ROBUST OBSERVER DESIGN FOR TIME-DELAY SYSTEMS: A RICCATI EQUATION APPROACH

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1 KYBERNETIKA VOLUME 35 (1999), NUMBER 6, PAGES ROBUST OBSERVER DESIGN FOR TIME-DELAY SYSTEMS: A RICCATI EQUATION APPROACH ANAS FATTOUH, OLIVIER SENAME AND JEAN-MICHEL DION In this paper, a ethod for /7 M observer design for linear systes with ultiple delays in state and output variables is proposed. The designing ethod involves attenuating of the disturbance to a pre-specified level. The observer design requires solving certain algebraic Riccati equation. An exaple is given in order to illustrate the proposed ethod. INTRODUCTION Many papers have dealt with the proble of stabilizing a linear syste with delays in state and/or input variables independently or dependently of the delay (see for exaple Furukawa et al [6], Lee et al [10], Niculescu et al [11], Choi et al [3] and the references therein). All these papers assue that the state variables are easurable entirely. However, this is not the case in practical situations. Few papers have considered the proble of designing an asyptotic observer for tie-delay systes. This proble has been solved by Watanabe et al [15], [16] by introducing a distributed tie-delay in the observer dynaic syste. Also, Pearson et al [12] and Raos et al [14] have built observers for delayed-state systes by coputing the unstable eigenvalues of the syste under consideration which, in soe case, ay be a very difficult nuerical proble [9]. In those papers the robustness issues have not been considered. Yao et al [17] have used a factorization approach in order to paraeterize an observer for tie-delay systes, however, no explicit procedure is given to obtain the observer. Soe papers have developed observer-based controllers for tie-delay systes (see for exaple Choi et al [2]), but the delay is considered in state variables only and the observer design requires the solution of a pair of algebraic Riccati-like equations. The authors have already obtained soe results on observer design for tie delay systes using two different approaches. The first one [4] uses the Razuikhin theore to give a sufficient condition for the existence of an Hoo observer. While the second one [5] gives soe criteria for the existence of an exponential observer.

2 754 A. FATTOUH, O. SENAME AND J.-M. DION This paper is concerned with the proble of Hoo observer design for linear systes with ultiple delays in state and output variables using the Krasovskii theore. An extended Luenberger-type observer is firstly proposed for the syste under consideration. The estiated error dynaics (which is the difference between the state of the syste and its estiate) is a ultiple delayed-state syste. In order to stabilize it and to get a pre-specified attenuation level between the disturbance and the estiated error, the technique proposed by Lee et al [10] for H^ stabilization of single delayed-state systes is generalized to the case of ultiple delays and used to study the stability of the estiated error syste. It leads to a Riccati equation to be solved. The paper is organized as follows. The proble stateent is presented in Section 1. Section 2 is devoted to the HQQ observer design. An exaple illustrates the proposed ethod in Section 3. The paper concludes with Section 4. Notation. M is the set of real nubers, M* is the set of real non-negative nubers, M n denotes the n diensional Euclidean space, M nx is the set of all n x real atrices, s is the Laplace variable, I n is the n x n identity atrix, 0 n is the n x n zero atrix, 2 is the space of square integrable functions on [0,oo), ^1,^2] is the space of continuous functions on [^1,^2] and. oo denotes the Hoo-nor defined as: T(s) oo ax {^(^0^)) : u ^}i (T) denotes the axiu singular value of the atrix T, j = y/^l and u denotes the frequency. For X G R nxn, the notation X > 0 (respectively, X > 0) eans that the atrix X is real syetric positive definite (respectively, positive sei-definite), X* denotes the adjoint atrix of X (X*(ju>) := X T ( JLJ)) } \i(x) denotes the i th eigenvalue of X. 1. PROBLEM STATEMENT Consider the following linear tie-delay syste: r x(t) = Ax(t) + Y7^Aix(t-ih) + Ed(t) < y(t) = Cx(t) + J2 =lcix(t-ih) + Fd(t) (1) k *(t) = 0(0; *e[-fc f o] where x(t) G R n is the state vector, y(t) G M p is the easured output vector, d(t) e R q is the 2 disturbance vector, <t>(t) G C[-/i,0] is the initial functional condition vector, h G -K + is the fixed, known delay duration, is a positive integer such that h represents the axial delay in the syste and A, A iy C, Cj, E and F, i = 1,...,, are real atrices with appropriate diensions. It should be noted that atrices E and F ay also include paraeter uncertainties or odelling errors. An extended Luenberger-type observer for syste (1) is given by the following

