A control engineer is fortunate to be able to work in a field where elegant mathematics often leads to a useful end result. Some of the most exciting examples of this aspect of control engineering have become widely known. For example Wiener filtering , has provided us a method of optimizing the design of constant coefficient linear filter to reduce the impact of noise. This work was thought to be so useful that it was classified during World War II and not published until 1948. In the sixties, Kalman filtering , provided a similar kind of a break through, but it was done in a state space setting. Both of these techniques were based on a knowledge of the spectral content of the disturbing signals, and both were aimed at estimation of signals in the presence of noise. Luenberger , developed a methodology for estimating unmeasured states and his method did not require any knowledge of stochastic processes. Luenberger’s state estimates are referred to as observers. Luenberger observers and Kalman filter both provide a mechanism for using estimates of unmeasured states in a linear feedback controller. In both cases there is a separation theorem available for the design of the estimator and controller. In the case of stochastic models, the optimal stochastic control , makes use of a Kalman filter to provide state estimates. The combined controller and estimator makes up an intermediate dynamical system which could be thought of as a compensator. This famous result becomes known as the “L.Q.G.”  result, meaning that it applied to linear systems with quadratic performance measures, and gaussian disturbances and initial conditions. Few would deny the mathematical elegance of this solved problem. Often the technique can lead to a useful end result. But the dimension of the Kalman filter used in the compensator could present practical difficulties with respect to implementation and this is still an issue thirty years later.
KeywordsAutomatic Control Kalman Filter Model Reduction Unmeasured State Luenberger Observer
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