Robust Tracking and Regulation Control of Uncertain Piecewise Linear Hybrid Systems

Transcription

1 ISIS Tech. Rept Robst Tracking and Reglation Control of Uncertain Piecewise Linear Hybrid Systems Hai Lin Panos J. Antsaklis Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA {hlin1, antsaklis.1}@nd.ed Abstract: In this paper, a class of ncertain, discrete-time, piecewise linear hybrid systems affected by both parameter variations and eterior distrbances is introdced. The robst tracking and reglation control problem for sch ncertain piecewise linear hybrid systems is investigated. The main qestion is whether there eists a controller sch that the closed-loop system ehibits desired behavior nder dynamic ncertainties and eterior distrbances. If a specified behavior can be forced on the plant by a control mechanism, then it is called attainable. We present a method for checking attainability that employs the predecessor operator and backward reachability analysis, and introdce a procedre for robst tracking and reglation controller design that ses linear programming techniqes. This procedre is then sed to solve the robst stabilization problem for ncertain piecewise linear hybrid systems. Keywords: Tracking and Reglation Control, Constrained Reglation, Uncertainty, Piecewise Linear/Affine Systems, Safety, Reachability, Controller Synthesis, Persistent Distrbance The partial spport of the National Science Fondation (NSF ECS & CCR ) is grateflly acknowledged. The first athor appreciates the spport from Center of Applied Mathematics Fellowship ( ), University of Notre Dame. 1

2 1 Introdction Hybrid Systems are heterogeneos dynamical systems of which the behavior is determined by interacting continos variable and discrete event dynamics. Hybrid systems have been identified in a wide variety of applications in control of mechanical systems, process control, atomotive indstry, power systems, aircraft and traffic control, among many other fields. Piecewise linear (affine) systems are recognized as an important model frame for hybrid systems and have been widely stdied in the literatre; see for eample [26, 13, 6, 16] and the references therein. The isses stdied inclde modeling [5, 26], stability [17, 10, 13], observability and controllability [6, 26] primarily. Piecewise linear systems arise often from linearizations of nonlinear systems. Notice, however, that for an important class of practical nonlinear systems with parameter variations the nmber of piecewise linear approimations may become prohibitively large. One way to stdy sch ncertain nonlinear systems in a systematic way is to se a bndle of parameterized linearizations, which cover the original ncertain nonlinear dynamics within a region. The model mismatching may be tackled by introdcing additive distrbances. This motivates s to stdy ncertain piecewise linear systems with distrbances as described in Section 2.1. The dynamic ncertainty and robst control of hybrid/switched systems is a highly promising and challenging field. However, the literatre on this topic is relatively sparse. Some of the contribtions inclde modelling ncertain hybrid/switched systems, reachability analysis, stability analysis and so on. For eample, implse differential inclsions were proposed as a modeling framework for ncertain hybrid/switched systems in [4], where some theoretic reslts for viability and invariance analysis in classical differential inclsions were etended to the implse differential inclsions. Reachability analysis reslts for ncertain hybrid/switched systems have appeared in [14, 19]. In addition, the safety and reachability problems for ncertain linear hybrid systems were stdied in [20] based on backward reachability analysis techniqes. There are also a few related pblications on the robst controller design. In [24], the athors gave an abstract algorithm, based on modal logic formalism, to design the switching among a finite nmber of continos variable systems. It was shown that the closed-loop system formed a hybrid atomata and satisfied certain specifications robstly. In [12], the athors proposed logically spervised switching mltiple controllers to control ncertain dynamical systems, while the closed-loop system forming a class of ncertain switched systems. The advantages of switching controllers over classical adaptive controllers were discssed in [12] as well. These advantages partially eplain the increasing interest in switched systems dring the past decade. For robst stabilization of ncertain switched systems, a qadratic stabilizing switching law was designed for polytopic ncertain switched systems based on LMI techniqes in [28]. Some related work on the indced gain 2

3 analysis for switched linear systems has appeared for eample in [11, 27]. In this paper, we investigate the tracking and reglation control problem for a class of discrete-time ncertain piecewise linear hybrid systems, which is affected by both parameter variations and eterior distrbances. The control objective is for the closed-loop system to ehibit certain desired behavior despite the ncertainties and distrbances. Specifically, given finite nmber of regions {Ω 0, Ω 1,, Ω M } in the state space, or goal is for the closed-loop system trajectories, starting from the given initial region Ω 0, to go throgh the seqence of finite nmber of regions Ω 1, Ω 2,, Ω M in the desired order and finally reach the final region Ω M and then remain in Ω M. This kind of specifications is analogos to the ordinary tracking and reglation problem in pre classical continos variable dynamical control systems. In addition, it also reflects the qalitative ordering of event reqirements along trajectories. In Section 2, discrete-time ncertain piecewise linear systems are defined, and the robst tracking and reglation control problem for sch ncertain piecewise linear system is qalitatively formlated. One of the main qestions is to determine whether there eist admissible control laws sch that the region-seqence can be followed. If sch an admissible control law eists, the region-seqence specification {Ω 0, Ω 1,, Ω M } is called attainable. The attainability checking is based on backward reachability analysis and symbolic model checking method discssed in Section 3. In Section 4, necessary and sfficient conditions for checking attainability are given. An optimization based method is proposed in Section 5 to design sch admissible control laws. In Section 6, the robst stabilization control problem for ncertain piecewise linear systems is formlated and solved as a specific application of the robst tracking and reglation control method developed here. Finally, conclding remarks are given. Note that some preliminary reslts were presented in [19, 20]. However, in this paper we propose a method for specification refinement and eplicitly deal with non-conveity in the controller design. Notation : ApolyhedroninR n is a (conve) set given by the intersection of a finite nmber of open and/or closed half-spaces in R n. A polytope is a closed and bonded (i.e. compact) polyhedron. A polyhedral set P will be presented either by a set of linear ineqalities P = { F i g i, i =1,,s}, compactlyp = { F g}, or by the dal representation in terms of its verte set { j }, denoted as vert{p}. A piecewise linear set consists of a finite nion of polyhedra. 2 Problem Formlation In this section, the discrete-time ncertain piecewise linear systems are defined and the robst tracking and reglation control problem is formlated. 3

