Analysis of Systems with State-Dependent Delay

Transcription

1 Analysis of Systems with State-Dependent Delay Matthew M. Peet Arizona State University Tempe, AZ USA American Institute of Aeronautics and Astronautics Guidance, Navigation and Control Conference Boston, MA August 20, 2013

2 Systems with State-Dependent Delay State-Dependent Delay arises which communication distance changes with time. Any Moving System. Sonar (Speed of Sound) EM (Speed of Light) Example: Position Measurement using Sonar xt () ( ) Propagation Distance: 2 ˆx(t) = x 0 + x(t) Delay is τ(t) = 2 x0+x(t) c. Feedback Dynamics: neglecting inertia... ( ẋ(t) = ax t 2x 0 c x 0 2 c x(t) ) M. Peet State-Dependent Delay: Introduction 2 / 18

3 Stability of Systems with State-Dependent Delay Consider the general class of systems with state-dependent delay: ( ( ) ) ẋ(t) = f x(t), x t g(x(t)) Note: State x(t) enters into the independent argument, t. Contrast to the fixed-delay case: ẋ(t) = f (x ( t τ )) Linear/Affine Form: ẋ(t) = A 0 x(t) + A 1 x(t τ 0 b T x(t)) Question: Given A R n, τ 0 > 0 and τ R n, Determine whether the system is stable. Estimate the rate of decay. M. Peet State-Dependent Delay: Introduction 3 / 18

4 Why State-Dependent Delay? The most common source of delay is Propagation Time. A time-delay system is the interconnection of an ODE with a transport PDE in the feedback channel. ẋ(t) = A 0 x(t) + A i x(t τ i ) ODE: The system G 1 ẋ 1 (t) = Ax 1 (t) + Bu 1 (t) y 1 (t) = Cx 1 (t) + Du 1 (t) [ ] [ [ ] A B A0 A1 A n ] G 1 (A, B, C, D) C D I 0 PDE: The system G 2 t x 2(t, s) = s x 2(t, s) x 2 (t, 0) = u 2 (t), y 2 (t) = [ x 2 ( τ 1 ) x 2 ( τ K ) ] T Of course, the solution is just x 2 (t, s) = u 2 (t s). G 2 y 3 y 2 y 1 u M. Peet State-Dependent Delay: Introduction 4 / 18

5 Systems with State-Dependent Delay Restrictions on the Model: For physical systems: Delay is always positive Delay is usually affine in the state Lower and upper bounds for the delay Other types of systems are often ill-posed or otherwise pathological [Verriest, 2013] Global Stability is not a well-defined problem for this model. Assumptions: In this talk, we use scalar systems of the form ẋ(t) = ax(t b cx(t)) where a, c R, b > 0. To ensure b + cx(t) > 0, we bound the state as x(t) d. Also leads to the bound b + cx(t) τm. Note: The extension to R n is not hard. M. Peet State-Dependent Delay: Introduction 5 / 18

6 Lyapunov-Krasovskii Functionals For linear fixed delay systems: ẋ(t) = Ax(t) + Bx(t τ) Theorem 1. The discrete-delay system is stable if and only if there exist continuous functions M and N such that V (φ) α φ and V (φ) 0 where V (φ) = τ [ ] T [ ] φ(0) φ(0) M(s) ds + φ(s) T N(s, θ)φ(θ)dsdθ φ(s) φ(s) τ τ Note: The functional is parameterized by unknown functions M and M. Problem: How to numerically compute M and N such that Answer: Convex Optimization V (φ) > α φ V (φ) < 0 M. Peet State-Dependent Delay: Introduction 6 / 18

7 Tractable or Intractable? Convex Optimization Problem: max bx subject to Ax C The problem is convex optimization if C is a convex cone. b and A are affine. Computational Tractability: Convex Optimization over C is, in general, tractable if The set membership test for y C is in P. x is finite dimensional. M. Peet State-Dependent Delay: Introduction 7 / 18

