Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability

Transcription

1 p. 1/1 Nonlinear Systems and Control Lecture # 19 Perturbed Systems & Input-to-State Stability

2 p. 2/1 Perturbed Systems: Nonvanishing Perturbation Nominal System: Perturbed System: ẋ = f(x), f(0) = 0 ẋ = f(x) + g(t, x), g(t, 0) 0 Case 1: The origin of ẋ = f(x) is exponentially stable c 1 x 2 V (x) c 2 x 2 V x f(x) c 3 x 2, x B r = { x r} V x c 4 x

3 p. 3/1 Use V (x) to investigate ultimate boundedness of the perturbed system Assume V (t, x) = V x g(t, x) δ, V (t, x) c 3 x 2 + f(x) + V x V x g(t, x) t 0, x B r g(t, x) c 3 x 2 + c 4 δ x = (1 θ)c 3 x 2 θc 3 x 2 + c 4 δ x 0 < θ < 1 (1 θ)c 3 x 2, x δc 4 /(θc 3 ) def = µ

4 p. 4/1 Apply Theorem 4.18 x(t 0 ) α 1 2 (α 1(r)) x(t 0 ) r c1 µ < α 1 2 (α 1(r)) δc 4 c1 < r δ < c 3 c1 θr θc 3 c 2 c 4 c 2 b = α 1 c2 1 (α 2(µ)) b = µ b = δc 4 c2 c 1 θc 3 c 1 For all x(t 0 ) r c 1 /c 2, the solutions of the perturbed system are ultimately bounded by b c 2

5 p. 5/1 Example ẋ 1 = x 2, ẋ 2 = 4x 1 2x 2 + βx d(t) β 0, d(t) δ, t 0 V (x) = x T Px = x T x (Lecture 13) V (t, x) = x 2 + 2βx 2 ( x 1x ) 16 x d(t) ( 1 8 x x ) δ x βk2 2 x 2 + x 8

6 p. 6/1 k 2 = max x T P x c x 2 = c Suppose β 8(1 ζ)/( 29k2 2 ) (0 < ζ < 1) V (t, x) ζ x δ 8 x (1 θ)ζ x 2, x 29δ 8ζθ def = µ (0 < θ < 1) If µ 2 λ max (P) < c, then all solutions of the perturbed system, starting in Ω c, are uniformly ultimately bounded by 29δ λ max (P) b = 8ζθ λ min (P)

7 p. 7/1 Case 2: The origin of ẋ = f(x) is asymptotically stable α 1 ( x ) V (x) α 2 ( x ) V x f(x) α 3( x ), V x k x B r = { x r}, α i K, i = 1, 2, 3 V (t, x) α 3 ( x ) + V x g(t, x) α 3 ( x ) + δk (1 θ)α 3 ( x ) θα 3 ( x ) + δk (1 θ)α 3 ( x ), x α < θ < 1 ( δk θ ) def = µ

8 Apply Theorem 4.18 µ < α 1 2 (α 1(r)) α 1 3 ( ) δk θ < α 1 2 (α 1(r)) δ < θα 3(α 1 2 (α 1(r))) k Example V (x) = x 4 Compare with δ < c 3 c1 θr c 4 c 2 ẋ = x 1 + x 2 V [ x ] x 1 + x 2 = 4x4 1 + x 2 α 1 ( x ) = α 2 ( x ) = x 4 ; α 3 ( x ) = 4 x x 2; k = 4r3 p. 8/1

9 p. 9/1 The origin is globally asymptotically stable θα 3 (α 1 2 (α 1(r))) k = θα 3(r) k rθ 1 + r 2 0 as r ẋ = x 1 + x 2 + δ, δ > 0 δ > 1 2 lim t x(t) = = rθ 1 + r 2

10 p. 10/1 Input-to-State Stability (ISS) Definition: The system ẋ = f(x, u) is input-to-state stable if there exist β KL and γ K such that for any initial state x(t 0 ) and any bounded input u(t) ( ) x(t) β( x(t 0 ), t t 0 ) + γ ISS of ẋ = f(x, u) implies BIBS stability sup t 0 τ t u(τ) x(t) is ultimately bounded by a class K function of sup t t0 u(t) lim t u(t) = 0 lim t x(t) = 0 The origin of ẋ = f(x, 0) is GAS

