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Politecnico di Torino Porto Institutional Repository Article Asymptotic controllability by means of eventually periodic switching rulesVol 49 2 2011 pp 476 497 Original Citation Bacciotti Andrea Mazzi Luisa 2011 Asymptotic controllability by means of eventually periodic switching rulesVol 49 2 2011 pp 476 497 In SIAM JOURNAL ON CONTROL AND OPTIMIZATION vol 49 n 2 pp 476


SIAM J CONTROL OPTIM c 2011 Society for Industrial and Applied Mathematics. Vol 49 No 2 pp 476 497,ASYMPTOTIC CONTROLLABILITY BY MEANS OF EVENTUALLY. PERIODIC SWITCHING RULES,ANDREA BACCIOTTI AND LUISA MAZZI. Abstract In this paper we introduce the notion of eventually periodic switching signal We. prove that if a family of linear vector elds satis es a mild nite time controllability condition. and for each initial state there exists a time dependent switching signal which asymptotically drives. the system to the origin possibly allowing di erent signals for di erent initial states then the. same goal can be achieved by means of an eventually periodic switching signal This enables us to. considerably reduce the dependence of the control law on the initial state In this sense the problem. addressed in this paper can be reviewed as a switched system theory version of the classical problem. of investigating whether or to what extent a nonlinear asymptotically controllable system admits. stabilizing feedback laws, Key words switched systems asymptotic controllability time dependent switching rules state. dependent switching rules stabilization, AMS subject classifications 93C10 93C65 93B27 93D20 34D05. DOI 10 1137 100798260, 1 Introduction This paper deals with the so called stabilization problem for.
switched systems In engineering literature the term switched system denotes a. special type of hybrid system whose time evolution is described by a continuous but. not necessarily di erentiable curve in the state space Examples of switched systems. can be frequently found in applications and technology Switched systems are often. de ned by assigning a family of vector elds fn x n N on R where N is some set. of indices and d is a positive integer and a piecewise constant function with values. in the index set N The function is called the switching rule It speci es the vector. eld which determines the system evolution at each instant The switching rule may. be either time dependent or state dependent although a uni ed treatment has been. attempted by some authors we prefer to keep the two options distinct at least at the. beginning Indeed the mathematical treatment of systems with a state dependent. switching rule encounters di culties which do not arise when the switching rule is. time dependent, A time dependent switching rule consists of a piecewise constant map t. 0 N It may or may not be dependent on the initial state Provided. that all the vector elds fn x are forward complete the trajectories of a switched. system with a time dependent switching rule exist for each initial state and are con. tinuable on the whole interval 0 Moreover they cannot exhibit chattering or. Fuller s phenomenon sometimes called Zeno phenomenon see 13 In particular if. the switching rule is the same for each initial state the trajectories can be determined. by solving a time varying di erential system, Received by the editors June 10 2010 accepted for publication in revised form December 12. 2010 published electronically March 22 2011 This work was supported by PRIN grant 20087W5P2K. from the Italian Ministry of University and Research. http www siam org journals sicon 49 2 79826 html, Dipartimento di Matematica del Politecnico di Torino C so Duca degli Abruzzi 24 10129 Torino. Italy andrea bacciotti polito it luisa mazzi polito it. ASYMPTOTIC CONTROLLABILITY 477, where f t x fn x on each interval on which t n However this simpli cation. is not possible if the switching rule is di erent for each initial state. A state dependent switching rule in its simpler version corresponds to a discon. tinuous feedback k x R N To compute the trajectories one needs to solve the. di erential system, whose right hand side is in general discontinuous Now it is well known that in this.
