Predictive Control Algorithm

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Acta Electrotechnica et Informatica No. 3, Vol. 7, 2007 1

ISSN 1335-8243 2007 Faculty of Electrical Engineering and Informatics, Technical University of Koice, Slovak Republic

PREDICTIVE CONTROL ALGORITHMBASED ON STATE SPACE MODELS OF DYNAMIC SYSTEMS

- INTERNET APPROACH

Anna JADLOVSK, Nikola KABAKOV, Iveta ZOLOTOVDepartment of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics,

Technical University of Koice, Letn 9, 042 00 Koice,tel. 055/602 4152, E-mail: [email protected], tel. 055/602 4217, E-mail: [email protected]

tel. 055/602 2551, E-mail: [email protected]

SUMMARYThe paper is focused on Predictive Control Algorithm design for time-invariant state space models of dynamic systems

with various dynamics, (mechanical system-helicopter, hydraulical system - three tanks with interaction). The predictive

control algorithm based on state space models is verified by simulation schemes in language Matlab/Simulink using

architecture of S-functions of the library PredicLib. The results are presented by created Internet applications using the

technology Matlab Web Server. Internet applications enable to simplify the usage of designed predictive algorithms for

testing their properties at various dynamics of MIMO non-linear systems without the proper installation language

Matlab/Simulink on clients PC.

Keywords: model based predictive control, dynamic system, state space model, equations of predictions, minimization ofquadratic criterion, conditions for constrains, virtual laboratory

1. INTRODUCTIONThe method of predictive control is a powerful

model-based approach for control of processes fromindustry, which enables user to handle differentcontrol requirements (limits, constraints).

Model Predictive Control (MPC) is the methodof control in which the goal is to compute such

a future control sequence, that the future systemoutput is driven as close as possible the referencetrajectory. This is accomplished by minimizingquadratic multistage cost function defined over the

prediction horizon [4], [5].The key features of MPC are:

an explicit using of a model of dynamic systemto predict the system output at the future time(horizon of prediction),

the calculation of a control sequence byminimizing of the cost function,

the receding strategy - calculation of the actualmanipulated variable in each control step.

The various MPC algorithms differ amongthemselves in the model used to represent the

process and the cost function to be minimized. ByGeneralized Predictive Control (GPC) algorithm

plant model is often in form of the rational transferfunction. In this case the solution of Diophantineequations is required to compute prediction of thesystem output [1], [2].

The predictive control algorithm presented in thispaper is based on the mathematical-physical modelsof the controlled dynamic systems in the state space

(mechanical system - helicopter, hydraulic system -three tanks with interaction). For using a state spaceformulation of the predictive control algorithm thestates of the system are necessary to be known.When, the states of the system are not available (are

not measured), then the states of the system areestimated by the linear state space observer based onKalman estimator [1].

The paper is focused on the predictive controlalgorithm design for time-invariant state spacemodels with various dynamics. The predictivecontrol algorithm is verified by simulation schemesin language Matlab/Simulink using architecture of

S-functions of the library PredicLib [7]. The resultsare presented by created an internet applicationsusing technology Matlab Web Server (MWS) [7],[10].

2. PREDICTIVE CONTROL BASED ONSTATE SPACE MODEL OF THE SYSTEM

The Model Predictive Control is a multistageapproach, combining feedforward and feedbackcontrol design.

The feedforward control design is represented bythe predictions based on the mathematical model ofthe dynamic system. This part is dominantcomponent of the control actions. The feedbackcontrol design from measured outputs serves forcompensation of some bounded model inaccuraciesand low external disturbances. The design consistsfrom local minimization of quadratic criterion (1) inwhich the predictions from the equations of the

predictions are involved. The minimization isrepeated in each time step [4].

The cost function for predictive controlcalculating fork-th step can be represented as

( )21 0

2

1 0

( ) ( | ) ( | )

( | )

p

p

Nn

MPC i i i

i j

Nm

l l

l j

J k y k j k r k j k

u k j k

= =

= =

= + + +

+ +

(1)


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2 Predictive Control Algorithm Based on State Space Models of Dynamic Systems Internet Approach


wherep

N is predictive horizon,i

,l

are the

positive weight coefficients for the output and theinput, ( | )

iy k j k+ is predicted system output value,

( | )i

r k j k + is required system output (reference)value fori-th output of the system.

