Delay-dependent robust H∞ control of uncertain linear systems with time-varying delays

An IntemabonalJoumal Available online at www.sciencedirect.com •

EI.SEVIF~R

computers& mathematics

with applications Computers and Mathematms with Apphcations 50 (2005) 13-32 www elsewer com/locate/camwa

Delay-Dependent Robust Control of Uncertain Linear Systems with Time-Varying Delays R. M. PALHARES* D e p a r t m e n t of Electronics Engineering Federal Umversity of M m a s Gerais A v A n t S m o Carlos 6627, 31270-010, Belo Horizonte, M G , Braml

palhares©cpdee, ufmg. br

C. D. CAMPOS Department of Electromcs and Telecommunications Engineering Pontifical Catholic Umverszty of Minas Geram Av. Dora Jos4 Gaspar, 30535-610, 500, Belo Horizonte, MG, Brazil campos~pucmznas, br

P. YA. EKEL G r a d u a t e P r o g r a m in Electrmal Engineering Pontffical Catholic University of Minas Geram Av. D o m Jos4 Gaspar, 30535-610, 500, Belo Horizonte, MG, Brazil

ekel©pucmznas, br M. C. R. LELES D e p a r t m e n t of Electronics Engineering Federal University of Minas Gerais A v A n t S m o Carlos 6627, 31270-010, Belo Horizonte, MG, Brazil

M. F. S. V. D'ANGELO Faculdade de Cl~ncla e Tecnologla de Montes Claros P r a ~ a d a Teenologia 77, S~o Jo~o, 39400-307, Montes Claros, M G , Braml

(Recewed January 2004; rewsed and accepted Apml 2005)

Abstract--This

paper proposes an optlmlzatmn-based approach to the robust 7-to~ control problem of uncertain contmuous or dmcrete-tlme linear tzme-mvariant systems with different tzme-varying delays m the state vector and control input of the dynamic equation and controlled output Sufficient delay-dependent conditions are derived for the control stabflmatmn problem, where both the sine of the tzme-varying delay and the raze of its time derivative (m the continuous-time case) play a crucml role for the closed-loop stabflzty with a guaranteed TLc¢ performance index. The solutmns that are proposed for the Hoo control problem are similar or less conservative, when compared to other recent approaches (~) 2005 Elsevier Ltd All rights reserved

Keywords--Tlme delay,

Robust stablhty, OptlmLzatlon

*Author to whom all correspondence should be addressed The authors are grateful to the Bramlian agencms CNPq and CAPES, and to the anonymous reviewers for their comments and suggestions that helped them improve the presentation of the paper. 0898-1221/05/$ - see front matter (~) 2005 Elsevmr Ltd All rights reserved doi 10.1016/j.camwa.2005 04 003

Typeset by .AA/~S-TEX

14

R

M

PALHARES

et al.

1. I N T R O D U C T I O N The control design problem for systems subject to time delay in the state vector and/or control input has received a lot of attention in the last decades. The increasing interest about this topic can be understood by the fact that time delays appear as an important source of instability or performance degradation in a great number of important engineering problems involving material, information or energy transportation. In the context of basic scmnces, it is not hard to find mathematical (non)linear models of real phenomena with time delay in circuits theory, economics, biology and mechanics, or more specifically, in chemical process, hydraulic, rolling mill, or computer controlled systems, and more recently Internet (or in a more general setting, time delays due to the signal commumcations in communication channels). Most of the approaches in literature concentrate on dealing with constant time delays. However, m some practical systems, time delays are varying in consequence of rapid (or even random) variations in transmission delays or motion of separated systems of master-slave type. In telecommunications systems, the presence of transmission delay variations is usually unavoidable, e.g., congestion-control algorithms, internet rerouting, mternet-based telerobotic systems, or network traffic [1-5]. Other classes of problems involving time-varying delays are biomedical and robotic systems [6] and cellular neural networks [7]. Besides the stability analysis of systems with time delay itself, many other subjects have been incorporated to the control design problem: parameter uncertainties, 7-{~ performance index, multiple time delays, delay-independent conditions, delay-dependent conditions, lumped and distributed delay cases, and optimization techniques involving LMI descriptions. Commonly, most of the approaches employ only the traditional delay-independent condition, in which the controller design is provided irrespective of the size of the time delay. On the other hand, it is well known that for systems in which the stability issue depends explicitly on the time delay, a delay-independent condition may not work. In fact, in order to overcome this difficulty, it is necessary to evidence the time delay m the control design to obtain a delay-dependent condition. The reader can find a great number of references about this topic in literature. The following references systemize the mare ideas, [8-20]. In the scenamo of continuous-time T{oocontrol, [10,16,21] deal with linear systems with constant time delays in the state. For systems with time-varying delays, [17,22] consider time delays in the state and [12] considers time delays in the state and control input. Also, [10,16,17,21] are delaydependent approaches whereas [12,22,23] are delay-independent approaches where no explicit information regarding the size of the time delay but only a fixed upper bound on the time derivative appears in the LMIs. Unlike the continuous-time case, there exist a few papers handling the robust ~oo control problem of discrete-time systems with time-varying delays in the state [24-26]. The LMIs approaches derived in [24-26] are of delay-dependent type and based on standard Lyapunov-Krasovskii functionals for discrete-time systems. A delay-independent approach for discrete-time systems is presented in [12]. In this paper, the main contribution is to state LMI sufficient delay-dependent conditions for the robust state-feedback control stabilization design, which guarantees an 7-{~ level of disturbance attenuation for both uncertain continuous or discrete-time state- and control-delayed systems with different time-varying delays. The parameter uncertainties dealt with are of polytopic type, which allows to extend the LMI results obtained for precisely known systems for a set where the vertices are elements of LMI type. The main results for continuous or discrete-time systems take as initial point a recent work [11], which is based on a new upper bound for the inner product of two vectors, and deals with the stabilization problem for systems with constant time delays. An early short version of this paper has appeared in [27] without addressing the problem of different time delays entering the state and control input vectors. Finally, for the continuous-time case one example is presented and the results are compared with recent approaches [10,16,17,19,21].