8 Parametrization of Lyapunov-Krasovskii Functionals Problem 1: Is the set of decision variables finite-dimensional? Decision variables are the functions M and N ( 0 [φ(0) ] ) T ( [φ(0) ] ) V (φ) = ( ) Z M(s)Z ds τ φ(t) φ(s) φ(s) + ( ) τ φ(t) ( ) T ( ) ( ) Z φ(s) N(s, θ)z φ(θ) dsdθ τ φ(t) Solution: Suppose M and N are polynomials of bounded degree. M(s) = c 1 T Z(s), N(s, θ) = c 2 T Z(s, θ) Problem 2: How to enforce positivity of the L-K functional? Need constraints on c 1 and c 2. M. Peet State-Dependent Delay: Introduction 8 / 18

9 Positivity Constraints for Polynomials Sum-of-Squares: p(x) 0 if p(x) = i g i (x) 2, denoted p Σ s Lemma 2 (Parametrization of Sums-of-Squares). Given multivariate polynomial p of degree 2d, p Σ s (p(x) 0 for all x R n ) if and only if there exists a positive matrix M S q such that p(x) = Z d (x) T MZ d (x) where z d is the vector of monomials of degree d or less. Lemma 3 (Polynomial Positivity on a Subset of R n ). Given polynomial p, p(x) 0 for all x {x : g i (x) 0} if there exist s i Σ s such that p(s) = s 0 (x) + g i (x)s i (x) i M. Peet State-Dependent Delay: Introduction 9 / 18

10 Positivity Constraints for Lyapunov Functionals Lemma 4. Suppose there exist S Σ s and polynomial T such that V (φ(0), φ(s), s) φ(0) 2 = S(φ(0), φ(s), s) + T (φ(0), s) with T (φ(0), s)ds = 0. Then τ(x(t)) for any φ C[ τ m, 0] We can tighten this a bit Restrict s [ τ(φ(s)), 0] Restrict φ(s) d ( ) V (φ(0), φ(s), s)ds αφ(0) 2 τ φ(t) {s, φ : g 1 (s, φ(s)) = s(s + τ(φ(s))) 0} {s, φ : g 2 (s, φ(s)) = d φ(s) 2 0} M. Peet State-Dependent Delay: Introduction 10 / 18

11 Positivity Constraints for Lyapunov Functionals Lemma 5. Suppose there exist S, S 1, S 2, S 3 Σ s and polynomial T such that V (φ(0), φ(s), s) φ(0) 2 = S + g 1 (s, φ(s))s 1 + g 2 (s, φ(s))s 2 + g 2 (s, φ(0))s 3 + T (φ(0), s) with T (φ(0), s)ds = 0. Then τ(x(t)) for any φ d. ( ) V (φ(0), φ(s), s)ds αφ(0) 2 τ φ(t) Where Recall S, S 1, S 2, S 3, T and V are the decision variables Represented using positive matrices. Constraints are equalities and matrix positivity (SDP). M. Peet State-Dependent Delay: Introduction 11 / 18

12 Positivity Constraints for Lyapunov Functionals Lemma 6. Suppose there exists a positive matrix Q 0 such that V 2 (φ(s), φ(θ), s, θ) = Z(s, φ(s)) T QZ(θ, φ(θ)) Then ( ) τ φ(t) for any φ C[ τ m, 0]. Positivity Constraints ( ) V 2 (φ(s), φ(θ), s, θ) ds dθ 0 τ φ(t) Constraints are equalities and matrix positivity (SDP). Convex Optimization Positive Lyapunov Functionals are represented using vectors and matrix positivity constraints. M. Peet State-Dependent Delay: Introduction 12 / 18

