Control Notes

Processes with lags or time constants (tau) longer than their dead times (td) are reasonably easy to tune. Most tuning rules work well for processes where tau > 2 td (lag dominant). The opposite is not true. Many tuning rules work very poorly when td > 2 tau (dead-time dominant).

Lag Dominant
When a process has a time constant that is much longer than the dead time, problems like overshoot and having to use high controller gains begin to appear. However, loops with long time constants still act in an intuitive way – if we add more control action we can make the process respond faster, like stepping down harder on the accelerator will get our car to the desired speed quicker.

Dead-Time Dominant
On the other side of the spectrum, when a process’ dead time is significantly longer than its time constant, it behaves much less intuitively – adding more control action does not make the process respond faster. For example, if your shower water is a little cold, opening the hot water tap a lot more is not going to get you to the right temperature any quicker, and it is going to have some serious side-effects.

I once saw several operators struggle to manually control the outlet temperature of a three-pass kiln. The kiln was a dead-time dominant process and its dead time was about 10 minutes long. The operators would notice the temperature is below set point and increase the firing rate. When they see no effect, they increase the firing rate more. And then some more, and more. Finally, when changes have made their way through the dead time, the temperature overshoots its set point by a large margin. Then the operators take the same actions and make the same mistakes in the opposite direction.

Needless to say, controller tuning also becomes difficult on dead-time dominant processes.

Tuning

Step response of a dead-time dominant process.

You will find that the Ziegler-Nichols tuning rules don’t work well at all on a dead-time dominant process. For example, the following process characteristics were measured from the step-response of a dead-time dominant process in the previous plot:

td = 0.276 minutes
tau = 0.013 minutes
gp = 0.89

Applying the Ziegler-Nichols tuning rules to this process gives the following controller settings: Kc = 0.05; Ti = 0.92 minutes. The result is an extremely sluggish control loop (see below).

Dead-time dominant loop tuned with the Ziegler-Nichols tuning rules.

Processes with time constants (tau) longer than their dead times (td) are reasonably easy to tune. Most tuning rules work well for processes where tau > 2 td (lag dominant). The opposite is not true. Most tuning rules work very poorly when td > 2 tau (dead-time dominant).

The Lambda tuning rules were designed for lag dominant processes and do not work all that well on dead-time dominant processes either. The Cohen-Coon tuning rules work much better than the Ziegler-Nichols rules, but they too aren’t the best tuning rule when the dead time is five or ten times as long as the time constant.

So what type of tuning rule will work well for controlling dead-time dominant processes? First, we need a lag-dominant controller, to make up for the absence of lag in the process. But if we just crank up the integral term, the loop will become unstable. So, second, we have to compensate by decreasing the controller gain.

The Cohen-Coon PI tuning rules will work reasonably well up to td = 2 tau, but it becomes sluggish after that. When td > 2 tau, it is better to use the dead-time tuning rule. It is as follows:

Kc = 0.36 / (gp * SM)
Ti = td / 3
No derivative.

SM is the stability margin and can be set to a value between 1 and 4. A value of 1 is equivalent to the 1/4-amplitude damping response. It is considered unsafe – the loop is very sensitive to changes in process conditions. A value of 2 or higher is recommended. It will reduce the overshoot, eliminate unnecessary cycling, and make the loop far more robust to changes in process conditions.

Hint: measure dead time in the same units of time as your controller’s integral setting. E.g. if your controller’s Ti setting is in minutes, measure td in minutes.

Notes:
– The tuning rules above are designed to work on controllers with interactive or non-interactive algorithms, but not controllers with parallel algorithms.
– Furthermore, they will work only on controllers with a controller gain setting and not a proportional band (found on Foxboro I/A controllers, for example).
– The rules assume the controller’s integral setting is in units of time (minutes or seconds), and not integral gain or rate (repeats per minute or repeats per second).

If your controller is different, parameter conversions will allow you to use these rules.

Applying the dead-time tuning rules to the process described above gives the following controller settings: Kc = 0.2; Ti = 0.092 minutes. The result is significantly better than what can be obtained with other tuning rules.

Dead-time dominant loop tuned with the Dead-Time tuning rules.

Better loop response can be obtained with a Smith Predictor, but this is more complex to implement and very sensitive to changes in process characteristics.

Stay tuned!

Jacques Smuts – Author of the book Process Control for Practitioners