Uncertain Linear Systems

15

l~egarding the discrete-time case, one example is also handled and the results are compared with [24,25]. The notation used in this paper is as follows. Ox(t) indicates ~(t) for continuous-time systems or x(t + 1) for discrete-time systems. L2 denotes both the spaces £2 (set of all square integrable and Lebesgue measurable functions defined on a given interval) or g2 (the discrete-time context of sequences). The boldface characters I and 0 denote, respectively, the identity and the null matrices of convenient sizes. T1,2(t) is used to denote Tl(t) and T2(t), the time-varying delays of the system. 2. D E L A Y - D E P E N D E N T

CONTINUOUS-TIME

CASE

Consider the following linear time-invariant dynamic system with time delay, Ox (t) = A x (t) + A d x (t -- 7-1 (t)) + B u (t) + B d u (t -- T2 (t)) + E w (t), z (t) = C x (t) + Cdx (t -- T1 (t)) + D u (t) + Ddu (t -- ~-2 (t)) + F w (t),

(1)

x (t) = ¢ ( t ) ,

where x(t) : R ~ ~n is the state vector, u(t) : ]~ --+ litm is the control input vector, w(t) • R ~ R p is the exogenous disturbance vector and z(t) . ]~ ~ ~q is the controlled output. ¢(t) is a given initial vector function which is continuous on the segment [-¢, 0), with V defined as max[Tl(t),72(t)], and vl(t) and T2(t) are the time-varying delays of the system. Assume perfect state measurement. Moreover, the time-varying delays satmfy the following conditions for continuous-time, 0 ~ T1,2 (t) ~ T1,2 < O~,

0 __<~-1,2(t) __
Vt C ~,

(2)

or the following one for discrete-time, 0 ~ 71,2(t ) ~ T1,2 < OO,

~/t e Z.

(3)

For control purposes, we consider the memorytess state-feedback control law, (t) = K x (t) .

Then, the closed-loop state-delayed system is given by Ox (t) = f i x ( t) + A d x (t -- ~-1 (t) ) + B d K x (t - ~'2 (t) ) + E w (t) ,

(4)

z (t) = C x (t) + Cdx (t - T1 (t)) + D d K x (t - "r2 (t)) + F w (t),

w i t h A ~= A + B K and C ~= C + D K . The main control problem to be addressed in this section is stated below. ( P ~ ) - - T H E ROBUST ~~cc CONTROL PROBLEM WITH TIME DELAY. Given scalars "Yl,2 and ql,2 satisfying (2) (or (3) for discrete-time systems) and scalar ~, > 0, determine a controller gain K such that the closed-loop state-delayed system (4) is robustly stable and ensures a prescribed 7/~ disturbance attenuation for any time-varying delays satisfying (2) (or (3) for discrete-time systems), namely, under zero imtial conditions and for any nonzero w E L2, [[Z[IL2 ~ "7 HW[[L2 ,

V (A, Ad, B, Bd, C, Cd, D, D d , E , F )

E 7),

where 7) is a polytopic set described by ~ vertices, 7)-~

~ ( A . . . . . ,F,); ~_>0;

(A,..,F)I(A,...,F)= •= 1

(~=1

.

(5)

~=1

In this situation, the closed-loop system is said to be robustly stable with ~, disturbance attenuation level. The above notion of robust stability for uncertain system means that (4) with w(t) = 0 is robustly stable if its trivial solution x(t) - 0 is globally uniformly asymptotically stable for all admissible uncertainties and the time delays ~-l.2(t) satisfy (2) or (3).

R M PALttARESet al.

16

3. A N A L Y S I S AND FOR CONTINUOUS-TIME

SYNTHESIS SYSTEMS

In the following are presented the delay-dependent LMI robust 7-/~ performance analysis and control design for continuous-time systems subject to different time delays in the state and control input. THEOREM 1. Conmder the closed-loop system (4) and let ~1,2, ¢1,2, and " / > 0 be given scalars, with "~L2 and gl,2 satlsfying (2). If there exist symmetric matrices X >- O, H1,2 >- 0, Q1,2 ~ 0,

and Z1,2 >- 0 and matrices V1,2 satisfying

-Tll,

T12z T13,

•

-Q1

o

elATZ1

e2.e~TZ2

C?

o

~AZZl

~A2Z=

Ca~

XE,

•

*

-Q2

o

elKTB~Z1

e2KTB~Z2

K T D d ,T

•

*

*

--'y2I

elE¢T Z1

e2E T Z2

FT

•

*

*

*

-elZ 1

0

0

•

*

*

*

*

-e2Z2

0

•

*

*

*

*

*

--I

v?

Zl-

v?

o,

v i = 1,...,~,

(6)

(7)

z,-

where

Tllz ~

XAt-}-ff[:X-I-TIH

1

-1-T2H2 q- gl

1

- } - g ? Ac-g2or-V T -1- i--¢1 Q1 +

1 1--¢2

Q2,

Tim g XAd~ - 171, T13, A XBd~K - V2, el e2 =

1 -- ¢1'

1-¢2"

Then, the closed-loop system is robustly stable with disturbance attenuation "y for any timevarying deIay satisfying (2). | PROOF Consider the Leibniz-Newton identity,

L

8 iJ(t) dt = ~(b) - v(a).

(8)

Then, system (4) can be rewritten as

2(t)=(fl+Ad+BdK)x(t)-Ad

£

2(a) da-BdK rl (t)

ft

2(a) da+Ew(t),

Jr-r2 (t)

(9)

z (t) = Cx (t) + Cdx (t - rl (t)) + D d K x (t - 72 (t)) + F w (t). Let the Lyapunov-Krasovskai functional be

v (x (t), x (t - n (t)) , . (t - 72 (t)), t) = 71 + v2 + va + v4 + vb,

(10)

Uncertain Linear Systems

17

with

V1

=

y~ -

V3 -

XT

(t) X x (t),

1 - ~1 (t~

- 1 -

~,(~)

1 L

4"L ( t )

1

/

v~ -

1 - +2 (t~----~

v5 -

-1 - -

1 ~-2 ( t )

2fiT

n (t)

~(,)

(OL) q l

L

x

(OL) d(~,

xT(a) Z2J:(a)dad/3,

_~(~) X T

(a) Q 2 x (a) da.

