Robust Non-Zenoness of Piecewise Affine Systems with Applications to Linear Complementarity Systems

Abstract Piecewise affine systems (PASs) constitute an important class of nonsmooth switching dynamical systems subject to state dependent mode transitions arising from control and dynamic optimization. A fundamental issue in dynamics analysis of switching systems pertains to the possible occurrence of infinitely many switchings in finite time, referred to as the Zeno behavior. There has been a growing interest in characterization of Zeno free switching systems. Different from the recent non-Zeno analysis of switching systems, the present paper studies non-Zeno properties of PASs subject to system parameter and/or initial state perturbations, inspired by sensitivity and uncertainty analysis of PASs. Specifically, by exploiting the geometry of polyhedral subdivisions and dynamical system techniques, this paper establishes a uniform bound on the number of mode switchings for a family of Lipschitz PASs under mild uniform conditions on system parameters and associated polyhedral subdivisions. This result is employed to show robust non-Zenoness of several classes of Lipschitz linear complementarity systems in different switching notions. The paper also develops partial results for robust non-Zenoness of non-Lipschitz PASs, particularly well-posed bimodal non-Lipschitz PASs

Switching and stability properties of conewise linear systems

Being a unique phenomenon in hybrid systems, mode switch is of fundamental importance in dynamic and control analysis. In this paper, we focus on global long-time switching and stability properties of conewise linear systems (CLSs), which are a class of linear hybrid systems subject to state-triggered switchings recently introduced for modeling piecewise linear systems. By exploiting the conic subdivision structure, the “simple switching behavior” of the CLSs is proved. The infinite-time mode switching behavior of the CLSs is shown to be critically dependent on two attracting cones associated with each mode; fundamental properties of such cones are investigated. Verifiable necessary and sufficient conditions are derived for the CLSs with infinite mode switches. Switch-free CLSs are also characterized by exploring the polyhedral structure and the global dynamical properties. The equivalence of asymptotic and exponential stability of the CLSs is established via the uniform asymptotic stability of the CLSs that in turn is proved by the continuous solution dependence on initial conditions. Finally, necessary and sufficient stability conditions are obtained for switch-free CLSs

Stabilizing Switched Linear Systems under Adversarial Switching

The problem of stabilizing discrete-time switched linear control systems using continuous input by the user and against adversarial switching by an adversary is studied. It is assumed that the adversary has the advantage in that at each time it knows the user\u27s decision on the continuous control input but not vice versa. Stabilizability conditions and bounds on the fastest stabilizing rates are derived. Examples are given to illustrate the results

Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's top

This paper studies local configuration controllability of multibody systems with nonholonomic constraints. As a nontrivial example of the theory, we consider the dynamics and control of a multibody spherical robot. Internal rotors and sliders are used as the mechanisms for control. Our model is based on equations developed by the second author for certain mechanical systems with nonholonomic constraints, e.g. Chaplygin's sphere and Chaplygin's top in particular, and the multibody framework for unconstrained mechanical systems developed by the first and third authors. Recent methods for determining controllability and path planning for multibody systems with symmetry are extended to treat a class of mechanical systems with nonholonomic constraints. Specificresults on the controllability and path planning of the spherical robot model are presented. Copyright © 2007 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/58647/1/1259_ftp.pd

Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications

In this paper, a unified framework is proposed to study the exponential stability of discrete-time switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples