Robust Stabilizing Control Laws For A Class Of Second .
B. Hu, X.Xu, A.N. Michel, P.J. Antsaklis, "Robust Stabilizing Control Law for a Class of Second-orderSwitched Systems," P roc o f t he A merican C ontrol C o nference , San Diego, CA, June 2-4, 1999.Proceedings of the American Control ConferenceSan Diego, California June 1999Robust Stabilizing Control Laws for a Class of Second-orderSwitched SystemsBo HuXuping XuAnthony N. MichelPanos J. AntsaklisDepartment of Electrical EngineeringUniversity of Notre DameNotre Dame, IN 46556, U.S.A.AbstractFor a class of second-order switched systems consistingof two linear time-invariant (LTI) subsystems, we show thatthe so-called conzc switching law proposed previously by thepresent authors is robust, not only in the sense that the control law is fIezible (to be explained further), but also in thesense that the Lyapunov stability (resp., Lagrange stability)properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since theconic switching laws always possess certain kinds of “quasiperiodic switching operations”.We also propose for a class of nonlinear second-orderswitched systems with time-invariant subsystems a switchingcontrol law which locally exponentially stabilizes the entirenonlinear switched system, provided that the conic switchinglaw exponentially stabilizes the linearized switched systems(consisting of the linearization of each nonlinear subsystem).This switched control law is robust in the sense mentionedabove.1. IntroductionSwitched systems are hybrid systems that consist of twoor more subsystems and are controlled by switching laws.These switching laws may be either supervised or unsupervised, time-driven or event-driven, and may be (logically)constrained or unconstrained. Many real-world processesand systems can be modeled as switched systems, includingchemical processes, computer controlled systems, switchedcircuits, and so forth.Recently, there has been increasing interest in the stability analysis of systems of this type (see, e.g., [l -[4], [SI-[lo]).The methodologies used in studying the qu itative properties of switched systems are very diverse. In [I] and [2],multiple Lyapunov functions are introduced and a result forthe stability of a switched system is established. In [8 , Linear Matrix Inequality (LMI) problems are formulated or thestability analysis of switched s stems consisting of linear subsystems. The LMI approach gee, e.g.,. [3] and [8]) proves tobe a very good way to determine su cient conditions for thestability of switched systems with affine subsystems. Otherrelated topics can be found in the survey paper [4] and thereferences therein.Another important issue is the synthesis problem on howt o derive stabilizing switching laws. Thus far, such resultsare quite rare, especially for high-order switched systems. In[8], a “region partition” procedure is mentioned, which is relevant in this regard. Actually, this problem was formulatedin [8] as an LMI problem. The partitioning is possible if asolution to the LMI problem can be obtained. In many cases,however, the LMI problem turns out t o be quite complicatedand the existence of a solution can not be guaranteed. In [9],conic switching laws were proposed to study second-ordera1’Supported in part by a Center of Applied Mathematics Fellowship, University of Notre Dame.*Corresponding author. Supported in part by an Alexandervon Humboldt Foundation Senior Research Award, Institut furNachrichtentechnik, Ruhr-Universitat Bochum, Germany.0-7803-4990-6/99 10.00 0 1999 AACClinear time-invariant switched systems, and for switched systems whose subsystems have unstable foci, both necessaryand sufficient conditions for stabilizability were established.This method can also be extended t o study switched systemsconsisting of LTI subsystems not necessarily with foci (see,[lo]). We point out that by following the procedure in [SI,it can be shown that for a given second-order switched system consisting of two linear time-invariant subsystems withunstable foci, the system still may or may not be stabilizable if the LMI problem has no solution. This reinforces thefact that the approach involving LMI yields only sufficientconditions. Clearly, necessary and sufficient conditions forsecond order LTI switched systems have advantages over