4 2.1 Uncertain Piecewise Linear Systems We consider discrete-time ncertain piecewise linear systems of the form (t +1)=A q (w(t))(t)+b q (w(t))(t)+e q d(t), t Z +, if P q (2.1) where (t) X R n, (t) U q R m, d(t) D q R r, w(t) W R v are the system state, the control inpt, the distrbance inpt, and the ncertainty parameter respectively. Let the finite set Q stand for the collection of discrete modes q. It is assmed that X, U q, D q and W are assigned polytopes for each mode q Q, andthatd q contains the origin. The partition of the state space X is given as a finite set of polyhedra {P q : q Q}, where P q X and q Q P q = X. The continos variable dynamics of each mode q is defined by the state matrices A q (w), B q (w) ande q.itisassmedthattheentriesofa q (w) andb q (w) are continos fnctions of w for every mode q. A possible evoltion of the ncertain piecewise linear systems from a given initial condition 0 X can be described as follows. First, there eists at least one discrete mode q 0 Q sch that 0 P q0 ; the mode q 0 is then called feasible mode for state 1 0. The net continos variable state is given by the transition 1 = A q0 (w) 0 + B q0 (w) + E q0 d for some w W, d D q0 and specific U q0. Then the above procedre is repeated for state 1 to determine the net possible state 2,andsoon. 2.2 Robst Tracking and Reglation In continos variable dynamical control systems, the tracking and reglation problem is a classical control problem. It can be formlated as follows: Given an initial region, a target region and a pipe connecting these two regions in the state space, design a control law so that all the trajectories starting from this initial region will be driven to the target region throgh the pipe. In the case of ncertain piecewise linear systems, the tracking and reglation problem can be formlated in an analogos fashion. However instead of connecting the initial region and the target region by a pipe, we se a seqence of connected polyhedral regions Ω i, which may be seen as inner approimations of the pipe specification (See Figre 1). The problem considered in this paper is to select feasible modes q(t) and to design admissible control signals (t) U q(t) sch that the closed-loop piecewise linear systems trajectories, starting from the given initial region Ω 0, go throgh the seqence of regions 1 In the definition of ncertain piecewise linear systems, it is not reqired that the partition P q has mtally empty intersections. Therefore, for the initial state 0 there may eist more than one feasible discrete modes. In sch cases, it is assmed that the crrent active mode, q 0, is randomly selected from these feasible modes. 4

5 Ω2 Ω3 ΩM Ω1 Ω0 Figre 1: Tracking and reglation control specification and its polyhedral region seqence approimation. Ω 1, Ω 2,, Ω M in the desired order, finally reach the final region Ω M andthenremain in Ω M, in spite of ncertainties and distrbances. Note that the region Ω i X does not necessarily coincide with the partitions P q in the definition of the ncertain piecewise linear systems (2.1). We propose to solve the robst tracking and reglation problem for the ncertain piecewise linear systems (2.1) in three stages. At the first stage, we check whether there eist feasible modes q(t) and admissible control signals, ((t)) U q(t), sch that the region-seqence specification is satisfied despite the ncertainties and distrbances. If there eists sch admissible control laws to satisfy the tracking and reglation specification, then the problem specification is called attainable. In order to check attainability, two different kinds of properties shold be checked, namely the direct reachability between two sccessive regions, Ω i and Ω i+1 for 1 i<m,andthe safety (or controlled invariance) for the final region Ω M. The analysis problems for safety and direct reachability are formlated as follows. Safety: Given a region Ω X, determine whether there eist admissible control laws sch that the evoltion of the system starting from Ω remains inside the region for all time, despite the presence of dynamic ncertainties and distrbances. Direct Reachability: Given two regions Ω 1, Ω 2 X, determine whether there eist admissible control laws sch that all states in Ω 1 can be driven into Ω 2 in finite steps withot entering a third region. If the problem specification is attainable, an appropriated controller is designed in Stage three. Otherwise, if the problem specification, which may be given as an initial inner polyhedral approimation of the pipe specification (see Figre 1), is not attainable, we propose a procedre in Section 4 to refine the original specification. The refined seqence of regions, if non-empty, are garanteed to be attainable. 5

6 The third stage is to design the controller to satisfy the tracking and reglation specification, if the specification is attainable or has been sccessflly refined. Similarly, the controller synthesis stage is also divided into two basic problems, that is safety control and direct reachability control. The two basic control problems are formlated as follows. Safety Controller Synthesis: Given a safe region Ω X, determine the admissible control laws to make the region Ω safe (controlled invariant); Direct Reachability Controller Synthesis: Given two regions Ω 1, Ω 2 X,whereΩ 2 is directly reachable from Ω 1, determine the admissible control laws sch that all the states in Ω 1 can be driven into Ω 2 in finite steps withot entering a third region. In the following sections, we will solve the robst tracking and reglation problem stage by stage. First, necessary and sfficient conditions for safety and direct reachability are given in Section 4. The safety and direct reachability checking are based on backward reachability analysis. In the net section, we will briefly discss the backward reachability analysis, which serves as one of the basic tools for the analysis that follows. 3 Robst Backward Reachability Analysis This section describes the reslts of backwards reachability analysis for the ncertain piecewise linear systems, which server as the fondation for checking safety, reachability and attainability. Forward/Backward reachability analysis has been sed widely in the stdy of hybrid systems [1, 24, 16], and a good deal of research effort has focsed on developing sophisticated techniqes drawn from optimal control, game theory, and comptational geometry to calclate or approimate the reachable sets for varios classes of hybrid systems [1, 9, 3, 16]. In this section, we will eactly calclate the robst one-step backward reachable set (defined below) for the ncertain piecewise linear systems (2.1). 3.1 Robst One-Step Predecessor Set The basic bilding block to be sed for backward reachability analysis is the robst one-step predecessor operator, which is defined below. Definition 3.1 The robst one-step predecessor set, pre(ω), is the set of states in X, for which admissible control inpts eist and drive these states into Ω in one step, despite distrbances and ncertainties, i.e. pre(ω) = {(t) X q(t) Q, (t) U q(t) : (t) P q(t), A q(t) (w)(t)+b q(t) (w)(t)+e q(t) d(t) Ω, d(t) D q(t),w W} 6