Taking the time-derivative of the functional (10), it follows that

y(-)=~+&+&+~+&,

(11)

with Vl :

2x T (t) x (A + A~ + B d n ) 2f (t) + 2~ T (t) X E ~ (t) -

f' XBdK L

2z T (t) X A d

(12)

~ (~) do~

dt--Tl(t)

- 2x T (t)

~ (a) da,

(13)

~2 (t)

~r2 -- 1 --T17"l(t)(t) :~T (t) Z l x (t) -

L

~1(t) ~T (o~) ZI:~ (o~) do~

7-1 (t) ~T (t) ZlX (t) 1 - "rl (t)

(14)

-- ~tL ~T (Oz)Zlx (oz) da,

(15)

1 ( t ) X T (t) QlX (t) V3 ~- 1 - "]'1 ~r4 -~ 1 --T2T2(t)(t) :~T (t) Z2:~ (t) T2 (t) -< 1- ÷~ (t)

__ f~

2fT

(t - 7"1 (t)) Q l x (t - -

7"1

(t)),

Lr2(t) ~T (00 Z2x (00 do~

:~T (t) Z2x (t)

(16)

~T (O~)Z2x (oz) dQ~, 1

(17)

x T (t) Q2X (t) -- XT (t -- 1-2 (t)) Q2 x (t -- T2 (~)).

Using the upper bound of the inner product of two vectors introduced in [11] for (12) and (13), it follows that (12) _< Wl(t)x T (t)Hlx(t)

+ 2xT(t)(V1 - XAd) -F

L (t)H2x(t)

L

xT (Oz)ZlX(Ol) da,

"~1

(13) < 72(t)x T

x(a) da

(18) (19)

18

R M PALHARES et

al.

f

t

+ 2xT(t)(V2 -- XBdK) Jt-~2(t) ~(~) d~

(20)

+

(21)

S(~)z~(~) as.

The above upper bounds hold if the matrices H1,2,171,2, and Z1,2 satisfy the constraints,

After straightforward manipulations of terms (14), (16), (18), and (20), elimination of terms (15), (17), (19), and (21), and taking into account ¢1,2 and ~1,2, the upper bounds to the size of the time delay and its variation rate, respectively, one may rewrite (11) as

x(t)

x(t-Tl(t))[

tz()<

1T

x(to~},(t))[

r

x(t)

l

|x(t--Tl(t))|

A~

,

(2s)

where FII11

Ac -~ [ !

H12

I'~13

1214

II22 H23 YI24 $

*

l-~33 1-134 '

*

1"I44

Hll ~- X 2 + 2~TX + ~IH1 + ~2H~ + V, + V ( + V2 + VT

1 Q2 + elfiTZlfI + e2ftTz2fi, Q1 4- I--~2

4H12 _a_XAd

-JrelAC Z1Ad 4. e2AT Z2Ad, H13 ~ XBdK - V2 + elAr Z1BdK + e2/tr Z2BdK, 1214 ~ XE 4. elfi-r Z1E + e2f~TZ2E, -- V1

II22 ~=-Q1 4. elA~ Z1Ad + e2ATdZ2Ad, H23 -~ el A TdZI BdK + e2A-~Z2BdK, II24 ~ elATdZ1E 4- e2AT Z2E, Has & -Q2 + elK-rB-~Z1BdK + e2K-rBTdZ2BdK, H34 -~ elKT B f ZIE + e2K'r B f Z2E, 1244 =~ elET Z1E + e2Er Z2E, e L -1 --ql' e2

--

i

--

q-'--'-~'

when H1,2, V1,2, and Z1,2 satisfy (22). Now, considering the 7400 performance index, J =

[z T (t) z (t) - 72w T (t) w (t)] dt,

and assuming, without loss of generality, zero initial conditions to (4) where V(-)lt=0 -- 0 and V(.)lt_~oo --* 0, the above index may be rewritten as ~-~ S

ZT (t) Z (t) -- 72~//T (t) W (t) 4- g (')

dt

Uncertain Linear Systems

19

and

[ ~(t) l |x(t- ~-i (t))|

:-
T

[

~(t)

I • (t - ~ (t)) ~'~ I • (t - ~-~ (t)) L ~ (t)

dr,

(26)

where Ac

A

OT c * * *

A~ +

OT Cd C~Cd * *

OT DdK

OT F

CTdDdK KTD-~DdK *

]

I

C~F KTD-~F [ ' -.y2I + F T F J

which is equivalent to the inequahty below, after applying the Schur's complement, T1

T2

*

-Q1

A~ A

,

,

0 -Q2 , *

XE 0 0 -72I *

*

*

T 3

e2ATZ2 elA-~Zi e2A-~Z2 elKT BT Z1 e2KT BT Z2 elETZi e2ETZ2 -eiZi 0

~m

elATZ1

*

cJ KTD f FT

0 0

--e2Z 2

-I

T1 ~ X - ~ + A T X +YlH1 + ~2H2 + V1 + V ? -[-1/2 --FV? +

T2 A XAd

1 1 -q

Qi +

I 1 -¢2

@2,

V1,

-

T3 ~ X B d K - V2, A e I -e 2 --

"Yl

1 -

ql'

1 - ¢2'

when H1,2, V1,2, and Z1,2 satisfy (22). Considering that the LMIs (6) ensure that Ac < 0 for the entire uncertain domain 7), one can conclude that J < 0 for all nonzero w(t) E £:5. Thus, system (4) is guaranteed to be robustly stable with 1/~ disturbance attenuation level 7 for any time delay condition that satisfy (2). | Based on the 7-fo~ stability analysis developed above the next theorem can be derived. It must be noticed that the/-/o~ analysis result presented in Theorem 1 is an extension of the new stability analysis result derived in [11]. THEOREM 2. Consider system (1) and let T1,2, Q,2 and 7 > 0 be given scalars, with "Yl,2 and q,2 satisfying (2) If there exist symmetric matrices Y >- O, Mi,2 ~- O, Wi,2 >- O, Ri,2 ~- O, and matrices L, N1,2, satisfying kI] 12~,

'~/13"~

* *

-W1 *

0 -W2

E¢ 0 0

e1~15~ elYA~ eiLTB~

e2YA~ e2LTB~

*

*

*

*

--elRX

0

0

* *

* *

* *

* *

* *

--e2R2 *

0 --I

" '~ff11"L

e2~16 ~

I,~/17:

YC~ LTDd ,m

o,

N1 N2 [NM~ZRT1y] ~0,- [M# LN2 YR~iY] ~°'-

V~=l,...,n,

(27)

(28)

R. M PALHARESet al.

20

where • n~ s A~Y + Y A ~ + B~L + L T B [ + ~iM1 + ~2M2 + Ni + N [ + N2 + N ~ + l _ q1l W~+ 1 1--~2W2, ~12.

