A Review of Stability Results for Switched and Hybrid Systems

Transcription

1 A Review of Stability Results for Switched and Hybrid Systems G. Davrazos and N. T. Koussoulas University of Patras, Electrical and Computer Engineering Department, Rio Patras, GREECE , Abstract--Hybrid and switched dynamic systems are of major research interest nowadays due to their use as models in many applications in computer science and systems control. One of the major problems in hybrid and switched dynamic systems is establishing their key property of stability, which also is important in controller design. Stability may prove also critical for real-time systems, embedded systems, and hybrid systems in general that arise in computer science problems where verification tests are undecidable. Relaxing demands in the search for a stability proof may be necessary for a specific problem. In this paper, we present a brief survey of stability results for hybrid and switched systems while taking into consideration the used model and the technique used for establishing stability. Emphasis is also given in results that take into consideration the interaction of continuous-time systems with the discrete-event part. Index terms--hybrid systems, switched systems, stability I. INTRODUCTION Hybrid and switched systems have numerous applications in control of mechanical systems, automotive industry, flight and air traffic control, switching power converters, process control, intelligent vehicle highway systems (IVHS), robotics, etc. Hybrid systems consist of a continuous-time and/or discrete-time process interacting with a logical or decision-making process. The continuous/discrete-time subsystem is represented as a set of differential/difference equations whereas the logical/decision making subsystem (supervisor) is represented as a finite automaton or a more general discrete event system, e.g. a Petri net. In the hybrid system context, the continuous/discrete time subsystem affects the discrete transitions of the supervisor and the supervisor affects the dynamic evolution of the continuous/discrete-time subsystem. Hybrid systems may arise in the hierarchical organization of complex systems too. Typical examples of this case include intelligent vehicle highway systems and robotics, especially in the supervision and coordination of multiple mobile robots. A switched system arises in two cases: One is when there are abrupt changes in the structure or the parameters of a dynamic system, which can be due to component failures or repairs, variations in the environmental disturbance or subsystems [41], and similar causes. Second, when a switched controller is used for a continuous system [37, 41]. The major reason why we choose a switched controller rather than a continuous one is that a switched controller can achieve better performance benefits [41].In [44] challenging problems for stability and control in the hybrid dynamic system framework are posed. Notions such as hybrid equilibrium stability, orbital stability, and trace stability for hybrid systems are introduced and definitions for them are given. We do not discuss in this paper the stability of hybrid systems whose discrete part possesses the Markovian property like [54], or the stability of systems that are composed of coupled differential and difference equations [10, 11] although some papers like [42, 43] are generic enough to deal with. One of the frequently used approaches in hybrid systems stability is to abstract away the high-level dynamics and concentrate only on the stability properties of the low-level dynamics. In the first part of our review (sec. II), we discuss Lyapunov stability while in the second part we categorize the papers according to the utilized hybrid system model. II. LYAPUNOV STABILITY Stability is a very old subject, dating back to the advent of the theory of differential equations (cf. original paper by Maxwell). The object of stability theory is to draw conclusions about the behavior of a system without actually computing its solution trajectories. In linear/nonlinear systems Lyapunov functions are used to investigate the stability of equilibrium points of the system or the invariance of certain sets. Poincaré mappings are used for studying stability by looking at the continuous states only at certain points in time ( discretization of the system trajectories). This technique is used mostly for the investigation of periodic stability. There are many notions of stability especially in nonlinear systems ranging from local stability to uniform global asymptotic stability. Problems of proving the different notions of stability for the nonlinear case are in general of extreme difficulty and even sometimes unsolvable. The solvability has to deal with the notion of stability we want to prove and the mathematical assumptions we have made for the system. It is true that the larger the number of assumptions we make the more limited the notion of stability. Of course, in the linear case things are not so difficult because of the strong effect of linearity,

