4 Estimation over Finite Time Interval

Abstract

In this chapter we shall consider the estimation problem over a finite interval of time. In formulating such problems we limit the allowable structure of the estimator so that the number of computations is kept within reason. In solving the problems presented here we primarily use the matrix version of the minimum principle [1] as our method of derivations. In certain cases it is preferrable to solve the problem using the innovations method [2, 3] and the orthogonal projection principle [4], so we shall introduce that principle as well, and it will be used extensively in future chapters. The importance of the material presented here is that we can apply our methods to non-stationary stochastic processes. This is the advance that Kalman Filtering made over Wiener filtering [6], only we are doing it in a reduced order state space setting. The systems considered can be time variable, as when one linearizes equations about a nominal trajectory which varies with time. Alternatively, we may look at stable systems during the time interval for which their initial conditions are having significant impact on the response. Or we may consider unstable systems for which stationary conditions are never met. Thus this chapter opens up many new possibilities, although we must still restrict ourselves to linear systems, described by state space equations.

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