Commutativity and asymptotic stability for linear switched DAEs Stephan Trenn joint work with Daniel Liberzon (UIUC) and Fabian Wirth (Uni W¨ urzburg) Technomathematics group, University of Kaiserslautern, Germany 50th IEEE Conference on Decision and Control and European Control Conference Orlando, Florida, USA, December 12th, 2011
Introduction Nonswitched DAEs Commutativity and stability Content Introduction 1 Systems class: definition and motivation Examples Nonswitched DAEs 2 Solution theory Consistency projector The matrix A diff Commutativity and stability 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability 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 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Motivation and question Why switched DAEs E σ ˙ x = A σ x ? modeling of electrical circuits with switches 1 DAEs E ˙ x = Ax + Bu with switched feedback controller 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 ( t ) = A ( t ) x ( t ) via 3 piecewise constant DAEs Question ? E p ˙ x = A p x asymp. stable ∀ p ⇒ E σ ˙ x = A σ x asymp. stable Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Commutativity and stability for switched ODEs Theorem (Narendra und Balakrishnan 1994) Consider switched ODE (swODE) x = A σ x ˙ with A p Hurwitz, p ∈ { 1 , 2 , . . . , p } and commuting A p , i.e. [ A p , A q ] := A p A q − A q A p = 0 ∀ p , q ∈ { 1 , 2 , . . . , p } (C) ⇒ (swODE) asymptotically stable ∀ σ . Sketch of proof: Consider switching times t 0 < t 1 < . . . < t k < t and p i := σ ( t i +), then x ( t ) = e A pk ( t − t k ) e A pk − 1 ( t k − t k − 1 ) · · · e A p 1 ( t 2 − t 1 ) e A p 0 ( t 1 − t 0 ) x 0 (C) = e A 1 ∆ t 1 e A 2 ∆ t 2 · · · e A p ∆ t p x 0 and ∆ t p → ∞ for at least one p and t → ∞ . Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Generalization to (swDAE) (swDAE) E σ ˙ x = A σ x Generalization - Questions Which matrices have to commute? What about the jumps? � 0 − 1 � − 1 0 � [ 0 1 �� � [ 0 0 �� Example 1: ( E 1 , A 1 ) = 0 0 ] , ( E 2 , A 2 ) = 1 1 ] , , 1 − 1 0 − 1 [ A 1 , A 2 ] = 0, but instability possible (see next slide) � 0 − 1 � 1 0 � [ 0 1 �� � [ 0 0 �� Example 2: ( E 1 , A 1 ) = 0 0 ] , ( E 2 , A 2 ) = 0 1 ] , , 0 − 1 1 − 1 [ A 1 , A 2 ] � = 0, but stability for all switching signals (see next slide) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Examples: jumps and stability Example 1: Example 2: �� 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 a Lyapunov function for all individuel modes Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Observations Solutions modes have restricted dynamics: consistency spaces switching ⇒ inconsistent initial values inconsistent initial values ⇒ jumps in x Stability common Lyapunov function not sufficient commutativity of A -matrices not sufficient stability depends on jumps Impulses switching ⇒ Dirac impulses in solution x Dirac impulse = infinite peak ⇒ instability Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Content Introduction 1 Systems class: definition and motivation Examples Nonswitched DAEs 2 Solution theory Consistency projector The matrix A diff Commutativity and stability 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Solutions for nonswitched DAE 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 ( SET , SAT ) = x 2 , , 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 . − 1 π 1 1 Consistency space: C ( E , A ) := im V Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Consistency projectors Observation � � ˙ � I � � J � � v � 0 v 0 = 0 N w ˙ 0 I w � v 0 � ∈ R n Consistent initial values: 0 � v 0 � � v 0 � arbitrary initial value R n ∋ Π �→ consistent initial value w 0 0 Definition (Consistency projector for regular ( E , A ) ) �� I 0 � J 0 Let S , T ∈ R n × n invertible with ( SET , SAT ) = � �� : , 0 N 0 I � � 0 I T − 1 Π ( E , A ) = T 0 0 Remark: Π ( E , A ) can be calculated easily and directly from ( E , A ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability The matrix A diff �� I 0 � J 0 � �� Let ( E , A ) be regular with ( SET , SAT ) = , N nilpotent , 0 N 0 I � I � 0 T − 1 consistency projector: Π ( E , A ) = T 0 0 Definition (Differential “projector”) � I � 0 Π diff ( E , A ) = T S 0 0 Theorem (Differential dynamic of DAE) x = Π diff x solves E ˙ x = Ax ⇒ ˙ ( E , A ) Ax � J � 0 A diff := Π diff T − 1 ( E , A ) A = T 0 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Content Introduction 1 Systems class: definition and motivation Examples Nonswitched DAEs 2 Solution theory Consistency projector The matrix A diff Commutativity and stability 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Stability result Consider again switched DAE: E σ ˙ x = A σ x Impulse freeness condition (IFC): ∀ p , q ∈ { 1 , . . . , N } : E p ( I − Π p )Π q = 0 Theorem (T. 2009) (IFC) ⇒ All solutions of E σ ˙ x = A σ x are impulse free Theorem (Main result) (IFC) ∧ ( E p , A p ) asymp. stable ∀ p ∧ [ A diff p , A diff q ] = 0 ∀ p , q ∈ { 1 , 2 , . . . , p } ⇒ (swDAE) asymptotically stable ∀ σ Interesting: no additional condition on jumps! Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
Introduction Nonswitched DAEs Commutativity and stability Sketch of proof From [ A diff p , A diff q ] = 0 ∀ p , q ∈ { 1 , 2 , . . . , p } (C) follows also [Π p , A diff [ A diff p ] = 0 ∧ [Π p , Π q ] = 0 ∧ p , Π q ] = 0 . Consider switching times t 0 < t 1 < . . . < t k < t and p i := σ ( t i +), then A diff x ( t ) = e A diff pk − 1 ( t k − t k − 1 ) Π p k − 1 · · · e A diff p 1 ( t 2 − t 1 ) Π p 1 e A diff pk ( t − t k ) Π p k e p 0 ( t 1 − t 0 ) Π p 0 x 0 (K) = e A diff 1 ∆ t 1 Π 1 e A diff 2 ∆ t 2 Π 2 · · · e A diff p ∆ t p Π p x 0 and ∆ t p → ∞ for at least one p and t → ∞ . Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Commutativity and asymptotic stability for linear switched DAEs
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