7 We can also define the one-step predecessor set nder the q-th mode, pre q (Ω), as the set of all states P q, for which an admissible control inpt U q eists and garantees that the system will be driven to Ω by the transformation A q (w) + B q (w) + E q d for all allowable distrbances and ncertainties. Proposition 3.1 The robst one-step predecessor set pre(ω) for an ncertain piecewise linear system can be compted as follows: pre(ω) = pre q (Ω) q Q Proof : Assme that q Q pre q(ω) is not empty. For all the states q Q pre q(ω), there eists at least one mode q Q sch that pre q (Ω) P q. Therefore, by the definition of pre q ( ), there eists control signal U q which drive the state into the set Ω nder the mode q despite the distrbances and ncertainties. So pre(ω) q Q pre q(ω). On the other hand, for the states / q Q pre q(ω), is not contained in any pre q (Ω) for every q Q. Therefore, nder all possible modes q Q, there always eists some distrbance or some ncertain parameter sch that no admissible control signal eists to drive the state into Ω. So pre(ω) q Q pre q(ω). Therefore, we only need to calclate the one-step predecessor set for each q-th sbsystem. 3.2 Predecessor Sets for Sbsystems In the seqel, we will focs on the linear constrained case. It is known that often in practice ncertainties enter linearly in the system model and they are linearly constrained [8]. To handle this particlar bt interesting case, we consider the class of polyhedral sets. Sch sets have been considered in previos works addressing the control of systems with inpt and state constraints. Their main advantage is that they are sitable for comptation. Therefore, in the seqel, we trn to polytopic ncertainty in A q (w) andb q (w) for every mode q Q. Withot loss of generality, we assme that v q A q (w) = wq k A k q,b q (w) = k=1 v q k=1 w k q B k q, where wq k 0and v q k=1 wk q = 1. The pair (A q(w),b q (w)) represents the model ncertainty which belongs to the polytopic set Conv{(A k q,bq k ), k =1,,v q } for each mode q Q. This is referred to as polytopic ncertainty and provides a classical description of model ncertainty. Notice that the coefficients wq k are nknown and possibly time varying. 7

8 The difficlty in calclating pre q (Ω) comes mainly from the fact that the region Ω is typically non-conve. Even if one starts with conve sets, the procedre dedces non-conve sets for piecewise linear systems after an one-step predecessor operation. Becase of the non-conveity, some of the linearity and conveity argments do not hold and etra care shold be taken. However, nder the polytopic ncertainty assmption, the calclation of the predecessor set for piecewise linear sets can be simplified, in view of the following proposition. Proposition 3.2 For polytopic ncertain piecewise linear systems, the robst one-step predecessor set for an assigned piecewise linear set Ω (may be non-conve) nder the q-th sbsystem can be determined from pre q (Ω) = v q k=1 pre k q(ω), where pre k q (Ω) stands for the one-step predecessor operator of the k-th verte state matri (A k q,bk q ) for 1 k v q, i.e. pre k q (Ω) = { P q U q : A k q + Bk q + E qd Ω, d D q }. Proof : See Appendi A. Therefore, we derived the relationship between the robst one-step predecessor operator for the polytopic ncertain systems, pre q ( ), and the one-step predecessor set of the verte dynamics, pre k q ( ) for k =1,,v q. From Proposition 3.2, it trns ot that the robst onestep predecessor set for a piecewise linear set Ω nder polytopic ncertain linear dynamics can be redced to the finite intersection of one-step predecessor sets corresponding to the dynamic matri polytope vertices, which have no parametric ncertainty. The predecessor set nder deterministic linear dynamics, pre k q (Ω), has been stdied etensively in the literatre and can be compted by Forier-Motzkin elimination [25] and linear programming techniqes; see for eample [8, 15] and the references therein. Proposition 3.3 The robst one-step predecessor set for a (non-conve) piecewise linear set Ω, pre(ω), can be written as a finite nion of polyhedra. Proof : When Ω is conve polyhedral set, it is trivially tre. When Ω is non-conve piecewise linear set, it was shown in [15] that pre k q(ω) can be written as a finite nion of polyhedra. Becase pre q (Ω) = k prek q (Ω), hence pre q(ω) can be written as a finite nion of polyhedra 8

9 by repeatedly applying the distribtive law. Therefore, the one-step predecessor set of nonconve piecewise linear set Ω, pre(ω) = q pre q(ω), can be written as a finite nion of polyhedra. Althogh the conveity is not preserved nder the one-step predecessor operation, the piecewise linearity remains nchanged in view of Proposition 3.3. Therefore, one can apply the predecessor operation recrsively, and this is eplored in the net section. 4 Safety, Direct Reachability and Attainability In this section, we first present necessary and sfficient conditions for checking the safety for a given region Ω X and the direct reachability between two given regions Ω 1 and Ω 2.Then a necessary and sfficient condition for checking the attainability of a given specification is presented. Finally, if the original specification can not be satisfied, a refinement procedre is given sch that the refined specifications, if not empty, are garanteed to be attainable. 4.1 Safety The following is an important, well-known geometric condition for a set to be safe (controlled invariant) [8]. Theorem 4.1 The set Ω is safe if and only if Ω pre(ω). Proof : The proof follows immediately from the definition of the predecessor set pre(ω). In general, a given set Ω is not safe. However, Ω may contain safety sbsets. In addition, it follows immediately from the definition that the nion of two safety sets is also safe. Therefore there eists a niqe safety sbset in Ω which contains all the safety sbsets in Ω. It is called maimal safety set in Ω. In order to calclate the maimal safety set in Ω, we introdce the one-step safety set of Ω as C 1 (Ω) = pre(ω) Ω. It follows from Proposition 3.3 that if Ω is a piecewise linear set then the one-step safety set C 1 (Ω) is also piecewise linear. Therefore, the one-step safety set operator can be sed recrsively to define i-step safety set C i (Ω) as follows. C i (Ω) = pre(c i 1 (Ω)) C i 1 (Ω), for i 2 The seqence of finite-step safety sets C i (Ω) has the following property. 9

