# Multivariable controllers balance competing objectives

11 March 2009

### Multivariable controllers differ from traditional single-variable controllers in that they can regulate more than one process variable by using more than one actuator at once.

##### Early application: spacecraft control

Doing so can be difficult if each actuator effects more than one process variable, but if those interactions can be quantified, the controller can determine the control efforts required to drive all of the process variables towards their respective setpoints simultaneously.

For example, an HVAC system responsible for maintaining the temperature and humidity in a conditioned space will find that the process variables are coupled. That is, they rise and fall together since condensing excess moisture out of the air requires cooling it, and adding moisture to the air requires an injection of hot steam. The trick is to chill the air and inject the steam in just the right combination.

Unfortunately, computing that ideal combination requires mathematical models and computational methods considerably more sophisticated than basic PID loops. The temperature and humidity can not be adequately controlled by two independent controllers operating in parallel. Each has to know what the other is doing or else any attempt to correct the temperature will disturb the humidity, which will in turn initiate humidity corrections that will disturb the temperature. Absent a co-ordinated effort, the two controllers would continue to fight each other in a never-ending cycle.

NASA engineers encountered a similar problem with the attitude control systems for their earliest spacecraft. They tried to control pitch, yaw and roll with three independent control loops. But since pitch causes yaw and yaw causes roll, the efforts of each controller affected the other two. The competing controllers ended up expending inordinate amounts of precious fuel during every manoeuvre.

The solution was found in the mathematical discipline of linear algebra, which can be used to quantify and compensate for the interactions between multiple actuators and process variables. Scores of multivariable control techniques based on the principles of linear algebra have ensued over the last 40 years, though very few have been applied outside of the aerospace, petrochemical and energy industries. They tend to be mathematically complex, but well worth the effort when coupled process variables are a problem.

Other benefits

Not only can multivariable controllers co-ordinate the efforts of multiple actuators simultaneously, they can optimise and constrain the overall control problem. That is, if the desired results can be achieved by more than one combination of control efforts, an optimising controller can select the combination that minimises a user-defined cost such as the discomfort of a room's occupants or the total volume of rocket fuel expended.

On the other hand, if it happens that the optimal control effort would require an impossible actuator position or an excessively high or low value for any of the process variables, a constraining controller can select the best combination of control efforts that comes closest to achieving the desired results without violating any of those constraints. Constraint management makes linear algebra-based multivariable controllers especially valuable in the petrochemical industry where the greatest profit is realised when processes are run with all of their operating conditions at their maximum allowable values.

A linear algebra-based multivariable controller can also determine if a particular selection of setpoints is even possible. In the HVAC example, high humidity and low temperatures can not generally be achieved simultaneously. A multivariable controller equipped with a thermodynamic model of the conditioned space would be able to deduce the controllability of the process and flag an unachievable combination of process variables.

—Vance VanDoren, Ph.D., P.E., Control Engineering