Controller Algorithms

An automatic controller compares a process measurement to a desired "setpoint" value and produces a "controller output" in response.

The first step in most algorithms is a "comparator" which calculates an "error":

The goal of the controller is thus to reduce the error to zero in an appropriate fashion.

Manual Control

Sometimes an operator will choose to adjust a manipulator based on intuition, experience, etc. Such an approach is inconsistent, and not practical in a situation where there are many control loops.

Almost all controllers will have an "auto/manual" switch, which can be used to toggle between open loop (controller disengaged) and closed loop (automatic control) operation.

On-Off Control

The simplest control algorithm is on-off control. This approach is employed by most home thermostats.

• "Bang-bang" action -- not smooth
• Usually need to design a "dead band" to prevent chattering

Proportional Control (P-only Control)

On-off control takes exactly the same action for small errors as it does for huge ones. The resulting cycling might be reduced if the controller was able to make big changes when error is big, but small changes when error is small.

• "Bias" (c₀ in Riggs) is the desired constant value when error is zero. Usually the bias is the nominal steady-state value of the signal to the final element. The bias term takes effect the moment the controller is switched on (placed in automatic) unless it is designed to provide "bumpless transfer".
• A dimensionless gain K_c^D=1 means that a 10% change in error will produce a 10% change in controller output. Dimensionless gains are the preferred approach whenever dealing with control signals; but when analyzing loops process variables may be used instead of signals, hence gains will not be dimensionless.
• Controllers are "tuned" by picking K_c to produce a desired output response. The primary effect of increasing K_c is faster response.
• Controllers are "direct acting" (output increases with increasing error) or "reverse acting" (control output decreases with increasing error). The selection of control action is based on understanding of the process and safety and operability considerations. It is directly linked with the "failure position" of the final elements. [Warning: Some books (notably Seborg, et al.) define direct and reverse acting the opposite of this definition.]

A P-only controller won't necessarily drive a process back to perfect setpoint. The permanent error which results is called "offset". This is because the gain of a process under proportional control has a (1+K_cK_p) term in the denominator.

Integral Control (Reset)

In many cases, would like to completely eliminate offset, so need a way to penalize persistent errors.

• Eliminates offset, since the integral will exist whenever there is error. Integral action is necessary to eliminate offset and any other steady state deviation from setpoint.
• Usually combined with proportional action to make a PI controller. "I only" will be very slow, since it won't act until the error has begun to accumulate.
• Integral action makes a loop more oscillatory

When a fixed error -- caused by calibration problems, valve saturation, overrides, etc. -- continues for a long time, the integrator can stick at a large value. This is called "reset windup". Most controllers have an "anti-reset windup" feature, where the error is calculated as the difference between setpoint and controller output rather than setpoint and process measurement, at least for the integral control calculations.

Derivative Control (Rate)

PI control is effected only by the magnitude and persistence of the error. There are times when it would be nice if could move the output faster when error is changing rapidly.

This is the algorithm for an "ideal" derivative action. Looking at it, you should suspect that it will not be easy to implement.

• A "D only" controller would be silly. It wouldn't be doing anything when the error wasn't changing.
• Derivative action is not used too much. Only between 5% (Riggs (2001) and 20% (Luyben (1990)) of loops include derivative. One reason is that it tends to amplify signal noise, and so is only used when it is important to reduce oscillation.
• Should not be used if large, rapid changes in manipulators are bad.
• A sudden change in setpoint, and hence in error, can make the derivative function suddenly very large. This "derivative kick" can be bad for a process, so most commercial controllers use the derivative of the measurement rather than the derivative of the error.

PID Control (Proportional-Integral-Derivative, Three Mode)

The PID controller is the workhorse of the process industries and around 90% of industrial loops are PI.
Shown is a "position form" of the PID algorithm, so called because it provides the control element with complete information as to where to position the manipulator.

References:

1. Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, p. 222- 226, 261-262.
2. Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 217-20, 230-38.
3. Seborg, D.E., T.F. Edgar, D.A. Mellichamp, Process Dynamics and Control, John Wiley, 1989, pp. 184-193.

R.M. Price
Original: 10/15/93
Modified: 2/3/94, 3/3/97, 3/16/98, 12/8/98; 7/1/2003, 8/10/2004

Copyright 1998, 2003, 2004 by R.M. Price -- All Rights Reserved