Delay-independent stability via a reset loop S. Tarbouriech & L. Zaccarian (LAAS-CNRS) Joint work with F. Perez Rubio & A. Banos (Universidad de Murcia) L2S Paris, 20-22 November 2012 L2S Paris, 20-22 November 2012 1 /
Outline 1 Overview of dynamical hybrid systems framework of Teel et al. 2 Problem statement and proposed solution 3 Numerical example 4 Concluding remarks L2S Paris, 20-22 November 2012 1 /
Goal of this work Context. Employ tools from Hybrid dynamical systems within control of Linear time-delay plants Not much work done in this context recent work [Liu, Teel CDC2012] Objective. Use hybrid loops or resets for stability recovery: start from a linear closed-loop where controller K stabilizes the plant P 0 without any time delay; recover (delay-independent) global exponential stability with time-delay plant P θ by enforcing suitable resets in the controller. u K p y P p u 0 K p P θ y p Reset rule x p L2S Paris, 20-22 November 2012 2 /
Hybrid techniques reach beyond limits of classical control We will design a hybrid loop to augment to the continuous-time controller by jump rules that recover stability lost due to the delay For this we use reset control systems formalism or, more generally hybrid dynamical systems formalism [Teel, Goebel, Sanfelice 2011] The use of hybrid systems is desirable due to their capability to: provide global asymptotic stability of closed-loops not stabilizable by continuous feedback (see e.g. [Hespanha et al, 1999, 2003]). guarantee a robustness with respect to small errors in the loop, which cannot be obtained using classical (i.e. with a continuous dynamics) controllers (see e.g. [Prieur 2005, Goebel and Teel, 2009]). improve the performance of linear systems in presence of disturbances [Becker et al, 2004, Nesic et al, 2008, Witvoet et al, 2007] More generally, hybrid systems can model a wider range of physical problems and provide improved observers [Allgower et al. 2007, Prieur et al. 2012] L2S Paris, 20-22 November 2012 3 /
Hybrid dynamical systems review: dynamics H = (C,D,F,G) n N (state dimension) C R 2 (flow set) D R 2 (jump set) F : C R 2 (flow map) G : D R 2 (jump map) C ẋ F(x) x + G(x) { ẋ F(x) x C H : x + G(x) x D D L2S Paris, 20-22 November 2012 4 /
Hybrid dynamical systems review: continuous dynamics H = (C,D,F,G) n N (state dimension) C R 2 (flow set) D R 2 (jump set) F : C R 2 (flow map) G : D R 2 (jump map) { ẋ F(x) x C H : x + G(x) x D x 2 { ẋ1 = x 2 ẋ 2 = x 1 +x 2 (1 x 2 1 ) 2 0 2 Van der Pol 4 2 0 2 4 x 1 L2S Paris, 20-22 November 2012 5 /
Hybrid dynamical systems review: discrete dynamics H = (C,D,F,G) n N (state dimension) C R 2 (flow set) D R 2 (jump set) F : C R 2 flow map) G : D R 2 (jump map) { ẋ F(x) x C H : x + G(x) x D {0,1} if x = 0 x + {0,2} if x = 1 {1,2} if x = 2 0 1 2 A possible sequence of states from x 0 = 0 is: (0 1 2 1) i i N L2S Paris, 20-22 November 2012 6 /
Hybrid dynamical systems review: trajectories x 3 C x 2 x0 x 1 x 4 x 5 x 6 x 7 { ẋ F(x) x C H : x + G(x) x D D L2S Paris, 20-22 November 2012 7 /
Hybrid dynamical systems review: hybrid time The motion of the state is parameterized by two parameters: t R 0, takes into account the elapse of time during the continuous motion of the state; j Z 0, takes into account the number of jumps during the discrete motion of the state. ξ(0, 0) ξ(5, 2) ξ(5, 0) ξ(5, 1) ξ(8, 2) ξ(8, 3) τ [0,5], ξ(τ,0) τ [5,8], ξ(τ,2) τ 8, ξ(τ,3) L2S Paris, 20-22 November 2012 8 /
Hybrid dynamical systems review: hybrid time E R 0 Z 0 is a compact hybrid time domain if J 1 E = ([t j,t j+1 ] {j}) j=0 where 0 = t 0 t 1 t J. E is a hybrid time domain if for all (T,J) R 0 Z 0 j (t 5,5) (t 4,4) (t 5,4) (t 3,3) (t 4,3) (t 2,2)=(t 3,2) (t 1,1) (t 2,1) E ([0,T] {0,1,...,J}) is a compact hybrid time domain. (t 0,0) (t 1,0) t L2S Paris, 20-22 November 2012 9 /
