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.
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.
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.
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!
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
Return to Start of Part 4: More Advanced
Concepts