A computational method for a class of jump linear quadratic systems

Abstract

A class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. An example is solved to illustrate the efficiency of the method.

A gradient flow approach for computing jump linear quadratic optimal feedback gains

“…Most of the algorithms proposed for solving this problem were based on the necessary conditions of optimality [8]. Recently a different approach for solving this problem has been considered in [10].…”

“…By seeking the best feedback control law depending both on the measurable output as well as the mode of the system, the authors in [10] convert the original stochastic control problem into an equivalent standard deterministic optimisation problem, which can be easily solved by any nonlinear programming technique.…”

“…In this paper, we consider exactly the same problem as [10] with the aim of finding an optimal feedback controller which depends on both the measurable output and the mode of the system. By using the idea of [9] and [11], a gradient flow based algorithm is derived for solving this problem.…”

“…By using the idea of [9] and [11], a gradient flow based algorithm is derived for solving this problem. This gradient flow method is better than the optimisation approach proposed in [10], because asymptotic stability of the system is automatically guaranteed by this gradient flow method without the necessity of solving an optimal parameter selection problem subject to complex stability constraints. Moreover, since the gradient flow always searches in the direction of decreasing cost of the objective functions and the cost at the boundary of the stability region is infinite, the search of the gradient flow algorithm never takes us to the boundary of the stability region.…”

“…Analysis of the Gradient Flow. Using the same technique as that is used in [10], it is easy to deduce that the gradient of J(F, P ) with respect to F is given by…”