10 Proposition 4.1 The seqence of finite step safety sets C i (Ω) is decreasing in the sense of C i (Ω) C i 1 (Ω), for i 1andC 0 (Ω) = Ω. The maimal safety set in Ω for polytopic ncertain piecewise linear system (2.1) is given by C (Ω) = C i (Ω). The proof is similar to the proof of Theorem 3.1 in [8], and it is omitted here. i=0 4.2 Direct Reachability Here we stdy the reachability problem for ncertain piecewise linear systems. It shold be emphasized that we are interested only in the case when reachability between two regions Ω 1 and Ω 2 is defined so that the state is driven to Ω 2 directly from the region Ω 1 in finite steps withot entering a third region. This is a problem of practical importance in hybrid systems since it is often desirable to drive the state to a target region of the state space while satisfying constraints on the state and inpt dring the operation of the system. The problem of deciding whether a region Ω 2 is directly reachable from Ω 1 can be solved by recrsively compting all the states that can be driven to Ω 2 from Ω 1 sing the predecessor operator. Given Ω 1 and Ω 2, define the robst one-step controllable set as the set consisting of all the states in Ω 1 for which there eist feasible modes and admissible control signals to drive sch states into Ω 2 in the net one step, for all allowable ncertainties and distrbances, i.e. K 1 (Ω 1, Ω 2 )={(t) Ω 1 q(t) Q, (t) U q(t), : (t) P q(t), A q(t) (w)(t)+b q(t) (w)(t)+e q(t) d(t) Ω 2, d(t) D q(t),w W} The robst one-step controllable set K 1 (Ω 1, Ω 2 ) can be compted as follows: K 1 (Ω 1, Ω 2 )=pre(ω 2 ) Ω 1 It follows from Proposition 3.3 that the robst one-step controllable set K 1 (Ω 1, Ω 2 )is piecewise linear if Ω 1 and Ω 2 are given as piecewise linear sets. Therefore, the robst one-step controllable set operator can be sed recrsively to define i-step controllable set K i (Ω 1, Ω 2 ) as follows. K i (Ω 1, Ω 2 )=K 1 (Ω 1, K i 1 (Ω 1, Ω 2 )), where i 1andK 0 (Ω 1, Ω 2 )=Ω 2. 10

11 With the introdction of the finite step controllable set from Ω 1 to Ω 2, the geometric condition to check the direct reachability can be given as follows. Theorem 4.2 Consider an ncertain piecewise linear systems and the regions Ω 1 and Ω 2. The region Ω 2 is directly reachable from Ω 1 in finite nmber of steps if and only if there eist finite integer N sch that Ω 1 N i=0 K i(ω 1, Ω 2 ). Proof : The proof follows immediately from the definition of the finite-step controllable set K i (Ω 1, Ω 2 ). 4.3 Attainability Given a finite nmber of regions {Ω 0, Ω 1,, Ω M }, the attainability for this seqence of regions is eqivalent to the following two different kinds of properties, namely the direct reachability from region Ω i to Ω i+1 for 0 i<mand the safety for the final region Ω M. Therefore, the following necessary and sfficient condition for attainability is derived. Theorem 4.3 The specification {Ω 0, Ω 1,, Ω M } is attainable if and only if the following conditions hold: First, Ω M is safe; and secondly the region Ω i+1 is directly reachable from Ω i, for i =0, 1,,M 1. Proof : The proof follows immediately from the attainability definition of a specification {Ω 0, Ω 1,, Ω M }. Combined with Theorem 4.1 & 4.2, it is straight forward to derive corresponding geometric conditions for attainability. Let s see a nmerical eample for illstration. Eample 4.1 Consider the discrete-time ncertain piecewise linear hybrid systems, { A 1 (w)(t)+b 1 (w)(t)+e 1 d(t), P 1 (t +1)= A 2 (w)(t)+b 2 (w)(t)+e 2 d(t), P 2. where P 1 = { R 2 100} and P 2 = { R , }. The verte matrices of polytopic ncertain A 1 (w) andb 1 (w) are ( ) ( ) A =,A = ( ) ( ) ( ) B =,B = ,E 1 =,

12 and the verte matrices of polytopic ncertain A 2 (w) andb 2 (w) are ( ) ( ) A =,A = ( ) ( ) ( ) B =,B = ,E 2 = Assme that the constraints of the continos control signal are given as U 1 = U 2 =[ 1, 1], while the bond of distrbance is d D 1 = D 2 =[ 0.1, 0.1]. First, we consider a region Ω 1 = { R 2 10} and calclate the one-step predecessor set of sch region, pre(ω 1 ), following the procedre given in Section 3. The set pre(ω 1 ) is plotted in the left part of Figre 2 as a nion of two polytopes. It can be seen that pre(ω 1 ) Ω 1, therefore the region Ω 1 is safe by Theorem 4.1. Net, we consider direct reachability to Ω 1 from another region Ω 0,givenasΩ 0 = { R , }. It is fond that Ω 0 3 i=0 K i(ω 0, Ω 1 ), therefore Ω 1 is directly reachable from Ω 0 in finite nmber of steps (less or eqal to three steps), by Theorem 4.2. The three-step controllable set from Ω 0 to Ω 1 is calclated and plotted in the right part of Figre one step predecessor set 50 Direct reachable set Figre 2: Left: The illstration of the one-step predecessor set pre(ω 1 ). Right: Three-step controllable set from Ω 0 to Ω 1. By Theorem 4.3, the attainability of the specification, {Ω 0, Ω 1 }, is garanteed. 4.4 Refinement of the Specification We have discssed how to check the attainability of a given region-seqence specification {Ω 0, Ω 1,, Ω M }. However, a given specification {Ω 0,, Ω M } may be not attainable. 12

