Stochastic linear quadratic control via random parameter‐dependent truncated balanced realization

Abstract

AbstractThis article deals with stochastic linear quadratic (LQ) control by using random parameter‐dependent truncated balanced realization (RPD‐TBR). The main contribution of the article is theoretical formulation to approximate high‐order stochastic LQ control problems, using the RPD‐TBR instead of the original RPD full state model. The whole approach uses the nonintrusive and intrusive generalized polynomial chaos (GPC) approximations to generate an RPD‐TBR and the corresponding reduced‐order stochastic LQ control formulation. The latter is then rewritten in the GPC space using an intrusive Galerkin‐type projection. Solving the reduced‐order Riccati equation for the resulting deterministic LQ problem enables to suitably approximate the solution of the original full‐order stochastic LQ control problem. The feasibility and effectiveness of the proposed approach are assessed by considering the LQ control of the tip‐tilt mirror for an Extremely Large Telescope‐class adaptive optics system, when the actuator's dynamics depends on an uncertain parameter with known distribution. Through numerical simulations, the proposed approach ability to perform LQ gain selection and performance evaluation for plants with probabilistic uncertainty more efficiently than the standard Monte Carlo approach is illustrated.