Canonical equations for boundary feedback control of stochastic distributed parameter systems

Abstract

The plant considered is described by a set of time-invariant linear partial differential equations with spatially generalized Wiener process disturbance of the state and point measurement. Boundary control is used to minimize the expected value of a functional that is quadratic in the state and control. The well-known separation into linear optimal state estimator and linear optimal deterministic controller is heuristically shown using an extension of the Caratheodory lemma to obtain sufficient conditions. Each resulting Hamilton-Jacobi equation is shown equivalent to a pair of linear canonical partial differential equations. Use of the canonical equations leads to exact analytical solutions under certain conditions by means of closed loop modes. This permits a paper and pencil analysis of the engineering properties of the control of a large class of practical distributed systems with boundary controls and point measurements. An example is given of the boundary control of a one space dimension diffusion equation with input noise at the other boundary and noisy measurement taken at an internal point. Because an analytical solution is obtained, the effect of the location of this internal measurement point can be analyzed.