Stability and Transient Performance of Discrete-Time ... - IEEE Xplore

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Stability and Transient Performance of Discrete-Time Piecewise Affine Systems Sanam Mirzazad-Barijough, Student Member, IEEE, and Ji-Woong Lee, Member, IEEE

Abstract—This paper considers asymptotic stability and transient performance of discrete-time piecewise affine systems. We propose a procedure to construct a nested sequence of finite-state symbolic models, each of which abstracts the original piecewise affine system and leads to linear matrix inequalities for guaranteed stability and performance levels. This sequence is in the order of decreasing conservatism, and hence gives us the option to pay more computational cost and analyze a finer symbolic model within the sequence in return for less conservative results. Moreover, in the special case where this sequence is finite, an exact analysis of stability and performance is achieved via semidefinite programming. Index Terms—Bisimulation, hybrid systems, linear matrix inequalities, Lyapunov functions, semidefinite programs, simulation.

I. INTRODUCTION

H

YBRID dynamical systems provide a unified framework for modeling a large variety of situations where both discrete and continuous dynamics interact with each other [1]–[4]. These systems appear in many different contexts such as switched electrical circuits [5], biological systems [6], networked control systems [7], and supervisory control problems [8]. Piecewise affine, or piecewise linear, systems are a special class of hybrid systems particularly useful for approximating complex nonlinear systems [9]. A piecewise affine system consists of a set of affine subsystems, which are coupled together via a switching logic that governs which subsystem is active at each instant of time. In the context of nonlinear system approximation, these affine subsystems are linearizations of a nonlinear system at different operating points. In addition, piecewise affine systems turn out to be equivalent to mixed-logic dynamical models [10], which are capable of modeling a large class of hybrid systems [11]. In this paper, we focus on the analysis of asymptotic stability and transient performance of discrete-time piecewise affine systems; see Section II for the precise definitions of our stability and performance notions. In spite of significant advances in the theory of piecewise affine systems, many of the important problems regarding these systems are still open mainly because they are NP-hard; for example, the problems of reachability and asymptotic stability are considered undecidable [12]–[14]. Manuscript received April 01, 2011; revised July 05, 2011 and August 25, 2011; accepted August 30, 2011. Date of publication September 22, 2011; date of current version March 28, 2012. Recommended for publication by Associate Editor P. Tabuada. The authors are with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802 USA (e-mail: sanam@psu. edu; [email protected]). Digital Object Identifier 10.1109/TAC.2011.2168994

Thus, it has been believed that, for practical applicability, solutions to these problems must involve approximation or conservatism [15]. On the contrary, our premise is that a combination of two main tools for the analysis of hybrid systems—namely, symbolic models and Lyapunov functions—leads to a novel result that does not involve approximation or inherent conservatism for a large class of systems. In the context of reachability and verification problems for hybrid systems, it is common to build discrete-state symbolic models that abstract hybrid dynamical systems [16]. The main objective of such an approach is to come up with a symbolic model, called a bisimulation, which is equivalent to the original hybrid system as far as reachability and verification problems are concerned. If a bisimulation exists, exact solution of these problems is possible. For the case where a bisimulation does not exist or it is hard to obtain, one can use the approximate metrics introduced in [17] to build a symbolic model, called an approximate bisimulation, which is approximately equivalent to the original system. Using approximate bisimulations, finitestate symbolic models can be constructed for a large class of hybrid systems that do not admit bisimulations [18]. However, while any analysis that relies on the existence of a bisimulation is inherently conservative, approximate bisimulations cannot be directly used to answer questions on asymptotic stability and transient performance. On the other hand, Lyapunov functions can be used to directly answer questions on asymptotic stability and transient performance [19]. As is well known, the stability of a piecewise affine system cannot be deduced from that of the individual subsystems [20]. Instead, usual approaches require the existence of common or multiple quadratic Lyapunov functions that are valid over all possible state trajectories [15], [21]–[24]. Other approaches compute lower-order piecewise affine Lyapunov functions using linear programming [25], [26], or higher order piecewise polynomial Lyapunov functions via, e.g., the sum-of-squares techniques [26], [27]. However, all these approaches are conservative. If these approaches fail to provide suitable Lyapunov functions, they do not give alternative, potentially better Lyapunov functions. To overcome the approximate or conservative nature of existing results involving symbolic models and Lyapunov functions, we propose in Section III that two subproblems be solved separately. The first subproblem is to obtain all switching sequences generated by the piecewise affine system without concerning their stability properties. To address this problem, an increasing family of switching structure-preserving state-space partitions is generated. This family gives rise to a nested sequence of symbolic models, each of which simulates the original

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MIRZAZAD-BARIJOUGH AND LEE: STABILITY AND TRANSIENT PERFORMANCE OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

piecewise affine system in the sense that it covers all switching behavior of the piecewise affine system. The second subproblem is to identify all exponentially stabilizing switching sequences among those obtained in the first subproblem. This problem is addressed using the switching path-dependent Lyapunov functions, which were introduced in [28] and enabled a convex characterization of all uniformly stabilizing sets of switching sequences for discrete-time linear systems [29]. The solutions to the two subproblems are then combined in order to obtain a semidefinite programming-based stability and performance analysis of discrete-time piecewise affine systems. The remarkable features of the proposed approach are as follows: • One is given the option to either stop partitioning the state space at some depth and settle for the analysis conditions given by the corresponding symbolic model, or partition the state space further with more computational cost paid in return for less conservative analysis. • Associated with each symbolic model is a system of linear matrix inequalities, which can be solved efficiently [21]. Minimization of a performance index subject to these inequalities gives rise to a semidefinite programming based stability and performance analysis. • If the sequence of symbolic models is finite, then the last symbolic model within the sequence is a bisimulation. That is, the associated system of linear matrix inequalities provides an exact analysis of the original piecewise affine system. • For a large class of piecewise affine systems, the proposed approach is not inherently conservative. That is, even if the sequence of symbolic models is infinite, the sequence tends to an infinite-state model, which is equivalent to the original system in some sense. See Section IV for details. Early conference versions of this work appeared in [30]–[33]. Before concluding the introduction, let us put our work in an alternative perspective. If we remove the state-dependence but allow nondeterminism in the switching logic, then the discrete-time piecewise linear system reverts to what is called the discrete linear inclusion. A traditional approach to examine the stability of discrete linear inclusions under arbitrary autonomous switching is to check simultaneous contractability of all possible state matrices [34]. This approach has evolved to improved versions of Lyapunov inequality-based (and more generally, linear matrix inequality-based) methods—see, e.g., [28], [35]–[37]—and extended to discrete linear inclusions under controlled switching [29], [38] and linear parameter-varying systems [39], [40]. In particular, as stated above, the path-dependent Lyapunov inequalities studied in [28], [36] give rise to exact, convex analysis of the stability of discrete linear inclusions. The present work is a result of our efforts to extend the Lyapunov theory for systems under arbitrary nondeterministic switching to that for systems under state-dependent deterministic switching. The sets of real numbers, positive integers, and nonnegative integers are denoted by , , and , respectively. The Euis denoted by . The clidean vector norm of given Euclidean norm induces the spectral norm of

