Problems in the optimal control of finite and infinite dimensional linear systems

A review of optimal control theory for linear systems with quadratic cost functions is presented. Some of the theoretical and practical limitations are discussed with special reference to distributed parameter systems. First a procedure is described for finding the optimal control by constructing a sequence of controllers that converges to the optimal; this method is valid for systems of infinite dimension provided that the operators in the state differential equation satisfy certain conditions. The proof is carried out both for the finite and infinite time interval and the connection is shown with the Riccati equation. The main problem in implementation is that one needs complete knowledge of the state at all times in order to build the optimal controller, this is almost certainly impossible for distributed parameter systems. When the state cannot be measured completely it is proved that an optimal control is realisable for time invariant finite dimensional systems. The problems of finding this control are then investigated and computational methods discussed. If the optimal control with complete knowledge of the state cannot be implemented, a method is presented whereby one can find bounds on the possible increase in the value of the cost function arising from the use of some sub-optimal control; several examples are considered. The constrained optimal control depends on the initial state and new optimisation criteria must be put forward to deal with the case in which the initial state is unknown; the most common consist of minimising the cost that can result from the worst initial state. It is then shown how the controllers designed according to these criteria may be improved by using one's limited observation at time zero to place some constraints on the initial state. The Liapunov matrix equation plays an important part in calculating the cost of any control so reducing the computational effort in its solution is useful. It is shown how this can be done and it is of special relevance for distributed parameter systems with their states expressed as an infinite series of eigenfunctions; the results are applied to a diffusion equation example. Finally, it is shown how optimal control theory may be applied to the design of proportional-integral-derivative controllers. This is done from two standpoints and the resulting controllers are shown to be identical, though the second method of proof is valid for infinite dimensional systems. The results are then applied to a simple example and to a distributed population dynamics system. The practicality of the methods of the thesis are applied to a system with realistic parameters; recommendations are made as to the best approaches