~ A m Y - N1,

~ia, ~ Bd,L - N2, @i~, S y A T + L T B T, ~16~ S y A T + L a B T, lwiT, & Y C [ + L T D , , el = 1 --ql ~

A

¢2

e2 -- 1 _ ~ 2 ,

then problem (P~) is solvable for any time-varying delay satisfying (2), with the control gain K = LY -i. |

PROOF. Pre- and post-multiplying the LMI (6) by diag

[ X - 1 , X - 1 , X - 1 , I, Z 1 1 , Z 2 1 , I],

with/1~ and C, replaced by A, + B~K and C, + D,K, respectively, one gets Tll, *

~712, - X - 1 Q 1 X -1

J?13z 0

E, 0

*

*

-X-1Q2X -i

0

,

•

,

---y2I

(30) e l ( X - 1 A T + X - 1 K T B T)

e2(X-1A T + X - 1 K T B T)

e i X - i A T,

e2X-IA T,

X --ICg, T

e l X - 1 K T BJ,

e2X-1KT B T"

y--I I~S /IT

x-icT

+X-iKTDT

40,

where

- e l Z ~ -i

0

0

*

-e2Z1 1

0

"~11~ --~ A , X -1 + X - 1 A { + B , K X -1 + X - 1 K T BT, + ¢ I X - 1 H 1 X -1 + T 2 X - 1 H 2 X -1 + X - 1 V 1 x - 1 + X - 1 V ? X

+ X-I--x-1vT

+ 1 --1 (1 X _ I Q 1 X _ 1+ 1 --1 ~2X _ I Q 2 X _ I ,

~f12, S A a X -1 - x - i v 1 x - i , ~f i3, S B d , K X -1 - X - 1 V 2 X - l , zx

"71 qi

el - - - - , - - , i --

e 2 ~---

1-(2

-1 + X - l y 2 x

-1

Uncertain Linear Systems

21

Pre- and post-multiplying (7) by diag IX -~,

x-l],

it follows

I X-~HIX -~ X-1V1x-1 ]

IX-iS2 X-1 X-1V2X-11 Lx_xv~x_l x_~z2x_x h O.

X_IV1TX_1 X_lZlX_ 1 ~--0,

Introducing the change of variables: Y - X -1, M1,2 - X-~H~,2X -~, N~,2 - X-~V~,~X -1, W1,2 = X-~QI,~X -1, R1,2 - Z~,~, and L - K X -~, the inequalities (27) and (28) are obtained. The last theorem ensures sufficient conditions for obtaining a robust 7-(~ controller. It must be noticed that the matrix inequalities (27),(28) are not LMIs, in fact Theorem 2 states a nonconvex problem. In order to overcome this one can either do the simple linearization R = Y, which turns (27),(28) into LMI conditions, but is somehow conservative, or choose to proceed with the same cone complementarity linearization algorithm, proposed in [28,29] (and similar to [11]) replacing (28) with S1,2J >-0, --

^

R1,2

~-0, (33)

$1,2

~ O,

-

•

;'- O,

71,

'2

-

~ 0. -

In this context, the following algorithm is proposed. 1. For ~1,2, ~1,2 and a disturbance attenuation level 7 > 0 given, with ¢1,2 and ;1,~ satisfying (2). Set k = 0. Find a feasible set of matrices ($1,2, 0 Y 0 ,R1,2, 0 ~01,2,]y0 ~0 satisfying 1,2,R1,2) (27) and (33). 2. Solve the problem, min

Trace {fk}

Subject to

(27) and (33),

(M,,2,W1,2,N1,2,L,S1,2,Y,R1,2,S1,2,Y1,2,R1,2) where sk

•

÷

÷

+

÷

3. If Trace {fk} ~ [(3n) + (3n)] and condition (28) holds, then there exists an ~o~ controller, K = L Y -1, which ensures the disturbance attenuation level 7. If (28) is not verified, set ~ + 1 = $1, $1k+l = S1, ]y~+l = ]~1, yk+l = y , /~+1 = /~1, R1k+' = R1, ~ + 1 = Se, S~ +1 = $2, ]Y2k+l = 1~2, 1~2 k+l =/~2, R~ +1 = R2 solutions of the optimization problem m Step 2. Set k = k + 1, if k < km~× (where km~x is the number of maximum iterations) return to Step 2, otherwise stop. Related to this algorithm, two paths can be implemented. The first one is concerned with finding the maximum time delay ¢. For that, an additional information must be introduced at Step 3, namely, if the condition (28) holds, then the time delay "~ can be increased returning to Step 2. The second one deals with the minimization of the disturbance attenuation level, 7, i.e., for ~ fixed, one can find the minimum 7 by implementing any line search algorithm and proceeding in the same way as m&cated at Steps 2 and 3. In all these situations, the upper bound to the time-derivative of the time-varying delay, ¢, is fixed. Moreover, if an initial -y is required, one can compute the minimum disturbance attenuation level, % by taking (27),(28) with the linear change of variables R = Y, which is an LMI problem for ¢ and ¢ given, with ¢ and q satisfying (2) Thus, solving this linearized problem can be seen as a starting point for the above algorithm Nevertheless, it must be noted that this linearized problem could be infeasible for increasing time delay, say Tm > ~, but for that same 7m, the algorithm above can work

22

R M PALHARES et al

4. A N A L Y S I S AND FOR DISCRETE-TIME

SYNTHESIS SYSTEMS

In the following are presented the delay-dependent LMI robust 7-(~ performance analysis and control design for discrete-time systems subject to different time delays in the state and control input THEOREM 3. Consider the closed-loop system (4) and let ¢1,2 and 7 > 0 be given scalars, with rI,2 satisfying (3). If there exist symmetric matrices X >- O, H1,2 >- O, Q1,2 >- O, and Z1, 2 >- 0 and matrices V1,2 satisfying "r I

-Y 1

-Y 2

0

~z]?X

,r2r2,Z2

C?