case solutions in classical or Carathe odory sense do not necessarily exist or are not. continuable Moreover chattering and Fuller s phenomenon may arise It turns out. that in general state dependent switching rules and time dependent switching rules. cannot be immediately converted into each other see 7. In this paper we address the open loop version of the stabilization problem bet. ter known in the classical control theory literature as the asymptotic controllability. problem We focus on a special class of time dependent switching rules called even. tually periodic switching rules already introduced in 6 We prove that if the family. of vector elds is linear and satis es a mild nite time controllability condition then. asymptotic controllability by means of generic time dependent switching rules im. plies asymptotic controllability by means of eventually periodic switching rules The. construction exploits the existence of a stable manifold of a discrete time dynamical. system associated to the given family of vector elds. The main feature of eventually periodic switching rules is that they exhibit a. strongly reduced dependence on the initial state Moreover they can be reinterpreted. in terms of state dependence Thus our results contribute to partially bridging the. gap between the notions of time dependent and state dependent switching rule In. fact we can recognize an analogy between the problem addressed in this paper and the. problem of investigating whether for a nonlinear system asymptotic controllability. implies feedback stabilization For sake of conciseness we do not report here the. details of this classical problem The reader can nd the precise statement in 18 and. some remarkable contributions in 8 1 17 9 We limit ourselves to point out that. the notion of sampled feedback law exploited in 8 combines both time dependence. and state dependence, This paper is organized as follows In section 2 we present the basic de nitions. and some preliminary facts to be used later In particular we introduce the discrete. time dynamical system associated to a periodic switching law In section 3 we expose. the main results The notion of nite time controllability we need in this paper is. introduced and discussed in section 4 The proof of the main results is given in. section 5 In section 6 we suggest the state dependent interpretation of eventually. periodic switching signals In section 7 we propose some simple results independent. of the controllability assumption ensuring the existence of a nontrivial stable manifold. for the associated discrete time system Finally section 8 contains some illustrative. examples and section 9 the nal comments The appendix is devoted to families of. nonlinear vector elds on compact manifolds in the spirit of classical geometric control. theory we prove some facts which are essential for the proof of the main results of. this paper, 2 Preliminary definitions In this paper we will be mainly concerned with. families of linear vector elds of R Let N 1 N where N 2 is a xed. integer and let us denote by F fn x n N a family of vector elds of R The. vector eld fn x is called the nth component of F,478 A BACCIOTTI AND L MAZZI. The family F is said to be linear if for each n N fn x An x where An is. a d d real matrix For each linear family F and each n N the curve n t x0. R Rd uniquely de ned by the conditions x An x and n 0 x0 x0 is called. the trajectory of the nth component issued from the initial state x0 R It is. represented as usual by n t x0 e x0, 2 1 Switching signals and switched trajectories Let N be equipped with. the discrete topology By switching signal we mean any right continuous piecewise. constant1 function 0 N The discontinuity points of a switching signal. form a nite or in nite possibly empty subset of the open half line 0 They are. called the switching times of We denote by I the set whose elements are t0 0 and. all the switching times of indexed in such a way that 0 t0 t1 t2 If the. set I is in nite then clearly limi ti The positive numbers i ti 1 ti. are called durations The number of switching times of in the interval 0 T T 0. is denoted s T, The set of all the switching signals is denoted by UN It possesses the so called.
concatenation property If 1 2 UN and T 0 then UN where. 1 t for t 0 T,2 t T for t T, Let UN and let a linear family F of R be given For each x0 R there is. a unique continuous curve t F t x0 0 R satisfying the conditions. F 0 x0 x0 and,F t x0 ti t ti F ti x0 t ti ti 1 ti I. We say that F t x0 is the switched trajectory of F issued from the initial. state x0 and corresponding to the switching signal It can be represented2 as. F t x0 e t ti A ti F ti x0,e t ti A ti e ti ti 1 A ti 1 et1 A 0 x0. for each t ti ti 1 and ti I We emphasize that the operator. x t x F t x,is linear and nonsingular for each t 0 and each. Remark 1 A switched trajectory of a linear family F can be reviewed as a. trajectory of a bilinear control system of the form x N n 1 un An x where the input u. is piecewise constant and takes value on the set 1 0 0 0 0 1 R. 2 2 Linear switched systems A linear switched system is de ned by a linear. family F of R together with a map R UN which assigns a switching signal. t x0 t to each point x0 R regarded as the initial state A linear switched. 1 This means that there are at most nitely many jumps in each compact interval. 2 Ofcourse a switched trajectory of a linear family can be represented in a much simpler form if. the matrices An commute however in this paper we do not make this assumption. ASYMPTOTIC CONTROLLABILITY 479, system is denoted by F The map is referred to as a time dependent switching.
