Recent Advances in Automatic Control, Information and Communications
Possible Way of Control of Multi-variable Control Loop by Using RGA
Method
PAVEL NAVRATIL, LIBOR PEKAR
Department of Automation and Control
Tomas Bata University in Zlin
nam. T.G. Masaryka 5555, 760 01 Zlin
CZECH REPUBLIC
pnavratil@fai.utb.cz, pekar@fai.utb.cz
Abstract: - The paper describes one of possible approaches to control of multi-variable control loops. In the
approach to control is used the so called RGA (Relative Gain Array) method, simple approach to ensure of
invariance of control loop and simple approach to a design of the so called primary controllers. The RGA serve
to determine the optimal input-output variable pairings especially for a multi-variable controlled plant. The so
called correction members are then generally considered for ensuring invariance of a control loop. Further, it is
considered that primary controllers are determined by arbitrary single-variable synthesis method for optimal
input-output variable pairings. Simulation verifications of the mentioned way of control are carried out for twovariable control loop.
Key-Words: - MIMO control loop, RGA, simulation, synthesis
Control methods of MIMO controlled plants can be
verified not only by using simulation tools, but also on
the laboratory models. Some multi-variable laboratory
models have been described in the literature, e.g.
helicopter model [5], [6], tank model [2], [7], etc.
Simulation experiments were performed, for
chosen MIMO controlled plant, in MATLAB/
SIMULINK software [8]. The MATLAB software
serves for programming and technical computing in
many areas. The SIMULINK software is part of the
MATLAB environment and serves to analyzing and
simulation of dynamics systems. It is possible to use
the MATLAB/SIMULINK software for education
and also for research [9], [10].
1 Introduction
Controlled plants with only one output variable
(controlled variable) which are controlled by a one
input variable (disturbance variable, manipulated
variable) are called as SISO (Single-Input SingleOutput, single-variable) controlled plants. However,
there are not a little cases where it is more than one
output variable controlled simultaneously by means
of more than one input variable, e.g. aircraft
autopilots, heat exchangers, chemical reactors,
distillation columns, helicopter, tank processes, steam
boilers, steam turbines, etc. [1], [2]. In these cases, it
means that there is larger numbers of dependent
SISO control loops. These control loops are complex
and have multiple dependencies and multiple
interactions between different input variables
(manipulated variables and disturbance variables) and
output variables (controlled variables). Mentioned
control loops are known as MIMO (Multi-Input
Multi-Output, multi-variable) control loop and
represent a complex of mutually influencing singlevariable control loops [1]. Special case of the MIMO
control loop is SISO control loop [3].
Multi-variable control methods have received
increased industrial interest [4]. It is often no easy to
tell when these control methods are necessary for
improved performance in practice and when usage of
simpler control structures are sufficient. Therefore, it
is useful to know functional limits and structure of
the whole control loop, i.e. controlled plant,
controller and separate signals in the control loop. [2]
ISBN: 978-960-474-316-2
2 Analysis and Control Design of
MIMO Control Loop
The controlled plant generally consists of m input
variables and n output variables. It is generally a
non-square controlled plant type n×m. It means,
there are three possible cases, i.e. m = n, m > n,
m < n. In the next part of the paper, it is mostly
considered that controlled plant have a same number
of input variables and output variables, i.e. m = n
(square controlled plant type n×n).
One of possible approach to analysis and control
design of MIMO control loop, for a controlled plant in
steady state, is using the so called the RGA (Relative
Gain Array) method [11], [12]. The RGA is useful for
MIMO controlled plants that can be decoupled. The
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Recent Advances in Automatic Control, Information and Communications
other approach to analysis and control design of
MIMO control loop can be found e.g. in [13], [14].
MIMO control loop with measurement of
disturbance variables (see Fig. 2), i.e. closed loop
transfer function matrix GW/Y (s) and disturbance
transfer function matrix GV/Y (s).
2.1 Description of the control loop
It is generally considered a MIMO control loop with
measurement of disturbance variables shown in the
Fig. 1 [1].
V(s)
GKC (s)
W(s)
E(s)
GR (s)
GSV (s)
UKC (s)
UR (s)
U(s)
GS (s)
GW/Y ( s )  I  GS ( s) GR ( s) GS ( s) GR ( s)
1
GV/Y (s)  I  GS (s) GR (s)
YSV (s)
YS (s)
2.2 The relative gain array
The RGA is a tool to the analysis of the interactions
between input variables uj and output variables yi
especially of a MIMO controlled plant. In other
words, the RGA is a normalized form of the gain
matrix that describes the influence of each input
variable on each output variable. Each element in the
RGA () is defined as the open control loop gain
divided by the gain between the same two variables
when all other loops are under so called "perfect"
control [11]. Further, it is considered a linear n×n
controlled plant.
11  1n 
Λ      
n1  nn 
ij 
(yi u j ) uk
(yi u j ) y
U j ( s)
Vk ( s )
;
control loop gain with all other control loop closed,
ij is the relative gains for the corresponding
variable pairings, i.e. element of the RGA.
Transfer function matrix of a controlled plant can
be written in the following form
k  1,..., m , m  n
Transfer function matrices of controller GR (s)
and correction members GKC (s) are considered in
the following forms
Y (s)  GS (s)U ( s)  U ( s)  GS1 (s)Y ( s)
U R ,i ( s )
E j (s)
, KC ij 
U KC ,i ( s )
Vk ( s )
;
(2)
i, j  1,..., n 
k  1,..., m , m  n
1
1
Λ(GS (s))  GS (s)  (GS (s))T  GS (0)  (GS (0))T
thus
The relations (1) and (2) can be used to build
other transfer function matrices that occur in the
ISBN: 978-960-474-316-2
(6)
where U(s) and Y(s) are n-dimensional vectors of
inputs and outputs variables and GS(s) is an n×n
transfer function matrix of the controlled plant.
From relations (5) and (6), it follows that the
RGA for the controlled plant GS(s) can be
determined as
where
Rij 
yl  0, l  1,.., n  l  i
other control loop open, (yi u j ) yl is the open
i, j  1,..., n 
 R11  R1n 
 KC11  KC1m 


