An averaging result for switched DAEs with multiple modes Stephan - PowerPoint PPT Presentation

An averaging result for switched DAEs with multiple modes Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany joint work with C. Pedicini, F. Vasca, L. Iannelli (Universit` a del Sannio, Benevento) 52nd IEEE Conference on Decision and Control, Florence, Italy Tuesday, 10th December 2013, TuC03, 17:40-18:00

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Contents What is “Averaging”? 1 Switched DAEs 2 Avaraging result for switched DAEs 3 Summary 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Averaging: Basic idea switched non-switched ≈ system average system fast switching Application Fast switches occurs at Pulse width modulation ”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 An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Simple example Example � − 2 � � 1 � 0 0 x = A σ x , ˙ A 1 = , A 2 = , σ : R → { 1 , 2 } periodic 0 1 0 − 2 switching frequency − → ∞ x 2 x 2 x 2 x 1 x 1 x 1 Fixed duty cycle for varying switching frequency (here 45 : 55) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Simple example Example � − 2 � � 1 � 0 0 x = A σ x , ˙ A 1 = , A 2 = , σ : R → { 1 , 2 } periodic 0 1 0 − 2 switching frequency − → ∞ x 2 x 2 x 2 x 1 x 1 x 1 Fixed duty cycle for varying switching frequency (here 55 : 45) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Averaging result for switched linear ODEs Consider switched linear ODE x ( t ) = A σ ( t ) x ( t ) , ˙ x (0) = x 0 with periodic σ : R → { 1 , 2 , . . . , M } and period p > 0 and let d 1 , d 2 , . . . , d M ≥ 0 with d 1 + d 2 + . . . + d M = 1 be the duty cycles of the switched system. Theorem ( Brockett & Wood 1974 ) Let the averaged system be given by x av = A av x av , ˙ x av (0) = x 0 and A av := d 1 A 1 + d 2 A 2 + . . . + d M A M . Then on every compact time interval: � x ( t ) − x av ( t ) � = O ( p ) . Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Content What is “Averaging”? 1 Switched DAEs 2 Avaraging result for switched DAEs 3 Summary 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Switched DAEs Modeling of electrical circuits with switches yields Switched differential-algebraic equations (DAEs) E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) ( swDAE ) Question Does a similar result also hold for switched DAEs? Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary A counterexample Consider E σ ˙ x = A σ x with �� � � �� �� � � �� 0 0 1 − 1 0 0 1 0 ( E 1 , A 1 ) = , ( E 2 , A 2 ) = , , 0 1 0 − 1 0 1 0 − 1 no switching slow switching fast switching Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary System class E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) ( swDAE ) Assumptions σ : [0 , ∞ ) → { 1 , 2 , . . . , M } periodic with period p > 0 W.l.o.g.: σ monotonically increasing on [0 , p ) and d k ∈ (0 , 1) is duty cycle for mode k ∈ { 1 , 2 , . . . , M } matrix pairs ( E k , A k ), k ∈ { 1 , 2 , . . . , M } , regular, i.e. det( sE k − A k ) �≡ 0 σ ( t ) 3 d 3 p 2 d 2 p 1 d 1 p t p 0 2 p Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Non-switched DAEs: Properties Theorem (Quasi-Weierstrass-form, Weierstraß 1868 ) ( E , A ) regular ⇔ ∃ T , S invertible: �� � � �� 0 0 I J ( SET , SAT ) = N nilpotent , , 0 0 N I Definition (Consistency projector) � � 0 I T − 1 Π ( E , A ) := T 0 0 Definition (Differential projector and A diff ) � � I 0 A diff := Π diff Π diff ( E , A ) := T S , ( E , A ) A 0 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Solution characterization of DAEs Theorem (Solution characterization, Tanwani & T. 2010 ) Consider DAE E ˙ x = Ax with regular matrix pair ( E , A ) and corresponding consistency projector Π ( E , A ) and A diff ⇒ x ( t ) = e A diff ( t − t 0 ) Π ( E , A ) x ( t 0 − ) ∈ C t ∈ ( t 0 , ∞ ) . x ( t 0 − ) Π ( E , A ) C x ( t 0 +) x ( t ) Remark: At t 0 the presence of Dirac-impulses is possible! Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Solution behavior for switched DAEs E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) ( swDAE ) with consistency projectors Π k and A diff k Theorem (Impulse freeness, T. 2009 ) All solutions of ( swDAE ) are impulse free, if ∀ k ∈ { 1 , 2 , . . . , M } : E k ( I − Π k )Π k − 1 = 0 , ( IFC ) where Π − 1 := Π M . Corollary All solutions of ( swDAE ) satisfying ( IFC ) are given by x ( t ) = e A diff ( t − t i ) Π i e A diff i − 1 ( t i − t i − 1 ) Π i − 1 · · · e A diff 2 ( t 3 − t 2 ) Π 2 e A diff 1 ( t 2 − t 1 ) Π 1 x ( t 1 − ) i Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Content What is “Averaging”? 1 Switched DAEs 2 Avaraging result for switched DAEs 3 Summary 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Condition on consistency projectors Assumption: commutative projectors ∀ i , j ∈ { 1 , . . . , M } : Π i Π j = Π j Π i ( C ) Lemma ( C ) ⇒ im Π 1 Π 2 · · · Π M = im Π 1 ∩ im Π 2 ∩ . . . ∩ im Π M Remark: im Π 1 ∩ . . . ∩ im Π M = C 1 ∩ . . . ∩ C M and obviously the averaged system, if it exists, can only have solutions within the intersection of the consistency spaces, hence the projector Π ∩ := Π 1 Π 2 · · · Π M plays a crucial role! In the example it was: Π 1 Π 2 = Π 1 � = Π 2 = Π 2 Π 1 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Main result E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) ( swDAE ) ∀ i , j ∈ { 1 , . . . , M } : Π i Π j = Π j Π i ( C ) Theorem (Averaging for switched DAEs) Consider impulse free ( swDAE ) with consistency projectors Π 1 , . . . , Π M satisfying ( C ) and A diff 1 , . . . , A diff M . The averaged system is x av = Π ∩ A diff ˙ av Π ∩ x av , x av (0) = Π ∩ x (0 − ) where Π ∩ = Π 1 Π 2 · · · Π M and A diff av := d 1 A diff + d 2 A diff + . . . + d M A diff M . 1 2 Then ∀ t ∈ (0 , T ] � x ( t ) − x av ( t ) � = O ( p ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes

What is “Averaging”? Switched DAEs Avaraging result for switched DAEs Summary Example Switch independent: 0 = v C 2 − Ri R i R Switch dependent: v C 1 v C 2 C 1 R C 2 open closed C 1 ˙ v C 1 = 0 , C 1 ˙ v C 1 + C 2 ˙ v C 2 = − i R , C 2 ˙ v C 2 = − i R , 0 = v C 1 − v C 2 , x = A σ x with x = ( v C 1 , v C 2 , i R ) ⊤ given by ⇒ switched DAE E σ ˙       0 0 0 0 -1 - R ( E 1 , A 1 ) = 0 0  , 0 0 0 C 1      0 0 0 0 -1 C 2       0 0 0 0 1 - R ( E 2 , A 2 ) = C 1 C 2 0  , 0 0 1      0 0 0 1 -1 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany An averaging result for switched DAEs with multiple modes