Gramians based model reduction for hybrid switched systems Y. Chahlaoui Younes.Chahlaoui@manchester.ac.uk Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) School of Mathematics The University of Manchester MOPNET 3 - Edinburgh Y.Chahlaoui Edinburgh, 20 Sept. 2010
Outline 1 Introduction 1 Model reduction: BT 2 Switched dynamical systems Switching logic Motivations 2 Model order Reduction of Switched Linear Systems (MoRSLS) Adequate choice of the measure of quality 3 MoRSLS scenarios Case 1: time-transmission switching Case 2: time-independent switching Stability under arbitrary switching Common Lyaponuv function problems Other algorithms under investigation Switched Quadratic Lyapunov Functions Stability analysis under restricted switching 4 Summary Y.Chahlaoui Edinburgh, 20 Sept. 2010 1/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 2/42 Time invariant model reduction idea N m N A B n p n  Ĉ m ˆB p C  = W T AV, ˆB = W T B, Ĉ = CV where n N P = VW T is a projector for W T V = I n y ŷ = T e (z)u := C(zI N A) 1 B }{{} T (z) Â) 1 ˆB u }{{} ˆT (z) Ĉ(zI n
Balanced Reduction (Moore 81) Lyapunov/Stein equations for system Gramians AP + PA T + BB T = 0 A T Q + QA + C T C = 0 APA T + BB T = P A T QA + C T C = Q With P = Q = Σ: Want Gramians Diagonal and Equal States Difficult to Reach are also Difficult to Observe Reduced Model  = W T n AV n, ˆB = W T n B, Ĉ = CV n PV n = W n Σ n, QW n = V n Σ n P = 0 x(τ)x(τ) T dτ = Q = 0 0 e AT τ C T Ce Aτ dτ e Aτ BB T e AT τ dτ Y.Chahlaoui Edinburgh, 20 Sept. 2010 3/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 4/42 Hankel norm error estimate (Glover 84) Why Balanced Trancation? Hankel singular values = λ(pq) Model reduction H error (Glover) y ŷ 2 (sum neglected singular values) u 2 Preserves Stability
Y.Chahlaoui Edinburgh, 20 Sept. 2010 5/42 Switched dynamical systems Hybrid system: Dynamical system that combines continuous with discrete-valued variables.
Example 1: Thermostat Y.Chahlaoui Edinburgh, 20 Sept. 2010 6/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 7/42 Example 2: Server system with congestion control Poincare map '
To think about... For a given initial condition we can have many solutions. With hybrid systems there are many possible notions of stability. Which one is the best? (engineering question, not a mathematical one) What type of perturbations do you want to consider on the initial conditions? (this will define the topology on the initial conditions) What type of changes are you willing to accept in the solution? (this will define the topology on the signals) Even for the simple case: Lyapunov, integral, Poincare (even for linear systems these definitions may differ: Why?) Y.Chahlaoui Edinburgh, 20 Sept. 2010 8/42
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Y.Chahlaoui Edinburgh, 20 Sept. 2010 10/42 Switched dynamical system II Continuous-time d dt x(t) = f i (x(t), u(t)), t R +, i I = {1,..., N} where: state x R n, control u R m, I an index set.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 11/42 Switching logic A logical rule orchestrating the switching between the subsystems. Usually described as classes of piecewise constant maps σ : R + I σ(t) has finite number discontinuities on any part of R + No-chattering requirement, the sliding-like motion can be easily identified before hand. i = σ(t) the active mode at time t In general, the switching logic is time-controlled, state-dependent, and with memory i = σ(t, x(t), σ(τ)) = σ(t+), τ < t Two cases time-transmission switching i = σ(t) = σ(t+), τ < t time-invariant switching i = σ(x(t), σ(τ)) = σ(t+), τ < t
