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On the stabilization and stability robustness against small delays of some damped wave equations

O. Morgul

1995
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IEEE Transactions on Automatic Control
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and the functions are solutions of (IO) for s E [a(t-),a(t+)]. Let us note that jumps m" ( a ( t , +)) -m" ( a ( t , -)), P" (a( tl+)) -P " ( a ( t p -) ) of the variables n~" ( a ( t ) ) , F ( a ( t ) ) coincide with jumps m(t,+) -m ( t , -) , P(t,+) -P ( t t -) of the solutions m ( t ) , P ( t ) of the equations with a measure (5). Thus, the optimal parameters m. '(a(t)), P " ( a ( t ) j are solutions of (5) everywhere in the considered time interval. This proves the theorem. V. CONCLUSIONS
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... m. V. CONCLUSIONS The minmax filtering equations over discrete-continuous observations follow from the minmax filtering equations over continuous observations given in [l] by virtue of replacing an absolutely continuous function U( t) by a bounded variation one in accordance with an observation equation. No additional computation is needed. The minmax filtering equations over discrete observations follow from the minmax filtering equations over continuous ones by virtue of transferring to discrete-continuous observations and assuming a bounded variation function ' U ( t) to be piecewise constant. The definition of a vibrosolution ensures the stability of the optimal estimate with respect to small variations of a bounded variation function ~( t ) and therefore observations. REFERENCES D. Bertsekas and I. B. Rhodes, "Recursive state estimation for a set membershp description of uncertainty," Theory of Optimal Systems with Generalized Controls (in Russian). Moscow: Nauka, 1988. __, "Filtering of finite-dimensional and infinite-dimensional diffusion processes on the basis of discrete-continuous observations," Soviet Phys. Dokl., vol. 34, no. 3, pp. 194-196, 1989. Yu. V. Orlov and M. V. Basin, "Some estimation problems over discretecontinuous observations," in Proc. 30th IEEE Con$ Decis. Contr., 1991, -, "On filtering of the Hilbert space-valued stochastic process over discrete-continuous observations," Abstract-In this note we consider a system which can be modeled by two different one-dimensional damped wave equations in a bounded domain, both parameterized by a nonnegative damping constant. We assume that the system is fixed at one end and is controlled by a boundary controller at the other end. We consider two problems, namely the stabilization and the stability robustness of the closed-loop system against arbitrary small time delays in the feedback loop. We propose a class of dynamic boundary controllers and show that these controllem solve the stabilization problem when the damping cuefMent is nonnegative and stability robustness problem when the damping coefficient is strictly positive. In recent years boundary control of flexible systems has become an active area of research. Most of the research in this area is concentrated on the problem of control and stabilization of conservative linear flexible systems (e.g., strings or beams without damping). Such systems have infinitely many eigenvalues on the imaginary axis and can be uniformly stabilized by using simple velocity feedback laws at their boundaries; see, e.g., [2] and [3]. It was shown, however, that these systems become unstable when arbitrary small time delays were introduced into the feedback laws; see, e.g., [5] and [6]. This lack of robustness and some other related results indicate that most of the conservative models in flexible structures are not well posed from the control theory point of view and possess potential limitations for the feedback design; see [8]. Recently in [7] it was argued that mathematical conservative models are never meant to represent physical systems for infinite time interval; hence any control theory based on these models should attempt to justify its conclusions by using an appropriately damped version of the corresponding conservative model. In this paper we consider two different damped wave equations both parameterized by a damping coefficient a 2 0. When a = 0, these models reduce to the standard conservative wave equation. To stabilize these systems, we propose a dynamic boundary control law. Following [7], we try to answer the following questions: i) Does the proposed control law stabilize the conservative model and improve the stability of the damped models? ii) Does the proposed control law robustly stabilize the damped models against small time delays in the feedback loop? In the following section we propose a class of dynamic boundary controllers to solve these problems. It should be emphasized that these controllers do not robustly stabilize the conservative wave equation against small time delays in the feedback loop. This note is organized as follows. In Section 11, we give two examples of the damped wave equation used in this note and propose a class of dynamic controllers to solve the problems stated above. In Section I11 we give stability results [i.e., answer to problem i)], and in Section IV we give robustness results [i.e., answer to problem ii)]. Finally we give some concluding remarks. _____ 1627 DAMPED MODELS We first consider the following damped wave equation (2) where (I 2 0 is a damping constant, v ( t ) is the boundary control input, and y ( t ) is the measured output. For simplicity, some coefficients are chosen to be unity. The system given by ( 1)-(2) is first introduced in [5] and later investigated in [l], [lo], and [7] . For a = 0, the system given by (1) and (2) reduces to the standard conservative wave equation with a boundary controller. To stabilize this system, the following simple controller and feedback law can be used where 1. > 0. It is known that the closed-loop system given by (1)-(4) is exponentially stable; see [5]. When the feedback law in (4) is replaced by ~( t ) = -y ( f -h ) , where the constant h > 0 represents a small time delay, the stability of the closed-loop system depends on k and rr. It is known that if 1p -L o k > p ( 5 ) then the closed-loop system is unstable for arbitrary small time delays h > 0. On the other hand, if the inequality in (5) is reversed, then there exists an h o > 0 such that for any h, 0 5 h 5 h o , the closed-loop system is L2-stable. This result is obtained in [5] by calculating the eigenvalues directly and in [ l ] and [lo] by using frequency domain techniques. The second example of the damped wave equation that we consider is the following (6) rc~(0.t) = o . l l~, ( 1 . t , +~, c l u~, , ( l where o 2 0 is a damping constant, c1 is either zero or one. This type of damping is not unnatural and is similar to Kelvin-Voight damping for the Euler-Bemoulli beam. The system (6) and (7), with n = 0, is first introduced and investigated in [7]. It can be shown that the closed-loop system given by (3), (4), (6), and (7) is exponentially stable (see Theorem 2 in Section 111). It was shown in [7], however, by direct eigenvalue calculations that the closed-loop system becomes unstable when the feedback law in (4) is replaced by ~( t ) = -y ( tlr), where the constant h > 0 represents a small time delay. In Section IV we will show that this instability could be predicted by considering the open-loop transfer function and could be eliminated by choosing c1 = 1 (see Corollary 2 in Section IV). We note that the case = 1 gives the natural boundary condition for (6), and this can be justified by considering the rate of change of the energy of the system. In Section IV we will show that even in the case Q = 0, by choosing appropriate dynamic boundary controllers, the instability with respect to time delays can be eliminated (see We propose the following dynamic boundary controllers to solve the stability problems stated above (8) where 2 E R", for some natural number TL, is the controller state, U is the controller input, A E R" is a constant matrix, b, c E R" are constant column vectors, d is a constant real number, and the superscript T denotes transpose. We first make the following assumptions concerning the controller Assumption 1: AU eigenvalues of A E Rnxn have negative real Assumption 2: (A, b ) is controllable, and (c, A) is observable. Assumption 3: d 2 0, and there exists a constant y, d 2 y 2 0, given by (8) and (9) throughout this work. Parts. such that the following holds Moreover, in case d > 0, we require > 0 as well. 0 We note that this type of controllers has been proposed for the stabilization of flexible structures. For the application to wave equation, see [ll] and [13], and to the Euler-Bernoulli beam, see u21. STABILITY RESULTS Let Assumptions 1)-3), stated above, hold. Then, it follows from the Meyer-Kalman-Yakubovich Lemma that given any symmetric positive definite matrix Q E R" " , there exists a symmetric positive definite matrix P E R" " , a vector q E R", and a constant E > 0 satisfying (12) moreover, in case d = 0 in (9) , we can take q = 0 and E = 1; see To analyze the systems considered in this paper, we first define the function space ' H as follows where the spaces L2 and H,k are defined as follows (16) Equations (l), (2), (8), and (9) together with feedback control law (17) where m = ( w U J~Z ) " E ' H, the operator A1 : ' H + 31 is a linear unbounded operator defined as H, k = {f E L21f,f',frr....,f(k' EL*,f(O) =O}. (4) can be written in the following abstract form Let Assumptions 1)-3) hold, let Q E R n x n be an arbitrary symmetric positive definite matrix, and let P E R"'", q E R" be the solutions of (12) and (13) where P is also a symmetric and positive definite matrix. We define the following "energy" norm in 7-1 We note that one can define an inner-product which induces the norm given above; hence without loss of generality we may assume that ' H is a Hilbert space. -1628 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40. NO. 9, SEPTEMBER 1995 Theorem I : Consider the system given by (17) , where the operator i) The operator A1 