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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 [1], 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 [2], 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 [3], 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 [4], 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.” [5] 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.

The static problem can only provide a limited amount of insight regarding the ideas behind the study of reduced order systems. The reason is that system order inherently has to do with how the state of a system evolves in time, i.e. with the order of the differential or difference equations considered. Some insight into how one proceeds, however, can be gained by looking at the static case, and one can get some idea of the difficulties involved. We shall begin by looking at a static control problem.

Much of the fundamental work done in the area of reduced order dynamic system optimization [1, 2] has been done in a setting which allows for a solution involving only algebraic equations rather than differential equations. This setting is in terms of time invariant linear systems driven by white noise processes. The basic assumption is that the processes considered are in steady state in a statistical sense, with a bounded constant second moment matrix. Technically, we refer to such systems as stationary stochastic processes [3]. The processes are dynamic in the sense that their states are moving with time. However, the statistics are constant.

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.

Up to this point we have not considered reduced order smoothing problems, where data over an entire interval may affect an estimate at any time during the interval. Such problems have the characteristics that they have non causal solutions and so may not be implemented in real time. Since this is the case, the reader may wonder why we would be interested in a reduced order sub optimal solution, as obtained in [1] instead of a full order optimal solution as presented in [2] and [3]. The answer is simply that complexity of the solution is still a factor, even when the signal processing is done off-line. If one has a state model of very high order, one does not want to be required to have a smoother of corresponding high order due to the high complexity of such a solution. The nicest situation one can have is when both the processing equations and design equations are of limited complexity. It should be noted, however, that there is a difference between the two categories even for off line processing, because the Riccati (design) equation is solved only once, but the smoothing (processing) equations could be used repeatedly on vast amounts of data.

Here we shall discuss finite time interval problems where the main idea is to control a system, and where estimation, if it is a part of the problem at all, is of secondary importance. We shall discuss problems with and without dynamic compensators. Dynamic compensators are important for the case when full state measurements are not available, so that an observer is useful [1]. They are also important for the case where only a noisy measurement of the observation is available and filtering of the noise is necessary. Such problems have been considered in [2, 3]. We will see that only under very special circumstances do such problems have elegant solutions. Such is the case when full order compensators are used, and the result known as the separation theorem [4], is probably the most elegant result in all of systems theory. We will begin our examination of Stochastic Control problems by examining the output feedback control problem which has been studied by Axsater [5].

In this chapter we will rederive some of the results that have been previously obtained in Chapter three using a concept of “reduced order innovation process.” The concept of “reduced order innovation process” we feel captures the qualitative essence of how useful information is extracted from the given measurements when a reduced order estimator is used.

The construction of estimators for linear two-point boundary value processe s (TPBVP’s) has been receiving greater attention in recent years [1–4]. This interest reflects a desire to estimate processes governed by ordinary differential equations with constraints at each end point of an interval. TPBVP’s occur frequently in physics and engineering. For example, partial differential equations in temporal steady-state often lead to TPBVP’s.

Flexible space structures are often modeled by a large set of second order differential equations. A Kalman filter designed for such a model might not be a very practical idea because of the high dimensionality required and the associated complexity of implementation. Here we derive a class of reduced order filters which reduce the complexity of the design and filtering process. As an added feature, the reduced order filter performance is shown to be completely insensitive to system parameters for an important class of problems.

In this chapter we shall address topics dealing with adding robustness to the design of reduced order filters. We will show how one can treat a broad range of noise or disturbance terms characterized as having bounded energy, rather than as having white noise characteristics. For such additive arbitrary disturbances, we show how to guarantee a bound on the ratio of the energy in the error to the energy in the disturbances. In [1], a rigorous derivation of the results contained here was developed. We will present here, a derivation based on game theory which more closely corresponds to the calculus of variations approach taken throughout this text, and follows the method suggested by Banavar and Speyer [2].