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A very significant gain in US national income can be realized through a more effective utilization of human resources, i.e., a gradual retirement system. A worker following a typical gradual retirement schedule can gain the equivalence of three more years in working lifetime. An econometric study shows that an aggregated average increase of one year in the working lifetime of each worker will produce a gain of well over 150 billion dollars annually in US national income. The gain is of sufficient magnitude to ameliorate the pending Social Security deficit as the baby boom generation approaches retirement. Employers would also gain from increased productivity, enhanced opportunities for innovation, and more effective utilization of productive capital on the factory floor.

The classic Hotelling (Ref. 1) model of exploitation of exhaustible resources assumes in its simplest form that the stock of the resource is known from the beginning. If there are no extraction costs, then the shadow prices associated with an optimal extraction policy rise at the rate of the market rate of interest. The only variable that has to be determined is the initial price, which then determines all future prices; the initial price depends on the interaction of demand (or utility) considerations with the initial stock. In a competitive world, prices would clearly have to rise at the rate of interest to keep resource holders indifferent between extracting the resource now and later.

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.

Specially designed filters to work between a finite resistance at one end and an open circuit or a short circuit at the other end can be made to duplicate exactly the attenuation and phase-shift characteristics of image parameter filters terminated in equal resistances. By doing so a 6-decibel flat gain and considerably higher frequency limit due to parasitic elements can be obtained without added cost. The design procedure follows exactly that of the well-known image parameter filters. A set of algebraic equations express the new element values in terms of the values for image parameter filters.

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.

Ascoli's t orem deals with continuous functions and states that the space of bounded, equi-continuous functions is compact. The present paper extends it to the measurable functions. Both the space of bounded qui-measurable functions, and its product with the space of bounded equicontinuous func ions re compact. (Author)

A fuzzy mapping from X to Y is a fuzzy set on X × Y. The concept is extended to fuzzy mappings of fuzzy sets on X to Y, fuzzy function and its inverse, fuzzy parametric functions, fuzzy observation, and control. Set theoretical relations are obtained for fuzzy mappings, fuzzy functions, and fuzzy parametric functions. It is shown that under certain conditions a precise control goal can be attained with fuzzy observation and control as long as the observations become sufficiently precise when the goal is approached.

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.

A fuzzy set algorithm is formulated for System Selection and Design Specification. To meet the requirements of a given application (Design Goal) the SSDS selects one among a finite number N of alternative systems and specifies the design parameters of the selected system optimally. Optimality is in the sense of highest membership function of the finally designed system in meeting the Design Goal. An application of the SSDS to electrical motor design is discussed.

The idea of guaranteed cost control can be easily extended to stochastic systems with uncertain parameters in plant dynamics. By a slight modification, the guaranteed error estimation can be formulated as a problem of guaranteed cost control. The system considered in this paper is described by a stochastic differential equation with an additive random input and a noisy measurement. The noise processes are assumed to be Gaussian, zero mean, independent with know variances. Furthermore the controlled plant has large parameter uncertainties known to be in some bounded sets. It is shown that with reasonable bounds on uncertain terms, the overall performance bounds are asymptotically tight in the sense that the solutions reduce to the optimal ones when uncertainties vanish.

Guaranteed cost control and estimation of fuzzy systems are obtained by the method of fuzzy dynamic programming. Three types of models are selected to describe the fuzzy systems: 1. Systems with uncertain parameters. 2. Systems with Donati's approximation in the norm. 3. Systems with extended approximation in the norm. In the latter two cases, minimax control is generally obtainable, and is shown to be the guaranteed cost control with lowest guaranteed cost. Using the concept of most likely estimation the estimation problem is formulated into a similar variational problem, and is solved by essentially the same analysis.

The lagging economy and high inflation of recent years can be quantitatively explained by a dynamic model which is based on well known structural equations and equations on price and wage reflecting collective bargaining and limited monopoly. Using the dynamic model, the effects of various economic policies are studied.

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}.

POWER COMPANIES are usually particular about the starting current and running power factor of an induction motor. However, induction-motor manufacturers cannot possibly supply a motor much better in these respects than the present National Electrical Manufacturers' Association Design C standards, even though ideal materials should be discovered in the future.