Stability of Linear Distributed Parameter Systems with Time-Delays

Transcription

1 Stability of Linear Distributed Parameter Systems with Time-Delays Emilia FRIDMAN* *Electrical Engineering, Tel Aviv University, Israel joint with Yury Orlov (CICESE Research Center, Ensenada, Mexico)

2 Outline Introduction Delay-Dependent Stability in a Hilbert Space: Linear Operator Inequalities Stability of the Heat Delay Equation: LMI conditions Stability of the Wave Delay Equation: LMI conditions Conclusions 1

3 Introduction Time-delay often appears to be a source of instability (Hale & Lunel, 1993). In the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g. Datko (SIAM 1988), Logemann et al. (SIAM 1996), Nicaise & Pignotti (SIAM 2006)). The stability analysis of Partial Differential Equations (PDEs) with delay is essentially more complicated than of ODEs. There are only a few works on Lyapunov-based technique for PDEs with delay. 2

4 The 2nd Lyapunov method was extended to abstract nonlinear time-delay systems in the Banach spaces in Wang (1994a, JMAA) and applied to scalar heat and scalar wave equations with constant delays and with the Dirichlet boundary conditions in Wang (1994b, JMAA), Wang(2006, JMAA). Stability and instability conditions for wave delay equations were found in (Nicaise & Pignotti, SIAM 2006). As. stability of PARABOLIC distributed parameter systems has been studied in (Fridman & Orlov, CDC 07). Generalization of these conditions to HYPERBOLIC systems is complicated. The objective of the present paper is to derive the exp. stability of general distributed parameter systems. A class of linear systems is considered, where a bounded operator acts on the delayed state. 3

5 The system delay is admitted to be unknown and time-varying. Sufficient exp. stability conditions are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. General methods for solving LOI have not been developed yet. Some finite dimensional approximations were considered in Ikeda, Azuma & Uchida (2001). Being applied to a heat/wave equation these conditions are represented in terms of standard finite-dimensional LMIs that appear to guarantee the stability of the 1-st/2-nd order delaydifferential equations. This reduction of infinite-dimensional LOIs to finite-dimensional LMIs is tight in the sense that the stability of the latter delay-differential equation is necessary for the stability of the heat/wave equation. 4

6 Problem Statement ẋ(t) = Ax(t) + A 1 x(t τ(t)), t t 0 (1) where x(t) H, H is a Hilbert space, delay τ(t) is a piecewise continuous function of class C 1 on each continuity subinterval and inf t τ(t) > 0, sup t τ(t) h (2) for some constant h > 0, A 1 is linear bounded operator, A is an infinitesimal operator, generating a strongly continuous semigroup T (t), and the domain D(A) is dense in H. Throughout, solutions of such a system are defined in the Caratheodory sense, i.e., Eq. (1) is required to hold almost everywhere only. Let the initial conditions x t 0 = ϕ(θ), θ [ h, 0], φ W (3) be given in the space W = C([ h, 0], D(A)) C 1 ([ h, 0], H). (4) 5

7 The initial-value problem (1), (3) is well-posed on [t 0, ) and its solutions can be found as mild solutions, i.e., as those of the integral equation x(t) = T (t t 0 )x(t 0 ) + t t0 T (t s)a 1 x(s τ(s))ds, t t 0. (5) In this talk we will consider 2 main examples: 1) heat ( parabolic eq); 2) wave (hyperbolic eq). Example 1: heat eq. z t (ξ, t) = az ξξ (ξ, t) a 1 z(ξ, t τ(t)), t t 0, 0 ξ π (6) with the constant parameters a > 0 and a 1, with the time-varying delay τ(t), and with the Dirichlet boundary condition z(0, t) = z(π, t) = 0, t t 0. (7) 6

8 The above boundary-value problem describes the propagation of heat in a homogeneous onedimensional rod with a fixed temperature at the ends in the case of the delayed (possibly, due to actuation) heat exchange with the surroundings. Here a and a 1 stand for the heat conduction coefficient and for the coefficient of the heat exchange with the surroundings, respectively, z(ξ, t) is the value of the temperature field of the plant at time moment t and location ξ along the rod. Heat eq. can be rewritten as ẋ(t) = Ax(t) + A 1 x(t τ(t)), t t 0 H = L 2 (0, π), A = a 2 ξ2 with the dense domain D( 2 ξ 2) = {z W 2,2 ([0, π], R) : z(0) = z(π) = 0}, and with the bounded operator A 1 = a 1. A generates an exponentially stable semigroup (see, e.g., Curtain & Zwart (1995) for details). 7