theexisting results in the literature.In the present paper, we study the robustness propertiesof the conic switchin control laws. For LTI switched sys(refer also t o Section 2) that the conictems, we know fromswitching control laws rely heavily on the switching information at the boundaries of certain conic regions. It has notbeen shown rigorously whether conic control laws can stillstabilize an entire switched system if the switching boundaries are not precisely reached when switching occurs. Alsonot answered is the question whether or not the stabilizingproperties will be preserved in the presence of perturbations,either vanishing or nonvanishing. The answers to the abovequestions are affirmative and are given below. We show inSection 3 that for LTI switched systems the conic switchinglaws are endowed with a certain kind of robustness property,either in the sense that these event-driven control laws havecertain flexibility on switching regions, or in the presence ofvanishing/nonvanishing perturbations, or a combination ofboth.In addition t o the above, in a more interesting problemwe ask whether or not we can determine conic switching lawsfor nonlinear switched systems and whether or not the conicswitching laws are still robust. We will show that the answerto this question is also affirmative. For a class of second-ordertime-invariant nonlinear switched systems whose linearizedsubsystems have unstable foci, we propose a conic switching law in Section 4 and show that this switching law notonly locally stabilizes the entire system, but also possessesrobustness properties similar to those discussed in Section 3.To demonstrate our results, we present some numericalexamples along with simulations in Section 5.For clarity of presentation, we will primarily address inthe present paper switched systems consisting of two subsystems. We point out, however, that similar results canalso be established for systems consisting of more than twosubsystems.Due to space limitations, all proofs are omitted. Readers interested in such technical details, should refer to thefollowing web site:http://uuv.nd.edu/-bhul/papers.html[jConic Switching Laws for LTISwitched Systems2.In the interests of completeness and clarity, we summarize in the present section the conic switching laws proposed2960Authorized licensed use limited to: UNIVERSITY NOTRE DAME. Downloaded on October 8, 2009 at 15:01 from IEEE Xplore. Restrictions apply.
B. Hu, X.Xu, A.N. Michel, P.J. Antsaklis, "Robust Stabilizing Control Law for a Class of Second-orderSwitched Systems," P roc o f t he A merican C ontrol C o nference , San Diego, CA, June 2-4, 1999.in [9]. For simplicity, we consider here only second-orderswitched systems consisting of two LTI subsystems with foci,even though the method is also applicable to the case of several subsystems that are not necessarily with foci. As in[lo], we say that a subsystem is of clockwise (counterclockwise) direction if starting from any nonzero initialcondition in the phase plane its trajectory is a spiral aroundthe origin in the clockwise (counterclockwise) direction.Consider two switched systems,k ( t ) A i ( t )k,( t ) A z z ( t ) ,(2.1)whose subsystems are both assumed to have unstable foci. )a nonzero point in the W2 plane, andLet z ( 2 1 , bedenote f i ( z ) Alz ( u I , u f) i ( z, ) A2z ( 3 , a 4 ) .Ex,Fig. 1 The angle 0;.We view z, f1 and f 2 as vectors in W2 and define the angle & , i 1 , 2 to be the angle between 3: and f i measured counterclockwise with respect to 3: (& is confined to- A 5 Oi A ) . Thus, Bi is positive (negative) if f;, as avector, is to the counterclockwise (clockwise) side of x (seeFig. 1). Also as in [9], we define the regionsIt can be shown that in R I , 0 3 , 0 5 we have 1011 2 1821 whilein Rz,0 4 , R6, 101 5 1021. Using the basic idea describedabove, we obtain t e conic switching law proposed in [9].bConic switching law: switch to subsystem 1 whenever thesystem state enters 0 1 , R3, 0 5 and switch to subsystem 2whenever the system state enters a2, 0 4 , 0 6 .The advantage of the conic switching law is shown in thefollowing theorem which concerns the stabilizability of theswitched system. Note that this result constitutes a necessary and sufficient condition as opposed to other resultsgiven in the literature.Theorem 2.1. Let 11 be a ray that goes through the origin.Let z o # 0 be on 11. Let z* be the point on 11 where the trajectory intersects 11 for the first time after leaving zo, whenthe switched system evolves according to the conic switching law. The switched system (2.1) with subsystems of thesame direction is asymptotically stabilizable if and only if11zf112 1)zo112 is realized by the conic switching law.0Case 2. T w o Subsystems of Opposite DirectionsAssume that subsystem 1 is of clockwise direction whilesubsystem 2 is of counterclockwise direction.The basic idea for determining an asymptotically stabilizing switching law is motivated by the following. Observethat in any conic region where 101I 102 I 2 'IF (see Fig.3 (b)),the following trajectory will be bounded, where the trajectory starts from z o in the conic region and evolves followingsubsystem 1 and then switches to another subsystem uponintersecting the boundaries of the conic region. This basicidea is formalized below.We introduce the following conic regions: o10 30 4R50 6Clearly, the interior of Ei, (E;,,)is the set of all points in theW2 plane where the trajectory of the ith subsystem would bedriven closer to (farther from) the origin if the subsystemevolves for sufficiently small amount of time starting fromthe point.To design stabilizing switching control laws, we identifythe following two different cases.V)U(0)@)Fig. 2 (a) Figure for Case 1. (b) Figure for Case 2.Case 1. T w o Subsystems of the S a m e DirectionWithout loss of generality, we assume that both subsystems of (2.1) are of clockwise direction.We now consider a switching law that asymptotically stabilizes the switched system. In other words, with our switching law, we desire to drive the trajectory closer and closerto the origin, i.e., Ilz(t)ll2 0 as t 3 00. It is intuitivethat we may want to try to associate with each point z E W2a subsystem such that the absolute value of angle B of thesubsystem is greater than 101 of another subsystem (Fig.2 (a)shows the case 1011 2 l&l). This basic idea is formalized inthe following.We define the following conic regions:RiQ3R40 5026 Els nE2,,,R2 E l , n E2 ,EIS n Ez, n ( 2 1 U z a 3 - ala4 5 0 } ,El, n E2, n { Z I U U - ala4 2 o},El,, r) Ez,, 0 { Z ) a z a s - ala4 I0},Elu n Ez, n { z I u u - a l a 4 2 0).oz El,, n2 0}, Els n E2, n {zla m- a l a 4 I 0}, El,, n Ez, n ( z l a z a 3 - a l a 4 1 O}, El,, f l E28 fl (z1aza3 - a l a 4 5 0). 1 2 0104 3The next result concerns the stabilizability of theswitched system.T h e o r e m 2.2. The switched system (2.1) with two subsystems of opposite directions is asymptoticall stabilizable ifand only if I n t ( R 1 ) U 1 7 4 0 3 ) U I n t ( R 5 ) # where Int(S-2)0denotes the interior of set R.Conic switching law: first, by following subsystem 1, forcethe trajectory into the interior of one of the conic regions0 1 , R3, &(there must be one available), and then switchto another subsystem upon intersecting the boundary of theregion so as to keep the trajectory inside the conic region.-V01 E n, E , , E18 n Eau n (3. Robustness Analysis of Conic Switching Laws for LTI Switched SystemsIn the present section we investigate the robustness problem of the previous proposed switching control law for LTIswitched systems. We focus our attention on switched systems consisting of two subsystems of opposite directions.Similar arguments can be applied to switched systems consisting of two subsystems of the same directions.We call the conic regions in Section 2, R I , 523, 0 5 , saferegions, since the existence of the interior of such regionsguarantees the existence of a stabilizing switching controllaw.The reason that the conic switching law applies lies inthe fact that there exists a safe region R (see Fig. 3) suchthat for every point X I E 11 C 80, by following an appropriate subsystem (for example, we assume subsystem A1 in thesubsequent discussion), the trajectory will intersect anotherboundary at 1 2 E 12 C 8R; then switch to another subsystem A2 until it intersects 11 again at 2 3 E 11. From [SI,weknow that if there exists a switching control law which stabilizes the entire switched system then the following condition2961Authorized licensed use limited to: UNIVERSITY NOTRE DAME. Downloaded on October 8, 2009 at 15:01 from IEEE Xplore. Restrictions apply.