*

-Q1

0

0

A-~,X ¢IATd,Z1

"PlF2,ZI

r2Ad,Z -7

C~,-r

* *

* *

-Q2 *

0 -721

Pa,X

firs,z1

¢2Fa,Z2

r4,

E?X

flE~Z1

~2E?Z2

F(

*

*

*

*

-X

0

0

0

* * *

* * *

* * *

* * *

* * *

-fiZz * *

0 -~2Z2 *

0 0 -I

zl l

--~ 0,

V, = 1 , . . . , n ,

z2l

(34)

,35,

where F1 ~ - X + elHI + ¢2H2 + VI + V~ + V2 + V~ + QI + Q2, ~(~

I) ~

Pa, ~ (Bd,K) T , a (D d*K ) r4, -~-

T,

Then, the closed-loop system is robustly stable with disturbance attenuation 7 for any timevarying delay satisfying (3). | PROOF. Consider the identity, t-1

Am (k) = x (t) - x (t - r (t)),

(36)

k=t-~-(t) where Am(k) ~=x(k + 1) - x(k). System (4) can be rewritten as t-1

t-1

k = t - r l (t)

k = t - r a (t)

(37)

z (t) = Cx (t) + CdZ (t - rl (t)) + DdKx (t - ru (t)) + Fw (t). Let the Lyapunov-Krasovskii functional be v (z ( t ) , z (t - q ( t ) ) , x (t - ~-2 ( t ) ) , ~ ( t ) , t ) = vl + v~ + va + v4 + v~, with

Vl = r

(t) x x (t), -1

t-1

Ax T (k) Z l i x (k), s=--rl (t) k=t+s

(as)

Uncertain Linear Systems --1 Z

v~:

23

t--i Z aJ(k)Z~x(k),

s=-r2(t) k=t+s V4

:

t-1 E

xT (]~) Q l x

(k),

k=t-r~ (t) t--1

v5 =

~

xT (k) Q ~ ( k )

k=t--r2(t)

Taking the difference of the functional (38) it follows that (39)

A v (.)= zxv~+ sv2 + Av3 + sv4 + Avs, with

t--1

-2x T(t)(A+Ad+BdK)TXAd

E

Ax(k)

(40)

k=t--'r I (t) t--1

-gzT(t)(A+Ad+BaK) TXBaK

E

Ax(k)

(41)

k=t-r2(t)

(42) (43) +2

Ax T (k) AJX (BdK) \k:t-~l(t) t--1

-2

E

Ax (k)

(44)

\k=t-~(t)

t--1

Ax -<(k) AdT+

~=t-~(t)

~

1

Ax T(k)(BdK) T XEw(t)

(45)

k=t-~(t)

+ 2x T (t) (A + Ad +BdK) TXEw(t) + w-c (t) E TXEw(t),

(46)

t-1

AV2 : rl (t) Ax T (t) Z, Ax (t) -

E

Ax -r (k) ZIAz (l~)

k=t-r~(t) (47)

<__7"1 (t) A x T (t) Z I A X (t) t-1

-

~

Ax ~ (k) z ~ x (k),

(48)

k=t--~l t-1

~v3 : ~2 (t) zXx T (t) z 2 / ~ (t) -

E

ZXx~ (k) z 2 ~ (k)

k=t-'r2 (t)

<_ ~2 (t) /Xx ~ (t) Z ~ A x (t)

(49)

t--1

-

Z

AxT (k) Z 2 ~ (k),

k=t--? 2 tg

4 :

/' Vs :

2g T (t) Q1 x (t) - x T (t - 7-1 (t)) Q1 x (t - 7"1 (t)), X T (t) O2X (t) -- X T (t -- ~ (t) ) O=X (t -- ~ (t) ) .

(5o)

24

M

R

PALHARES

et al

Using the upper bound of the inner product of two vectors introduced in [11] for (40) and (41) it follows that

(40) ___(T1 (t) xT (t) Hlx (t)

[(

+2= T(t) v l - ~+Ad+BdK

'-1

XA~

~

/,~(k)

(51)

k=t--Tl(t) t--1

+ ~

~x T (k) Z~a~(k),

(52)

k=t---71

(41) _ ~2 (t) x T (t) H2~ (t)

t-1

+2xT(t)

1/2

(fI+Aa+BdK) T k=t-r2 (t)

t-1

+ ~

A x m (k) z~A~ (k).

(54)

k=t- ~2

The above upper bounds hold if the mamces

HL2, 1/1,2,and Z1,2 satisfy the constraints

After straightforward manipulattons of terms (42)-(47), (49), (51), and (53), elimination of terms (48), (50), (52), and (54), and taking into account %2, the upper bounds to the size of the time delay, one may rewrite (39) as

T Ix (t x(t) - ",-1 (t)) zxv (.) _< - ~-2 (t)) (t) where

Ad zx 211

zs

x(t) ] x (t - ~-1 (t)) /

Ad x (tw;2) (t))] ,

'~'!1 212 213 ~14 "~'22 "~'23 '~'24

AT XA-

*

233

234

*

*

244

'

X + ¢II-II+ ~2H2 + V1 + V~ + V2 + V• +QI +Q2

@¢1 (n -I)

T Z1 (A - I ) + ¢2 (A - I ) T Z2 (A - I ) ,

~'12

-V1 + ~T XAd + ¢1 (\A --I]~ ZlA~ +¢2 (\ A - I )T Z2Ad,

213

-V2 + SIT X B d K + ~1 ( A -

214 A ATXE -[- ¢1 (A - I )

T

I) T Z 1 B d K

ZIE +

+ f2

('4 - I) T Z 2 B d K ,

¢2(fI - I )T Z2E,

222 ~- -Q1 + A-~XAd + "T1A-~Z1Ad + ¢2A-~Z2Ad, ~'23 7~24 = AJ X E + ¢1A~ Z1E + ~A~ Z2E, "~"33 ~---Q2 + (BdK) T X B g K + 71 (BdK) T Z1BdK + ~2 (BdK) T Z2BdK, :~34 ~=(BdK) T X E + 71 (BdK) T Z1E + ~2 (BdK) T Z2E, ~'~44 ~- ET X E + ~IET Z1E + "T2ETZ2E,

(56)

Uncertain Linear Systems

25

when HL2, V1,2 e Zi,2 satisfy (22). Now, considering the 7-/00 performance index,

= ~ [ZT (t) Z (t) -- "72WT (t) W (t)], t=0 and assuming, without loss of generality, zero initial conditions and stability to system (4), it implies V(')]t=0 = 0 and Y(.)lt-~o~ -~ e, with e -~ 0 if w(t) = 0 or e < co if w(t) # 0. In this way, the above index may be rewritten as

(5s)

J ~ ~ [ZT (t) Z (t) -- "y2wT (t) ~ (t) + AV (.)], t=0 moreover,

[~ x(t) oo (t (t)) -

J <- Z t=0

T

-cT C * Ad & Ad + , ,

CTCd CTdCd * •

-

T1 (t))

-

1

(59)

Ad x(t-~2(t))J' w (t)

(t)

where

I

X (t

~-~

(t - ~. (t))

CTDdK C~DdK KTDTdDdK •

cTF CTd F KTD~F --72I + F T F

which is equivalent to the inequality below, after applying the Schur's complement,

-I~l * * , /~ Lx d-~- ,

-V1 * • •

-V2 0 --Q2 , •

0 0 0 -72I *

ATx ATd X F3X ETx -X

*

*

*

*

*

--T1 Z1

0

*

*

*

*

*

*

-~2Z2

0

*

*

*

*

*

*

*

-I.

-Qi

"rlF2Z1

~2F2Z2

~IA~Z1

¢2ATdZ2

~T CTd ¢11~3Z1 ~2I~3Z2 I~4 ¢IETZi ¢2ETZ2 F T 0 0 0 0

with F1 & - X + "rlH1 +¢2H2 +V1 + V ~ +V2 + V ~ + ~ 1 + Q 2 , F2 ~

t,

(A - I)T ,

F3 = ( B d K )

T

,

~' (DdK) T F4 = when HL2 , V1,2, and Zi,2 satisfy (22). Considering that the LMIs (34) ensure that Ad -< 0 for the entire uncertain domain 7), one can conclude that ff < 0 for all nonzero w(t) C ~2- Thus, system (4) is guaranteed to be robustly stable with 7-/o0 disturbance attenuation level 7 for any time delay condition that satisfy (3). I Next, the discrete-time synthesis version is presented and the same cone algorithm indicated in the last section can be used. THEOREM 4. Consider system (1) and let ~1,2 and 7 > 0 be given scalars, with ~i,2 satisfying (3) I f there exist s y m m e t r i c matrices Y >- O, M1,2 >- 0, W1,2 ~- 0, R1,2 ~" 0, and matrices L, N~,2,

R M PALHARES et al

26

satisfying "~11 *

-N1 -W 1

-N2 0

0 0

~15~ ~T)2~,

"Yl(I~16 ~ "F1~T~2~,

"Y2~1~17z "Y2(]}2~

~1~18~

YC T T

,

•

*

*

-Y

*

*

*

*

* *

* *

*

*

*

*

0

0

0

* *

-fiR1 *

0 -f2R2

0 0

*

*

$

.,v, [ M4 yR_~ly] ~'- O, -

L N~

where

L N2

_y+.YlM1 _l._.P2M2 _I._N1 +NIT

(i)11 a

T -~0,

V~=l,...,t~,

(60)

--I

yR;l~l. ]

(61)

~- O, -

_I_N2 -.l--N : -I-W 1 -FW2,

YA~ + LTB T

i]~15 ~ ~

~ia~ ~ YA~ + L T B~ - Y, ~in ~ YA~ + LT B~ - Y,

~is, g YC~ + LT D T ~2~ =^

YA~

then problem ( p r ) is solvable for any time-varying delay satisfying (3), with the control gain

K = L Y -1.

l

PROOF. Pre- and post-multiplying the LMI (34) by diag [X -1, X -1, X -1, I, X -1, Z~-i, Z~-i, I], with .4, and C, replaced by A, + B~K and C~ + D,K, respectively, one gets F1 , ,

-X-1V1 x - 1 - X - i Q 1 X -1 ,

-X-1V2 x-1 0 -X-1Q2X -i

0 0 0

~2, T X -1 A& X - i K T Bd ~T

*

*

*

--,T2I

,

*

*

*

*

*

*

*

*

*

*

*

*

*

*

$

*

*

*

E T -X

-I

(63) "Y1(f'2~ -- X - 1 )

~I X - 1 A ~

~2 (f'2~ - X - 1 )

e2X-1A T&

X-1C~ + X-iKTD~ X-iC~

.71X-1KTBT

¢2X-1KTBT

0

0

0

-¢iZ -i

0

0

*

--'Y2 Z2 -1

0

*

*

--I

y--1 R"TDT

-4 O,

Uncertain Linear Systems

27

where

F1 A - X -1 + f l X - 1 H I X -1 -1-f2X-1H2X -1 + X-1V1X-1 -t- X - 1 V ? X -1 -}- X-1V2 x - 1 ~- X - 1 V / X -1 ~- X-1Q1X-1 j- X-1Q2 X - l , F2, A X - 1 A ~ + X - 1 K T B ~ . Pre- and post-multiplying (35) by dlag [X -1, x - l ] , it follows t h a t

X-IHIX-I

x-lgl x-1 ]

X-1V2X -1 ]

[ X-1H2X -1

X-iV~X-1 x_lzlx_l] >-o,

[x_IKx_I x_iz2x_l

Introducing the change of variables, Y : X -1, M1,2 -- X-1H1,2X -1, N1,2 = X-1V1,2X -1, |

W1,2 ~- X-1Q1,2 X - l , ~t~1,2= Z-1~,2,and L - K X -1, the inequalities (60) and (61) are obtained 5. E X A M P L E S

EXAMPLE 1. Consider the same example as shown in [16,17,19,21], with the following matrices for system (1).

[0° 01]' C=[0

1],

[1 ;11 0

Cd=[O

9 '

01,

[00] '

'

Dd=O,

D=0.1,

[1]

1 '

F=O.

