Pairing the Manipulated and Controlled Variables

The obvious question is how to pair the manipulated and controlled variables. One manipulated variable can potentially affect more than one controlled variable, resulting in interactions among the control loops. The concern is that adjusting a manipulated variable to move one controlled variable in the desired direction could cause another variable to move away from its set point. It is clear that a proper design of the control loops is necessary to prevent such a scenario. Toward this goal, certain strategies should be followed as described below. 

The following method, based on steady-state gains, is recommended for pairing the manipulated and controlled variables (Bristol, 1966). The gains may be obtained from steady-state plant data or from a steady-state process simulator. The steady-state gain is defined as 

The closed-loop gain A is the gain in the i-j loop when all the other loops are closed, that is, their controlled variables are held at their set points  

The relative gains are defined as the ratios of the open-loop to closed-loop gains  

A process with n controlled variables and n manipulated variables is characterized by an n by w matrix of relative gains, the relative gain matrix. 

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To sum up, the key is to pair the manipulated and controlled variables such that the relative gain parameter is positive and as close to one as possible. 

Analyze a MIMO system with relative gain array, and assess the pairing of manipulated and controlled variables. 

There are valuable insights that can be gained from using the classical transfer function approach. One decision that we need to appreciate is the proper pairing of manipulated and controlled variables. To do that, we also need to know how strong the interaction is among different variables. 

Figure 10.12. Block diagram of a 2 x 2 servo system. The pairing of the manipulated and controlled variables is not necessarily the same as shown in Fig. 10.11.

With this scenario, the system may eventually settle, but it is just as likely that the system in Fig. 10.12 will spiral out of control. It is clear that loop interactions can destabilize a control system, and tuning controllers in a MIMO system can be difficult. One logical thing that we can do is to reduce loop interactions by proper pairing of manipulated and controlled variables. This is the focus of the analysis in the following sections. 

After proper pairing of manipulated and controlled variables, we still have to design and tune the controllers. The simplest approach is to tune each loop individually and conservatively while the other loop is in manual mode. At a more sophisticated level, we may try to decouple the loops mathematically into two non-interacting SISO systems with which we can apply single loop tuning procedures. Several examples applicable to a 2 x 2 system are offered here. 

A 100 kmol/h stream containing 50% mole benzene and 40% mole toluene is sent to a 12-stage distillation column on the sixth stage from the top. The column pressure is 100 kPa, with a total condenser and a reboiler. The distillate is the benzene product with a specification of 6.0% mole toluene, and the bottom is the toluene product with a specification of 6.0% mole benzene. These specifications will be met by manipulating the reflux rate and the reboiler heat duty. It is required to determine the best pairing between the manipulated and controlled variables. 

Distillation columns have four or more closed loops—increasing with the number of product streams and their specifications—all of which interact with each other to some extent. Because of this interaction, there are many possible ways to pair manipulated and controlled variables through controllers and other mathematical functions with widely differing degrees of effectiveness. Columns also differ from each other, so that no single rule of configuring control loops can be apphed successfully to all. The following rules apply to the most common separations. 

Requires K12K21 = 0. Open-loop gain is the same as the closed-loop gain. The controlled variable (or loop) / is not subject to interaction from other manipulated variables (or other loops). Of course, we know nothing about whether other manipulated variables may interact and affect other controlled variables. Nevertheless, pairing the i-th controlled variable to they-th manipulated variable is desirable. 

As pointed out earlier, the problem with pairing on the basis of avoiding interaction is that interaction is not necessarily a bad thing. Therefore, the use of the RGA to decide how to pair variables is not an effective tool for process control applications. Likewise the use of the RGA to decide what control structure (choice of manipulated and controlled variables) is best is not effective. What is important is the ability of the control system to keep the process at setpoint in the face of load disturbances. Thus, load rejection is the most important criterion on which to make the decision of what variables to pair, and what controller stmcture is best. 

Interaction is unavoidable between the material and energy balances in a distillation column. The severity of this interaction is a function of feed composition, product specification, and the pairing of the selected manipulated and controlled variables. It has been found that the composition controller for the component with the shorter residence time should adjust vapor flow, and the composition controller for the component with the longer residence time should adjust the liquid-to-vapor ratio, because severe interaction is likely to occur when the composition controllers of both products are configured to manipulate the energy balance of the column and thereby "fight" each other. 

The next question comes back to the meaning of the RGA, and how that may influence our decision in pairing manipulated with controlled variables. Here is the simple interpretation making use of (10-36) and (10-37) ... 

Control engineers know that it takes one manipulated variable for each measured variable we wish to control to setpoint. When the number of controlled variables equals the number of manipulated variables we pair up the different variables and use PI controllers for regulation. Sometimes we are fortunate to have more manipulated variables than control specifications. We can then optimize the use of the manipulators while controlling to setpoint (e.g., valve position control). Sometimes, however, the number of control objectives exceeds the number of available manipulators and we cannot control all variables to setpoint. This is when the concept of partial control is useful. 

Pressure is often considered the prime distillation control variable, Pressure affects condensation, vaporization, temperatures, compositions, volatilities, and almost any process that takes place in the column. An unsatisfactory pressure control often implies poor column control. Pressure is therefore paired with a manipulated stream that is most effective for providing tight pressure control. When the top product is liquid, this stream is almost always the condensation rate when the top product is vapor, this stream is almost always the top product rate (see Sec. 17.2.). 

Control system design consists of two steps control stmcture selection includes the choice of suitable manipulated and measured variables as well as their pairing design and parameterization of some control algorithm defines the computation of the required values of the manipulated variables from the measurements and given set-points. Let us first focus on the control structure selection problem. 

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