Inverse Nyquist Array INA

Rosenbrock (Computer-Aided Control System Design, Academic Press, 1974) was one of the early woikers in the area of multivariable control. He proposed the use of INA plots to indicate the amount of interaction among the loops. 

In a SISO system we normally make a Nyquist plot of the total openloop transfer function B. If the system is closedloop stable, the (— 1,0) point will not be encircled positively (clockwise). Alternatively, we could plot l/Gu B). This inverse plot should encircle the (— 1, 0) point negatively (counterclockwise) if the system is closedloop stable. See Fig. 16.4. 

An INA plot for an Nth-order multivariable system consists of N curves, one for each of the diagonal elements of the matrix that is the inverse of the 

Then the sum of the magnitudes of the off-diagonal elements in a given row of the Q matrix is calculated at one value of frequency and a circle is drawn with this radius. This is done for several frequency values and for each diagonal 

The resulting circles are sketched in Fig. 16.5 for the n plot. The circles are called Gershgorin rings. The bands that the circles sweep out are Gershgorin bands. If all the off-diagonal elements were zero, the circles would have zero radius and no interaction would be present. Therefore the bigger the circles, the more interaction is present in the system. 

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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... 

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