Averaging for non-homogeneous switched DAEs Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany joint work with E. Mostacciuolo, F. Vasca (Universit` a del Sannio, Benevento, Italy) 54th IEEE Conference on Decision and Control, Osaka, Japan Wednesday, 16th December 2015, WeB10.4, 14:30-14:50
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Contents What is “Averaging”? 1 Explicit solution formulas for switched DAEs 2 Averaging result 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Averaging: Basic idea switched non-switched ≈ σ system averaged system fast switching Application Fast switches occurs at Modulations (pulse width, amplitude, frequency) ”Sliding mode“-control In general: fast digital controller Simplified analyses Stability for sufficiently fast switching In general: (approximate) desired behavior via suitable switching Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Periodic switching signal Switching signal σ : R → { 1 , 2 , . . . , M } has the following properties piecewise-constant and periodic with period p > 0 duty cycles d 1 , d 2 , . . . , d M ∈ [0 , 1] with d 1 + d 2 + . . . + d M = 1 switched non-switched σ, p ≈ system averaged system x σ, p x av fast switching Desired approximation result On any compact time interval it holds that � x σ, p − x av � ∞ = O ( p ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Known results x = A σ x + B σ u , ˙ x (0) = x 0 x (0 − ) = x 0 E σ ˙ x = A σ x , with averaged system with average system x av = Π ∩ A diff x av = A av x av + B av u , ˙ x av (0) = x 0 x av (0 − ) = Π ∩ x 0 ˙ av Π ∩ x av , where A av = � M i =1 d i A i and B av = � M av = � M i =1 d i B i . where A diff i =1 d i A diff . i No further conditions required! Not always working! Additional assumptions needed on so called consistency projectors. References References Homogeneous case: Brocket & Wood 1974 Two modes: Inhomogenous case: Iannelli, Pedicini, T. & Vasca 2013 ECC Ezzine & Haddad 1989 Arbitrarily many modes: Numerous generalizations ... Iannelli, Pedicini, T. & Vasca 2013 CDC Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Switched DAEs with inhomogenity E σ ˙ x = A σ x + B σ u Trivial Counter Example Canonical question ( E 1 , A 1 , B 1 ) = (0 , 1 , 1) ? ( E 2 , A 2 , B 2 ) = (0 , 1 , − 1) Averaging for B σ = 0 ⇒ Averaging for B σ � = 0 Solution of example with duty cycles d 1 = d 2 = 0 . 5: x u t − u Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Contents What is “Averaging”? 1 Explicit solution formulas for switched DAEs 2 Averaging result 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Non-switched DAEs: Basic definitions Theorem (Quasi-Weierstrass form, Weierstraß 1868 ) ( E , A ) regular : ⇔ det( sE − A ) �≡ 0 ⇔ ∃ S , T invertible: �� � � �� I 0 J 0 ( SET , SAT ) = , , N nilpotent 0 0 N I Can easily obtained via Wong sequences ( Berger, Ilchmann & T. 2012) Definition (Consistency projector) � � I 0 T − 1 Π := T 0 0 Definition (Differential and impulse projector) � � � � I 0 0 0 Π diff := T Π imp := T S , S 0 0 0 I Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Explicit solution formula for DAEs For E ˙ x = Ax + Bu with regular ( E , A ) let A diff := Π diff A , B diff := Π diff B , E imp := Π imp E , B imp := Π imp B . Theorem (Explicit DAE solution formula, T. 2012) Every solution x of E ˙ x = Ax + Bu with regular ( E , A ) is given by � t n − 1 x ( t ) = e A diff t Π x − e A diff ( t − s ) B diff u ( s ) d s − � ( E imp ) ℓ B imp u ( ℓ ) ( t ) , x − 0 ∈ R n 0 + 0 ℓ =0 Corollary ( B imp = 0 case) If B imp = 0 , then x solves E ˙ x = Ax + Bu