Time-Varying LQR Control with Matrix Smoothing for Continuous-Time Switched Systems Under Time-Dependent Switching

This paper addresses the analysis and control of continuous-time linear switched systems within the linear time-varying (LTV) framework. A correspondence between switched and LTV system representations is first established, enabling the use of LTV tools for stability and controllability analysis under arbitrary switching sequences. Two state-feedback synthesis approaches are considered. The first is a switched LQR design formulated via Linear Matrix Inequalities using Multiple Lyapunov Functions. The second is a continuous-time LTV-LQR obtained by solving the differential Riccati equation. To mitigate the discontinuities associated with hard switching, a matrix smoothing strategy based on the partition of unity is introduced. This approach generates continuous approximations of the plant matrices utilized exclusively for control synthesis in the time-varying Riccati framework, while the actual plant retains its hard-switched dynamics. Numerical simulations for representative switched systems illustrate that the proposed smoothed LTV formulation leads to improved closed-loop behavior compared to the switched LQR baseline, including reduced control discontinuities and lower values of the quadratic performance index for the considered scenarios. The results indicate that the method is suitable for applications where abrupt control variations are undesirable.