# Module 6: Multivariable Systems

## Introduction

The course so far has covered a number of different techniques for determining the controller parameters for a PI controller. It should be noted that these were for single feedback control loops.

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.

## Analysis of Interaction

Let us consider first the simple black box process below which has two inputs - u1, u2 - and two outputs - y1, y2 - which are paired together as shown:

A change in u1 has two effects:

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

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

## Definition of the Relative Gain Array

The relative gain array can be defined as

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.

## Calculation of the Gain Matrix

The first step is to calculate the gain matrix. This can be thought of as the Open Loop Gain Matrix. This gives an indication of the influence that each input has on each output.

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

What we need to know is

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

Thus the gain matrix can be denoted by

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

Note that some entries in the Gain Matrix may be negative. This means that a negative gain has to be used in the control loop associated with that entry.

## Calculation of the Relative Gain Array

The gain matrix above gives some insight into which pairings have the most influence on each other and whether positive or negative gains are needed in the final controllers. The next stage of the analysis takes into account the interaction when the loops are closed.

Consider the following questions:

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

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

## Interpreting the Results

Now that we know how to calculate the RGA we can make some comments:

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. uj has no effect on 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!

## Alternative Method for 2X2 System

From a linearised steady state model

y1 = K11u1 + K12u2

y2 = K21u1 + K22u2

Open Loop Gain:

K11

For closed loop gain solve model for y2 = 0:

y1 = K11u1 - (K12K21/K22)u1

Closed Loop Gain:

K11 - (K12K21/K22)

Therefore

In the 2X2 case this is the only element that has to be calculated. The rest of the matrix can be determined by remebering that
• Rows sum to 1
• Columns sum to 1

## Strategies for Reducing Loop Interaction

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

## Tuning Multiloop Controllers

• 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