Controllability of switched differential-algebraic equations

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1 Controllability of switched differential-algebraic equations Ferdinand Küsters Markus G-M Ruppert Stephan Trenn Fachbereich Mathematik Technische Universität Kaiserslautern Postfach Kaiserslautern Germany Abstract We study controllability of switched differential algebraic equations We are able to establish a controllability characterization where we assume that the switching signal is known The characterization takes into account possible jumps induced by the switches It turns out that controllability not only depends on the actual switching sequence but also on the duration between the switching times Keywords: switched systems differential-algebraic systems; descriptor systems; controllability; Introduction In this paper we study controllability of switched differential-algebraic equations (switched DAEs) of the form E σ ẋ = A σ x + B σ u () where σ : R 2P} P N is a piecewiseconstant switching signal (in particular we assume that in each compact interval only finitely many switches occur) This class of systems plays an important role when modeling systems containing algebraic constraints together with sudden structural changes (eg electrical circuits) see 5 We assume that σ is given (and known) and not available for control hence here we view () as a linear time-varying dynamical system and adapt the notion of controllability in the behavioral sense Controllability of time-varying linear DAEs has been studied for example in but all these approaches assume smooth coefficient matrices and are therefore not applicable to study switched DAEs Controllability of switched ordinary differential equations (ODEs) is studied extensively in the literature however most references view the switching signal as a control input see eg and the references therein There are surprisingly few references concerned with the notion of controllability for a given switching signal; the apparently only exception is 5 which focuses on periodic switching signals Hence a controllability characterization for an external switching signal will go beyond the address: trenn@mathematikuni-klde (Stephan Trenn) state-of-the-art even for switched ODEs Some preliminary results concerning the controllability of switched DAEs are presented in the conference contribution 9; unfortunately this publications contain some errors (in particular the equivalence claimed in 9 Prop 3 is not correct cf Remark 36) It should also be noted that presents some duality between controllability and observability for switched linear ODEs (were the switching signal is seen as an input) and in the context of smoothly time-varying DAEs duality of observability and controllability was studied in 4 It is not clear whether any reasonable duality notion also holds for linear switched DAEs (); in particular the connection between the controllability characterization presented here and the already known observability characterization for switched DAEs 2 is a topic of future research Our main contribution is the controllability characterization for switched DAEs given in Theorem 3 which heavily builds on the controllability characterization for the single-switch case in Theorem 35 Mathematical preliminaries like distributional solutions and controllability for non-switched DAEs are revisited in Section 2 We use the following notation: The real and natural numbers are denoted by R and N respectively The indicator function for some interval I R is denoted by I ie I (t) = if t I and zero otherwise The restriction of a function f : R R to some interval ab) R is f ab) := ab) f ; this restriction is also well defined for piecewise-smooth distributions (see Section 23) The space AB R n for matrices A R n n B R n m denotes the larges A-invariant subspace which Preprint submitted to Elsevier September 9 24