9 Example 2: Wave equation z tt (ξ, t) = az ξξ µ 0 z t (ξ, t) a 0 z(ξ, t) a 1 z(ξ, t τ(t)), t 0, 0 ξ π, z(0, t) = z(π, t) = 0, t t 0. (8) Eqs (8) describe the oscillations of a homogeneous string with fixed ends in the case of the delayed (possibly, due to actuation) stiffness restoration. Here a - elasticity coefficient, µ 0, µ 1 - dissipation coefficients, a 0, a 1 - the restoring stiffness coefficients, z(ξ, t) is the deflection of the string and z t (ξ, t) its velocity at time moment t and location ξ along the string. 8

10 Introduce the operators A = 0 1 a 2 ξ 2 a 0 µ 0, A 1 = [ 0 0 a 1 0 ] D( 2 ξ 2) = {z W 2,2 ([0, π], R) : z(0) = z(π) = 0}, Then (8) can be represented as ẋ(t) = Ax(t) + A 1 x(t τ(t)), t t 0 in the Hilbert space H = L 2 (0, π) L 2 (0, π) with the infinitesimal operator A, possessing the domain D(A) = D( 2 ξ 2) L 2(0, π) and generating an exp. stable semigroup (see, e.g., Curtain & Zwart (1995)). 9

11 Our aim is to derive exp. stability criteria for (1), (2). The stability concept under study is based on the initial data norm in the space W = C([ h, 0], D(A)) C 1 ([ h, 0], H) defined as φ W = Aφ(0) + φ C 1 ([ h,0],h) (9) Suppose x(t, t 0, φ) denotes a solution of (1), (3) at a time instant t t 0. System (1) is said to be exponentially stable with a decay rate δ > 0 if K 1: x(t, t 0, φ) 2 Ke 2δ(t t 0) φ 2 W t t 0. (10) 10

12 Exponential Stability in a Hilbert Space Given a continuous functional V : R W C([ h, 0], H) R, (11) its V along (1) is defined as follows: V (t,φ, φ) = lim sup s s [V (t+s, xt+s (t,φ), ẋ t+s (t, φ)) V (t, φ, φ)]. Notation: x t = x(t + θ), θ [ h, 0]. Lemma Let δ, β, γ and a continuous functional V : R W C([ h, 0], H) R such that the function V (t) = V (t, x t, ẋ t ) is absolutely continuous for x t, satisfying (1), and β φ(0) 2 V (t, φ, φ) γ φ 2 W, V (t, φ, φ) + 2δV (t, φ, φ) 0. (12) Then (1) is exponentially stable with the decay rate δ and (10) holds with K = γ β. 11

13 In a Hilbert space D(A), consider V (t, x t,ẋ t ) = x(t), P x(t) + t t h e 2δ(s t) x(s), Sx(s) ds +h 0 t h t+θ e 2δ(s t) ẋ(s), Rẋ(s) dsdθ + t t τ(t) e 2δ(s t) x(s), Qx(s) ds, τ d < 1 (13) P : D(A) H is a linear operator, P > 0, R, Q, S L(H), R, Q, S 0 and x, P x γ P [ x, x + Ax, Ax ], x, Qx γ Q x, x, x, Rx γ R x, x, x, Sx γ S x, x x D(A) and some positive γ P, γ Q, γ S, γ R. (14) We use Cauchy-Schwartz (Jensen s) inequality: R L, R > 0, scalar l > 0 and x L 2 ([a, b], H) the following holds: l l 0 x(s), Rx(s) ds l 0 x(s)ds, R l 0 x(s)ds. 12

14 We obtain conditions in 2 forms: 1) by substituting the right side of (1) for ẋ(t); 2) by using descriptor approach (Fridman SCL 2001). Theorem 1 (1) is exp. stable with the decay rate δ if LOI 1 is feasible in D(A) D(A) D(A): Φ h = e 2δh + h 2 Φ 11 0 P A A 1 P 0 0 R 0 R 0 (S + R) R R R 2R+(1 d)q holds provided that A RA 0 A RA A 1 RA 0 A 1 RA 1 0, Φ 11 = A P + P A + 2δP + Q + S. (15) Theorem 1 gives delay-dependent conditions (h-dependent) even for δ 0. For S = R = 0 we obtain the following quasi delayindependent conditions, which coincide for ODE with (Mondie & Kharitonov, TAC 2005): 13

