ICICI014P - Widjiantoro, Predictive Control

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International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc.,

August 3 -5, 2005, Bandung, Indonesia

89

Nonlinear Model Predictive Control Containing

Neural Model and Controller for MIMO Process

Bambang L. Widjiantoro1)

, The Houw Liong2)

, Yul Y. N3)

, and Bambang SPA3)

1)

Jurusan Teknik Fisika FTI ITS Surabaya

Kampus ITS Sukolilo Surabaya Jatim Indonesia

E-mail: [email protected])

Departemen Fisika ITB Bandung3)

Departemen Teknik Fisika ITB Bandung

Jl. Ganesha 10 Bandung

Abstract Most of industrial processes are

characterized as multi input multi output (MIMO)

and nonlinear processes as well as influenced by

the disturbances that are detrimental to it. These

conditions give difficulties in the design of a proper

control system, which may overcome the above

characteristics. One of model based control

strategy that can be applied to improve the control

system performance is Nonlinear Model Predictive

Control (NMPC), which requires nonlinear model

allowing analog characteristics between the model

and its respective process

In view of nonlinear model predictive controls

restriction, it relies on the nonlinear model

derivation, which often requires simultaneous

mathematical equations, complex and somewhat

difficult, in particular for MIMO and nonlinear

processes. Apparently other restriction deals with

solution of nonlinear optimization, as consequence

of nonlinear model application, which often yields

local minima in its solution or even divergent along

with iterative time consuming computation.

The research proposes a new structure and

algorithm of nonlinear model predictive control by

integrating neural networks as nonlinear model and

controller. Nonlinear model can be developed using

neural networks without involving the complex

mathematical equations and detail information of

the process. The neural networks controller will be

able to eliminate a nonlinear optimization solution.

The current control signal is given by the output

neural networks controller, instead as the solution

of the optimization problems and no nonlinear

optimization must be solved at every sampling time.

The neural networks controller is also able to

generate multivariable control signals directly

without iterative computation. Therefore, the

computational time for determining control signal

can be reduced. The type of neural networks in the

research is multilayer perceptron (MLP), which

comprises input layer, hidden layer with tangent

hyperbolic activation function and output layer with

linear activation function.

The proposed Nonlinear Model Predictive Control

(NMPC) is applied on cascaded tanks process to

control its level. The results show that the proposed

NMPC is able to yield good control system

performance and to overcome process

characteristics mentioned above. The process

output can follow the desired set point adequately

well and generate the smooth control signals

profile. Moreover, with the new NMPC scheme, the

requirements of nonlinear model to represent the

MIMO process can be simplified and computation

time to obtain the control signals can be reduced.

Keywords Nonlinear Model Predictive Control,

optimization, neural networks, multilayer

perceptron (MLP), MIMO process.

I. INTRODUCTION

Most of industrial processes are

characterized as multi input multi output

(MIMO) and nonlinear processes as well as

influenced by the disturbances that are

detrimental to it. These conditions give

difficulties in the design of a proper control

system, which may overcome the above

characteristics. Application of conventional

control system often produces worsecontrol performances since it depends on

the assumptions such as a linearization and

restricted of operation range. Attempts for

increasing the control performances have

been done by developing model based

control system, where a model is used

explicitly in determining the control signal.

One of model based control strategy that


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International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005

Proc., August 3 -5, 2005, Bandung, Indonesia

90

can be applied to improve the control

system performance is Nonlinear Model

Predictive Control (NMPC) which requires

nonlinear model allowing analog

characteristics between the model and its

respective process. The nonlinear model

predictive control algorithm offers thepotential for improved process control

performance (Henson, 1998).

The neural networks offer many

advantages to build up the nonlinear model

of the process. Their ability in nonlinear

mapping between input output is the

potential benefits in the development of

nonlinear model. The nonlinear model can

be built without complicated mathematical

equations and detail information about the

process.

When the predictive control strategy is

based on nonlinear model, independently

of the nature of model, the prediction

equation cannot be solved explicitly, as in

the linear case, and an iterative solution of

the performance function evaluating the

future behaviour of the system is required.