3 Robust Observer Design for Tie-Delay Systes: A Riccati Equation Approach 755 dynaical syste: (x(t) = Ax(t) + Z? =l Aix(t-ih)-L(y(t)-y(t)) ( \ y(t) = Cx(t) + ZT=iCii(t-ih) where x(t) E JR n is the estiated state of x(t) } y(t) G JR P is the estiated output of y(t) and L is the observer gain atrix of appropriate diension to be designed. The estiated error, defined as e(t) = x(t) x(t), obeys the following dynaical syste, obtained fro equations (1) and (2): e(t) = (A- LC) e(t) + ^(A { - LC { ) e(t - ih) + (E - LF) d(t). (3) -=i Reark 1. In this paper no control input is considered in the syste (1) since the sae control input can be considered in the syste (2). Those two ters cancel each other in the estiated error syste (3). The next definition introduces the concept of j-observer throughout this paper. which will be used Definition 2. Given a positive scalar 7, the syste (2) is said to be a j-observer for the associated syste (1) if the solution of the functional differential equation (3) with d(t) = 0 converges to zero asyptotically and, under zero initial condition, the Hoo nor of the transfer function between the disturbance and the estiated error is bounded by 7, that is: 1. lit-^oo e(*) -> 0 for d(t) = 0, 2. T e( i(.s) oo < 7 under zero initial condition, where T e d(s) is given by: T e d(s) = sl n -(A- LC) - T(AІ - Ld) e ' i=l (E-LF). The purpose of this paper is to design a constant gain atrix L such that syste (2) is a 7-ofaserver for the associated syste (1) for soe positive scalar THE OBSERVER DESIGN In this section, a sufficient condition for the existence of a 7-observer for tie-delay systes of the for (1) is provided. Furtherore, an explicit calculation of the observer gain atrix L is given. Firstly, a result concerning the Hoo asyptotic stability of ultiple tie-delay systes is stated in Proposition 3. This result is an extension of the ain result of Lee et al [10] concerning the Hoo stabilization of single delayed-state systes. Then, this Proposition will be used to derive the ain result of this paper in Theore 4.

4 756 A. FATTOUH, O. SENAME AND J.-M. DION Proposition 3. Consider the following linear ultiple delayed-state syste: J x(0 = Ax(t) + Z?=iAix(t-ih) + Dd(t) \ x(t) = 4>(t); t [-h,0] (4) where x(t) G -R n is the state vector, d(t) G JR q is the 2 disturbance vector, <j>(t) G C[ /i,0] is the initial functional condition vector, h G M* is the fixed delay duration (could be unknown) and A } Ai and D, i = 1,.,.,, are real atrices with appropriate diensions. Given a positive scalar 7, if there exists two syetric positive definite atrices Q and P such that A T P + PA + P ( JT AiQ- l Aj \p + Q+ \l n + PDD T P < 0 (5) then the ultiple delayed-state syste (4) is asyptotically stable for any value of the delay and the following inequality holds: 8І n -A-~ГAiЄ- ai V 1»=i -sih \ JJ <7- (6) Proof. See Appendix A. Using the above Proposition, the gain atrix L can be designed such that the syste (2) is a ^-observer for syste (1) sis follows. Theore 4. Consider the tie-delay syste (1). Given a positive scalar 7, there exists a ^-observer of the for (2) for the syste (1) if the following algebraic Riccati equation A T P + PA + 2PІ J 2 ~Ţ2AІAJ + EE T --C т e»=1 I P -\[I 2 E C < CT + FFT \ t = l > ) P c + \ + ln=0 has a syetric positive definite solution P for soe positive scalar 6. In this case the required observer gain is given by L = -P~ l C T. e (7) (8) Proof. Suppose that Riccati equation (7) has a syetric positive definite solution P for a given 7 and soe appropriate positive scalar e. Choosing L = \P~ l C T,

