Input-output analysis of relay servo systems

The LPRS method presented in the previous chapter allows one to build a linearized model of the averaged motions in a relay servo system. This is done by analyzing the system response to an external constant input. We have shown that the discontinuous system reacts to a constant input essentially like a linear system (if the averaged values of the variables are considered). Therefore, the relay nonlinearity acts as a certain equivalent gain with respect to the averaged values of the respective signals. This happens due to the “chatter smoothing” phenomenon.

It is normally assumed in describing function analysis [8, 50] that if the input is not a constant but a slowly varying signal (the meaning of the term “slow” was discussed previously), then the concept of the equivalent gain (incremental gain for DF analysis) of the relay is still applicable, and the value of the equivalent gain remains the same as in the analysis of the system response to a constant input. In the course of this analysis, the relay is replaced with the equivalent gain while the linear part remains unchanged. To be able to use the concept of the equivalent gain for non-constant input signals, we need to assume the inputs are slow compared to the oscillations. According to this assumption and the equivalent gain concept, the external input is propagated through the relay without any lags or delays, which results in the equivalent gain being a real number (not a complex one). There are some other approaches similar to the DF method that use the same assumption [71, 87]. However, in the setup for the equivalent gain derivation that was used above (when the input to the system was constant), the equivalent gain could only be obtained as a real number. Because we assume that the inputs are slow compared to the oscillations, we also assume the equality of the equivalent gain at a varying input to the one at a constant input.