13 For eample, if we set Ω 0 = { R , } and N =3in the previos eample, then N i=0 K i(ω 0, Ω 1 )doesnotcoverω 0. Therefore, not every state in Ω 0 can be driven to Ω 1 in N or less steps according to Theorem 4.2. However, we still can specify a sbset contained in Ω 0,schas N i=0 K i(ω 0, Ω 1 ) Ω 0, for which admissible control laws eist and drive the states into Ω 1 in N or less steps. Based on this idea, we can determine a seqence of sbsets of the previos nattainable region-seqence specification, and the obtained sbset seqence is attainable by constrction. This process is referred to as refinement of the specification. A refinement algorithm is now given: Algorithma 4.1 Refinement of the Specifications INPUT: {Ω 1,, Ω M }; Ω M = C (Ω M ) for i = M 1,, 1 Ω i = N j=0 K j(ω i, Ω i+1 ) end OUTPUT: { Ω 1,, Ω M } Here N stands for the pper bond of steps allowed to be taken to reach the sccessive region. The basic idea is to sbstitte the region Ω M with C (Ω M ) so as to satisfy the safety reqirement of the final region. And Ω i is replaced with a finite-step controllable sbset to satisfy the reachability reqirements. Therefore, the refined specification { Ω 1, Ω2,, ΩM } is attainable if the refinement process can terminate sccessflly and retrn non-empty Ω i for all 1 i M. Note that the determination of C (Ω M ) algorithm may fail to terminate in finite nmber of steps, and that the refined specifications may contain non-conve piecewise linear sets. 5 Controller Synthesis The hybrid tracking and reglation control problem considered in this section is to select feasible modes q(t) and to design admissible control signals, (t) U q(t), sch that the state trajectory (t) goes throgh the regions, namely Ω 0,Ω 1,Ω 2, in the specified order and the closed-loop system satisfies some reqirements, sch as seqencing of events and evental eection of actions. In Section 4, we specified the conditions for eistence of sch control laws so that the closed-loop system satisfies safety, reachability and attainability specifications. Here we design sch control laws based on optimization techniqes. As it was discssed in the previos section, the attainability controller synthesis problem can be divided into two basic problems, that is safety controller synthesis and direct reachability controller 13

14 synthesis. In the following, we first present a systematic procedre for the controller design for these two basic cases. Then a procedre for attainability controller synthesis is given. 5.1 Safety First, we consider the safety controller synthesis for the terminating region, Ω M. Withot loss of generality, we assme that Ω M is a connected piecewise linear set (may be nonconve), and we specify a polytope P M contained in the interior of Ω M. Let s assme that P M = { G M g M }. We define the cost fnction J M : Q W U q R + as J M (q, w, ) = G M (A q (w) + B q (w)) = v q k=1 w k [G M (A k q + Bk q )] where stands for the infinite norm 2. The cost fnction J M is in fact the Minkowski fnction indced from the the conve polytope P M [8]. Becase Ω M is assmed to be safe, Ω M pre(ω M )= q pre q(ω M ). Therefore, for any Ω M, there eists at least one mode q Q sch that pre q (Ω M ). We call sch mode feasible for state. The control signal can be selected as the soltion to the following min-ma optimization problem for one of sch feasible modes q: min U q ma J M(q, w, ) w W s.t. pre q (Ω M ) The constraint pre q (Ω M ) in the above optimization problem means that the admissible control signal U q mst keep the state trajectory inside Ω M despite the ncertainties and distrbances along the mode q. The eistence of sch control signals comes from the safety of Ω M and the feasibility of mode q for crrent state by assmption. Therefore, we can always select an admissible control signal for a feasible mode q. Theoptimalactionof the controller is one that tries to minimize the maimm cost, to try to conteract the worst distrbance and the worst model ncertainty. In the following, we will describe step by step how to solve the above min-ma optimization problem for the controller design. First, we assme that Ω M is conve, and it can be represented by Ω M = { F M θ M }. In this case, the constraint pre q (Ω M ) in the above optimization problem can be 2 For a disconnected (non-conve) piecewise linear set, we may describe the set as a finite nmber of disjointed connected piecewise linear sets, and indce different cost fnctions for each of the disjointed piecewise linear sets. 14

15 represented as: A q (w) + B q (w) + E q d Ω M F M (A q (w) + B q (w) + E q d) θ M F M (A q (w) + B q (w)) θ M δ, where δ is a vector whose components are given by δ j =ma d Dq Fj T E qd, andfj T is the j-th row of matri F M. The constraint F M (A q (w) + B q (w)) θ M δ, which holds for all w W, contains infinite nmber of constraints. However, according to Proposition 3.2, these infinite nmber of constraints are in fact eqivalent to a finite nmber of constraints corresponding to the verte matrices of polytopic ncertain A q (w) andb q (w). Therefore, the previos min-ma optimization problem can be rewritten as: s.t. min U q ma J M(q, w, ) w W F M [A 1 q + Bq 1 ] θ M δ F M [A 2 q + B2 q ] θ M δ F M [A vq q + Bq vq ] θ M δ U q The min-ma optimization problem is sally difficlt to solve, so we go one step frther to eqivalently transform it into a pre minimization problem by introdcing an ailiary variable z [7], s.t. min U q z J M (q, w, ) z F M [A 1 q + Bq 1 ] θ M δ F M [A 2 q + B2 q ] θ M δ F M [A vq q + Bq vq ] θ M δ U q Notice that the constraint J M (q, w, ) z shold be hold for all parameter ncertainties, and it introdces infinite nmber of constraints again. Becase of the linearity and conveity of J M, the constraints J M (q, w, ) z can be redced into a finite nmber of linear constraints based on similar argments for constraints F M (A q (w) + B q (w)) θ M δ above. Then the original min-ma optimization problem can be eqivalently redced to the 15