5 Robust Observer Design for Tie-Delay Systes: A Riccati Equation Approach 757 Riccati equation (7) can be rewritten as follows: A T P + PA + IP I T 2 J2 AiAj + EE T ] P - PLC - C T L T P + 2^ PL ( EO.Of ) L T P + 2PLFF T L T P + i±---/ n = 0 ОГ (.4 - LCfP + P(A -LC) + PWY J A' AT + EE T \P+!±---/ + j 2 PL(J2CiCT)L T P + PLFF T L T P ( Using the following inequality [7]: i=i \ Y J C i Cj)L T P- 1 2P{Y,AiAj\ P-PLFF T L T P-PEE T P (9) i=i i=i / -XX T - YY T < XY T + YX T where X and Y are any two atrices with suitable diensions, it follows that -PL (zo.of) L T P - P (f>* T ) P and < PL(^C i Aj\p + p{y j A i C T \L T P (10) -PLFF T L T P - PEE T P < PLFE T P + PEF T L T P. (11) By (10) and (11), equation (9) leads to the following inequality: (A - LC) T P + P(A - LC) + ÝP\У J AiAj ) P + PEE T P + үpl ПГ dca L T P + PLFF T L T P - үpl (jr daj\ P - үp (j2 AІCA L T P - PLFE T P - PEF T L T P + Ц^/ < 0 or (A - LC) T P + P(A - LC) + үp ^(AІ-LCO^AІ-LCІ) 1 P L»'=l + I n + ^I n + P(E-LF)(E-LF) т P<Q. (12)

6 758 A. FATTOUH, O. SENAME AND J.-M. DION Applying now the "Bounded Real Lea" (see Appendix B) leads to (A - LC) T P + P(A - LC) + т 2 / 5 ^(AІ-LCMAІ-LCІY : = 1 + / + 4I«+ P(E ~ LF ) ( E ~ LF ) Tp < 0 (13) for soe P = P T > 0 with P > P (Note that P = P if (12) is a strict inequality). Using now Proposition 3, inequality (13) directly corresponds to (5) applied on estiated error syste (3) with Q = -\ln- By proposition 3, it follows that syste (3) is asyptotically stable independently of the delay and */ - (A - LC) - J2(Ai - Ld) c ' i=i 1 i-i (E - LF) <T ì.e. This ends the proof. IITedOOHoo < 7- It should be noted that Theore 4 provides a delay independent criterion for the existence of a 7-observer. However, the delay value ust be known for the observer ipleentation. Reark 5. Soe sufficient conditions for the existence of a syetric positive definite solution to the Riccati equation (7) are presented. These conditions link the existence of a 7-observer (given in Theore 4) with soe stabilizability properties on the tie-delay syste. Let us define the following atrices: K -C т Є J2dcт + ғғ т t=l H := [jax, -ya 2,... ya, E\. c- l -±^i n Using the above definitions, Riccati equation (7) can be rewritten as: A T P + PA- 2PHH T P + K = 0 (H) with A := -A. For a given 7 > 0, if there exists a positive scalar e such that K is syetric and positive definite, then there exists a full rank atrix J JR nxn such that K = J T J. In this case, Riccati equation (14) has a syetric positive definite solution P if (A,H) is stabilizable and (J, A) is detectable (see [18] Corollary 13.8, p. 338). Now this solution P is syetric positive definite since K is syetric positive definite. Since J has full rank then (J, ^4) is always detectable. If (A,H) is stabilizable, then (A,H) is also stabilizable. So Riccati equation (7) has a syetric positive definite solution P if:

7 Robust Observer Design for Tie-Delay Systes: A Riccati Equation Approach 759 (i) there exists a positive scalar e such that K is syetric positive definite, (ii) the pair (A y H) is stabilizable. 3. ILLUSTRATIVE EXAMPLE In order to illustrate the applicability of the proposed ethod, let us consider the following tie-delay syste: i(t) = ' " 0 x(ł) + " * 0 x(t-l) " 0 1 z{t-2) + đ(t) y(t) 0 0 " " 1 0 " 0 1 x(ť) x(t-l) x(t-2) + d(t). Note that this tie-delay syste is weakly observable but is not strongly observable as (C, A) is not observable (see Lee and Olbrot [9] for the definitions). For this exaple two "/-observers will be constructed for different values of 7 using the proposed ethod. (i) 7 = 1: in this case, solving Riccati equation (7) for e = 0.1 gives: p = and the corresponding gain is: L = The estiated errors are shown in Figure 1 for soe initial condition <+>(t). This figure shows that the estiated errors converge asyptotically to zero for d(t) = 0. In Figure 2, the singular value of T e d(j u ) is traced versus the frequency. It shows that T e d(ju;) oo is less than one (less than 0.27 in this case). (ii) 7 = 0.1: solving Riccati equation (7) for e = gives: p = the corresponding gain is: L = The estiated errors are shown in Figure 3 for the sae initial condition <j>(t) as above. This figure shows that the estiated errors converge asyptotically to zero for d(t) = 0. In Figure 4, the singular value of TedU^) is traced versus the frequency. It shows that T e «-.(ju;) oo is less than 0.1 (less than in this case).

8 760 A. FATTOUH, O. SENAME AND J.-M. DION <->,«.( ),< ) Fig. 1. Estiated errors for 7=1. Fig. 2. <r(t e d(jw)) versus u;. 01 (-),«.(- )«2 « "'-, 1-03 /"~ '-'P^ Ì 1 / ; -» 1 * ' 1 1 / \ l / \ii/ : * ' -0.5 Fig 3. Estiated errors for 7 = O.L Fig 4. <r(ted(jw)) versus w. Note that in the both cases, the curve between 1 and 0 sec. represents the initial function condition of the given syste. For siulation purposes, an initial value at tie t = 2 sec. is used to generate an initial value function on / G [ 1,0]. It should be noted that the sufficient conditions given in Reark 5 are not satisfied for this exaple although soe efficient "{-observer can be constructed. 4. CONCLUSION In this paper, a ethod for a ^-observer design for tie-delay systes has been developed. The design ethod not only stabilizes the observer but also guarantees an Hoo nor bound between the disturbance and the estiated error. The designed observer uses the past values of the estiated state but the observer gain atrix is designed independently of the delay (i.e. the presented ethod is a delayindependent one but the delay ust be known for the construction of the observer). This observer gain is obtained by solving a odified Riccati equation. Sufficient conditions for solving this odified Riccati equation have also been provided.

9 Robust Observer Design for Tie-Delay Systes: A Riccati Equation Approach 761 APPENDIX A: PROOF OF PROPOSITION 3 The proof is divided into two parts: in the first part, the asyptotic stability of the syste (4) is deonstrated, and, in the second part, the inequality (6) is proved. Consider that for a given 7 > 0 there exists Q = Q T > 0 and P = P T > 0 such that Riccati equation (5) holds. First part. Define the Lyapunov functional V as follows: V(t) := x T (t) Px(t) + J2 ~ T W QxW) <Wi=i - t J<- ih It is easy to verify that there exist two positive scalars /?i and /3 2 such that Pi\\x(t)\\ 2 <V(t)<0 2 \\x(t + 6)\\\ (15) The corresponding Lyapunov derivative along the trajectory of x(t) is given by where dv(t) dt = x*(t)ax(t) (16) x(ť) := x(t) x(t - h) x(t - 2Һ) x(t h) A := A T P + PA + Q PAx PA 2 A T P -Q 0 A\P 0 -Q PA 0 0 AІP 0 0 Q Since \l n + PDD T P > 0, inequality (5) iplies that A T P+PA+P[A 1} 4... A ]di*g{q- l } [A u A 2,..., A ] T P+Q<0 (17) where diag{q" 1 } denotes a x block-diagonal atrix with eleents Q~ l. Using the Schur copleent [1] and (17) it follows that A is negative definite. Hence àv(t) <-A dt i (.4) x(ť) 2 (18) Cobining (15) and (18), one can prove the asyptotic stability of syste (4) using the Krasovskii theore (see Appendix C).

