The aim of this module is to introduce a method which deals with multivariable systems:

*What happens when there are a number of control loops which interact
with each other?*

To solve this problem
we use what is known as the **Relative Gain Array**. This allows
us to match up variables that have the biggest effect on each other,
without having undesirable effects on the others.

A change in * u1* has two effects:

**Direct Effect**on, the measured variable of that loop**y1****Indirect Effect**on, via the control loop interaction**y2**

Two potential problems arise from this process interaction:

- It may destabilise the closed loop system
- It tends to make controller tuning more difficult

What is needed is a method of pairing up measured variables and manipulated variables so as to reduce this interaction. Selecting control loop pairings is not always clear cut. Incorrect pairings can result in:

- Poor control performance
- Reduced stability margins

The **Relative Gain Array** (RGA) provides:

- A measure of process interaction
- An indication of control loop pairings

where

It is easier to think of calculating this in two stages

- Calculate the
**Gain Matrix** - Calculate the
**RGA**

Both these stages are described in turn below.

Let us return to the two input, two output process from before:

- What is the response in
when*y1*is altered but*u1*is kept constant? Note that*u2*is allowed to change also.*y2* - What is the response in
when only*y2*is altered?*u1* - What is the response in
when only*y1*is altered?*u2* - etc

Thus the gain matrix can be denoted by

This can be extended logically for 3X3 systems and so on.

Consider the following questions:

- How does
respond to a step change in*y1*only but at the same time keeping*u1*constant?*y2* - How does
respond to a step change in*y2*only but while keeping*u1*constant?*y1* - How does
respond to a step change in*y1*only but while keeping*u2*constant?*y2* - etc

These can be used along with the gain matrix above to form the
**Relative Gain Array**.

Firstly note that only steady state information is required. You will see later in the experiment associated with this section that the controller parameters do not really matter it is the final steady state values that are important. It is controller independent which means that perfect control is assumed.

Next note that the RGA is normalised so that each row and each column sums to 1.0. This actually makes it easier to calculate since

- In 2X2 case only have to evaluate 1 element
- In 3X3 case only have to evaluate 4 elements

Finally note that the elements are dimensionless and so independent of units.

The next question is what does it all mean? How do we interpret the results?

- = 1
- Open loop gain and
closed loop
gain are identical and interaction does not affect the pairing of
.*uj-yi* - = 0
- Open loop gain is zero,
i.e.
has no effect on*uj*.*yi* - 0 < < 1
- Closed loop interaction
increases gain. Interaction is most severe when = 0.5.
- > 1
- Closed loop
interaction reduces
gain. Higher values indicate more severe interaction.
- < 0:
- Closed loop gain is in opposite direction from open loop gain. Avoid!

Therefore, considering all the above, the strategy is to match the variables where is nearest to 1 while avoiding the which are zero or negative!

*
*

y2 = K21u1 + K22u2

**
Open Loop Gain:
**

For closed loop gain solve model for ** y2 = 0**:

*
*

**
Closed Loop Gain:
**

Therefore

- Rows sum to 1
- Columns sum to 1

*Detune*one or more control loops- Choose different manipulated or controlled variables
- Consider a
*Decoupling Controller* - Consider a multivariable controller

- Tune single loops with all other loops in manual
- Close all loops, if performance is OK then so nothing
- If not OK then detune less critical loops by reducing controller gains and increasing integral times

OR if one loop is clearly more important:

- Tune important loop first
- Leave first loop in automatic and tune other loops keeping gain low to avoid adverse effects on important loops