The switched trajectory of F issued from any point x0 R and corresponding. to the switching signal x0 will also be called a trajectory of F. A switched system for which is constant i e the same switching signal t is. applied for each initial state x0 will be simply written as F. In what follows we denote by the Euclidean norm of a vector or the Frobenius. norm of a matrix Moreover we write Sr x R x r r 0 The following. lemma will be used later, Lemma 1 Let F be a linear family of R Then there exist 0 and 1. with the following property For each switching signal each T 0 each t 0 T. and each x0 R,1 F t x0 k 1 e T x0,where k s T, Proof As is well known for each n N there exist n R and n 0 such that. etAn n e n t,for each t 0 and R,Let max 1 1 n and max 0 1 n Let t0 t1 t2. be the sequence of switching times of If T t0 t1 then 1 is obvious Assuming. that 1 is true for T tk tk 1 Then it is not di cult to prove that it is true also. for T tk 1 tk 2 The result follows by induction, 2 3 Asymptotic controllability Given a linear family of vector elds F it. is interesting to characterize those switching maps if any such that for each initial. P1 limt F t x0 x0 0, As is well known this problem is not trivial since even if all the matrices An are.
Hurwitz i e all their eigenvalues lie in the open left complex plane it may happen. that the trajectory corresponding to some switching signals and some initial states. diverges 13 20, Definition 1 The linear family of vector fields F is said to be asymptotically. controllable if there exists a switching map such that property P1 holds for each. x0 R In this case we also say that is an AC switching map for F. Definition 2 The linear family of vector fields F is said to be uniformly asymp. totically controllable if there exists a switched signal t such that property P1 holds. for each x0 R with x0 In this case we also say that is a UAC switching. signal for F, The notion of asymptotic controllability sometimes shortened to asycontrollabil. ity see 18 is classical It means that all the initial states can be eventually driven. toward the origin but di erent switching signals might be required for di erent ini. tial states On the contrary uniform asymptotic controllability means that the same. switching signal works for all the initial states 3. 3 In 20 6 asymptotic controllability and uniform asymptotic controllability are respectively. termed pointwise stabilizability and consistent stabilizability. 480 A BACCIOTTI AND L MAZZI, Clearly if F is uniformly asymptotically controllable then it is asymptotically. controllable but the converse is false in general This is shown by an example in 20. p 58 and also by examples presented later in this paper. Remark 2 Asymptotic controllability has been merely de ned here in terms. of the attraction property P1 However the vector elds of F being linear P1. automatically implies stability as is speci ed in the following proposition. Proposition 1 If a linear family of vector fields F is asymptotically controllable. then it is possible to find an AC switching map such that besides P1 the following. additional property holds,P2 0 0 such that x0 implies F t x0 x0 t 0. We report the proof of Proposition 1 for the reader s convenience although similar. arguments can be found in 19 and in 21, Proof For each p S1 the asymptotic controllability assumption yields a switch.
ing signal p and a time Tp such that, Since x F Tp x p is a nonsingular linear map there exists a neighborhood. Up of p such that, for each x Up The sets Vp Up S1 form a relatively open covering of S1 By. the compactness argument we can extract a nite covering Vp1 VpL For each. 1 L rename p T Tp and let O x 0 x x Vp It is,clear that 1 L O R 0 If x O then. 2 F T x x F T x x,Moreover as long as t 0 T by Lemma 1 we have. 3 F t x k 1 e T x,where T max T1 TL and k max s 1 T1 s L TL.