2 which makes local stability of the system to imply global stability. Besides Lyapunov stability, Lagrange stability is another kind of stability, which has been studied for dynamical systems. In hybrid and switched systems Lagrange stability has not been studied extensively except for a few works that we consider in our review. The main approaches used in the literature for hybrid and switched systems are Lyapunov theory and Poincaré mappings, with the Lyapunov approach being the dominant one. In Lyapunov theory, there are two main directions either by using Lyapunov function or by reformulating the definitions of stability in the sense of Lyapunov and then proving stability using similar methodology to books for linear/nonlinear control. If we choose the Lyapunov function approach we can use either the technique using a common Lyapunov function or the more general technique using multiple Lyapunov functions. The different ways of proving stability using Lyapunov approach are summarized in Fig. 1. Common Lyapunov function Lyapunov Stability Stability Multiple Lyapunov FIGURE 1. Lagrange Stability Poincaré Mappings Modifying A. Common Lyapunov Function First in our review we will present the stability results that they use the common Lyapunov function approach for proving stability for hybrid and switched systems. In [3] we find a method for proving stability and controller design for switched nonlinear systems based on linearization. As it is stated, if the differential equations corresponding to the linearization of the initial switched nonlinear system are (asymptotically) stable and share the same Lyapunov function, then the initial system is (asymptotically) stable around the equilibrium point in which the linearization took place. In [12], the fact that the Lyapunov function will decrease along any hybrid system trajectory is used for proving hybrid system stability. The common Lyapunov function technique is also discussed in brief in [15,16]. Controller synthesis methodology that guarantees the existence of a single Lyapunov function for the closed loop hybrid system is proposed in [47]. The idea behind this methodology is based on controller design for simultaneous common pole placement for all of the subsystems. An approach similar to common Lyapunov function is used in [49] for a hybrid control algorithm. In [50], common Lyapunov function is used for proving stability for switched systems under synchronous or asynchronous controller switching. The fact that a common Lyapunov function guarantees stability under arbitrary switching has led researchers to look for conditions under which a common Lyapunov function exists. Lie algebra is used for formulation of conditions in [1, 28, 29]. The main idea is that if all matrices A i of a switched linear system have a solvable Lie algebra, then a common Lyapunov function exists and the system is stable under arbitrary switching. In [45] it is shown that a common Lyapunov function exists when the stability matrices A i commute pair wise. A sufficient condition for the existence of a common Lyapunov function when the constituent linear timeinvariant system A i of the switching linear system has triangular structure is proposed in [51] and exponential stability is proved under this condition. Similar results using common Lyapunov function and products of matrices for proving hybrid system stability are presented in [13]. Common Lyapunov function techniques for proving stability for switched and hybrid systems are limited to these kinds of systems with linear subsystems. Switched and hybrid systems with nonlinear subsystems are not examined because of the difficulty to find a common Lyapunov function for all nonlinear subsystems. The common Lyapunov function technique suffers also from another limitation: It is not hard to construct examples with hybrid systems, which are stable but they do not have a common Lyapunov function. These examples show us that the fact of non-existence of common Lyapunov function doesn t imply that the hybrid or the switched system is unstable. B. Multiple Lyapunov functions The multiple Lyapunov functions technique is more common. The idea behind this technique is that if we have Lyapunov functions for each continuous subsystem then all we need is to put restrictions on switching to guarantee stability. Firstly we will present the most useful results (in our opinion) for switched and hybrid systems stability analysis. In [55], one Lyapunov function V i is found for each linear vector field f i and (asymptotic) stability is guaranteed by requiring the energy at the consecutive times just before switching to be a decreasing sequence. Another stability result

3 proposed by Branicky in [5-8] is applicable to hybrid and switched systems with nonlinear vector fields in the general case: Each vector field is assumed to be globally Lipschitz and have the origin as an equilibrium point. It is also assumed that there are a finite number of vector field switchings in finite time to avoid the Zeno phenomenon. By introducing multiple Lyapunov functions, which are positive definite and continuously differentiable, stability is guaranteed by requiring the energy not to increase when no vector field switchings occur and to be a non-increasing sequence at the consecutive times when switching to different vector fields. The stability results by Ye, Michel and Hou in [21,61,62] are very general in the sense that they can be applied to different types of systems that exhibit some kind of hybrid behavior. These results can be considered as an extension of the