GR (s)       GKC (s)   

 
Rn1  Rnn 
KCn1  KCnm 
i, j  1, .., n
u k  0, k  1,.., n  k  j
(yi u j ) uk is the open control loop gain with all
(1)
where
YSV ,i ( s )
(5)
where
l
SV 11  SV 1m 
 S11  S1n 


G S (s)       G SV (s)   

 
 SVn1  SVnm 
S n1  S nn 
, SV ij 
GSV (s)  GS (s)GKC (s) (4)
Y(s)
The description of the figure is following, i.e.
matrices GS (s), GR (s), GSV (s) and GKC (s) denote
transfer function matrices of a controlled plant,
controller, measurable disturbance variables and
correction members. Signal Y(s) denotes the Laplace
transform of the vector of controlled variables, U(s)
is the Laplace transform of the vector of manipulated
variables, V(s) is the Laplace transform of the vector
of measurable disturbance variables, W(s) is the
Laplace transform of the vector of setpoints and E(s)
is the Laplace transform of the vector of control
error (E(s) = W(s) - Y(s)).
They are considered transfer function matrices of
a controlled plant GS (s) and measurable disturbance
variables GSV (s) in forms
YS ,i ( s )
(3)
These transfer function matrices can be use at
control design and also to ensuring invariance of the
MIMO control loop.
Fig. 1 - Basic scheme of MIMO control loop with
measurement of disturbance variables
S ij 
1
ij  [GS (0)]ij  [GS1 (0)] ji
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(7)
Recent Advances in Automatic Control, Information and Communications
where s = j →  = 0 → steady state,  operator
implies an element by element multiplication
(Hadamard or Schur product).
In the [12] is presented the procedure to calculate
RGA for non-square transfer function matrix by
using the pseudoinverse.
There are some important properties and rules to
understanding and analyzing the RGA, i.e.
 Separate elements ij are dimensionless and so
independent of units.
 The sum of all the elements ij of the RGA (5)
across any row, or any column will be equal to 1
n
n
i 1
j 1
 ij   ij  1
2.3 Invariance of control loop
The invariance of the MIMO control loop can be
ensured if the transfer function matrix GV/Y (s) (4) is
zero. It can be possible if the following relation is valid
1
G KC ( s)  G S ( s) G SV ( s)
In the next part of the paper, it is considered an
approach which ensures the so called approximate
invariance of control loop. It means that influence of
disturbance variables is generally eliminated only
partially (e.g. only in steady state). Further it is
considered the diagonal elements of the transfer
function matrix GSV (s) (1b) and GS (s) (1a) have a
dominant influence. The influence of the other
elements of the transfer function matrix GSV (s) and
GS (s) is omitted at design of correction members
KC. So that invariance of the control loop is ensured
only for diagonal elements. The influence of
separate elements of the transfer function matrix
GSV (s) and GS (s) can be verified by using the RGA
tool. In this case it is considered that corresponding
number of SISO branched control loops with
measurement of a disturbance variable is used.
Connection of all SISO branched control loops is
the same and they differ in separate transfer
functions and variables (see Fig.2). [1].
(8)
The relation simplifies calculation of elements
ij, e.g. in 2×2 case, only 1 element must be
calculate to determine all elements, in 3×3 case,
only 4 elements must be calculate to determine
all elements, etc.
 Each row in the RGA represents one output
variable yi and each column in the RGA
represents one input variable uj. The
interpretation of the determined elements ij in
the RGA can be classified as follows
 ij = 1: This implies that uj influences yi
without any interaction from the other
control loop.
 ij = 0: This means uj has no effect on yi.
 0 < ij < 1: This indicates that control pair uj yi is influenced by the other control loops.
 ij > 1: The positive value of the RGA
indicates that the control pair yi - uj
represents dominant control loop. Others
control loops have an influence on the
control pair in the opposite direction.
 ij < 0: This means that the control pair yi - uj
causes instability of the control loop.
