The Multi-Objective Optimization Problems of Markovian Jump Systems Closed by General Controllers: A Mode-Separation Approach

This paper studies the multi-objective optimization problems (MOPs) of Markovian jump systems (MJSs) closed by general controllers. Firstly, the linear quadratic regulator (LQR) problem of MJSs closed by a general controller with sampled or synchronous modes is studied and estimated by a mode-separation approach, whose traditional algebraic Riccati equations (AREs) are removed. Meanwhile, two traditional situations about mode-dependent and -independent controllers are contained as special ones. Secondly, an MOP deeply depending on mode separations is proposed and solved by the non-dominated sorting whale optimization algorithm (NSWOA) such that the minimum value of LQR problem and the maximum expectation value of summed mode dwell times in the same mode separation are simultaneously optimized. Particularly, in order to further improve the estimation effect, a single-objective optimization problem (SOP) is presented and computed by applying the deep deterministic policy gradient (DDPG) technique, whose optimal controller in addition to its best mode separation is given in detail. Thirdly, similarly but more generally, another controller with its mode and state both sampled is constructed to realize the LQR problem, whose effects are better than some existing methods and can also be improved by solving some optimization problems. Finally, a simulation is used to verify the effectiveness and superiority of the proposed methods. Note to Practitioners—This paper addresses the LQR problem for MJSs, commonly used in automation and control with uncertainty and frequent mode switching. Traditional LQR methods rely on mode-dependent controllers and AREs, leading to high computational costs and real-time demands, especially in industrial settings. To overcome these challenges, this paper proposes a general controller design and an MOP, replacing AREs with a mode-separation strategy, simplifying the process and reducing costs. A controller with both mode and state sampling is also introduced to achieve the LQR objective in complex industrial environments. Although effective, further research is needed for its application in large-scale systems. Future work will focus on expanding its applicability.