Hybrid dynamical systems review: solution Formally, a solution satisfies the flow dynamics when flowing and satisfies the jump dynamics when jumping ξ t 1 t 2 t 3 t 4 t 1 2 3 j 4 L2S Paris, 20-22 November 2012 10 /
Goal of this work (recall) Context. Employ tools from Hybrid dynamical systems within control of Linear time-delay plants Not much work done in this context recent work [Liu, Teel CDC2012] Objective. Use hybrid loops or resets for stability recovery: start from a linear closed-loop where controller K stabilizes the plant P 0 without any time delay; recover (delay-independent) global exponential stability with time-delay plant P θ by enforcing suitable resets in the controller. u K p y P p u 0 K p P θ y p Reset rule x p L2S Paris, 20-22 November 2012 11 /
Plant and controller dynamics is linear Continuous-time linear plant with time-delay θ in the state: P θ { ẋp = A p x p +A pd x p (t θ)+b p u p y p = C p x p Continuous-time controller stabilizing P 0 is: K { ẋc = A c x p +B c y p u p = C c x c u K p y P p u 0 K p P θ y p Reset rule x p L2S Paris, 20-22 November 2012 12 /
Problem Formulation Continuous-time closed-loop with delay without resets may be unstable: { ẋp = A p x p +A pd x p (t θ)+b p C c x c ẋ c = A c x c +B c C p x p Define the augmented state x = x p x c x p (t θ) = x pd Problem: Determine K p, C, D augmenting the above closed loop as } ẋ p = A p x p +A pd x pd +B p C c x c x C ẋ c = A c x c +B c C p x p x p + } = x p x D = K p x p x + c with the goal to recover global asymptotic stability of the origin. Recall: C, D are the flow and jump sets. L2S Paris, 20-22 November 2012 13 /
Delay-independent results require assumption on P θ P θ must be good enough to allow for Lyapunov-Krasovskii functionals Assumption LK. Given the matrices in P θ, there exist two symmetric positive definite matrices P p and Q, a positive scalar ǫ p and a hybrid gain K p such that [ (Ap +B p C c K p ) ] P p +P p (A p +B p C c K p )+Q P p A pd A pd P p Q [ ] Pp 0 2ǫ p 0 Q Meaning of Assumption LK: delay-independent stability of the linear time-delay plant stabilized by K p : ż(t) = (A p +B p C c K p )z(t)+a pd z(t θ) = A K z(t)+a pd z(t θ) z dynamics coincides with the closed loop dynamics with delay when clamping x c = K p x p L2S Paris, 20-22 November 2012 14 /
Convex formulation helps optimizing P p, K p, Q, ǫ p Lemma LK Assumption LK holds with K p κ max if and only if there exist two symmetric positive definite matrices Q p and S, a positive scalar ǫ p and a matrix X such that: X κ max, Q p I [ Qp A p +X C cb p +A p Q p +B p C c X +S A pd Q p ] Q p A pd S 2ǫ p [ Qp 0 0 S One then can build the following solution to Assumption LK K p = XQp 1, Q = Qp 1 SQp 1, P p = Qp 1 Maximizing ǫ p is a Generalized Eigenvalue Problem (qausi-convex) Recall that x c K p x p induces exponentially stable dynamics ż(t) = (A p +B p C c K p )z(t)+a pd z(t θ) = A K z(t)+a pd z(t θ) L2S Paris, 20-22 November 2012 15 / ]
Problem solution: use resets to go back to z dynamics Solution: define the coordinate ξ = [x p x pd x c K p x p ] T and select K p = K p from Lemma LK and for any ǫ c > 0 fix P p A K +Q/2 P p A pd P p B p C c ǫ p P p 0 0 C = He 0 Q/2 0 ξ ξ ξ 0 ǫ p Q 0 ξ 0 0 ǫ c I 0 0 0 P p A K +Q/2 P p A pd P p B p C c ǫ p P p 0 0 D = He 0 Q/2 0 ξ ξ ξ 0 ǫ p Q 0 ξ 0 0 ǫ c I 0 0 0 Recall hybrid scheme where x = [x p x c x pd ] T } ẋ p = A p x p +A pd x pd +B p C c x c ẋ c = A c x c +B c C p x p x p + } = x p = K p x p x + c x C x D Interpretation: if x D, jumps ensure x c + = K p x p (i.e., back to the good z dynamics: ż = (A p +B p C c K p )z +A pd z d L2S Paris, 20-22 November 2012 16 /