Control pairs yi - uj whose input and output
variables have positive RGA elements (ij) and their
values are close to one are considered as the optimal
control pairs [14]. If the value of ij fulfils above
mentioned general rule the control of the control loop
for the control pair yi - uj is possible. For other values,
the control can become difficult because the
interaction rate is too high. Then, it is possible to
determine the parameters of the so called primary
controllers via classical SISO synthesis methods for
the optimal control pairs yi - uj gained by using the
RGA pairing method [15].
The RGA pairing method has also some
shortcoming, i.e. the RGA method ignores process
dynamics. If the transfer function has very large
time delay or time constant relative to the others,
steady state RGA analysis provide an incorrect
recommendation. In this case it is then preferable to
use the so called the RNGA (relative normalized
gain array) pairing method.
ISBN: 978-960-474-316-2
(9)
v
SV
KC
w
y
u
e
R
S
Fig. 2 - SISO branched control loop with measurement
of disturbance variable v
Transfer function of correction members KC is
determined by using the following relation
KCii 
SV ,ii
S ii
KCij  0
i   1, ,n , S ii  0
(10)
i, j   1, ,n , i  j
where SV,ii are separate elements of transfer matrix
GSV (s), Sii are separate elements of transfer function
matrix GS (s).
In the case that diagonal elements of transfer
function matrix GSV (s) or GS (s) have not dominant
influence then described approach to solution of
invariance of the control loop may not ensure the
desired behaviour.
The above mentioned approach for ensuring
invariance of control loop by using SISO branched
control loops with measurement of disturbance
variables can also be used for a reduction of
interactions of separate non-dominant control loops,
i.e. a reduction of influence of non-dominant
elements of the transfer function matrix of
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Recent Advances in Automatic Control, Information and Communications
controlled plant GS (s) in the MIMO control loop.
Such the control loop is then called decoupling
control loop. In this case, it is considered that
separate non-diagonal elements of the transfer
function matrix of controlled plant GS (s) represent
measurable disturbance variables.
two output variables. The Laplace transform of a
vector of an output variable is given by the
following relation
Y ( s )  G S ( s ) U ( s )  G SV ( s )V ( s )
where Y(s), U(s) and V(s) is the Laplace transform of
the vector of controlled variables (Y(s) = [Y1, Y2]T),
manipulated
variables
(U(s) = [U1, U2]T)
and
measurable disturbance variables (V(s) = [V1, V2]T).
Mentioned equation (12) can be described in the
following form
2.4 Control design of control loop
One of the possible approaches to control of MIMO
control loops is described in the following part. This
approach uses analysis of the interactions between
input variables and output variables, i.e. the RGA
method. It can be generally divided a solution this
problem into several parts, i.e. determination of
parameters of primary controllers then ensuring
invariance of control loop and also ensuring at least
partial decoupling control loop, i.e. partial reduction
of the influence of non-dominant elements (nonoptimal control pairs) of the transfer function matrix
of controlled plant GS (s) in the MIMO control loop.
The so called primary controllers are designed by
any synthesis method of SISO control loops.
Parameters of primary controllers are determined for
optimal control pairs gained by using the RGA (5),
(7) for the controlled plant GS(s).
Invariance of control loop is ensured by using
SISO branched control loops with measurement of
disturbance variables and by determination of
correction members KC via relation (10). These
correction members are determined only for
diagonal elements of transfer function matrix GSV (s)
and GS (s).
For ensuring decoupling control loop is possible
to use the approach described in paragraph 2.3. In
this case, the relation (11), derived from (10), is
used, i.e.
7.5