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. For , we by (resp. ) to mean that is symmetric write and positive definite (resp. nonnegative definite). The -byidentity matrix is denoted by , or simply whenever its dimension is understood. The -by- matrix of zeroes is denoted , or simply if its size is understood. by II. PROBLEM FORMULATION A. Definitions Let (1a) where

, , and

, are given. Let (1b)

be a partition of the Euclidean space into nonempty convex polyhedral cells such that, along with and for , each is an intersection of half-spaces that are either closed or open. In general, the cells are not necessarily open or closed. The pair defines the discrete-time piecewise affine system described by

(2a) for

and

, where the switching sequence is such that if

(2b)

for . Each initial state generates a , a state sequence switching sequence , and an output sequence , , , and for . where If for some , then the piecewise affine system is said to be in mode at time . Given a switching sequence for , we have

for ,

with , where the state transition matrix is defined by the matrix product if if

and the vector

by

We are concerned with analyzing the stability and perfor. Our mance of the discrete-time piecewise affine system stability notion requires that the initial states generate a set of

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switching sequences that in turn give rise to a uniformly exponentially convergent set of state transition matrices. Definition 1: Let . The piecewise affine system is said to be uniformly exponentially stable on if there and such that, for all switching sequences exist generated by initial states according to (2), we have (3a) (3b)

for ,

with

, and (3c)

as

. Definition 2: A set of switching sequences is said to be uniformly exponentially stabilizing for if there exist and such that (3a) is satisfied for all , with and for all . In this case, each is said to be uniformly exponentially stabilizing for . The relation between the two definitions above is as follows. If the switching sequence generated by an initial state is uniformly exponentially stabilizing for , and if the affine for all modes that appear infinitely many times terms in , then the piecewise affine system is uniformly ex; conversely, if the piecewise affine ponentially stable on is uniformly exponentially stable on for system , then the switching sequence generated by some is uniformly exponentially stabilizing. Our stability condition is stronger than asymptotic stability (even if we do not assume the affine terms are zero at the is unioutset). That is, if the piecewise affine system is formly exponentially stable on , then the system asymptotically stable on in the sense that all state seaccording to quences generated by initial states as . On the other hand, our (2) satisfy notion of stability, in a sense, imposes an additional robustness requirement against unforeseen perturbations of the state at arbitrary time instants [38]. In particular, if the affine terms , then it is stronger than merely requiring and such that the existence of (4)

and

has

The state matrices and are rotation matrices which bring and into and in at most five steps the states in without changing their Euclidean norm, whereas the matrices and decrease the Euclidean norm of the states in cells and by a factor bounded above by 0.8. Thus, if we choose and , the piecewise is exponentially stable and satisfy (4) for affine system and . However, as is shown in [42], no all switching sequence is uniformly exponentially stabilizing for , and hence the system is not uniformly exponentially stable on any nonempty subset of . On the other hand, our performance criterion requires not only that the state sequence converges to the origin over all initial states of our interest, but also that the squared sum of the output sequence is uniformly within some desired level. Such a requirement on the transient performance can be interpreted as a relative-stability requirement, which ensures the system is “sufficiently” stable. be bounded and let . The Definition 3: Let is said to satisfy uniform perforpiecewise affine system mance level on if there exists an such that (5) for all output sequences according to (2).

generated by initial states

B. Problem Statement Our objective is to construct a sequence of finite-state symbolic models, each of which covers the behavior of the original piecewise affine system and gives rise to a linear matrix inequality condition for guaranteed stability and performance. Conversely, we also require that, the further one goes down the sequence, the less conservative the associated analysis condition is, and that exact analysis is achieved whenever the sequence turns out to be finite. III. SOLUTION APPROACH: TWO SUBPROBLEMS

and for all . This point is illustrated by for all adapting an example in [41] to our case. Example 1: Consider the system with , where has

A. Separation Approach In order to fulfill our objective stated in the previous section, we propose that the stability and performance analysis problems be separated into two subproblems. In terms of stability analysis, these subproblems are described as follows: 1) The first subproblem is to obtain all switching sequences generated by the piecewise affine system without concerning their stability properties. The solution to this problem is obtained by constructing a nested sequence of symbolic models by successively partitioning the

MIRZAZAD-BARIJOUGH AND LEE: STABILITY AND TRANSIENT PERFORMANCE OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

state space according to the underlying state-dependent switching structure. Each symbolic model within the sequence in turn induces a superset of all switching sequences generated by the piecewise affine system. 2) The second subproblem is to identify all stabilizing switching sequences among the set of switching sequences induced by such a symbolic model. The solution to this problem is obtained by checking the feasibility of a system of linear matrix inequalities, associated with which is a quadratic, but switching path-dependent, Lyapunov function. In the context of performance analysis, additional linear matrix inequalities are needed in order to guarantee a desired performance level as well as stability. This, however, affects the second subproblem only and the proposed separation approach remains valid. The remainder of this section is devoted to addressing the two subproblems in the context of stability analysis. B. Subproblem 1: Switching Structure-Preserving Switching Sequences This subsection presents a recursive procedure to construct a nested sequence of symbolic models such that each symbolic model in the sequence represents a superset of all switching sequences generated by initial states according to (2). , each For a given shall be called a switching path of length , or simply an -path. Given polyhedra , which partition the state , define space

for . Suppose that, for some we have obtained polyhedra , in this manner. Then define

