Stability of uniformly bounded switched systems and observability - PowerPoint PPT Presentation

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Stability of uniformly bounded switched systems and observability Philippe JOUAN Universit´ e de Rouen, LMRS, CNRS UMR 6085 Joint work with Moussa Balde, Universit´ e de Dakar Workshop on switching dynamics & verification IHP, Paris, France, January 28-29, 2016. Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Table of contents Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Uniformly bounded linear switched systems ◮ Finite collection B 1 , B 2 , . . . , B p of d × d matrices. ◮ They share a weak quadratic Lyapunov function P , i.e. B T i P + PB i ≤ 0 for i = 1 , . . . , p . ◮ We can assume P = Id , so that: B T i + B i ≤ 0 for i = 1 , . . . , p The linear switched system d X ∈ R d , dt X = B u ( t ) X u ( t ) ∈ { 1 , 2 , . . . , p } is stable. Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Switching laws ◮ A switching law, or input, is a piecewise constant and right-continuous function from [0 , + ∞ ) to { 1 , . . . , p } . ◮ For such a switching law u , the trajectory from X is denoted by Φ u ( t ) X . ◮ The ω -limit set, for a given initial point X , is: � Ω u ( X ) = { Φ u ( t ) X ; t ≥ T } T ≥ 0 Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Two loci ◮ K i = { X ∈ R d ; X T ( B T i + B i ) X = 0 } � = � X � ◮ V i = { X ∈ R d ; � � e tB i X � ∀ t ≥ 0 } It is the largest B i -invariant subspace of K i . These loci were previously defined (See Serres, Vivalda, Riedinger, IEEE 2011) Let u be a switching law: ◮ For any X ∈ R d the ω -limit set Ω u ( X ) is contained � p i =1 K i . ◮ For certain classes of inputs (non-chaotic inputs) Ω u ( X ) is contained � p i =1 V i . Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Pairs of Hurwitz matrices The linear switched system ˙ X ∈ R d X = B u ( t ) X is defined by a pair of Hurwitz matrices B 0 , B 1 ∈ M ( d ; R ) assumed to satisfy B T i + B i ≤ 0 i = 0 , 1 . Problem Find (necessary and) sufficient conditions for the switched system to be GUAS. Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Asymptotic stability The switched system being linear is GUAS ( G lobally U niformly A symptotically S table) if and only if for every switching law u the system is globally asymptotically stable, that is ∀ X ∈ R d Φ u ( t ) X − → t → + ∞ 0 . Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Asymptotic stability The switched system being linear is GUAS ( G lobally U niformly A symptotically S table) if and only if for every switching law u the system is globally asymptotically stable, that is ∀ X ∈ R d Φ u ( t ) X − → t → + ∞ 0 . It was proved in [B.J. SIAM 2011] that the switched system is GUAS as soon as � K 1 = { 0 } K = K 0 But this condition is not necessary. It is possible to build GUAS systems for which dim K = d − 1 regardless of d . Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Hurwitz matrices and observability Theorem (Characterization of Hurwitz matrices) B is a d × d-matrix s.t. B T + B ≤ 0 and K = ker( B T + B ) . According to the orthogonal decomposition R d = K ⊕ K ⊥ , B writes � A − C T � B = (1) C D with A T + A = 0 and D T + D < 0 . Then B is Hurwitz if and only if the pair ( C , A ) is observable. Example � 0 1 � Assume B in the previous form, A = and C nonzero. − 1 0 Then B is Hurwitz for any D (that satisfies D T + D < 0). Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Sketch of the proof Consider the linear system: � ˙ x = Ax x ∈ K (Σ) = y ∈ K ⊥ y = Cx If (Σ) is not observable, then there exists x ∈ K , x � = 0, such that Ce tA x = 0 for all t ∈ R . − C T � e tA x � A � � x � � we get e tB Since B = = C D 0 0 This does not tend to 0 and B is not Hurwitz. Conversely if B is not Hurwitz, its limit trajectories lie in K and verifie Ce tA x = 0. Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Convexification For λ ∈ [0 , 1] we consider the matrix B λ = (1 − λ ) B 0 + λ B 1 . Fundamental space K = K 0 ∩ K 1 . Lemma For all λ ∈ (0 , 1), K λ = ker( B T λ + B λ ) = K . According to the orthogonal decomposition R d = K ⊕ K ⊥ , B λ writes − C T � A λ � B λ = λ , C λ D λ with A T λ + A λ = 0 for λ ∈ [0 , 1], and D T λ + D λ < 0 for λ ∈ (0 , 1). Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system The associated bilinear system We consider the bilinear controlled and observed system: � ˙ x = A λ x (Σ) = y = C λ x where λ ∈ [0 , 1], x ∈ K , and y ∈ K ⊥ . Definition The system (Σ) is said to be uniformly observable on [0 , + ∞ [ if for any measurable input t �− → λ ( t ) from [0 , + ∞ [ into [0 , 1] , the output distinguish any two different initial states, that is ∀ x 1 � = x 2 ∈ K m { t ≥ 0; C λ ( t ) x 1 ( t ) � = C λ ( t ) x 2 ( t ) } > 0 , where m stands for the Lebesgue measure on R , and x i ( t ) for the solution of ˙ x = A λ ( t ) x starting from x i , for i = 1 , 2 . Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Main result Theorem The linear switched system � A i − C T � ˙ i X = B u ( t ) X where B i = C i D i is GUAS if and only if the bilinear system � ˙ x = A λ x (Σ) = y = C λ x is uniformly observable on [0 , + ∞ [ . Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Sketch of the proof If the system is not GUAS, we obtain a limit trajectory ψ � t ψ ( t ) = ℓ + B λ ( s ) ψ ( s ) ds 0 where t �− → λ ( t ) is a measurable function from [0 , + ∞ into [0 , 1]. It is obtained using a weak- ∗ limit of t �− → B u ( t ) on some sequence of intervals [ t k , + ∞ [. We show that ψ ( t ) is in K and writes ψ ( t ) = ( φ ( t ) , 0) according to the decomposition R d = K ⊕ K ⊥ . � A λ ( t ) φ ( t ) � Its derivative d dt ψ ( t ) = B ( t ) ψ ( t ) = is also in K , so C λ ( t ) φ ( t ) that C λ ( t ) φ ( t ) = 0 for almost every t ∈ [0 , + ∞ [ . Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Observability of the bilinear system For λ ∈ [0 , 1], x ∈ K , and y ∈ K ⊥ � ˙ x = A λ x = (1 − λ ) A 0 + λ A 1 (Σ) = y = C λ x = (1 − λ ) C 0 + λ C 1 ◮ The trajectories are contained in spheres, because the A i ’s are skew-symmetric. Philippe JOUAN, Universit´ e de Rouen Switching & Observability

Uniformly bounded linear switched systems Stability of pairs of Hurwitz matrices Observability of the bilinear system Observability of the bilinear system For λ ∈ [0 , 1], x ∈ K , and y ∈ K ⊥ � ˙ x = A λ x = (1 − λ ) A 0 + λ A 1 (Σ) = y = C λ x = (1 − λ ) C 0 + λ C 1 ◮ The trajectories are contained in spheres, because the A i ’s are skew-symmetric. ◮ A trajectory x ( t ) on I = [0 , T ] or I = [0 , + ∞ [ that is contained in S k − 1 = { x ∈ K ; � x � = 1 } and satisfies C λ ( t ) x ( t ) = 0 for almost every t ∈ I is a NTZO trajectory ( N on T rivial Z ero O utput) or a bad trajectory. Philippe JOUAN, Universit´ e de Rouen Switching & Observability