The relative gain array

The RGA provides engineers with a quantitative comparison of how one control loop will affect others. Since it compares the affect of a manipulated variable on one controlled variable at a time, the problem is broken into more manageable segments. The disadvantage of the RGA method lies in the fact that it provides no information on the controller stability in dynamic situations, i.e. during process disturbances. 

1 Calculating the relative gain array with experiments 

Let us now consider mi as a candidate input to pair with yi. To evaluate this choice against the alternative of using m2, the system must undergo two experiments. 

Experiment 1 step change in m, with ail loops open 

Experiment 2 step change in mj with loop 1 open 

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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 ... 

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)... 

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 ... 

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. 

Define the relative-gain array for a process with two inputs and two outputs. Extend the definition to a process with N inputs and N outputs. 

Consider a process with the following transfer functions Hi2(s) = H2 (s) = 0 and Hu(s), H22(s) 0. Show that the relative-gain array is given by... 

What are the properties of a relative-gain array How many relative gains do you need to compute in order to specify completely the relative-gain array of a process with (a) three inputs and three outputs, and (b) N inputs and N outputs ... 

The idea of the relative-gain array and its use for selecting the control loops can be found in the original paper by Bristol ... 

VI.13 Identify the proper couplings between inputs and outputs of the systems in Problem VI. 11. Use the relative-gain array methodology so that the resulting loops have the minimum possible steady-state interaction. 

The relative-gain array method, which determines how the controlled and manipulated variables should be coupled to yield control loops with minimal interaction... 

Arrange the four relative gains An, Xl2, A2i, and A22 into a matrix form, which is known as the relative-gain array ... 

The relative-gain array is a square matrix, which implies that the number of manipulated variables is equal to the number of controlled outputs. Now, suppose that we have a process with two outputs and three possible manipulations, m i, m2, and m3. There are three possible pairs of manipulated variables (mi, m2), (m2, m3), and (m3, mi). Therefore, we can form three different relative-gain arrays ... 

Example 24.4 Select the Loops Using the Relative-Gain Array... 

Explain how you can use the relative-gain array to select the loops with minimum interaction. Why would you avoid coupling an output y, with a manipulated variable m, if Xv < 0 Does kv < 0 imply that another relative gain is larger than 1 or not Explain. 

Therefore, the following discussions of the relative gain array (RGA) and decoupling are quite brief We include them not because they are all that useful, but because they are part of the history of multivariable control. You should be aware of what they are and what their limitations are so that when you see them being misapplied (which, unfortunately, occurs quite often) you can be knowledgeably skeptical of the conclusions drawn. 

All of the linear C R measures use the approximations, F s and which describe the effects of the control variables and disturbances, respectively, on the process outputs. A commonly used controllability measure is the relative-gain array (RGA Bristol, 1966), which relies only on F s. The disturbance condition number (DCN Skogestad and Morari, 1987) and the disturbance cost (DC Lewin, 1996) are resiliency measures that require a disturbance model, in addition to P s. These C R measures are especially useful in... 

The issue of this kind of control configuration has been investigated using frequency dependent formulations of measures such as the condition number, the Relative Gain Array by Bristol (1966) and the Relative Disturbance Gain by Stanley et al (1985). This paper will focus on discussing the dynamic control structure on the heat pump section and how each dynamic control structure affects on the stability of the integrated distillation column. 

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