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

本文给出了线性时滞系统多项式矩阵Riccati方程解和最优调节器的一种计算方法。首先，研究环上的矩阵Riccati方程。特别地，还研究了Laurent多项式环上的矩阵Riccati方程。然后，我们讨论了环上系统和延时系统。其次，考虑了不依赖于时滞的稳定性和点稳定性。研究了一类与时滞无关的时滞系统的镇定问题。将时滞系统视为实多项式环上的系统，研究了相应的洛朗多项式环上的矩阵Riccati方程。采用多项式矩阵Riccati方程方法求解系统的镇定问题。给出了任意域上有理转移矩阵的最小状态空间实现过程。该过程基于Smith-McMillan形式，并导致Jacobson范式的状态转移矩阵。最后，研究了线性时滞系统的观测反馈抗扰问题。将时滞系统视为实多项式环上的系统，并在几何方法的框架内表述该问题。然后，得到了问题可解的充分必要条件。

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.