Y.Chahlaoui Edinburgh, 20 Sept. 2010 12/42 Motivations Many systems cannot be asymptotically stabilized by a single smooth feedback control law. Intelligent control by switching between different controllers to achieve stability and improve transient response. Most important basic problems: Find conditions that guarantee the stability of the switched system for any switching signal. Identify those classes of stabilizing switching signals. Construct a stabilizing switching signal. Events detection is very important. All techniques are currently intractable. A reduced model will make feasible computational investigation. Especially: Formal verification
Adequate choice of the measure of quality Formal verification of hybrid systems A typical safety verification problem consists of determining if any trajectory of (Σ) starting in a given set of initial conditions S enters a given set of unsafe states B. Reduction based on an upper bound are not suitable Instead use ( y(t) ŷ(t) = y(t) ŷ(t) = 0 ) 1 y(t) ŷ(t) 2 2 dt sup y(t) ŷ(t) t [0, ) Balanced truncation like method could be considered. Y.Chahlaoui Edinburgh, 20 Sept. 2010 13/42
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Model reduction of switched linear systems (MoRSLS) Hypothesis Each subsystem is considered to be asymptotically stable, controllable and observable. No restriction on the switching signals: arbitrary switching. Different model order reduction scenarios Y.Chahlaoui Edinburgh, 20 Sept. 2010 16/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 17/42 Scenario 1 Switching path is known a priori {(t 1, σ 1 ), (t 2, σ 2 ),..., (t N, σ N )} Use BT with the Gramians N P = Q = ti i=1 ti N i=1 t i 1 e A i τ B i B T i e AT τ i Ci T t i 1 e AT i τ dτ C i e A i τ dτ
Y.Chahlaoui Edinburgh, 20 Sept. 2010 17/42 Scenario 1 Use BT with the Gramians N ti P = e A i τ B i Bi T e AT τ i dτ = t i 1 Q = i=1 ti N i=1 t i 1 e AT i τ C T i C i e A i τ dτ = N α i P i i=1 N β i Q i i=1 N α i =1 i=1 N i=1 β i =1 P i Q i ti = e A i τ B i Bi T t i 1 e AT i τ dτ ti = e AT τ i Ci T C i e A i τ dτ t i 1 or or or or tn = = t0 0 tn t 0 e A i τ B i B T i e AT i τ dτ = e A i τ B i B T i e AT i τ dτ e AT i τ Ci T C i e A i τ dτ = e AT i τ C T C i e A i τ dτ 0 i
MoRSLS: scenario 2 Reduce each subsystem independently and keep the same switching rule Y.Chahlaoui Edinburgh, 20 Sept. 2010 18/42
MoRSLS: scenario 2a Y.Chahlaoui Edinburgh, 20 Sept. 2010 19/42
MoRSLS: scenario 2b Y.Chahlaoui Edinburgh, 20 Sept. 2010 20/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 21/42 MoRSLS: scenario 3 - Stability under arbitrary switching The main question Whether the switched system is stable when there is no restriction on the switching signals. Assumptions It is necessary to require that all the subsystems are asymptotically stable. Also sufficient if: A i being pairwise commutative (A i A j = A j A i for all i, j I) (Narendra & all (1994) and Zhai & all (2002)) Ai symmetric (A T i = A i for all i I) (Zhai & all (2004)) Ai normal (A T i A i = A i A T i for all i I) (Zhai & all (2006)) The stability of the switched system is guaranteed under arbitrary switching if there exists a common Lyapunov function for all the subsystems. One has to solve a very large LMI. PA i + A T i P < 0, i I
Y.Chahlaoui Edinburgh, 20 Sept. 2010 22/42 Simple case: Shorten and Narendra (99, 03) A necessary and sufficient condition for the existence of a common quadratic Lyapunov function can be based on the stability of the matrix pencil formed by the pair of subsystems state matrices: γ α (A 1, A 2 ) = αa 1 + (1 α)a 2, α [0, 1]. γ α (A 1, A 2 ) is said to be Hurwitz if two modes A 1 and A 2 in R 2 2. Real (λ (γ α (A 1, A 2 ))) < 0, for all 0 α 1.