13 The Derivative Now Recall the dynamics: ẋ(t) = ax(t τ 0 + bx(t)) τ(x(t)) = τ 0 + bx(t) With Lyapunov Functional V (t) = ) ( )V 1 (x(0), x(s), s ds+ τ x(t) The derivative is ( ) τ x(t) ) ( ) V 2 (x(s), s, x(θ), θ ds dθ τ x(t) ) V (t) = ( ) V 3 (x(t), x(t τ(x(t))), x(t + s), s ds τ x(t) ) + ( ) ( )V 4 (x(t + s), x(t + θ), s, θ ds dθ τ x(t) τ x(t) where V 4 (x θ, x ξ, θ, ξ) = θ V 2(x θ, x ξ, θ, ξ) + ξ V 2(x θ, x ξ, θ, ξ). (1) M. Peet State-Dependent Delay: Introduction 13 / 18

14 The Derivative ẋ(t) = ax(t τ 0 + bx(t)) τ(x(t)) = τ 0 + bx(t) ) V (t) = ( ) V 3 (x(t), x(t τ(x(t))), x(t + s), s ds τ x(t) ) + ( ) ( ) V 4 (x(t + s), x(t + θ), s, θ ds dθ τ x(t) τ x(t) V 3 (x t, x τ, x s, s) = 1 τ(x t ) (abx τ 1)V 1 (x t, x τ, τ(x τ )) + 1 τ(x t ) V 1(x t, x t, 0) + ax τ x V 1(x t, x s, s) θ V 1(x t, x s, θ) + (2abx τ 2)V 2 (x τ, x s, τ(x t ), θ) + 2V 2 (x t, x s, 0, θ) M. Peet State-Dependent Delay: Introduction 14 / 18

15 Stability Test Suppose there exist S i Σ s for i = 1, 2, 3, L i Σ s for i = 1, 2, 3, 4, some ɛ j > 0 for j = 1, 2, polynomial R 1 (x 0, θ), R 2 (x 0, x 1, θ) and matrices M, N > 0, such that 1) V 1 (x 0, x 2, θ) + R 1 (x 0, θ) ɛ 1 x i=1 S i(x 0, x 2, θ)g i (x 0, x 2, θ) Σ s, 2) V 3 (x 0, x 1, x 2, θ) + R 2 (x 0, x 1, θ) ɛ 2 x i=1 L i(x 0, x 1, x 2, θ)g i (x 0, x 1, x 2, θ) Σ s, 3) V 2 (x 2, x 3, θ, ξ) = Z T d (x 2, θ)mz d (x 3, ξ), 4) V 4 (x 2, x 3, θ, ξ) = Z T d (x 2, θ)nz d (x 3, ξ), 5) τ(x 0) R 1(x 0, θ)dθ = 0, 6) τ(x 0) R 2(x 0, x 1, θ)dθ = 0, Then the system is asymptotically stable for all x t Ω, where Ω is defined as Ω := {x t C : x t τ 0 /(2b)}. M. Peet State-Dependent Delay: Introduction 15 / 18

16 Numerical Validation xt () x 0 ẋ(t) = ax(t τ 0 + bx(t)) a= 0.1 a= 0.5 a= 1 τ 0 =0.1 b [4e-4, 1] b [1e-4, 1] b [2e-4, 1] τ 0 =0.5 b [6e-4, 2] b [3e-4, 2] b [3e-3, 0.02] τ 0 =1 b [7e-4, 3] b [8e-4, 2] b [3e-3, 0.02] Table: The minimum and maximum stable values of b for a fixed a and τ 0. M. Peet State-Dependent Delay: Introduction 16 / 18

17 Numerical Validation xt () x 0 ẋ(t) = ax(t τ 0 + bx(t)) x(t) t Figure: Simulation Results using a = 0.1, b = 1, τ 0 = 6, and initial condition φ(θ) = 0.5 sin(θ) M. Peet State-Dependent Delay: Introduction 17 / 18

18 Conclusions: A Difficult Problem: Lyapunov Stability Test Convexifies the problem Relies on SDP. Complexity depends on Accuracy. Numerical Code Produced: Not currently posted Must generalize to multidimensional systems Practical Implications The effect is small, but finite. Future Work Joint Positivity Controller Synthesis Will be Available for download at M. Peet State-Dependent Delay: Conclusion 18 / 18