STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING H LENS, M E BROUCKE, AND A ARAPOSTATHIS 1 Introduction The obective of this paper is to explore necessary and sufficient conditions for stability of a class of switched linear systems in the plane The switched systems we consider consist of several linear subsystems with a conic switching law specifying the active subsystem at each point in R There is interest to understand the structure of such systems both because of existing applications in mechanical systems, switched power converters, and other fields [4], and also to promote further applications The review article [3] discusses three classes of stability questions for switched systems The maority of results are based on common Lyapunov functions or multiple Lyapunov functions Lyapunov theory typically only provides sufficient conditions for stability and, further, multiple Lyapunov function theory requires knowledge of the switched system traectories to be able to perform the analysis Also, while switched linear systems are indeed nonlinear systems if viewed as a whole, multiple Lyapunov function theory does not exploit the linearity of the subsystems A direct approach based on a qualitative analysis of the vector fields was developed by Xu and Antsaklis [7] in order to obtain necessary and sufficient conditions for asymptotic stabilizability and to construct stabilizing control laws based on conic switching for two linear subsystems Boscain [1] obtained necessary and sufficient conditions for stability of a switched linear system with two subsystems and arbitrary switching between them Holcman and Margaliot [] obtained a necessary and sufficient condition for stability of two homogeneous subsystems with arbitrary switching by constructing an appropriate common Lyapunov function Our goal is to find an intrinsic stability criterion based solely on the eigenvalues and eigenvectors of the subsystems and on the switching boundaries, in analogy with the canonical linear system theory We show this is possible for D systems with conic switching More precisely, if the plane is partitioned into a set of disoint convex cones with one linear system active in each cone and if there are no visible eigenvectors, then there is an equivalent linear system whose stability determines the stability of the switched system The eigenvalues of the equivalent system are said to be the characteristic values of the switched system Further, the eigenvalues and the eigenvectors of the subsystems can be used to provide a qualitative classification of linear switched systems, ust as in the canonical linear system theory The key notion in this classification is that of visible eigenvectors We assume throughout that the switched system exhibits no chattering Similar ideas can be found in Date: May 15, 004 The second author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC The third author was supported in part by the Office of Naval Research through the Electric Ship Research and Development Consortium, in part by the National Science Foundation under Grants ECS-01807 1
H LENS, M E BROUCKE, AND A ARAPOSTATHIS the paper [6] They obtain a necessary and sufficient condition for stability of D switched linear systems with conic switching (SLSCS by calculating the gain of a Poincare map They also identify the relevance of visible eigenvectors In this paper we go one step further by obtaining explicit expressions for the characteristic values of the switched system Roughly speaking, for a switched system there are two mechanisms that lead to stability or instability One is the effect of the time-average of the eigenvalues of the individual linear components on each partition weighted by the fraction of the time that traectories spent on each partition The other is induced by the non-commutativity of the individual linear maps The expressions obtained in this paper distinguish between the two components and thus shed some new light on the issue of stability The paper is organized as follows In Section we present two preliminary results needed for the ensuing discussion: the effect of dominant eigenvectors on the direction of the flow in R, and a result on traectories escaping convex cones in R d In Section 3 we give our main result of computing the characteristic values of a switched linear system In Section 4 we classify SLCSC according to visible eigenvectors Preliminaries Two preliminary results are needed First, we define two types of eigenvectors to be used later: dominant and visible Second, we prove a general result in R d about traectories escaping convex cones Definition 1 Given a D linear system with real, distinct eigenvalues, we define the eigenvector corresponding to the greater eigenvalue to be the dominant eigenvector The other eigenvector is the non-dominant eigenvector Lemma 1 Traectories of a D linear system with real, distinct eigenvalues not starting on either eigenvector turn from the non-dominant to the