On linear quadratic optimal control for time-varying multimodal linear systems with time-triggered jumps

This letter details the solution to the linear quadratic (LQ) optimal control problem over a finite interval for time-varying multimodal linear systems with time-triggered jumps. By multimodal, we mean the possibility for the system state to change dimension after every jump. To this end, we introduce the multimodal jumping differential Riccati equation (MJDRE) and we show the equivalence between the solvability of the optimal control problem and the existence of a finite solution of the MJDRE. The MJDRE can be used to compute optimal tracking gains for hybrid system with state-triggered jumps, whose state dimension changes after each jump (multimodal hybrid system). This is demonstrated, in simulation, on a 2DOF dual-mass spring-damper system.