Relay feedback with self-tuning negative hysteresis for first-order plants with dead time

When using relays to tune PID regulators, thresholds yielding positive hysteresis are said to give the best data since they reveal the part of the frequency response of the plant that lies on the third quadrant. However, this is valid only if the open-loop frequency response will be moved clockwise. Besides, limit cycles grow with positive hysteresis. For smaller limit cycles, negative hysteresis is needed. Moreover, if the open-loop frequency response is going to be moved counter-clockwise, data supplied by negative hysteresis is more meaningful. Plants with dead time demand counter-clockwise movement, which is what the dead-time compensator does. These explanations partially lean on the describing function concept. Yet, this concept does not work well for first-order plants with dead time since they provide non-sinusoidal relay inputs without the proper average value and amplitude unless relay input thresholds are properly tuned. The paper promotes the use of self-tuning negative hysteresis and is devoted solely to SISO cases. We seek smaller limit cycles when relay output levels may not become closer together