Optimal Control of Non-Autonomous Switched Linear Systems: Linear Matrix Inequality Approach

Switched linear system is an important class of switched systems which can model a wide range of nonlinear systems. In this paper, optimal control, stability and robustness of these systems are discussed. Continuous control inputs and discrete switching signal are the design variables to obtain the lower bound of a given cost function. In the proposed method, first, we assign a quadratic Lyapunov function to each subsystem such that must satisfy a set of inequalities to reach the lower bound of the cost. Second, to guarantee the exponential stability of overall system, these quadratic functions also must satisfy the conditions of presented exponential stability theorem. To obtain Lyapunov functions, state-feedback control inputs, switching signal and lower bound of the cost function, two sets of obtained inequalities are converted to a set of Linear Matrix Inequalities (LMIs) that must be solved. Moreover, optimality of the lower bound is proved. Finally, robustness of presented method against norm bounded time-varying uncertainties is shown. Several examples illustrate the efficiency of presented method.