Estimation of basins of attraction for controlled systems with input saturation and time-delays Benjamin Biemond, Wim Michiels Department of Computer Science, KU Leuven 0 – Outline 1 Introduction 2 Basin of attraction estimate for delay-free system 3 Time-delay system 4 Example 5 Conclusion Benjamin Biemond 2/18 1 – Outline 1 Introduction 2 Basin of attraction estimate for delay-free system 3 Time-delay system 4 Example 5 Conclusion Benjamin Biemond 3/18 1 – Introduction 4/18 Saturation Plant delay delay Controller Sat(u) Saturation: u piecewise affine no global performance/ stabilisation of control system ⇒ only trajectories attracted from bounded basin of attraction Benjamin Biemond 1 – Basin of attraction 5/18 Definition (Basin of attraction) Subset in state space of all initial conditions with trajectories that converge to the origin Delay systems: initial functions xτ : [−τ, 0] → Rn Approximated by sublevelsets of Lyapunov-Krasovskii functional {xτ ∈ AC([−τ, 0], Rn )| V (xτ ) ≤ γ}, with dV dt <0 Non-delayed polynomial systems: sublevelset of Lyapunov function provides arbitrarily close estimate of basin of attraction [Giesl ’08] Existing results for saturation and delays: Polytopic overapproximation of saturation function [Fridman, Dambrine ’09; Gomes da Silva et.al., ’11, Tarbouriech et.al, ’11] → linear techniques Benjamin Biemond 1 – Model and approach 6/18 Sat(u) x(t) ˙ = Ax(t) + Bsat(Kx(t − τ )) u In this presentation: 1 Find basin of attraction estimate for delay-free system Lyapunov function Vnd : Rn → R≥0 , piecewise quadratic → exploits piecewise affine characteristic of differential equation 2 Benjamin Biemond 1 – Model and approach 6/18 Sat(u) u x(t) ˙ = Ax(t) + Bsat(Kx(t − τ )) In this presentation: 1 Find basin of attraction estimate for delay-free system Lyapunov function Vnd : Rn → R≥0 , piecewise quadratic → exploits piecewise affine characteristic of differential equation 2 Study delayed system Lyapunov-Krasovskii functional V : AC([−τ, 0], Rn ) → R≥0 , with Z 0 V (xτ ) =Vnd (xτ (0)) + x˙ τ (s)T Qx˙ τ (s)ds −τ Z 0 Z + −τ Benjamin Biemond θ 0 x˙ τ (¯ s)K T B T RBK x˙ τ (¯ s)d¯ sdθ, 2 – Outline 1 Introduction 2 Basin of attraction estimate for delay-free system 3 Time-delay system 4 Example 5 Conclusion Benjamin Biemond 7/18 2 – Basin of attraction estimate for delay-free system x(t) ˙ = Ax(t) + Bsat(Kx(t)) sat(y) = sign(y) min(|y|, 1) x ∈ Rn , K ∈ R1×n , A has eigenvalues in LHP Continuous Lyapunov function T (x − ss ) Ps (x − ss ), Kx > 1 Vnd (x) = xT P0 x, |Kx| ≤ 1 (x + ss )T Ps (x + ss ), Kx < −1 Free parameters ss , with Kss < 1, and P0 . Basin of attraction estimate: {x ∈ Rn | Vnd (x) ≤ γ} with γ = sup{¯ γ | Vnd (x) ≤ γ¯ ⇒ V˙ nd (x) ≤ 0} Benjamin Biemond 8/18 2 – Basin of attraction estimate for delay-free system x(t) ˙ = Ax(t) + Bsat(Kx(t)) sat(y) = sign(y) min(|y|, 1) x ∈ Rn , K ∈ R1×n , A has eigenvalues in LHP Continuous Lyapunov function T (x − ss ) Ps (x − ss ), Kx > 1 Vnd (x) = xT P0 x, |Kx| ≤ 1 (x + ss )T Ps (x + ss ), Kx < −1 Free parameters ss , with Kss < 1, and P0 . Basin of attraction estimate: {x ∈ Rn | Vnd (x) ≤ γ } with Stronger requirement: γ = sup{¯ γ | Vnd (x) ≤ γ¯ ⇒ V˙ nd (x) ≤ −Vnd (x)} Benjamin Biemond 8/18 2 – Basin of attraction estimate for non-delayed system γ = sup{¯ γ | Vnd (x) ≤ γ¯ ⇒ V˙ nd (x) ≤ −Vnd (x)} Quadratically constraint quadratic problem min γ = x∈Rn (x − ss )T Ps (x − ss ) Kx − 1 ≥ 0 (x − ss )T L(x − ss ) + 2(x − ss )T Ps (Ass + B) ≥ 0, with L = Ps A + AT Ps + Ps . s.t. Non-convex constraint solved using KKT-conditions → roots of polynomial equation Benjamin Biemond 9/18 3 – Outline 1 Introduction 2 Basin of attraction estimate for delay-free system 3 Time-delay system 4 Example 5 Conclusion Benjamin Biemond 10/18 3 – Time-delay system 11/18 x(t) ˙ = Ax(t) + Bsat(Kx(t)) +B (sat(Kx(t − τ )) − sat(Kx(t))) | {z } | {z } ω Fnd Use Lyapunov-Krasovskii functional: Z 0 V (xτ ) =Vnd (xτ (0)) + x˙ τ (s)T Qx˙ τ (s)ds −τ Z 0 Z + −τ For almost al t, dVnd dt 0 x˙ τ (¯ s)(BK)T RBK x˙ τ (¯ s)d¯ sdθ, θ = ∇Vnd Fnd +∇Vnd Bω. | {z } ≤−Vnd Now, use existing techniques to find expressions/upper bounds for terms ∇Vnd Bω and derivatives of quadratic terms in Lyapunov-Krasovskii functional. Benjamin Biemond 3 – Overapproximations 12/18 Z 0 Z 0Z 0 V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+ x˙ τ (¯ s)(BK)TRBK x˙ τ (¯ s)d¯ sdθ −τ −τ θ | {z } | {z } =:w =:W Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02] Z t ∇Vnd Bω ≤ τ δ1 Vnd + t−τ Benjamin Biemond x(s) ˙ T (BK)T R1 (BK)x(s)ds ˙ 3 – Overapproximations 12/18 Z 0 Z 0Z 0 V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+ x˙ τ (¯ s)(BK)TRBK x˙ τ (¯ s)d¯ sdθ −τ −τ θ | {z } | {z } =:w =:W Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02] Z t ∇Vnd Bω ≤ τ δ1 Vnd + x(s) ˙ T (BK)T R1 (BK)x(s)ds ˙ t−τ dw =x(t) ˙ T Qx(t) ˙ − x(t ˙ − τ )T Qx(t ˙ − τ) dt + x(t) ˙ T P3 (−x(t) ˙ + Ax(t) + Bsat(Kx(t − τ ))) Benjamin Biemond 3 – Overapproximations 12/18 Z 0 Z 0Z 0 T V (xτ ) = Vnd (xτ (0))+ x˙ τ (s) Qx˙ τ (s)ds+ x˙ τ (¯ s)(BK)TRBK x˙ τ (¯ s)d¯ sdθ −τ −τ θ | {z } | {z } =:w =:W Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02] Z t ∇Vnd Bω ≤ τ δ1 Vnd + x(s) ˙ T (BK)T R1 (BK)x(s)ds ˙ t−τ dw =x(t) ˙ T Qx(t) ˙ − x(t ˙ − τ )T Qx(t ˙ − τ) dt + x(t) ˙ T P3 (−x(t) ˙ + Ax(t) + Bsat(Kx(t − τ ))) Use sat(Kx(t − τ )) = αKx(t − τ ), α ∈ [0, 1] Benjamin Biemond 3 – Overapproximations 12/18 Z 0 Z 0Z 0 T V (xτ ) = Vnd (xτ (0))+ x˙ τ (s) Qx˙ τ (s)ds+ x˙ τ (¯ s)(BK)TRBK x˙ τ (¯ s)d¯ sdθ −τ −τ θ | {z } | {z } =:w =:W Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02] Z t ∇Vnd Bω ≤ τ δ1 Vnd + x(s) ˙ T (BK)T R1 (BK)x(s)ds ˙ t−τ dw =x(t) ˙ T