The joint spectral radius for semigroups generated by switched - PowerPoint PPT Presentation

The joint spectral radius for semigroups generated by switched differential algebraic equations Stephan Trenn ∗ and Fabian Wirth ∗∗ ∗ Technomathematics group, University of Kaiserslautern, Germany ∗∗ Department for Mathematics, University of W¨ urzburg, Germany SIAM Conference on Applied Linear Algebra Valencia, Spain, 18.06.2012

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Content Introduction 1 Evolution operator and its semigroup 2 Converse Lyapunov theorem and Barabanov norm 3 Conclusions 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Switched DAEs Linear switched DAE (differential algebraic equation) (swDAE) E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) or short E σ ˙ x = A σ x with switching signal σ : R → { 1 , 2 , . . . , P } piecewise constant, right-continuous locally finitely many jumps (no Zeno behavior) matrix pairs ( E 1 , A 1 ) , . . . , ( E P , A P ) E p , A p ∈ R n × n , p = 1 , . . . , P ( E p , A p ) regular, i.e. det( E p s − A p ) �≡ 0 impulse-free solutions (but jumps are allowed!) Question Growth rate and extremal norms for E σ ˙ x = A σ x ∀ σ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Solution formula Theorem ( A diff and Π , Tanwani & T. 2010) Let ( E , A ) be regular and consider E ˙ x = Ax on [0 , ∞ ) ⇒ ∃ unique consistency projector Π and unique flow matrix A diff : x = A diff x x (0) = Π x (0 − ) ˙ on (0 , ∞ ) Furthermore, A diff Π = Π A diff . Corollary (Solution formula for switched DAE) Any solution of the switched DAE E σ ˙ x = A σ x has the form x ( t ) = e A diff k ( t − t k ) Π k e A diff k − 1 ( t k − t k − 1 ) Π k − 1 · · · e A diff 1 ( t 2 − t 1 ) Π 1 e A diff 0 ( t 1 − t 0 ) Π 0 x ( t 0 − ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Switched ODEs with jumps Corollary x solves E σ ˙ x = A σ x on [0 , ∞ ) ⇔ x solves switched ODE with jumps x = A diff ˙ p i x on [ t i , t i +1 ) x ( t i ) = Π p i x ( t i − ) , i ∈ N � where 0 = t 0 , t 1 , . . . , are the switching times of σ and σ [ t i , t i +1 ) ≡ p i � Impulse freeness assumption Above solution characterization only valid when switched DAE produces no Dirac impulses in x . Theorem (Impulse freeness characterization, T. 2009) E σ ˙ x = A σ x has only impulse free solutions ∀ σ ⇔ ∀ p , q ∈ { 1 , . . . , P } : E q ( I − Π q )Π p = 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Evolution operator Consider in the following switched ODE with jumps x = A i x on [ t i , t i +1 ) ˙ x ( t i ) = Π i x ( t i − ) , i ∈ N where 0 = t 0 < t 1 < t 2 < . . . and � A Π = Π A , Π = Π 2 � � ( A i , Π i ) ∈ M ⊆ � ( A , Π) compact Solutions: x ( t ) = e A k ( t − t k ) Π k e A k − 1 ( t k − t k − 1 ) Π k − 1 · · · e A 1 ( t 2 − t 1 ) Π 1 e A 0 ( t 1 − t 0 ) Π 0 x ( t 0 − ) Definition (Set of all evolutions with fixed time span t ≥ 0 ) � k � k � � � e A i τ i Π i � S t := � ( A i , Π i ) ∈ M , τ i = t , τ i > 0 , τ k ≥ 0 � � i =0 i =0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Semi group property Lemma (Semi group) The set � S := S t t > 0 is a semi group with S s + t = S s S t := { Φ s Φ t | Φ s ∈ S s , Φ t ∈ S t } Need commutativity to show “ ⊆ ”: e A τ Π = e A ( τ − τ ′ ) e A τ ′ ΠΠ = e A ( τ − τ ′ ) Π e A τ ′ Π for any ( A , Π) ∈ M and 0 < τ ′ < τ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Exponential growth bound Definition (Exponential growth bound) For t > 0 the exponential growth bound of E σ ˙ x = A σ x is ln � Φ t � λ t ( S t ) := sup ∈ R ∪ {−∞ , ∞} t Φ t ∈S t Definition implies for all solutions x of E σ ˙ x = A σ x : � x ( t ) � = � Φ t x (0 − ) � ≤ � Φ t � � x (0 − ) � ≤ e λ t ( S t ) t � x (0 − ) � Difference to switched ODEs without jumps λ t ( S t ) = ±∞ is possible! All jumps are trivial, i.e. Π p = 0 ⇒ λ t ( S t ) = −∞ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Infinite exponential growth bound Example: �� 0 � � 1 �� �� 0 � � − 1 �� 0 − 1 0 0 ( E 1 , A 1 ) = ( E 2 , A 2 ) = , , 0 1 0 − 1 1 1 0 − 1 � x � � x � x 2 x 1 t t � k � � � 1 1 1 1 For small dwell times: Φ t ≈ (Π 1 Π 2 ) k = = 2 k − 1 1 1 1 1 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Existence of exponential growth rate Theorem (Boundedness of S t ) S t is bounded ⇔ the set of jump projectors is product bounded Reminder: � k � k � � � e A i τ i Π i � S t := � ( A i , Π i ) ∈ M , τ i = ∆ t , τ i > 0 , τ k ≥ 0 � � i =0 i =0 Theorem (Exponential growth rate well defined) Let the jump projectors be product bounded and not all be trivial, then the (upper) Lyapunov exponent � Φ t � λ ( S ) := lim t →∞ λ t ( S t ) = lim t →∞ sup t Φ t ∈S t of the semi-group S is well defined and finite. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Connection to the generalized spectral radius Oberservation: x solves switched ODE ⇔ x ( t + 1) ∈ { Φ x ( t ) | Φ ∈ S 1 } Definition (Generalized spectral radius) For k ∈ N define the discrete growth rate � Φ k Φ k − 1 · · · Φ 1 � 1 / k . ρ k ( S 1 ) := sup Φ i ∈S 1 The generalized spectral radius is ρ ( S 1 ) := lim k →∞ ρ k ( S 1 ) . ln � Φ � Clearly, ln ρ k ( S 1 ) = sup Φ ∈S k = λ k ( S k ) and therefore k λ ( S ) = ln ρ ( S 1 ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Contents Introduction 1 Evolution operator and its semigroup 2 Converse Lyapunov theorem and Barabanov norm 3 Conclusions 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations

Introduction Evolution operator and its semigroup Converse Lyapunov theorem and Barabanov norm Conclusions Converse Lyapunov theorem for switched DAEs Consider again E σ ˙ x = A σ x (swDAE) with corresponding semigroup S t . (swDAE) uniformly exponentially stable ∃ M ≥ 1 , µ > 0 : � x ( t ) � ≤ Me − µ t � x (0 − ) � : ⇔ ∀ t ≥ 0 ⇒ λ ( S ) ≤ − µ < 0. Definition (Lyapunov norm) For ε > 0 define e − ( λ ( S )+ ε ) t � Φ t x � ~ x ~ ε := sup sup t > 0 Φ t ∈S t Theorem (Converse Lyapunov theorem, T. & Wirth 2012) (swDAE) is uniformly exponentially stable ∀ σ ⇒ V = ~ · ~ ε is Lyapunov function for sufficiently small ε > 0 In particular: V (Π x ) ≤ V ( x ) for all projectors Π Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The joint spectral radius for semigroups generated by switched differential algebraic equations