Branicky result. The nonlinear vector fields are assumed to be Lipschitz continuous and (like in Branicky s work) have the origin as equilibrium point. It is also assumed that there is a finite number of vector field switchings in finite time. Multiple Lyapunov functions are again introduced. The requirement in Branicky s work that the energy is not allowed to increase when no vector field switchings occur, is weakened by the condition that the energy only has to be bounded by a continuous function, which is zero at the origin. Results for generalized hybrid dynamical systems are also presented in [42,43] using the multiple Lyapunov functions approach. In these papers, Lyapunov stability is shown for a general hybrid system model whose state is defined on an arbitrary metric space and evolves along some notion of generalized abstract time. Petterson in [56-58] proposes theorems for stability, asymptotic stability, and exponential stability in the sense of Lyapunov using multiple Lyapunov functions. By using piecewise quadratic Lyapunov function candidates and replacing the regions where the different stability conditions have to be valid by quadratic inequality functions, followed by S- procedure, the problem of verifying stability is turned into an Linear Matrix Inequalities (LMI) problem. Formulation of stability conditions as LMI has the advantage that stability can be verified using mathematical tools and more specifically Matlab s LMI toolbox. In the same works, Petterson introduces a novelty in the existing theorems for hybrid system stability using multiple Lyapunov functions: the local Lyapunov function measuring the system s energy is the same for different discrete states or several local Lyapunov functions are used for measuring the system s energy for the same discrete state. This is contrary to existing results where either the same Lyapunov function measures the system s energy for all discrete states or one local Lyapunov function measures the system s energy for the same discrete state. In [14-16] Fierro uses multiple Lyapunov functions to formulate a theorem for asymptotic stability. According to this theorem, the basic conditions to be satisfied are that each Lyapunov function for each subsystem must be non-increasing for each continuous subsystem (meaning that each subsystem is stable) and that the energy of the system prior to the next switching instant decreases with respect to the energy of this system just prior to the previous switching instant: Vi ( x( t 1)) V ( x( t )) ( x) j 1 j+ i j j φ. + Non-smooth Lyapunov functions are used in [37] for designing hybrid controllers. Controllers are selected corresponding to the smallest Lyapunov function. In [23] the search for piecewise quadratic Lyapunov functions is stated as a convex optimization problems in terms of LMI. Rubensson in [52,53] proposes results for proving hybrid system stability using discrete-time Lyapunov techniques. Throughout his work, it is assumed that exponential stability at the discrete switching times of the hybrid system is a sufficient measure for the exponential stability of the underlying continuous subsystems. Thus far, the common approach was to introduce multiple Lyapunov functions, one Lyapunov function associated to continuous subsystem and then prove stability. In the proposed discrete-time approach, the discrete models relate switch points to discrete transitions. A unique Lyapunov function is introduced for each transition. The conditions of the proposed theorems are formulated as LMI problems for easier computational solution using Matlab. Koutsoukos in [25,26] uses piecewise linear Lyapunov functions for stability analysis of switched systems. Analysis is carried out using Lyapunov-like functions for each subsystem. These Lyapunov functions are pieced together in a certain manner to compose a Lyapunov function that guarantees that the energy of the overall system decrease to zero along the state trajectories of the switched system. These multiple Lyapunov functions correspond to conic partitions of the statespace, which are computed efficiently using developed algorithms. This methodology is used for computing switching laws that guarantee the stability of the switched system. In [18,19], researchers use the results of Pettersson and Johansson for proving hybrid system stability using fundamental cycles. The main result in [18,19] states that instead of searching for a Lyapunov function and examining stability for all switching sequences, we only need consider searching over a potentially smaller sized set of fundamental cycles. In [63,64] researchers propose a methodology for designing switching controllers for systems with changing dynamics that is a mobile or a walking robot. The main consideration is that the switching controllers plus the system must exhibit stable behavior. In stability analysis, multiple Lyapunov functions are used for deriving stability conditions for systems that evolve in embedded manifolds. In [36] stability results from Branicky [5-8], Ye, Michel, Hou [21,61,62] and Johansson [23] are

