Simultaneous and sequential control design for discrete-time switched linear systems using semi-definite programming

The control of switched linear discrete-time systems occurs in multiple engineering fields, where it has been used to deal with complex and non-linear systems. This paper presents two strategies to design control laws for discrete-time switched linear systems, whilst guaranteeing asymptotic stability of the closed loop. Firstly, an arbitrary switching signal is considered. In this scenario a common quadratic Lyapunov function is used for stability, but subsystem Lyapunov functions are employed to improve local subsystem performance. Secondly, a constrained switching signal, associated with subsystem lower dwell time bounds is studied. In this case, a decrease in Lyapunov cost is achieved by design, based on dwell time constraints only, thus removing the need for both a common quadratic Lyapunov function or direct stable switches. It is shown in both cases that the control design problems can be formulated as one or a sequence of semi-definite programming problems, and therefore can be solved efficiently. Finally, two examples are provided in order to illustrate the different techniques presented.