Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach

Transcription

1 Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach Michael Malisoff (LSU) Joint with Frederic Mazenc and Silviu-Iulian Niculescu 0/6

2 Motivation 1/6

3 Motivation Systems with feedback controls are common in engineering. 1/6

4 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. 1/6

5 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. 1/6

6 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. Discontinuous delays arise, e.g., in networked control systems. 1/6

7 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. Discontinuous delays arise, e.g., in networked control systems. Frequency domain and Lyapunov-Krasovskii ideas do not apply. 1/6

8 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. Discontinuous delays arise, e.g., in networked control systems. Frequency domain and Lyapunov-Krasovskii ideas do not apply. Time-varying systems arise from linearizing around solutions. 1/6

9 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. Discontinuous delays arise, e.g., in networked control systems. Frequency domain and Lyapunov-Krasovskii ideas do not apply. Time-varying systems arise from linearizing around solutions. Emulation methods typically require small feedback delays. 1/6

10 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. Discontinuous delays arise, e.g., in networked control systems. Frequency domain and Lyapunov-Krasovskii ideas do not apply. Time-varying systems arise from linearizing around solutions. Emulation methods typically require small feedback delays. Reduction model and prediction lead to distributed terms. 1/6

11 Motivation Systems with feedback controls are common in engineering. The controls are state dependent parameters we must choose. They must often be functions of time-lagged values of states. Discontinuous delays arise, e.g., in networked control systems. Frequency domain and Lyapunov-Krasovskii ideas do not apply. Time-varying systems arise from linearizing around solutions. Emulation methods typically require small feedback delays. Reduction model and prediction lead to distributed terms. Z. Artstein, I. Karafyllis, M. Krstic, S. Niculescu, P. Pepe,... 1/6

12 Our Contractiveness Lemma 2/6

13 Our Contractiveness Lemma Lemma: Let T > 0 be a constant, w : [ T, ) [0, ) admit a sequence {v i } and positive constants v a and v b such that v 0 = 0, v i+i v i [v a, v b ] for all i 0, w be continuous on [v i, v i+1 ) for all i 0 with finite left limits at each v i, and d : [0, ) [0, ) be piecewise continuous. 2/6

14 Our Contractiveness Lemma Lemma: Let T > 0 be a constant, w : [ T, ) [0, ) admit a sequence {v i } and positive constants v a and v b such that v 0 = 0, v i+i v i [v a, v b ] for all i 0, w be continuous on [v i, v i+1 ) for all i 0 with finite left limits at each v i, and d : [0, ) [0, ) be piecewise continuous. Assume there is a constant ρ (0, 1) such that w(t) ρ w [t T,t] + d(t) for all t 0. (1) 2/6

15 Our Contractiveness Lemma Lemma: Let T > 0 be a constant, w : [ T, ) [0, ) admit a sequence {v i } and positive constants v a and v b such that v 0 = 0, v i+i v i [v a, v b ] for all i 0, w be continuous on [v i, v i+1 ) for all i 0 with finite left limits at each v i, and d : [0, ) [0, ) be piecewise continuous. Assume there is a constant ρ (0, 1) such that Then w(t) ρ w [t T,t] + d(t) for all t 0. (1) w(t) w [ T,0]e ln(ρ) T t + 1 (1 ρ) 2 d [0,t] for all t 0. (2) 2/6

16 Our Contractiveness Lemma Corollary: Let X : [0, ) [0, ) be piecewise C 1 and admit constants g 0, a s 0, T > 0, and δ (0, 1) and piecewise continuous functions a : [0, ) [ a s, ), b : [0, ) [0, ), and λ : [0, ) [0, ) such that and Ẋ(t) a(t)x(t) + b(t) e t t T a(m)dm + t t T sup s [t g,t] X(s) + λ(t) t g (3) b(m)e t m a(l)dl dm δ t T + g. (4) Assume that lim t p X(t) R at each p 0. Then X(t) X [0,T +g] e ln(δ) T +g (t T g) + TeTa s λ [g,t] (1 δ) 2 t T + g. (5) 2/6

17 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (6) 3/6

18 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (6) Assumption 1: The functions A, B, and h are bounded and piecewise continuous. Also, there exist a bounded piecewise continuous p 1 : [0, ) R, constants p 2 > 0 and p 3 > 0, and a C 1 function P : [0, ) R n n such that P(t) is symmetric and positive definite for all t, such that V (t, z) = z P(t)z satisfies p 2 z 2 V (t, z) p 3 z 2 V t (t, z) + V z (t, z)[a(t) + B(t)]z p 1 (t)v (t, z) (7a) (7b) for all t 0 and z R n. 3/6

19 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (3) Assumption 2: There exist T > 0 and δ (0, 1) such that r(t) = 2 P(t)B(t) p 2 is such that we have t t h(t) [ A(m) + B(m) ] dm, (8) e t t T p 1(m)dm + t t T r(m)e t m p 1(l)dl dm δ (GA) for all t 2(T + h ). 4/6

20 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (3) Theorem: If (3) satisfies Assumptions 1-2, then it is uniformly globally exponentially stable to 0. 5/6

21 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (3) Theorem: If (3) satisfies Assumptions 1-2, then it is uniformly globally exponentially stable to 0. Decay rates p 1 (t) can take positive and negative values. 5/6

22 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (3) Theorem: If (3) satisfies Assumptions 1-2, then it is uniformly globally exponentially stable to 0. Decay rates p 1 (t) can take positive and negative values. Takes into account the case where B(t)x(t h(t)) has a stabilizing effect. 5/6

