Observer-Based Control of Piecewise-Affine Systems

This paper presents a new synthesis method for both state and dynamic output feedback control of a class of hybrid systems called piecewise-affine (PWA) systems. The synthesis procedure delivers stabilizing controllers that can be proven to give either asymptotic or exponential convergence rates. The synthesis method builds on existing PWA stability analysis tools by transforming the design into a closed-loop analysis problem wherein the controller parameters are unknown. More specifically, the proposed technique formulates the search for a piecewise-quadratic control Lyapunov function and a piecewise-affine control law as an optimization problem subject to linear constraints and a bilinear matrix inequality. The linear constraints in the synthesis guarantee that sliding modes are not generated at the switching. The resulting optimization problem is known to be NP hard, but suboptimal solutions can be obtained using the three iterative algorithms presented in the paper. The new synthesis technique allows controllers to be designed with a specified structure, such as a combined regulator and observer. The observers in these controllers then enable switching based on state estimates rather than on measured outputs. The overall design approach, including a comparison of the synthesis algorithms and the performance of the resulting controllers, is clearly demonstrated in four simulation examples. ∗Paper provisionally accepted for publication in the International Journal of Control. †Funded in part by a PRAXIS XXI grant. ‡Author to whom all correspondence should be addressed: 37-391, 77 Massachusetts Ave., Cambridge, MA 02319. Tel: (617) 253-3267; Fax: (617) 258-5940.