About Polynomial Instability of Linear Switched Systems Paolo Mason (CNRS / Laboratoire des Signaux et Systèmes) (Joint work with Yacine Chitour and Mario Sigalotti) Porquerolles, October 28th, 2010 P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 1 / 19
Linear Switched Systems Linear Switched System (continuous time) : x ∈ R n , A ( t ) ∈ A ⊂ R n × n . ( S ) x ( t ) = A ( t ) x ( t ) ˙ A ( · ) = any meas. function [ 0 , + ∞ ) → A ; referred as a switching law. A compact. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 2 / 19
Linear Switched Systems Linear Switched System (continuous time) : x ∈ R n , A ( t ) ∈ A ⊂ R n × n . ( S ) x ( t ) = A ( t ) x ( t ) ˙ A ( · ) = any meas. function [ 0 , + ∞ ) → A ; referred as a switching law. A compact. Example : A = { A 1 , A 2 } or A = { λ A 1 + ( 1 − λ ) A 2 : λ ∈ [ 0 , 1 ] } Remark : wlog A can be taken convex P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 2 / 19
Stability and Lyapunov exponent Maximal Lyapunov Exponent of A defined as � 1 � ρ ( A ) = sup lim sup t log � x ( t ) � . A ( · ) , x ( 0 ) t →∞ P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 3 / 19
Stability and Lyapunov exponent Maximal Lyapunov Exponent of A defined as � 1 � ρ ( A ) = sup lim sup t log � x ( t ) � . A ( · ) , x ( 0 ) t →∞ ρ ( A ) < 0 ( S ) Uniformly Globally Asymptotic Stable ( UGAS ) → Uniformly Exponentially Stable ( UES ), i.e. ∃ M , λ > 0 s.t. ∀ x ( 0 ) ∈ R n , t ≤ 0, A ( · ) � x ( t ) � ≤ Me − λ t � x ( 0 ) � . ρ ( A ) = 0 ( S ) stable: all trajectories are bounded and there exists one traj. not converging to 0, ( S ) marginally unstable: ∃ unbounded traj. with non-expon. growth. ρ ( A ) > 0 ( S ) unstable: ∃ traj. going to ∞ exponentially. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 3 / 19
In general it is not easy to verify if a given switched system is stable at the origin, even for the simplest class of linear switched system i.e. single-input systems. x ∈ R 2 , u ( t ) ∈ [ 0 , 1 ] x = u ( t ) A 1 x + ( 1 − u ( t )) A 2 x ˙ where A 1 , A 2 are 2 × 2 matrices. 1 x = A 1 x ˙ 0.75 x = u ( t ) A 1 x + ( 1 − u ( t )) A 2 x ˙ 0.5 0.25 -0.4 -0.2 0.2 0.4 0.6 0.8 1 -0.25 0.5 -0.5 -1.5 -1 -0.5 0.5 1 -0.5 x = A 2 x ˙ 0.6 0.4 -1 0.2 -0.5 0.5 1 1.5 -0.2 -0.4 ASYMPTOTICALLY STABLE UNSTABLE P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 4 / 19
Two-dimensional systems The case of two-dimensional systems x ∈ R 2 , u ( t ) ∈ [ 0 , 1 ] x = u ( t ) A 1 x + ( 1 − u ( t )) A 2 x ˙ has been completely solved by U. Boscain (2002), who gave easily verifiable necessary and sufficient conditions. → methods based on the notion of worst trajectory worst trajectory : forms the smallest angle exponential stability instability with the exiting radial direction P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 5 / 19
General case x ∈ R n , A ( t ) ∈ A x ( t ) = A ( t ) x ( t ) ˙ In general, (i.e. n ≥ 3) notion of worst trajectory not valid anymore (linked to Jordan Separation theorem). Classical to seek a Lyapunov function (e.g. polynomial, etc.). But, P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 6 / 19
General case x ∈ R n , A ( t ) ∈ A x ( t ) = A ( t ) x ( t ) ˙ In general, (i.e. n ≥ 3) notion of worst trajectory not valid anymore (linked to Jordan Separation theorem). Classical to seek a Lyapunov function (e.g. polynomial, etc.). But, Theorem [U. Boscain, Y. Chitour, P. Mason] For each exp. stable linear switched system, ∃ common polynomial Lyapunov function. Degree of this common polynomial Lyapunov function is not uniformly bounded over all exp. stable linear switched system. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 6 / 19
Reduction to ρ ( A ) = 0 Computation of ρ ( A ) VERY HARD in general and even numerically. Up to a translation, A ❀ A − ρ ( A ) Id reduce to case ρ ( A ) = 0. ⇒ Study of the case ρ ( A ) = 0 crucial to understand stability properties. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 7 / 19
Invariant flags Definition A irreducible if ∄ V nontrivial subspace of R n invariant wrt all matrices of A P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 8 / 19
