Switched differential algebraic equations: Jumps and impulses Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Research seminar at IIT Delhi, 29/03/2017, 11:30
Introduction Switched DAEs: Solution Theory Observability Summary Contents Introduction 1 Switched DAEs: Solution Theory 2 Definition Review: classical distribution theory Restriction of distributions Piecewise smooth distributions Distributional solutions Impulse-freeness Observability 3 Definition The single switch result Calculation of the four subspaces C − O − and O − + O imp + Summary 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Motivating example t ≥ 0 t < 0 i i + + u ( · ) v u ( · ) v L L − − L d inductivity law: d t i = v 0 = v − u switch dependent: 0 = i Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Motivating example t ≥ 0 t < 0 i i + + u ( · ) v u ( · ) v L L − − x = [ i , v ] ⊤ x = [ i , v ] ⊤ � 0 � L � � 0 � � � L � � 0 � � 0 � 0 1 0 1 x = ˙ x + u x = ˙ x + u 0 0 0 1 − 1 0 0 1 0 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Motivating example t ≥ 0 t < 0 i i + + u ( · ) v u ( · ) v L L − − E 1 ˙ x = A 1 x + B 1 u E 2 ˙ x = A 2 x + B 2 u on ( −∞ , 0) on [0 , ∞ ) → switched differential-algebraic equation Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Solution of circuit example t < 0 t ≥ 0 v = u i = 0 L d v = L d d t i = v d t i Solution (assume constant input u ): v ( t ) i ( t ) u t t 0 0 δ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Observations t ≥ 0 t < 0 i i + + u ( · ) v u ( · ) v L L − − Observations x (0 − ) � = 0 inconsistent for E 2 ˙ x = A 2 x + B 2 u unique jump from x (0 − ) to x (0 + ) derivative of jump = Dirac impulse appears in solution Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Dirac impulse is “real” Dirac impulse Not just a mathematical artifact! Drawing: Harry Winfield Secor, public domain Foto: Ralf Schumacher, CC-BY-SA 3.0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Contents Introduction 1 Switched DAEs: Solution Theory 2 Definition Review: classical distribution theory Restriction of distributions Piecewise smooth distributions Distributional solutions Impulse-freeness Observability 3 Definition The single switch result Calculation of the four subspaces C − O − and O − + O imp + Summary 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Definition Switch → Different DAE models (=modes) depending on time-varying position of switch Definition (Switched DAE) Switching signal σ : R → { 1 , . . . , N } picks mode at each time t ∈ R : E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) (swDAE) y ( t ) = C σ ( t ) x ( t ) + D σ ( t ) u ( t ) Attention Each mode might have different consistency spaces ⇒ inconsistent initial values at each switch ⇒ Dirac impulses, in particular distributional solutions Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Definition Switch → Different DAE models (=modes) depending on time-varying position of switch Definition (Switched DAE) Switching signal σ : R → { 1 , . . . , N } picks mode at each time t ∈ R : E σ ˙ x = A σ x + B σ u (swDAE) y = C σ x + D σ u Attention Each mode might have different consistency spaces ⇒ inconsistent initial values at each switch ⇒ Dirac impulses, in particular distributional solutions Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Distribution theory - basic ideas Distributions - overview Generalized functions Arbitrarily often differentiable Dirac-Impulse δ is “derivative” of Heaviside step function ✶ [0 , ∞ ) Two different formal approaches Functional analytical: Dual space of the space of test functions 1 (L. Schwartz 1950) Axiomatic: Space of all “derivatives” of continuous functions 2 (J. Sebasti˜ ao e Silva 1954) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Distributions - formal Definition (Test functions) C ∞ := { ϕ : R → R | ϕ is smooth with compact support } 0 Definition (Distributions) D := { D : C ∞ → R | D is linear and continuous } 0 Definition (Regular distributions) � f D : C ∞ f ∈ L 1 , loc ( R → R ): → R , ϕ �→ R f ( t ) ϕ ( t )d t ∈ D 0 Definition (Derivative) Dirac Impulse at t 0 ∈ R D ′ ( ϕ ) := − D ( ϕ ′ ) δ t 0 : C ∞ → R , ϕ �→ ϕ ( t 0 ) 0 � ∞ � R ✶ [0 , ∞ ) ϕ ′ = − ϕ ′ = − ( ϕ ( ∞ ) − ϕ (0)) = ϕ (0) ( ✶ [0 , ∞ ) D ) ′ ( ϕ ) = − 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Multiplication with functions Definition (Multiplication with smooth functions) α ∈ C ∞ : ( α D )( ϕ ) := D ( αϕ ) E σ ˙ x = A σ x + B σ u (swDAE) y = C σ x + D σ u Coefficients not smooth ∈ C ∞ Problem: E σ , A σ , C σ / Observation, for σ [ t i , t i +1 ) ≡ p i , i ∈ Z : E σ ˙ x = A σ x + B σ u ∀ i ∈ Z : ( E p i ˙ x ) [ t i , t i +1 ) = ( A p i x + B p i u ) [ t i , t i +1 ) ⇔ y = C σ x + D σ u y [ t i , t i +1 ) = ( C p i x + D p i u ) [ t i , t i +1 ) New question: Restriction of distributions Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Desired properties of distributional restriction Distributional restriction: { M ⊆ R | M interval } × D → D , ( M , D ) �→ D M and for each interval M ⊆ R D �→ D M is a projection (linear and idempotent) 1 ∀ f ∈ L 1 , loc : ( f D ) M = ( f M ) D 2 � � supp ϕ ⊆ M ⇒ D M ( ϕ ) = D ( ϕ ) ∀ ϕ ∈ C ∞ : 3 0 supp ϕ ∩ M = ∅ ⇒ D M ( ϕ ) = 0 ( M i ) i ∈ N pairwise disjoint, M = � i ∈ N M i : 4 � D M = D M i , D M 1 ˙ ∪ M 2 = D M 1 + D M 2 , ( D M 1 ) M 2 = 0 i ∈ N Theorem ([T. 2009]) Such a distributional restriction does not exist. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Proof of non-existence of restriction Consider the following (well defined!) distribution: d i := ( − 1) i � D := d i δ d i , i + 1 i ∈ N ϕ -1 / 2 -1 / 4 0 1 / 3 1 Restriction should give � D [0 , ∞ ) = d 2 k δ d 2 k k ∈ N Choose ϕ ∈ C ∞ such that ϕ [0 , 1] ≡ 1: 0 1 � � D [0 , ∞ ) ( ϕ ) = d 2 k = 2 k + 1 = ∞ k ∈ N k ∈ N Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
Introduction Switched DAEs: Solution Theory Observability Summary Dilemma Switched DAEs Distributions Examples: distributional solutions Distributional restriction not possible Multiplication with non-smooth Multiplication with non-smooth coefficients coefficients not possible Or: Restriction on intervals Initial value problems cannot be formulated Underlying problem Space of distributions too big. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations: Jumps and impulses
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