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In their fundamental paper Kalman and Bucy solved the problem of explicitly designing the optimal estimator of the state of a certain type of Markov process at the time T + h ( h ⩾ 0) given observations of a certain number of linear combinations of the states for a certain observation time (0, T ). The problem with h = 0 is currently known as the filtering problem and the one with h > 0 as the prediction problem. Below is given a solution of the smoothing or interpolation problem (0 A ⩽ T ), which is left open in their paper. This problem has been tackled by many authors, but it is believed that the method of the present paper has some points of interest.

The present paper is closely related to an earlier publication wherein the present authors proposed a mathematical formulation for continuous nonlinear minimal-variance filters that led to stochastic differential equations satisfying known existence and uniqueness conditions for such equations. The necessity for the proposed reformulation of the problem arose from the use of the white-noise model to represent the physical system. The adoption of such a model allows certain formal simplicities in the derivation of filter equations, but these simplicities must be paid for by new complexities introduced by the requirement for mathematical rigor, as discussed in the previous paper. It is now shown that the modified formulation, while necessary for assured mathematical validity, can be dispensed with in the context of a computational mechanization.

This paper presents a survey of some recent results in the fields of nonlinear filtering and stochastic control resulting from the use of Gaussian sum approximations to certain density functions. When trying to estimate and control the state of a nonlinear stochastic dynamic system from noisy measurement data, the a posteriori density of the state conditioned on the measurement data contains all of the available information about the system state. If these densities were available, it should be possible to develop “optimal” estimation and control policies. The well-known Bayesian recursion relations describe the evolution of this a posteriori density in terms of given a priori densities, the system dynamics and measurement equations and the available data. Unfortunately, it is seldom possible to solve these closed forms for the required a posteriori densities. Over the last few years it has been shown that when the a priori densities are approximated by a sum of Gaussian densities with positive weighting coefficients that linear non-Gaussian problems can be solved analytically and that new natural solutions to nonlinear problems are available. Recently this method has been extended to dual deterministic optimal control problems.

Hierarchical Control is an attractive solution when the optimization of a large scale system is requested. This paper introduces a consistent approach to the coordination problem in the dynamic case. Two basic transformations on the system afford real decomposition into subproblems by means of the generalized Lagrangian. Two specific hierarchical structures fulfilling efficiently the INPRE and INBAL principles are deduced. This method gives a convenient realization algorithm in both cases. Moreover, these principles are shown, in a sense, to be dual.

This paper deals with a new method for optimizing control of multimodal systems by using fuzzy automata. Two classes of fuzzy automata corresponding to the Mealy and Moore type of ordinary automata are formulated in which the transition may be executed on the basis of the membership function between these two states, and these automata are used as the optimizing controller for a global search of the optimum of a multimodal systems. The concepts of the higher order transition in the automata and the partition of the domain of objective function are introduced to the method of optimizing control, and the control systems can hold the true optimum at small hunting loss without keeping any local optimum. The operation of self-organization is performed to optimize the system in the automata.

This paper establishes two concepts of identifiability for the unknown parameters of linear, constant, stable, discrete-time dynamic systems. Necessary and sufficient conditions are obtained for the system parameters to be identifiable in the deterministic sense (Theorems 1-2) and stochastically identifiable (Theorem 3). These conditions are shown to be significant in studying the asymptotic properties of maximum likelihood estimators (Corollary 3.1). Generalizations of these concepts to include the identification of the system's initial state are also discussed (Theorems 2a and 3a).

The derivation of formulas for sequential estimation in the presence of unestimated uncertain parameters is the subject of earlier papers. It is shown herein that the estimation procedure involves the involuntary estimation of the uncertain parameters.