Switched and Hybrid Systems

Design procedures for stable switched linear systems

A major focus of the hybrid systems group is the analysis and design
of switched linear systems. Our research  is focussed in three main
directions: (i) determining necessary and sufficient conditions for the
uniform global asymptotic stability for switched linear systems under arbitrary
switching conditions; (ii) determining the existence of stabilising switching
sequences for switched linear systems; and (iii) applications. Our recent
work has been motivated by our interest in communication systems and has
led to work on the stability of time-varying positive linear systems.

Necessary and sufficient conditions for stability under arbitrary switching

We have focussed on determining tractable conditions for the existence of global Lyapunov functions. Here, we have considered a number of classical problems: (i) analytic conditions for the existence of quadratic Lyapunov functions; (ii) analytic conditions for the existence of non-quadratic Lyapunov functions; and (iii) numerical conditions for the existence of Lyapunov functions. Recent results obtained include time-domain versions of the Circle Criterion; an extension of the Circle Criterion for non-linear switched systems; analytic conditions for the existence piecewise linear Lyapunov functions; design procedures for stable pole-placement for SISO switched systems; and a describing function for switched linear system.

Determining stabilising switching sequences

Recently, we have addressed the design issues of determining stabilising switching sequences for switched unstable systems. The aim is to develop state-feedback switching strategys which (i) make the switched systems stable; (ii) reduce the switching frequency in contrast with the existing switching laws; and (iii) are robust against (time-varying and nonlinear) system perturbations. Results obtained include a dwell-time state/output-feedback switching strategy for a class of switched systems which are not necessarily quadratically stabilisable. The analysis is based on a combined technique from continuous-time Lyapunov approach and discrete-time Lyapunov method.

Synthesis of switched linear control systems

We have focussed on developing a synthesis framework for switched linear control systems which extends the standard linear theory. When both the control input and switching sequence are design variables, we have considered the following problems: (i) system structural decomposition of a switched system by means of controllability and observability; (ii) feedback equivalence and feedback reduction for multi-input switched systems; and (iii) constructive stabilising design procedures based on the canonical forms. More synthesis problems are under current investigation.

Applications

We are also actively involved in the design of switched control systems. To-date, we have, in collaboration with industrial partners, participated in the design of switched speed control systems; 4-wheel steering systems; switched wind-turbine controllers; switched anti-locked braking systems; and in the design of switched congestion controllers for communication systems.

Group personnel

Robert Shorten
Oliver Mason
Selim Solmaz