A new switching signal x0 can now be de ned for each initial state x0 0 by. concatenation according to the following procedure. 1 if x0 O 0 then apply the input 0 restricted on the interval 0 T 0. 2 set x1 F T 0 x0 0,3 replace x0 by x1 and repeat the previous steps. Taking into account both 2 and 3 and using induction it is not di cult to. F t x0 x0 k 1 e T x0,for each t 0 The conclusion easily follows. ASYMPTOTIC CONTROLLABILITY 481, 2 4 Periodic switching signals In this paper we are interested in switching. rules whose structure is required to satisfy particular restrictions. A periodic switched system is a pair F such that is periodic for F More. precisely a switching signal t is said to be periodic of period T for F if there. exist a string of real numbers 0 H where H is an integer H 1 and a string. of indices n1 nH N such that,t nh for t h 1 h for each h 1 H. t t T for t T, The points h mT with h 1 H and m 0 1 coincide with the.
switching times provided that n1 n2 n2 n3 nH n0 Note that is. constant when H 1, Definition 3 The linear family F is said to be periodically asymptotically con. trollable if it is uniformly asymptotically controllable by means of a periodic switching. Remark 3 In 20 it is proven that periodic asymptotic controllability and uniform. asymptotically controllability are actually equivalent. Asymptotic controllability can be achieved by means of high frequency periodic. switching signals see 22 20 under the assumption that for some integer H 1. and some indices n1 nH N there exists a Hurwitz convex combination of the. matrices An1 AnH This assumption was originally introduced in 16 to prove. the existence of state dependent stabilizing switching rules see 5 for an extension. to nonlinear systems, Asymptotic controllability can also be achieved by means of switching signals. with large dwell time see Lemma 2 of 15 provided that all the matrices An are. Hurwitz The case of a pair of planar oscillators has been studied in 3. 2 5 Eventually periodic switching signals As already mentioned there. exist families of linear vector elds not uniformly asymptotically controllable and. hence not periodically asymptotically controllable but which can be asymptotically. driven toward zero by applying a di erent switching signal for each initial state We. are especially interested in switching rules of the following type. Definition 4 A switching map R UN is said to be eventually periodic. of period T if there exist a periodic switching signal t of period T and a map. T0 R 0 such that for each x0 R,x0 t t T0 x0 t T0 x0. The switched system F is said to be eventually periodic if the map is. eventually periodic 4, In other words a switched system is eventually periodic if it behaves as a periodic. one after an initial transient interval depending on the initial state. Definition 5 We say that a linear family F is eventually periodically asymp. totically controllable if there exists an eventually periodic map such that property. P1 holds for each x0 R, In the remaining part of this section we illustrate a systematic way to construct.
eventually periodic AC switching maps compare with the rst example of section 8. 4 Eventually periodic switching maps have been introduced in 6 with the name of near periodic. switching rules,482 A BACCIOTTI AND L MAZZI, Let F be a linear family of R Let t be any periodic switching signal of. period T With the notation introduced above we associate to F the linear. discrete dynamical system,4 xk 1 T xk F T xk k N, Of course if the origin is an asymptotically stable xed point for the associated. discrete dynamical system 4 then is a UAC switching signal for F 20 22 As is. well known the origin is asymptotically stable for 4 if and only if all the eigenvalues. of the linear operator T lie in the unit disc of the complex plane If d. counting multiplicity among the eigenvalues of T meet this condition then. the initial states which are asymptotically driven to the origin along the solutions of. 4 form a dimensional invariant subspace of R called the maximal stable subspace. of 4 and denoted Ws A subspace W R will be called a stable subspace of 4 if. it is invariant and contained in Ws, Now the construction of eventually periodic AC switching maps for F can be. easily achieved if the following conditions are met. C1 there exists a periodic switching signal whose associated discrete dynamical. system 4 has a nontrivial stable subspace W, C2 for each x0 R x0 0 there exists a switching signal depending on x0. such that the corresponding switched trajectory of F issued from x0 intersects. W in nite time, 3 The main results In order to state the main results of this paper we need.
