Exact methods of determining the steady-state oscillations in feedback systems incorporating one nonlinear element of the ON-OFF type have been developed by Hamel and Tsypkin. It is shown that these methods are not generally applicable. A general method is developed which overcomes these limitations and suitable stability margins are derived. F EEDBACK systems incorporating one nonlinear element of the ON-OFF type have received considerable attention in literature. Kochenburger'*' has used the describing function method for the analysis and synthesis of systems of this type. The main disadvantage of this method is that it is only approximate and ,not generally applicable. Tsypkin and Hame13'4 have developed exact methods using the frequency response and time domain approach respectively. However, it will be shown that these latter methods are not applicable to systems where the period of oscillation depends on initial conditions. The objective of the following discussion will be to derive a general method which overcomes these limitations and to derive suitable stability margins.
CONDITIONS FOR STEADY-STATE OSCILLATIONSThe system to be discussed is shown in Fig. 1. L(s) is the transfer function of the linear element and the output of the ON-OFF nonlinear element N is fl depending on whether t(~) = -V(T) ? 0 (see Fig. 2).The system may be a feedback control system with a relay for the ON-OFF element, or it may be a linear feedback network with a very high gain amplifier in the forward loop operating under large signal conditions. The amplifier is then represented by the nonlinear element N and t,he saturation limits of the amplifier by fl.The response U(T) to the sequence of steps can be written as
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