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A new approach to robust control of hybrid systems over infinite time.pdf

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1292 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 A New Approach to Robust Control of Hybrid Systems Over Infinite Time Andrey V. Savkin and Robin J. Evans Abstract— The paper presents a new approach to output feedback robust control synthesis problems for hybrid dynamical systems. The hybrid system under consideration is a composite of a continuous plant -control problem and a discrete-event controller. An output feedback on an infinite time interval is considered.
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  1292 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 A New Approach to Robust Control of Hybrid Systems Over Infinite Time Andrey V. Savkin and Robin J. Evans  Abstract—  The paper presents a new approach to output feedbackrobust control synthesis problems for hybrid dynamical systems. Thehybrid system under consideration is a composite of a continuous plantand a discrete-event controller. An output feedback      -control problemon an infinite time interval is considered. The main results are given interms of the existence of suitable solutions to a dynamic programmingequation and a Riccati algebraic equation of the      -filtering type.These results show a connection between the theories of hybrid dynamicalsystems and robust and nonlinear control.  Index Terms— Controller switching,      control, hybrid systems, Ric-cati equations, robust control. I. I NTRODUCTION Hybrid dynamical systems (HDS) have attracted considerableattention in recent years (see, e.g., [1]–[4]). In general, HDS’s arethose that consist of a logical discrete-event decision-making systeminteracting with a continuous-time process. A simple example isa climate control system in a typical home. The on/off nature of the thermostat is modeled as a discrete-event system, whereas thefurnace and air-conditioner are modeled as continuous-time systems.Some other examples include transmissions and stepper motors [3],computer disk drives [1], robotic systems [4], higher-level flexiblemanufacturing systems [5], and intelligent vehicle/highway systems[6].In this paper, we consider robust control problems for a newclass of HDS’s. The HDS under consideration consists of a linearcontinuous-time plant with the control and disturbance inputs and adiscrete-event controller. There are several theoretically interestingand practically significant problems concerning the use of switchedcontrollers. In some situations it is possible to design several con-trollers and then switch between them to provide a performanceimprovement over a fixed controller. In other situations the choiceof linear or nonlinear controllers available to the designer is limitedand the design task is to use the available set of controllers in anoptimal fashion. This latter problem is the type we consider in thispaper and includes, for example, the optimal switching between gearsin a gearbox and the optimal switching between heating and coolingmodes of operation in an air-conditioning plant. The controller isdefined by a collection of given nonlinear output feedback controllerswhich are called basic controllers. Then, our control strategy is arule for switching from one basic controller to another. The controlgoal is to achieve a level of performance defined by an integralperformance index. This integral performance index is similar to therequirement in standard      -control theory (see, e.g., [7]–[10]). Weobtain a solution for an output feedback problem on an infinite timeinterval. The main result is given in terms of the existence of suitablesolutions to a Riccati algebraic equation of the      -filtering type and Manuscript received August 22, 1995. This work was supported by theAustralian Research Council.A. V. Savkin is with the Department of Electrical and Electronic Engineer-ing, University of Western Australia, Nedlands, Perth, WA 6009, Australia.R. J. Evans is with the Department of Electrical and Electronic Engineeringand Cooperative Research Center for Sensor Signal and Information Process-ing, University of Melbourne, Parkville, Victoria 3052, Australia.Publisher Item Identifier S 0018-9286(98)05813-9. a dynamic programming equation. If such solutions exist, then it isshown that they can be used to construct a corresponding controller.Riccati algebraic equations have been widely studied in controltheory, and there exist reliable methods for obtaining solutions. Thesolution to dynamic programming equations has been the subject of much research in the field of optimal control theory. Furthermore, in[11] a method for obtaining numerical solutions has been proposedfor dynamic programming equations of the type considered in thepaper. Since dynamic programming equations and      -type Riccatiequations are well known in the modern robust and nonlinear controltheories (see, e.g., [7], [9], [10], [12]), this paper shows that thesetheories when suitably modified, provide a framework for studyingHDS.II. P ROBLEM  S TATEMENT We consider the linear system defined on the infinite time interval                                                                      (1)where              is the  state ,               and             are the disturbance inputs ,              is the  control input  ,             is the  controlled output  ,             is the  measured output  , and        and    are given matrices. Controlled Switching:  Suppose we have a collection of givennonlinear output feedback controllers                                  111                   (2)where     1      1   111      1   are given continuous functions from     to       such that            111         Thecontrollers in (2) are called  basic controllers . We will consider thefollowing class of output feedback controllers. Let      be a giventime. Let         and     1   be a function which maps from theset of the output measurements      1         to the set of symbols         111      Here      1         denotes the set of all possible measuredoutputs       on the interval        Then, for any sequence of suchfunctions             we will consider the following dynamic nonlinearoutput feedback controller:                111                               where            1         (3)As above, our control strategy is a rule for switching from one basiccontroller to another. Such a rule constructs a symbolic sequence           from the output measurement    1    The sequence           is called a switching sequence.  Notation:  In this paper, 1   denotes the standard Euclidian norm.Also, for any                     denotes the Hilbert space of squareintegrable vector-valued functions defined on        Definition 2.1:  Consider system (1). Let             and            be given matrices. Suppose that there exist constants     and      and a function           such that                    forall            , and for any vector            there exists a controller of the form (2) and (3) such that the following conditions hold.1) For any initial condition     and disturbance inputs     1     1              the closed-loop system (1)–(3) hasa unique solution which is defined on        0018–9286/98$10.00  󰂩  1998 IEEE  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 1293 2) For any solution       to the closed-loop system (1)–(3) with     1     1               as     3) The inequality                              0             0                        0                   (4)holds for all         and for all solutions to the closed loopsystem with any disturbance inputs     1     1              Then, the output feedback robust control problem defined by thematrices    and    is said to have a solution via controlled switchingwith the output feedback basic controllers (2).  Remark:  The problem defined in Definition 2.1 is similar to the     -control problem with transients (see, e.g., [9]) with a specialclass of output feedback controllers.III. T HE  M AIN  R ESULT Our solution to the above problem involves the following Riccatialgebraic equation:                        0                  0          (5)Also, we consider a set of state equations of the form                     0                                          (6)  Notation:  Let    1   be a given function from       to     and let           be a given vector. Introduce the following cost function:                    1                         0            0                    0        (7)Then            1  1       1           1                                       (8)where the supremum is taken over all solutions to the system (32) with    1                             and initial condition           Now we are in a position to present the main result of this paper. Theorem 3.1:  Consider system (1) and the basic controllers (2).Let             and             be given matrices. Suppose that           is controllable,      is observable, and     1  1  is defined by (8). Then, the following statements are equivalent.1) The output feedback robust control problem defined by thematrices    and    has a solution via controlled switching withoutput feedback basic controllers (2).2) There exists a constant         and solution        to the Riccatiequation (5) such that the matrix   1                 0        is stable and the dynamic programming equation               111                 1  (9)has a solution            such that        and                          for all              Furthermore, suppose that condition 2) holds and let          be anindex such that the minimum in (9) is achieved for             ,and let      1   be the solution to the (6) with the initial condition           Then the controller (2) and (3) associated with theswitching sequence            , where   1               , solves the outputfeedback robust control problem defined by the matrices       and   with           0                        0            and      1         1   In order to prove this theorem, we will use the following lemma.  Lemma 3.1:  Let               and                 be givenmatrices,         be a given number,            be a given vector,          be a given constant, and     1   and     1   be given vectorfunctions. Suppose that the solution    1   to the Riccati equation                                  0                   0       (10)with initial condition       0    is defined and positive definiteon the interval        Then, condition                                 0             0                    0              0               (11)holds for all solutions to the system (1) with    1       1   and    1       1  if and only if                                          0                 0        0          0               (12)for all             , the cost function (7), and the solution to the equation                      0                                                 (13)with    1       