2 contains imb ie AB = imb AB A 2 B A n B 2 Mathematical preliminaries 2 Regular matrix pairs Here we restrict our attention to the non-switched DAE Eẋ = Ax + Bu (2) for matrices EA R n n and B R n m We call (xu) a (classical) solution of (2) if x is continuously differentiable and (x u) satisfy (2) Note that it is possible to define weaker notions of solutions; however we will consider distributional solutions later on anyway which encompasses all other possible solution concepts For existence and uniqueness of solutions regularity of the matrix pair (EA) is important: Definition 2 (Regularity) A matrix pair (E A) R n n R n n is called regular if and only if det(se A) is not the zero polynomial The following important well known characterizations for regularity hold (see eg 5 Thm 632) Proposition 22 (Regularity characterizations) The following statements are equivalent: (i) (EA) is regular (ii) There exist invertible ST R n n such that (EA) is transformed into a quasi-weierstrass form (QWF): ( ) I J (SETSAT ) = (3) N I where N is a nilpotent matrix (iii) The DAE Eẋ = Ax + Bu has a solution for all sufficiently smooth inputs u and each corresponding solution is uniquely determined by the initial value x() The transformation matrices S and T for obtaining the QWF (3) can easily be calculated via the Wongsequences 7: V := R n V i+ := A (EV i ) i = 2 W := } W i+ := E (AW i ) i = 2 (4) where MS := Mx x S } and M S := x Mx S } denote the image and preimage respectively of S under the matrix M It is easily 2 seen that these sequences of subspaces converge after finitely many steps; denote these limits as V(EA) := V i and W(EA) := W i (5) i N i N Choosing full rank matrices V and W such that imv = V(EA) and imw = W (EA) the QWF can be obtained via the transformation matrices : T = VW and S = EVAW The following projectors play an important role in the analysis of DAEs and were first introduced in Definition 23 (Consistency differential impulse projectors) Consider the regular matrix pair (E A) with QWF (3) The consistency projector is I Π (EA) := T T The differential projector is Π diff I (EA) := T S The impulse projector is Π imp (EA) := T S I In all three cases the block structure corresponds to the block structure of the QWF Furthermore let A diff := Π diff (EA) A B diff := Π diff (EA) B Eimp := Π imp (EA) E Bimp := Π imp (EA) B Note that only the consistency projector is a projector in the usual sense; the differential and impulse projectors are not idempotent Furthermore the definitions do not depend on the specific choice of the transformation matrices S and T An important feature for DAEs is the so called consistency space defined as follows: Definition 24 (Consistency space) Consider the DAE (2) then the consistency space is } x R n classical solution x of Eẋ = Ax with x() = x

3 and the augmented consistency space is } x R n classical solution (xu) of Eẋ = Ax + Bu with x() = x Proposition 25 (3) Consider the DAE (2) then the consistency space is V(EA) and the augmented consistency space is V (EAB) := V(EA) Eimp B imp 22 Controllability of nonswitched DAEs For DAEs (2) there are many different notions of controllability see eg the survey 2 here we will only consider controllability in the behavioral sense (or equivalently R-controllability): Definition 26 (Controllability) The DAE (2) is called controllable if and only if for all solutions (x u ) and (x 2 u 2 ) of (2) there exists a solution (x 2 u 2 ) and T > such that (x 2 u 2 ) ( ) = (x u ) ( ) (x 2 u 2 ) (T ) = (x 2 u 2 ) (T ) The controllability subspace is } T > (xu) solution of (2): C (EAB) := x R x() = x x(t ) = It is well known (see eg 2 Lem 23) that the controllability space is equal to the reachability space ie } T > (xu) solution of (2): C (EAB) = x R x() = x(t ) = x The following controllability characterizations are also well known (for the specific formulation used here see 3 Cor 45) Proposition 27 Consider the regular DAE (2) and the notation from (5) and Definition 23 (i) (2) is controllable if and only if A diff B diff = V (EA) (ii) The controllability space of (2) is given by C (EAB) = A diff B diff E imp B imp In particular the DAE is controllable if and only if the controllability space equals the augmented consistency space ie C (EAB) = V (EAB) 3 For the analysis of controllability the following Kalman controllability decomposition (KCD) is useful Proposition 28 (KCD 3 Cor 46) Consider a regular DAE (2) with the notation from (5) and Definition 23 Choose full column rank matrices V V 2 W W 2 as follows: imv = A diff B diff imw = E imp B imp V(EA) imv 2 W(EA) imw 2 Then T = V W V 2 W 2 and S = EV AW EV 2 AW 2 are invertible and transform the DAE into a KCD: ( ) E E (SETSATSB) = 2 A A 2 B E 22 A 22 where (E A B ) = ( I J N I BJ B N ) with nilpotent