Theorem Let A 1, A 2 be two Hurwitz matrices in R 2 2. The following conditions are equivalent: 1 there exist a CQLF with A 1, A 2 as the two subsystems; 2 the matrix pencils γ α (A 1, A 2 ) and γ α (A 1, A 1 2 ) are Hurwitz; 3 the matrices A 1 A 2 and A 1 A 1 2 do not have any negative real eigenvalues. This result is difficult to generalize to higher dimensional systems. Y.Chahlaoui Edinburgh, 20 Sept. 2010 23/42
Let A 1, A 2 be two Hurwitz matrices in R n n. Theorem A necessary condition for the existence of a CQLF is that the matrix products A 1 [αa 1 + (1 α)a 2 ] and A 1 [αa 1 + (1 α)a 2 ] 1 do not have any negative real eigenvalues for all 0 α 1. Theorem If rank(a 2 A 1 ) = 1. A necessary and sufficient condition for the existence of a CQLF for the switched system with A 1, A 2 as the two subsystems is that the matrix product A 1 A 2 does not have any negative real eigenvalues. Equivalently, the matrix A 1 + γa 2 is non-singular for all γ [0, + ). Y.Chahlaoui Edinburgh, 20 Sept. 2010 24/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 25/42 Theorem Shorten and Narendra (2002) Let A 1, A 2,..., A N be a finite number of Hurwitz matrices in R 2 2 with a 21i 0 for all i. A necessary and sufficient condition for the existence of a CQLF is that a CQLF exists for every 3-tuple of systems {A i, A j, A k }, i j k, for all i, j, k {1,..., N}.
Common Lyaponuv function problems The standard interior point methods for LMIs may become ineffective as the number of modes increases. Liberzon & all (2004) proposed an interactive gradient decent algorithm, which could converge to the CQLF in finite number of steps. But the convergence is in the sense of probability one, and it is a numerical method. Algebraic conditions remain challenging task (Liberzon (2003) and Liberzon & all (1999)). Complex matrix pencils conditions Converse Lyaponuv theorems Recently, in Laffey & Smigoc (2007), A tensor condition was introduced as a necessary condition for the existence of a CQLF for the general case, i.e., a switched system consisting of a finite number of n-th order LTI systems. Alternatively, Liberzon & al. (1999) proposed a Lie algebraic condition for switched LTI systems, which is based on the solvability of the Lie algebra generated by the subsystems state matrices. Y.Chahlaoui Edinburgh, 20 Sept. 2010 26/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 27/42 Other algorithms under investigation The system ẋ = A i x, i = 1,..., N has a common Lyaponuv function V (x) := x T P N x where A T 1 P 1 + P 1 A 1 = I, A T i P i + P i A i = P i 1, i = 2,..., N. P i is the unique symmetric positive definite matrix solution of the corresponding equation.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 28/42 Other algorithms under investigation The system ẋ = A i x+b i u, has a common Lyaponuv function i = 1,..., N V (x) := x T P N x where A T 1 P 1 + P 1 A 1 = B 1 B T 1 or ɛ 2 B 1 I, A T i P i + P i A i = ɛ 2 B i P i 1, i = 2,..., N. P i is the unique symmetric positive definite matrix solution of the corresponding equation.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 29/42 Other algorithms under investigation The system ẋ = A i x+b i u, i = 1,..., N has a common Lyaponuv function where V (x) := x T P N x A T 1 P 1 + P 1 A 1 = ɛ 2 I, A T i P i + P i A i = ɛ 2 P i 1, i = 2,..., N. ɛ = ɛ B1,..., B N P i is the unique symmetric positive definite matrix solution of the corresponding equation.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 30/42 Switched Quadratic Lyapunov Functions (Daafouz & all. (2002) and Fang & all. (2004)) A less conservative class of Lyapunov functions are the Switched Quadratic Lyapunov Functions: 1 Every subsystem is stable, P i = P T i 0, i I 2 P i are patched together based on the switching signals σ(t) to construct a global Lyapunov function as V (t, x(t)) = x T (t)p σ(t) x(t).
