Control Notes

The dynamic response of self-regulating processes can be described reasonably accurately with a simple model consisting of process gain, dead time and lag (time constant). The process gain describes how much the process will respond to a change in controller output, while the dead time and time constant describes how quickly the process will respond.

Although the dead time and time constant both seem to describe the same thing, there are several fundamental differences between how dead time and time constant affects a control loop. The first difference is that dead time describes how long it takes before a process begins to respond to a change in controller output, and the time constant describes how fast the process responds once it has begun moving.

Measuring the Dead Time and Time Constant of a Process

Let’s begin with the measurement of dead time and time constant of a self-regulating process. Typically, one will place the controller in manual control mode, wait for the process variable to settle down, and then make a step change of a few percent in the controller output. At first the process variable does nothing (dead time) and then it begins changing (time constant) until finally it settles out at a new level.

Measuring Dead Time and Time Constant

To measure the dead time and time constant, draw a horizontal line at the same level as the original process variable. We’ll call this the baseline. Then find the maximum vertical slope of the process variable response curve. Draw a line tangential to the maximum slope all the way to cross the baseline. We’ll call this crossing the intersection.

– The process dead time is measured along the time axis as the time spanned between the step change in controller output and the intersection.

Next, measure the total change in process variable. Then find the point on the process response curve where the process variable has changed by 0.63 of the total change in process variable. We’ll call this point P63.

– The process time constant is measured along the time axis as the time spanned between the intersection (described previously) and P63.

Dead Time versus Time Constant

We can draw a chart with a continuum of dead time through time constant (see figure below). Processes woth dynamics consisting of pure dead time will be on the left and pure lag (time constant) on the right. In the middle the process dead time will equal its time constant.

We’ll find that flow loops and liquid pressure loops fall just about in the middle of the continuum, because their dead time and time constant are almost equal. Gas pressure and temperature loops will be located more toward the right – they are lag (time constant) dominant. Serpentine channels in water treatment plants and conveyors with downstream mass meters will appear on the left side – they are dead-time dominant.

Level loops should actually be treated differently, but can be approximated on the continuum by replacing the time constant with their residence time (time they will take to fill or empty out at full flow rate.) Most level loops will be located far to the right, having relatively short dead times.

The ratio of dead time to time constant affects the controller modes and tuning rules we use, the controllability of the process, and the minimum possible loop settling time.

A continuum from pure Dead Time to pure Lag

Controller Modes

The derivative control mode works well where process variables continue to move in the same direction for some time, i.e. lag-dominant processes. Derivative control does not work well on processes where the process variable changes sporadically – typically processes with relatively short time constants, located in the middle and to the left on the continuum.

Applicability of Tuning Rules

Most tuning rules will work on lag-dominant processes. However, the Ziegler-Nichols rules have only a narrow range of applicability. Lambda / IMC tuning rules apply to a broader spectrum of processes, while Cohen-Coon has the widest coverage. The Dead-Time tuning rule, applies to processes on the left, as its name implies.

Controllability

Lag-dominant loops are easier to control than dead-time-dominant loops. Operators find that lag-dominant processes respond much more intuitively than dead-time-dominant processes and are easier to control in manual mode.

Loop Settling Time

When tuning a loop for the shortest possible settling time, one finds that there is a minimum limit on settling time. If you tune the controller any tighter, the loop will begin oscillating. The minimum settling time depends mostly on the amount of dead time in a control loop, and will be between two and four times the length of the dead time. The ratio of time constant to dead time determines where the minimum settling time falls between two and four times the process dead time.

Stay tuned!

Jacques Smuts – Author of the book Process Control for Practitioners