N and full controllability space ie C (E A B ) = R rankv +rankw 23 Distributional solutions The switched DAE () usually will not have classical solutions because each mode of the switched DAE given by the DAE E i ẋ = A i x + B i u might have different (augmented) consistency spaces which enforce jumps in the state-variable x We therefore utilize the piecewisesmooth distributional solution framework as introduced in 3 ie x and u are vectors of piecewise-smooth distributions given by f C pw T R is D pwc := D = f D + D t discret t T : D t T t spanδ t δ t δ t } where Cpw denotes the space of piecewise-smooth functions f D denotes the regular distribution induced by f and δ t denotes the Dirac impulse with support t} For a piecewise smooth distribution D = f D + t T D t D pwc three types of evaluation at time t are defined: left sided evaluation D(t ) := f (t ) right sided evaluation D(t + ) := f (t + ) and the impulsive part Dt := D t if t T and Dt = otherwise It can be shown (see eg 4) that the space D pwc can be equipped with a multiplication in particular the multiplication of a piecewise-constant function with a piecewise-smooth distribution is well defined and the switched DAE ()

4 can be interpreted as an equation within the space of piecewise-smooth distribution Hence the following solution behavior (depending on σ) is well defined: B σ := (xu) D n+m pwc Eσ ẋ = A σ x + B σ u } and restrictions of x and u to intervals are well defined as well In this solution framework existence and uniqueness of solutions for () can be established: Proposition 29 (5 Cor 652) Consider the switched DAE () with regular matrix pairs (E p A p ) for all p = 2P and assume that the piecewise constant σ is constant on ( ) Then for all inputs u D m pwc there exists a solution x D n pwc of () and each solution is uniquely determined by x( ) In the following we will call () a regular switched DAE if the assumptions of Proposition 29 are satisfied Note that in particular the jumps and Dirac impulses induced by switches are uniquely determined in this solution framework (in contrast to other approaches where for example additional glueing conditions are imposed see eg 8) Furthermore a side effect of the considered distributional solution framework is the possibility to consider impulsive controls ie we allow for Dirac impulses in u It is a well known fact that for classical ODE systems any controllable states can be controlled to each other in arbitrarily short time; the following result shows that it is even possible to control two states instantaneously to each other when allowing impulsive controls: Lemma 2 (Instantaneous control) Consider the (nonswitched) DAE (2) with corresponding controllability space C (EAB) Then for all x x C (EAB) there exists a (distributional) solution (x u) of (2) with x( ) = x and x( + ) = x Proof Due to linearity it suffices to show that any controllable state x C (EAB) is reachable ie there exists an input u D m pwc such that the corresponding solution x D n pwc satisfies x( ) = and x( + ) = x Without restriction we can assume that the DAE is already in KCD as in Proposition 28 hence it remains to show that for any v R n w R n 2 where n := dim A diff B diff and n 2 := dim E imp B imp we can choose u such that the solutions of v = Jv + B J u v( ) = Nẇ = w + B N u w( ) = satisfy v( + ) = v w( + ) = w First note that (see eg 4 5) w( + n 2 ) = i= N i B N u (i) ( + ) Due the properties of the KCD dim NB N = n 2 we can find u u u n 2 R m such that n 2 i= n 2 i= N i B N u i = w Set u on (ε) (for some ε > ) equal to the polynomial t i i! ui then we achieve w( + ) = w as required To construct u such that v( + ) = v we make the following ansatz for the impulsive parts of u and v at t = : n u = i= u i δ (i) n and v = i= v i δ (i) These impulsive parts must obey the differential equation v = Jv + B J u which can be rewritten as Hence v( + ) = Jv + B J u v = Jv + B J u v n 2 = Jv n + B J u n v n = v( + n ) = i= J i B J u i Due to the KCD we have dim JB J = n and we can always achieve v( + ) = v by choosing u accordingly (which can be done independently from the choice of u on the interval (ε)) Remark 2 The proof of Lemma 2 also shows that the controllability space C (EAB) does not change when considering the DAE Eẋ = Ax + Bu within the piecewise-smooth distributional solution framework 3 Controllability of switched DAEs 3 Controllability definition Definition 3 (Controllability) The switched DAE () is called controllable if and only if the corresponding

