13.5 Properties and Application of the RGA

Sum of Rows and Columns Property of the RGA

Notice that Equation (13.24) yields the following relationships

This property for 2 x 2 systems is general for n x n systems. That is, each row of the RGA sums to 1.0 and each column of the RGA sums to 1.0.

Then, for a 2 x 2 system, only one relative gain must be calculated to find the entire array,

Equation 13.25

For a 3 x 3 system, only four of the nine elements would need to be calculated. In practice, they are all calculated simultaneously using (13.23).

Use of RGA to Determine Variable Pairing

It is desirable to pair output i and input j such that l_ij is as close to 1 as possible.

Example 13.2: RGA for Variable Pairing

Consider a relative gain array,

We would pair y₁ with u₁ and y₂ with u₂ in this case.

However, if

we would pair y₁ with u₂ and y₂ with u₁.

Two Input–Two Output Systems

It is easy to show the two possible cases for the relative gain array of a 2 x 2 system. If there are an odd number of positive elements in G(0), then

That is, for an odd number of positive elements in the process gain matrix, all the relative gains will be between 0 and 1. If there are an even number of positive elements in G(0), then

This means that for an even number of positive elements in the process gain array, all the relative gains will be outside the 0–1 range.

Implications for the Sign of a Relative Gain

First of all, we can state that if l₁₁ was negative, we would not want to pair input 1 with output 1. Let us use the following reasoning. Assume that the open-loop gain between input 1 and output 1 (g₁₁) is positive; this implies that the controller gain is also positive. A negative relative gain tells us that the gain between input 1 and output 1 with output 2 perfectly controlled (that is, loop 2 closed) is negative. This means that if loop 2 is closed, we need a negative controller gain on loop 1. However, we noted before that if loop 2 is open, we need a positive controller gain on loop 1. It is not desirable to have a control system where you have to change the sign of the controller gain depending on whether the other loops are open or closed, so the bottom line is the following:

We do not want to pair output i and input j if l_ij is negative.

The results for a 2 x 2 system indicate that if l₁₁ > 1 then l₁₂ = l₂₁ < 0. Also, if 0 < l₁₁ < 1, then 0 < l₁₂ < 1. Let us think about the differences between having a l₁₁ greater than 1 and having a l₁₁ less then 1.

If l₁₁ > 1, then .

This tells us that the gain between u₁ and y₁ is larger with loop 2 open than with loop 2 closed.

• If we design the u₁-y₁ controller with loop 2 open, then close loop 2, we expect somewhat sluggish behavior, since the effective process gain will be lower than the u₁-y₁ gain.

However,

If we design the u₁-y₁ controller with loop 2 closed, then if we have to open loop 2 for some reason, we could have destabilizing behavior, since the u₁-y₁ gain will be higher than when the controller was tuned.

If 0 < l₁₁ < 1, then .

This tells us that the gain between u₁ and y₁ is smaller with loop 2 open than with loop 2 closed.

If we design the u₁-y₁ controller with loop 2 open, then close loop 2, we expect more aggressive behavior, since the effective process gain will be higher than the u₁-y₁ gain.

However,

If we design the u₁-y₁ controller with loop 2 closed, then if we have to open loop 2 for some reason, we expect sluggish behavior, since the u₁-y₁ gain will be lower than when the controller was tuned.

Midchapter Summarizing Remarks

You must be careful when tuning SISO loops in a multiple input–multiple output (MIMO) system, because the other loops in the system can greatly affect the gain between the input-output variables under consideration.

Also, note that we have only been concerned with static (steady-state) effects. Dynamic interaction has a major effect when control systems are tuned. Our experience is that one rarely reverses a pairing decision based on the steady-state relative gain, when one considers the dynamic effects.

The primary result from a dynamic interaction analysis is to recognize additional interaction affects above the steady-state ones and either accept the degradation in control system performance, or use a "true" multivariable control system design technique rather than separate SISO controllers. Sometimes the interaction and other problems will be so severe that only one loop can be closed.