The optimal estimation theory of the Wiener-Kalman filter
is extended to cover the situation in which the number of memory elements
in the estimator is restricted. A method, based on the simultaneous
diagonalisation of two symmetric positive definite matrices, is given
which allows the weighted least square estimation error to be minimised.
A control system design method is developed utilising this
estimator, and this allows the dynamic controller in the feedback path
to have a low order. A 12-order once-through boiler model is constructed
and the performance of controllers of various orders generated by the
design method is investigated. Little cost penalty is found even for
the one-order controller when compared with the optimal Kalman filter
system. Whereas in the Kalman filter all information from past
observations is stored, the given method results in an estimate of the state variables which is a weighted sum of the selected information
held in the storage elements. For the once-through boiler these weighting
coefficients are found to be smooth functions of position, their form
illustrating the implicit model reduction properties of the design
Minimal-order estimators of the Luenberger type also generate
low order controllers and the relation between the two design methods
is examined. It is concluded that the design method developed in this
thesis gives better plant estimates than the Luenberger system and, more
fundamentally, allows a lower order control system to be constructed.
Finally some possible extensions of the theory are indicated.
An immediate application is to multivariable control systems, while
the existence of a plant state estimate even in control systems of very
low order allows a certain adaptive structure to be considered for systems
with time-varying parameters.
Submitted in partial fulfilment of the requirements for the award of
Doctor of Philosophy of the Loughborough University of Technology.