A Principled Approximation for Optimal Control of Semi-Markov Jump Linear Systems using Pseudo-Markovianization

High-performance control of semi-Markov jump linear systems requires an accurate model for theunderlying stochastic jump process. In practice, however, a reasonable compromise between the controlquality and computational costs should be made and the tractability of the control design problem hasto be established. This paper considers the problem of finite-horizon optimal quadratic control of semi-Markov jump linear systems and investigates how the modeling quality of the underlying jump processmay affect the control performance. The problem is first examined using the phase-type distributionapproach by approximating an arbitrary semi-Markov jump linear system with fully observable jumpsby a Markov jump linear system with partially observable jumps. Then, through a process called pseudo-Markovianization, a technique for low-order approximation of the jump process is proposed that modelsa semi-Markovian process by a Markov-like model with possibly negative transition rates. It is shownthat in modeling of the holding-time distributions the probabilistic interpretation of the model does notneed to be preserved. The flexibility provided by the technique enables us to obtain a more accurate, yet low-order approximate model for the jump process for control design. Several examples are given throughout the paper to demonstrate the strengths and effectiveness of the technique.