5 solution behavior B σ is controllable in the behavioral sense on some interval (T ) ie (x u )(x 2 u 2 ) B σ (x 2 u 2 ) B σ : (x 2 u 2 ) ( ) = (x u ) ( ) (x 2 u 2 ) (T ) = (x 2 u 2 ) (T ) For s < t the (st)-controllability space is } C σ (st) := x R n (xu) B σ : x(s + ) = x x(t ) = Remark 32 If a regular switched DAE () is considered the controllability definition can be reformulated as follows: For all consistent x (ie x V (EAB)) there exists an input u defined on some interval T ) such that the solution x with initial condition x( ) = x satisfies x(t ) = Furthermore it is easy to see that u can always be chosen such that u is identically zero on ε) for some sufficiently small ε > We first like to highlight the relationship between the two controllability spaces defined so far: Lemma 33 Consider the regular switched DAE () and assume that for s < t the switching signal σ is constant on (st) Let p := σ(s + ) then C (st) σ = C (Ep A p B p ) Proof It is clear that C σ (st) C (Ep A p B p ) because the same input u defined on (st) can be used (cf Remark 2) To show the converse we have to show that any x C (Ep A p B p ) can actually be achieved by the switched system ie there exists (xu) B σ with x(s + ) = x This was already shown in Lemma 2 with the help of an impulsive input defined on ss + ε) for some arbitrarily small ε > Due to the invariance property of C (Ep A p B p ) it follows that x((s + ε) + ) C (Ep A p B p ) and we can now choose u on (s+εt) such that x(t ) = This shows x C (st) σ 32 The single switch case From a practical point of view we have to assume that the control action starts at a certain point in time Without restriction we therefore assume that the input u can only be chosen by us on the interval ) This does not exclude that the input u is non-zero on ( ) we just can t influence it in the past (in fact this is already implied by Definition 3) 5 We will now consider the simplest nontrivial switched system where the switching signal has only one switch at the time t s : σ ts} t < t s (t) := (6) t t s The controllability properties are very different for the two cases t s = and t s > because if t s = then the input cannot take advantage of the controllability properties of the mode before the switch In fact any controllable switched system with switching time t s = will also be controllable with switching time t s > but not vice versa! Example 34 Consider a switched DAE () with switching signal (6) and (E A B ) = ( ) (E A B ) = ( ) For given input u the solution of the switched DAE is given by ( ) ( ) x x x = = ( ts ) u ts ) x 2 u ( ts ) u(t s ) ts ) for some x R If t s = we cannot choose u(t s ) If u( ) this means that x 2 (t) for all t > ie the switched system is not controllable If however t s > then we are free to choose u(t s ) = as well as u ts ) = hence x ts ) = which shows that now the switched system is controllable For convenience let C p := C (Ep A p B p ) and analogously for other matrices and spaces indexed by the corresponding matrix pair or triple We are now ready to formulate our first main result: Theorem 35 (Characterization of controllability: Single switch case) Consider the regular switched DAE () with switching signal (6) If t s = then () is controllable if and only if Π C V If t s > then () is controllable if and only if C + Π C V (7) Furthermore for s < t s < t ( ) C (st) = C σ ts} + e Adiff (t s s) Π C V

6 Proof Case t s = Assume the switched DAE is controllable then for all consistent x( ) = x V there exists an input u defined on T ) such that x(t ) = This implies x( + ) C Without restriction we can assume that u = and u (i) ( + ) = for all i N (cf Remark 32) Then (see eg 5 Thm 65) x( + ) = Π x( ) and therefore x = x( ) Π C which shows V Π C To show the converse let x( ) = x V and choose u equal to zero on ε) for some ε > Then x( + ) = Π x Π V Π Π C C and consequently x(ε ) C hence there exists u defined on εt ) such that x(t ) = Case t s > Assume the switched DAE is controllable then for all consistent x( ) = x V there exists an input u such that x(t ) = If T t s this implies already V C and we are done hence assume now T > t s Then x(t s + ) C Without restriction we assume