,

(6) . Thus, by induction, we for all path lengths and for all -paths . It is inferred from (6) that is the set of all states in which will evolve to a state in in one step. In other words, is the set of all initial states such that, according to whenever the state sequence is generated by for . Now define (2), we have

for obtain

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A sequence of directed graphs is naturally associated with , . For path length , let be the directed are the modes graph defined as follows. The nodes of such that is nonempty, and there is a directed if and only if edge from node to node in is nonempty. Similarly, for a path length , a switching of length is a node of path a directed graph denoted by if and only if is to nonempty; there is a directed edge from node in if and only if is node . nonempty and The sequence of directed graphs , , generated in , , this manner, along with the state space partitions , defines a nested sequence of symbolic models . These finite-state models abstract the original infinite-state piecewise affine model given by (2). With the state-space parti, , understood, we will simply say that the ditions are the symbolic abstractions of the piecewise rected graphs . For any directed graph whose nodes are affine system -paths, define has a directed edge from to

for all

Then is the set of all switching sequences generated . For each path length , it is readily seen that by generates all switching sequences that generates (i.e., ), and that generates all switching seof the piecewise affine quences that initial states generate according to (2). Therefore, in terms system of the terminology in algorithmic approaches to the analysis of simulates both dynamical systems [43], the symbolic model and for each . The following example il, lustrates the construction of state-space partitions and directed graphs . with , where Example 2: Consider the system has

and

has

and Solving the obvious linear vector inequalities, we obtain for . Each is a family of polyhedral cells that form a partition of the state space. These polyhedral cells are intersections of half-spaces, and hence their interiors are obtained by solving linear vector inequalities. for all and for all Since , we have that is a finer is for each . state-space partition than

where, for instance,

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D

G

G

Fig. 1. Switching structure-preserving state-space partitions and associated directed graphs for Example 2. Directed graphs are obtained from and . Each symbolic model ( ) simulates the piecewise affine system ( . (b) . (c) ). The polytope will be used later in Example 9. (a) (d) . (e) . (f) . (g) .

D D

G

G

G

D ;G

S; D

Other cells in and also , are obtained in a simare used to generate ilar manner. The state-space partitions has three a nested sequence of graphs. For example, since , , , the directed graph will have nonempty cells three nodes; namely, nodes 1, 2, and 3. The cell is parti, , in , and so node 1 has tioned into three cells a self-loop and two edges directed out to nodes 2 and 3. The are obtained similarly. The directed graph other edges in generates a superset of all switching sequences that are generaccording to (2). For example, ated by initial states (1, 1, 2, 1, ) and (2, 2, 1, 2, ) are among the switching seand , respectively. quences generated by initial states in The state-space partitions and associated directed , , and are shown in Fig. 1. This example will graphs be revisited later in Examples 6 and 9. C. Subproblem 2: Stabilizing Switching Sequences Suppose the first subproblem has been solved. Given a directed graph obtained as in the previous subsection for some , the second subproblem is to identify all stabilizing . To switching sequences among the sequences generated by address this problem, we need the notions of admissible sets and minimal sets of switching paths introduced in [29]. To simplify ,a notation, we will use the convention that ; if , then is a switching dummy mode, if as usual. path of length Definition 4: Let . A nonempty set is said to be an admissible set of -paths there exists an integer if for each and a switching path such that and for . If the only admissible set of -paths is itself, then is said to be a minimal satisfying set of -paths. . If is an admissible (resp. minDefinition 5: Let imal) set of -paths, and if there exists an indexed family of , , such that matrices (7)

P

D

D

D D.

, then is said to be an -admissible for all (resp. -minimal) set of -paths. If is an admissible set of -paths, then each switching path leads to itself via the switching paths in . Whether a in switching sequence is uniformly exponentially stabilizing or not is determined by the switching paths that appear infinitely many times in . The set of all such switching paths of any length is necessarily admissible. If this set is -admissible, then there exists a quadratic, switching path-dependent Lyapunov function [28], and hence is uniformly exponentially stabilizing for . More generally, if is an -admissible set of -paths, then yields a set of switching sequences for all

(8)

which is uniformly exponentially stabilizing [28], [36]. Conversely, if is a uniformly exponentially stabilizing set such that the for , then there exists a path length largest admissible subset of (9) is -admissible [28], [36]. In general, if the largest admissible subset of (9) is -admissible for some , then the same is true is incremented to . Thus, the path length will when be taken to be larger than without loss of generality. It is readily seen that each admissible (resp. -admissible) set of -paths is a finite union of minimal (resp. -minimal) sets, but that a union of -minimal sets is not necessarily -admissible. Also, if a minimal set is not -minimal, then none of the admissible sets containing is -admissible [29]. Thus, the search for all admissible (and hence -admissible) sets can be based on identifying all minimal sets. As (8) associates each with a periodic switching sequence, which is minimal set unique up to a time shift, a minimal set consisting of switching , paths , is -minimal if and with only if the spectral radius of the matrix product is less than one.

MIRZAZAD-BARIJOUGH AND LEE: STABILITY AND TRANSIENT PERFORMANCE OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

Consider a set of one-paths given by , where are written as to simplify notation. The one-path 11 leads to itself in one step. The one-path 12 leads to 23, which leads to 31, which leads back to 12. Proceeding in this manner, we see that every leads to itself via the switching paths in switching path in . Thus, is an admissible set of switching paths of length one. However, this set is not minimal because there are other , , and admissible sets such as that are strict subsets of . In fact, none , , and of the strict subsets of any of the three sets is admissible, and so , , and are minimal sets of . one-paths. Clearly, we have Now, let be as in Example 2. For the set to be -admis, and sible, the Lyapunov inequalities (7) defined over hence given by

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Example 3:

and must be feasible. It turns out that these inequalities are not feais not -admissible. On the other hand, insible and hence , so is equalities (7) are feasible over -admissible. Since and are subsets of an -admissible set, they are -minimal; their -minimality can also be shown and are less by verifying that the spectral radii of is less than one, so than one. The spectral radius of is -minimal as well. Note that, even though is a union of -minimal sets, it is not -admissible. As in (8), let for all Then being -admissible implies that the set of switching sequences is uniformly exponentially stabilizing for . Conversely, it is readily seen that the set

of -paths is -admissible for and hence for all . of -paths We are only interested in the admissible sets , so that that occur in

where . One may wish to obtain the -path of as in [36] for all and then find all extensions minimal sets of -paths based on . Let . -path extension of is the directed graph The such that there exists a directed edge from to in if and only if the following holds. There exists a directed edge from to in ; there exists a directed edge from to in ; and . For in Fig. 1 and its one-path extenexample, the directed graph sion are shown in Fig. 2. Further extensions , can be obtained in a similar manner to obtain the minimal of

G =G

G . 2(G ) is G G .