to formulate a special controllability assumption about the family F. Let x y R We say that y is reachable from x in time t 0 if there exists a. switching signal such that F t x y We denote by RF T x the set of all. points y of R which are reachable from x in some time t T. If y RF T1 x and z RF T2 y then z RF T1 T2 x this follows. immediately from the concatenation properties, Definition 6 The family F of linear vector fields is said to be radially control. lable if for each pair of points x y R x 0 y 0 there exist 0 and T 0. such that y RF T x, The notion of radial controllability will be discussed in section 4 where we also. explain how it can be checked in practice, Theorem 1 Let F be a linear family of R Assume that it is radially control. lable and asymptotically controllable Then for each x0 R x0 0 there exists a. periodic switching signal such that the discrete dynamical system 4 associated to. F has a nontrivial stable subspace containing the point x0. Theorem 2 Let F be a linear family of R Assume that it is radially control. lable and asymptotically controllable Then F is eventually periodically asymptotically. controllable Moreover an eventually periodic AC switching map can be found in. such a way that property P2 is fulfilled as well, The proof of Theorems 1 and 2 will be given in section 5 it relies crucially on the. results of nonlinear nature presented in the appendix. Remark 4 Theorems 1 and 2 can be easily extended to families of linear systems. F An x n N for which the index set N is not necessarily nite provided that for. some 1 and some 0 the inequality etAn e t holds for each t R and. each n N In particular Theorem 2 remains true when the index set N is a compact. topological space,ASYMPTOTIC CONTROLLABILITY 483, Remark 5 Let us remark that with respect to asymptotic controllability even.
tually periodic asymptotic controllability considerably reduces the dependence of the. control policy on the initial state Actually after the initial transient we need to drive. the system to reach the stable subspace of the associated discrete dynamical system. the control becomes periodic and can be implemented in an automatic way. 4 Radial controllability In this section we discuss the notion of radial con. trollability In particular we show that the radial controllability of a family of linear. vector elds on R can be checked by looking at the global controllability of a family. of in general nonlinear vector elds on a d 1 dimensional compact manifold. To each family F fn x An x n N of linear vector elds of R we associate. a family F formed by the in general nonlinear vector elds. 5 fn x An x, where x R x 0 and t denotes transposition For each n N fn x is. homogeneous of degree one5 and analytic on R 0 It is immediate to verify that. fn x is tangent to any sphere Sr r 0 In fact fn x is the projection of An x on. the tangent space of the sphere Sr with r x at the point x. By virtue of homogeneity we can limit ourselves to r 1 Let the vector elds. fn p be the restriction of the vector elds fn x to the d 1 dimensional sphere S1. and let F fn p n N, Proposition 2 F is radially controllable if and only if F is globally controllable. Proof For any n N let t be a solution of x An x and let o t be a. solution of p fno p By direct computation we see that if o 0 0 0. then o t t t for each t R Now assume that F is radially controllable. and let p q S1 There exists a switching signal which steers p to q for some 0. Clearly the same switching law applied to F o steers p to q Vice versa if F o is globally. controllable for each pair z y R we can nd a switched signal steering p z z. to q y y Of course the same switching law applied to F steers z to y for. Thus the problem is reduced to test controllability of a family of vector elds on. a d 1 dimensional manifold Combining Proposition 2 with Propositions 6 and 7. see the appendix we get the following corollary, Corollary 1 Let F be a linear family of R If F is radially controllable then. there exist T 0 and S such that for each pair of points z y R z 0 y 0. there exist 0 such that y RF T z for some T T In addition a switching. signal steering z to y in time T can be found with s T S. In fact the following proposition provides an important additional information. Proposition 3 Let F be a radially controllable linear family of R Let T 0. and S be as in the statement of Corollary 1 Then there exists a real number 0. enjoying the following property for each pair of points z y R z 0 y 0 there. exist a positive number and a switching signal such that. 5 This means that for each a R 0 one has fn ax afn x. 484 A BACCIOTTI AND L MAZZI,and y F T z with T T s T S. Proof By Corollary 1 there exist a positive number and a switched signal. which connects z to y in time T T such that s T S In other words we can. 7 y e K AnK e 1 An1 z, for some integer K S some indices n1 nK N and some positive durations.