At every sampling time, the current

manipulated variable is calculated using an

optimization procedure, which determines

the optimal profile control actions that

minimize the objective function. Hence,

the use of nonlinear model allows the

development of predictive control strategy

for nonlinear dynamic process, but it also

implies that at every sampling time a

nonlinear optimization problem should be

solved. This is a potential drawback of

those control strategy because the solution

of optimization problem are usually

computational laborious, particularly in

large processes. Most nonlinearoptimization algorithms use some form of

search technique to scan the feasible space

of the objective function until the extreme

point is located. This search is generally

guided by calculation of the objective

function and/or its derivatives and it

implies a large computational effort

because several iterations should be carried

out to reach the extreme point. As

consequence, it worthwhile developing

nonlinear model predictive control strategy

that require less computational effort.

Some researchers (Rohani et.al, 1999 and

Fernholz et.al, 2001) have used neuralnetworks based nonlinear model predictive

control with MISO (Multi Input Single

Output) structure for MIMO process. This

structure, of course, requires the numerous

models to represent the entire process and

also need large memory.

The research proposes a new structure and

algorithm of nonlinear model predictive

control by integrating neural networks as

nonlinear model and controller. Nonlinear

model can be developed using neuralnetworks without involving the complex

mathematical equations and detail

information of the process. The neural

networks controller will be able to

eliminate a nonlinear optimization solution.

The current control signal is given by the

output neural networks controller, instead

as the solution of the optimization

problems and no nonlinear optimization

must be solved at every sampling time. The

neural networks controller is also able to

generate multivariable control signals

directly without iterative computation.

Therefore, the computational time for

determining control signal can be reduced.

The paper is organized as follows. Section

two describes the concept of nonlinear

model predictive control. Section three

discusses about the development of

nonlinear process model using neural

networks. Section four describes about the

development of neural networks controllerin nonlinear model predictive control

algorithm. Section five will illustrate the

simulation of the proposed control strategy

to MIMO process.


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II. NONLINEAR MODEL PREDICTIVE

CONTROL (NMPC)

Model predictive control (MPC) refers to a

class of control algorithms in which a

dynamics process model is used to predict

and optimize process performance. In thebeginning, the MPC algorithm is based on

linear dynamic models and therefore is

referenced by term linear model predictive

control. Although often unjustified, the

assumption of process linearity greatly

simplifies model development and

controller. Many processes are sufficiently

nonlinear to preclude the successful

application of linear model predictive

control algorithm. This has led to the

development of nonlinear model predictive

control in which a more accurate nonlinearmodel is used for process prediction and

optimization.

The concept of Nonlinear Model Predictive

Control (NMPC) is depicted in figure 1

(Garcia et.al, 1989).

Controlsignals,u(k)

Reference trajectory

Model

Setpoint,r

Process output,

y(k)

1k Nt + Nut k + 2k Nt +

(k)y

Figure 1. The concept of NMPC

The idea behind model predictive control

(MPC) is at each iteration to minimize the

objective function of the following type:

( ) ( ) = == = +++=n

l

N

kl

m

j

N

Nijj

u

)kt(u)it(y)t(r)t(J1 1

21

2

2

1

(1)

N1 and N2 denote minimum and maximum

prediction horizon.

Nu denotes control horizon.

denotes weighting factor for controlsignal.

y is the prediction of future process output

from the model.

n and m denote the number of output

process and signal control, respectively.

III. THE DEVELOPMENT OF

NONLINEAR PROCESS MODELUSING NEURAL NETWORKS

A multi layer perceptron (MLP) has been

chosen for modelling purposes of MIMO

process. The basic MLP networks is

constructed by ordering the neurons in

layers, letting each neuron in a layer take

as input only the outputs of units in the

previous layer or external inputs. If the

networks have two such layers of neurons,

it refers to as a two layer networks, if it has

three layers it is called a three layer

networks and so on. Figure 2 illustrates the

example of three layer networks including

input layer, hidden layer and output layer.

The mathematical formula expressing what

is going on in the MLP networks takes the

form:

+

+=

= =

hn

1j0,i

n

1l0,jll,jjj,iii Wwwf.WFy

(2)

Cybenko (1989) has shown that all

continuous function can be approximated

to any desired accuracy with neural

networks of one hidden layer of hyperbolic

tangent hidden neuron and a layer of linear

output neuron.

