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USE OF DYNAMIC MATRIX CONTROL IN SIMULATION OF HEAT
SYSTEM
Stanislav Talaš2, Vladimír Bobál
1,2 and Adam Krhovják
2
Tomas Bata University in Zlin 1Centre of Polymer Systems, University Institute
2Department of Process Control, Faculty of Applied Informatics
T. G. Masaryka 5555
760 01 Zlin
Czech Republic
E-mail: [email protected]
KEYWORDS
Model Predictive Control, Dynamic Matrix Control,
MATLAB/SIMULINK, Simulation, Heat Exchanger.
ABSTRACT
This paper demonstrates the use of Model Predictive
Control (MPC) to system control. Dynamic Matrix
Control (DMC) method was chosen and its functionality
was verified by a simulation of system control based on
a real laboratory model. A control algorithm and the
simulation were realized in MATLAB/SIMULINK
program environment. Results have proven capabilities
of DMC method to control stable oscillatory and non-
minimum phase systems. Additionally, real model pa-
rameters were tested with a demonstration of a possibil-
ity of tuning by a ratio of weighting values form objec-
tive function.
INTRODUCTION
Considering the scientific area of process control, it
targets at present tendency of satisfying demands of the
maximal productivity of the highest quality products at
the lowest cost possible. With the power of the modern
computing technology an approach of finding optimal
results in reasonable time was made possible.
Advanced methods popular in industries with slow and
large dimensional systems are predictive control meth-
ods (Qin & Badgwell, 2003). These techniques com-
monly contain an internal model for system behavior
predictions. Gained information is further used to calcu-
late a sequence of control inputs by minimizing a sum
of squares between the desired and predicted trajecto-
ries. Therefore an optimal output is received in refer-
ence to the minimal error, eventually to the change of
control inputs.
Development in this area started in 1980s with the pub-
lication of DMC method (Cutler & Ramaker, 1980).
Original purpose of DMC was focused on multivariable
constrained control problems, mainly occurring in
chemical and oil industry. The influence of DMC
caused its widespread use in world’s major industrial
companies (Morari & Lee, 1999).
Over the time there was a vast development of the DMC
algorithm, its modifications and possibilities of applica-
tion. (Garcia & Morshedi, 1986) provided a utilization
of a quadratic algorithm for an efficient handling of
constraints, tuning and robustness. (Shridhar & Cooper,
1997) suggested a tuning strategy of DMC parameters
for SISO systems, followed by an approach in case of
MIMO systems (Shridhar & Cooper, 1998). (Dougherty
& Cooper, 2003) described an approach to tune the
parameters of the basic DMC algorithm for the case of
integrating processes. In occurrence of nonlinear pro-
cesses (Dougherty & Cooper, 2003) suggested a new
adaptive control strategy using the output of multiple
linear DMC controllers to maintain the performance
over a wide range of operational levels.
The purpose of this paper is to give an insight on abili-
ties of DMC options for the control of stable processes,
primarily in the area of tuning its performance by
changing the weight ratio of the optimization process
between the output error and a demand of action value.
The paper is organized in the following way. General
principles of MPC are presented first, followed by the
description of specific properties of the DMC method.
Basic functionality and characteristics are introduced.
The final section presents the implementation of the
simulation into the MATLAB/SIMULINK program
environment and its results.
MODEL PREDICTIVE CONTROL
Predictive control is an approach to control a process
trough optimization. The main principle is in the predic-
tion of future process outputs based on the inner model
of the process. The goal of the control algorithm is to
find such a vector of input values that the output of the
model is optimal along the defined time area called
horizon. To ensure robustness and stability an approach
using feedback called the receding horizon strategy is
often applied. From the vector of input values only the
first value is used as an increment ∆u(k) added to the
previous input giving the current input value u(k). In the
next step the entire procedure is repeated with new
process output values.
The area of the optimization is defined by values of
horizons representing the amount of sampling periods
from the current time into the future. Values of horizons
N1 and N2 limit the area, where the divergence between
the desired and the output value is minimized. The hori-
zon Nu limits the distance of steps where the action
value is minimized.
