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Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Workshop on Hybrid Dynamic Systems, 16:20-16:50, Thursday, July 29, 2010, Waterloo, Ontario, Canada

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Contents Introduction 1 Existence & uniqueness of solutions 2 Lyapunov functions for non-switched DAEs 3 Switching & asymptotic stability 4 Summary 5 Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Switched DAEs DAE = Differential algebraic equation Homogeneous switched nonlinear DAE E σ ( t ) ( x ( t ))˙ x ( t ) = f σ ( t ) ( x ( t )) ( swDAE ) or short E σ ( x )˙ x = f σ ( x ) with switching signal σ : R → { 1 , 2 , . . . , N } piecewise constant locally finite jumps subsystems ( E 1 , f 1 ) , . . . , ( E N , f N ) E p : R n → R n × n , f p : R n → R n smooth, p = 1 , . . . , N linear case: E p ∈ R n × n , f p = A p ∈ R n × n , p = 1 , . . . , N Questions Existence and nature of solutions? ? E p ( x )˙ x = f p ( x ) asymp. stable ∀ p ⇒ E σ ( x )˙ x = f σ ( x ) asymp. stable Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Example (linear) Example (linear, i.e. E σ ˙ x = A σ x ): � 0 � � 0 � � 1 � � − 1 � 1 − 1 1 − 1 ( E 1 , A 1 ) : x = ˙ x ( E 2 , A 2 ) : x = ˙ x 0 0 1 − 1 0 0 1 0 non-switched: switched: x 2 x 2 x 1 x 1 More (linear) examples in [Liberzon & T., IEEE Proc. CDC 2009] Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Observations Solutions Subsystems have constrained dynamics: Consistency spaces Switching ⇒ Inconsistent initial values Inconsistent initial values ⇒ Jumps in x Stability Common Lyapunov function not sufficient Overall stability depend on jumps Impulses Linear case: switching ⇒ Dirac impulses in solution x Dirac impulse = infinite peak ⇒ Instability Nonlinear case: f (Dirac impulse)? Undefined. Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Contents Introduction 1 Existence & uniqueness of solutions 2 Lyapunov functions for non-switched DAEs 3 Switching & asymptotic stability 4 Summary 5 Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Assumptions on subsystems Consider non-switched DAE: E ( x )˙ x = f ( x ) Definition (Consistency space) C ( E , f ) := { x 0 ∈ R n | ∃ (classical) solution x with x (0) = x 0 } Time invariance: x solution ⇒ x ( t ) ∈ C ( E , f ) ∀ t Assumptions on non-switched DAE A1 f (0) = 0, hence 0 ∈ C ( E , f ) A2 C ( E , f ) is closed manifold (possibly with boundary) in R n A3 ∀ x 0 ∈ C ( E , f ) ∃ unique solution x : [0 , ∞ ) → R n with x (0) = x 0 and x ∈ C 1 ∩ C ∞ pw Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Linear case Linear case: A1 , A2 trivially fulfilled Lemma (Linear case and A3 ) ( E , A ) fulfills A3 ⇔ matrix pair ( E , A ) is regular, i.e. det( sE − A ) �≡ 0 Theorem (Linear switched case: Existence & Uniqueness, [T. 2009]) E σ ˙ x = A σ x with regular matrix pairs ( E p , A p ) has unique solution for any switching signal and any initial value. Impulses in solution Above solutions might contain impulses! Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Assumption A4 Consider switched nonlinear DAE E σ ( t ) ( x ( t ))˙ x ( t ) = f σ ( t ) ( x ( t )) ( swDAE ) with consistency spaces C p := C ( E p , f p ) Assumption A4 A4 ∀ p , q ∈{ 1 , . . . , N } ∀ x − 0 ∈ C p ∃ unique x + 0 ∈ C q : x + 0 − x − 0 ∈ ker E q ( x + 0 ) Motivation: x + 0 − x − 0 jump at switching time x in direction x + 0 − x − Dirac impulse in ˙ 0 A4 ⇒ no Dirac impulse in product E σ ( x )˙ x A4 ⇒ unique jump with above property Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Existence & Uniqueness of solutions Definition (Solution) x ∈ C ∞ pw is called solution of swDAE : ⇔ E σ ( x ) D ( x D ) ′ = f σ ( x ) D within the space of piecewise-smooth distributions [T. 2009] Theorem (Existence & uniqueness of solutions) (swDAE) + A1 - A4 has unique solution for all (consistent) initial values Remark (Consistency projectors) A4 induces unique map Π q : � p C p → C q such that x ( t +) = Π q ( x ( t − )) for all solutions of (swDAE) with σ ( t +) = q . Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary A4 for the linear case Lemma (Linear consistency projector) Choose invertible S p , T p ∈ R n × n such that �� I 0 � � J p 0 �� ( S p E p T p , S p A p T p ) = , 0 N p 0 I with N p nilpotent, then � � I 0 T − 1 Π p = T p p 0 0 Theorem (Linear equivalent of A4 ) A4 ⇔ ∀ p , q : E q (Π q − I )Π p = 0 Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Contents Introduction 1 Existence & uniqueness of solutions 2 Lyapunov functions for non-switched DAEs 3 Switching & asymptotic stability 4 Summary 5 Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability

Introduction Existence & uniqueness of solutions Lyapunov functions for non-switched DAEs Switching & asymptotic stability Summary Reminder: Lyapunov function for ODEs V : R n → R n is called Lyapunov function for ˙ x = f ( x ) : ⇔ V is positiv definite and radially unbounded ˙ V ( x ) := ∇ V ( x ) f ( x ) < 0 for all x � = 0 ˙ V ( x ) < 0 ⇔ V decreases along solutions No reference to solutions Definition of Lyapunov function does not refer to any solutions. Definition (Lyapunov function for non-switched DAE) V : C ( E , f ) → R n is called Lyapunov function for E ( x )˙ x = f ( x ) : ⇔ L1 V is positive definite and V − 1 [0 , V ( x )] is compact L2 ∃ F : R n × R n → R ≥ 0 ∀ x ∈ C ( E , f ) ∀ z ∈ T x C ( E , f ) : ∇ V ( x ) z = F ( x , E ( x ) z ) L3 ˙ V ( x ) := F ( x , f ( x )) < 0 ∀ x ∈ C ( E , f ) \ { 0 } Stephan Trenn (joint work with Daniel Liberzon) Institute for Mathematics, University of W¨ urzburg, Germany Switched Differential Algebraic Equations: Solution Theory, Lyapunov Functions, and Stability