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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 method. 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.