The linear-quadratic optimal regulator for descriptor systems

In this paper we investigate the linear-quadratic optimal regulator problem for the continuous-time descriptor system Ex = Ax + Bu where E is, in general, a singular matrix. The infinite-horizon version of this problem was previously solved by Cobb [5] by a quite different approach. We solve a somewhat more general finite-horizon problem by applying Hamiltonian minimization to derive the optimal trajectory of the vector consisting of the concatenated descriptor, codescriptor, and control vectors. From this trajectory the optimal feedback gain relating the control and descriptor variable can be computed. By transforming to a coordinate system which can be computed by performing a singular value decomposition of E, we derive a Riccati differential equation whose solution gives the optimal cost. The steady-state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters. In general there does not exist a unique optimal feedback gain but a linear variety whose dimension is equal to the number of inputs times the rank deficiency of E. A more complete version of this paper has been submitted for publication [3].