The first term in the performance criterion refersto the square variation of the predicted systemoutput from the desired reference trajectory, whilethe second term is added in order to limit thecontroller output, greater yields less activecontroller output.

The predictive control algorithm design is basedon the linearized equations of discrete state spacemodel of the dynamic system, which will be used tocompute of the predictor.

Let us consider linear discrete state space modelof the dynamic system

( 1) ( ) ( )

( ) ( ) ( )

k k k

k k k

+ = +

= +

x Ax Bu

y Cx Du(2)

where ( )kx denotes the state vector (dimension n),

( )ky denotes the system outputs or measurements

to be controlled (dimension n) and ( )ku (dimensionm) denotes the system inputs (or the controlleroutputs), A , B , C are the matrices defining thestate space model. In generally for the real systemsthe matrix 0=D .

The model maps interval of one sampling period.Principle of the equations of predictions isexpression of future values of the outputs ( )ky from

the current measured state ( )kx as follows

1

1

( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

( ) ( ) ( )

( 1) ( ) ( ) ( )

( 1)

p p

p p

N N

p

p

N N

p

p

k k

k k k

k k k

k N k k

k N

k N x k k

k N

=

+ = +

+ = +

+ = + +

+ + + + = + +

+ + +

y Cx

x Ax Bu

y CAx CBu

x A x A Bu

Bu

y CA CA Bu

CBu

M

L

L

(3)

Equations (3) can be usefully written in the matrixnotation

( )k= +y Gu Sx (4)

where

( 1) ( )( 2) ( 1)

,

( ) ( 1)p p

k kk k

k N k N

+ + + = =

+ +

y uy u

y u

y u

M M

2

1 2

,

p p pN N N

= =

CB 0 0 CA

CAB CB 0 CAG S

0

CA B CA B CB CA

L

L

M M O M

L

This is the base of the predictive control newunknown values are obtained from actual values.

2.1.Computing of an optimal controlFor computing of an optimal control the costfunction (1) can be written in matrix notation:

( ) ( ) T T

MPCJ = +y r M y r u Lu (5)

where

( 1) ( 1)

( 2) ( 2),

( ) ( )

( 1)

( 2)

( )

p p

p

k k

k k

k N k N

k

k

k N

+ + + + = =

+ +

+ + =

+

r

r r M

r

L

L

L

M M M O M

L

L

L

M M O M

L

0 0

0 0

0 0

0 0

0 0

0 0

[ ]( )1 2( ) nk j diag + = L[ ]( )1 2( ) mk j diag + = L

for 0, 1, ,p

j N= K .

The predictor written by the matrix notation (4) canbe substituted in the cost function (5) for an optimalcontrol computing. We obtain:

( ) ( )( ) ( )T

MPC

T

J k k= + +

+

Sx Gu r M Sx Gu r

u Lu

(6)

After simplifying the equation (6) the cost functioncan be written in the following form:

0

1

2T T

MPCJ = + +u Hu b u f (7)

where

( )

( )

( ) ( )0

2

2 ( )

( ) ( )

T

TT

T

k

k k

= +

=

=

H G MG L

b Sx r GM

f Sx r Sx r

The minimum of the cost functionMPC

J (7) can be

found by making gradient ofMPC

J equal to zero,which leads to:


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( ) ( )11 ( )T T k

= = + u H b G MG L G M Sx r (8)

The result of the equation (8) is the trajectoryconsisting of optimal control inputs, where the firstm of them are applied on the system and is given by:

( )( ) ( )k k= u K r Sx (9)

where K is the first m-rows of the matrix

( )1

T T

+G MG L G M .For the dynamic system with the constrains forcontroller output value or the system output value,vector of the optimal control u is calculating byOptimization Toolbox function quadprog oflanguage Matlab:

min max( , , , , , )T

CONquadprog=u H b L v U U

wheremin max U u U and CON L u v .

The vectors minU and maxU are the column vectorsthose elements are minimal and maximal values of

( )ku . With using the matrix CONL and the vector v can be defined a system of inequalities that insure soconstrain conditions will be satisfied [2], [3].

2.2. Implementation of predictive controlalgorithm based on the state space model toMatlab

The algorithm of predictive control for MIMOdynamic system was designed with using S-functions in programmable environmentMatlab/Simulink.

The algorithm for the calculating of the controlinput value in k-th step:

1.reading of the matrices A , B , C of the linearizeddiscrete state space model (2) of dynamicsystem, the reference vector trajectory rand theestimated state vectorxe(k),

2.creating of the weight matrices M and L ,3.calculate the matrix of the free response S and

the matrix of the force response G,

4.if required constrains for the values of the controlinput ( )ku , then continue by the step 8,

5.calculating of the feedback gain of the controlmatrix K,

6.calculating of the controller output by( ) ( ( ))k k=

eu K r Sx ,

7.continue by the step 12,8.creating of the vectors minU and maxU ,9.calculating Hessian matrix H and the vector

T

b ,10. min max( , ,[ ],[ ], , )Tquadprog=u H b U U ,11. ( )ku consists from the first m-rows of the

control vector u ,

12.the application of the control signal u(k) on thesystem input.

3. SIMULATION VERIFICATION OF MPCALGORITHM FOR DYNAMIC SYSTEMS

Designed MPC algorithm based on the statespace model was applicated on simulation models asmechanical system-helicopter and hydraulic system-three tanks with interaction.

For verification of the predictive algorithm bydesigned control structure on Fig. 1. in languageMatlab/Simulink was created model of helicopter asS-function in the state space for an equilibrium pointwhere as non-linear mathematical-physical model ofhelicopter was used model, which is derived in [6].Functional block of MPC controller is included inthe library PredicLib [3].

[W2(:,1),1*W2(:,3:7)]

yaw angle

reference trajectory

[W1(:,1),W1(:,3:7)]

pitch angle

reference trajectory

delta Ur

delta Us

delta theta

delta psi

helicopter modelSystem output

ref(k)

x(k)u(k)

State space

predictive

controllery(k)

u(k)xe(k)

Discrete

Kalman

estimator

Fig. 1 The control simulation scheme of the

predictive control based on state space model of thesystem

The mechanical system-helicopter has twodegrees of freedom, the rotation of the helicopter

body with respect to the horizontal axis (pitch angle ) and the rotation around the vertical axis (yawangle ), which are measured by two sensors. The

helicopter can move from -170,170 in yaw, and

from -45,45 in pitch.

The inputs to the model are the voltagesR

U and

SU affecting the main and the rear rotor. Both the

inputs are constrained between 10V and10V .For linearization of the nonlinear differentialequations of the helicopter at an equilibrium point

by 0 = and 0 = the following measurements

were done at input voltages 1.7RE

U V= ,3

SEU V= of the steady state of the lab helicopter

model [6]. At the voltagesRE

U andSE

U in aequilibrium point the system parameters are:

57 [ / ]RE rad s = [ ]0E rad = 0 [ / ]

Erad s

= 100 [ / ]SE rad s = (10)

0 [ ]E

rad = 0 [ / ]E

rad s =


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where

R RE R S SE S

R RE R S SE S

U U U U U U

= + = +

= + = +

We can find linear-time-invariant (LTI) modelby linearizing the non-linear differential equationsfrom [6] with the values of an equilibrium point andsystem parameters (10) by Jacobian matrices

, ,

,

E E E E

i i

C C

j jx u

= = = =

= =

x x u u x x u u

f fA B (11)

The LTI continuous model of the helicopter can nowbe expressed as

( ) ( ) ( )

( ) ( )C C

C

t t t

t t

= +

=

x A x B u

y C x

&(12)

where ( )tx , ( )tu , ( )ty are the state, the input and

the output vectors.The matrices

CA ,

CB and

CC of the state space

helicopter model (12) have structure

11 11

31 32 33 34 32

44 42

61 64 66 61

12

35

0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0,

0 0 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0

0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0

0 0 0 0 0 1

C C

C

A B

A A A A B

A B

A A A B

C

C

= =

=

A B

C

and their elements are defined in [6] and also inCyberVirtLab [8].

Further the state vector ( )tx and the input vector

( )tu are

[ ] [ ]1 2

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

T

R S

T T

R S

t t t t t t t

t u t u t U t U t

=

= =

x

u

where

( )R

t - deviation from the main rotors angularvelocity in the equilibrium point

( )t - pitch angle,

( )t - pitch velocity,

( )S

t - deviation from the rear rotors angularvelocity in equilibrium point,

( )t - yaw angle,( )t - yaw velocity,

( )R

U t - deviation from the voltage equilibriumpoint for the main rotor,

( )S

U t - deviation from the voltage equilibriumpoint for the rear rotor.

The sensors are measuring the two states ( )t

and ( )t . Taking derivative of ( )t and ( )t , we

also have the values for ( )t and ( )t . Thephysical value of the measured output will be:

1( )y t - voltage from pitch sensor measuring [ ]V ,

2 ( )y t - pitch velocity [rad/sec],

3 ( )y t - voltage from yaw sensor measuring [ ]V ,

4 ( )y t - yaw velocity [rad/sec] .

The discretization of the linearized model of thehelicopter (12) can be done in the language Matlabusing function c2d at the chosen sampling time

ST .

Parameters of the simulation for MPC algorithmas:

the equilibrium point of the helicopter model, the sampling period

ST ,( 0.01

ST s= ),

the prediction horizonP

N for the output of the

system, (we consider 10P

N = when theconstrains on the output of the predictivecontroller are defined),

the constrains for controller output value minRU ,maxRU , minSU , maxSU ,

the weight coefficients of matrices M and L ofthe cost function

MPCJ (1)

can be defined by an Internet form [8], which isincluded in CyberVirtLab [7].

The results of the tracking of the referencetrajectories 1 3( ), ( )r t r t by the systems outputs

1 3( ) ( ), ( ) ( )y t t y t t = = using MPC algorithm areon Fig. 2, whereas on the output of the system haveeffect disturbance in form of the white noise thatsimulates of the measurement error.

Fig. 2 The tracking of the reference trajectories

1 3( ), ( )r t r t by real outputs of the system 1 3( ), ( )y t y t


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The output of MPC controller an optimal controlsignals 1 2( ), ( )u t u t are on Fig. 3.

Fig. 3 The MPC controller outputs control signals

1 2( ), ( )u t u t

The results of MPC strategy with the constrainson the control inputs, which are illustrated on Fig. 2and Fig. 3 (with added white noise to the output ofthe system) shows on the tracking of the referencetrajectories when was used discrete linear Kalmanestimator [3].

In a similar way is MPC algorithm applicated onhydraulic system of the three tanks with aninteraction whereas for computing of the predictor(4) was used linearized model of MIMO system inthe state space. Unlike from the mechanical system

helicopter, which is unstable with quick dynamics,the hydraulic system is stable with slow dynamic.The model of the hydraulic system consists from

three cylindrical tanks with the same cylindrical tankbottom area F, which are interconnected at thebottom by pipes. The outflows from the tanks arecontrolled by relative open of value

iV .

The outflow from the first tank is the inflow tothe second one, the picture of the system is showedon Internet - CyberVirtLab [9].For changing levels 1( )H t , 2 ( )H t and 3 ( )H t holds:

11 1 2

22 2 3

33 3 4

( ) ( ) ( )

( )( ) ( )

( )( ) ( )

dH tF M t M t dt

dH tF M t M t

dt

dH tF M t M t

dt

=

=

=

(13)

From the physics (the part of the hydraulics) isknown the equation for calculation the mass flow

between tanks ( is thedensity of the liquid):

( )112 . ( ) ( )

( ) ( )

1,...,3

i i i

i vi i

i

f g H t H tM t k f u t

F

i

+

=

=

(14)

where

vik is constructing constant of the outlet of the

cylindrical tank,( )( )if u t is characteristic function of the valve iV ,

1( )M t [ / ]kg s is the mass inflow to cylindrical tank,

( )iM t [ / ]kg s is i-th mass flow between tanks

through the valvei

V ,

( )i

H t [ ]m denotes the water level in the i-thcylindrical tank,

F 2[ ]m is the cylindrical tank bottom area,

if

2[ ]m is the area outflow of the i-th cylindricaltank,

( )i

u t [ ]mm is the rises of the output outlet of the

cylindrical tank, for 1,...,3i = .

