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: talas@fai.utb.cz

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 talas@fai.utb.cz.

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 bobal@fai.utb.cz.

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.