4 applied to the most widespread model for modeling hybrid dynamical system that is hybrid automata. In [20], the notion of average dwell time is introduced and exploited for designing supervisors for switched systems. Dwell time supervisors force every candidate controller to remain in the loop for at least τ d time units, thus guaranteeing a fixed dwell time of τ d. Stability of the whole system is guaranteed and multiple Lyapunov functions are used in analysis. Stability properties of switched systems consisting of both stable and unstable subsystems using piecewise Lyapunov functions (multiple Lyapunov functions) with an average dwell-time approach [20,28] are investigated in [65]. In the proposed switching law, the total activation ratio between Hurwitz stable subsystems and unstable subsystems is required to be no less than a specified constant which is computed using the desirable stability degree of the switched system. In [4] Bishop proposes methods for normalization of Lyapunov functions for controlled switching of hybrid systems. The primary contribution is the development of techniques by which Lyapunov normalization can be carried out in such a way as to encode relative system information and generate switching surfaces based on containment of invariant sets for the component subsystems. In [59], stability analysis exploits the concept of passivity in a hybrid system framework. Common quadratic, convex homogeneous, homogeneous piecewise quadratic Lyapunov functions are used for the analysis. C. Modifying Theorems A common approach followed in many works for showing stability is to form and prove modified definitions and theorems for hybrid system stability according to some factors, for example, to the used hybrid system model. Below we present some papers that follow this approach. Definitions for Lyapunov stability and methodology for proving that a hybrid system is stable using a Lyapunov function are modified for hybrid systems which are described using the Extended Duration calculus with infinite intervals (abbreviated EDC i ) in [22]. Structural properties of the high level discrete event driven dynamics are considered as important as those of the low level continuous dynamics and this fact is exploited in [66] for proving hybrid stability contrary to the existing results where high level dynamics are abstracted away and the analysis focuses only on low level dynamics. In that work, matrix polytopes are used for taking into consideration the stability of the low level continuous dynamics. In [48] Lyapunov s stability theorem via linearization (Lyapunov s indirect method) is generalized to hybrid systems. Notions like hybrifold (hybrid manifold) and hybrid flow are introduced and used for proving the theorem for Lyapunov s indirect method in a hybrid system framework. The methodology, which is developed in [24] for showing stability, is quite different from what we are used to. Using linear programming for computing viable state (state from which a trajectory can be continued for infinite time) stability properties is investigated. A new method for showing stability for linear switched systems based on vector projection theory is presented in [60] and from what is stated it is advantageous over the multiple Lyapunov function method because it is more amenable to numerical computation and avoids the uncertainty in statespace partition and Lyapunov function construction. In [39, 40, 46] existence and stability of periodic trajectories are studied for hybrid dynamical systems which are modeled using differential automata. Theoretical results are applied to a switched server. Exponential stability for homogeneous switched systems is proved in [2] assuming homogeneity and Lipschitzian condition for the switched vector fields. In [38] we find a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems. According to this condition, global asymptotic stability of each subsystem and the pair wise commutation of the vector fields that define the subsystems are sufficient for the global asymptotic stability of the switched system. Notions such as minimum (maximum) holding time, which is defined as the required engagement time of each subsystem at least (at most) are introduced and used in [27] for proposing stability properties for switched systems. Redundancy of each engaged system is a new notion, which is introduced and used in the formulation of stability properties. [30-35] are some papers from the same research group with interesting and in our opinion useful results. Analytically in [34] a generalized Petri Net model for hybrid dynamic system is proposed and new sufficient conditions for Lyapunov stability are derived. These conditions require only the Lyapunov function to be non-increasing along each subsequence of switching contrary existing results [21,61,62], which require the Lyapunov function to be non-increasing along the whole sequence. The theoretical results are applied to a switched server system, which is modeled with generalized Petri net model. In [30] necessary and sufficient conditions are derived which require only the Lyapunov function to be non-increasing only along one subsequence of switching using a generic model for modeling hybrid systems. Theoretical results are applied to a switched server system. In [32] stability of impulsive hybrid systems is studied using the same result as above while in [31] stability and robust stability of switched systems is studied using generalized matrix measure. In [33] the same methodology (identification of a non-increasing subsequence of the Lyapunov function) as well as a