Fig. 2. Directed graph in Fig. 1 and its one-path extension Each minimal set of switching paths of length one that occur in , and vice versa. (a) . (b) associated with an elementary cycle in

G

sets of switching paths of lengths , that occur . in However, as will be seen in the next section, it suffices for our only. Moreover, to obtain the minimal purposes to consider -paths that occur in , one does not need to consets of explicitly. It is because, due to the following propostruct -paths in sition, obtaining all minimal sets of amounts to finding all elementary cycles (i.e., cycles where no . node, except for the first and the last ones, appears twice) in , there exists a one-to-one correProposition 1: For spondence between the family of minimal sets of -paths and the family of elementary cycles in . that occur in -paths that occur Proof: Let be a minimal set of . Associated with is a and a switching in such that path and such that are distinct for . Here, the -paths and contained in are distinct for because, such that otherwise, there would exist and , so that the set is admissible and is a proper subset of , which contradicts minimality of . that the Thus, it follows from sequence of nodes

(10) forms an elementary cycle in . Conversely, suppose a sequence of the form (10), , is an elementary cycle in . Then with and we have that that the switching paths are all dis. Then the -paths tinct for are all distinct for , . Thus, and satisfy is a minimal set of -paths occurring in . The elementary cycles of a directed graph can be obtained by using Johnson’s algorithm [44]. The following is a simple example that illustrates Proposition 1. Example 4: Let be as in Example 2. The directed graph for this example and its one-path extension are shown in Fig. 2. There are five elementary cycles in ; namely, 1 1, 2 2, 3 3, 1 2 1, and 1 3 1. From these elementary cycles we obtain all five minimal sets

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of one-paths that occur in : {11}, {22}, {33}, {12, 21}, and {13, 31}. Indeed, one can verify that these minimal sets are . precisely those one can obtain from

implies

, , . Thus, we obtain

and

IV. MAIN RESULTS: COMBINATION OF TWO SUBPROBLEMS In this section, the solutions of the two subproblems are combined in order to analyze the stability and performance of piecewise affine systems. The results will be discussed in four subsections. Stability analysis of piecewise affine systems is given in Sections IV-A and IV-B. Sections IV-C and IV-D discuss performance analysis.

which implies and

is uniformly exponentially stabilizing for

,

A. Stability Analysis via Simulations Let and be as in (1). Suppose the first few state-space , and directed graphs , are obpartitions and , tained as in Section III-B. Given define

(11) be the maximal admissible set of Then, let -paths contained in : is a (12)

minimal set of

-paths in are associated with the union The of all irreducible, or strongly connected, subgraphs (i.e., subgraphs where there exists a directed path between any two pair that are reachable from in . of nodes) of Thus, is relevant to stability analysis. On the other contains all -paths that are reachhand, the set in , and will be used in later sections able from for the purpose of performance analysis. and . The pieceTheorem 1: Let is uniformly exponentially stable on wise affine system if is -admissible and if when. ever Proof: Let ; let de, which is finite. Choose an ininote the cardinality of and generate a switching sequence tial state according to (2). Then there exist such that with and such that the following holds : for all if otherwise. (13) Since is -admissible, a result based on switching path-dependent Lyapunov functions [36, Lemma 2] immediand , indepenately implies that there exist whenever there is no dent of , such that such that . Also, since whenever , we have when there is no such that . In with , then we have general, if , for some with . . Then (13) Let

which implies

is bounded with

Since ,

, and are independent of as long as , we conclude that the system is uniformly ex. ponentially stable on The following algorithm is based on Theorem 1, and gives a procedure to obtain a sequence of sets such and such that the piecewise affine system that is uniformly exponentially stable on for each . Algorithm 1: Put and . from and . Step 1) Construct of all with Step 2) Obtain the union such that is -admissible and for . . Step 3) Let and go to Step 1. Step 4) Increment to Example 5: Consider the system with , where has

and

has

The first few state-space partitions and directed graphs are shown in Fig. 3. The partitions and give rise to the graph , from which we obtain , , . Among these sets, only is -admissible. and . Thus, we determine that the system Moreover, is uniformly exponentially stable on . The stability of the system on and cannot be determined at this point. . The graph , which Next, let us consider the case of

MIRZAZAD-BARIJOUGH AND LEE: STABILITY AND TRANSIENT PERFORMANCE OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

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Fig. 3. Switching structure-preserving state-space partitions and associated directed graphs for Example 5. The system (S ; D ) is uniformly exponentially stable on the area shaded in light gray, and not uniformly exponentially stable on any subset of the area shaded in dark gray. For each path length L, the stability test based on the symbolic model (D ; G ) is inconclusive on the area not shaded in D . However, the size of this area decreases in L. The polytope P will be used later in Example 8. (a) D . (b) D . (c) D . (d) D . (e) D . (f) G . (g) G . (h) G . (i) G .

