In this pap e we prove in a constructive way, the ee ale b e we e pie- a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot be e asilydely from those of the comp tline subsyste Inste we propose practical nume te base onmixe te line programming. Keywords--- Hybrid syste controllability,obse ability, pie line syste pie a#ne syste mixe teline programming I. Introducti In recent yearsb oth control and computer science haveb een attractedb y hybridsystem [1], [2], [23], [25], [26],b ecause they provide a unified framework fordescribgARB( cesses evolving accordingto continuous dynamics, discrete dynamics, and logic rules. The interest is mainly motivatedb y the large variety of practical situations, for instance real-time systems, where physical processes interact with digital controllers. Several modelingformalisms h...

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

Abstract — In this paper we consider dynamical systems which are driven by “events ” that occur asynchronously. It is assumed that the event rates are fixed, or at least they can be bounded on any time period of length T. Such systems are becoming increasingly important in control due to the very rapid advances in digital systems, communication systems, and data networks. Examples of such systems include, control systems in which signals are transmitted over an asynchronous network; distributed control systems in which each subsystem has its own objective, sensors, resources and level of decision making; parallelized numerical algorithms in which the algorithm is separated into several local algorithms operating concurrently at different processors; and queuing networks. We present a Lyapunov-based theory for asynchronous dynamical systems and show how Lyapunov functions and controllers can be constructed for such systems by solving linear matrix inequality (LMI) and bilinear matrix inequality (BMI) problems. Examples are also presented to demonstrate the effectiveness of the approach.

We propose a notion of passivity for hybrid systems. Our work is motivated by problems in haptics and teleoperation where several computer controlled mechanical systems are connected through a communication channel. To account for time delays and to better react to user actions it is desirable to design controllers that can switch between different operating modes. Each of the interacting systems can be therefore naturally modeled as a hybrid system. A traditional passivity definition requires that a storage function exists that is common to all operating modes. We show that stability of the system can be guaranteed even if different storage function is found for each of the modes, provided appropriate conditions are satisfied when the system switches.

In this paper, we introduce a new class of Lyapunov functionals for analyzing hybrid dynamical systems. This class can be thought of as a generalization of the Lyapunov functional introduced by Yakubovich for systems with hysteresis nonlinearities which incorporates path integrals that account for the energy loss or gain every time a hysteresis loop is traversed. Hence, these Lyapunov functionals capture the path-dependence of the “stored energy ” in hybrid dynamical systems and are therefore less conservative over previously published approaches in analyzing such systems. More importantly, we show that searching over the proposed class of Lyapunov functionals to prove some specification (e.g., stability) can be cast as a semidefinite program (SDP), which can then be efficiently solved (globally) using widely available software. Examples are presented to show the effectiveness of this class of Lyapunov functionals in analyzing hybrid dynamical systems.

This work builds on the stability analysis of piecewisea #ne systems reported in #1# and extends it to obtain a new synthesis tool for output feedback controllers. The proposed technique relies on formulating the search for a piecewisequadratic Lyapunov function and a piecewise-a#ne controller as a Bilinear Matrix Inequality. This can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities which can be solved numerically very e#ciently. Akey point in this design technique is that it can be used to design controllers with di#erent structures depending on the number of constraints that are added. In particular, it is shown that a controller with the structure of a regulator and estimator can be designed so that switching based on state estimates rather than on the output can be performed. It is also shown that many other desired features can be included in the design, such as boundedness of the control signals. Furthermore, the applicabilit...

This paper presents a method for stability analysis of switched and hybrid systems using polynomial and piecewise polynomial Lyapunov functions. Computation of such functions can be performed using convex optimization, based on the sum of squares decomposition of multivariate polynomials. The analysis yields several improvements over previous methods and opens up new possibilities, including the possibility of treating nonlinear vector fields and/or switching surfaces and parametric robustness analysis in a unified way.

This paper establishes a framework to automate the generation of a piecewise-affine approximation for a class of nonlinear systems. The results enable a completely automated procedure for designing an output feedback controller using a method described previously. To perform the piecewise-affine approximation it is assumed that a uniform rectangular grid is provided for the domain of the nonlinearity. Given this grid, there are three basic steps of the algorithm presented in this paper. First, a simplicial partition of the grid is found that provides a partition of the entire state space into polytopic cells. Then, the piecewise-affine approximation of the nonlinearity within each cell of the partition is derived. Finally, a polytopic description of the cells and a parametric description of the boundaries between cells is created. Each cell of the partition will have a closed loop equilibrium at a point that must be specified. This paper presents an optimization objective that can be used to automate the selection of these equilibrium points. All steps in this process are completely automated, which representsakey step in the development of control synthesis tools for piecewise-a#ne approximations of nonlinear systems.

This paper investigates haptic interaction with virtual environments composed of objects with diverse dynamic properties. Passivity is often used for stability analysis of haptic systems. We demonstrate that when the dynamics of the virtual environment during the interaction changes, the approaches using traditional notion of passivity fail. We then show that a recently proposed notion of passivity for hybrid systems can be used to design stable interaction strategies for such systems.

In a series of papers, the authors have developed a method for analysis of piecewise linear systems. The idea is to use Lyapunov functions that are piecewise quadratic. Such Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. This paper presents two approaches for improving the efficiency of these computations. It is shown that by splitting the analysis computations into two distinct steps, one can decrease the computations with roughly 50% essentially without introducing conservatism. By using ellipsoidal boundings rather than polyhedral descriptions of the operating regimes, it is possible to reduce the computations even further. Combined, the two approaches allow the computation times to be reduced with an order of magnitude compared to previous formulations. However, it is shown that the use of ellipsoidal cell boundings in the S-procedure introduces conservatism in comparison with analysis based on polytopic region descriptions. An explicit formula for the minimal volume ellipsoid containing a simplex is also given, together with a complete proof.