Feedback control of linear diffusion processes

Abstract

We consider the class of linear diffusion processes described by the generalized heat equation ∂u/∂t+ Au = F which relates the temperature distribution u(x, t) of a body Ω in n-dimensional space to the applied heat sources F(x, t). The operator A is a time-invariant, symmetric differential operator with discrete, semi-bounded spectrum. The control is provided by M heat sources in or on the body and the temperature is measured by P sensors located at various points along the body. A modal feedback control is found by applying known state variable control methods to a truncated modal approximation of the above distributed parameter system. The controller is a combination of a state estimator and a linear feedback law and it controls N of the modes of the process. Although the rest of the process modes are not controlled the effect of the controller spilling some of its energy into this residual (uncontrolled) mode system must be accounted for. Our main result is that this control spillover does not change the essential dissipative nature of the diffusion process but only increases the desired control response time by a predictable amount. Therefore, it is possible to use well known finite dimensional control techniques to produce a practical modal feedback controller for a linear diffusion process and the control spillover due to the infinite dimensional nature of the process can be accounted for without changing the form of the controller.