Control Design for Ultimate Boundedness of Linear Switched Systems With Controlled and Uncontrolled Subsystems

Abstract

Motivated by the Satellite Formation Keeping Control Problem, this paper presents a theoretical framework for designing controllers for the ultimate boundedness of linear switched systems consisting of both controlled (and thus stable) and uncontrolled (unstable) subsystems. It is shown that if the dwell time in the controlled mode (i.e ‘on time’) and that of the uncontrolled mode (’off time’) are chosen judiciously, ultimate boundedness of the switched system is guaranteed. Towards this direction explicit formulae for the switching times between controlled and uncontrolled modes are provided as a function of the parameters of the ultimate boundedness region. Using Lyapunov theory, a control design procedure is presented that achieves a good trade off between the ratio of ‘off’ and ‘on’ times and the size of the ellipsoidal boundedness regions which are representative of the system performance. The proposed control design technique has useful applications in mechanical and aerospace systems. The importance of this theory and possible extensions of these concepts are discussed.