Characterization of Limit Cycles in Symmetric-Send-on-Delta Based Control Systems. A Describing Function Approach

In this paper we study the limit cycles that may appear in closed loop PID control systems with symmetric send on delta sampling schemes. The analysis is based on the describing function (DF) technique. The DF of the Symmetric-Send-on-Delta sampler is obtained, leading to a plot that is formed by several lines, each one corresponding to a number of output level crossings of the output during the oscillation. The intersection of those lines with the open loop Nyquist plot of the system allows to predict the existence of limit cycles, their stability and the frequency and amplitude of oscillations. As the DF plot has several lines, several intersections may occur, and then several stable limit cycles are possible. Some examples have been presented with one, two and three different limit cycles to validate the limit cycle prediction strategy proposed.