 S11 S12   s 2  5s  6
GS (s)  

0.5
S 21 S 22  
 s 2  3s  2
6

s  5s  6 
1.5

2
s  3s  2 
2
 0.85
 SV 11 SV 12   s 2  5s  6
GSV (s)  

0.2
SV 21 SV 22  
 s 2  3s  2
(14)
0.65 
s 2  5s  6  (15)
0.6

2
s  3s  2 
Step responses of transfer function matrices
GS (s) and GSV (s) are shown in the following figures
(see Fig. 3, Fig. 4).
Step Response
(11)
To: Out(1)
Amplitude
From: In(1)
To: Out(2)
i , j   1, , n , S ii  0, i  j
(13)
Transfer function matrices GS(s) and GSV(s) are
considered in these forms
From: In(2)
1.5
1
0.5
0
1
0.5
0
0
2
4
6
8
100
Time
2
4
6
8
Step Response
Amplitude
From: In(1)
From: In(2)
0.2
0.1
0
0.4
0.2
0
0
2
4
6
8
100
Time
2
4
6
8
10
Fig. 4 - Step response of transfer matrix GSV(s) (15)
3.2 Control of two-variable control loop
The approach to MIMO control described in the
paragraph 2.4 is used for control of the two-variable
control loop. First transfer functions of primary
controllers for optimal control pairs (5), (7) are
determined then approximate invariance of control
loop is ensured by using (10) and finally decoupling
control loop is solved by using (11).
3 Simulation Verification of Described
Approach to Control
3.1 Description of two-variable controlled plant
It is considered two-variable controlled plant, i.e.
controlled plant having a two input variables and
ISBN: 978-960-474-316-2
10
Fig. 3 - Step response of transfer matrix GS(s) (14)
The relation (11) is used for determination of
parameters of the so called auxiliary controllers RPij.
In this case it is considered that aside from diagonal
elements of transfer function matrix GS (s), i.e. Sij
have non-dominant influence and diagonal elements
of transfer function matrix GS (s), i.e. Sii have
dominant influence. Modified version of the MIMO
branched control loop with measurement of
disturbance variables, where the auxiliary controllers
are used, is shown in Fig. 5.
To: Out(1)
S ii
Y1 
U1 
V1 
Y   GS (s) U   GSV ( s) V 
 2
 2
 2
To: Out(2)
RPij 
S ij
(12)
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Recent Advances in Automatic Control, Information and Communications
To determination of optimal control pairs of the
controlled plant (14) was used (7), hence
 S (0) S12 (0)   S11 (0) S12 (0) 
Λ(GS (0))   11


S 21 (0) S 22 (0) S 21 (0) S 22 (0)
The scheme of modified two-variable control
loop is considered according to Fig. 5.
T
SV12
(16)
v1
then
 11 12   1.3636  0.3636



 21 22   0.3636 1.3636 
(17)
which means that optimal control pairs are y1 - u1
and y2 - u2. These optimal control pairs
corresponding transfer functions S11 and S22 (1a)
for which are designed primary controllers of the
transfer function matrix of GR(s), i.e. in this case
transfer functions R11 and R22 (2a). Parameters
these controllers (PI controllers) were determined
via the method of balance tuning [16]. To use this
method it was necessary to modify transfer
functions S11 and S22 as follows
S11, x
e1
w1
KC11
S12
RP12
SV11
u1
R11
y1
S11
SV21
v2
w2
e2
KC22
S21
RP21
SV22
u2
R22
y2
S22
Fig. 5 - Modified two-variable branched control
loop with measurement of disturbance variables
1.25
0.75