5 method of construction of the continuous function are presented and extended for uncertain quasiperiodic hybrid dynamic systems. Switched controller is used for stabilization of single input bilinear discrete time systems in [35]. Lyapunov function approach is used for analyzing stability especially modified for this kind of systems. D. Poincaré mappings Poincaré mappings are also a useful technique for studying stability especially of limit cycles and periodic trajectories. This technique is used for studying hybrid system stability in [53]. E. Lagrange stability Thus far, we reviewed research works referring mainly to Lyapunov stability. However, some research works [17,42,43] use the notion of Lagrange stability. In [17] Hassibi et al. use a candidate Lyapunov function which consists of a piecewise quadratic term which depends on the discrete state. These Lyapunov functionals can be computed by solving semi-definite programs and by basic network optimization. In [42,43] theorems for uniform boundedness and uniform ultimate boundedness are given. Lagrange stability for switched and hybrid systems is also studied in [5,6,8] using Iterated Function Systems. III. CLASSIFICATION ACCORDING TO THE USED MODEL In this part, we categorize the papers according to the hybrid system model. Almost all papers in our review use a generic model for the hybrid system, where the vector field is either linear or nonlinear. The generic hybrid system model that is commonly used is of the form x ( t) = f i ( x( t)), where i is selected in some way by the supervisor so as some assumptions like finite switches in finite time to be satisfied or of the form x ( t) = f ( x( t), m( t)), m( t) = ϕ( x( t), m( t )) where x(t) is the continuous system state and m(t) the discrete-event system state. Some papers dealing with hybrid system stability use more specific hybrid system models. Specifically, in [12] hybrid systems are constructed as a combination of discrete state systems and continuous state systems. In [14,16], the hybrid system is composed of a finite state machine or a Petri net interacting with a continuous system described by differential equations. In [18,19], the utilized hybrid system model is a programmable timed Petri net. Extended duration calculus with infinite intervals is used for modeling hybrid systems in [22]. In [34] generalized Petri nets is introduced and used for modeling hybrid systems. Logic is used in [49] for modeling the discrete event part and differential equations for the continuous part. Differential automata are used for modeling hybrid dynamic systems in [39,40]. A finite automaton is used for high level discrete event dynamics and differential equations are used for low-level continuous state dynamics in [66]. A table summarizing all this information appears at the end of the paper. IV. CONCLUSIONS We presented a survey of the main stability results for hybrid and switched dynamic systems that exist in literature. The research works were presented according to the utilized methodology for proving stability in the sense of Lyapunov while attention was also given to alternative stability definitions. Another classification was performed according to the used hybrid system model (cf. table at the end of the paper). It is evident that almost all research works use generic hybrid and switched models when dealing with stability. It is evident that there is a plethora of stability approaches and relevant results, which is a testimony to the importance of this topic in both theory and practice. The final word has not been written yet. ACKNOWLEDGMENT This research was supported by the Greek GSRT and European Social Fund under the PENED 99 project 99ED4. REFERENCES [1] Agrachev A. A., Liberzon D., Lie-algebraic conditions for exponential stability of switched systems Proc. 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7 Generic Discrete Discrete Event Purely Purely Linear Nonlinear Common Multiple Modifying Other Paper Switched Hybrid Lyap. Fn Lyap. Fns Theorems Remarks 1 * * * * 2 * * * * 3 * * * * 4 * * * 5 * * * * * +Lagrange 6 * * * * * +Lagrange 7 * * * * * 8 * * * * * +Lagrange 9 REVIEW 10 Differential + Difference Equations 11 Differential + Difference Equations 12 * * DSS * 13 * * * * 14 * * FSM/PN * * 15 * * FSM/PN * * 16 * * FSM/PN * 17 * * * +Lagrange 18 * * FSM * 19 * * PTPN * 20 * * * * 21 * * * * 22 * * EDC i * Linear 23 * affine * * 24 * * * * 25 * * * * * 26 * * * * * * 27 * * * * 28 REVIEW 29 * * * * 30 * * * * 31 * * * * 32 * * * * 33 * * * * 34 * * GPN * 35 * Bilinear * * 36 * * * Hybrid Automata * 37 * * * * * 38 * * * * Differential 39 * * Automata * Differential Automata * Dwell Time 40 * * 41 REVIEW 42 * * * * * * * Generic 43 * * * * * * * Generic 44 Setting up and Definition of Problems for Hybrid Systems 45 * * * * 46 * * * * 47 * * * * 48 * * * * 49 * * Logic * 50 * * * * * 51 * * * * 52 * * * * *

8 Purely Purely Paper Switched Hybrid Linear Nonlinear Generic Discrete Discrete Event Common Lyap. Fn Multiple Lyap. Fns Modifying Theorems 53 * * * * * 54 Hybrid Systems with Markovian Property 55 * * * * 56 * * * * * 57 * * * * * 58 * * * * * 59 * * * * 60 * * * * * 61 * * * * * 62 * * * * * 63 * * * * * 64 * * * * * 65 * * * * Finite 66 * * Automaton * Other Remarks +Poincare Mappings