is obtained from

and

, gives

, , , and . It is readily seen that the sets and are -admissible, and is uniformly exponentially stable hence that the system . On the other hand, we can also deon termine that the system is not uniformly exponentially stable because where the on any nonempty subset of is not less than one. We cannot decide spectral radius of and at this point. Inthe stability of the system on crementing the path length once more, we consider the case of and obtain , , , and , where only and are -admissible. Hence, the system is uniformly ex. Proceeding in ponentially stable on this manner, we obtain , and so on. If , then we conclude that the we terminate Algorithm 1 at system is uniformly exponentially stable on ,

and that the system is not uniformly exponentially stable on any . As shown in Fig. 3(d), the stability nonempty subset of test at path length is inconclusive on the cells and . The following corollary shows that, for a large class of piecewise affine systems, Algorithm 1 is nonconservative in the sense that, in the limit, the algorithm partitions the state space into a stable part and a nonstable part. , Corollary 1: Suppose that, for some mode is bounded, the origin belongs to the polyhedral cell the interior of , the spectral radius of is less than one, and . Let , where , the affine term are the sets generated by Algorithm 1. Then the piecewise affine is uniformly exponentially stable on for each system , asymptotically stable on , and not uniformly expo. nentially stable on any nonempty subset of is uniformly exProof: By Theorem 1 the system for each . Each ponentially stable on

satisfies for some , so the system is asymptoti. It remains to show that the system is not cally stable on uniformly exponentially stable on any part of . such that We will first prove that there exists an , where denotes the interior of a set. Let be such that . Since the spectral radius of is less than one and since , the boundedness of implies that there exists an such that for whenever . That is, the node all in the directed graph has only one outgoing edge, which forms a self loop. Hence, Theorem 1 implies . Moreover, since there exists a sufficiently small ball with such that for all whenever , for all and since such a ball must satisfy , we conclude that for some . To show the desired result by contradiction, suppose there exsuch that is uniformly exponenists a nonempty for all . tially stable on and such that Let be such that , and let be such that . Choose an arbitrary and generate and for according to (2). Since is uniformly exponentially stable on and since , there exists a , which pos, such that sibly depends on for all . Because for all , we have the following. No matter what directed path

one takes from is guaranteed to reach in that have

in , steps a node of the in . Thus, we . Then, as

one form such must

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we have . However, this , and hence contradicts our assumption implies on . As in Corollary 1, suppose that, for some bounded cell in , the origin is in the interior of , the spectral radius of is . Then it is ensured that less than one, and the affine term belongs to if and only if it generates an initial state a switching sequence which is eventually constant at mode . This condition, as assumed in Corollary 1, is sufficient but not necessary for the nonconservatism of Algorithm 1. For example, a nonconservative stability analysis is possible even if the origin is at the boundary of two or more cells [30], [32]. Nevertheless, it is a condition commonly imposed in the analysis of piecewise affine systems [15], [26]. Also, whenever a polyhedral cell contains the origin in its interior, it can be taken to be bounded without loss of generality (because one can always partition into a finite number of polyhedra and redefine the state-space partition ). As suggested by Example 1, however, the corollary is not valid if the stability condition (3) is replaced with that of exponential stability (4). B. Stability Analysis via Bisimulation In this section, we will investigate the situation where the polyhedra , generated by Algorithm 1 satisfy for some , and show that is the maximal set on which the piecewise affine system is uniformly exponentially stable. . A cell is said Definition 6: Let -invariant if to be for some . If every is -invariant, then the state-space partition is said to be -invariant. is invariant if it is as fine as or, In other words, equivalently, if each node in the directed graph has exactly simuone outgoing directed edge. In this case, not only , but also the system lates the piecewise affine system simulates in the sense that the set is equal to the set of all switching sequences generated by initial states according to (2). That is, according to the terminology in algorithmic approaches to the analysis of dynamical is a finite-state bisimulation systems, the symbolic model [16] of , and it enables efficient stability and performance is -invariant for analysis. As we will see below, if , then the set is finite and all switching sesome are eventually periodic in the sense defined quences in in [45]; in particular, an exact stability analysis based on reduces to computing the spectral radii of a finite number of matrix products. and . A switching sequence Definition 7: Let in is said to be -eventually periodic if

for all . Lemma 1: Suppose that . Then, for each node

is

-invariant for some in , there exists a unique

satisfying switching sequence . Moreover, this is -eventually periodic with . Proof: Choose a node in . Since is -invariant, there is a unique outgoing edge from this node, and this edge is directed to a node of the . Then, since is -inform variant, there is a unique outgoing edge from this node, and . Prothis edge is directed to a node of the form ceeding in this manner, we conclude that there exists a unique satisfying . Since each node in has exactly one outgoing edge, and since the is bounded by , this must be number of nodes in -eventually periodic with . is -invariant for some . If the Suppose piecewise affine system is not uniformly exponentially stable , but if the affine terms ason any subset of sociated with the periodic part of the corresponding switching sequence are all zero, then one cannot conclude that none of converges to the origin. This is the states because, even though (3) implies (4) in this case, the converse is not necessarily true. Nevertheless, the following lemma inthat condicates that, even in this case, the states in verge exponentially fast to the origin reside within a set of measure zero. Without the invariance assumption, this lemma is not valid. is -invariant for some Lemma 2: Suppose that . Let be a node in ; let satisfy . If is not uniformly exponenwith tially stabilizing for , then there is no and nonempty interior such that (4) is satisfied for all for all . Proof: To prove the desired result by contradiction, , , and suppose there exist with nonempty interior such that (4) holds for all and . Since the interior of is nonempty, there such that one can choose linexists an open ball from . Since is early independent vectors -invariant, Lemma 1 says that there exists a unique satisfying . This is necessarily -eventually periodic for some and . Choose a and an . is a linear combination of , we have Since for some . Since for each , we have whenever . -evenFor this to hold, the switching sequence being for all . Also, as tually periodic implies that , there exists a such holds regardless of . Thus, that

Since is arbitrary, and since the aperiodic part of contains a finite number of modes, this contradicts the assumption that is not uniformly exponentially stabilizing for .