After inducting the equation (14) to the equation

(13) whereas we consider, that function ( )( )if u t will be linear, we obtain the non-linear differentialequations (15) describing dynamics of the changinglevels ( )

iH t in the tanks:

1 1 21 11 1

1 1

2 ( ) ( )( ) ( )( )

v

f g H t H tdH t M t k u t

dt F F

=

1 1 221 1

2

2 2 32 2

2

2 ( ) ( )( )( )

2 ( ) ( ) ( )

v

v

f g H t H tdH tk u t

dt F

f g H t H t k u tF

=

(15)

2 2 332 2

3

3 33 3

3

2 ( ) ( )( )( )

2 ( )( )

v

v

f g H t H tdH tk u t

dt F

f g H tk u t

F

=

For the set-point state of the equations (15), that iswhen ( ) / 0

idH t dt = , for 1,..., 3i = and by expansion

to Taylor series for the set-point

10 20 30 10 20 30[ ]SP = H H H u u u we obtain linearizedstate space model (12),where the structure of the matrices

CA ,

CB ,

CC is

11 12 11

21 22 23 21 22

32 33 32 33

0 0 0

, 0

0 0

1 0 0

0 1 0 ,

0 0 1

C C

C

A A B

A A A B B

A A B B

= =

=

A B

C

the state vector [ ]1 2 3( ) ( ) , ( ), ( )T

t H t H t H t = x

and the control vector [ ]1 2 3( ) ( ) ( ) ( )T

t u t u t u t = u .


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The MPC algorithm was applicated on thehydraulic system, which is created as S-function,and verified by modified control structure on Fig. 1in the language Matlab/Simulink.

The parameters of the simulation for MPCalgorithm as:

the equilibrium point of the hydraulic system, the sampling period

ST ,( 0.2

ST s= ),

the prediction horizonP

N for the output of the

system, ( 5P

N = ),

the weight coefficients of the matrices M andL , of the cost function

MPCJ (1)

can be defined by an Internet form [9], which isincluded in CyberVirtLab [7].

The presentation of the results of MPC strategy

is illustrated on Fig. 4 and Fig. 5.The plots on Fig. 4 compare the reference outputof the system ( )

ir t and the actual output of the

closed loop system ( )i

H t .

Fig. 4 The graphic representation of the referenceoutputs ( )

ir t and the real outputs of the hydraulic

system ( )i

H t

The plot in Fig.5 shows the results of MPCalgorithm an optimal control signals 1( )u t , 2( )u t ,

3( )u t .

Fig. 5 The MPC controller outputs control signals

1( )u t , 2( )u t , 3( )u t

4. CONCLUSIONThis paper presents the approach of an internet

verification MPC algorithm based on the state spacemodel of the dynamical system (mechanical system -helicopter and hydraulic system) by the simulationschemes in language Matlab/Simulink using thetechnology Matlab Web Server.

These created an internet applications can beused on the exercises of courses as Theory Optimaland Adaptive Systems, Control and ArtificialIntelligence and Control and Process Visualization,which are lectured on the Department of Cyberneticsand Artificial Intelligence of Faculty of ElectricalEngineering and Informatics, Technical Universityof Koice.

An internet applications enable to simplify theusage of designed MPC algorithms for the testingtheir properties at the various dynamics of MIMOnon-linear systems without the proper installationsimulation language Matlab/Simulink on the clients

PC. The authors of this paper have attempted to anintroduce the technology Matlab Web Server assuitable means of motivating students within thecourses such as Control Theory and Visualization bythe simulation verification of an optimal algorithmdesign in Matlab/Simulink environment.

ACKNOWLEDGMENTS

This research has been supported by the ScientificGrant Agency of Slovak Republic under project

Vega No.1/2183/05 Multiagent Hybrid Control ofLarge Scale Systems and under project KegaNo.3/4230/06 Monitoring and Supervised Controlof Simulated Processes Virtual Laboratory IIRS.


