Stability of Periodic Motions and Synthesis of Relay Sampled Data Control Systems

This article is devoted to research and design of relay systems with control of data sampling. It is shown that the time sample has a significant effect on the parameters of periodic oscillations. We propose an exact method for analyzing periodic modes in digital self-oscillatory control systems with a two-position relay element and a linear piecewise-linear part is proposed. The proposed approach extends the phase hodograph method to the class of systems operating in discrete time. Two approaches have been developed to assess the stability of periodic motions in such systems. In the first approach, a discrete representation of a plant is considered and areas of stability are defined for each possible limit cycle. The sampling of the control system causes a delay in the switching of the relay in a batch mode in comparison with the continuous case. The second approach assumes the replacement of a discrete system by an equivalent continuous system with a time delay. Further, the asymptotic orbital stability of self-oscillations in a relay control system (RCS) with a delay is estimated. We consider the linearization of relay systems with digital control of the input signal. It is also shown that when linearizing a relay element in a digital RCS using a useful signal, the relay transfer ratio will belong to a certain range of values. Synthesis of corrective devices for relay control systems with regard to digital implementation has been reviewed. At the stage of optimization of parameters of the relay control system, the sample is taken into account. The model example demonstrates an advantage in the synthesis of digital technologies. It is shown that when optimizing the controller parameters with regard to time discretization, it was possible to provide the desired frequency of self-oscillations, which ensures the best accuracy of the tracking mode.

self-oscillations, relay control, discrete systems, limit cycle, phase locus method, relay-pulse systems

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