Proceedings 28th European Conference on Modelling and Simulation ©ECMS Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani (Editors) ISBN: 978-0-9564944-8-1 / ISBN: 978-0-9564944-9-8 (CD)
Figure 1: Receding horizon strategy
Calculation of the optimal output consists of a free re-
sponse prediction describing the system behavior in the
case of a constant input and the forced response with a
reaction on a suggested series of inputs. Based on the
superposition principle, the sum of these responses
results in the future output prediction.
Several methods of MPC are used in practice; the main
differences are in the description of the controlled pro-
cess and in the objective function.
The Figure 2 shows a layout of the predictive control
and a data transfer.
Figure 2: Basic structure of model predictive control
The optimization process is based on the minimization
of values involved in control. Their mutual relations are
formed by an objective function. The general expression
of an objective function is
,)1()()()(ˆ)(1
222
1
uN
i
N
Ni
ikuiikwikyiJ (1)
where δ(i) and λ(i) are weighting values, usually con-
stants representing a ratio of the minimization between a
divergence of output from the desired value and a
change of the action value (Normey-Rico & Camacho,
2007).
Dynamic matrix control
DMC method is based on a step function limited to first
N points. It is assumed that the process is stable and the
disturbance prediction is stable all over the horizon
equal to actual difference between measured value ym(k)
and its estimation k).
This method is often used in industry; some of the main
reasons are an easy identification of the internal model
and a possible addition of constraints into the minimiza-
tion criterion. The disadvantage is an unsuitable use for
unstable systems due to the step function in the internal
model.
Predicted values along the horizon are given by the
following equation
)(ˆ)()(ˆ
1
iknjikugiky
Nu
j
j
, (2)
where gj represents the system step response value in
time j and disturbance predictions (k + i) are consid-
ered constant along the horizon.
By defining the free response
,)()()()(
1
jkuggkyikf
Nu
j
jji
(3)
predicted values are computed as
.)()()(ˆ
1
ikfjikugiky
i
j
j
(4)
By expanding expression (4) to the number of future
values given by the control horizon Nu an equation for
the prediction of future outputs is created
11ˆ SyHuGuy , (5)
where
.
,
,
,
)(ˆ
)(...)2()1(
)1(...)1()(
)(ˆ...)1(ˆˆ
1
1
kyy
Nkukukuu
Nkukukuu
Nkykyy
u
T
u
T
u
T
(6)
In equation (5) S is an unitary vector with size Nu × 1
and G, H are matrices with size Nu × Nu
,
1
1
1
,0
00
11
12
1
SG
ggg
gg
g
NuNu
.
)(
)(
)(
)(
)(
)(
)(
)(
)(
2
2
1
22
24
23
11
13
12
NuNu
NuNu
NuNu
NuNu
gg
gg
gg
gg
gg
gg
gg
gg
gg
H
(7)
The output prediction is calculated using the sum of the
free response and the forced response like
fGuy ˆ , (8)
where f is the free response given by equation
11 SyHuf . (9)
Optimization is performed by finding the minimal value
of the objective function (1) accomplished by a differen-
tiation with respect to the vector of the action variable u
and equating to zero. The shape of the control law is
).(1
wfQGQGQGu
TT (10)
Where matrices Qδ and Qλ represent the weighting val-
ues from the objective function and are formed by iden-
tity matrices with value of δ respectively λ on the main
diagonal. From the calculated array only the first value
∆u(k) is used in the current step. In each of the follow-
ing steps the calculation is repeated (Camacho & Bor-
dons, 2004), (Haber et al., 2011).
