MULTI-OBJECTIVE PREDICTIVE CONTROL: APPLICATION

FOR AN UNCERTAIN PROCESS

Anes Bedoui

Laboratoire d’Analyse et Commande des Systèmes (ACS), ENIT, Tunisie.

Faouzi Bouani, Mekki Ksouri

Institut National des Sciences Appliquées et de Technologie, Tunis, Tunisie

Keywords: Predictive control; multi objective optimization; laboratory process; weighting functions method.

Abstract: This paper deals with the application of the Multi Objective Generalized Predictive Control (MOGPC) to

level control in a laboratory process. The major characteristic of the considered plant is that the manual

draining vane can take many positions causing changes in plant dynamics and strong disturbances in the

process. The controller is based on a set of Controlled Auto Regressive Integrated Moving Average

(CARIMA) model. The Recursive Least Squares (RLS) algorithm is used to estimate each model

parameters. The control law is obtained by minimizing a multi objective optimization problem. The

weighting sum approach is considered to formulate the control problem as a single criterion optimisation

one. The real time control system implementation confirms the opportunity of using the MOGPC scheme to

an uncertainty system.

1 INTRODUCTION

The Generalized Predictive Control (GPC) principle

consists in calculating the control input by the

minimization of a cost function over a future time

horizon under certain process constraints (Clarke et

al. 1987). Since the constraints on the input and the

output signals can be explicitly taken in account by

the GPC, this approach of control has attracted the

attention of many control researchers and industrials

(Boucher and Dumur 1996, Ben Abdennour et al.

2001).

Dynamics of industrial plants are usually not

completely known and are subject to change from

time to time. The complexity of industrial process

makes difficult their representation by only one

model. Consequently, the strategy which consists to

characterize the system with several models, every

model possessed its own validity domain, has been

developed (Brian and Bequette 2001). The strategy

of multi model control suffers from the difficulty of

determination model’s validity especially in noisy

systems. Another technique can be used to handle

nonlinear systems is the robust control design.

Robust controllers explicitly consider the

parametric variation in the process model for

calculating the control law (Gutierrez and Camacho

1995, Oliveira et al. 2000, Brdys and Chang, 2002).

The introduction of the uncertainty parameters leads

to the resolution of a min-max optimisation problem

which is hard to solve (Ramirez et al. 2002).

This paper presents the application of multi

objective predictive controller to an uncertain plant.

The major characteristic of the considered plant is

that the manual draining vane can take many

positions causing changes in plant dynamics and

strong disturbances in the process. Each operating

region can be modelled with a CARIMA model. The

Recursive Least Squares (RLS) algorithm is used to

estimate the model parameters. The control law is

obtained by minimizing a multi objective

optimization problem. The weighting sum approach

is considered to formulate the control problem as a

single criterion optimisation one.

This paper is organized as follows. Section 2

presents the description of the process. Section 3 is

reserved to the multi criteria generalized predictive

control based on a set of CARIMA model; the use of

the weighting functions method is also described.

233

Bedoui A., Bouani F. and Ksouri M. (2005).

MULTI-OBJECTIVE PREDICTIVE CONTROL: APPLICATION FOR AN UNCERTAIN PROCESS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 233-238

DOI: 10.5220/0001181602330238

Copyright

c

SciTePress

Section 4 gives the results obtained in real time from

the water level regulation. The final section of the

paper presents the conclusion.

2 PROCESS DESCRIPTION

The process is schematically depicted in figure 1.

The main goal is to control in closed loop the level

in tank 1 by adjusting the liquid flow rate with the

electric actuator pump. The sampling time period is

fixed to 4s. While exploiting different step

responses, we give in figure 2 the steady state

characteristic. Then, the relation between the level

and the flow rate is non linear. It’s well known that

the capacity dominated process can be described by

a first order linear differential equation about a

desired operating level (William et al. 2000).

With the considered process, the manual

draining vane can take different positions then the

system can be modeled with an uncertainty first

order model. In order to identify the model

parameters, we have recorded two files of measures

giving the evolution of the water level for a shape of

crenel control. The obtained data, for the first and

the second positions of the draining vane, are

respectively represented in Figure 3 and Figure 4.

The presence of a numeric model is a necessary

condition for the development of the predictive

control, since it permits to calculate the predicted

output on a finished horizon. Consider the single

input single output process, which may be described

by the CARIMA model as follows (Clarke et al.