that the input u is identically zero on t s t s +ε) for some ε > and it follows that x(t s ) Π x(t+ s )} Π C Because of linearity we have x(t s ) = x(t s ) + x(t s ) where x is the solution of the switched DAE with zero input and initial value x( ) = x and x is the solution of the switched DAE with zero initial condition and the input from above This implies (invoking Lem 3) e Adiff t s x = x(t s ) = x(t s ) x(t s ) Π C + C (8) ie e Adiff t s V Π C +C Since V is e Adiff invariant (this easily follows from the fact that A diff p Π imp p = ) and e Adiff t s is invertible we have e Adiff t s V = V which shows necessity To show the converse let x V and denote by x the solution of the switched DAE with zero input and initial value x Then x(ts ) V and we can choose x C such that x(t s ) x Π C (9) Furthermore we can choose an input u defined on t s ) such that the solution x of the corresponding switched DAE with zero initial value satisfies x(t s ) = x By linearity the solution x of the switched DAE with initial 6 value x and the above input u satisfies x(t s ) = x(t s ) + x(t s ) (9) Π C Choosing u on t s t s +ε) identically zero for some ε > we have x(t + s ) = Π x(t s ) C and we can finally choose u on t s + εt ) for some T > t s + ε such that x(t ) = which shows that the switched DAE is controllable Formula for C (st) σ ts} Let x C (st) σ ts} then x V and we find an input u defined on s t) such that the corresponding solution x of () satisfies x(t ) = With the same arguments as in the first part of the t s > -case we can conclude that e Adiff (t s s) x C + Π C and therefore ( ) x C + e Adiff (t s s) Π C V To show the converse subspace inclusion let x (C + e Adiff (t s s) Π C ) V Denote by x the solution of the switched DAE with zero input and initial value x then x(t s ) ( C + Π C ) V Similar as in the second part of the t s > -case we can define an input u on t s t) such that x(t ) = for some T > t s which concludes the proof Remark 36 We can write (7) also as A diff Bdiff Eimp B imp + Π Adiff Bdiff V E imp B imp where Π C = Π A diff Bdiff follows from the facts that A diff Bdiff imπ and imπ E imp B imp = } A further simplification can be done by substracting E imp B imp on both sides resulting in the sufficient condition A diff Bdiff + Π Adiff Bdiff V () A necessary condition can be obtained from (7) by adding any complementing space of E imp B imp within resulting in the condition W W A diff Bdiff + Π Adiff Bdiff = Rn () Despite their simple transformations the three condi-

7 tions (7) () and () are not equivalent in fact only the following holds () (7) () and we will show with the following two examples that the converse implications are not true Example 37 ((7) ()) Consider the switched DAE () with switching signal (6) for t s > and (E A B ) = ( ) (E A B ) = ( ) Since the matrices are already in QWF the following can simply be read off: V = im A diff = B diff = E imp = B imp = Π = I A diff = B diff = Hence C + Π C = im + im = R 2 ie (7) holds and the switched system is controllable However () is not satisfied: A diff Bdiff + Π Adiff Bdiff = } + im im = V Example 38 (() (7)) Consider the switched DAE () with switching signal (6) for t s > and ( ) (E A B ) = ) (E A B ) = ( Since the matrices are already in QWF we can easily read of the following: V = im W = im A diff = B diff = E imp = B imp = Π = I A diff = B diff = Hence C + Π C = im + im im + im = V ie the switched system is not controllable However () is satisfied W A diff Bdiff + Π = im Adiff Bdiff + } + im = R General switching signal We now consider an arbitrary piecewise constant switching signal σ which is constant on ( ) Without restriction we can relable the matrices such that t < t σ(t) = (2) k t t k t k+ ) where t < t < t 2 < are the switching times of σ Without restriction we can assume that t = (if there was no switch at t = we can artificially insert a switch to the same mode) Lemma 39 Consider the regular switched DAE () with switching signal (2) then for all l > k > : C (t k t l ) σ = (C k + e Adiff k (t k t k ) Π k C (t kt l ) σ ) V k Proof The proof follows exactly the same line of arguments as the proof for C (st) σ ts} in Theorem 35 We are now ready to present the controllability characterization for general switched DAEs Theorem 3 Consider the switched DAEs () with switching signal (2) and corresponding