The linear-quadratic optimal regulator for descriptor systems: Discrete-time case

Abstract

In this paper the linear-quadratic optimal regulator problem for the discrete-time descriptor system Ex k + 1 = Ax k + Bu k , where E is, in general, a singular matrix, is investigated. The continuous-time version of this problem was treated elsewhere. The discrete-time theory is largely parallel to the continuous-time theory but there are several important differences. One is that in the discrete-time case a fairly general problem can be solved with singular cost matrices. The key to the solution, as for the continuous-time problem, is a coordinate system which can be computed by performing a singular value decomposition of E; this is called an SVD coordinate system. The optimization problem is solved by applying Hamiltonian minimization to compute the optimal trajectory of the vector consisting of the concatenated descriptor, codescriptor, and control vectors. In the general (singular cost) case this trajectory is computed by recursively solving a discrete-time Riccati equation. From the optimal trajectory the optimal descriptor variable feedback gain relating the control to the descriptor variable can be computed. Several Riccati difference equations are derived, all of which have the same solution; this solution can be used to compute the optimal trajectory and cost. For non-singular problems, even in the finite-horizon case, the optimal feedback gain can be computed non-recursively by solving an eigenvalue-eigenvector problem formulated from the system parameters in the SVD coordinate system. The optimal gain for the infinite-horizon (steady-state) problem, on the other hand, can be computed from the solution of an eigenvector problem involving the untransformed system parameters.