The Riccati Equation And Its Application In Max-plus Algebra

Riccati equations are widely considered as playing a very important role in modern control theory and applications. Due to the important applications in real life, many scholars have made a systematic study of Riccati equation. In addition, the max-plus algebra has significance in research work and also has important application value on discrete systems and control. Many problems of manufacture, communication, traffic,etc, can be studied with max-plus algebraic method.We study a class of Riccati equation in max-plus algebra in this paper. We will give the definition of a class of Riccati equation in max-plus algebra. In order to study the max-plus algebra Riccati equation, we also define Lyapunov equation and other related concepts in max-plus algebra.We discuss the solvability of max-plus algebra Riccati equation by studying the max-plus algebra Lyapunov equation in this paper. We get a sufficient condition for the solvability of Riccati equation and an analytical expression of solutions. In addition, we give the range of eigenvalues of weak discriminant matrix when the solutions of Riccati equation have no zero diagonal elements. At the same time, we prove that the sum of the two solutions of Riccati equation is also a solution of this equation. Finally, we try to study the applications of Riccati equations in optimal control for a class of maxplus discrete event systems. We prove the relationship between the solutions of Riccati equation and the output-nulling subspaces of the corresponding system. Then we find the minimal cost function of a class of discrete max-plus linear systems by studying Riccati equation’s solutions.