Switched behaviors with impulses A unifying framework Stephan Trenn ∗ and Jan C. Willems ∗∗ ∗ Technomathematics group, University of Kaiserslautern, Germany ∗∗ Department of Electrical Engineering, K.U. Leuven, Belgium 51st IEEE Conference on Decision and Control Tuesday, December 11, 2012, 14:20–14:40, Maui, USA
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Contents Introduction 1 Distributional behaviors 2 Main result: Autonomy characterization 3 Conclusions 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions The usual modeling approach with inputs and outputs Usual modeling using inputs and outputs: System x = Ax + Bu ˙ Input u Output y y = Cx + Du Drawbacks of this approach: Separating external signals as inputs and outputs Example: Electrical circuit with “wires sticking out” Is the current or the voltage at the wires an input? Algebraic constraints have to be eliminated Example: First principles modeling of electrical circuit contains Kirchhoff laws as algebraic constraints Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions The behavioral approach Behavioral approach ↔ describe system by set of trajectories: B = { w : R → R q | w fulfills system laws } � d � R � � � � = ( w ) = 0 w d t System Signals w � d � R ( w ) = 0 d t Kernel representation via matrix polynomials Let R ( s ) ∈ R p × q [ s ] be a polynomial with matrix coefficients: R ( s ) = R 0 + R 1 s + R 2 s 2 + . . . R d s d , R 0 , R 1 , . . . , R d ∈ R p × q The associated differential operator is given by � d � � w ( t ) + . . . + R d w ( d ) ( t ) � R ( w ) = t �→ R 0 w ( t ) + R 1 ˙ w ( t ) + R 2 ¨ d t Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Switched systems viewed as time-varying systems Definition (Switched system) System description changes suddenly at certain times = time-varying system with “piecewise-constant” descriptions Time-varying behaviors Instead of R ( s ) ∈ R p × q [ s ] consider R ( s ) ∈ map( R → R p × q )[ s ], i.e. R ( s ) is a polynomial with matrix function coefficients: R ( s ) = R 0 ( · ) + R 1 ( · ) s + R 2 ( · ) s 2 + . . . R d ( · ) s d and the associated differential operator is given by � d w ( t )+ . . . + R d ( t ) w ( d ) ( t )) � R ( w )( t ) = R 0 ( t ) w ( t )+ R 1 ( t ) ˙ w ( t )+ R 2 ( t ) ¨ d t Kernel representation of time-varying behavior still: � d � R � � � � B = w ( w ) = 0 d t Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Global kernel representation Global kernel representation � d Here R d t )( w ) = 0 should hold on the whole time axis R , in particular at the switching times! Major difference to all previous approaches, where differential equations should only hold between the switches and the switching times are treated separately, see e.g. Geerts & Schumacher: “Impulsive-smooth behaviors in multimode systems”, Automatica 1996 Rocha, Willems, Rapisarda & Napp: “On the stability of switched behavioral systems”, last year’s CDC Bonilla & Malabre: “Description of switched systems by implicit representations”, next talk Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Example i L i L + + u v L L v L u L − − constant input: u = 0 ˙ inductivity law: L d d t i L = v L switch dependent: 0 = v L − u 0 = i L Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Example i L i L + + u v L L v L u L − − w = [ u , i L , v L ] ⊤ 1 0 0 0 0 0 1 0 0 0 0 0 ˙ w = 0 0 L 0 w + 0 0 -1 ˙ w = 0 0 L 0 w + 0 0 -1 0 0 0 -1 0 1 0 0 0 0 1 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Example i L i L + + u v L L v L u L − − w = [ u , i L , v L ] ⊤ 1 0 0 0 0 0 ˙ w = 0 switch closed on [0 , 1): 0 L 0 w + 0 0 -1 0 0 0 - ✶ [0 , 1) 1- ✶ [0 , 1) ✶ [0 , 1) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Solution of example u = 0 ˙ ⇔ u constant on whole time axis Inductivity law L d d t i L = v L holds globally (switch independent) v L ( t ) i L ( t ) u t t 1 1 δ t s Switch open on ( −∞ , 0): i L = 0 ⇒ v L = 0 i L ( t ) = u Switch closed on [0 , 1): v L = u ⇒ L t unique jump in w at t = 0 Switch open on (1 , ∞ ): i L = 0 ⇒ v L = 0 unique jump in w at t = 1 and Dirac impulse at t = 1 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Requirements for switched behavior framework Requirements extrapolated from example Solutions exhibit jumps Jumps are uniquely determined (no additional jump map is required) Solutions contain Dirac impulses Dirac impulses are also uniquely determined Jumps and impulses can be handled by distributional solution space, however the definition � d � R � w ∈ D q � � B = d t )( w ) = 0 w ,. . . , w ( d ) with requires multiplication of the distributions w , ˙ piecewise-constant coefficient matrices! Multiplication with non-smooth coefficients A general multiplication of distributions with non-smooth coefficient is not well defined! Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Piecewise-smooth distributions Way out: Consider smaller space of piecewise-smooth distributions Definition (Piecewise smooth distributions D pw C ∞ ) f ∈ C ∞ � pw , � � D pw C ∞ := � T ⊆ R locally finite , f D + D t � i δ ( i ) � ∀ t ∈ T : D t = � n t i =0 a t t ∈ T � t f D D t i − 1 D t i +1 D t i t i − 1 t i t i +1 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Switched behavior well defined Time-varying behavior with piecewise-smooth coefficient matrices: � d � R � w ∈ ( D pw C ∞ ) q � � � B = ( w ) = 0 d t where R ( s ) ∈ ( C ∞ pw ) p × q [ s ] well defined. Fuchssteiner multiplication D pw C ∞ even allows definition of multiplication of two distributions ⇒ we can consider general distributional behaviors: � d � R � w ∈ ( D pw C ∞ ) q � � � B = ( w ) = 0 d t where R ( s ) ∈ ( D pw C ∞ ) p × q [ s ] Dirac impulses in coefficient matrices Why should one need Dirac impulses in the coefficient matrices? Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
Introduction Distributional behaviors Main result: Autonomy characterization Conclusions Impulsive systems Definition (Impulsive system) Let t 0 < t 1 < t 2 < . . . be the impact times. An impulsive system is given by x ( t ) = Ax ( t ) + Bu ( t ) ˙ for t ∈ ( t k , t k +1 ) x ( t k +) = J k x ( t k − ) for k = 0 , 1 , 2 , . . . Theorem For x ∈ ( D pw C ∞ ) n and J ∈ R n × n : x = ( J − I ) δ 0 x ˙ ⇔ x (0+) = Jx (0 − ) and constant otherwise Corollary x solves impulsive ODE ⇔ x solves distributional ODE � ( J k − I ) δ t k ) x + Bu =: A x + Bu with A ∈ ( D pw C ∞ ) n × n x = ( A + ˙ k Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched behaviors with impulses - a unifying framework
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