Some aspects of optimum control in a simple distributed parameter system.
thesisposted on 19.11.2015, 08:58 by William Norman. Taberner
The Problem. The problem discussed in this thesis concerns optimal control of temperature within a finite slab of material, subject to external disturbances on one face, by applying control signals to the opposite face. Throughout the treatment is one-dimensional and the process (i.e. the slab) is assumed to be linear. introduction (Part 1). The problem is stated explicitly, and the general field of distributed-parameter control and the techniques used for solution of distributed-parameter optimal problems are reviewed. Standard Methods (Part2). The standard methods of solving "diffusion-type" problems in the time and frequency domains are described and typical solutions presented. Steady State Control in the Presence of Sinusoidal Disturbances (Part 3). Optimal control steady state solutions for the external control drive when the disturbances are sinusoidal are obtained by using the technique of slab sub-division. This reduces the problem to state variable form, and then Pontryagin's maximum principle is used on the set of first order "state variable" equations. Solutions are obtained analytically for a very simple approximation, and then analogue computer solutions of the state variable and adjoint equations for more elaborate sub-division of the slab are presented. These solutions are then synthesised into a general control equation. Finally a pure hill-climbing technique is used to generate an optimal solution. Steady State Control in the Presence of Random Disurbances (part 4). Optimal steady state solutions for the required external control drive when the disturbances are random are obtained using the technique of slab sub-division. The random disturbance treated is stationary band-limited Gaussian white noise. For a simple slab sub-division a Pontryagin analysis is used. For a more elaborate sub-division of the slab, digital computer solutions are sought via the Hamilton-Jacobi-Riccati method.