Minimal representation of matrix valued white stochastic processes and U–D factorisation of algorithms for optimal control

Abstract

Two different descriptions are used in the literature to formulate the optimal dynamic output feedback control problem for linear dynamical systems with white stochastic parameters and quadratic criteria, called the optimal compensation problem. One describes the matrix valued white stochastic processes involved, using a sum of deterministic matrices each one multiplied by a scalar stochastic process that is independent of the others. Another, that is more general and concise, uses Kronecker products instead. This article relates the statistics of both descriptions and shows their advantages and disadvantages. As to the first description, an important result that comes out is the minimum number of matrices multiplied by scalar, independent, stochastic processes needed to represent a certain matrix valued white stochastic process, together with an associated minimal representation. As to the second description, an important result concerns the generation of all Kronecker products that represent relevant statistics. Both results facilitate the specification of statistics of systems with white stochastic parameters. The second part of this article further exploits these results to perform an U–D factorisation of an algorithm to compute optimal dynamic output feedback controllers (optimal compensators) for linear discrete-time systems with white stochastic parameters and quadratic sum criteria. U–D factorisation of this type of algorithm is new. By solving several numerical examples, the U–D factored algorithm is compared with a conventional algorithm.