11.8: Heuristic Tuning

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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For many practical systems, tuning of a PID controller may proceed without any system model. This is especially pertinent for plants which are open-loop stable, and can be safely tested with varying controllers. One useful approach is due to Ziegler and Nichols (e.g., Bélanger, 1995), which transforms the basic characteristics of a step response (e.g., the input is \(1(t)\)) into a reasonable PID design. The idea is to approximate the response curve by a first-order lag (gain \(k\) and time constant \(\tau\)) and a pure delay \(T\):

\[ P(s) \, \simeq \, \frac{k e^{-Ts}}{\tau s + 1} \]

The following rules apply only if the plant contains no dominating, lightly-damped complex poles, and has no poles at the origin:

\(\quad \text{P}\)	\(k_p = 1.0 \tau / T\)
\(\quad \text{PI}\)	\(k_p = 0.9 \tau / T\)	\(k_i = 0.27 \tau / T^2\)
\(\quad \text{PID}\)	\(k_p = 1.2 \tau / T\)	\(k_i = 0.60 \tau / T^2\)	\(k_d = 0.60 \tau\)

Note that if no pure time delay exists (\(T = 0\)), this recipe suggests the proportional gain can become arbitrarily high! Any characteristic other than a true first-order lag would therefore be expected to cause a measurable delay.