15 Corollary Given δ > 0, (1) is exp. stable with the decay rate δ for all delays with τ(t) d < 1 if P > 0 and Q 0 subject to (14) such that the LOI [ (A+ δ) ] P + P(A + δ)+q P A 1 A 1 P 0 (1 d)qe 2δh (16) holds in D(A) D(A). The inequality (10) is satisfied with K = max{γ P, hγ Q }/β. Differently from the finite dimensional case, the feasibility of the strict LOIs for h = 0 (or δ = 0) does not necessarily imply the feasibility of these LOIs for small enough h (δ) because h 2 (δ) is multiplied by the operator, which may be unbounded. 14

16 It may be difficult to verify the feasibility of LOI 1, if the operator that multiplies h 2 (and depends on A) in Φ h is unbounded. To avoid this, we will derive the 2-nd form of LOI by the descriptor method (Fridman, SCL 2001), where the right-hand sides of the expressions 0 = 2 x(t), P 2 [Ax(t) + A 1x(t τ(t)) ẋ(t)], 0 = 2 ẋ(t), P 3 [Ax(t) + A 1x(t τ(t)) ẋ(t)] (17) with some P 2, P 3 L(H) are added into the right-hand side of V. 15

17 LOI 2 via descriptor method 0 Φ d11 Φ d12 0 P 2 A 1 + Re 2δh Φ d22 0 P 3 A 1 (S + R)e 2δh Re 2δh [2R + (1 d)q]e 2δh holds, where (18) Φ d11 = A P 2 + P 2 A + 2δP + Q + S Re 2δh, Φ d12 = P P 2 + A P 3, Φ d22 = P 3 P 3 + h2 R. (19) 16

18 Stability of a Scalar Delay Heat Equation z t (ξ, t) = az ξξ (ξ, t) a 0 z(ξ, t) a 1 z(ξ, t τ(t)), t t 0, 0 ξ π (20) with the constant parameters a > 0 and a 0, a 1, with the time-varying delay τ(t), and with the Dirichlet boundary condition z(0, t) = z(π, t) = 0, t t 0. (21) Here we apply the descriptor method LOIs. The boundary-value problem (20), (21) can be rewritten as (1) in the Hilbert space H = L 2 (0, π) with A = a 2 ξ 2 a 0 with the dense domain D( 2 ξ 2) = {z W 2,2 ([0, π], R) : z(0) = z(π) = 0}, (22) and with the bounded operator A 1 = a 1. A generates an exponentially stable semigroup (see, e.g., Curtain & Zwart (1995) for details). 17

19 Delay-independent conditions are derived by using Lyapunov-Krasovskii functional V = p π 0 z 2 (ξ, t)dξ +q t t τ(t) π0 e 2δ(s t) z 2 (ξ, s)dξds (23) with some constants p > 0 and q > 0. Then P = p, Q = q Integrating by parts and taking into account boundary conditions, we find that for x D(A) x, (A P + P A)x = 2a π 0 pzz ξξ dξ = 2a π 0 pz 2 ξ dξ 2a π 0 pz 2 dξ where the last inequality follows from the Wirtinger s Inequality (Wang, 1994 JMAA) Let z W 1,2 ([a, b], R) be a scalar function with z(a) = z(b) = 0. Then b a z2 (ξ)dξ (b a)2 π 2 b a (z (ξ)) 2 dξ. (24) 18

20 We thus obtain that the LOI is satisfied provided that the following LMI [ ] q 2(a + a0 )p a 1 p a 1 p (1 d)qe 2δ(s t) < 0 (25) is feasible. LMI (25) with δ = 0 is feasible iff a + a 0 > 0, a 2 1 < (a + a 0) 2 (1 d). 19