MIRZAZAD-BARIJOUGH AND LEE: STABILITY AND TRANSIENT PERFORMANCE OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

Combining the two preceding lemmas, we obtain the following theorem. Theorem 2: Suppose that is -invariant for some . Let . Then for each node of , there exists a unique switching sequence sat. This is necessarily isfying -eventually periodic with . The piecewise affine system is uniformly exponentially stable on if satisfies the and only if every such with following: is less than one; (1) the spectral radius of (2) the affine terms . , then Moreover, if 1) does not hold for a given with nonempty interior such that (4) there is no and . is satisfied for all Proof: The system is uniformly exponentially if and only if it is uniformly exponentially stable on for all such that stable on . However, by Lemma 1, there exists a unique satisfying , is uniformly exponentially stable on so the system if and only if it is uniformly exponentially . If is -eventually periodic, then stable on for all , is so Theorem 1 and Lemma 1 imply that the system if and only if 1) uniformly exponentially stable on and 1) hold true. The rest of the theorem is immediate from Lemma 2. Example 6: We will apply Theorem 2 to the system considered in Example 2. Fig. 1 shows that the state–space partition

is -invariant. Among the eight nonempty polyhedral cells in , five cells , , , , and intersect in Fig. 1 shows that the the polytope . The directed graph initial states in these cells generate switching sequences (1, 2, 1, 2, ), (1, 1, 2, 1, 2, ), (2, 1, 2, 1, ), (3, 1, 2, 1, 2, ), and (2, 2, 1, 2, 1, ), respectively. These switching sequences are all eventually periodic, and modes 1 and 2 alternate in their is periodic parts. Since the spectral radius of the product , we determine that the less than one, and since system is uniformly exponentially stable on . has , Example 7: Suppose the system

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-invariant. We have . Since none of the mahas spectral radius less than one, the system trices is not uniformly exponentially stable on any part . Even though all initial states within of the state space converge to the origin, is a and hence has empty interior. proper subspace of C. Performance Analysis via Simulations In this section, we will generalize the stability analysis in Section IV-A to a semidefinite programming-based transient and be as in (1), and performance analysis. Again, let , and suppose the first few state-space partitions , are obtained as in Section III-B. directed graphs Given and , recall that the sets and of -paths are as defined in (11) and (12), respectively. be the solution to Lemma 3: Let

with

. If

are such that

with

, then , . Proof: The result appears in, e.g., [46] and is immediate . from the definition of Given a polytope (i.e., bounded convex polyhedron) , the following theorem presents a system of linear matrix inequalities that are sufficient for the piecewise affine system to satisfy a uniform performance level . By running the semidefinite program of minimizing subject to these inequalities, one can obtain the best possible performance bound (or more precisely, its -path that the symbolic model extension ) gives us. If one wishes to obtain a better bound, , provided then one can obtain the next symbolic model that the computational cost of doing so is not prohibitive, and use it to obtain a potentially smaller value of . Theorem 3: Let and ; let be a polyis uniformly exponentope. The piecewise affine system tially stable on and satisfies uniform performance level on if the following hold true for each node of with : , 1) There exist indexed families of , and , , such that

and

(14a) (14b) for all

It is readily seen that we have for all 2, 3, and hence that the initial state-space partition

,

, (14c)

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for implies

. Lemma 3, along with (14a)–(14d), then

(14d) for all

(16)

, and for

, so that

(14e) for all vertices of 2) The affine terms

. and

whenever . be such that . Proof: Let Then (14a) and (14b) hold over all -paths in , -paths and block (1, 1) of (14c) and (14d) hold over all . Thus, as is a finite set, conin such that dition 2) implies that there exist ,

and satisfying . It follows from [47, Corollary 12], specialized to is uniformly expostability analysis, that the system . Because this holds whenever nentially stable on , and because there are a finite number of , the system is uniformly exponentially such stable on . To prove that the given uniform performance level is satisfied on , let be such that and choose any satisfying . Define

(17) for

. Since is stable on , the limit exists. Moreover, inequality (14e) implies the exsuch that istence of an

for all

for . For , equation

with

whenever is a vertex of the polytope . Taking over all satisfying the maximum of , we obtain that (5) is satisfied whenever is a vertex of . Finally, by the second inequality in (16), the right, and therefore the hand side of inequality (17) is convex in satisfies the uniform performance level . system Example 8: Consider the system given in Example 5 with additional system parameters

, consider the Lyapunov

(15) , such that the

with the terminal condition solution

The polytope indicated in Fig. 3 has vertices at , , , and . The sets obtained in Example 5 are shaded in light gray in Fig. 3(a)–(e), respectively. and since the system is uniformly exponentially Since stable on , the system satisfies a uniform performance level on . We have and for (18)

for and

, where if

if . Rewrite (15) as

The vertices of for which we need to check the are , , feasibility of (14e) with , for ; , , , for ; and so on. For each satisfying (18), we from the graph in Fig. 3(i). Also, obtain , , , ,

MIRZAZAD-BARIJOUGH AND LEE: STABILITY AND TRANSIENT PERFORMANCE OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

, , . By running the and subject to (14) for each semidefinite program of minimizing satisfying (18), and by taking the maximum of the , we determine that the resulting over all such on . system satisfies uniform performance level This performance level can potentially be improved by consid. ering

Then, for , Lyapunov equation

with

and for

with the terminal condition solution

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, consider the

, such that the

D. Performance Analysis via Bisimulation As we have seen earlier, if the state-space partition is -invariant for some , then the directed graph (or, more precisely, the pair ) defines a finite-state . In particular, for each node bisimulation of in , all initial states generate a unique eventually periodic switching sequence. As in the stability analysis presented in Section IV-B, an efficient and exact performance analysis is achieved in this case. is -invariant for some Theorem 4: Suppose that . Let and let be a polytope. The piecewise is uniformly exponentially stable on and affine system satisfies uniform performance level on if and only if the of with following hold true for each node : , 1) There exist indexed families of , and , , such that (14a) and (14b) hold for all , such that (14c) and (14d) , and hold for all . such that (14e) holds for all vertices of and whenever 2) The affine terms . Proof: Sufficiency is immediate from Theorem 3. To is uniformly prove necessity, suppose that the system exponentially stable on and satisfies the uniform performance level on . Let be such that . is -invariant, by Lemma 1 there exists a Since unique, eventually periodic switching sequence such that . This is -eventually periodic for some and . Let and , , be the state and output sequences associated with . occurs infinitely Each switching path many times in , so in order for the state and output sequences to satisfy (3) and (5), respectively, the affine terms associated -paths in must be zero. with each mode in the and , This implies condition 2) holds true. For define

for , where if and if . Since the system is and hence on uniformly exponentially stable on , the switching sequence is uniformly exponentially stabilizing for . Thus, for , and we have each for . Moreover, since the system satisfies the uniform performance level on , there exists an such that

for all

. Since the

switching paths are , , satisfy distinct, and since the matrices for all , we can choose a sufficiently small and write

for , so that condition 1) holds true. Example 9: Let us revisit the system Example 2 with additional system parameters

considered in

The polytope shown in Fig. 1 has vertices at , , , and [5/4 1]. It was shown in Example 6 that the is -invariant, and that the system state-space partition is uniformly exponentially stable on . We have for (19)

and

if

if

.

