# Gain scheduling handles nonlinear processes

## A majority of feedback control techniques, including the venerable PID algorithm, relies on the principle of linearity that guarantees a Y % change in the process variable following an X % change in the control effort. The ratio or gain between X and Y will be fixed, whether the process is currently running at maximum capacity, minimum capacity, or somewhere in between.

06/01/2002

A majority of feedback control techniques, including the venerable PID algorithm, relies on the principle of linearity that guarantees a Y % change in the process variable following an X % change in the control effort. The ratio or gain between X and Y will be fixed, whether the process is currently running at maximum capacity, minimum capacity, or somewhere in between. A controller need only know the value of that gain and the speed at which the process moves to select its control efforts appropriately.

Unfortunately, not all processes are strictly linear. Even the classic linear system comprised of a weight hanging from a spring will respond less and less to forces applied to the mass as the spring is stretched (or compressed) to the limits of its travel.

 The purple curve shows how the pH of a solution responds to the addition of a base. Process gain is much higher in range B than in ranges A and C.

## A pH control example

On the other hand, even nonlinear processes can be approximated as linear if X and Y are small enough. Consider, for example, a chemical process where a base is added to a solution to increase its pH. As the figure shows, the process reacts much more dramatically to the addition of the base when the pH is already in range B. A controller attempting to raise the pH all the way from range A to range C would proceed much too aggressively through range B if it assumed that the entire process were governed by a single low gain. Conversely, it would be much too conservative in ranges A and C if it assumed that the high gain of range B prevailed throughout.

The classical solution to this problem is to operate the process entirely within one range or another (that is, keep X and Y low) or use a control algorithm that varies its gain as the process gain varies. If the variations in the process gain can be observed or inferred while the controller is in operation, it is fairly simple to update the controller's gain accordingly. This is often accomplished with a gain schedule- a look-up table that gives the controller gain appropriate for the current operating range as indicated by the value of the process variable.

In the pH control example, the gain schedule would have three entries corresponding to each of the three pH ranges. Each controller gain would be set according to a separate tuning test executed while the process operates in the corresponding pH range.

## Variations on the theme

Several variations on this basic theme have been applied to related control problems. For example, indicators other than the process variable have been used to select the appropriate gain from the schedule. A sunlight sensor could be used to select the appropriate gain for controlling the room temperature in a glass walled office building. When the sun is shining brightly, the controller would select a less aggressive gain for heating control or a more aggressive gain for cooling.

Entries in the gain schedule can also be set automatically by a controller that can tune itself. It only would need to perform its tuning test the first time the process variable enters a particular range, then simply retrieve the computed gain whenever the process variable returns to that range. This is a form of adaptive control .

Sometimes process nonlinearities are sufficiently well understood to permit use of a characterizer . A characterizer amplifies or attenuates the process variable as necessary to make the process look linear to the controller. It accomplishes the same end as a gain schedule, but requires only a single controller gain rather than separate gains for each operating range.

Comments? E-mail Vance VanDoren at controleng@msn.com

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