An Approximation of the Riccati Equation in Large-Scale Systems With Application to Adaptive Optics

The problem of finding linear optimal controllers and estimators, like linear quadratic regulators or Kalman filters (KFs), is solved by means of a matrix Riccati equation. A bottleneck of such an approach is that the numerical solvers for this equation are computationally intensive for systems with a high number of states, making it difficult if not impossible to apply optimal (minimum-variance) control and/or estimation methods to large-scale systems. A specific example is adaptive optics (AO) system for the next generation of extremely large telescopes, for which the number of states to be estimated by a KF is in the order of the tens of thousands, making the numerical solution of the Riccati equations problematic. In this paper, we show that for a special class of state-space systems, the discrete-time algebraic Riccati equation can be simplified with an approximation which leads to a closed-form solution, which can be computed more quickly and used as an alternative to standard numerical solvers. The class of systems for which this approximation holds includes a class of models widely employed in AO, namely autoregressive (AR) models of order 1 or 2 (AR1 and AR2). We verify a posteriori the accuracy and applicability of the proposed solution.