1 K with 1 K T Let us de ne S 1 e T is independent. of z y and it remains only to prove that 6 holds With the same notation as in. Proposition 1 from 7 we have,y nK n1 e nK K e n1 1 z. K e 1 K z S 1 e T z,as required,5 Proof of the main results. 5 1 Proof of Theorem 1 We prove that for each w 0 there exists a periodic. switching signal of some period T 0 such that for the associated discrete dynamical. system one has, with 0 1 2 In other words we prove that can be found in such a way that w. is a real eigenvector of the linear operator T corresponding to a real eigenvalue. lying inside the open unit disc It follows immediately that the one dimensional. subspace generated by w is a nontrivial stable subspace of the linear map T. Let w R w 0 Since F is asymptotically controllable there exists a switch. ing signal 1 depending on w such that limt F t w 1 0 In particular. there exists a time T1 0 such that, where z F T1 w 1 and 0 is the number in the statement of Proposition 3. Recall that depends only on F, Since the system is radially controllable we can use again Propositions 1 and 3.
to nd a switched signal 2 a time T2 0 and a positive number such that. w F T2 z 2 with T2 T and, Let us now de ne a periodic switching law whose period is T T1 T2 in. such a way that,1 t if t 0 T1,2 t T1 if t T1 T, The switching law is well de ned because of the concatenation property We. clearly have,w F T2 z 2 F T2 F T1 w 1 2 F T w T w,ASYMPTOTIC CONTROLLABILITY 485. Moreover by virtue of 9 and 10,from which we easily get 1 2 as desired. 5 2 Proof of Theorem 2 According to Theorem 1 condition C1 is ful lled. by F Let be a switched signal periodic for F and let w be such that 8 holds. with some 1 2 By virtue of the radial controllability assumption for each x0 0. there exist a switching signal x0 and a positive time T0 x0 such that. F T0 x0 x0 x0 w0, where w0 is parallel to w We get an eventually periodic AC switching map by setting.
x0 t t 0 T0 x0,t T0 x0 t T0 x0, All the switched trajectories of F satisfy property P1. Finally we prove that the eventually periodic switching signal 11 satis es also. property P2 Recall that by Proposition 6 we can choose x0 in such a way that. s x0 T x0 S and T0 x0 T By repeating similar arguments as in the proof of. Proposition 1 we have,12 F t x0 x0 F t x0 x0 S 1 e T x0. for t 0 T0 x0 Hence in particular w0 S 1 e T x0 Analogously for. t T0 x0 T0 x0 T,F t x0 x0 F t T0 x0 F T0 x0 x0 x0,F t T0 x0 w0. H 1 e T w0,where T is the period of and H s T, By construction if we set w1 F T w0 then w1 is parallel to w0 and. Now let t T0 x0 T T0 x0 2T As before we have,F t x0 x0 F t T0 x0 T w1 H 1 e T w1 H 1 e T w0.
The reasoning can actually be iterated on each interval of the form T0 x0. mT T0 x0 m 1 T Taking into account 12 we nally get. F t x0 x0 S H 2 e T T x0,Property P2 is easily achieved with S H 2 e T T. 6 State dependent switching rules Let the linear family F of R be given. The construction of state dependent switching rules is frequently achieved in the lit. erature on the base of the following procedure see for instance 4. 486 A BACCIOTTI AND L MAZZI, 1 Find a nite family of open pairwise disjoint subsets of R 1 L such. 2 Associate an index n N to each region,3 De ne x n whenever x. Typically this can been accomplished by the aid of a Lyapunov like function. if F is quadratically stabilizable see 13 and hence in particular if there exists a. Hurwitz convex combination of the matrices A1 AN In the latter case the regions. 1 L are conic, As already mentioned in the introduction this procedure leads to a system of. equations with discontinuous right hand side for which the existence of Carathe odory. solutions and a fortiori switched solutions is not sure To overcome the di culty. one can try to introduce hysteresis in the systems as in 16 and 13 or to resort. to generalized Filippov or Krasowski solutions as in 2 However sometimes an. approach based on a eventually periodic switching law might be preferable. Recall that although not reproducible in general by means of a purely static. memoryless feedback law an eventually periodic switching map has a reduced depen. dence on the initial state Remark 5 Under certain circumstances this allows us to. identify appropriate switching loci with associated appropriate indices. Let be a periodic switching signal With the notation of section 2 let. 1 T e H AnH e 1 An1, where h h h 1 h 1 H and 1 H T Assume that the discrete.