)(f1

)(f2

)(F1

)(F2 1

2

3

0,iw

0,iW

1y

2y

Input layer Hidden layer

Output layer

j,iw j,iW

Figure 2. Multilayer perceptron


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In order to determine the weight values, a

set of examples of how the outputs, iy ,

should relate to the inputs, i must be

available. The task of determining the

weights from these examples is called

training or learning. The aim of training

procedure is an adjustment of the weights

to minimize the error between the neural

networks output and the process output

(also called by target). A learning

algorithm is associated with any change in

the memory as represented by the weights;

learning does not in this sense imply a

change in the structure of the memory.

Therefore, learning can be regarded as a

parametric adaptation algorithm. The

learning algorithm is Levenberg-Marquardt

method. This algorithm requires theinformation of gradient and Hessian

matrices. The convergence will generally

be faster than for the back-propagation

algorithm. The detail derivation of

Levenberg-Marquardt method can be seen

in Norgaard et.al (2000).

IV. THE DEVELOPMENT OF NEURAL

NETWORKS CONTROLLER

In the nonlinear model predictive control

(NMPC), the controller has a task toreplace the optimization problem for

generating the control signal. Thus, there is

no nonlinear optimization when the neural

networks used in NMPC strategies. The

following section will derive the algorithm

of the neural networks controller in NMPC.

The algorithm was derived and modified

from Galvan et.al (1997) and Norgaard

et.al (2000).

The derivation of control algorithm using

neural networks controller can be viewedas the analog of equation (2). Hence, the

control signal u(k) minimizing the

objective function in equation (1) can be

written as follows :

)nmspm

spU,,U,Y,,Y,Y,Y,,YgU LLL 111=

(3)

where g is a mapping function that

determines the solution of optimization

problem given by the objective function in

equation (1) and y is the model output.

If the function g was known, the expressiongiven by equation (3) would provide at

every time k the control signal that must be

applied to the system to reach the desired

control objective, when a predictive control

strategy is employed. The problem of

finding an expression for the function g can

now be interpreted as functional

approximation problem. Based on

approximation capabilities of neural

networks, one can decide to approximate

the functional g by neural networks

controller.

The learning of the predictive neural

controller consists in determining the

weight parameter set (Wc) such that the

control law, given by equation (3),

provides a predictive performance

controller. The training procedure is carry

out in on line approach or called by

specialized training. In specialized training,

the neural networks controller is trained to

minimize the objective function so thesystem output will follow the reference

signal closely. The learning algorithm used

in this research was back-propagation

algorithm.

Suppose Wc be a weight parameter of the

neural networks controller and set point

remains constant along the prediction

horizon, the gradient of the error function

given by equation (1) with respect to the

parameter can be written as follows:

( ) = = = =

+

+++=

m

1j

N

Ni

n

1l

N

1k c

k,lk,l

c

jjj

c

2

1

u

W

UU2

W

)it(y)it(y)it(r2

W

)t(J

(4)

with: )kt(uU lk,l +=

Based on the above description, the neural

predictive controller parameters can be


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updated at every sampling time using the

following learning rule:

c

oldc

newc

W

JWW

+= (5)

V. SIMULATION

In the following section, the containing

neural model and controller will be applied

to the MIMO process. The proposed

control system is applied to the cascade

tanks, which adopted from Babuska

(1998). The cascade tanks is depicted in

figure 3.

X

1 2

3 4

h1 h2

h3 h4

q1 q2X

Figure 3. The cascade tanks

The goal of control system is to maintain

level in tanks 1 and 2 by adjusting the flow

rate q1 and q2.

The first step in designing of NMPC is to

develop the process model by system

identification. Figure 4 illustrates the input-

output data.

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

(x10

l/s)

-5

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

0 100 200 300 400 500 600 700 800 900 10000

1

2

0 100 200 300 400 500 600 700 800 900 10000

1

2

Input - output data of cascade tanks

Cacah

(m)

(m

)

(m)

(m)

(x10

l/s)

-5 q1

q2

h1

h2

h3

h4

Figure 4. Input output data

The system identification with neural

networks algorithm was done in NARX

(Nonlinear Auto Regressive with

eXogenous input) or series parallel model

structure. The history length for both the

input signal and the output signal was 4. It

means that the input spaces consist thepresent and three past values of the input-

output signal. Thus, the input layer has 24

variables as the input of regressor. The

hidden layer consists of 8 neurons with

hyperbolic tangent function. The result of

neural networks model in identi-fication of

the process is shown in figure 5. From this

figure, it can be seen that the neural model

can identify and anticipate behaviour of the

process satisfactorily.