1     1       1   and initial condition           The proof of Lemma 3.1 is given in the Appendix. Proof of Theorem 3.1 [ 1)    2)]:  If condition 1) holds, then forany          the controller (2) and (3) corresponding to      is a solution to the following output feedback finite time interval     -control problem:   1       1     1           1   0                                                                         (14)where      is a constant such that condition (4) holds. Now considerthe disturbance input    1  0     1    Then, we have    1       and    1       Therefore, from (14) we have   1      1           1      0               0                                                             1294 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 Hence, it follows from [12, Lemma 5] that the solution      1   to theRiccati equation                                      0          0                     0       with initial condition          is defined and positive definite on        Since condition (11) with         and          holds forany         and    1   and    1   connected by the controller (2) and(3), Lemma 3.1 implies that (12) holds. From (12) we obtain that         0              (15)Furthermore, we can take        and obtain from the continuity of the solutions to the Riccati equations with respect to the parameterson a finite-time interval and condition (15) that there exists a constant        that for any     the solution     1   to the Riccati equation(10) with initial condition         is defined and positive definiteon        and        0              (16)Now let      1   be the solution to (10) with initial condition         Then, well-known properties of the Riccati equation (10) (see, e.g.,[13]) imply that                        and                             (17)Relations (16) and (17) imply existence of the limit     1                  and     0       Now it is clearthat      is a solution to (5). Also, since            as     ,we have that      is the minimum constant solution. Hence,      isa stabilizing solution, i.e., the matrix              0         isstable (see, e.g., [13]).Now let      be a constant such that (4) holds and         Wehave proved above that there exists a solution to (10) with initialcondition        and            as      (18)Let    be the class of all controllers of the form (2) and (3) and let        be fixed. Introduce for any         111    the followingfunction:           1       1           1           1                                         (19)where the supremum is taken over all solutions to (13) with ini-tial condition               According to the theory of dynamicprogramming (see, e.g., [7] and [14])      1   satisfies the equations                                       111                 1  where            1  1       1               1                                             (20)and the supremum is taken over all solutions to the system (13)with    1                                  and initial condition              Now we prove that there exists a function    1      such that      and                              (21)Indeed, condition (12) of Lemma 3.1 together with the definition(19) imply that                                          (22)It can be seen from (19) that                          1                                    1              (23)Now consider an input         for           Then,       for          and for all    1       and the inequality (23) implies                                         0                       (24)for the solutions to (13) with        and         Since      is astabilizing solution, condition (18) implies that    1   is a stabilizingsolution to the Riccati equation (10). Hence, system (13) with       and        is exponentially stable. From this and the inequalities(22) and (24), we have that there exists a constant      such thatcondition (21) holds with                              Also, it followsfrom (19) that                             (25)Hence, from (21) and (25) we have existence of the limit          1                             Clearly     1   is a solutionto the equation              111                1   Now conditions (18) and (21) imply existence of the limit           1                , and      1   is a solution to equation (9).Also, it follows from the above that                          and        This completes the proof of this statement. 2)    1):  Equation (9) implies that for the controller associatedwith the switching sequence            defined by (9), we have                                   0                         0                   Furthermore, Lemma 3.1 implies that (11) holds with         0                      0            and           Hence, condition (4) holdswith           0      and           Conditions 1) and 2) of Definition 2.1follow immediately from inequality (4), assumption         , andobservability of        This completes the proof of the theorem.  