For thin example, the time-derivative of the time delay is ~1 = 0, so q = 0, as exactly considered in the above references, where the LMIs approaches had also been developed to deal with linear systems with time-varying delays. Searching for the m a x i m u m time delay for which there exists an 7-{00 stabihzing controller, in [16], the m a m m u m delay achieved is 71161 = 1.28s, with 0'[161 -- 0 18 and K[~61 = [0

-130.38].

Now, in [17] (using Theorem 5), one can find as maximal time delay 7117] = 1.408 s and 3'[17] = 106.1506, with K[17] = [-156.36

-1439.66].

In [19] (using T h e o r e m 2), one can find the same results as presented in [17] On the other hand, applying the approach presented here, even for the time delay T -- 5 0 s one is able to find a stabilizing controller,

KNonconvex= [--102.4825

-136.66],

with 7 = 22 (although some computational effort is required). However, as can be seen, increasing the value of the m a x i m u m time delay implies a degradation of the disturbance attenuation level

28

R

10

i

M

i

PALHARES

i

et

al.

i

i

i

i

7.5 5 2.5 ,-.;.=

0 2.5 5 7.5 10

I

I

r

I

i

I

I

20

40

60

80 t

100

120

140

160

Figure 1 Evolution of the state variab]es of the closed-loop system in Example 1, for ~- = 5 0s, considering the control KNonconvcx. I

30

20

10

0 j

10

0

I

I

I

2

4

6

I

I

I

I

8 10 12 14 t Figure 2 State evolution of the closed-loop system on Example 1 with K[19] and simulated for the time-delay ~- = 5 0s

1 x

/

16

\

0

1

I

I

I

S

10

15

I

i

i

I

20 25 30 35 t Figure 3 State evolution of the closed-loop system m Example 1, simulated for the time-delay r = 1.408 s with K[19] (sohd line), and KN . . . . . vox (dashdot hne) which has been obtained for ~- = 5.0 s.

40

F i g u r e 1 d e p i c t s t h e t i m e - r e s p o n s e of t h e s y s t e m we c o n s i d e r e d above, for t h e control g a i n gNonconvex, closing t h e loop, and t h e t i m e d e l a y ~- = 5.0 s. As a second i l l u s t r a t i o n , F i g u r e 2 depicts t h e t i m e - r e s p o n s e of t h e closed-loop s y s t e m w i t h t h e s t a t e - f e e d b a c k control gain K119], w h e n c o n s i d e r i n g t h e t i m e - d e l a y 7- = 5.0 s. N o t i c e t h a t these controllers do not e n s u r e t h e closed-loop s t a b i l i t y for t h a t size of t i m e - d e l a y Finally, F i g u r e 3 depicts t h e t i m e - r e s p o n s e of t h e closed-loop s y s t e m for two cases. T h e first one takes into a c c o u n t t h e control gain K[19] o b t a i n e d for t h e m a x i m a l t i m e d e l a y ¢[19] = 1.408 s, w h i c h is r e p r e s e n t e d by t h e solid line for t h e two s t a t e variables. T h e second one considers t h e control g a i n KN . . . . . . . . . o b t a i n e d for t h e t i m e delay T = 5.0S, h o w e v e r s i m u l a t e d for t h e s a m e T = 1.408 s, a n d r e p r e s e n t e d by t h e d a s h d o t line. It is clear t h a t b e t t e r p e r f o r m a n c e was o b t a i n e d w i t h / ( N o . . . . re×

Uncertain Linear Systems

29

5

45 4i

3.5

3

2.5

2

1.5

1

0.5

0

0

10

20

30

40

50 Tempo

60

70

80

90

100

(s)

Figure 4 Time-varying delay histogram, Example 2

EXAMPLE 2. Consider the same example as in [25], with matrices Ca = 0 and F = 0, where the discrete-time system with time-varying delay is given by

•

0

o.1 ]x (t - ~ (t))

lx•

0.02

~ +

2d

0.5

0.3 w(t),

B

E

z(t)=[1 3]z(t)+ l~u(t). C

D

Figure 4 depicts the histogram of the time-varying delay, and let ~ = 5 s be the upper bound for the time-varying delay. With ~ fixed, Table 1 presents the minimum disturbance attenuation level, % obtained from the approach proposed in this paper, and two other ones in [24,25]. From Table 1 one can note the improvement in terms of the disturbance attenuation level 7-/o~, for ~ fixed. Table 1 Comparatwe table for the proposed approach and two other ones, Example 2.

Approach (Proposed)

[2~] [24]

~"

Gain

I 9703

[0.3793- 1 1163]

2 2388

[ 0 4046 --0.1762 ]

2.2399

[o 4044 -o 1762]

The simulation results for the proposed approach are depicted in Figures 5 and 6 with zero initial conditions and the varying-time delay evolution in Figure 4. The disturbance signal is

30

R M PALHARES ¢t al.

2 1.5

II

1

05

0.5

15

1l0

210

310

4=O

i

510 Tempo (s)

70

80

9O

IO0

Figure 5 State evolution of the closed loop system, Example 2.

41

'

2~

I I

0

4

0

lo

2o

r

30

~'0

do

Tempo(s)

6'0

~ - 70 ~

" 80

90'

I O0

Figure 6 Evolution of the disturbance signal, w (dotted hne), and the controlled output signal, z (sohd hne), Example 2.

defined as w(t) =

2,

if 10 < t < 30,

-2,

if 6 0 < t < 8 0 ,

0,

otherwise.

The state evolution is presented in Figure 5, and Figure 6 illustrates the relation between the disturbance signal and the controlled output signal It is easy to note that V = 1.9703 holds, for this disturbance signal

Uncertain Linear Systems

31

On the other hand, suppose that the disturbance attenuation level, 7, is fixed, say " / = 5, and one looks for the maximum upper bound to the time-varying delay. Table 2 shows the maximum upper bound, y, allowed when considering the proposed approach and two other ones in [24,25]. It is clear that the proposed approach relaxed the upper bound Table 2 Maximum upper bounds to the time-varying delay allowed for 7 -- 5, Example 2. Approach

"~

Gain

(Proposed)

1 × 106s

[--0.9850 --0.8813]

[25] [24]

172s

[--0 7492 1 0387]

172s

[--0 7485 1 0373]

6. C O N C L U S I O N S This paper addressed the robust 7/o0 control problem for uncertain linear systems with different time-varying delays in the state and input vectors. Both the size of the time-varying delay as well as its tlme-derivative were considered in the approach. Further, a delay-dependent sufficient condition was presented allowing to handle the robust control problem in an LMI-based iterative algorithm. As a particular characteristic of the proposed approach, the original system was not increased as other technique in the literature called descriptor system approach [16]. The behavior and efficmncy of the control design approach had been illustrated by means of two examples for continuous and dlscrete-txme systems.