Invariant flags Definition A irreducible if ∄ V nontrivial subspace of R n invariant wrt all matrices of A Maximal invariant flag for A : { 0 } = E 0 � E 1 � · · · � E k − 1 � E k = R n where E i invariant wrt each A ∈ A , ∄ V invariant wrt A such that E i − 1 � V � E i . A 11 A 12 · · · 0 A 22 A 23 · · · Coordinate system 0 0 A 33 A 34 · · · → A = , ∀ A ∈ A adapted to the flag . ... ... ... . . 0 · · · · · · 0 A kk P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 8 / 19
Invariant flags Definition A irreducible if ∄ V nontrivial subspace of R n invariant wrt all matrices of A Maximal invariant flag for A : { 0 } = E 0 � E 1 � · · · � E k − 1 � E k = R n where E i invariant wrt each A ∈ A , ∄ V invariant wrt A such that E i − 1 � V � E i . A 11 A 12 · · · 0 A 22 A 23 · · · Coordinate system 0 0 A 33 A 34 · · · → A = , ∀ A ∈ A adapted to the flag . ... ... ... . . 0 · · · · · · 0 A kk Call A i = { A ii : A ∈ A} . Then A i irreducible and ρ ( A ) = max i ρ ( A i ) P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 8 / 19
Marginal stability / instability Assume from now on ρ ( A ) = 0 → marginal stability or instability Theorem (N. Barabanov) If A irreducible, then ∃ a norm v : R n → [ 0 , + ∞ ) s. t.: v ( x ( t )) ≤ v ( x ( 0 )) for every switching law A ( · ) and initial cond. x ( 0 ) ; ∃ traj. x ( · ) s. t. v ( x ( t )) ≡ v ( x ( 0 )) ∀ t ≥ 0 , ∀ x ( 0 ) . Thus if A irreducible the system is marginally stable! P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 9 / 19
Marginal stability / instability Assume from now on ρ ( A ) = 0 → marginal stability or instability Theorem (N. Barabanov) If A irreducible, then ∃ a norm v : R n → [ 0 , + ∞ ) s. t.: v ( x ( t )) ≤ v ( x ( 0 )) for every switching law A ( · ) and initial cond. x ( 0 ) ; ∃ traj. x ( · ) s. t. v ( x ( t )) ≡ v ( x ( 0 )) ∀ t ≥ 0 , ∀ x ( 0 ) . Thus if A irreducible the system is marginally stable! Otherwise, if { 0 } = E 0 � E 1 � · · · � E k − 1 � E k = R n maximal invariant flag then from the block form and variation of constant we get � x ( t ) � ≤ C ( 1 + t k − 1 ) � x ( 0 ) � In principle the system could be unstable with polynomial growth. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 9 / 19
Resonance Definition Consider a reducible switched system A and denote by A 1 , . . . , A k the subsystems corresponding to a maximal invariant flag. We say that A i 1 , . . . , A i r of A ( i 1 . . . i r distinct) are in resonance if they satisfy: ( a ) ρ ( A i 1 ) = . . . = ρ ( A i r ) = 0 with v i j ( · ) corresponding Barabanov norms; ( b ) ∃ A ( · ) in A with associated switching laws A i j i j ( · ) in A i j and corresp. trajectories γ i j ( · ) of A i j such that v i j ( γ i j ( t )) = const for every t > 0 and for j = 1 , . . . , r . P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 10 / 19
Resonance Definition Consider a reducible switched system A and denote by A 1 , . . . , A k the subsystems corresponding to a maximal invariant flag. We say that A i 1 , . . . , A i r of A ( i 1 . . . i r distinct) are in resonance if they satisfy: ( a ) ρ ( A i 1 ) = . . . = ρ ( A i r ) = 0 with v i j ( · ) corresponding Barabanov norms; ( b ) ∃ A ( · ) in A with associated switching laws A i j i j ( · ) in A i j and corresp. trajectories γ i j ( · ) of A i j such that v i j ( γ i j ( t )) = const for every t > 0 and for j = 1 , . . . , r . Theorem (Y. Chitour, P.M., M. Sigalotti) Let A be convex compact. Assume that the linear switched system associated with A is marginally unstable. Then A is reducible and, for any maximal invariant flag, it admits two subsystems A i j , j = 1 , 2 , in resonance. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 10 / 19
Consequences Simplest nontrivial case of reducible systems: � A 0 � A 1 A 0 � A 1 � A 0 = A 1 = A = conv { A 0 , A 1 } , 11 12 11 12 , . A 0 A 1 0 0 22 22 Assume A 0 , A 1 Hurwitz and ρ ( A ) = 0. P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 11 / 19
Consequences Simplest nontrivial case of reducible systems: � A 0 � A 1 A 0 � A 1 � A 0 = A 1 = A = conv { A 0 , A 1 } , 11 12 11 12 , . A 0 A 1 0 0 22 22 Assume A 0 , A 1 Hurwitz and ρ ( A ) = 0. n = 2 , 3 A marginally unstable ⇒ 0 eigenvalue of A 0 or A 1 ⇒ contradiction P. Mason (CNRS / L2S) Polynomial instability of switched systems October 28th, 2010 11 / 19
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