Qx(t) ˙ − x(t ˙ − τ )T Qx(t ˙ − τ) dt + x(t) ˙ T P3 (−x(t) ˙ + Ax(t) + Bsat(Kx(t − τ ))) ≤x(t)(Q ˙ − P3 − τ2 P3 R2−1 P3 )x(t) ˙ − x(t ˙ − τ )T Qx(t ˙ − τ) Z t R2 + x(s) ˙ T(BK)T BK x(s)ds ˙ + max x(t) ˙ TP3 (A + αBK)x(t) 2 α∈[0,1] t−τ Benjamin Biemond 3 – Overapproximations 12/18 Z 0 Z 0Z 0 V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+ x˙ τ (¯ s)(BK)TRBK x˙ τ (¯ s)d¯ sdθ −τ −τ θ | {z } | {z } =:w =:W with R = R1 + R22 : Z t dW =− x(s) ˙ T (BK)T RBK x(s)ds ˙ + τ x(t) ˙ T (BK)TRBK x(t) ˙ dt t−τ Benjamin Biemond 3 – Overapproximations 12/18 Z 0 Z 0Z 0 V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+ x˙ τ (¯ s)(BK)TRBK x˙ τ (¯ s)d¯ sdθ −τ −τ θ | {z } | {z } =:w =:W with R = R1 + R22 : Z t dW =− x(s) ˙ T (BK)T RBK x(s)ds ˙ + τ x(t) ˙ T (BK)TRBK x(t) ˙ dt t−τ 0 P3 (A + αBK) 0 dV Ψ 0 z(t) ≤(− + τ δ1 )Vnd + max z(t)T 0 dt α∈[0,1] 0 0 −Q with Ψ = −P3 + τ( 12 P3 R2−1 P3 + K T B T (R1 + 12 R2 )BK) and z(t) = col(x(t), x(t), ˙ x(t ˙ − τ )). Benjamin Biemond 3 – Sufficient conditions 13/18 Theorem If there exist P3 , R1 , R2 , S ≺ 0 and scalar δ1 > 0 such that 1 (− + τ δ1 )S 2 P3 (A + αBK) ≺ 0, 1 T Ψ 2 (A + αBK) P3 −δ1 Pi Pi ≺ 0, S ≺ P0 , S ≺ 12 Ps . Pi −R1 with α ∈ {0, 1} and i ∈ {0, s}, then all trajectories with initial condition in {xτ ∈ AC[−τ, 0]| V (xτ ) ≤ γ } are attracted towards the origin. Benjamin Biemond 4 – Outline 1 Introduction 2 Basin of attraction estimate for delay-free system 3 Time-delay system 4 Example 5 Conclusion Benjamin Biemond 14/18 4 – Example 15/18 T 0 1 0 −0.25 5.00 1.11 Let A = ,B= ,K= , P0 = −0.2 0.05 1 −0.2 1.11 10.74 T We select = 0.05 and ss = 0.65 −0.05 4 3 2 x2 1 0 −1 −2 −3 −4 −6 −4 −2 0 2 4 x1 Solution to matrix inequality found for τ = 0.015 Benjamin Biemond 6 Black: Vnd , gray: xT P0 x 5 – Outline 1 Introduction 2 Basin of attraction estimate for delay-free system 3 Time-delay system 4 Example 5 Conclusion Benjamin Biemond 16/18 5 – Conclusion 17/18 Presented novel Lyapunov-Krasovskii functional exploiting piecewise affine nature of delay differential equation Attains larger basin of attraction estimate for delay-free case Sufficient condition for attraction for fixed delay Fridman, E. and Dambrine, M. (2009) Control under quantization, saturation and delay: An LMI approach Automatica, 45, 2258-2264. Fridman, E. (2002). Effects of small delays on stability of singularly perturbed systems. Automatica, 38(5), 897-902. Giesl, P. (2008). Construction of a local and global Lyapunov function using radial basis functions. IMA Journal of Applied Mathematics, 73(5), 782-802. Tarbouriech, S., Garcia, G., Gomes da Silva Jr., J.M. and Queinnec, I. (2011). Stability and stabilization of linear systems with saturating actuators. Springer- Verlag, London. Benjamin Biemond Thank you for your attention Benjamin Biemond
© Copyright 2024 ExpyDoc