Quadratic and Non-Quadratic Stability Criteria for Switched Linear Systems

This thesis deals with the stability analysis of switched linear systems. Such systems are characterised by a mixture of continuous dynamics and logic-based switching between discrete modes. This system class appears in a large variety of control systems and has a wide field of applications in the modern industrial society. While the application of switched control systems can be very beneficial, their stability analysis is often complex. Even though, switched system analysis has been studied extensively in the last decade, few analytical tools for stability have been developed. To date, the most common stability tools are based on numerical optimisation that provide little insight into the (in)stability properties of the process. The objective of this thesis is to develop stability tools that are readily applicable and provide some support for the design process of controllers for switched systems. In this thesis stability criteria for hybrid systems that resemble many of the classical stability results for linear time-invariant systems are derived. The principal tool employed for stability analysis of the systems is Lyapunov theory: both quadratic and non-quadratic Lyapunov functions are used to derive compact eigenvalue conditions that guarantee stability of certain classes of switched systems. New results for second order switched systems are developed, a describing function technique for switched systems is presented, and pole-placement techniques for stabilising switched singleinput single-output (SISO) PID control structures are derived. In addition, the results also lead to new more compact versions of well known SISO stability criteria for nonlinear systems of the Lur’e type. In this context, we show that well known criteria such as the Circle Criterion, Popov Criterion, and the KYP lemma can be evaluated in a compact manner.