generates a CO-semigroup T(t) of contractions in 3-1, (for the terminology of the semigroup theory, the reader is referred to, e.g., [14]). ii) For a = 0, d = 0, the semigroup T ( t ) is asymptotically stable, i.e., the solutions of (17) asymptotically converge to zero. iii) For a + d > 0, the semigroup T ( t ) is exponentially stable. i) We first define the following new "energy" of the system AI is given by (18) . Assume that Assumptions 1)-3) are satisfied. Proof: where E(t) is given by (20) . Note that due to boundary condition (2) at the fixed end, the integral term in (21) can be embedded in E ( t ) . By differentiating (21) with respect to time, we obtain where in deriving the first equation we used integration by parts, (l), (2), (8), and (9), to obtain the second equation we used (4), (12), and (13). It follows from (22) that the operator A1 is dissipative. It can be shown that the operator X I -A1 : 31 + 'H is onto for X > 0 (see [ I l l and [13] for similar calculations). Hence from Lumer-Phillips theorem we conclude that A1 generates a CO-semigroup of contractions on 7-L; see [14]. ii) See [ l l ] and [13]. iii) For a = 0 and d > 0, see [ll]. Hence we consider the case n > 0, rl 2 0. It is known that the operator A1 has compact resolvent when a = 0; see 1111. Since the terms containing a can be considered as a bounded perturbation to this operator, it can easily be shown that the operator A1 has compact resolvent for a > 0 as well. This implies that the operator A1 has point spectrum. By using (22) it can be shown that A1 cannot have an eigenvalue on the imaginary axis. Since A1 has point spectrum, it follows that the imaginary axis belongs to the resolvent of Al. To obtain an estimate of the resolvent on the imaginary axis, let y = ( f h r ) ' E ' H be given. We have to find 711 = (p T z ) T E D ( A , ) such that By using (18) in (23) , after some straightforward calculations we conclude that II(jw1 -A1)-'II < ,m for w sufficiently large (see [11] for similar estimates). Since the imaginary axis belongs to the resolvent set p ( A1 ) of the operator A I , and since for each X E p ( AI ), the operator ( X I -A1 is compact, it follows that for any R < x, the following estimate holds sup II(juJ1-AI)-lI/ < m. (24) ,ja By combining these results we conclude that estimate (24) holds for all w'. Hence, it follows from a result of [9] that the Go-semigroup 0 Now we consider the system given by (6) and (7) , with the controller (8) and (9), and the feedback law (4). This system can T ( t ) generated by the operator AI is exponentially stable. be written in the following abstract form m = d;?m,m(O) E 3-1 (25) where m = ( w w~z ) ? ' E 'H , the operator A2 : 'H + 'H is a linear unbounded operator defined as where ( p~z )~ E 31. The domain D(&) of the operator d 2 is defined as D(A2) := {(p r z )T E 'HIP E H:. r E Hi, z E R"; p,(l) + a a r , ( l )c T z + dr(1) = O}. (27) Theorem2: Consider the system given by (25). Assume that i) The operator A2 generates a CO -semigroup T ( t ) of contractions Assumptions l)-3) are satisfied. Then we have the following: for each one of the following cases: i.1) for a = 1, i.2) for (Y = 0 and d = 0, i.3) for (Y = 0 and d > 0, provided that cTb is sufficiently large or a is sufficiently small. ii) For a = 0, d = 0, the semigroup T ( t ) is asymptotically stable, i.e., the solutions of (25) asymptotically converge to zero. iii) For a + d > 0, the semigroup T ( t ) is exponentially stable. Proof: ii) iii) For case i.l), consider the "energy" E ( t ) given by (20) . By differentiating (20) with respect to time and by using (4), (6)-(9), it can be shown that E 5 0, hence the operator A 2 is dissipative for the case i.1). For case i.2), we again consider the "energy" E ( t ) given by (20). Note that in this case since d = 0, without loss of generality we can take p = 0 and E = 1 in (12) and (13) ; see [15, p. 1321. By differentiating (20) with respect to time and by using (4), (6)-(9) and some straightforward inequalities, it can be shown that E 5 0, hence the operator A2 is dissipative for case i.2). Finally, for case i.3), we choose the following "energy" function where E ( t ) is given by (20) . By differentiating (28) with respect to time, by using (4), (6)-(9), and following the analysis for the case i.2), it can be shown that E2 can be made negative if a is sufficiently small or cTb is sufficiently large. Provided that, we conclude that operator A2 is dissipative for the case i.3). It can easily be shown that in all cases the operator X I -A;? : see [l I] for similar results). It then follows from the Lumer-Phillips Theorem that the operator A2 generates a CO semigroup of contractions in 'H. See [ll]. The case a = 0, d > 0 was proved in [ll]. Hence, we consider the case (1 > 0 and d 2 0. It is known that for the uncontrolled case (i.e., (6) and (7) with 11 O), the resulting system generates an exponentially decaying analytic semigroup. Since the controller given by (8) and (9) is essentially finite dimensional, it can be shown that the operator A2 generates an