EXPERIMENTAL LABORATORY HEAT
EQUIPMENT
A scheme of the laboratory heat equipment Pekař et al.,
2009) is described in Figure 4. The heat transferring
fluid (e. g. water) is transported using a continuously
controllable DC pump (6) into a flow heater (1) with
max. power of 750 W. The temperature of a fluid at the
heater output T1 is measured by a platinum thermome-
ter. Warmed liquid then goes through a 15 meters long
insulated coiled pipeline (2) which causes the signifi-
cant delay (20 – 200 s) in the system. The air-water heat
exchanger (3) with two cooling fans (4, 5) represents a
heat-consuming appliance. The speed of the first fan can
be continuously adjusted, whereas the second one is of
on/off type. Input and output temperatures of the cooler
are measured again by platinum thermometers as T2,
respective T3. The platinum thermometer T4 is dedicated
for measurement of the outdoor-air temperature. The
laboratory heat equipment is connected to a standard PC
via technological multifunction I/O card MF 624. This
card is designed for the need of connecting PC compati-
ble computers to real world signals. The card is de-
signed for standard data acquisition, control applications
and optimized for use with Real Time Toolbox for
SIMULINK. The MATLAB/SIMULINK environment
was used for all monitoring and control functions.
Figure 4: Scheme of laboratory heat equipment
SIMULATION CONTROL OF BASIC DYNAMICS
In order to verify the control algorithm a simulation
scheme was created in the SIMULINK environment as
can be seen in Figure 3. To test the general control ca-
pabilities of DMC, three systems representing different
dynamics were chosen. These systems are described
with following continuous transfer functions:
Stable non-oscillatory system
154
2)(
21
ss
sG .
Stable oscillatory system
124
2)(
22
ss
sG
.
Non-minimum phase system
154
210)(
23
ss
ssG
.
Discrete versions with sampling time T0 = 2s are
21
211
12076,04728,01
0821,07419,0)(
zz
zzzG
21
211
24834,06806,01
3679,07859,0)(
zz
zzzG
21
211
37780,10980,11
0821,07419,0)(
zz
zzzG
Figure 3: Simulation scheme in SIMULINK program environment
Considering systems dynamics, the sizes of the control
horizon Nu and the maximum horizon N2 were set to 30
steps, while the minimum horizon N1 remained 1. Con-
trol performances for corresponding systems are
demonstrated with the desired trajectory with the shape
of the step change known ahead by the control algo-
rithm. Results can be seen in the following figures.
Figure 5: Control of stable non-oscillatory system G1
Figure 6: Control of stable oscillatory system G2
Figure 7: Control of non-minimum phase system G3
Results have proven that the DMC algorithm is able to
control stable, oscillatory and even non-minimum phase
systems.
Simulation control of a heat system
The following simulation model is based on identifica-
tion results of a laboratory heat-system. The real system
is constructed as a circulation of water warmed by a
heater and cooled by fans creating a stable circuit with
large time constants. In order to concentrate the focus to
the topic, the traffic delay caused by a fluid transport is
not included in the following simulation.
This laboratory model was identified as a stable second
order system with continuous expression
52
5
410075.403017.0
10664.2001614.0)(
ss
ssG
and its discrete version with sampling time of 60 sec-
onds
21
21
4
1636.0097.11
0281.00719.0)(
zz
zzzG
Values of horizons were set to 30 steps for the control
horizon Nu as well as the maximum horizon N2, the
minimum horizon N1 was set to 1. As a tuning mecha-
nism to gain a suitable precision with an appropriate
action value, several different settings of weighting
values δ(i) and λ(i) ratio were examined.
The precision of the control process was determined by
an integral of a squared error (ISE) criterion.
dtteISE
02)( (11)
Due to large time constants of the model a step function
was selected as a shape of the desired trajectory, in
order to prove the ability of DMC to control the system
optimally.
Figure 8: Reference and output values with weight pa-
rameters ratio 1:1
-2
0
2
4
6
8
10
12
14
0 20 40 60 80
w, u
, y
time [s]
y - process outputw - reference signalu - controller output
-2
0
2
4
6
8
10
12
14
0 20 40 60 80
w, u
, y
time [s]
y - process outputw - reference signalu - controller output
-2
0
2
4
6
8
10
12
14
0 20 40 60 80
w, u
, y
time [s]
y - process outputw - reference signalu - controller output
-2
0
2
4
6
8
10
12
14
0 2000 4000 6000 8000
w, y
time [s]
y - process output
w - reference signal
Figure 9: Controller output value with weight parame-
ters ratio 1:1
Figures 8 and 9 demonstrate the control performance in
the case of process optimization evenly distributed be-
tween the divergence of the output from the desired
value and the change of the action value. This setting
has achieved a value of the error criterion ISE = 7005.