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REFERENCES

[1] Jadlovsk, A: Modeling and Control of DynamicProcesses Using Neural Networks, FEI TUKoice, Informatech Ltd., 2003, 173 p., ISBN 80-88941-22-9, (in Slovak)

[2] Jadlovsk, A. and Kabakov, N.: ApplicationParameter Estimation in the Algorithm ofAdaptive Generalized Predictive Control for

Non-linear Process, AT&P Journal plus, No.7/2005 Artificial Intelligence in Practise, pp. 29-34, ISSN 1336 - 5010

[3] Kabakov, N.: Predictive Control of Non-linearProcesses, Thesis to Dissertation Exam,Department of Cybernetics and ArtificialIntelligence, Faculty of Electrical Engineeringand Informatics, Technical University of Koice,53 pages, 2006, (in Slovak)

[4] Kozk, .: Characteristics of Chosen Methodsand Algorithms of Predictive Control, AT&PJournal plus No. 3/2002, pp. 15-27, ISSN 1335 2237, (in Slovak)

[5] Richalet, J.: Industrial Application of ModelBased Predictive Control, Automatica Vol.29,

No.5/1993, pp.1251-1274[6] Unneland, K.: Application of Model Predictive

Control to a Helicopter Model, Master Thesis,Norgwegian University, p.73, 2003

[7] http://cyberneticsmws.fei.tuke.sk/MatlabWebServer_welcome - Virtual Laboratory CyberVirtLab

[8] http://cyberneticsmws.fei.tuke.sk/cybervirtlab/Helikoptera - I/O form for helicopter model

[9] http://cyberneticsmws.fei.tuke.sk/cybervirtlab/Hydraulicky_system - I/O form for hydraulic model

[10] Zolotov, I. and Ligu, J. and Jadlovsk, A.:Remote and Virtual Lab CyberVirtLab,Proceedings of the 17-th EAEEIE Conference onInnovation in Education Engineering, Craiova,2006, Romania, pp.339-342, ISBN 978-973-742-350-4

BIOGRAPHIES

Anna Jadlovsk (doc., Ing., PhD.) received MSc.degree in Technical Cybernetics from the Faculty ofElectrical Engineering of the Technical UniversityKoice in 1984 and PhD degree in Automatization

and Control in 2001 from the same University. Shedefended her habilitation thesis Modeling andControl of Dynamic Processes Using ArtificialIntelligence in 2004. She works at the Department ofCybernetics and Artificial Intelligence, Faculty ofElectrical Engineering and Informatics, Technical

University in Koice as Associate Professor. Shelectures Simulation Systems, Introduction to Non-linear Systems, Control and Artificial Intelligence.Her main research activities include problemsadaptive and optimal controlespecially predictivecontrol with constrains for non-linear processesusing neural networks (Intelligent Control Design).She is an author of the articles and the contributions

published in the journals and internationalconference proceedings. She is also an co-author offour monographs.

Nikola Kabakov (Ing.) was born on 16.8.1980. He

received Ing. (MSc.) degree in Control Engineeringand Automation from the Faculty of ElectricalEngineering of the Technical University Koice in2004. Since September 2004 he has been as PhD.student in the Department of Cybernetics anArtificial Intelligence, Faculty of ElectricalEngineering and Informatics at TU in Koice. Hisresearch activities include predictive control of non-linear processes using methods of artificialintelligence.

Iveta Zolotov (doc., Ing., CSc.) graduated (MSc.)at the Department on the Technical Cybernetics TU

of Koice in 1983. Her CSc. thesis (1997) dealt withhierarchical image representations. She defended herhabilitation thesis Visualization as an Aided Tool inControl and Decision Making in 2001. She works atthe Department of Cybernetics and ArtificialIntelligence, Faculty of Electrical Engineering andInformatics, Technical University in Koice asAssociate Professor. She teaches Control andProcess Visualization and Computer Vision. Herspecialization is focused on simulation and modelingof technological processes, their monitoring andcontrol, supervisory control and data acquisitionsystems, human-machine interfaces and their design,

virtual and remote laboratories, as well as ImageSegmentation. She is an author of articles andcontributions published in journals and internationalconference proceedings. She is also co-author of onemonograph.

Predictive Control Algorithm

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