analytic semigroup when b = 0. The term multiplying b can be considered as a perturbation, and it can easily be shown that A2 generates an analytic semigroup when 1 2 E2(t) = E ( t ) + -adW:(l.t) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 9, SEPTEMBER 1995 1629 IV. llbll is sufficiently small (see [14, p. 80-811). Note that for any k > 0, we can rescale b and c as b = kb, F = $c without changing the transfer function g ( s ) given by (11). Hence, without loss of generality we can assume that llbll could be selected as small as desired; hence the operator $12 generates an analytic semigroup. Since the semigroup T ( t ) generated by A2 is a contraction semigroup, it follows that the imaginary axis belongs to its resolvent set and that estimate (24) holds for the operator Az, for all U ; see [14, pp. 61-62]. Therefore, from [9] it follows that the semigroup T ( t ) generated by Az is exponentially decaying. ROBUST STABILITY WITH RESPECT TO SMALL TIME DELAYS In this section we analyze the stability of the systems (l), (2) and (6), (7) together with the controller (8), (9) and the delayed feedback law u ( t ) = -y(th ) . To analyze input-output stability of this system, we use the frequency domain approach. The terminology used here is borrowed from [lo]. Let H ( s ) denote the transfer function of a single-input/single-output plant between its input U and its output y. H ( 5 ) is said to be well posed if it is bounded on some right-half plane and is said to be regular if it has a limit at +m along the real axis. If we apply the unity feedback and set U = Ty, where T is the new input, then the closed-loop transfer function between T and y becomes Go(s) = H ( s ) ( l + H ( s ) ) -' . When there is a small time delay by F in the feedback loop, the new transfer function G'( s) from r to y becomes G'(s) = H ( s ) ( 1 + a-'"H(s))-'. We say that Go is robustly stable with respect to delays if there is an EO > 0 such that for any E E [0, FO], G' is &-stable. If this property does not hold, then arbitrary small time delays destabilize Go. Let the transfer function H ( s) be meromorphic (i.e., analytic except at its poles) on the half plane CO = {s E C I R e { s } > 0). Let B denote the (discrete) set of poles of H in CO, and let y* be defined as Theorem 4: Let H ( s ) be a regular transfer function and assume that Go = H ( l + H)-' is Lz-stable. Let 7 * be defined as in (29). i) If y' < 1, then Go is robustly stable with respect to delays. ii) If y* > 1, then Go is not robustly stable with respect to delays. U Now consider the system given by (l), (2), (8), and (9). An easy calculation shows that the (open loop) transfer function H ( s) from U to y is Proofi See [lo]. For a different version of this result, see [4]. where g ( s ) is given by (11) (see also [ l ] and [lo] for the case g ( s ) = k, where k > 0 is a constant). Since the system is exponentially stable for the case n + d > 0 (see Theorem l), it follows that Go is Lzstable; hence Theorem 4 is applicable. Note that when d = 0, both g ( s ) and H (s) are strictly proper. As is shown below, this is important for the stability robustness with respect to small delays. Corollary I : Consider the system given by (l), (2), (8), and (9). Let the assumptions of Theorem 1 be satisfied. Assume that a > 0. i) If d < a, then Go is robustly stable with respect to time ii) If d > $$, then Go is not robustly stable with respect to delays, time delays. Proufi From the formulation it is obvious that Theorem 4 is applicable; hence we need to compute y* given by (29). Note that I g(s) I i s bounded on CO and g(s) = d + o ( l / s ) for large s. By using this and the results of [lo], it can be shown that To see this, following [lo], first note that for s E CO, we have and I s/(s+ a) 15 1. This shows that y* 5 d s . To prove the reverse inequality, we choose sn = l / n + j ( 2 n + l)?r/2 for R E N. It can easily be shown that lim,L-m H(s,) = d& which proves that (31) is satisfied. The claims of Corollary 1 now follows from Theorem 4. 0 Remark I : This result has been known for the nondynamic controller case (i.e., when g ( s ) = k, where k > 0 is a constant); see [5], [l], [lo], and [7]. Hence Corollary 1 can be considered as a generalization of the similar results presented in the references mentioned above. Note, however, that Corollary 1 is still valid when d = 0, in which then case i) is trivially satisfied, hence the corresponding Go is always robustly stable with respect to small time delays for all a > 0. Moreover by Theorem 1, for the case d = 0, the closed-loop system is exponentially stable for a > 0 and is asymptotically stable for n = 0. Hence, the controller given by (8) and (9) solves the problems stated in the introduction. Moreover, for the case d = 0, both the corresponding controller transfer function g ( s ) and the open-loop transfer function H i s ) are strictly proper; see (11) and (30). These points are important for actual implementation U Next, we consider the system given by (6)-(9). An easy calculation I 1e--2(s+a) 15 1 + e -z a , and I 1 + e-'('+') I> 1e -z a , of g ( s ) and for the well posedness of the model; see [8]. shows that the open-loop transfer function from U to y is s g ( s ) 1e-28 where ;3 is given by and g( s) is given by (1 1). Since the system is exponentially stable for the case a + d > 0 (see Theorem 2), it follows that Go = H ( 1 + is L zstable; hence Theorem 4 is applicable. We have the following corollary. Corollary 2: Consider the system given by (6)-(9). Let the conditions in Theorem 3 are satisfied. Assume that a > 0. i) If o = 1, then -* = 0, hence Go is robustly stable with respect ii) If Q = 0 and d = 0, then y* = 0, hence Go is robustly stable iii) If cv = 0 and d > 0, then y* = +cc, hence Go is not robustly Proof From the formulation it is obvious that Theorem 4 is applicable, hence we need to compute y* given by ( 29). For s E CO, it follows from (33) that B E CO as well; hence we have I 1-a-" 15 2 for 5 E CO. Next we show that infseco I l+eCzP I> 0. To show this, first we define the set CM = {s E CO\ 1 s )> M } for ,If > 0. From (33) it follows that for I s I sufficiently large we have dfl, hence one can easily show that I 1 + e P z M I> 1 -e--for s E Cbf, provided that M is sufficiently large. An easy calculation also shows that all zeroes of 1 + e-z8 = 0 and (33) are in the left half of the complex plane and are all bounded away from the imaginary axis. Hence it follows easily that inf,~c,-c-, I 1 + e-" I> 0, for otherwise there must be a zero in CO -Cllr which is a contradiction. From these arguments it follows that is bounded on CO. Since to small time delays. with respect to small time delays. stable with respect to small time delays. 1630 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40. NO. 9, SEPTEMBER 1995 g(s) is bounded on CO and g(s)d + o(l/s) for 1 s I sufficiently large, it follows from (29) and (32) that for cy = 1, we have y* = 0. For the case cy = 0 and d = 0, note that g ( s )o ( l / s ) for large s, hence we have y* = 0. For the case cy = 0 and d > 0, it follows that > * -6 for large s, hence we have 7* = +os. Now, Corollary 2 follows from Theorem 4. 0 Remark 2: Example 2 was first introduced in [7]. The controller proposed in [7] was nondynamic, i.e., (8)-(9) are not present and the controller was given by (3). It can be shown that the conclusions of Theorem 2 and Corollary 2 are valid in this case as well; hence, Theorem 2 and Corollary 2 can be considered as a generalization of similar results presented in [7]. Moreover, as stated in case ii) of Corollary 2, the use of strictly proper controllers (i.e., d = 0) 0 eliminates the instability due to small time delays. V. CONCLUSION In this paper we considered two different damped wave equations, both parameterized by a damping constant a 2 0. When a = 0, these equations reduce to the standard conservative wave equation. We assumed that the system is fixed at one end and is controlled at the other end. We studied two problems: stabilization of these models for a 2 0 and robust stabilization against small time delays in the feedback law for a > 0. To solve these problems we posed a class of dynamic boundary controllers. Under some assumptions, one of which is the strict positive realness of the controller transfer functions, we obtained various stability results. In particular we showed that the proposed controllers stabilize the models considered for a 2 0 and that robustly stabilizes the same models against small time delays in the feedback loop for a > 0. The examples presented here clearly indicates that while strict positive realness of the controller transfer functions is important for stability, the strict properness is important for robustness against small time delays (for the case a > 0). Finally, the ideas presented here can be extended to other flexible structures, such as flexible beams under various modeling assumptions. This will be the subject of a forthcoming paper. REFERENCES J. Bontsema and S. A. deVries, "Robusmess of flexible structures against small time delays." in Proc. example on the effect of time delays in boundary feedback stabilization of wave equations," SIAM J. Contr. Optim." vol. 24, pp. 152-156, 1986. R. Datko, "Two examples of ill-posedness with respect to small time delays in stabilized elastic systems," robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop," Institut fiir Dynamische Systeme, Universitat Bremen, Tech. Abstract-A queueing system consisting of two parallel heterogeneous servers is considered. Customers can arrive at discrete-valued instants and, upon their arrivals, they are immediately routed to one of the server buffers. The interarrival times are assumed to be integer, independent, identically distributed random variables, whereas the service times of the servers am assumed to be integer and deterministic. The optimization problem considered is the minimization of the customer mean flow time over an idnite horizon. The existence of a stationary optimal policy with a switchover structure is established.

doi:10.1109/9.412634
fatcat:gks3c5x7dfhdpj6pjdkfqwtwyi