16 following linear programming problem, if the control constraints U q is given as a polytope. s.t. min U q z G M [A 1 q + B1 q ] z G M [A 2 q + Bq 2 ] z G M [A Nq q + Bq Nq ] z G M [A 1 q + B1 q ] z G M [A 2 q + B2 q ] z G M [A vq q + Bq vq ] z F M [A 1 q + Bq 1 ] g M δ F M [A 2 q + B2 q ] g M δ F M [A vq q + Bq vq ] g M δ U q where z stands for a colmn vector of proper dimension and with all elements being z. Hence the admissible control signal can be designed for each feasible mode q of the crrent state by solving a linear program of the above form. The feasibility of the linear program is garanteed by the safety of Ω M and the feasibility of mode q, i.e. pre q (Ω M ). Note that there may be more than one modes feasible for the crrent state. For sch case, a linear program is solved for each feasible mode, and the active mode and control action is selected as the pair that retrn the minimal vale of the cost fnction. Notice that the vales for cost fnctions of different mode q are comparable, becase they are vales of Mikowshki fnctions indced from the same conve set P M. After the control action is applied and the system evolves along the active mode, the net step state is garanteed to be contained inside Ω M, namely the safety of Ω M is garanteed. Then the controller synthesis process repeated for the new state to design net step control signals. In the following, we will deal with the case when Ω M is a non-conve piecewise linear set. As it is shown in Proposition 3.3, pre q (Ω M ) can be written as a finite nion of polyhedra. Assme pre q (Ω M )= s i=1 Φ i. For each feasible mode q, pre q (Ω M )= s i=1 Φ i, then there eists at least one i, for 1 i s, sch that Φ i. The admissible control signal U q, which keeps the state trajectory remaining in Ω M despite the ncertainties and distrbances along the mode q, can be designed throgh the following min-ma optimization problems. min U q ma J M(q, w, ) w W s.t. Φ i 16

17 Becase of the conveity of Φ i, for i =1,,s, each min-ma optimization problem can be redced into a linear programming problem as shown for the case of conve Ω M.Itisclear that at least one of these linear programs is feasible if q is a feasible mode for, andthe soltion is an admissible control that satisfies the safety specification. It is worth pointing ot that the above partition of the non-conve set pre q (Ω M ) into finite nmber of conve sets Φ i does not imply that each Φ i is safe. In fact, we only rely on the property that if Φ i pre q (Ω M ) then admissible control signals to keep the net state remaining inside Ω M (not Φ i )alongthemodeq do eist. Therefore, at each step the admissible hybrid control law is designed by solving a finite nmber of linear programs. The nmber of linear programs to be solved depends not only on the nmber of feasible modes for crrent state bt also on how many polytopic cells Φ i of the (non-conve) region pre q (Ω M ) contains. The nmber of linear constraints for each linear programming problem is determined by the nmber of verte matrices, namely v q, and the nmber of faces of the polytopic sbregion Φ i. Hence, the online optimization based controller synthesis method may become complicated and inefficient for large dimensional practical hybrid systems. We will retrn to this isse in the conclsion part. 5.2 Direct Reachability Net, we consider the reachability between two sccessive regions Ω i and Ω i+1. The control objective is to drive every state in Ω i to Ω i+1 withot entering a third region. Similarly, we specify a polytope P c Ω i+1, which can be represented as P c = { G C g C }. We define the cost fnctional, J C : Q W U q R + as J C (q, w, ) = G C (A q (w) + B q (w)) = v q k=1 w k [G C (A k q + Bk q )] Becase Ω i+1 is direct reachable from Ω i, so that for all states in Ω i,therealwayseist discrete mode q, and admissible control signal U q sch that the net state remains in Ω i Ω i+1, for all allowable distrbances and ncertainties. Therefore, there always eists q Q to make the following min-ma optimization problem feasible: min U q ma J C(q, w, ) w W s.t. pre q (Ω i Ω i+1 ) This optimization problem is of the same type as the one stdied in the safety controller synthesis. Based on similar argments as in the safety controller synthesis, the above min-ma optimization problem can be redced into a finite nmber of linear programming problems. 17

18 5.3 Tracking Attainable Specifications Assme that the given tracking and reglation specification {Ω 0, Ω 1,, Ω M } is attainable. Or task in this section is to select feasible modes q(t) and to design admissible control signals, (t) U q(t), sch that for all the initial states 0 contained in Ω 0 will be driven into Ω 1, withot violating state and inpt constraints, then into Ω 2 and so on. Finally, the state trajectory will reach the final region Ω M and stay there. The controller design procedre is now described: For initial condition 0 Ω 0, determine the feasible modes for 0 as act( 0 )={q Q 0 pre q (Ω 0 Ω 1 )}. Note that act( 0 )is a non-empty finite set by reachability assmption. For each feasible mode, we employ the reachability controller design procedre for Ω 0 and Ω 1. This can be done by solving a finite nmber of linear programs. And for each feasible mode q, at least one of the linear programs is feasible and retrns an admissible control signal U q. This claim comes from the assmed direct reachability from Ω 0 to Ω 1. The active mode q and control signal ( 0 )is selected among these admissible control signal pairs as the one that retrns the smallest vale of the cost fnction, which represents the best effort being taken to reach Ω 1. Then, the control signal is applied and its corresponding feasible mode q is followed, and the net state 1 is garanteed to be contained in Ω 0 or Ω 1, for all possible distrbances and ncertainties. If 1 does not reach Ω 1, then the reachability controller design procedre for Ω 0 and Ω 1 is taken again to obtain the net step feasible mode and admissible control signal. If, after some steps, (t) isdrivenintoω 1, then the reachability controller design procedre for Ω 1 to Ω 2 is invoked. This procedre is repeated and finally (t) reaches Ω M, for which the safety controller synthesis procedre is followed to obtain the active modes and admissible control signals. Similar to the case of reachability controller synthesis, the non-emptiness of the feasible modes, and the feasibility of the finite nmber of linear programs can be garanteed for the safety controller synthesis of Ω M. The following algorithm describes the controller design procedre that garantees the attainability for an attainable specification described by {Ω 0, Ω 1,, Ω M }. Algorithma 5.1 Attainability Control INPUT: {Ω 0, Ω 1,, Ω M }; for i =0, 1,, M 1, while (t) Ω i and (t) / Ω i+1 Design reachability controller from Ω i to Ω i+1 end end Design safety controller for Ω M OUTPUT: q, 18