11 p. 11/1 Theorem (Special case of Thm 4.19): Let V (x) be a continuously differentiable function such that V α 1 ( x ) V (x) α 2 ( x ) x f(x, u) W 3(x), x ρ( u ) > 0 x R n, u R m, where α 1, α 2 K, ρ K, and W 3 (x) is a continuous positive definite function. Then, the system ẋ = f(x, u) is ISS with γ = α 1 1 α 2 ρ Proof: Let µ = ρ(sup τ t0 u(τ) ); then V x f(x, u) W 3(x), x µ

12 p. 12/1 Choose ε and c such that V x f(x, u) W 3(x), x Λ = {ε V (x) c} Suppose x(t 0 ) Λ and x(t) reaches Ω ε at t = t 0 + T. For t 0 t t 0 + T, V satisfies the conditions for the uniform asymptotic stability. Therefore, the trajectory behaves as if the origin was uniformly asymptotically stable and satisfies x(t) β( x(t 0 ), t t 0 ), for some β KL For t t 0 + T, x(t) α 1 1 (α 2(µ))

13 p. 13/1 x(t) β( x(t 0 ), t t 0 ) + α 1 1 (α 2(µ)), t t 0 x(t) β( x(t 0 ), t t 0 ) + γ ( sup u(τ) τ t 0 ), t t 0 Since x(t) depends only on u(τ) for t 0 τ t, the supremum on the right-hand side can be taken over [t 0, t]

14 p. 14/1 Example ẋ = x 3 + u The origin of ẋ = x 3 is globally asymptotically stable V = 1 2 x2 V = x 4 + xu = (1 θ)x 4 θx 4 + xu (1 θ)x 4, x ( u θ ) 1/3 0 < θ < 1 The system is ISS with γ(r) = (r/θ) 1/3

15 p. 15/1 Example ẋ = x 2x 3 + (1 + x 2 )u 2 The origin of ẋ = x 2x 3 is globally exponentially stable V = 1 2 x2 V = x 2 2x 4 + x(1 + x 2 )u 2 = x 4 x 2 (1 + x 2 ) + x(1 + x 2 )u 2 x 4, x u 2 The system is ISS with γ(r) = r 2

16 p. 16/1 Example ẋ 1 = x 1 + x 2 2, ẋ 2 = x 2 + u Investigate GAS of ẋ 1 = x 1 + x 2 2, ẋ 2 = x 2 V (x) = 1 2 x x4 2 V = x x 1x 2 2 x4 2 = (x x2 2 )2 ( ) x 4 2 Now u 0, V = 1 2 (x 1 x 2 2 )2 1 2 (x2 1 + x4 2 ) + x3 2 u 1 2 (x2 1 + x4 2 ) + x 2 3 u V 1 2 (1 θ)(x2 1 + x4 2 ) 1 2 θ(x2 1 + x4 2 ) + x 2 3 u (0 < θ < 1)

17 p. 17/1 1 2 θ(x2 1 + x4 2 ) + x 2 3 u 0 if x 2 2 u or x 2 2 u and x 1 θ θ if x 2 u ( ) 2 u θ θ ρ(r) = 2r ( ) 2r θ θ ( 2 u θ ) 2 V 1 2 (1 θ)(x2 1 + x4 2 ), x ρ( u ) The system is ISS

18 p. 18/1 Find γ For x 2 x 1, For x 2 x 1, min { 1 4 x 2, V (x) = 1 2 x x (x2 1 + x2 2 ) 1 4 x x2 1 = 1 2 x2 1 V (x) 1 16 (x2 1 +x2 2 ) (x2 2 +x2 2 )2 = 1 4 x4 2 V (x) 1 16 x 4} V (x) 1 2 x x 4 α 1 (r) = 1 4 min { r 2, 1 4 r4}, α 1 1 (s) = { γ = α 1 1 α 2 ρ α 2 (r) = 1 2 r r4 2(s) 1 4, if s 1 2 s, if s 1