Relative-Gain Matrix

The relative-gain matrix, or relative-gain array (RGA), was originally su ested by Bristol as a means of determining the steady-state interaction between process variables. Shinskey and McAvoy have been assiduous in exploring the 2 p]ications to distillation columns. One of the most ludd and concise treatments we have seen is that of Ray. The implication is that by proper pairing of variables one may arrive, in some instances, at a control loop structure that promises less interaction than other feasible structures. 

Like other linear techniques, the relative-gain array assumes that the principle of superposition holds. In the published papers and books there is also an assumption, not implicit in the mathematics, that the steady-state gains give a true indication of interaction. For real systems dynamics effects may x just the opposite of steady-state effects and may be dominant. Further, since a distillation column is apt to be just one equipment piece in a sequence of process steps, there are usually fewer choices of control system structure than 

Some work has been done on defining a dynamic RGA. See reference 6. 

Bottom composition controlled by boilup (usually steam flow). 

Base level controlled by bottom product withdrawal. 

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Shinsky (1979) uses a relative gain matrix to select which variables to manipulate and measure. Again dynamics are ignored. The method allows one to find which variables influence which others the most if they were put into a feedback control loop. 

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. 

A quick inspection of these expressions shows that the terms in each column or row of the relative gain matrix add up to unity. This is true for matrices of any order n ... 

Equation 16.21 states that the terms of the relative gain matrix are evaluated by multiplying each term of the steady-state open-loop gain matrix by the corresponding term of the transpose of the inverse of that matrix. 

The sum of each row and each column is unity-this Is one of the features of the relative-gain matrix. In a 2 by 2 matrix such as (7.7), it is only necessary to solve for one element, the others being equal or complementary. 

Since all coefficients are positive, two of the terms in the relative gain matrix must be negatlve-which two depending on whether ag > bf. To allow inspection of the properties of this process, let a = b = f = 0.5 and g = 1.0 then the matrix appears as ... 

The coupling in this example is constant, i.e., only constants appear in the matrix, because the mathematical model of the process is linear. Observe how the coefficients in the model fall into place in the relative-gain matrix, corresponding to the transformation procedure involving Eq. (7.13). 

Two liquids are mixed to a controlled density and total flow. Construct a relative-gain matrix for the system using mi and to represent the manipulated flows of streams whose densities are pj and p - let F be total flow and p the density of the blend. Assume that the volumes are additive. 

In a given distillation column, a 1 percent increase in distillate flow D causes distillate composition y to decrease by 0.8 percent, and bottoms composition X to decrease by 1.1 percent. Under the same conditions, a 1 percent increase in steam flow Q causes y to increase by 0.3 percent and x to decrease by 0.2 percent. Calculate the relative-gain matrix. 

Prove that a relative-gain matrix may be prepared from inverted closed-loop gains as well as open-loop gains, as described in the paragraph following Eq. (7.13). Illustrate this with the 2 by 2 matrix given in Eq. (7.16). 

Feedback control over the quality of two products leaving a tower encounters severe coupling. It is not often tried and has, under certain circumstances, failed altogether. Derivation of the relative-gain matrix will reveal the reasons behind the difficulty. 

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