Questions: Process Model Identification and Relay Autotuning

Question 1

In a relay autotuning test, the relay switches between ±d and the resulting process output oscillates with amplitude a. Using the describing function approximation, what is the ultimate gain K_u?

• A. K_u = πa / (4d)
• B. K_u = 4d / (πa)
• C. K_u = d / a
• D. K_u = 2d / a

Answer: (B)

The describing function of a relay with amplitude d, evaluated at oscillation amplitude a, gives an effective gain of 4d/(πa). At the limit cycle, the loop gain equals 1: this relay gain times the process gain at the critical frequency must equal 1. Solving for the process gain (which is K_u) gives K_u = 4d/(πa). The factor of π comes from the Fourier series of a square wave — the fundamental harmonic has amplitude 4/π times the relay amplitude.

Question 2

Why does a relay feedback system naturally oscillate at the phase crossover frequency (the frequency where the open-loop phase is −180°), rather than at some other frequency?

• A. The relay is tuned in advance to that frequency based on a process model
• B. The phase crossover frequency is where the process gain is highest, so oscillations grow largest there
• C. Sustained oscillation requires loop gain = 1 and loop phase = −180°; the relay automatically satisfies the phase condition, so the system settles at the only frequency where both conditions hold
• D. The relay's switching speed is physically matched to the natural frequency of the process

Answer: (C)

A sustained limit cycle requires two simultaneous conditions: loop gain = 1 and loop phase = −180°. The relay is a sign-inverting nonlinear element — its switching behavior introduces an effective −180° phase inversion (like a sign flip in feedback). A sinusoidal oscillation can persist only at the frequency where the process itself also contributes −180° of phase, making the total loop phase −360° (≡ 0°) or equivalently where the process phase is −180°. At any other frequency, the phase relationship causes the oscillation to die out or grow; only the phase crossover frequency supports a stable limit cycle.

Question 3

The relay autotuning test is safer than finding the ultimate gain by manually increasing proportional gain because the relay limits the amplitude of the process excitation.

• T. True
• F. False

Answer: True

This is the key practical advantage. In the classical Ziegler-Nichols closed-loop method, the operator increases proportional gain until the system marginally oscillates — the process can swing widely during this experiment, and the operator must intervene quickly if instability develops. In contrast, the relay limits the controller output to ±d (a designer-chosen fraction of the control range), so the process output oscillates by approximately ±a. The test runs in a few oscillation cycles, often unattended, with predictable and bounded excitation — which is why commercial PID controllers implement it as a push-button 'autotune' feature.

Question 4

Relay autotuning requires an explicit mathematical model of the process (transfer function or state-space) to determine the critical frequency before the experiment begins.

• T. True
• F. False

Answer: False

This is precisely what relay autotuning avoids. Traditional PID tuning using Ziegler-Nichols requires knowing K_u and T_u — which requires either an explicit model or a dangerous manual gain-increase experiment. Relay autotuning determines these parameters from the limit cycle itself: the period of oscillation gives T_u (and hence ω_u), and the oscillation amplitude combined with the relay amplitude gives K_u via the describing function. No prior model is needed, which is the key advantage for automated commissioning of industrial controllers.

Question 5

What is the 'describing function' approximation in the context of relay autotuning, and why is it necessary for the analysis?

Model answer: The describing function is a method for approximately analyzing nonlinear elements (like a relay) in frequency-domain terms. A relay is nonlinear — its output is a square wave, not a sinusoid — so standard linear frequency-response analysis cannot be directly applied. The describing function approximates the relay as an equivalent linear gain at the fundamental frequency of its output, equal to 4d/(πa) where d is the relay amplitude and a is the amplitude of the input sinusoid. This approximation is valid when the process acts as a low-pass filter that attenuates higher harmonics of the square wave, leaving primarily the fundamental frequency in the output. The approximation enables us to use linear analysis (the condition loop gain = 1 at phase crossover) to relate the observable oscillation parameters to K_u.

The describing function is the mathematical bridge that allows the nonlinear relay to be handled with linear tools. Its limitation is that it assumes higher harmonics are negligible — if the process passes harmonics significantly, the identified K_u and T_u will be inaccurate. For most industrial processes (which are low-pass in nature), this assumption holds well enough for practical PID tuning.