An Inversion-Free Iterative Algorithm With a Scalar Tuning Parameter for Coupled Riccati Matrix Equations Arising in LQ Optimal Control of Markov Jump Systems

In this article, iterative algorithms are investigated to solve the Riccati algebraic matrix equations arising in the context of linear quadratic (LQ) optimal control of discrete-time Markov jump systems. By using the matrix inversion lemma, an equivalent form for the considered coupled Riccati matrix equation is constructed, and then two intermediate variables are introduced to further construct a new equivalent form in order to avoid the operation of matrix inversion. With the aid of the obtained equivalent form, an inversion-free iterative algorithm (IFIA) is developed to obtain the positive definite solutions of the discrete coupled Riccati matrix equation via the principle of fixed points. To improve the convergence performance, the latest updated information is exerted in the presented algorithm. Furthermore, the convergence analysis is conducted by mathematical induction for the obtained IFIA. At the end, a numerical example is given to demonstrate the effectiveness of the proposed algorithm by making comparisons with existing results.