if, and only if, x solves x = A diff x + B diff u , x (0) = Π x − x − 0 ∈ R n ˙ 0 , Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Solution behavior of switched DAEs Consider the switched DAE E σ ˙ x = A σ x + B σ u with regular matrix pairs ( E i , A i ). Distributional solutions Existence and uniqueness of solutions is guaranteed, however only within a distributional solution framework in particular, Dirac impulses may occur in x Here we are only interested in the impulse-free part x − x [ · ] of the (distributional) solution x . The effects of Dirac impulses for averaging are discussed this evening 17:40 here. Theorem (Switched DAEs and switched ODEs with jumps) Assume B imp = 0 . Then x solves switched DAE ⇔ x − x [ · ] solves i x ( t ) = A diff σ ( t ) x ( t ) + B diff ˙ σ ( t ) u ( t ) , ∀ t / ∈ { t k | t k is k-th switching time of σ } x ( t + k ) x ( t − k ) = Π σ ( t + k ) , k = 0 , 1 , 2 , . . . , i.e. x solves switched DAE ⇔ x − x [ · ] solves switched ODE with jumps Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Contents What is “Averaging”? 1 Explicit solution formulas for switched DAEs 2 Averaging result 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Known averaging result Theorem (Homogeneous case, Ianelli, Pedicini, T. & Vasca 2013 ) Consider homogeneous switched DAE E σ ˙ x = A σ x with regular matrix pairs ( E i , A i ) . If Π i Π j = Π j Π i then the averaged system is given by x av = Π ∩ A diff x av (0) = Π ∩ x − ˙ av Π ∩ x av , 0 where A diff av := d 1 A diff + d 2 A diff + · · · + d M A diff Π ∩ = Π M Π M − 1 · · · Π 1 , M , 1 2 i.e. on every compact interval contained in (0 , ∞ ) we have � x σ, p − x av � ∞ = O ( p ) . Condition on consistency projector can be relaxed ( Mostacciuolo, T. & Vasca 2016 ) to the assumption that ∀ i ∈ { 1 , 2 , . . . , M } im Π ∩ ⊆ im Π i , ker Π ∩ ⊇ ker Π i Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Main result We have seen that B imp = 0 is necessary for the relationship i x solves switched DAE ⇔ x − x [ · ] solves switched ODE with jumps It is also sufficient to ensure averaging: Theorem (Averaging for inhomogeneous switched DAEs) Consider switched DAE E σ ˙ x = A σ x + B σ u with regular ( E i , A i ) , p-periodic switching signal σ and Lipschitz continuous u. Assume furthermore B imp = 0 ∀ i ∈ { 1 , . . . , M } , i Π i Π j = Π j Π i ∀ i , j ∈ { 1 , . . . , M } . Then the average system is given by x av = Π ∩ A diff av Π ∩ x av + Π ∩ B diff x av (0) = Π ∩ x − ˙ av u , 0 where Π ∩ = Π M Π M − 1 · · · Π 1 , A diff av := d 1 A diff 1 + . . . + d M A diff M and B diff av := d 1 B diff + . . . + d M B diff M , 1 i.e. on every compact set contained in (0 , ∞ ) we have � x σ, p − x av � ∞ = O ( p ) . Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
What is “Averaging”? Explicit solution formulas for switched DAEs Averaging result Illustrative example S 1 S 2 R 1 L i L v C 1 v C 2 u C 1 C 2 R 2 With x = ( v C 1 , v C 2 , i L ) ⊤ we have the following four DAE descriptions: S 1 closed S 1 closed S 1 open S 1 open S 2 open S 2 closed S 2 closed S 2 open � C 1 0 0 � C 1 C 2 0 � C 1 C 2 0 � C 1 0 0 � � � � E 1 = E 2 = E 3 = E 4 = 0 C 2 0 0 0 L 0 0 0 0 C 2 0 0 0 0 0 0 0 0 0 L 0 0 0 � 0 � 0 � 0 0 1 � − 1 � � 0 − 1 � 0 0 � 1 R 2 0 R 2 0 − 1 A 1 = 0 A 2 = A 3 = A 4 = 0 − 1 R 2 0 R 2 − 1 0 − R 1 1 − 1 0 1 0 − R 1 1 − 1 0 0 0 1 0 0 1 � 0 � 0 � 0 � 0 � � � � B 1 = , B 2 = , B 3 = , B 4 = . 0 1 0 0 1 0 0 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Averaging for non-homogeneous switched DAEs
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