Y.Chahlaoui Edinburgh, 20 Sept. 2010 31/42 Theorem Fang & all. (2004) If there exist positive definite symmetric matrices P i = Pi T R n n and matrices F i, G i R n n (i I), satisfying [ Ai Fi T + F i A T ] i P i A i G i F i Gi T A T i Fi T P j G i Gi T 0 for all i, j I, then the switched discrete linear system is asymptotically stable. The LMI can be replaced either by [ Pi A T ] i P j 0 or by P j A i P j [ Pi A i G i Gi T A T i P j G i Gi T ] 0 However, it is worth pointing out that the switched quadratic Lyapunov function method is still a sufficient only condition.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 32/42 Necessary and Sufficient Stability Conditions The asymptotic stability problem for switched linear systems with arbitrary switching is equivalent to the robust asymptotic stability problem for polytopic uncertain linear time-variant systems, for which several strong stability conditions exist. x k+1 = A(k)x k where A(k) A = Conv{A 1, A 2,..., A N }. Here, Conv stands for convex combination. Lemma (Bauer & all. 1993) The linear time-variant system above is robustly asymptotically stable if and only if there exists a finite integer n such that A i1 A i2... A in < 1 for all n-tuple A ij {A 1, A 2,..., A N }, where j = 1,..., n.
Theorem (Lin & all. 2005) A switched linear system x k+1 = A σ(k) x k, where A σ(k) {A 1, A 2,..., A N }, is asymptotically stable under arbitrary switching if and only if there exists a finite integer n such that A i1 A i2... A in < 1 for all n-tuple A ij {A 1, A 2,..., A N }, where j = 1,..., n. Y.Chahlaoui Edinburgh, 20 Sept. 2010 33/42
Proposition The following statements are equivalent: 1 The switched linear system x k+1 = A σ(k) x k, where A σ(k) {A 1, A 2,..., A N }, is asymptotically stable under arbitrary switching; 2 the linear time-variant system x k+1 = A(k)x k where A(k) A = Conv{A 1, A 2,..., A N }, is robustly asymptotically stable; 3 there exists a finite integer n such that A i1 A i2... A in < 1 for all n-tuple A ij {A 1, A 2,..., A N }, where j = 1,..., n. Y.Chahlaoui Edinburgh, 20 Sept. 2010 34/42
Y.Chahlaoui Edinburgh, 20 Sept. 2010 35/42 Stability analysis under restricted switching Slow switching Multiple Lyapunov Functions
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Y.Chahlaoui Edinburgh, 20 Sept. 2010 38/42 Piecewise Quadratic Lyapunov Functions (Pettersson & al. 2001) A Lyapunov-like function V i (x) = x T P i x needs to satisfy the following two conditions 1 There exist constant scalars β i α i > 0 such that α i x 2 V i (x) β i x 2 hold for all x Ω i. Consider a quadratic Lyapunov-like function candidate, and require that α i x T Ix x T P i x β i x T Ix holds for any x Ω i. That is { x T (α i I P i )x 0 x T (P i β i I )x 0 holds for all x Ω i.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 39/42 Piecewise Quadratic Lyapunov Functions (Pettersson & al. 2001) 2 For all x Ω i and x 0, V i (x) < 0. Equivalently, P i = Pi T such that x T [A T i P i + P i A i ]x < 0 for x Ω i. 3 Switching Condition: x T P j x x T P i x for x Ω i,j Ω i Ω j
Y.Chahlaoui Edinburgh, 20 Sept. 2010 1/42 S-procedure (Pettersson & al. 2002) Let consider Ω i = {x x T Q i x 0}, Ω i,j = {x x T Q i,j x 0} Theorem The switched system is (exponentially) stable if there exist matrices P i = Pi T and scalars α > 0, β > 0, µ i 0, ν i 0, ϑ i 0, and η i,j 0, such that αi + µ i Q i P i βi ν i Q i A T i P i + P i A i + ϑ i Q i I P j + η i,j Q i,j P i are satisfied.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 41/42 References Lin H. & Antsaklis P.J., Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results. IEEE Trans. Automatic Control, vol. 54, no. 2, 2009. Sun, Z. & Ge, S. S. Switched linear systems. Control and design. Communications and Control Engineering. London: Springer., 2005.
Y.Chahlaoui Edinburgh, 20 Sept. 2010 42/42 Summary Switched dynamical systems Model reduction of switched dynamical systems Stability analysis under arbitrary switching Necessary and sufficient conditions for asymptotic stability A balanced truncation-like method for model reduction of switched dynamical systems Experiments are encouraging http://www.maths.manchester.ac.uk/ chahlaoui