19 The eample below illstrates the methods developed in this paper. Eample 5.1 Consider the same eample setp as in the previos section. We have defined a specification, {Ω 0, Ω 1 }, and checked its attainability. Here we will design the controller to satisfy the tracking and reglation specification. Assme initial condition 0 =[16, 36] T, which is contained in Ω 0 bt not contained in Ω 1. First, design the direct reachability control signal for 0. Select the cost fnction as the indced Minkowski fnction of the set P C = { R 2 10} for each mode q =1, 2. Then solve the optimization problem w.r.t. the cost fnction for each mode nder constraints pre q (Ω 0 Ω 1 ). Following the procedre developed above, the active mode and control signal can be designed by solving a finite nmber of linear programs. In this eample, we need to solve two linear programs with 26 linear ineqality constraints in and z each (16 of these constraints are indced from the cost fnction, 8 constraints come from predecessor constraint, and 2 are control constraints). Here it is shown that the active mode is q = 2 and the control signal = 1.00 U 2 retrn the best control effort. Therefore, if the control signal is applied and the mode q = 2 is followed, the net state 1 is garanteed to be contained in Ω 1 Ω0 despite ncertainties and distrbances. If we simlate the state evoltion nder nominal condition, i.e. setting D = {0} and choosing the epicenter of the state matri 1 2 (A1 q + A 2 q) for each mode, then we obtain the net step state 1 =[ 2.42, 16.67] T,which is contained in Ω 0 Ω1 as epected. Becase 1 / Ω 1, the reachability reglation design is repeated again, also by solving two linear programs with 26 linear ineqality constraints. After one more step, we obtain 2 =[5.13, 5.09] T which is contained in Ω 1. Then a safety reglation controller is designed. The cost fnction is indced from the region Ω 1, becase of its conveity. Similarly, we obtain active modes and control signals by solving two linear programs at each step. Simlation reslts nder nominal assmptions are shown in Figre 3. The seqence of selected active mode and admissible control signals of the controller is also illstrated in Figre 3. 6 Practical Stabilization Several classes of control problems with practical importance can be stdied in the framework of tracking and reglation control for ncertain hybrid systems. In this section, we will consider the stabilization problem for ncertain piecewise linear systems and solve it by transforming it into a tracking and reglation control problem. First, we formlate the stabilization problem. Becase of parameter variations and eterior distrbances, it is only reasonable to stabilize the ncertain piecewise linear system within a neighborhood region of the eqilibrim. This is called practical stabilization or 19

20 Tracking and Reglation Control seqence of the active discrete modes σ(t) seqence of the continos control signal (t) Figre 3: Left: Simlation for closed-loop nominal plant (assming d = 0). Right: The active mode seqence q and control signals of the controller. ltimate bondedness control in the literatre; see for eample [8] and the references therein. Formally, practical stabilizability can be defined as follows. Definition 6.1 The discrete-time ncertain piecewise linear system (2.1) is practically stabilizable or niformly ltimate bonded in the polytope Ω, which contains the origin in its interior, if and only if for every possible initial condition (0) = 0 X, there eists T ( 0 ) > 0 and admissible control laws, sch that (t) Ω for t T ( 0 ). 2 Ω0 Ω1 1 Figre 4: The hybrid systems ltimate bondedness control as a reglation problem. Figre 4 illstrates the definition of the practical stabilization above. It is worthy to point ot that the region X may be obtained as an oter-approimation of the bonded 20

21 region (Ω 0 in the figre) containing all the initial conditions for a semi-global asymptotic practical stabilization problem, while Ω inner-approimates the small neighborhood of the origin (Ω 1 in the figre) as illstrated in Figre 4. Therefore, any semi-global asymptotic practical stabilization problem can be stdied in the framework of tracking and reglation control, and two kinds of problems are considered here: For a given polytope Ω with the origin contained in its interior, is the discrete-time ncertain piecewise linear system (2.1) practically stabilizable in Ω? If yes, then design the practical stabilization control law. If no, find the practically stabilizable region for Ω and design the stabilization control law. In the following, we will show that the above two practical stabilization problems can be formlated and solved as tracking and reglation problems. For the first problem, the ncertain piecewise linear system is practically stabilizable in Ω if Ω is a safety set (or has non-empty maimal safety sbset C (Ω)) and Ω (or C (Ω)) is direct reachable from all possible initial states in X, i.e. the region-seqence specification {X, Ω} (or {X, C (Ω)}) is attainable by definition. Therefore, the first problem can be solved by checking the attainability of {X, Ω} (or {X, C (Ω)}). By Theorem 4.2, the direct reachability from X to the target region Ω (or C (Ω)) is verified if there eists a finite integer N sch that the nion of finite-step controllable set N i=0 K i(x, Ω) (or N i=0 K i(x, C (Ω))) eqals to the set X. One interesting observation is that it is not necessary to take the nion of the finite step controllable sets for direct reachability checking, becase the i-step controllable K i (X, C (Ω)) has the following non-decreasing property. Proposition 6.1 The seqence of mltiple-step controllable sets K i (X, C (Ω)) is non-decreasing in the sense of K i (X, C (Ω)) K i 1 (X, C (Ω)), where i 1andK 0 (X, C (Ω)) = C (Ω). Proof : Prove by mathematical indction. First, K 0 (X, C (Ω)) = C (Ω) pre(c (Ω)) X = K 1 (X, C (Ω)). Secondly, if we assme K i (X, C (Ω)) K i+1 (X, C (Ω)). Then, K i+1 (X, C (Ω)) = pre(k i (X, C (Ω))) X pre(k i+1 (X, C (Ω))) X = K i+2 (X, C (Ω)). The non-decreasing property of the seqence K i (X, Ω) also holds for a safety set Ω. Becase of the above proposition and the reslts in Section 4, we can derive the following answer for the first problem. 21

