RMP Lecture Notes

Deviation Variables

Most real process variables are functions of time. Typically, values fluctuate around a "normal" value, sometimes slightly higher, sometimes lower. This "long time" value is one aspect of "steady-state".

When we are controlling a system, we want to make small compensatory changes to flows, etc., to try and pull the process back to setpoint. It makes more sense to calculate the change needed rather than calculating the new valve position from scratch every time. This is made easier if we keep track of how a variable differs from its steady-state value instead of tracking its total value.

We can define any variable x as the sum of two parts: the average or steady-state value and the deviation or perturbation from that value.

In control system analysis, we are typically more interested in the deviation variable (a.k.a. perturbation variable), so it is common practice to rewrite systems in terms of deviation variables.

Many analysis techniques (such as Laplace transforms) are limited to linear systems. Linear systems have certain big advantages when using perturbation variables:

constant terms in many ODEs vanish
if we use perturbation variables and linearize around the steady-state, initial conditions are zero

These may be best illustrated by examples.

EXAMPLE: Take the equation

into deviation variables. Begin by noticing that Example

Then split the original variables into deviation form, simplify, and see what happens. Example

The additive constant "biasing" terms have vanished. Also notice that using deviation variables means that we do not need a value for the steady-state positions of x and m to solve the equations.

References:

Coughanowr and Koppel, Process Systems Analysis and Control, McGraw-Hill, 1965, pp. 67-70.
Luyben, W.L. Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 171-76.
Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 74-77.
Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 151-53.
Seborg, D.E., T.F. Edgar, D.A. Mellichamp, Process Dynamics and Control, John Wiley, 1989, pp. 86-93.

R.M. Price
Original: 10/18/93
Modified: 1/24/97, 2/2/98; 5/16/2003