On the Singularly Perturbed Matrix Differential Riccati Equation

In this paper, the finite-time optimal control problem for time-invariant linear singularly perturbed systems is considered. The reduced-order pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed differential matrix Riccati equation of dimension n1+ n2into the regular differential matrix Riccati equation pure-slow of dimension n1and the stiff differential matrix Riccati equation pure-fast of dimension n2. A formula is derived that produces the solution of the original singularly perturbed matrix Riccati differential equation in terms of solutions of the pure-slow and pure-fast reduced-order differential matrix Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear systems independently in pure-slow and pure-fast time scales. An example for a catalytic fluid reactor model has been include to demonstrate the utility of the method.