COMPUTATION OF SOLUTIONS TO POLYNOMIAL MATRIX RICCATI EQUATIONS AND OPTIMAL REGULATOR FOR TIME-DELAY SYSTEMS

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This research paper gives a method for computing solutions of polynomial matrix Riccati equations and optimal regulator for linear time delay systems. First, matrix Riccati equations over a ring are studied. In particular, matrix Riccati equations over Laurent polynomial rings are also investigated. Then, we discuss systems over ring and time delay systems. Next, stability of independent of delay and pointwise stability are considered. A problem of stabilization independent of delay for time-delay systems is investigated. Time-delay systems are regarded as systems over the ring of real polynomials, and the corresponding matrix Riccati equations over Laurent polynomial ring are studied. Polynomial matrix Riccati equation approach is considered for the stabilization problem. We derive a procedure for a minimal state space realization of a rational transfer matrix over an arbitrary field. The procedure is based on the Smith-McMillan form and leads to a state transition matrix in Jacobson normal form. Finally, The problem of disturbance rejection by observation feedback for linear time delay systems is investigated. The time delay systems are regarded as systems over the ring of real polynomials, and the problem is formulated within the framework of a geometric approach. Then, a necessary and sufficient condition for the problem to be solvable is obtained.