dominant eigenvector Definition We define an eigenvector to be visible if it lies in the cone in which its subsystem is active Theorem Let K be a closed convex cone in R d, and suppose K does not contain a subspace of R d Suppose no eigenvectors of A R d d lie in K Then for any initial condition x 0 K, x 0 0, the traectory of the system ẋ = Ax eventually leaves the cone K; that is, for each x 0 K, x 0 0, there exists t 0 R such that e At 0 x 0 / K Proof We prove the theorem by contradicting the hypothesis Suppose that for some initial condition x 0 K, e At x 0 K, for all t 0 Let K denote the maximal invariant set under the semigroup {e At } contained in K Then K, and since the dynamics are linear, it is evident that K is also a closed convex cone By [5, Lemma 1], K contains an eigenvector of A, leading to a contradiction 3 Characteristic values of D switched systems In this section we compute the characteristic values of D SLSCS We start with the case when each subsystem has complex eigenvalues and then show all other cases can be mapped to this case The method is to compute the growth of traectories after one cycle around the origin (or half-cycle and use this parameter to obtain the asymptotic behavior of the switched system
STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING 3 Let A R have a pair of complex eigenvalues λ ± ω Let K be a closed cone generated by the pair of vectors {u 1, u } R, and suppose that ẋ = Ax governs the dynamics in K We assume the system is oriented so that a traectory starting at u 1 moves counterclockwise and exits the cone crossing the boundary generated by u The condition for this is simply u T 1 AT Ju 1 0, where J = ( 0 1 1 0 Note that A takes the form: A = P ( λi + ωj P 1 (31 Letting z = P 1 x, the system transforms to ż = (λi + ωjz and the cone K is mapped to the cone K generated by the vectors { P 1 u 1, P 1 } u Let Θ(, be the angle in radians between two vectors in R in the counterclockwise sense The time τ that it takes the system ż = (λi + ωjz to transverse the sector K is τ = Θ( P 1 u 1, P 1 u (3 ω This is of course the same time that it takes the original system ẋ = Ax to transverse the sector K Remark 31 The cosine (or sine of the angle Θ ( P 1 u 1, P 1 u can be calculated in closed form from P, u 1 and u However, the inverse cosine function is needed for the calculation of the time τ Now let { u 1,, u n, u n+1 } be an ordered (say counterclockwise set of unit vectors in R such that u n+1 = u 1 Let { K 1,, K n } be a set of cones (either closed, half-closed, or open that form a partition of R such that K i is generated by { u i, u i+1 } On each K i we have the dynamics ẋ = A i x We assume that each A i R has a pair of complex eigenvalues λ i ± ω i, and that all traectories move counterclockwise (no sliding modes Then, each A i has the form A i = P i ( λi I + ω i J P 1 i, i = 1,, n (33 Define τ i = Θ( Pi 1 u i, Pi 1 u i+1, ω i i = 1,, n, (34 n τ = τ i, (35 i=1 u i = P 1 i u (36 Theorem 31 The traectory x(t of the switched system grows asymptotically as e (µ+ωt where µ = α + log β τ (37 ω = π τ, (38
4 H LENS, M E BROUCKE, AND A ARAPOSTATHIS and α = n i=1 τ iλ i τ β = un n u n 1 n 1 =1 u The expressions for α and β are independent of the choice of the P i s (39 u +1 (310 Remark 3 Since the stability of the SLSCS is determined by the complex numbers µ±ω, we call them the characteristic values of the switched system Proof Let B i = λ i I + ω i J Assume x 0 = u 1 We have x(τ = P n e Bnτn Pn 1 P n 1e B n 1τ n 1 Pn 1 1 P 1e B 1τ 1 P1 1 x 0 For some constant β n, x(τ = P n β n u n 1 = β n, so we must determine β n We have β n u n 1 = β n u n 1 for some constant β n 1 Hence we have that = e λnτn Pn 1 P n 1 e B n 1τ n 1 Pn 1 1 P 1e B 1τ 1 P1 1 x 0 = e λnτn Pn 1 P n 1 β n 1 un n 1 = e λnτn β n 1 u n n, β n = e λnτn un n u n 1 β n 1 Continuing this argument inductively we obtain β = e λ τ u u +1 β 1 β 1 = e λ 1τ 1 u 1 1 u 1 =,, n We obtain log β x(τ = e [α+ τ ]τ from which the first result follows The expression for ω can be obtained directly Next we must show these expressions, in particular β, are independent of the choice of P i s Suppose there exists Q P such that A = Q(λI + ωjq 1 = P (λi + ωjp 1 Then P 1 Q(λI + ωj = (λi + ωjp 1 Q (311 Let T = P 1 Q By multiplying out (311 we find T has the form [ ] t1 t T = t t 1 By direct calculation and using P 1 = T Q 1, one can verify that for any u R P 1 u = T Q 1 u,
STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING 5 where T = t 1 + t Thus, we have u u +1 = P 1 u P 1 u +1 1 T Q u = T Q 1 u +1 = Q 1 u Q 1 u +1 Hence β is independent of the choice of the transforming matrices Remark 33 Note that if the matrices {A i } commute pairwise, then β = 1 Therefore, in this case stability results if the time-average of the mean eigenvalue is negative (α < 0 Likewise, if λ i = 0 for all i, then stability depends only on β, which does not depend on the eigenvalues of the individual matrices A i Example 31 Consider a SLSCS with two linear systems [ ] [ 0 0 1/ A 1 = A 1/ 0 = 0 The switching boundaries are u 1 = [1 0] T and u = [0 1] T The eigenvalues of A 1 are ± and of A, ± The characteristic values of the SLSCS computed using (39-(310 are 04413 + 31 Real Eigenvalues In this section we show that the asymptotic behavior of a linear system with real distinct or real repeated eigenvalues inside a cone that does not contain the system s eigenvectors is equivalent to the asymptotic behavior of a system with complex eigenvectors Each such subsystem in the SLSCS can be replaced by its asymptotically equivalent system with complex eigenvalues, so that we only need the formulas (37-(38 to compute the characteristic values of an arbitrary D SLSCS To this end, let A R have distinct real eigenvalues λ 1 > λ, let K be a closed cone generated by the pair of vectors {u, u } R, and suppose that ẋ = Ax governs the dynamics in K Further, we assume that K contains no eigenvectors of A and the system is oriented so that the traectory starting at u moves counterclockwise and( exits the cone crossing the boundary generated by u λ1 0 The matrix A has the form A = P P 0 λ 1 so we can map to the canonical system ż(t = ] ( λ1 0 z(t (31 0 λ Define the unit vectors ũ = P 1 u, = P 1 u (313 Since K contains no eigenvectors of A it has to be the case that ũ and have the same sign (component wise By Theorem, the time τ it takes the system to traverse the cone is finite If we define the growth of the system as it tranverses the cone to be e λτ then we have ( (ũ (ũ e λ 1 τ 0 1 0 e λ τ ũ = e λτ 1 (314 Solving (314 we obtain τ = log ( ũ 1 ũ 1ũ λ = λ 1 + λ, (315 λ 1 λ (316 + 1 ( τ log ũ 1 ũ 1ũ
6 H LENS, M E BROUCKE, AND A ARAPOSTATHIS In the case of repeated eigenvalues coupled in a Jordan block, ie, A = P ( λr 1 P 0 λ 1, r instead of (314, we solve (e λrτ τe λrτ to obtain 0 e λrτ τ = ũ 1 ũ 1 ũ (ũ 1 ũ = 1 ũ ũ λ = λ r + 1 τ log ( ũ (ũ = e λτ 1, (317 det (ũ 1 ũ 1 ũ, (318 (319 It can be shown as before that the formulas (315-(316 and (318-(319 do not depend on the particular choice of the transforming matrix P The asymptotically equivalent system with complex eigenvalues is ( λ ω ẋ = P P ω λ 1 x, ( where ω = Θ P 1 i u,p 1 i u τ Example 3 Consider a SLSCS with two linear systems [ ] [ 4 3 0 1 A 1 = A 6 5 = 3 The switching boundaries are u 1 = [1 0] T and u = [0 1] T The eigenvalues of A 1 are 1, and of A, 1, The characteristic values of the SLSCS are 1 + 3993 ] 4 Classification of D SLSCS Linear systems are usually categorized according to their eigenvalues and eigenvectors into nodes, foci, centers, and saddles Since SLSCS are essentially nonlinear, such a classification is not possible, but if we redefine the categories for linear systems and add a few categories, one can characterize the behavior of SLSCS that consist of two subsystems In the following, by stable we mean in the sense of Lyapunov Also, we assume that eigenvectors do not lie on switching boundaries By stable eigenvector, we mean an eigenvector with associated real eigenvalue that is stable Definition 41 We define a D SLSCS with two subsystems to be (1 a stable (unstable focus if traectories starting in R \ {0} are stable (unstable and orbit the origin ( a stable (unstable node if traectories starting in R \ {0} are stable (unstable and do not orbit the origin (3 a saddle if traectories starting in R \ L are unstable, where L is a line through the origin (4 pseudostable if traectories starting in R \ L are stable, where L is a line through the origin (5 semistable if there is a cone K such that traectories starting in K are stable and traectories starting in R \ K are unstable
STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING 7 (6 a double saddle if traectories starting in R \ L 1 L are unstable, where L 1, L are lines through the origin 41 Stability Categories SLSCS can be categorized according to the number of visible eigenvectors The number of visible eigenvectors is important since we can infer which cases in Definition 41 are at all possible 411 No visible eigenvectors This is the case studied in Section 3 The switched system is a focus and its stability is determined by its characteristic values as given in Theorem 31 41 One visible eigenvector If there is one visible eigenvector, that eigenvector must be dominant, otherwise the cone would be negatively invariant and by Theorem, the other cone has a finite escape time, which together would result in chattering If the visible eigenvector is dominant, all traectories will eventually approach that eigenvector The SLSCS is therefore a node It is stable if the visible eigenvector is stable, and unstable in the other case 413 Two visible eigenvectors In this case, at least one of the vectors must be dominant otherwise both cones would be negatively invariant, indicating chattering If there is a double eigenvector visible, the behavior is equivalent to one of the first two cases below (1 Both are stable All traectories will approach the dominant (or one of them, if they are both dominant eigenvector or remain on an eigenvector Since that will in any case be a stable eigenvector, the system is a stable node ( Both are unstable The reverse is true and the system is an unstable node (3 One is stable There are three possible cases: (a Both dominant In this case they have to lie in different cones, which are both positively invariant Therefore, traectories starting in the cone with the stable eigenvector will be stable, those starting in the other cone will be unstable The system is thus semistable (b Stable dominant, unstable non-dominant All traectories except those that start on the unstable eigenvector will eventually approach the stable eigenvector and therefore be stable The system is pseudostable (c Unstable dominant, stable non-dominant All traectories except those that start on the stable eigenvector will eventually approach the unstable eigenvector and therefore be unstable The system is a saddle 414 Three visible eigenvectors As in the case of one visible eigenvector, the single eigenvector must be dominant for the system not to chatter This implies the following possible cases: (1 All are stable All traectories will approach a dominant eigenvector or remain on one of the eigenvectors Since that will in any case be a stable eigenvector, the system is a stable node ( All are unstable The reverse is true and the system is an unstable node (3 Two are stable The unstable vector will always be dominant, hence it will attract traectories starting in a cone bounded by the nearest eigenvectors or switching boundaries One of the stable vectors will also be dominant, attracting the rest of the traectories The SLSCS will therefore be semistable
8 H LENS, M E BROUCKE, AND A ARAPOSTATHIS (4 One is stable There are two possible cases: (a single stable The single eigenvector is dominant, so will attract traectories starting in a surrounding cone; the same holds for the dominant of the unstable eigenvectors The system is semistable (b single unstable The stable vector will necessarily be non-dominant Therefore, all traectories except those starting on the stable eigenvector will be attracted to one of the unstable vectors Hence the system is a saddle 415 Four visible eigenvectors This case is special in the sense that all eigenvectors are visible Thus, there will always be two dominant and two non-dominant eigenvectors visible From that we can immediately infer that it is not allowed that the two non-dominant eigenvectors lie next to each other, otherwise chattering would occur The first two cases are rather trivial: (1 All are stable All traectories will approach a dominant eigenvector or remain on one of the eigenvectors Since that will in any case be a stable eigenvector, the system is a stable node ( All are unstable The reverse is true and the system is an unstable node (3 Three are stable This means that there will be exactly one dominant eigenvector that is stable and one that is unstable Analogously to the case with three visible eigenvectors, this means that the system is semistable (4 Two are stable If stable eigenvectors alternate with unstable eigenvectors, ie both subsystems are saddles, both dominant eigenvectors will be unstable Therefore all traectories that do not start on either of the stable eigenvectors are unstable Hence the system is a double saddle If the subsystems are a stable and an unstable node, there will be a dominant eigenvector that is stable and one that is unstable As before, this implies that the system is semistable (5 One is stable Both dominant eigenvectors will be unstable Hence the system is a saddle References [1] U Boscain Stability of planar switched systems: the linear single input case SIAM J Control Optim, vol 41, no 1, pp 89 11, 00 [] D Holcman and M Margaliot Stability analysis of second-order switched homogeneous systems SIAM J Control Optim, vol 41, no 5, pp 1609 165, 003 [3] D Liberzon, AS Morse Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, vol 19, no 5, pp 59 70, October 1999 [4] AS Morse, ed Control Using Logic-Based Switching Springer-Verlag, 1997 [5] M Pachter and DH Jacobson Observability with conic observation set IEEE Trans Automatic Control, vol AC-4, pp 63 633, August 1979 [6] M Pachter and DH Jacobson The stability of planar dynamical systems linear-in-cones IEEE Trans Automatic Control, vol AC-6, pp 587 590, April 1981 [7] X Xu and PJ Antsaklis Stability of second-order LTI switched systems International Journal of Control vol 73, no 14, pp 161 179, 000
STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING 9 Dept of Electrical and Computer Engineering, University of Toronto, Toronto ON M5S 3G4 Canada E-mail address: hlens@gmxde Dept of Electrical and Computer Engineering, University of Toronto, Toronto ON M5S 3G4 Canada E-mail address: broucke@controlutorontoca Dept of Electrical and Computer Engineering, University of Texas, Austin Tx 7871, USA E-mail address: ari@eceutexasedu