Stability Properties of Switched Dynamical Systems A Linear Matrix Inequality Approach

Doctoral thesis, 2003

The dynamical properties of many natural phenomena are traditionally described by smooth differential equation models. The use of automated control structures imposes new behaviors on the controlled system. Actions are often organized in a logical manner. Switched dynamical models are a natural extension to smooth differential equation models. They provide an intuitive description of many concepts in control theory. Particularly useful is the ability to characterize the presence of a logical behavior in the system models. System analysis is usually performed either in continuous time or in discrete time utilizing uniform sampling. These concepts are well known and understood in the literature. This work is concerned with the analysis of switched system models. With the switched models there is a new concept of event-driven discrete time, evolving according to logical events in the switch structure. The system properties can not be directly translated between this asynchronous time and the continuous time-scale. Methods are presented that take into account the relation between the two time domains. The property of stability forms a base for most analysis methods in control system theory. Stability conditions for switched systems are derived using Lyapunov theory, where an abstract measure on system energy is used to conclude the system's evolution. It is shown how suitable Lyapunov energy functions are formulated. New developments in optimization and software implementations make it possible to find the Lyapunov functions by solving convex optimization problems. The method relies on problem formulations with linear matrix inequalities. The presented results are mainly based on switched linear systems, but extensions are made to include nonlinear dynamics in a linear parameter-varying framework. The behavior of switched systems with a stationary operating point can be considered to be well understood. This thesis pays special attention to the periodic behaviors that often arise in switched dynamical systems. We consider in particular the stable behavior of a periodic trajectory, referred to as a switched limit cycle. It is shown how Lyapunov theory can be used to state stability conditions in form of linear matrix inequalities. Non-conservative estimations are made on system performance, stated as an exponential convergence rate.