An Approximation of the Riccati Equation in Large-Scale Systems With Application to Adaptive Optics
The matter of finding linear optimal controllers and estimators, like linear quadratic regulators or Kalman filters (KFs), is solved by suggests that 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 variety of states, creating it troublesome if not impossible to use optimal (minimum-variance) control and/or estimation ways to large-scale systems. A specific example is adaptive optics (AO) system for the subsequent generation of extraordinarily large telescopes, for which the amount of states to be estimated by a KF is in the order of the tens of thousands, creating the numerical solution of the Riccati equations problematic. During this paper, we tend to show that for a special class of state-house systems, the discrete-time algebraic Riccati equation can be simplified with an approximation that results in a closed-kind resolution, that will be computed additional quickly and used as an alternate to standard numerical solvers. The category of systems for that this approximation holds includes a category of models widely used in AO, specifically autoregressive (AR) models of order 1 or two (AR1 and AR2). We tend to verify a posteriori the accuracy and applicability of the proposed answer.
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