REFERENCES 1 N. Xi and T J Tarn, Stablhty analysm of non-time referenced Internet-based telerobotm systems, Robotzcs and Autonomous Systems a2 (2-3), 173-178, (2000) 2 H S. Hu, L.X Yu, P.W Tsm, and Q. Zhou, Internet-based robotic systems for teleoperatmn, Assembly a u t o m a U o n 21

(2), 143-151, (2001)

3. R Oboe, Force-reflecting teleoperatlon over the Internet The JBIT project, Proceedzngs of the IEEE 91 (3), 449-462, (2003) 4 S Mumr and W J Book, Control techmques and programming issues for time delayed, Journal of Dynamzc Systems Measurement and Control 125 (2), 205-214, (2003) 5 F Blanchmi, R Lo Clgno and R Tempo, Robust rate control for integrated servmes packet networks, IEEEA C M Transactwns Networkzng 10 (5), 644-652, (2002) 6 F. Wu and K.M. Grlgorladm, LPV systems with parameter-varying time delays. AnMysls and control, Automatzea 37 (2), 221-229, (2001) 7. D M Zhou and J.D Cao, Globally exponential stablhty conditions for cellular neural networks with timevarying delays, Apphed Mathematzcs and Computatwn 131 (2-3), 487-496, (2002). 8 J -P Richard, Time-delay systems: An overvmw of some recent advances and open problems, Automat*ca 39 (10), 1667-1694, (2003) 9. V I Skorodinskfi~ Iteratlonal method of constructxon of Lyapunov-Krasovskn functxonals for linear systems with delay, Automatwn and Remote Contzol 51 (9), 1205-1212, (1990) 10 C.E deSouza and X. L1, Delay-dependent robust ~o~ control of uncertain hnear state-delayed systems, Automatzca 35 (7), 1313-1321, (1999) 1I Y S Moon, P Park, W H. Kwon and Y S. Lee, Delay-dependent robust stabilizatmn of uncertain statedelayed systems, [nternatwnal Journal of Control 74 (14), 1447-1455, (2001) 12 J H Kam and H B Park, T/~ state feedback control for generalized continuous/discrete time-delay system, Automatzca 35, 1443-1451, (1999) 13 C E de Souza, R M PMhares and P L D Peres, Robust "H~ filtering for uncertain hnear systems with multiple tlme-v~rylng state delays, IEEE Transactzons on S~gnal Processing 49 (3), 569-576, (2001) 14 M S Mahmoud, Robust Control and Fdtemng for Tzme-Delay Systems, Control Engzneemng Seines, Marcel Dekker, Inc., New York, (2000) 15 F Zheng and P M Frank, Robust control of uncertain dlstrabuted delay systems wath applmation to the stabdizataon of combustion in rocket motor chambers, Automatzea 38, 487-497, (2002) 16. E Frldman and U. Shaked, A descriptor system approach to 7~oo control of linear tame-delay systems, IEEE Transactwns on Automatzc Control 47 (2), 253-270, (2002). 17 E Fridman and U. Shaked, Stability and ~ control of systems wath time-varying delays, Proceedings of the 15th IFAC Tmennzal World Congress, Barcelona, Spaln, (July 2002)

32

R M PALHARESet al

18. R M Palhares, C.E. deSouza and P.L.D Peres, Robust ~oc filtering for uncertain discrete-time state-delayed system, IEEE Transactzons on Szgnal Processzng 49 (8), 1696-1703, (2001) 19 E. Fndman and U Shaked, Delay-dependent stability and 7-/00 control. Constant and time-varying delays, Internatwnal Journal of Control 76 (1), 48-60, (2003). 20 A Hauram, H H Michalska and B Boulet, Robust output feedback stabilization of uncertain time-varying state-delayed systems wlth saturating actuators, Internatzonal Journal of Control 77 (4), 399-414, (2004) 21 E Fndman and U Shaked, New bounded real lemma representations for time-delay systems and their apphcations, IEEE Transactwns on Automatzc Control 46 (12), 1973-1979, (2001). 22. J.H. Kim, Robust mixed ~ 2 / ~ control of time-varying delay systems, Internatwnal Journal of Systems Sczenee 32 (11), 1345-1351, (2001) 23 Y Y. Cao, Y X Sun, and J. Lam, Delay-dependent robust T/oo control for uncertain systems with time-varying delays, IEE Proceedings--Control Theory and Apphcations 145 (3), 338-344, (1998). 24 S -H. Song, J -K Sim, C.-H. Ylm and H.-C. Kim, "Hoo control of discrete-time linear systems with timevarying delays m state, Automatica 35, 1587-1591, (1999). 25 K T. Kim, S H Cho, K.H Bang and H.B Park, ~o~ control for discrete-time linear systems with time-varying delays in state, Proceedzngs of the 27eh Annual Conference of the IEEE Industmal Electronzes Society, 707711, (2001) 26 K T Klm, S H Cho, J.K. Klm and H.B Park, ~ controller design for dmcrete-time hnear systems with time-varying delays m state, In Proceedzngs of the ~0 th IEEE Conference on Deezszon and Control, 14461447, Orlando, FL, (2001) 27. R.M. Palhares, C.D. Campos, M.C.R. Leles, P.I. Ekel and M.F.S.V. D'Angelo, Delay-dependent 7-g~ control stabilization of hnear systems with time-varying delays, In Proceedings of the ~th IFAC Workshop on TzmeDelay Systems, Rocquencourt, France, (2003). 28 O.L. Mangasarian and J S. Pang, The extended linear complementarity problem, S I A M Journal of Matmx Analys*s and Apphcatwns 2, 359-368, (1995) 29 L. E1 Ghaom, F. Oustry and M. Air Rami, A cone complementarity linearization algorithm for state outputfeedback and related problems, IEEE Transaetzons on Automatzc Control 42 (8), 1171-1176, (1997)