The vertices of for which we need to check the are then , feasibility of (14e) with , for ; , , , , for ; and so on. From the graph in Fig. 1(g), we deduce for each satisfying (19). Also, , , we have , , and . Minimizing subject to (14) with

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for each satisfying (19) and taking the max, we conclude imum of the resulting over all such satisfies uniform perforthat the piecewise affine system mance level on , and that it does not satisfy any . performance bound V. CONCLUSION In this paper, we bridged two disparate paths to the analysis of hybrid systems—namely, symbolic model-based and Lyapunov function-based approaches—via linear matrix inequalities, and presented a scalable stability and performance analysis of piecewise affine systems in the discrete-time domain. The presented analysis gives rise to an increasing sequence of systems of linear matrix inequalities, through which the engineer is able to make a tradeoff between conservatism and computational cost. That is, if one wishes to enlarge a given stability region or improve a given performance bound, one is allowed the option to pay an increased computational premium in return for tighter, less conservative analysis. Moreover, for a fairly large class of systems, the conservatism associated with the presented analysis becomes arbitrarily small as one moves down the associated sequence of linear matrix inequalities. This property can be put in contrast with Lasserre’s approach to nonconvex polynomial optimization [48], where the degree of relaxation, rather than the amount of conservatism, becomes arbitrarily small as one goes down a sequence of linear matrix inequalities. We also showed that, if a finite-state bisimulation is attained, our stability and performance analysis problems can be addressed with great ease. In this case, using a version of the classical bisimulation algorithm [16] would yield the same result. Nevertheless, our approach is distinct from the classical approach because it is specialized to the stability analysis of piecewise affine systems, and because, as discussed above, it allows one to obtain a correct (but not necessarily exact) result without achieving bisimulation. One does not in general know whether a given piecewise affine system admits a bisimulation or how much computation will be needed to attain one, if any. However, at least for some special classes of hybrid systems, these questions can be answered in a conservative manner [12], [49]–[51]. We expect that some of these results can be translated to our problem setting, and that our simulation/bisimulation approach will also benefit from such a translation. A future research objective is to apply the analysis results established in this paper to real-world examples. A major challenge in this objective is to obtain “small” piecewise affine models of realistic nonlinear systems, so that the computational cost involved in the stability and performance analysis is substantially reduced and yet the key features of the original systems are preserved. For example, a realistic nonlinear model of a typical power electronic circuit involves hundreds of state variables, and, unless the dimension of the state space is significantly reduced, most of the hybrid systems approaches currently available in the literature, including ours, will not be applicable. Another future research direction is to extend the analysis results to controller synthesis. One potential scenario in this direction is, as suggested in [52], to obtain a feedback controller so that the closed-loop system admits a finite-state bisimulation.

A first step in this scenario would be to characterize all feedback controllers, under which the closed-loop piecewise affine system leads to an invariant state-space partition. Then the next step would be to obtain the “best” one among these controllers that guarantees closed-loop stability and a minimal performance bound. A vital question in doing so is how to modify the separation approach discussed in Section III by integrating the state and control spaces in a natural manner. REFERENCES [1] P. J. Antsaklis, X. D. Koutsoukos, and J. Zaytoon, “On hybrid control of complex systems: A survey,” Eur. J. Autom., vol. 32, no. 9–10, pp. 1023–1045, 1998. [2] R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proc. IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000. [3] M. Johansson, Piecewise Linear Control Systems. Secaucus, NJ: Springer-Verlag, 2003. [4] R. Goebel, R. Sanfelice, and A. Teel, “Hybrid dynamical systems,” IEEE Control Syst. Mag., vol. 29, no. 2, pp. 28–93, Apr. 2009. [5] W. P. M. H. Heemels, M. K. Çamlibel, A. J. van der Schaft, and J. M. Schumacher, “Modelling, well-posedness, and stability of switched electrical networks,” in Hybrid Systems: Computation and Control, ser. Lecture Notes in Computer Science, O. Maler and A. Pnueli, Eds. Berlin, Germany: Springer-Verlag, 2003, vol. 2623, pp. 249–266. [6] R. Alur, C. Belta, F. Ivançic´, V. Kumar, M. Mintz, G. Pappas, H. Rubin, and J. Schug, “Hybrid modeling and simulation of biomolecular networks,” in Hybrid Systems: Computation and Control, ser. Lecture Notes in Computer Science, M. Di Benedetto and A. Sangiovanni-Vincentelli, Eds. Berlin, Germany: Springer-Verlag, 2001, vol. 2034, pp. 19–32. [7] W. Zhang, M. S. Branicky, and S. Phillips, “Stability of networked control systems,” IEEE Control Syst. Mag., vol. 21, no. 1, pp. 84–99, Feb. 2001. [8] A. S. Morse, “Supervisory control of families of linear set-point controllers part I. exact matching,” IEEE Trans. Autom. Control, vol. 41, no. 10, pp. 1413–1431, Oct. 1996. [9] E. Sontag, “Nonlinear regulation: The piecewise linear approach,” IEEE Trans. Autom. Control, vol. 26, no. 2, pp. 346–358, Apr. 1981. [10] A. Bemporad, G. Ferrari-Trecate, and M. Morari, “Observability and controllability of piecewise affine and hybrid systems,” IEEE Trans. Autom. Control, vol. 45, no. 10, pp. 1864–1876, Oct. 2000. [11] A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407–427, 1999. [12] T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya, “What’s decidable about hybrid automata?,” in Proc. 27th Annu. ACM Symp. Theory of Comput., Ser. STOC ’95, New York, 1995, pp. 373–382, ACM. [13] V. D. Blondel and J. N. Tsitsiklis, “Complexity of stability and controllability of elementary hybrid systems,” Automatica, vol. 35, no. 3, pp. 479–489, 1999. [14] V. D. Blondel, O. Bournez, P. Koiran, C. H. Papadimitriou, and J. N. Tsitsiklis, “Deciding stability and mortality of piecewise affine dynamical systems,” Theoret. Comput. Sci., vol. 255, no. 1–2, pp. 687–696, 2001. [15] G. Ferrari-Trecate, F. A. Cuzzola, D. Mignone, and M. Morari, “Analysis of discrete-time piecewise affine and hybrid systems,” Automatica, vol. 38, no. 12, pp. 2139–2146, 2002. [16] R. Alur, T. Henzinger, G. Lafferriere, and G. Pappas, “Discrete abstractions of hybrid systems,” Proc. IEEE, vol. 88, no. 7, pp. 971–984, Jul. 2000. [17] A. Girard and G. J. Pappas, “Approximation metrics for discrete and continuous systems,” IEEE Trans. Autom. Control, vol. 52, no. 5, pp. 782–798, May 2007. [18] A. Girard, G. Pola, and P. Tabuada, “Approximately bisimilar symbolic models for incrementally stable switched systems,” IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 116–126, Jan. 2010. [19] A. Rantzer and M. Johansson, “Piecewise linear quadratic optimal control,” IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 629–637, Apr. 2000. [20] R. M. Jungers, The Joint Spectral Radius, Theory and Applications. Berlin, Germany: Springer-Verlag, 2009.

