The application of optimal control theory to a class of extremum control systems

This thesis constitutes a contribution to the theory of extremal control systems which come within the automatic control field of learning in engineering science.
An extremal control system comprises an extremal process (which is to be controlled) and an extremal controller. The class of extremal control systems referred to in the title of the thesis refers to those systems in which the extremal process concerned can represented by a particular form of mathematical model.
Investigation of the application of optimal control theory to this class of extremal control system is motivated by two basic requirements. The first of these originates in the control of complex industrial processes, particularly in the direct digital control situation. Here there are a number of possible applications for extremal control, and since costs involved are generally high, the requirement is generally for extremal controllers maintaining as low a system operating cost as possible, rather then for controllers with a simple structure. Hence there is a need for optimal extremal controllers.
The second requirement for establishing optimal extremal controllers lies in the design of simpler extremal controllers. The performance of the optical extremal controller is required as a basic on which to judge the simpler controllers and decide whether any more significant improvement can be made.
The major part of this investigation of the application of optimal control theory to extremal control systems is concerned with the simplest possible case, where no account is taken of input or output lags or noise, or of multiple inputs. Since no previous optical extremal controllers known this is the natural starting point.
A rigorous analysis of state variables, probability nu sufficient statistics for this simplified case is presented, and this leads on to a new approach to the application of dynamic programming which in turn results in an original functional recurrence equation. Analysis of this functional recurrence equation leads to s numerical solution procedure including many checks. This eventually establishes the optical extremal controller.
Simulation techniques are used to confirm the performance of the optimal extremal controller, and hybrid computing facilities used to show that it can be implemented on a small on-line computer and used in a direct digital control situation without suffering from interfacing effects. Thus for the first time there is now available an optimal extremal controller, and moreover it can be confidently expected to perform optimally in a practical situation.
Comparison of the performance of the optimal extremal controller with the performance of simpler controllers shows that, in the simplified case, there is still room for considerable improvement on the simple controllers, and a quantitative measure of just how much improvement might be possible is now available.
An initial investigation of the application of optimal control theory to more complex processes, involving noise, lags and multiple inputs, is presented and this shows that theoretical difficulties are likely to prevent further optimal extremal controllers from being established. In these cases there is therefore a requirement to establish closely optimal controllers. One approach would be just to use the optimal controller from the simplified case in these more complex situations. Simulations are presented establishing the performance of the optimal extremal controller, and a suboptimal extremal controller, controlling not the process for which they were designed but a similar process involving output noise. These simulations show that these controllers are certainly not optimal in the more complex situation. A different approach must therefore be taken to establishing closely optimal controllers in the more complex situation. The thesis finishes with a discussion of how this might be achieved.