consistency projector Π k controllability space C k augmented consistency space V k for each mode k N } For each l N define the following sequence of subspaces: C l l := C l C l k := C k + e Adiff k (t k t k ) Π k Ck l k = l2 Then () is controllable if and only if there exists l N such that Π C l V Proof First note that for any matrix M and subspaces S T with imm T we have that M (S T ) = M S Hence because imπ k V k we can utilize Lemma 39 to show inductively that C (t kt l+ ) σ = C l k V k Finally we can use the same argument as in the t s = - case of the proof of Theorem 35 to show the claim The dependencies of the switching times is crucial and in general cannot be avoided as the following example shows

8 Example 3 (Dependency on the switching time) Consider the switched DAE () with switching signal (2) and (E A B ) = ( ) (E A B ) = ( ) (E 2 A 2 B 2 ) = ( ) (E k A k B k ) = (E 2 A 2 B 2 ) k > 2 For t < t 2 the input is zero and thus the system cannot be controlled For t t 2 the solution is given by x(t) = cos(t2 t )x u(t) The system is controllable if and only if cos(t 2 t ) = ie t 2 = t + π( + 2k)k N 4 Conclusion We have presented a characterization of controllability for switched DAEs This is the first result for controllability of switched DAEs and many important questions remain open Duality and the connection between observability and controllability is a topic of ongoing research Another question is whether the dependencies on the switching times can be relaxed resulting in necessary or sufficient conditions for controllability Finally a long term goal is the construction of stabilizing controllers taking into account the special controllability features of switched DAEs References Thomas Berger Achim Ilchmann and Stephan Trenn The quasi-weierstraß form for regular matrix pencils Lin Alg Appl 436(): Thomas Berger and Timo Reis Controllability of linear differential-algebraic systems - a survey In Achim Ilchmann and Timo Reis editors Surveys in Differential-Algebraic Equations I Differential-Algebraic Equations Forum pages 6 Springer-Verlag Berlin-Heidelberg 23 3 Thomas Berger and Stephan Trenn Kalman controllability decompositions for differential-algebraic systems Syst Control Lett 24 in press 4 Stephen L Campbell Nancy K Nichols and William J Terrell Duality observability and controllability for linear timevarying descriptor systems Circuits Systems Signal Process (4): Jelel Ezzine and AH Haddad Controllability and observability of hybrid systems Int J Control 49(6): Achim Ilchmann and Volker Mehrmann A behavioural approach to time-varying linear systems Part : General theory SIAM J Control Optim 44(5): Achim Ilchmann and Volker Mehrmann A behavioural approach to time-varying linear systems Part 2: Descriptor systems SIAM J Control Optim 44(5): Paula Rocha Jan C Willems Paolo Rapisarda and Diego Napp On the stability of switched behavioral systems In Proc 5th IEEE Conf Decis Control and European Control Conference ECC 2 Orlando USA pages Markus G-M Ruppert and Stephan Trenn Controllability of switched DAEs: The single switch case In PAMM - Proc Appl Math Mech volume 4 Wiley-VCH Verlag GmbH 24 to appear Zhendong Sun and Shuzhi Sam Ge Switched linear systems Communications and Control Engineering Springer- Verlag London 25 Aneel Tanwani and Stephan Trenn On observability of switched differential-algebraic equations In Proc 49th IEEE Conf Decis Control Atlanta USA pages Aneel Tanwani and Stephan Trenn Observability of switched differential-algebraic equations for general switching signals In Proc 5st IEEE Conf Decis Control Maui USA pages Stephan Trenn Distributional differential algebraic equations PhD thesis Institut für Mathematik Technische Universität Ilmenau Universitätsverlag Ilmenau Ilmenau Germany 29 4 Stephan Trenn Regularity of distributional differential algebraic equations Math Control Signals Syst 2(3): Stephan Trenn Switched differential algebraic equations In Francesco Vasca and Luigi Iannelli editors Dynamics and Control of Switched Electronic Systems - Advanced Perspectives for Modeling Simulation and Control of Power Converters chapter 6 pages Springer-Verlag London 22 6 Chi-Jo Wang Controllability and observability of linear time-varying singular systems IEEE Trans Autom Control 44():9 95 October Kai-Tak Wong The eigenvalue problem λt x + Sx J Diff Eqns 6: