Relative gain array steady state

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)... 

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. 

VI.14 Consider the processes with input-output relationships given in Problem VI.12. Using Bristol s relative-gain array, select the loops with the minimum steady-state interaction. 

Using Bristol s relative-gain array, select the control loops with minimum steady-state interaction. 

Shows how to use quantitative analysis with steady-state and dynamic relative gain arrays (RGA and DRGA) to select control loop pairings reliably and to use the IMC model-based approach to provide preliminary tuning of single-loop PI controllers. 

After defining the sets of controlled and manipulated variables, steady state controllability indices like Niederlinski index, relative gain array, Morari resiliency index, condition number are determined (e.g. Luyben, 1990). The evaluation of the... 

The relative gain array (RGA) is used in order to measure the process interaction and provide a tool in the design of multi-loop control systems. The RGA is a matrix composed of elements called The element in the ith row and jth column is the ratio of the steady state gain between the /th controlled variable and the yth manipulated variable when all other manipulated variables are constant divided by the steady state gain between the same two variables when all other process variables are constant as shown in the following expression ... 

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... 

These preliminary decisions leave four controlled variables and four manipulated variables that can be analyzed using a 4 X 4 relative gain array. For the core plant model (mass balance equations only) and values of the operating parameters given in Appendix I, the steady-state gain matrix is... 

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