23 Assumptions and Theorem ẋ(t) = A(t)x(t) + B(t)x(t h(t)) (3) Theorem: If (3) satisfies Assumptions 1-2, then it is uniformly globally exponentially stable to 0. Decay rates p 1 (t) can take positive and negative values. Takes into account the case where B(t)x(t h(t)) has a stabilizing effect. If p 1 is a nonnegative constant and P, A and B are constant, then (GA) simplifies to this averaging condition: 1 t sup h(m)dm t 2(T + h ) T t T p 2 δ e Tp 1 2 PB ( A + B ) T (9) 5/6

24 Conclusions 6/6

25 Conclusions We covered a large class of systems that arises in engineering. 6/6

26 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. 6/6

27 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. Contractiveness circumvents Lyapunov-Krasovskii methods. 6/6

28 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. Contractiveness circumvents Lyapunov-Krasovskii methods. Our analogs cover uncertain coefficient matrices A and B. 6/6

29 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. Contractiveness circumvents Lyapunov-Krasovskii methods. Our analogs cover uncertain coefficient matrices A and B. Interesting examples are amenable to our new techniques. 6/6

30 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. Contractiveness circumvents Lyapunov-Krasovskii methods. Our analogs cover uncertain coefficient matrices A and B. Interesting examples are amenable to our new techniques. Mazenc, F., M. Malisoff, and S.-I. Niculescu, Stability and control design for time-varying systems with time-varying delays using a trajectory based approach, SIAM Journal on Control and Optimization, 55(1): , /6

31 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. Contractiveness circumvents Lyapunov-Krasovskii methods. Our analogs cover uncertain coefficient matrices A and B. Interesting examples are amenable to our new techniques. Mazenc, F., M. Malisoff, and S.-I. Niculescu, Stability and control design for time-varying systems with time-varying delays using a trajectory based approach, SIAM Journal on Control and Optimization, 55(1): , Thank you for your attention! 6/6

32 Backup Slides to Use if Time Allows or Questions Warrant 1/4

33 Backup Slides to Use if Time Allows or Questions Warrant Two Examples From: Mazenc, F., M. Malisoff, and S.-I. Niculescu, Stability and control design for time-varying systems with time-varying delays using a trajectory based approach, SIAM Journal on Control and Optimization, 55(1): , /4

34 Example 1 ẋ(t) = x(t) + b(t)x(t 1), (10) with x R, and b periodic of period 1 defined by (i) b(t) = 0 when t [0, c) and (ii) b(t) = d when t [c, 1]. 2/4

35 Example 1 ẋ(t) = x(t) + b(t)x(t 1), (10) with x R, and b periodic of period 1 defined by (i) b(t) = 0 when t [0, c) and (ii) b(t) = d when t [c, 1]. Let d > 1 and c (0, 1) be any constants such that (1 c)d < 1 e 1 2. (11) 2/4

36 Example 1 ẋ(t) = x(t) + b(t)x(t 1), (10) with x R, and b periodic of period 1 defined by (i) b(t) = 0 when t [0, c) and (ii) b(t) = d when t [c, 1]. Let d > 1 and c (0, 1) be any constants such that (1 c)d < 1 e 1. (11) 2 This example does not seem to be covered by Razumikhin or Lyapunov-Krasovkii functions. 2/4

37 Example 1 ẋ(t) = x(t) + b(t)x(t 1), (10) with x R, and b periodic of period 1 defined by (i) b(t) = 0 when t [0, c) and (ii) b(t) = d when t [c, 1]. Let d > 1 and c (0, 1) be any constants such that (1 c)d < 1 e 1. (11) 2 This example does not seem to be covered by Razumikhin or Lyapunov-Krasovkii functions. Our theorem applies with V (x) = 1 2 x 2 and u = 0. 2/4

38 Example 2 Let l > 0 be arbitrary and k N be an odd integer and consider where x is valued in R, and ẋ(t) = x(t h(t)), (12) h(t) = max { } 0, l sin k (t). (13) 3/4

39 Example 2 Let l > 0 be arbitrary and k N be an odd integer and consider where x is valued in R, and ẋ(t) = x(t h(t)), (12) h(t) = max { } 0, l sin k (t). (13) This example does not seem to be covered by Razumikhin or Lyapunov-Krasovkii functions. 3/4

40 Example 2 Let l > 0 be arbitrary and k N be an odd integer and consider where x is valued in R, and ẋ(t) = x(t h(t)), (12) h(t) = max { } 0, l sin k (t). (13) This example does not seem to be covered by Razumikhin or Lyapunov-Krasovkii functions. Our theorem applies, without any restriction on l > 0. 3/4

41 Example 2 Let l > 0 be arbitrary and k N be an odd integer and consider where x is valued in R, and ẋ(t) = x(t h(t)), (12) h(t) = max { } 0, l sin k (t). (13) This example does not seem to be covered by Razumikhin or Lyapunov-Krasovkii functions. Our theorem applies, without any restriction on l > 0. For each T > 0, there is a k such that Assumption 2 is satisfied. 3/4

42 Conclusions We covered a large class of systems that arises in engineering. We incorporated delays h(t) without requiring ḣ(t) < 1. Contractiveness circumvents Lyapunov-Krasovskii methods. Our analogs cover uncertain coefficient matrices A and B. Interesting examples are amenable to our new techniques. Mazenc, F., M. Malisoff, and S.-I. Niculescu, Stability and control design for time-varying systems with time-varying delays using a trajectory based approach, SIAM Journal on Control and Optimization, 55(1): , Thank you for your attention! 4/4