21 Delay-Dependent Conditions Choose the LKF of the form V (t, z t, z t s) = (p 1 p 3 a) π 0 z 2 (ξ, t)dξ +p 3 a π 0 zξ 2 (ξ, t)dξ + π 0 [ r 0 h t t+θ e 2δ(s t) z 2 s (ξ, s)dsdθ +s t t h e 2δ(s t) z 2 (ξ, s)ds +q ] t t τ(t) e 2δ(s t) z 2 (ξ, s)ds dξ with p 1 > 0, p 3 > 0, s > 0, r > 0 and q 0. Then the operators in LOI take the form P = p 3 (a 2 ξ 2 + a) + p 1, R = r, Q = q, S = s, P 3 = p 3, P 2 = p 2 > 0 and p 2 δp

22 We integrate by parts and apply Wirtinger s Inequality (Wang, 1994 JMAA): Let z W 1,2 ([a, b], R) be a scalar function with z(a) = z(b) = 0. Then b b (b a z2 a)2 (ξ)dξ π 2 a (z (ξ)) 2 dξ. (26) We show that P > 0 and that the LOI of Th.2 is feasible if the following LMI is feasible φ 11 φ 12 0 φ 14 2p 3 + h 2 r 0 p 3 a 1 (s + r)e 2δh re 2δh φ 44 φ 11 = 2(a + a 0 )p 2 + 2δp 1 + q + s re 2δh, φ 12 = p 1 p 2 (a + a 0 )p 3, φ 14 = p 2 a 1 + re 2δh, φ 44 = [2r+(1 d)q]e 2δh <0, 21

23 Remark The same LMIs appear to guarantee the scalar time-delay equation ẏ(t) + (a + a 0 )y(t) + a 1 y(t τ(t)) = 0 (27) to be exp. stable. System (27) corresponds to the first modal dynamics (with k = 1) in the modal representation of the Dirichlet boundary-value problem for the heat equation y k (t) + (a + a 0 )k 2 y k (t) + a 1 y k (t τ(t)) = 0, k = 1, 2,... projected on the eigenfunctions of the operator 2 ξ 2 ( this operator has eigenvalues k 2 ). The stability of the heat eq. implies the stability of ODE (27). Thus the reduction of infinite-dimensional LOI to finite-dimensional LMIs is tight, since the stability of (27) is necessary for the stability of the heat equation. The above is consistent with the frequency domain analysis in the case of constant delays (Huang & Vanderwalle, SIAM 2004). 22

24 Remark The descriptor-based LOIs (LMIs) are affine in the system operators (coefficients). Consider now the systems under question with the uncertain operators (coefficients) from the uncertain polytope, given by M vertices. By the arguments of (Boyd et al. 1994), the uncertain systems are exp. stable if the corresponding LOIs (LMIs) in the vertices are feasible. Example Consider the controlled heat eq. z t (ξ, t) = z ξξ (ξ, t) + rz(ξ, t) + u, z(0, t) = z(l, t) = 0, (28) where ξ (0, l), t > 0 and where r is uncertain parameter satisfying r β with given β. It was shown in (Rebiai & Zinober, IJC 1993) that for l = 1 the state-feedback as. stabilizes (28). u = γz(ξ, t), γ > ( β 2π )2 23

25 By our method u = γz(ξ, t), γ > β π 2 Since β π 2 ( β 2π )2, our method guarantees exp. stabilization via a lower gain. Consider next l = π, β = 0.1 and the feedback u = z(ξ, t τ(t)) with the uncertain delay τ(t) [0, h], τ d < 1 This is a polytopic system reached by choosing r = ±0.1. We verify the feasibility of delay-dependent LMIs in the two vertices corresponding to r = ±0.1. We use LMI toolbox of Matlab and find the maximum values of h for which the system remains as. stable: d = 0.5, h = 2.04; unknown d, h = As shown before, the latter results correspond also to the stability of with r 0.1. ẏ = ( 1 + r)y(t) y(t τ(t)), 24

26 Exp. Stability of the Wave Equation Consider the wave equation z tt (ξ, t) = az ξξ µ 0 z t (ξ, t) µ 1 z t (ξ, t τ(t)) a 0 z(ξ, t) a 1 z(ξ, t τ(t)), t 0, 0 ξ π (29) with the Dirichlet boundary condition (21). Introduce the operators A = 0 1 a 2 ξ 2 a 0 µ 0, A 1 = [ ] 0 0 a 1 µ 1 (30) where the domain D( 2 ξ2) is determined by (22). Then (21), (29) can be represented as (1) in the Hilbert space H = L 2 (0, π) L 2 (0, π) with the infinitesimal operator A, possessing the domain D(A) = D( 2 ξ 2) L 2(0, π) and generating an exp. stable semigroup (see, e.g., Curtain & Zwart (1995)). 25