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

0 100 200 300 400 500 600 700 800 900 10000

1

2

0 100 200 300 400 500 600 700 800 900 10000

1

2

Cacah

dash : model output

soid : process output

h1

h2

h3

h4

Figure 5. Identification of neural model

After developing the nonlinear process

model, the next step is to determine the

neural networks controller to generate the

optimal control signal of the process. The

structure of the neural networks controller

was also multi layer perceptron (MLP),

with input, hidden and output layer, and

used the back-propagation learning

algorithm to up date the weights of neuralcontroller.

The neural controller should be trained in

several times (epoch) until the process

output close to the desired set point and

will give the minimum value of objective

function. Then, the best value of weights

was chosen as the neural networks


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controller to generate the control signal for

the process. Simulation of NMPC

containing neural model and controller is

illustrated in figure 6, while figure 7 shows

the control signals that generated by neural

controller.

0 500 1000 1500 2000 2500 300015

20

25

30

35

40

45

0 500 1000 1500 2000 2500 300078

80

82

84

86

88

Performance of NMPC Containing Neural Model and Controller

Output #1

Output #2

Samples

(cm)

(cm)

Figure 6. Performance of the control

system

The good set point tracking was achieved

in figure 8. The process output can track

the set point without offset. Moreover, the

neural networks controller can also

generate the smooth control signal for the

process.

0 500 1000 1500 2000 2500 3000-0.1

0

0.1

0.2

0.3

0.4

0 500 1000 1500 2000 2500 30000.2

0.4

0.6

0.8

1

Control signal

Samples

q1(x1e-5l/s)

q2(x1e-5l/s)

Figure 7. Control signals of NMPC

VI. CONCLUSION

The algorithm of NMPC containing neural

model and controller for MIMO process

has been presented in the paper. With this

structure, the requirements of nonlinear

model to represent the entire process canbe simplified using single model and

solution of nonlinear optimization

procedure to generate the control signals

can be eliminated.

The proposed control system was applied

to the cascade tanks and can yield the good

control system performances.

REFERENCES

[1]. Babuska R (1998), Fuzzy Modelling andIdentification Toolbox Users Guide,

http://lcewww.et.tudelft.nl/~babuska.

[2]. Carlos E. Garcia, David M. Prett, ManfredMorari, (1989), Model Predictive Control:Theory and Practice a Survey, Automatica,

25, pp. 335-348.

[3]. Cybenco G (1989), Approximation bySuperpositions of A Sigmoidal Function,Mathematics of Control, Signals and Systems,

2(4), 303 314.

[4]. Fernholz G, Rossman V, Engell S, BredehoeftJ.P (2001), System Identification Using Radial

Basis Function Nets for Nonlinear Model

Predictive Control of A Semibatch ReactiveDistillation Column, Internal Report,

Laboratory of Process Control, University of

Dortmund.

[5]. Galvan I.M, Zaldivar J.M (1998), Applicationof Recurrent Neural Networks in BatchReactors. Part II: Nonlinear Inverse and

Predictive Control of The Heat Transfer Fluid

Temperature, Chemical Engineering and

Processing, 37, 149 161.

[6]. Henson,M.A (1998), Nonlinear ModelPredictive Control: Current Status and Future

Directions, Computers and Chemical

Engineering, 23, pp. 187-202.

[7]. Norgaard M, Ravn O, Poulsen N.K, HansenL.K (2000), Neural Networks for Modellingand Control of Dynamic Systems, Springer

Verlag.

[8]. Rohani S, Haeri M, Wood H.C (1999),Modelling and Control of A ContinuousCrystallization Process Part II: Model

Predictive Control, Computers and Chemical

Engineering, 23, 279 286.

ICICI014P - Widjiantoro, Predictive Control

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