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The fundamental equation that describes limit cycles in nonlinear sampled-data systems was derived. The equivalence of limit cycles with finite pulsed systems having a periodically varying sampling-rate was observed, and the methods of analysis applied to the latter were extended to obtain these limit cycles with the aid of final value theorem. This fundamental equation is modified and simplified under certain assumptions as it can be applied to systems both with and without integrators. The limitation on the longest period of saturated and unsaturated oscillation is investigated and the critical gain for their existence is derived, starting from the modified fundamental equation. Also, the stability of limit cycles and the equilibrium point is considered, based on Neace's method. Various kinds of non-linearities, namely, pulse-width modulation, relay saturating amplifier with linear zone and quantized level amplifier are discussed. Self-excited oscillations are mainly examined, as well as the possible existence and stability of limit cycles, however, the method can be extended to forced oscillations.
Various techniques are available for the analysis of nonlinear sampled-data systems. Most of these methods use either the phase plane approach or the describing function technique. Since the performance of such a system is described, at sampling instants, by means of a difference equation, an approach based on the difference equation would seem to be both natural and direct. The principle of complex convolution for a transform is explained and its geometrical interpretation is given. It is shown how the application of the convolution transform is both direct and simple with respect to solving nonlinear difference equations when the equation is given in scalar form. Dependence of the convergence of the solution on the initial value and the degree of nonlinearity is pointed out. It is concluded that for difference equations of second order and higher, this method involves too much laborious computation to justify its use. A simple method is presented for examining free oscillations in a sampled-data system containing either relay or a saturating amplifier. In addition, a certain analytical technique, analogous to that for differential equations, is developed to investigate the stability of forced oscillations for certain types of nonlinear difference equations. (Author).
The minimal time control problem for some pulsewidth-modulated-sampled-data (PWM) systems with second order plants is treated. Both linear and non-linear plants are considered; all having one thing in common: the optimal switching locus (OST) for a continuous relay system with the same plants is a monotonic curve in the phase plane. First it is shown how, by a reverse time mapping, it is possible to associate systematically with each initial state a finite canonical sequence which specifies the optimal PWM control over the entire transient process during which the state is taken to the origin. Then it is shown how, given an arbitrary initial state its associated canonical sequence can be constructed on the basis of the observed value of the state. A brief description of an electromechanical computer capable of implementing minimal time control for some of the systems considered is also given. (Author).

AFOSR.

United States. Air Force. Office of Scientific Research

Published: 1950

Total Pages: 1190


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