Figure 10: Reference and output values with weight
parameters ratio 10:1
Figure 11: Controller output value with weight parame-
ters ratio 10:1
Figures 10 and 11 illustrate the case of the increased
weight parameter of the action value difference. Figure
11 clearly presents an increased smoothness of the con-
troller output trajectory implying lower demands on a
physical actuator. On the other hand, the precision of
tracking the desired output value has significantly de-
creased, as can be seen in Figure 10, rising the value of
the error criterion ISE = 13390.
Figure 12: Reference and output values with ratio 1:10
Figure 13: Controller output value with weight parame-
ters ratio 1:10
Figures 12 and 13 show the case when the ratio is in-
creased towards the divergence of the output from the
desired trajectory, providing a high precision perfor-
mance with a value of the error criterion ISE = 3920.
This comes with a disadvantage of sharp changes in the
action value and a considerably high peak value.
CONCLUSION
The paper presents results of the simulated control of
the laboratory heat system by the DMC. Outcomes of
simulations present the possibilities of tuning the per-
formance of predictive control methods. A significant
impact of the weight ratio was proven, as well as con-
-5
0
5
10
15
20
25
30
0 2000 4000 6000 8000
u
time [s]
u - controller output
-2
0
2
4
6
8
10
12
14
0 2000 4000 6000 8000
w, y
time [s]
y - process output
w - reference signal
-5
0
5
10
15
20
25
30
0 2000 4000 6000 8000
u
time [s]
u - controller output
-2
0
2
4
6
8
10
12
14
0 2000 4000 6000 8000w
, y
time [s]
y - process output
w - reference signal
-5
0
5
10
15
20
25
30
35
40
0 2000 4000 6000 8000
u
time [s]
u - controller output
nection between a precision of the output value and a
dynamics of the action value as can be seen on Table 1.
Table 1: ISE comparison of weight parameters ratios
λ : δ weight ratio ISE criterion value
1 : 1 7 005
10 : 1 13 390
1 : 10 3 920
Results have confirmed the ability of DMC to provide a
high quality control, furthermore with use of weight
values also to determine an amount of the required ac-
tion value and the precision of the output value.
ACKNOWLEDGEMENT
This article was created with support of Operational
Programme Research and Development for Innovations
co-funded by European Regional Development Fund
(ERDF), national budget of Czech Republic within the
framework of the Centre of Polymer Systems project
(reg. number: CZ.1.05/2.1.00/03.0111).
REFERENCES
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AUTHOR BIOGRAPHY
STANISLAV TALAŠ studied at the To-
mas Bata University in Zlín, Czech Repub-
lic, where he obtained his master degree in
Automatic Control and Informatics in
2013. He now attends PhD. study in the
Department of Process Control, Faculty of Applied
Informatics of the Tomas Bata University in Zlín. His e-
mail address is [email protected].
VLADIMÍR BOBÁL graduated in
1966 from the Brno University of Tech-
nology, Czech Republic. He received his
Ph.D. degree in Technical Cybernetics
at Institute of Technical Cybernetics,
Slovak Academy of Sciences, Bratisla-
va, Slovak Republic. He is now Profes-
sor at the Department of Process Control, Faculty of
Applied Informatics of the Tomas Bata University in
Zlín, Czech Republic. His research interests are adap-
tive and predictive control, system identification and
CAD for automatic control systems. You can contact
him on email address [email protected].
ADAM KRHOVJÁK studied at the To-
mas Bata University in Zlín, Czech Repub-
lic, where he obtained his master degree in
Automatic Control and Informatics in
2013. He now attends PhD. study in the Department of
Process Control, Faculty of Applied Informatics of the
Tomas Bata University in Zlín. His research interest
focus on modeling and simulation of continuous time
technological processes, adaptive and nonlinear control.
He is currently working on programming simulation
library of technological systems.