Idea behind proof of (hybrid) global exponential stability Recall definition of flow C and jump D sets: P p A K +Q/2 P p A pd P p B p C c ǫ p P p 0 0 C = ξ He 0 Q/2 0 ξ ξ 0 ǫ p Q 0 0 0 ǫ c I 0 0 0 D = (R n R nc R n )\C Consider the Lyapunov-Krasovskii functional (with suitable λ > 0): t V(x t ) = x pp p x p + x p(τ)qx p (τ)dτ +λ(x c K p x p ) (x c K p x p ) t θ During flows, since x C, from Assumption KL (for small λ > 0): V(x t ) ǫ x 2, for some ǫ > 0 ξ Across jumps, since x + p = x p, x + c = K p x p, we get from the blue guy): V(x + t ) V(x t ) 0 Stability proof uses Hybrid Lyapuonv Krasovskii theorem (next slide) L2S Paris, 20-22 November 2012 17 /
Lyapunov-Krasovskii thm for a class of hybrid systems For a state x R n, x denotes the Euclidean norm of x and x t θ := max x(t s) s [0,θ] Theorem LK Assume that there exists a functional V(x t ), class K functions α 1, α 2 and a positive definite function ρ such that α 1 ( x ) V(x t ) α 2 ( x θ ) V(x t ) ρ( x ), x C V(x + t ) V(x t ) 0, x D and assume that the solutions exhibit persistent flow. Then the origin of the hybrid time-delay system is GAS. Alternative formulations appeared in [Banos et al. 2009, Guo et al. 2012] Exponential stability holds for homogeneous dynamics L2S Paris, 20-22 November 2012 18 /
Numerical example: problem data The plant is defined by the following data [ ] [ 2 0 1 0 A p =, A 0 0.9 pd = 1 1 ], B p = [ 1 1 The controller is defined by the following data [ ] [ ] 1 0 1 A c =, B 1 0 c =, C 0 c = [ 0 1 ] ], C p = [ 1 1 ] The continuous-time closed-loop system with no delay is exponentially stable with θ = 0. The continuous-time closed-loop system with delay system is unstable for θ > 1.6 L2S Paris, 20-22 November 2012 19 /
Numerical example: selection of the reset parameters Plot of the maximized ǫ p versus the bound κ m imposed on K p. Three simulations cases ( ). κ m ǫ p K p 0.081 3.6 10 4 0.0809 1 0.5976 0.9974 2.65 0.7977 2.6474 Need K p sufficiently large to ensure ǫ p > 0, namely GAS of the origin of the hybrid closed loop L2S Paris, 20-22 November 2012 20 /
Example: Plant state response with θ = 2! L2S Paris, 20-22 November 2012 21 /
Example: Controller state response with θ = 2 L2S Paris, 20-22 November 2012 22 /
Concluding Remarks Summary: This work combines concepts from the recent framework for hybrid dynamical systems with ideas from time-delay systems Use of hybrid loops (controller jumps) allows to recover stability which may be destroyed by time delay A hybrid Lyapunov Krasovskii theorem is instrumental for this goal Academic example illustrated the proposed approach Perspectives: Extend to nonlinear systems (using Lyapunov) Determine delay-dependent compensation schemes using Lyapunov Razumikin approach L2S Paris, 20-22 November 2012 23 /
References Delay-Independent Stability of Reset Systems, IEEE Trans. Aut. Contr. 54(2) 341 346, 2009. O. Beker, C.V. Hollot and Y. Chait. Fundamental properties of reset control systems. Automatica 40(6), 905-915, 2004. F. Fichera, C. Prieur, S. Tarbouriech, L. Zaccarian. Improving The Performance of Linear Systems by Adding a Hybrid Loop: The Output Feedback Case, American Control Conference, Montreal, Canada, June 2012. R. Goebel, R. Sanfelice and A. R. Teel. Hybrid dynamical systems. IEEE Control Systems Magazine 29(2), 28-93, 2009. D. Nesic, L. Zaccarian and A.R. Teel. Stability properties of reset systems. Automatica 44(8), 2019-2026, 2008. C. Prieur, S. Tarbouriech, L. Zaccarian. Guaranteed stability for nonlinear systems by means of a hybrid loop, 8th IFAC Symposium on Nonlinear Control Systems, Bologna (Italy), September 2010. C. Prieur, S. Tarbouriech, L. Zaccarian. Lyapunov-based hybrid loops for stability and performance of continuous-time control systems, Automatica, to appear. L2S Paris, 20-22 November 2012 /