dynamical system associated to 1 has a nontrivial stable subspace W1. Then it is possible to prove that for each h 2 H,Wh e h 1 Anh 1 e 1 An1 W1. is a stable subspace for the discrete dynamical system associated to the operator. h e h 1 Anh 1 e 1 An1 e H AnH e h Anh, Note that h corresponds to the same periodic signal as 1 translated of a. quantity 1 h Assume further that the subspaces Wh are pairwise transver. sal and associate the index nh with the subspace Wh An eventually periodic AC. switching map whose periodic part coincides with can be redescribed according to. the following steps, 1 transient initial interval starting from any initial state x0 0 drive the. system to hit one of the subspaces Wh, 2 steady state behavior when the system trajectory hits Wh switch on the. nh component, Note that during the transient interval the system is operated in open loop.
However according to the results of section 4 the length of the transient interval can. be predicted During the steady state the control procedure can be implemented. automatically To compare our approach with the more traditional one sketched. above the reader may nd useful to look at the examples of section 8. ASYMPTOTIC CONTROLLABILITY 487, 7 Existence of stable subspaces Note that in the statement of Theorems 1. and 2 we do not impose any assumption about the asymptotic behavior of the single. components of F Constructing UAC switching signals is trivial if there exists at. least one n N such that An is Hurwitz In this case indeed one can take t n. Hence the natural motivation of Theorems 1 and 2 apparently relies on the case. where none of the components of F is asymptotically stable However other reasons. of interest might come from certain practical applications Indeed it may happen that. switching among two or more components is compulsory The admissible switching. rules might have a partially xed structure in the sense that the activation of the. various components must obey to a preassigned sequence while other details such as. durations i e the times elapsed between two consecutive switches are available for. design In such a situation we know that a bad choice of the durations can lead to. instability for certain initial states even if some or all the matrices An are Hurwitz. To avoid similar drawbacks the ideas developed in the proof of Theorems 1 and 2 can. be fruitfully applied, In this section we limit ourselves for simplicity to consider the case where all the. components of F must be cyclically activated following a prescribed order each one. for a nonvanishing interval of time More precisely we present some simple conditions. which allows us to predict the existence of a nontrivial stable subspace for the discrete. dynamical system de ned by an operator of the form. 13 1 N e N AN e 1 A1, we simply write instead of 1 N when the string 1 N is clear from the. We emphasize that the criteria of this section are independent of the radial con. trollability assumption, Proposition 4 Let F be a linear family of R with index set N 1 N. Assume that for some 1 N n 0 n 1 n 1 the matrix A n n An. has at least one eigenvalue with negative real part Then there exists a sequence of. positive durations 1 N such that the discrete dynamical system defined by the. operator 13 has a nontrivial stable subspace, Proof Let n n T for each n N and some T 0 For su ciently small T.
there exists a matrix C T such that, Such a matrix C T can be represented by the Baker Campbell Hausdor ex. pansion 23,C T n An T G T T 2, where G T is bounded Recall that the eigenvalues depend continuously on the el. of a matrix By taking a small enough T and using the assumption that. n n A n has at least one eigenvalue with negative real part we arrive at the con. clusion that C T has at least one eigenvalue with negative real part as well The. statement easily follows, Remark 6 As already recalled the much stronger assumption that for some choice. of 1 N the matrix A is Hurwitz has been used in 16 in order to construct. 488 A BACCIOTTI AND L MAZZI, state dependent switching rules and in 22 see also 20 in order to construct high. frequency periodic UAC switching signals, The assumption of Proposition 4 is ful lled in particular if for some index n the.
