Global Practical Stabilization of Discrete-time Switched Affine Systems via Switched Lyapunov Functions and State-dependent Switching Functions

Abstract

This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via switched Lyapunov functions with the objectives of achieving less conservative stability conditions and less conservative size of the ultimate invariant set of attraction. The main contribution is to propose a state-dependent switching controller synthesis that guarantees simultaneously the invariance and global attractive properties of a convergence set around a desired equilibrium point. This set is constructed by the intersection of a family of ellipsoids associated with each of switched quadratic Lyapunov functions. The global practical stability conditions are proposed as a set of Bilinear Matrix Inequalities (BMIs) for which an optimization problem is established to minimize the size of the ultimate invariant set of attraction. A DC-DC buck converter is considered to illustrate the effectiveness of the proposed stabilization and controller synthesis method.

Practical Stability Analysis and Switching Controller Synthesis for Discrete-time Switched Affine Systems via Linear Matrix Inequalities

“…The ellipsoid E(P, R * ) in Relation (17) with R * ≤ 1 is an invariant set of attraction for the system (2) under switching rule (8) when e(k) ∈ E(P, 1) and switching rule (19) when e(k) / ∈ E(Q, e c , 1). Thus, system (2) is globally practically asymptotically stable in the sense of Def.…”

“…A review of recent results on the practical stabilization of switched systems without common equilibria or in the particular case of switched affine systems can be found in [16] and [17]. The main differences are basically in the level of conservatism and level of computational complexities [18].…”

“…The approach in [16] was extended in [17] using switched centralized quadratic Lyapunov functions. Reference [26] extended the existing works [8], [10], [11], [17] employing single Lyapunov functions via a common shifted quadratic Lyapunov function and a shifted ellipsoid parameterized independently.…”

“…The approach in [16] was extended in [17] using switched centralized quadratic Lyapunov functions. Reference [26] extended the existing works [8], [10], [11], [17] employing single Lyapunov functions via a common shifted quadratic Lyapunov function and a shifted ellipsoid parameterized independently. These previous studies based on the common Lyapunov functions imply the existence of a Schur stable convex combination of the subsystems matrices as a necessary condition.…”

“…These previous studies based on the common Lyapunov functions imply the existence of a Schur stable convex combination of the subsystems matrices as a necessary condition. In the case of using switched Lyapunov functions [11], [17], this requirement is relaxed; however, the resulting BMI problems still contain many bilinear terms and offer a high degree of computational complexity. Similar to [16], in the present work, we use the switched centralized quadratic Lyapunov functions and relax the limitation of the existing stable convex combination of the subsystems matrices.…”