Controllability and observability are not dual for switched DAEs - PowerPoint PPT Presentation

Controllability and observability are not dual for switched DAEs Stephan Trenn joint work with Ferdinand K¨ usters (Fraunhofer ITWM) Technomathematics group, University of Kaiserslautern, Germany SciCADE 2015 , Potsdam Friday, 18.09.2015, 11:00

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Contents A counter example 1 Adjoint systems for switched DAEs 2 Dual systems for switched DAEs 3 Observability, Determinability, Controllability, Reachability 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Naive dual of a switched DAE Switched DAE Dual for switched DAE? E ⊤ p = A ⊤ σ p + C ⊤ E σ ˙ x = A σ x + B σ u σ ˙ σ u d y d = B ⊤ y = C σ x σ p Classical dual [Cobb ’84] Non-switched DAE E ⊤ ˙ p = A ⊤ p + C ⊤ u d E ˙ x = Ax + Bu y d = B ⊤ p y = Cx Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability An example Solution E σ ˙ x = A σ x + B σ u , y = C σ x on ( −∞ , 1): on [1 , 2): on [2 , ∞ ): x 1 ( t ) = x 0 ∀ t ∈ R x 1 = 0 + 0 · u ˙ x 1 = 0 + 0 · u ˙ x 1 = 0 + 0 · u ˙ 1 x 2 ( t ) = ✶ [1 , ∞ ) ( t ) x 0 0 = x 2 0 = x 1 − x 2 x 2 = 0 ˙ 1 y ( t ) = ✶ [2 , ∞ ) ( t ) x 0 y = 0 y = 0 y = x 2 1 ⇒ observable E ⊤ p = A ⊤ σ p + C ⊤ y d = B ⊤ σ ˙ σ u d , σ p Solution p 1 ( t ) = p 0 p 1 = 0 + 0 · u d ˙ p 1 = p 2 + 0 · u d ˙ p 1 = 0 ˙ ∀ t ∈ R 1 � t 0 = p 2 0 = − p 2 p 2 = u d ˙ p 2 ( t ) = ✶ [2 , ∞ ) u d y d = 0 y d = 0 y d = 0 2 ⇒ not controllable Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Some remarks concerning duality Switched DAEs are special time-varying DAEs: E ( t ) ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y = C ( t ) x ( t ) whose dual is not (c.f. Balla & M¨ arz ’02, Kunkel & Mehrmann ’08) E ( t ) ⊤ ˙ p ( t ) = A ( t ) ⊤ p ( t ) + C ( t ) ⊤ u d ( t ) y d = B ( t ) ⊤ x ( t ) For time-varying systems, adjoint system and dual system have to be distinguished, here: dual = time-inverted adjoint Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Contents A counter example 1 Adjoint systems for switched DAEs 2 Dual systems for switched DAEs 3 Observability, Determinability, Controllability, Reachability 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Adjointness for linear ODEs Linear ODE Adjoint of linear ODE p = − A ⊤ p − C ⊤ u a x = Ax + Bu ˙ ˙ y a = B ⊤ p y = Cx Input-State-Output-maps Adjoint maps Input-map: Adjoint of input-map: u ( · ) �→ g ( · ) := Bu ( · ) p ( · ) �→ y a ( · ) := B ⊤ p ( · ) Adjoin of input-state-map: Input-state-map: � � � � � � � � p T , h ( · ) �→ p (0) , p ( · ) x 0 , g ( · ) �→ x ( T ) , x ( · ) p = − A ⊤ p − h , p ( T )= p T p solves ˙ x ( · ) solves ˙ x = Ax + g , x (0)= x 0 Adjoint of state-output-map: State-output-map: u a ( · ) �→ h ( · ) := C ⊤ u a ( · ) x ( · ) �→ y ( · ) := Cx ( · ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Classical adjointness conditions Behavior: B ( A , B , C ) := { ( u , x , y ) | ˙ x = Ax + Bu , y = Cx } Theorem (van der Schaft ’91) ( u a , p , y a ) solves adjoint system ⇔ following adjointness condition holds d t ( p ⊤ x ) − y ⊤ d a u + u ⊤ a y = 0 ∀ ( u , x , y ) ∈ B ( A , B , C ) ( A ) In terms of behaviors: { ( u a , p , y a ) | ( A ) holds } = B ( − A ⊤ , − C ⊤ , B ⊤ ) B ( E ( · ) , A ( · )) := { x | E ( · ) ˙ x = A ( · ) x } Adjointness condition for E ( t ) ˙ x ( t ) = A ( t ) x ( t ) , Balla & M¨ arz ’02 d d t ( p ⊤ E ( · ) x ) = 0 , ∀ x ∈ B ( E ( · ) , A ( · )) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Adjointness for switched DAEs B ( E , A ) := { x | E ˙ x = Ax } B ( A , B , C ):= { ( u , x , y ) | ˙ x = Ax + Bu , y = Cx } Adjointness for ˙ x = Ax + Bu , y = Cx Adj. for E ( · ) ˙ x = A ( · ) x ∀ ( u , x , y ) ∈ B ( A , B , C ) : ∀ x ∈ B ( E ( · ) , A ( · )) : d t ( p ⊤ x ) − y ⊤ d a u + u ⊤ d t ( p ⊤ E ( · ) x ) = 0 , d a y = 0 Adjointness condition for switched DAEs and adjoint behavior With B σ := { ( u , x , y ) | E σ ˙ x = A σ x + B σ u , y = C σ x } let adjointness condition be: d t ( p ⊤ E σ x ) − y ⊤ d a u + u ⊤ a y = 0 ∀ ( u , x , y ) ∈ B σ ( A σ ) Furthermore, a behavior B ⊆ { ( u a , p , y a ) } is called a behavioral adjoint of B σ : ⇔ ( A σ ) holds ∀ ( u , x , y ) ∈ B σ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Behavioral adjoint representation Theorem Consider d t ( p ⊤ E σ ) = − p ⊤ A σ − u ⊤ d a C σ , ( adj ) y ⊤ a = p ⊤ B σ Then B a σ := { ( u a , p , y a ) | ( u a , p , y a ) satisfies ( adj ) } is a behavioral adjoint of B σ . Attention Switched DAE and ( adj ) are equations in a certain distribution space In this space only non-commutative multiplication is defined, in particular p ⊤ A σ � = ( A ⊤ σ p ) ⊤ ( adj ) is not causal Piecewise-constant E σ is differentiated → Dirac impulses occur in coefficient matrices Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Problem: Adjoint is not a switched DAE Fundamental problem d d t ( p ⊤ E σ ) = − p ⊤ A σ − u ⊤ a C σ , ( adj ) y ⊤ a = p ⊤ B σ is not a switched DAE, in particular: Solution theory? Controllability, observability? Time-inversion Problems can be resolved by considering time-inversion and recalling dual = time-inverted adjoint Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Contents A counter example 1 Adjoint systems for switched DAEs 2 Dual systems for switched DAEs 3 Observability, Determinability, Controllability, Reachability 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs

A counter example Adjoint systems for switched DAEs Dual systems for switched DAEs Observability, Determinability, Controllability, Reachability Time-inversion and T -dual Definition (Time inversion for distributions) For T ∈ R let T T : D → D denote the time-inversion at T on the space of distributions D , i.e. for all test functions ϕ ∈ C ∞ and all distributions 0 D ∈ D : T T ( D )( ϕ ) := D ( ϕ ( T − · )) Convention: s = T − t and σ := σ ( T − · ) � Definition ( T -dual of switched DAE) Let B a σ be a behavioral adjoint of switched DAE. The T -dual behavior of the switched DAE is B T -dual := { ( u d , z , y d ) | ( u a , p , y a ) = ( T T ( u d ) , T T ( z ) , T T ( y d )) ∈ B a σ } σ Question Representable as switched DAE? Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Controllability and observability are not dual for switched DAEs