Optimal control of discrete-time linear fractional order systems with multiplicative noise

TL;DR

This paper develops two methods for solving finite horizon linear quadratic optimal control problems for discrete-time linear fractional systems affected by multiplicative noise, using dynamic programming and the principle of optimality.

Contribution

It introduces a novel expanded-state model approach and an algorithm directly based on the fractional system for optimal control of stochastic fractional systems.

Findings

Both methods produce linear optimal controls computable by software.

Numerical examples demonstrate the effectiveness of the proposed methods.

Abstract

A finite horizon linear quadratic(LQ) optimal control problem is studied for a class of discrete-time linear fractional systems (LFSs) affected by multiplicative, independent random perturbations. Based on the dynamic programming technique, two methods are proposed for solving this problem. The first one seems to be new and uses a linear, expanded-state model of the LFS. The LQ optimal control problem reduces to a similar one for stochastic linear systems and the solution is obtained by solving Riccati equations. The second method appeals to the Principle of Optimality and provides an algorithm for the computation of the optimal control and cost by using directly the fractional system. As expected, in both cases the optimal control is a linear function in the state and can be computed by a computer program. Two numerical examples proves the effectiveness of each method.