The locus of a perturbed relay system (LPRS) theory

As we considered in the previous chapter, the motions in relay servo systems are normally analyzed as motions in two separate dynamic subsystems: the “slow” subsystem and the “fast” subsystem. The “fast” subsystem pertains to self-excited oscillations or periodic motions. The “slow” subsystem deals with forced motions caused by an input signal or by a disturbance, a non-zero initial conditions component of the motion, and usually pertains to the averaged (over the period of the self-excited oscillation) motion. The two dynamic subsystems interact with each other via a set of parameters: the results of the solution of the “fast” subsystem are used by the “slow” subsystem. This decomposition of the dynamics is possible if the external input is much slower than the self-excited oscillations, which is normally the case. Exactly as in the DF method, we shall proceed from the assumption that the external signals applied to the system are slow in comparison to the oscillations.

Consider again the harmonic balance equation (1.19). Using the formulas for the negative reciprocal of the DF (1.20) and the equivalent gain of the relay (1.23), we can rewrite formula (1.19) as follows: