Stability and Robust Control of Time-Varying Systems Using Linear Matrix Inequalities

This article provides a theoretical contribution by presenting a detailed study on the application of Linear Matrix Inequalities (LMIs) in time-varying systems. Specifically, it addresses a set of LMI conditions that ensure the stability of such systems. To derive these conditions, Lyapunov’s Direct Method is employed, and the stability is characterized using class- $\mathcal {K}$ functions. The parallel distributed compensation technique is utilized to guarantee stability for both open-loop and closed-loop systems. Furthermore, the study establishes sufficient LMI conditions for global exponential stability in linear time-varying systems and presents exponential stability conditions for nonlinear time-varying systems. Finally, the robust control method is validated through the controller design of a parametrically excited pendulum. In addition, a comparative analysis between the robust method and the control method based on the Lyapunov–Floquet transformation is presented.