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[21] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, ser. Studies in Applied Mathematics. Philadelphia, PA: SIAM, Jun. 1994, vol. 15. [22] M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 475–482, Apr. 1998. [23] M. Johansson and A. Rantzer, “Computation of piecewise quadratic Lyapunov functions for hybrid systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 555–559, Apr. 1998. [24] G. Feng, “Stability analysis of piecewise discrete-time linear systems,” IEEE Trans. Autom. Control, vol. 47, no. 7, pp. 1108–1112, Jul. 2002. [25] P. Grieder, M. Kvasnica, M. Baotic´, and M. Morari, “Stabilizing low complexity feedback control of constrained piecewise affine systems,” Automatica, vol. 41, no. 10, pp. 1683–1694, 2005. [26] P. Biswas, P. Grieder, J. Löfberg, and M. Morari, “A survey on stability analysis of discrete-time piecewise affine systems,” in Proc. 16th IFAC World Congr., Prague, Czech Republic, Jul. 2005. [27] S. Prajna and A. Papachristodoulou, “Analysis of switched and hybrid systems-beyond piecewise quadratic methods,” in Proc. Amer. Control Conf., Jun. 2003, vol. 4, pp. 2779–2784. [28] J.-W. Lee and G. E. Dullerud, “Uniform stabilization of discrete-time switched and Markovian jump linear systems,” Automatica, vol. 42, no. 2, pp. 205–218, 2006. [29] J.-W. Lee and G. E. Dullerud, “Uniformly stabilizing sets of switching sequences for switched linear systems,” IEEE Trans. Autom. Control, vol. 52, no. 5, pp. 868–874, May 2007. [30] J.-W. Lee, “Separation in stability analysis of piecewise linear systems in discrete time,” in Hybrid Systems: Computation and Control, ser. Lecture Notes in Computer Science, M. Egerstedt and B. Mishra, Eds. Berlin, Germany: Springer-Verlag, 2008, vol. 4981, pp. 626–629. [31] S. A. Krishnamurthy and J.-W. Lee, “A computational stability analysis of discrete-time piecewise linear systems,” in Proc. 48th IEEE Conf. Decision Control, Dec. 2009, pp. 1106–1111. [32] S. Mirzazad-Barijough and J.-W. Lee, “On stability characterization of discrete-time piecewise linear systems,” in Proc. Amer. Control Conf., Jul. 2010, pp. 916–921. [33] S. Mirzazad-Barijough and J.-W. Lee, “On stability and performance analysis of discrete-time piecewise affine systems,” in Proc. 49th IEEE Conf. Decision Control, Dec. 2010, pp. 4244–4249. [34] T. Ando and M. Hsiang Shih, “Simultaneous contractibility,” SIAM J. Matrix Anal. Applicat., vol. 19, no. 2, pp. 487–498, 1998. [35] P.-A. Bliman and G. Ferrari-Trecate, “Stability analysis of discrete-time switched systems through Lyapunov functions with nonminimal state,” in Proc. IFAC Conf. Anal. Design Hybrid Syst., 2003, pp. 325–330. [36] J.-W. Lee and P. P. Khargonekar, “Detectability and stabilizability of discrete-time switched linear systems,” IEEE Trans. Autom. Control, vol. 54, no. 3, pp. 424–437, Mar. 2009. [37] A. A. Ahmadi, R. Jungers, P. A. Parrilo, and M. Roozbehani, “Analysis of the joint spectral radius via Lyapunov functions on path-complete graphs,” in Proc. 14th Int. Conf. Hybrid Syst.: Comput. Control, Ser. HSCC ’11, M. Caccamo, E. Frazzoli, and R. Grosu, Eds., New York, 2011, pp. 13–22, ACM. [38] J.-W. Lee and G. E. Dullerud, “Supervisory control and measurement scheduling for discrete-time linear systems,” IEEE Trans. Autom. Control, vol. 56, no. 4, pp. 873–879, Apr. 2011. [39] J. Daafouz and J. Bernussou, “Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties,” Syst. Control Lett., vol. 43, no. 5, pp. 355–359, 2001.

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Sanam Mirzazad-Barijough (S’06) received the B.S. and M.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2006 and 2008, respectively. She is currently pursuing the Ph.D. degree in the Department of Electrical Engineering, Pennsylvania State University, University Park. Her research interests are in hybrid and power systems, and power electronics.

Ji-Woong Lee (S’98–M’02) received the B.S. degree in electronic engineering from Sogang University, Seoul, Korea, in 1990, the M.S. degree in electrical engineering from the University of Maryland at College Park in 1996, and the M.S. degree in mathematics and the Ph.D. degree in electrical engineering, both from the University of Michigan, Ann Arbor, in 2002. He has held postdoctoral positions at the University of Illinois at Urbana-Champaign and at the University of Florida, Gainesville. He is currently an Assistant Professor in the Department of Electrical Engineering at the Pennsylvania State University, University Park. His research interests are in hybrid, stochastic, and decentralized systems, and statistical learning.