Jump Linear Quadratic Gaussian Control in Continuous Time

Abstract

This paper is concerned with the optimal quadratic control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The systems are also subject to gaussian input and measurement noises. This Jump Linear Quadratic Gaussian (JLQG) optimal control problem can be used to consider the control of systems which are subject to abrupt changes in their structure and components and also with disturbances on actuators and sensors. The solution of the continuous-time Jump Linear Quadratic (JLQ) problem, (the JLQG problem with complete state information and no input noise) is known. However in many applications, the plant state is available only through noisy observations on the output channel. In this paper, the optimal solution for the JLQG problem in finite time is given. This solution is based on a separation theorem. The optimal state estimator is sample path dependent (it may depend upon past as well as the current values of the jump parameter). For the infinite time JLQG problem, the optimal solution is also obtained. If the plant parameters are constant in each value of the underlying jumping process, then the controller part of the compensator converges to a time invariant control law (which depends on the jump parameter). However the filter portion of the optimal infinite time horizon JLQG compensator is not time invariant. A suboptimal filter which does converge to a steady-state solution (under certain stochastic stabilizabilty and observability conditions) is also derived.