Relative gain array

The most popular and widely used technique for determining the best controller pairing is the relative gain array (RGA) method (Bristol, On a New Measure of Process Interaction, IEEE Trans. Auto. Control, AC-11, 133, 1966). The RGA method provides two important items of information ... 

A SISO eontroller was designed (for mean erystal size), even though relative gain array analysis showed possible interaetions between all of the three eontrol... 

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

You may not find observing the process gain matrix satisfactory. That takes us to the relative gain array (RGA), which can provide for a more quantitative assessment of the effect of changing a manipulated variable on different controlled variables. We start with the blending problem before coming back to the general definition. 

Example 10.4. Evaluate the relative gain array matrix for the blending problem. The complete relative gain array matrix for the 2 x 2 blending problem is defined as... 

There are several notable and general points regarding this problem, /.< ., without proving them formally here. The sum of all the entries in each row and each column of the relative gain array A is 1. Thus in the case of a 2 x 2 problem, all we need is to evaluate one element. Furthermore, the calculation is based on only open-loop information. In Example 10.4, the derivation is based on (10-25) and (10-26). 

We can now state the general definition of the relative gain array, A. For the element relating the i-th controlled variable to they-th manipulated variable,... 

The relative gain array can be derived in terms of the process steady state gains. Making use of the gain matrix equation (10-32), we can find (not that hard see Review Problems)... 

For your information, relative gain array can be computed as the so-called Hadamard product, Ay = KjjKrH, which is the element-by-element product of the gain matrix K and the transpose of its inverse. You can confirm this by repeating the examples with MATLAB calculations. 

Interaction among control loops in a multivariable system has been the subject of much research over the last 20 years. Various types of decouplers were explored to separate the loops. Rosenbrock presented the inverse Nyquist array (INA) to quantify the amount of interaction. Bristol, Shinskey, and McAvoy developed the relative gain array (RGA) as an index of loop interaction... 

The degrees of interaction between different sets of control loops controlling a MIMO plant can usually be determined by the use of the relative gain array technique due to BRISTOL 36. This approach can be illustrated by considering the process shown in Fig. 7.74 which has two inputs and two outputs. The procedure is as follows ... 

The two possible control configurations for a system with two inputs and two outputs are shown in Fig. 7.77. One example of this is illustrated in Fig. 7.73 where the overhead and bottoms product compositions of a distillation process are controlled using the reflux and steam-to-reboiler flowrates respectively as the manipulated variables. Theoretically, we could employ the reflux flowrate to control the bottoms product composition and the steam-to-reboiler flowrate to control the overhead product composition. It is possible to determine which configuration produces the least interaction by forming the system relative gain array A, where ... 

Reserve control module Reserve data acquisition module Relative gain array... 

Relative gain array parameter, wavelength Relative gain array Deviation variable Density... 

A related idea in process control which has received much interest recently is the analysis of interactions among states, outputs, and controls. The analytical technique used in many commercial applications is the relative gain array (Bristol,... 

This would avoid the necessity of actual controller synthesis, which is obviously unattractive and could be quite time-consuming. One approach for control evaluation discussed earlier is the relative gain array (72), (74). No actual synthesis of the controller is required in these algorithms. The development of such screening tools is still in its infancy but appears to be quite promising for concurrent design/control evaluation. Such techniques, if simple to use, would be immediately acceptable for use by major engineering firms and the process industries. 

In the 1960s and 1970s, the relative gain array (RGA) [19] was the only systematic tool available for control structure selection. Its simplicity, practical success, and lack of theoretical basis were disturbing to many academics. Since then, the RGA has been largely vindicated its range of applicability has been defined, its limitations are well understood, and its... 

The subscript m denotes constant values for all manipulations except m, (i.e., all loops open), while subscript y indicates that all outputs except y, are kept constant by the control loops (i.e., all loops are closed). Similarly, the relative-gain array is given by... 

The relative-gain array indicates how the inputs should be coupled with the outputs to form loops with the smaller amount of interaction. But the persisting interaction, although it is the smallest possible, may not be small enough. Example 24.5 demonstrated this aspect clearly. In such a case, the two control loops still affect each other s operation very seriously, and the overall control system is characterized as unacceptable. 

---