13 Modifications of Relay Feedback Methods

13.1 Process Activation Method Using Pulse Signals¹

This section introduces a closed-loop process activation method to reduce the harmonics and obtain more accurate frequency-response data of the process. It can also successfully remove the effect of the input nonlinearity by using the disturbance rejection technique. The method combines 10 pulses to generate one period of the relay signal. The 10 pulses are combined in an optimal way by solving a constrained nonlinear optimization problem to minimize the harmonics. In the implementation, the closed-loop process activation method uses the optimal solution obtained without continuing to solve the optimization problem any more. So, the implementation of the proposed method is almost as simple as that of the previous methods. Let us call it Relay_PS.

The signal generated by Relay_PS (Je et al., 2009) has five pulses in the half-period, as shown in Figure 13.1. The first cycle is activated by the conventional relay feedback method. First, the relay output (equivalently, process input) u(t) = αd is entered until the process output y(t) deviates from the initial value. After that, one cycle is determined as follows: u(t) = −αd when y(t) ≥ 0 and u(t) = αd when y(t) < 0. Here, d is the magnitude of the multi-pulse signal and α < 1 is introduced for a smooth transit from the conventional relay mode to the multi-pulse mode without changing the fundamental frequency term, which will be explained ...