B. Hu, X.Xu, A.N. Michel, P.J. Antsaklis, "Robust Stabilizing Control Law for a Class of Second-orderSwitched Systems," P roc o f t he A merican C ontrol C o nference , San Diego, CA, June 2-4, 1999.is satisfied: z 3 q z l for some constant 0 q 1. Fromthis, we know that if such a switching control law exists, itexponentially stabilizes the entire switched system.3.1. Robustness for switchings onlyRobustness Question 1: In view of the previous discussion, it is required that switchings occur exactly at timeswhen a trajectory intersects 11 or Z2. Can this requirement bemade more flexible? This gives rise to the following question:are there any marginal conic regions RI and R2 that include11 and 1 2 , respectively (see Fig. 4), so that any switchinghappening inside these two regions will lead to exponentialstability?x1Fig. 3 Switching happens on the boundaryof the safe region R: 11 and I z .In the present section, we first study switched systemsdescribed byk ( t ) A;z(t), i 1,2,where A1 and AZ are of opposite directions. Without lossof generality, we assume the following conic switching law:for any z o E Rz,subsystem A2 is first activated until thetrajectory intersects 1 1 , and then proceeds following the conicswitching law described above.Before going further, we introduce three lemmas. Thesepreliminary results are frequently resorted to in the subsequent qualitative analysis.1 t.1.L e m m a 3.1. Let k ( t ) A z (. t.) be a LTI system with focus,where A :pX;(3.1)The solution with z(to) zo#0has the following properties: If a 0 and ,f3 0, thenthe solution z ( t ) eA(t-tO)zo is a logarithmic spiral thatconverges to the origin clockwise; If LY 0 and p 0, thenthe solution z(t) eA(t-tO)zO is a logarithmic spiral thatconverges to the origin counterclockwise; If a 0 and 0,then the solution z ( t ) eA(t-*o)zo is a logarithmic spiralthat diverges to 00 clockwise; If a 0 and p 0, thenthe solution z ( t ) eA('-'0)z0 is a logarithmic spiral that0diverges to m counterclockwise.L e m m a 3.2. Consider an autonomous system k ( t ) Az(t) g(z(t))(3.2)where g E C, i.e., g is continuous, and g ( 0 ) 0. For an initialvalue %(to) z o # 0, Iet T 0 denote the time required forthe solution of system k ( t ) A z ( t ) to move between tworays I1 and 12 once (see, e.g., Fig. 1) and let T AT 0denote the time period the solution of k ( t ) A z ( t ) g ( z ( t )takes to move between two rays 11 and 12 once (if possible].We have the following properties:(i) There exists a constant vo 0 such that when 0 Y YO, then for every E 2 0, whenever 11g(z)11 5 1 1 1 1e is satisfied, there exists a constant K 0 so that whenthe trajectory is outside the disc BK,,it proceeds alon aspiral-like curve similarly to the solution of k ( t ) Azrt).Furthermore, if E 5 1, for a trajectory outside the disc BK&there exist two constants C1, C2 0 (independent of v) suchthat lAT( 5 C1v C z G . 0, then there exists a constant(ii) If lim, oTO 0 such that when 0 T T O , each solution startinginside B, {z E R2 : llzll T } goes towards the outside ofB, (or converges to 0) along a spiral-like curve similar to the0solution of k ( t ) A z ( t ) . e L e m m a 3.3. For systems described by k ( t ) A z ( t )g ( z ( t ) ) and initial condition ( t o , z o ) , if 11g(z)11 5 vllzllE,then it is true that Fig. 4 Switching occur within the conic regionsRI and R2.It is clear from Fig. 4 that such marginal regions arecharacterized by angles B;j 0, i, j 1 , 2 , and in fact fori 1,2, R, {z E w21z (Tcose,Tsine)T,-eil e 4 2 , O T m}. We need to establish the existence of O i jthat guarantees the robustness of the switching control law.To answer the above questions, for solutions beginningfrom any initial condition (to, zo), we assume that the trajectory follows subsystem A2 for ti - t o time until it switchesat xi eAZ(tl-to)zo E R I . Then it follows subsystem A1for t2 - tl time until it switches at xi e A 1 ( t z - t l ) z ; E R?.Next, it switches back to subsystem A2 for t3 - t i time untilit arrives at z eAz(t3-tZ)z; E R I , and so forth.Assume that from any point z1 E Z I , it takes TI timeto arrive at 2 2 eAIT1zl E 12 while following subsystemA and it takes T2 time to return to 11 at 2 3 eAZTZz2 eAZTZeAiT1z1E I I . Clearly, TI and T2 are independent ofthe choice of 2 1 . As before, we assume that z 3 q z l forsome constant 0 q 1. That is, e A Z T Z e A I T 1 z l q z l ,which implies that q is an eigenvalue of matrix e A z T 2 e A l T 1 .It is also clear that there exist quantities At,, At21, At22,At3, which might be negative, and points 5 2 1 , 6 2 2 E 12 andz 1 , 2 3 E 11 such that z1 eAlAtlz; E Zl, z; eAiAtz1z21,z22 eAzAtzz x 2 E 1 2 , 2 3 e A z ( T z A t 3 ) z 2 2 E 11, 213 e A Z A t 3 x 3 , where t 2 - t i At1 T IAt,,, t3 - t z AAt22 TZ At3. Denoting 0 (811,012,021,022)(see Fig.4), we have the following Observation: There exists a nonnegative continuous function c(e) 2 0 satisfying lime o c(0) c(0) 0 such thatmax(lAt11, ( A t (At221,l,lAt3l)5c(0).(3.3)This observation follows intuitively from the idea given inthe proof of Lemma 3.2.Now due to the quasi-periodicity of the switching law(i.e., it switches back and forth for almost the same periodsof time 2'1 and T2, respectively), it suffices to show that thereexist switching regions RI, RZ (i.e., 811, 012, 8?1,.021 i0 )such that no matter when the switchings occur within regionsRI and R2, it is true that11zL115q111z:Il with a constant0 q1 1.To see this, we computez e A z ( t 3 - t z ) e A ( t z - t i ) ; eAz(Atzz Tz Ats)eAi ( A t i T i A t z i )21- e A z ( A t z z A t 3 ) e A z T z e A eiA i ( A t i A t z i ) z 1- eA (Atzz At3)eA TzeA Ti e A z ( A t A t 3 ) e A T e(A, A lT( Al t i A t z ) - I) .- eAz ( A t z z A h ) e A eAiT TI z 1 e A ( A t A t 3 ) e A T e A T l( I - e A I A t l X.2962Authorized licensed use limited to: UNIVERSITY NOTRE DAME. Downloaded on October 8, 2009 at 15:01 from IEEE Xplore. Restrictions apply.(3.4)
B. Hu, X.Xu, A.N. Michel, P.J. Antsaklis, "Robust Stabilizing Control Law for a Class of Second-orderSwitched Systems," P roc o f t he A merican C ontrol C o nference , San Diego, CA, June 2-4, 1999. eAz(Atzz Ats)eA T2eA1T1(eA1(At1 At21) - 1)xiof the switched systems in the presence of perturbations,including both vanishing and nonvanishing perturbations.- qeAz(Atzz At )eAiAti2'1 , A z ( A z z A ) A z(1T z- eAA i AT' - )xi eAz(Atzz At3)eA2TzeAITl ( e A l (At1 AtZ1) - I) .,where I denotes the identity matrix. It is now not difficultto see that there exists E 0 such that whenmax{(Atll, ( A t z l I ,(AtzzI,l A t 3 ( } 5 E , we have1 l A z ( A t z 2 A t 11 5) e1 , A Z ( A Z Z A ) (1AZTe AZi A tAi T )I1 52,9,( e Ai (At1 A h ) - 411 I ?.Therefore, Ilzill 5 (q
Bo Hu Xuping Xu Anthony N. Michel Panos J. Antsaklis Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556, U.S.A. Abstract For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conzc switching law proposed previously by the
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