22 Proposition 6.2 For an given polytope Ω with origin contained in its interior, the discretetime ncertain piecewise linear system (2.1) is practically stabilizable in Ω, if and only if Ω has non-empty maimal safety sbset C (Ω), and there eist finite N sch that the N-step controllable set K N (X, C (Ω)) X. Proof : To prove the sfficiency, it is assmed that Ω has non-empty maimal safety sbset C (Ω). Note that for safety set Ω, C (Ω) = Ω. Also it is assmed that there eist a finite integer N sch that the N-step controllable set K N (X, C (Ω)) X. Therefore, for all the initial states 0 X K N (X, C (Ω)), there eist feasible mode q and admissible control signal that drive 0 into C (Ω) with at most N steps according to the definition of K N (X, C (Ω)). Hence, there eists T ( 0 ) N, sch that T (0 ) C (Ω) Ω. As long as the state trajectory starting from 0 reaches C (Ω) Ω, feasible modes and control signals eist to make the trajectory remain inside C (Ω) Ω. This is becase of the safety of C (Ω). Therefore, the discrete-time ncertain piecewise linear system is practically stabilizable in Ω. The necessity can be proven by contradiction. Assme that the discrete-time ncertain piecewise linear system is practically stabilizable in Ω. If Ω does not have any safe sbsets, i.e. C (Ω) =, then the state trajectory can not stay inside Ω, which is a contradiction. On the other hand, if there does not eist a finite integer N to make K N (X, C (Ω)) X, then, according to Proposition 6.1 and Theorem 4.2, some initial states in X can not be driven into C (Ω) in finite nmber of steps. In addition, even if these states might be driven into Ω \C (Ω), the point is that these trajectories will eventally escape the set Ω in the ftre, which is implied by the maimality of the maimm safety set C (Ω) Ω. For either case, it always leads to a contradiction. If we fail to find a finite N sch that K N (X, C (Ω)) X, then we consider the second problem, i.e. determining the attractive region to Ω, in fact C (Ω). Similar to the refinement of the specification in Section 4.4, the attractive region of the second problem can be given as K N (X, C (Ω)) for N large enogh. Therefore, the practical stabilization problem of the ncertain piecewise linear system can be transformed into a tracking and reglation problem. The stabilization control law maybe designed by solving the optimization based controller synthesis problem developed in the previos section. Eample 6.1 If we reformlate the previos two eamples in the following way. Consider a semi-global asymptotic practical stabilization problem for the ncertain piecewise linear hybrid systems. Assme that all the initial conditions are contained inside the region X =Ω 0, and that the target origin neighborhood region is given by Ω = Ω 1. 22

23 First, calclate the maimal safety set in Ω. Becase Ω is a safety set by itself, the maimal safety set C (Ω) = Ω. Secondly, check whether all the bonded initial conditions in X canbedrivenintoc (Ω) in finite steps. Becase the non-decreasing property, it is not necessary to take the nion of these finite-step controllable sets. It was obtained that X K 3 (X, C (Ω)). Therefore, according to Proposition 6.2, the discrete-time ncertain piecewise linear system is practically stabilizable in Ω. The stabilization control laws may be designed as the Eample 5.1 shows. However, as it has been pointed ot, the termination of stabilization procedre described above is not garanteed. To overcome sch difficlties, in [22, 23], we focsed on a sbclass of sch ncertain hybrid systems, ncertain switched systems, and proposed a new robst stabilization procedre based on set-indced Lyapnov fnctions and conve analysis. 7 Conclsions In this paper, the tracking and reglation control problem for polytopic ncertain piecewise linear systems was formlated and solved. The eistence of a controller sch that the closedloop system follows desired seqence of regions nder ncertainties and distrbances was stdied first. And a procedre for refinement of the specifications was presented. Then, based on the novel notion of attainability for the desired behavior of ncertain piecewise linear systems, we presented a systematic procedre for robst controller design by sing linear programming techniqes. The robst stabilization problem for ncertain piecewise linear systems was then formlated and solved in the framework of robst tracking and reglation control developed here. The contribtions of this paper primarily concern the eplicit consideration of the timevariant parametric ncertainties and persistent eterior distrbances in hybrid systems. In addition, the eact calclation of the backward predecessor sets and controller design by linear programming techniqes gives s necessary and sfficient conditions for safety, direct reachability and attainability analysis and effective methods for hybrid controller synthesis. The predecessor operator, attainability checking and controller design methods for piecewise linear hybrid systems have been implemented in a Matlab toolbo called HyStar [18], which is available pon reqest from the athors. Note that a new version capable of dealing with parameter ncertainties has also been developed. As it has been pointed ot, there is one difficlty of the online optimization-based controller, namely that the online comptational brden may be very heavy for certain cases. Hence, an alterative method was developed in [21] to constrct eplicit state feedback control laws for safety controller synthesis. The feasible control law was given as piecewise linear 23