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This paper introduces frame recursive processing as a new algoritlun for processing of blurred or unblurred pictorial information with additional noise. It gives an improved image which approaches optimum in the least mean square error sense. The method represents a new direction in two-dimensional digital filtering from the current trend of using generating equations and a Kalman filter which requires artificial introduction of a causal order of data points. Applications include two-dimensional image restoration, three-dimensional image reconstruction from two-dimensional cross sections, and real-time image processing of a moving object. In all cases the optimum linear processor utilizes all available information on the second statistical moments to give the least mean square error, and is realized by frame recursive processing in successive approximation with an exponentially decaying error. A fast hardware realization of the frame processor is also proposed.

Halkin has given a derivation of the discrete maximum principle using a convexity requirement. An example given in this paper shows that incorrect results may be obtained when Halkin's convexity requirement is not met. There are, however, systems that do not satisfy the convexity requirement, but for which there is still a maximum principle. The discrete maximum principle is rederived with a requirement, directional convexity, that is weaker than convexity and which considerably extends its applicability. Though convexity has appeared to be basic in the development of optimal control theory, it is only the weaker property of directional convexity which is required for much of the development.

Kalman's concept of optimum filtering is interpreted in such a way that it can be applied to both nonlinear and linear systems with Gaussian or non-Gaussian statistics. The essential idea is to synthesize a generalized Kalman filter in two stages, a) propagation and b) measurement and correction. The analysis of each stage is independent of the other, and the generalized results are quite simple. In the present paper, the application of the above result is confined to linear systems: 1) Optimum filtering and interpolation of randomly sampled signals; for the interpolation problem it is assumed that the signals are measured exactly at the sampling instant. 2) Optimum filtering of related continuous and randomly sampled signals. 3) Optimum control of randomly sampled linear systems with quadratic cost criterion.

IN AN A-C series motor, the flux pulsation in the field is predominantly at line frequency. The field iron loss can be computed easily and represented by a loss component in the magnetizing current. However, due to the rotation of the armature, the gap flux is rotating relative to the armature, as well as pulsating. The armature iron loss depends on both the line frequency and the speed. Part of the iron loss is supplied electrically by a loss component in the magnetizing current, while the remaining part is supplied mechanically as a load on the shaft. The accurate calculation of motor performance is dependent upon the correct determination and also the correct allocation of these losses.

Two closely related problems 1) adaptive curve fitting and 2) suboptimization can be solved by using the generalized maximum principle. A given curve is approximated by a sequence of arcs in such a way that the duration covered by an arc is automatically shortened at places where the curve is more curly. For the suboptimization problem, a discrete version of the Euler-Lagrange equation is obtained.

Adaptive control systems can be classified according to the response time of the adaptive loop T a relative to that of the main servo loop T m . The response time T a cannot be smaller than the time needed for making the required measurement on which the adjustments are based. In a slow adaptive system, T_{a}\ggT_{m} , and the adjustments are made according to the estimated situation (statistical parameters of the inputs changing plant parameters, etc.) or the estimated performance (error, cost, intermediate parameters etc.). If T_{a}\llT_{m} , usually a reference model is chosen and the system is forced to conform to the reference model. Sometimes the reference model is dimensionless so that the fastest response can be obtained. In both cases, T_{a}\ggT_{m} and T_{a}\llT_{m} , the system can be analyzed by introducing suitable approximations. The condition of stability for the adaptive loop is derived in general terms. If T_{a}\approxT_{m} , both analysis and synthesis become difficult. The concept of "dual control" is introduced, but not developed in the paper.

The multiperiod optimal control problem is formulated and solved using dynamic programming.

The peak tracking problem can be reduced to a Kalman falter problem [1] with the additional variable of the excursion amplitude c , which is then obtained by maximizing the expected peak. In the special case where the parameters do not change, the method yields two tracking procedures depending on the criterion used: 1) Tracking for a limited time and then settling for the parameter value so determined. It is shown that the expected error is proportional to t-1, where t is the tracking time [2]. 2) A procedure which agrees with the Kiefer-Wolfowitz stochastic approximation method [3]. It is shown further that the expected total reduction in peak value (due to error and hunting loss) is proportional to t^{-1/2}.