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[276h] - Probably the Best Simple PID Tunings Rules in the World

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Abstract:

Is it possible today to publish a paper on simple PID tuning rules at a "advanves in process control" session at an AIChE meeting? I am making a try because I think the rules presented in this paper are most useful both for practical applications and for teaching.

The aim of this paper is to present analytic tuning rules which are as simple as possible (so that the can be easily memorized) and still result in a good closed-loop behavior. Although the PID controller has only three parameters, it is not easy, without a systematic procesure, to find good values (tunings) for them. In fact, a visit to a process plant will usually show that a large number of the PID controllers are poorly tuned.

The objective of this paper is to provide simple model-based tuning rules that give insight into how the tuning depends on the process parameters based on very simple process information. These rules may then be used to assist in retuning the controller if, for example, the production rate is changed. Another related objective is that the rules should be so simple that they can be memorized.

There has been previous work along these lines; most noteworthy the early paper by Ziegler and Nicholds (1942) and the IMC PID-tuning paper by Rivera, Morari and Skogestad (1986). The Ziegler-Nichols tunings result in a very good disturbance response for integrating processes, but are otherwise known to result in rather aggressive tunings (e.g. Tyrues et al. (1992), and also give poor performance for processes with a dominant delay. On the other hand, the IMC PID tuning rules are known to result in poor disturbance response for integrating processes (e.g. Chien and Fruenhauf, 1990). but generally give very good responses for setpoint changes. The tuning rules presented in this paper are based on the IMC rules of Rivera et al. (1996), but the rules have been simplified and the integral term modified to improve disturbance rejection for integrating processes. Furthermore, rather than deriving separate rules for each transfer function we instead start by approximating the process by a first-order plus time delay processes (e.g. using the ``half method''), and then use a single tuning rule. This is much simpler and appears to give controller tuning with comparable performance.

The first step is to approximate the process as a first or second order process with effective delay,
g(s) = {k'\over (s + 1/\tau_1)(\tau_2 s + 1)} e^{-\theta s}
where k' is the initial gain and \theta is the effective delay. The following PID-tunings are recommended for the cascade form PID controller:
K_c = {1 \over k'} {1 \over \tau_c + \theta}
\tau_I = \min\{\tau_1, 4 (\tau_c + \theta)\}
\tau_D = \tau_2 \label{taudsis} \eeq

The original IMC tuning rules yield $\tau_I = \tau_1$, that is, the integral time is selected so as to exactly cancel the dynamics correpsonding to the diminant (first-order) time constant $\tau_1$. However, this gives a very sluggish response to input (load) disturbances for slow ($\tau_1$ large) or integrating processes. Therefore, for such processes it is suggested to use a lower integral time, and the recommended value just avoids the slow oscillations that would otherwise result by using ``too much'' integral action for such a process.

The optimal value of the tuning parameter $\tau_c$ is determined by a trade-off between fast speed of response and good disturbance rejection on the one hand (which are favored by a small value of $\tau_c$) and stability, robustness issues and small input usage on the other hand (which are favored by a large value of $\tau_c$)

The main limitation on achieving a fast closed-loop response is the time delay. Selecting the desired response time equal to the time delay, \tau_c = \theta, gives a reasonably fast response with moderate input usage and good robustness margins. With $\tau_I = \tau_1$ the system always has a gain margin (GM) of 3.14, a phase margin (PM) of $61.4^o$, and the tunings provide time delay error robustness of 214%. As expected, the robustness margins are somewhat poorer for ``sluggish'' processes, where we in order to improve the disturbance response use $\tau_I = 8 \theta$. For example, for an integrating process the suggested tunings give a a gain margin of 2.96, a phase margin of $46.9^o$, and a maximum allowed time delay error of 149%.

In some cases, especially for processes with a small time delay, the choice $\tau_c=\theta$ may be unnecessary ``aggressive'', and we may want to increase $\tau_c$ or equivalently decrease $K_c$. However, there is a minimum controller gain needed to achieve satisfactory disturbance rejection. Let $y_max$ denote the maximum allowed output error, and $u_max$ denote the input change needed to reject the largest expected disturbances. Then it is shown in the paper that for acceptable disturbance rejection we must choose K_c \ge u_max / y_max If this minimum controller gain is larger than the the controller gain given above, then the process is not controllable (at least not with PID control with reasonably robust tunings).

The problem of obtaining the effective delay can be set up as a parameter estimation problem. However, our objective is not ``optmality'' but ``simplicity'', so we choose to use a much simpler approach where we simply add all the ``neglected'' small time constants to the effective delay, except for the largest which we distribute evenly to the delay and the time constant using the ``halve method''. This extremely simple rule has been applied to numerous examples, including the examples of Astrom (Automatica, 1998) and Isaksson (Automatica, 1999) and leads to very good final PID tunings.

The tuning rules presented give invalueable insights, for example, into how we must change the tuning parameters in response to changes in the process:

1. If the maximum output deviation is too large then the gain should be increased.
2. If the settling time is too large then the integral time should be reduced.
3. If the oscillations are too large and these have a period shorter than the integral time, then the gain should be reduced or the integral time increased.
4. If the oscillations are too large and these are slow with a period more than about three times the integral time, then the gain should be increased or the integral time increased.
5. Level loops are frequently poorly tuned resulting slow oscillatiions with period P. A simple rule for retuning the controller for an integrating process is prestented: To avoid `slow'' oscillations the product of the controller gain and integral time should be increased by at least a factor 0.1 (P/\tau_I)^2.

Discussions with Professor David Clough from the University of Colorado at Boulder are gratefully acknowledged.

For a full (and rather long) paper please see: www.chembio.ntnu.no/users/skoge/publications/2001/tuningpaper_reno/

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