27 Delay-Ind. Conditions are derived for µ 1 0. V = ap 03 π0 z 2 ξ (ξ, t)dξ + π 0 v T (ξ, t)p 0 v(ξ, t)dξ + t t τ(t) π0 v T (ξ, s)e 2δ(s t) Qv(ξ, s)dξds, [ ] p01 p P 0 = 02, P p 02 p 03 w =P0 + diag{ap 03, 0} > 0, Q 0, v T (ξ, t) = [z(ξ, t) z t (ξ, t)]. Then 2 P = diag{ ap 03 ξ 2, 0} + P 0, Q 0. Corollary from delay-ind. conds In a particular case where a = 1, a 0 = a 1 = 0, the wave eq. is exp. stable for all τ d < 1 if µ 2 1 < (1 d)µ2 0 Remark. The condition 0 µ 1 < µ 0 for the stability of the wave eq. with constant delay and a = 1, a 0 = a 1 = 0 and was obtained by Nicaise & Pignotti (SIAM 2006), where it was shown that if µ 1 µ 0, there exists a sequence of arbitrary small delays that destabilize the system. 26

28 Delay-Dependent Conds: µ 1 = 0 We apply LOI of Theorem 1. Since the delay appears only in u, we choose V as follows: V = ap π0 3 zξ 2(ξ, t)dξ + π 0 [z(ξ, t) z t (ξ, t)]p [ ] 0 z(ξ, t) dξ z t (ξ, t) + [ π 0 hr 0 t h t+θ zt 2(ξ, s)e2δ(s t) dsdθ +s t t h z 2 (ξ, s)e 2δ(s t) ds +q ] t t τ z 2 (ξ, s)e 2δ(s t) ds dξ, [ ] [ ] p1 p P 0 = 2 ap3 + p, P p 2 p w = 1 p 2 > 0. 3 p 2 p 3 where r > 0, s > 0, q 0. Then the operators P, Q, R in LOI 1 are given by P = ap 3 2 ξ 2 + p 1 p 2 > 0, p 2 p [ ] 3 q 0 Q = 0, 0 0 R = diag{r, 0} 0, S = diag{s, 0} 0. 27

29 Denote C δ = [ δ 1 a a 0 µ 0 + δ ]. (31) LOI is feasible if the following LMIs are satisfied: p 2 p 3 δ, ] [ 0 re 2δh + a 1 0 (s + r)e 2δh re 2δh (2r + (1 d)q)e 2δh φ w 0 P w [ where ] < 0, (32) φ w = C T δ P w + P w C δ + diag{q + s re 2δh, h 2 r}. The same LMIs appear to guarantee the stability of ODE with delay z(t) = C 0 z(t) + A 1 z(t τ(t)), z(t) R 2. This ODE governs the first modal dynamics of the modal representation of the Dirichlet boundary-value problem. 28

30 Example Consider the controlled wave equation z tt (ξ, t) = 0.1z ξξ (ξ, t) 2z t (ξ, t) + u, (33) with boundary condition (21), t t 0, 0 ξ π, 0 τ h, τ d < 1. Applying LMI to the open-loop system we find that (33) with u = 0 is exp. stable with the decay rate δ = Considering next a delayed feedback u = z(ξ, t τ(t)) and verifying the LMI, we find that the closedloop system is exp. stable with a greater decay rate δ = 0.8 for all 0 τ(t)

31 Conclusions A general framework is given for exp. stability of linear distributed parameter systems in a Hilbert space with a bounded operator acting on the delayed state. Stability conditions are derived in terms of LOIs in the Hilbert space. In the case of a heat/wave scalar equation with the Dirichlet boundary conditions, these LOIs are reduced to finite-dimensional LMIs by applying new Lyapunov-Krasovskii functionals. The simplicity and the tightness of the results are the advantages of the new method, which can be extended to vector case and to Neumann boundary conditions. Boundary value control of the heat/wave will be presented on CDC08. 30

32 As it happened with LMIs in the finite dimensional case, LOIs are expected to provide effective tools for robust control of distributed parameter systems. 31