matrix An has at least one eigenvalue with negative real part Indeed we can take. a convex combination with n 1 for each n n so that n n An can be viewed. as a small perturbation of An On the other hand the discrete dynamical system. associated to 13 may have a nontrivial stable subspace even if all the matrices An. and their convex combinations have all their eigenvalues in the open right half plane. this happens for instance in the reversed time version of Example 1. It can be proven that if the matrices A1 AN are symmetric then the existence. of an index n such that the matrix An has at least one eigenvalue with negative real. part is a necessary condition for asymptotic controllability 20 Combining these. observations we obtain the following result, Corollary 2 Let F be a linear family of R such that each matrix An is. symmetric If F is asymptotically controllable then there exists a sequence of positive. durations 1 N such that the discrete dynamical system defined by the operator. 13 has a nontrivial stable subspace, The assumption of Corollary 2 can be slightly relaxed by asking that for each. Proposition 5 Assume that all the matrices A1 AN are Hurwitz Then for. each sequence of positive durations 1 N the discrete dynamical system defined. by the operator 13 has a nontrivial stable subspace. In fact the conclusion of Proposition 5 can be proved to be valid under the. following slightly weaker assumption For each n 1 N tr An 0 where tr. denotes the trace of the matrix An, Proof Recall that for each t R and each square matrix A det etA etr tA. e If tr An 0 then,0 det etAn 1,for each t and each n N Hence. det det e N AN e 1 A1 det e N AN det e 1 A1 1, On the other hand det 1 d where 1 d are the nonnecessarily.
distinct eigenvalues of It follows that at least one eigenvalue of lies in the. interior of the unit disc of the complex plane, Proposition 5 implies that a family F whose components are all asymptotically. stable cannot be completely destabilized by applying a periodic switching signal. 8 Examples To illustrate the problems investigated in this paper the related. notions and the approach based on the associated discrete dynamical system 4 we. revisit below the example of a pair of linear systems both with spiral con guration. Although the same example can be found elsewhere we point out many new aspects. and details, Example 1 Consider in R the family F formed by the pair of vector elds. f1 x A1 x f2 x A2 x where6, 6 The diagonal entries of A and A have been taken equal to 1 4 since this value is convenient. for numerical simulations but in fact any 0 works,ASYMPTOTIC CONTROLLABILITY 489. A1 and A2 are both completely unstable It is easy to check that there exist no. Hurwitz convex combinations of A1 and A2 In fact the system is not quadratically. stabilizable Nevertheless the switching rule, is of the type considered at the beginning of section 6 It gives rise to an asymptotically.
stable system of ordinary di erential equations with discontinuous right hand side. Note that in spite of the discontinuities in this case switched trajectories exist for. each initial state and for every t R and are unique. It is also possible to verify that F is not uniformly asymptotically controllable. the necessary condition in 20 p 59 is not met Now we show how an eventually. periodic AC switching map can be constructed for F. Let 0 be xed We rst consider the periodic switching law of period 2 such. that t 1 for t 0 and t 2 for t 2 The fundamental matrices for. A1 and A2 are respectively,cos sin 1 cos sin,e A1 1 e 4 e A2. sin cos sin cos,and their product is,1 2 sin2 sin cos. 14 e A2 e A1 e 12,1 sin cos 1 2 1 sin, In order to analyze the stability of the periodic switched system of period 2. de ned by F and we study the stability of the associated discrete dynamical system. de ned by 14 Therefore let us consider the characteristic equation of 14. 2 1 2 1 2 2, Since e 1 for any 0 at least one eigenvalue 1 lies out of the unit circle. To check whether the eigenvalue 2 lies out of the unit circle too we may check the. behavior of the inverse matrix e A2 e A1 1 whose eigenvalues are 1 1. 1 and 2 The,characteristic equation of the inverse matrix is.
16 2 e 2 2 sin e 0, Since e 1 by the Schur Cohn lemma 12 both of the eigenvalues of the. inverse matrix lie in the unit circle if and only if. 17 e 2 2 sin2, Therefore 2 lies out of the unit circle if and only if 17 is veri ed while it. belongs to the unit circle if the inequality is reversed. Inequality 17 is veri ed if either,2 sin2 0 2 1,a 2 1 2 2 1 1.

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