# Multivariable Controllers Balance Performance with Cost

## Controllers that juggle multiple process variables are neither simple nor common, but they can handle some of the most complex control problems.

This tutorial is the last in a series of four on the fundamentals of process control. Part 1, in February, examined PID control . Part 2, in May, presented the Smith Predictor control strategy. Part 3, in September, compared sampled and continuous control algorithms .

Most process controllers perform a familiar drill–measure the variable of interest, decide if its value is acceptable, apply a corrective effort if necessary, and repeat. Industrial controllers generally make their decisions according to the ubiquitous proportional, integral, derivative (PID) algorithm that relies on a single sensor for the measurement and a single actuator for the corrective action.

This single-variable-control routine works very well for a wide variety of control problems with process variables that can be manipulated independently. For example, the temperature inside a hot water heater is controlled by internal heating elements, while the temperature of the room outside is controlled by the building’s heating, venting, and air conditioning (HVAC) system. Changing one temperature does not generally change the other, so each can be controlled by its own independent thermostat.

The problem gets much trickier when the control system is required to achieve multiple objectives all at once using multiple actuators that affect all of the process variables simultaneously. Consider a commercial HVAC system designed to regulate the relative humidity of the room air as well its temperature. Lowering the room temperature raises the relative humidity since cold air can’t hold as much moisture. Conversely, injecting steam into the room raises not only the humidity, but the temperature as well. The temperature and humidity are said to be coupled, since changing one changes the other.

**Coordination is the key**

Because of this coupling effect, the controller that regulates the steam injector and the thermostat that regulates the chiller must work together if either is to achieve its objective. Better still, the two controllers could be integrated to achieve a combined objective–the comfort of the room’s occupants–rather than separate target values for their respective process variables. Such a controller would have the leeway to choose from several equally comfortable combinations of temperature and humidity. Given the thermodynamic properties of the room and the cost of steam and electricity, the controller should be able determine which combinations are achievable and which would be cheapest to achieve.

This is a classic example of multivariable control. By balancing the actions of several actuators that each affect several process variables, a multivariable controller tries to maximize the performance of the process at the lowest possible cost. Multivariable controllers are most common in the aeronautical, energy, and petrochemical industries. In a distillation column, for example, there can be hundreds of tightly coupled temperatures, pressures, and flow rates that must all be coordinated to maximize the quality of the distilled product. A jet aircraft control system must coordinate the plane’s engines and flight control surfaces to keep it flying on the course dictated by the pilot.

A multivariable control system can also take into account the cost of applying each control effort and the potential cost of not applying the correct control effort. Costs can include not only financial considerations, such as energy spent vs. energy saved, but safety and health factors as well.

Remember Chernobyl? The failure of that nuclear reactor was blamed in part on operators who prevented the control system from doing its job. The cost was catastrophic by every measure.

**Multivariable control techniques**

So how does a multivariable controller do all this? There are just a few basic multivariable control techniques, but oddly enough, PID isn’t one of them. The PID algorithm is by far the most popular technique for single variable control, but applying PID control to a multivariable process is not simply a matter of installing another controller for each additional process variable.

The traditional PID algorithm does not account for the effects of coupling nor for the cost of applying a control effort. Its only objective is to correct deviations in a single process variable. However, if control costs are negligible, and if the process variables can somehow be decoupled, then multiple PID controllers can be combined to regulate multivariable processes.

*Figure 1: Some multivariable processes can be* decoupled *so that each process variable responds to only one actuator. This two variable decoupler, for example, could be applied to the HVAC problem mentioned earlier. If the temperature in the room is defined as process variable 1 (PV1) and the humidity is defined as process variable 2 (PV2), then box P21 represents the effect that a change in temperature has on the humidity (the relative humidity rises when the temperature drops). Conversely, box P12 represents the effect that a change in humidity has on the temperature (the temperature rises when hot steam is used to raise humidity). In order to negate these effects, the decoupler D12 must cause a cooling action whenever steam is added to the air stream, whereas decoupler D21 must cut back the steam injection whenever the chiller is commanded to lower the room temperature.*

Figure 1 shows how a simple process with two controllers (C1 and C2) and two process variables (PV1 and PV2) can be decoupled so that each controller ends up affecting only one process variable. The decouplers (D21 and D12) are designed to cancel the cross-over effects (P21 and P12) that each controller has on the other process variable. They allow the controllers–even single variable PID controllers–to operate as if each was in control of its own independent process (P11 or P22).

Unfortunately, decoupling works only if the cross-over effects are either very weak or very well understood. Otherwise, the decouplers will not be able to negate the cross-over effects completely. Decoupling can also fail if the behavior of the process changes even slightly after the decouplers have been designed and implemented.

**Minimum variance control**

A minimum variance control algorithm is generally much more effective for multivariable control. Variance is a measure of how badly a process variable has changed from its setpoint over a period of time. It is computed by periodically squaring the measured error between the two values and adding the results into a historical total. For a multivariable process, the overall variance is a weighted sum of the variances computed for each individual process variable.

A minimum variance controller coordinates all of its control efforts so as to minimize the overall variance. It can also minimize the cost of control by treating each control effort as if it were another process variable with a setpoint of zero. The weighting factors used for the overall variance calculation can be chosen to dictate how much emphasis the controller places on eliminating errors vs. minimizing control efforts. In the HVAC example above, the controller can be designed to be more or less aggressive depending on the relative benefits of reducing energy expenditures vs. keeping the room’s occupants comfortable.

Minimum variance controllers can also impose absolute limits or constraints on the control efforts and the process variable errors. The HVAC control system would have to constrain its efforts so that the valve in the steam injector is never asked to open more than 100%. Conversely, the humidity in the room would have to be constrained to remain within tolerable limits at all times no matter how expensive the control costs might be.

Unfortunately, the benefits of multivariable control come at a price. The mathematical formulation of even the simplest minimum variance control algorithm is tedious and much more complex than the theory of PID control. It’s no wonder that the PID algorithm remains the champion of all control techniques.

*For more information, contact Dr. Vance J. VanDoren , VanDoren Industries, 3220 State Road 26W, West Lafayette, IN 47906; Tel: 317/497-3367, ext. 8262; Fax: 317/497-4875*