Modelling dynamic systems with differential equations
3.3 Chemical process models
3.3.4 Interaction and the Relative Gain Array
The Wood-Berry distillation column model is an example of a multivariable, interacting process.
This interaction is evident from the presence of the non-zero off diagonal terms in Eqn.3.24. A quantitative measure of the extent of interaction in a multivariable process is the Relative Gain Array (RGA) due to Bristol, [35]. The RGA is only applicable to square systems (the number of manipulated variables equal the number of controlled variables), and is a steady-state measure.
The RGA, given the symbol Λ, is an (n× n) matrix where each element is the ratio of the open loop gain divided by the closed loop gain. Ideally, if you are the sort who do not like interactions, you would like Λ to be diagonally dominant. Since all the rows and all the columns must sum to 1.0, then a system with no interaction will be such that Λ = I.
The open loop steady-state gain is easy to determine. Taking the Wood-Berry model, Eqn.3.24,
Figure 3.14: Distillation interactions are ev-ident when we step change the distillate and bottoms setpoints independently.
as an example, the final steady-state as a function of the manipulated variable u, is yss= Gssu=
12.8 −18.9 6.6 −19.4
u (3.29)
The ijth element of Gssis the open loop gain between yiand uj. The Gssmatrix can be evaluated experimentally or formed by applying the final value theorem to Eqn3.24. Now the closed loop gain is a function of the open loop gain, so only the open loop gains are needed to evaluate the RGA. Mathematically the relative gain array, Λ, is formed by multiplying together Gssand G−⊤ss elementwise,3
Λ= Gss⊗ G−1ss
⊤
(3.30) where the special symbol ⊗ means to take the Hardamard product (also known as the Schur product), or simply the elementwise product of two equally dimensioned matrices as opposed to the normal matrix multiplication.
In MATLAB, the evaluation of the relative gain array Λ is easy.
1 >>Gss = [12.8, -18.9; 6.6, -19.4] %Steady-state gain from Eqn.3.29
>>L = Gss.*inv(Gss)' %See Eqn.3.30. Don’t forget the dot-times (.*)
which should return something like Λ=
2.0094 −1.0094
−1.0094 2.0094
Note that all the columns and rows sum to 1.0. We would expect this system to exhibit severe interactions, although the reverse pairing would be worse.
The usefulness of the RGA is in choosing which manipulated variable should go with which control variable in a decentralised control scheme. We juggle the manipulated/control variable parings until Λ most approaches the identity matrix. This is an important and sensitive topic in
3Note that this does not imply that Λ = Gss
G−1ss ⊤
, i.e. without the Hardamard product. See [179, p456] for further details.
process control and is discussed in more detail in [191, pp494–503], and in [179, p457]. The most frequently discussed drawback of the relative gain array, is that the technique only addresses the steady state behaviour, and ignores the dynamics. This can lead to poor manipulated/output pairing in some circumstances where the dynamic interaction is particularly strong. The next section further illustrates this point.
The dynamic relative gain array
As mentioned above, the RGA is only a steady-state interaction indicator. However would could use the same idea to generate an interaction matrix, but this time consider the elements of the transfer function matrices as a function of frequency by substituting s = jω. This now means that the dynamic relative gain array, Λ(ω), is a matrix where the elements as functions of ω.
Consider the (2× 2) system from [181]
which has as its distinguishing feature significant time delays on the diagonal elements. We could compute the dynamic relative gain array matrix using the definition
Λ(s) = G(s)⊗ G(s)−T (3.32)
perhaps using the symbolic toolbox to help us with the possibly unwieldy algebra.
Listing 3.1: Computing the dynamic relative gain array analytically
>>syms s
>>syms w real
3
>>G=[2.5*exp(-5*s)/(15*s+1)/(2*s+1), 5/(4*s+1); ...
1/(3*s+1), -4*exp(-5*s)/(20*s+1)] %Plant from Eqn.3.31
>>RGA = G.*inv(G') %Λ(s), Eqn.3.32
8 >>DRGA = subs(RGA,'s',1j*w) %Substitute s = jω
>> abs(subs(DRGA,w,1)) %Magnitude of the DRGA matrix at ω = 1 rad/s ans =
0.0379 0.9793
13 0.9793 0.0379
You can see that from the numerical values of the elements of Λ at ω = 1 rad/s that this system is not diagonally dominant at this important frequency. Fig.3.15validates this observation.
An alternative numerical way to generate the elements of the DRGA matrix as a function of frequency is to compute the Bode diagram for the multivariable system, extract the current gains, and then form the RGA from Eqn.3.30.
Listing 3.2: Computing the dynamic relative gain array numerically as a function of ω. See also Listing3.1.
G = tf({2.5, 5; 1 -4}, ...
2 {conv([15 1],[2,1]),[4 1]; [3 1],[20 1]}, ...
'ioDelayMatrix',[5,0; 0,5]) %Plant from Eqn.3.31 [¬,¬,K] = zpkdata(G); sK =sign(K); %get sign of the gains [Mag,Ph,w] = bode(G); %Compute Bode diagram
7 DRGA = NaN(size(Mag));
for i=1:length(w)
K = Mag(:,:,i).*sK; %Gain, including signs DRGA(:,:,i) = K.*(inv(K))'; %Λ(ω) = K⊗ K−T
12 end
OnDiag = squeeze(DRGA(1,1,:)); OffDiag = squeeze(DRGA(1,2,:));
semilogx(w, OnDiag,'-',w,OffDiag,'--'); %See Fig.3.15
The trends of the diagonal and off-diagonal elements of Λ are plotting in Fig. 3.15. What is interesting about this example is that if we only consider the steady-state case, ω = 0, then Λis diagonally dominant, and our pairing looks suitable. However what we should be really concentrating on are the values around the corner frequency at ω ≈ 0.1 where now the off-diagonal terms start to dominate.
Figure 3.15: The diagonal and off-diagonal el-ements of the RGA matrix as a function of
frequency. 10
−3 10−2 10−1 100 101
0 0.2 0.4 0.6 0.8 1
frequency λ1,1
λ1,2
For comparison, Fig. 3.16shows the dynamic RGA for the (3× 3) OLMR distillation column model from Eqn.3.25. In this case the off-diagonal elements do not dominate at any frequency.
Figure 3.16: The elements of the (3× 3) RGA
matrix from Eqn.3.25. 10
−3 10−2 10−1 100 101
−1.5
−1
−0.5 0 0.5 1 1.5 2 2.5
frequency
elements of Λ(ω)
Diagonal elements
Off−diagonal elements λ11
λ22 λ33
λ12 λ13
λ23