How to Solve Singular Discrete-Time Riccati Equations

Abstract

There exists a solution to the discrete time algebraic Riccati equation giving closed loop eigenvalues inside or on the unit circle, assuming the system is stabilizable. This solution is always unique. Numerical methods, like the sorted generalized Schur form method and the Kleinman iteration, often fail in case of zeros on the unit circle or lack of left invertibility. It is here suggested how such singular cases could be made regular by reduction operations on the system matrix pencil [−zI + A, B; C, D]. The solution may be discontinuous with respect to parameter variation, but the reduction approach seems numerically appealing. This is demonstrated using simple illustrative examples. The solution is the largest symmetrical matrix satisfying a corresponding LMI. To obtain a feasible solution for interior-point methods it is also necessary to do a reduction, for some problems even further than what is described in this paper.