Stability of switched DAEs Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Workshop Architecture Hybride et Contraintes , Paris June 4th 2012, 14:00
Introduction Nonswitched DAEs Distributional solutions Stability results Contents Introduction 1 Systems class: definition and motivation Examples Nonswitched DAEs 2 Solutions: Consistency and underlying ODE Lyapunov functions Distributional solutions for switched DAEs 3 Reminder: classical distribution theory Piecewise-smooth distributions Stability results 4 Impulse freeness Arbitrary switching Slow switching Commutativity Lyapunov exponent and converse Lyapunov theorem Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Switched DAEs Switched linear 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 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 Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Motivation and questions Why switched DAEs E σ ˙ x = A σ x ? modeling of electrical circuits with switches 1 DAEs E ˙ x = Ax + Bu with switched feedback 2 u ( t ) = F σ ( t ) x ( t ) or u ( t ) = F σ ( t ) x ( t ) + G σ ( t ) ˙ x ( t ) approximation of time-varying DAEs E ( t )˙ x = A ( t ) x via 3 piecewise-constant DAEs Question ? E p ˙ x = A p x asymp. stable ∀ p ⇒ E σ ˙ x = A σ x asymp. stable ∀ σ Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Example 1: jumps and stability Example 1a: Example 1b: �� 0 � � 1 �� �� 0 � � 1 �� 0 − 1 0 − 1 ( E 1 , A 1 ) = , ( E 1 , A 1 ) = , 0 1 0 − 1 0 1 0 − 1 �� 0 � � − 1 �� �� 0 � � 1 �� 0 0 0 0 ( E 2 , A 2 ) = , ( E 2 , A 2 ) = , 1 1 0 − 1 0 1 0 − 1 x 2 x 2 x 2 unstable!!! x 1 x 1 x 1 Remark: V ( x ) = x 2 1 + x 2 2 is Lyapunov function for all subsystem Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Example 2: impulses in solutions i L i L + + u L u u L u L L − − u = 0 ˙ constant input: L d inductivity law: d t i L = u L switch dependent: 0 = u L − u 0 = i L Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Example 2: impulses in solutions i L i L + + u L u u L u L L − − x = [ u , i L , u L ] ⊤ 1 0 0 0 0 0 1 0 0 0 0 0 ˙ x ˙ x 0 L 0 x = 0 0 1 0 L 0 x = 0 0 1 0 0 0 − 1 0 1 0 0 0 0 1 0 Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Solution of example L d d t i L = u L , 0 = u L − u or 0 = i L u constant, i L (0) = 0 � 1 , t < t s switch at t s > 0: σ ( t ) = 2 , t ≥ t s u L ( t ) i L ( t ) u t t t s t s δ t s Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Observations Solutions modes have constrained dynamics: consistency spaces switches ⇒ inconsistent initial values inconsistent initial values ⇒ jumps in x Stability common Lyapunov function not sufficient stability depends on jumps Impulses switching ⇒ Dirac impulse in solution x Dirac impulse = infinite peak ⇒ instability Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Contents Introduction 1 Systems class: definition and motivation Examples Nonswitched DAEs 2 Solutions: Consistency and underlying ODE Lyapunov functions Distributional solutions for switched DAEs 3 Reminder: classical distribution theory Piecewise-smooth distributions Stability results 4 Impulse freeness Arbitrary switching Slow switching Commutativity Lyapunov exponent and converse Lyapunov theorem Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Solutions for unswitched DAEs Consider E ˙ x = Ax . �� 0 4 0 � − 4 π − 4 0 � �� Theorem (Weierstrass 1868) ( E , A ) = , − 1 4 π 0 1 0 0 0 0 0 − 1 − 4 4 ( E , A ) regular ⇔ x 3 ∃ S , T ∈ R n × n invertible: �� � � �� I 0 J 0 x 2 ( SET , SAT ) = , , 0 N 0 I N nilpotent, T = [ V , W ] x 1 Corollary (for regular ( E , A ) ) x solves E ˙ x = Ax ⇔ x ( t ) = Ve Jt v 0 � 0 4 � � − 1 − 4 π � V = , J = 1 0 V ∈ R n × n 1 , J ∈ R n 1 × n 1 , v 0 ∈ R n 1 . π − 1 1 1 Consistency space: C ( E , A ) := im V Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Consistency projector Observation � � � � � � � � I 0 v ˙ J 0 v = 0 N w ˙ 0 I w � � v 0 Consistent initial value: , because N ˙ w = w ⇔ w ≡ 0 0 � v 0 � � v 0 � Π arbitrary initial value �→ consistent initial value w 0 0 Definition (Consistency projector for regular ( E , A ) ) �� I 0 � J 0 Let S , T ∈ R n × n be invertible with ( SET , SAT ) = � �� , : 0 N 0 I � � I 0 T − 1 Π ( E , A ) = T 0 0 Remark: Π ( E , A ) can be calculated easily and directly from ( E , A ) (via the Wong sequences) Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Lyapunov functions for regular ( E , A ) Definition (Lyapunov function for E ˙ x = Ax ) ⊤ > 0 on C ( E , A ) and P = P ⊤ > 0 solutions of Q = Q A ⊤ PE + E ⊤ PA = − Q (generalize Lyapunov equation) Lyapunov function V : R n → R ≥ 0 : x �→ ( Ex ) ⊤ PEx V monotonically decreasing along solutions: � � � � ⊤ PE ˙ � � ⊤ PEx d d t V x ( t ) = Ex ( t ) x ( t ) + E ˙ x ( t ) = x ( t ) ⊤ E ⊤ PAx ( t ) + x ( t ) ⊤ A ⊤ PEx ( t ) = − x ( t ) ⊤ Qx ( t ) < 0 Theorem (Owens & Debeljkovic 1985) E ˙ x = Ax asymptotically stable ⇔ ∃ Lyapunov function Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Contents Introduction 1 Systems class: definition and motivation Examples Nonswitched DAEs 2 Solutions: Consistency and underlying ODE Lyapunov functions Distributional solutions for switched DAEs 3 Reminder: classical distribution theory Piecewise-smooth distributions Stability results 4 Impulse freeness Arbitrary switching Slow switching Commutativity Lyapunov exponent and converse Lyapunov theorem Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
Introduction Nonswitched DAEs Distributional solutions Stability results Distribution theory - basics Distributions - overview generalized functions arbitrarily often differentiable Dirac impulse δ 0 is “derivative” of unit jump ✶ [0 , ∞ ) Two different formal approaches functional analytical: dual of the space test functions 1 (L. Schwartz 1950) axiomatic: space of all “derivatives” of continuous functions 2 (J. Sebasti˜ ao e Silva 1954) Stephan Trenn (joint work with Daniel Liberzon) Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Stability of switched DAEs
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