[276h]  Probably the Best Simple PID Tunings Rules in the
World
Author
Information:
 Sigurd
Skogestad (speaker)
 Norwegian University of Science
and Technology (NTNU)
 Chemical
Engineering
 Trondheim, B7491
 Norway
 Phone: +47 7359
4154
 Fax: +47 7359
4080
 Email: skoge@chembio.ntnu.no


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 closedloop 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 modelbased
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
PIDtuning paper by Rivera, Morari and Skogestad (1986). The
ZieglerNichols 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 firstorder 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 PIDtunings
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 (firstorder) 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 tradeoff 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 closedloop 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/