e 0.228s S 22, x 
e 0.414s (18)
0.6565s  1
1.147s  1
3.3 Simulation verification of control loop
then
R
G R ( s )   11
0
0 
R22 
The MATLAB/SIMULINK software [8] is used for
simulation verification of proposed control method
of the two-variable control loop (see Fig. 5).
Simulation courses of the two-variable control
loop with utilization of chosen SISO synthesis
method, which is used at design of parameters of
primary controllers, are presented in the following
figures (see Fig. 6, Fig. 7). The following parameters
were chosen and used at all simulation experiments
[10, 80]
 time vector of setpoints (tw1, tw2):
 vector of setpoints (w1, w2):
[0.7, 0.7]
 time vector of disturbances (tv1, tv2): [40, 120]
 vector of disturbances (v1, v2):
[0.4, 0.4]
 total time of simulation (tS):
150
0.05
 time step (k):
0.6s  0.2383
s
(19)
0.9913s  0.8261
R22 
s
R11 
Beside above mentioned methods to determine of
parameters of primary controllers is possible to use
also other SISO synthesis methods, e.g. Ziegler
Nichols methods, Cohen-Coon method, the method
of desired model, Whiteley method, the pole
placement method, etc. [1], [17].
Correction members KC were determined from
relation (10), where was assumed a dominant
influence of diagonal elements of transfer function
matrix GSV (s)
0  0.1133 0 


KC 22   0
0.4
(20)
1.5
y1, u1, w1, v1
 KC
G KC ( s )   11
 0
Influence diagonal elements of the GSV (s) (15)
was verified by using (7), i.e. determination of the
RGA for the GSV (s)
 1.3714  0.3714
(0))  
 (21)
 0.3714 1.3717 
RP12   0
0.8



0  0.3333 0 
ISBN: 978-960-474-316-2
w1 - setpoint
v1 - disturbance variable
1
0
50
100
150
100
150
time
T
1
Decoupling control loop was ensured by using of
non-dominant elements of transfer function matrix
GS(s) (14). Parameters of the auxiliary controllers
RP were determinate via relation (11), i.e.
 0
G RP ( s)  
 RP21
u1 - manipulated variable
0.5
-0.5
0
y2, u2, w2 , v2
Λ(GSV (0))  GSV (0)  (GSV
1
y1 - controlled variable
0.5
y2 - controlled variable
u2 - manipulated variable
w 2 - setpoint
v2 - disturbance variable
0
-0.5
0
50
time
Fig. 6 - Simulation course of control loop with
utilization method of balance tuning without the
use of auxiliary controllers RP12 and RP21
(22)
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Recent Advances in Automatic Control, Information and Communications
y1, u1, w1, v1
1
y1 - controlled variable
u1 - manipulated variable
w 1 - setpoint
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Control Systems, 2nd edition. Institution Of
Engineering And Technology, 1988.
[14] M. T. Tham, An Introduction to Decoupling
Control, University Newcastle upon Tyne, 1999.
[15] J. Zakucia, Control of the Quadruple-Tank
Process, CTU in Prague, diploma work (in
Czech).
[16] P. Klan and R. Gorez, Balanced Tuning of PI
Controllers. European Journal of Control,
Vol. 6, No. 6, 2000, pp. 541-550.
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v1 - disturbance variable
0.5
0
-0.5
0
50
100
150
100
150
time
y2, u2, w2 , v2
1
y2 - controlled variable
u2 - manipulated variable
0.5
w 2 - setpoint
v2 - disturbance variable
0
-0.5
0
50
time
Fig. 7 - Simulation course of control loop with
utilization method of balance tuning with the use of
auxiliary controllers RP12 and RP21
It is obvious from the simulation courses of
control loop shown in the Fig. 6, Fig. 7 and from
other simulation experiments that the proposed
control method can be used to control of a MIMO
control loop. Using this control method it was
possible to ensure approximate invariance of control
loop (see Fig. 6, Fig. 7) via correction members KCij
(20) and also decoupling control loop (see Fig. 7
compare to Fig. 6) via auxiliary controllers RPij (22).
4 Conclusion
The goal of the paper was to describe one of the
possible approaches to control of MIMO control
loops. The control method uses the RGA tool for
ensuring optimal control pairs for which they are
determined parameters of the primary controllers via
classical SISO synthesis methods. Further, it is used
SISO branched control loop with measurement of
disturbance variable for ensuring approximate
invariance of control loop. The approach used for
ensuring invariance of control loop is also used to
ensure decoupling control loop. Simulation
verification of the control method of control was
presented on the two-variable control loop.
The proposed approach to control is valid under
the following condition, i.e. the proposed control
method is considered for multi-variable controlled
plant with same number input and output variables.
The future work will be focused on simulation
verification of the control method for simulation
model of the concrete multi-variable controlled plant.
Acknowledgement
This work was supported by the European Regional
Development Fund under the project CEBIA-Tech
No. CZ.1.05/2.1.00/03.0089.
References:
[1] J. Balate, Automatic Control, 2nd edition, BEN,
Praha, 2004, 664 pp. (in Czech)
ISBN: 978-960-474-316-2
96