Remarks: 1) It can be shown, using the methodology of [15], that robuststability of the closed-loop system (1)–(3) in the sense of Definition 2.1 implies input-to-state stability: for any initialcondition     and disturbance inputs     1     1               the corresponding solution    1   to the closed-loop system be-longs to              Here             denotes the Banach spaceof measurable vector-valued functions defined and essentiallybounded on        2) The disturbance rejection problem (4) is equivalent to some ro-bust stabilization problem (see, e.g., [16]) where the underlyinglinear system is described by the state equations (1). However,in this case,       and       are the uncertainty inputs, and       is  IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 9, SEPTEMBER 1998 1295 the uncertainty output. The uncertainty inputs       and       arerequired to satisfy a certain integral quadratic constraint. Then,Theorem 3.1 gives a solution for the corresponding problem of robust stabilization via output feedback switching control (2)and (3).A PPENDIX Proof of Lemma 3.1:  Given an input–output pair      1      1   ,if condition (11) holds for all vector functions    1     1    and    1  satisfying (1) with    1       1   and such that                              (26)then, substitution of (26) into (11) implies that (11) holds if and onlyif            1                 (27)for all    1                         where           1   is defined by           1  1      0              0                     0                         0                 0        (28)and    1   is the solution to (1) with disturbance input    1   andboundary condition             Now consider the following minimization problem:     1                     1  (29)where the minimum is taken over all    1   and    1   related by (1)with the boundary condition              This problem is a linearquadratic optimal tracking problem in which the system operates inreverse time.We wish to convert the above tracking problem into a trackingproblem of the form considered in [17] and [18]. In order to achievethis, first define        to be the solution to the state equations                               (30)Now let        1         0          Then, it follows from (1) and (30) that        satisfies the state equations                           (31)where            Furthermore, the cost function (28) can berewritten as           1               1      0               0                      0                                    0                            0                   (32)where                    0           Equations (31) and (32) nowdefine a tracking problem of the form considered in [17] where     1      1   and     1   are all treated as reference inputs. In fact, the onlydifference between this tracking problem and the tracking problemconsidered in the proof of the result of [18] is that in this paper wehave a  sign indefinite  quadratic cost function.The solution to this tracking problem is well known (e.g., see [17]).Indeed, if the Riccati equation (10) has a positive-definite solutiondefined in       with initial condition       0    then theminimum in     1                        1  will be achieved for any        1    and     1    Furthermore, as in[18], we can write     1                        1       0                  0         0           0                                  (33)where       1   is the solution to state equations                       0                                                          with initial condition             Now let      1  1        1       1   Using the fact that             0          it follows that (33) can berewritten as     1                     1      0                 0        0          0                           where      1   is the solution to state equations (6) with initial condition           From this we can conclude that condition (11) with agiven input–output pair      1      1   is equivalent to the inequality(12).R EFERENCES [1] A. Gollu and P. P. Varaiya, “Hybrid dynamical systems,” in  Proc. 28th IEEE Conf. Decision and Control , Tampa, FL, 1989.[2] P. J. Antsaklis, J. A. Stiver, and M. Lemmon, “Hybrid systems modelingand autonomous control systems,” in  Hybrid Systems , R. L. Grossman,A. Nerode, A. P. Ravn, and H. Rishel, Eds. New York: Springer-Verlag, 1993.[3] R. W. Brockett, “Hybrid models for motion control systems,” in  Essaysin Control , H. L. Trentelman and J. C. Willems, Eds. Boston, MA:Birkhauser, 1993.[4] A. Back, J. Guckenheimer, and M. Myers, “A dynamical simulationfacility for hybrid systems,” in  Hybrid Systems , R. L. Grossman, A.Nerode, A. P. Ravn, and H. Rishel, Eds. New York: Springer-Verlag,1993.[5] S. B. Gershwin, “Hierarchical flow control: A framework for schedulingand planning discrete events in manufacturing systems,”  Proc. IEEE  ,vol. 77, no. 1, pp. 195–209, 1989.[6] P. P. Varaiya, “Smart cars on smart roads: Problems of control,”  IEEE Trans. Automat. Contr. , vol. 38, pp. 195–207, 1993.[7] T. Basar and P. Bernhard,      -Optimal Control and Related Min-imax Design Problems: A Dynamic Game Approach . Boston, MA:Birkh¨auser, 1991.[8] I. R. Petersen, B. D. O. Anderson, and E. A. Jonckheere, “A firstprinciples solution to the nonsingular      control problem,”  Int. J. Robust and Nonlinear Control , vol. 1, no. 3, pp. 171–185, 1991.[9] P. P. Khargonekar, K. M. Nagpal, and K. R. Poolla, “      controlwith transients,”  SIAM J. Control and Optimization , vol. 29, no. 6, pp.1373–1393, 1991.[10] M. R. James and J. S. Baras, “Robust      output feedback controlfor nonlinear systems,”  IEEE Trans. Automat. Contr. , vol. 40, pp.1007–1017, 1995.[11] A. V. Savkin, R. J. Evans, and I. R. Petersen, “A new approach torobust control of hybrid systems,” in  Hybrid Systems, III—Verificationand Control , Lecture Notes in Computer Science 1066, T. A. Henzinger,R. Alur, and Ed. D. Sontag, Eds. New York: Springer-Verlag, 1996,pp. 553–562.
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