10 762 A. FATTOUH, O. SENAME AND J.-M. DION Second part. Let us define the positive definite atrix S as follows: S := -(A T P + PA + P[A 1,A 2,...,A ]dizg{q- l } [A 1,A 2,...,A ] T P then + Q + -^I n + PDD T P) V A T P + PA + P[A 1, A 2,..., A^di^Q-'U^, A 2,...,A ] T P + Q+\l n + PDD T P + S = 0. (19) Define the following atrices W(ju) := P[A U A 2,..., A ] diagfq- 1 }.,. [A u A 2,..., A ] T P + Q-^2 AjPe ju ' ih - J2 PA^^ «=i X(ju) := [juin-a-ytai*-* 1 = 1»=i ч- 1 jivih Note that W(ju) is non-negative definite for all u G M. Using the above definition, equation (19) can be rewritten as (X*(ju,))~ 1 P + PX-\JLO) - W(jio) - \l n - PDD T P -5 = 0. (20) V Pre and post ultiplying (20) by D T X*(ju>) and X(ju>)D respectively iplies that D T X*(ju>) PD + D T PX(ju)D - D T X*(ju>)PDD T PX(ju) D - I n = -/ + D T X*(ju>) (WO) + \l n + s\ X(ju) D. It follows that -(/ - D т PX(-jш) D) т (I n - D т PX(jш) D) = -/ + D T X*(jш) {W(jш) + S} X(jш) D 1 + ^D т X*(jш)X(jш)D. (21) 7 The left hand side of the above equation is non-positive definite for all u M. Define T(ju) := X(ju) D, then equation (21) iplies and -7 n + D T X*(ju) {W(ju) + 5} X(ju) D + \T*(JLJ)T(JU) < 0 7 T*(ju)T(ju) < y 2 In-7 2 D T X*(ju>){W(juj) + S}X(ju>)D

11 Robust Observer Design for Tie-Delay Systes: A Riccati Equation Approach 763 for all u e JR. Since W(ju>) and S are non-negative and positive atrices for all u E M respectively then T*(ju)T(ju)< 7 2 I n for all w G -K, i.e. \\( S I n -A-Y /^e-' ih r 1 D\\oo<J. 1 = 1 This ends the proof. APPENDIX B: THE BOUNDED REAL LEMMA [13] The following stateents are equivalent: (i) There exists a syetric positive definite atrix P such that A T P + PA + PBB T P + C T C < 0. (ii) The Riccati equation A T P + PA + PBB T P + C T C = 0 has a solution P > 0. Furtherore, if these stateents hold then P < P. APPENDIX C: THE KRASOVSKII THEOREM [8] Consider the retarded functional differential equation x(t) = f(t,x t ), t>t 0, x t (0) = x(t + 0) (22) x t0 (e) = <f>(e); ee[-h,o]. Assue that <j>(6) C[ h,0], the operator f(t,<f>) is continuous and Lipschitzian in 0and/(*,O) = 0. If there exist a continuous functional V(t,<f>) such that "1(110(0)11) < V(t, <(>) < u;2( W) ), V < -w 3 (\\x(t)\\) where uv,(r), r > 0, are soe scalar, continuous, nondecreasing functions such that Wi(0) = 0 and Wi(r) > 0 for r > 0. Then the trivial solution of (22) is uniforly asyptotically stable (see [8] Theore 5.3, p. 73). (Received Deceber 11, 1998.)

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