1987):

)()()()(

11

kuqBkyqA ∆=∆

−−

, (1)

where y(k) is the output signal and u(k) is the input

signal

. The term

1

1

−

−=∆ q

corresponds to an

integral action which permits the annulment of the

permanent regime error.

)(

1−

qA

and

)(

1−

qB

are

polynomials of degrees n

a

and n

b

in backward shift

operator q

-1

:

a

n

na

qaqaqA

−

−−

+++= ...1)(

1

1

1

, (2)

b

n

nb

qbqbqB

−

−−

++= ...)(

1

1

1

. (3)

For each operating region, a local CARIMA model

is determined. The model parameters are identified,

off-line, by using the (RLS) algorithm:

)()1(

ˆ

)()( kkkyke

T

φθ

−−=

(4)

)()()()1(

ˆ

)(

ˆ

kekkPkk

φθθ

+−=

(5)

)()1()(1

)1()()()1(

)1()(

kkPk

kPkkkP

kPkP

T

T

φφ

φφ

−+

−−

−−=

(6)

where e(k) is the prediction error;

[]

T

nbna

bbaak ......)(

11

=

θ

is the parameter vector;

[]

T

nbkukunakykyk )(...)1()(...)1()( −−−−−−=

φ

is the observation vector and P(k) is the covariance

matrix.

The input output data, considered in this work,

belong to two different working points. These data

given by figures 3 and 4 lead, respectively, to the

following models:

[]

T

k 0372.09864.0)(

1

−=

θ

and

[]

T

k 0301.09803.0)(

2

−=

θ

(7)

Personal computer

pump

Tank 2

Tank 1

Tank Level

Data

acquisition

device

Manual draining

vane

Figure 1: Laboratory process of level control

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

234

Figure 2: Non linear steady state characteristic

Figure 3: The water level y(k) and the flow rate u(k) (first

position of the draining vane).

Figure 4: The water level y(k) and the flow rate u(k)

(second position of the draining vane)

3 CONTROL AND DESIGN

For systems that present several modes of working,

different models can be built which are specific to

every mode of particular working of the system. We

consider the set of models:

{}

)(,...),(

1

kkM

n

θθ

=

(8)

where n is the number of possible models. A

multicriteria optimisation problem can be formulated

as follows:

()

n

U

JJ ...,,min

1

∆

(9)

where J

i

is formulated by using the model )(k

i

θ

.

3.1 Single criterion GPC

The objective of the generalized predictive control

results in the minimization of the criterion under the

following analytic relation (Clarke et al. 1987):

()()

()

()()

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∑∑

+∆++−+=

=

−

=

2

1

1

0

2

2

/

ˆ

2

1

N

j

u

N

j

c

jkujkykjkyJ

λ

(10)

where N

2

is the prediction horizon, N

u

is the control

horizon,

λ

is the control increments weighting

factor, )(ky

c

is the set point,

()

kjky /

ˆ

+ ,

],1[

2

Nj ∈

, is the j step ahead predicted output and

)(ku

∆ is the control increment.

The minimization of the criterion requires the

computation of the predicted output over the

prediction horizon i.e.:

()

kjky /

ˆ

+ ,

],1[

2

Nj ∈

. This

can be achieved by the model of the process. The j

step ahead predicted output is given by the following

relation (Clarke et al. 1987, Ben Abdennour et al.

2001).

)/()/()/(

ˆ

kjkykjkykjky

a

f

+++=+

(11)

where

)1()/( −+∆=+ jkuQkjky

j

f

and

()

)(1)/( kyGkuRkjky

jja

+−∆=+

where Q

j

, R

j

and G

j

are polynomials solutions of

Diophantine equations.

On a prediction horizon N

2

and on a control horizon

N

u

, it is possible to transcribe equation (11) under

matrix shape:

a

UHYHUQY ∆++∆=

21

ˆ

(12)

where

Y

ˆ

is the vector of the predicted output,

U

∆

is

the vector of the present and the future control

increments. The vector

Y

is formed by the present

MULTI-OBJECTIVE PREDICTIVE CONTROL: APPLICATION FOR AN UNCERTAIN PROCESS

235

value and the old values of the output, the vector

a

U∆

is formed by the old increments of the control,

matrices

H

1

, H

2

and

Q

are formed, respectively, by

the coefficients of polynomials

j

G ,

j

R and

j

Q .

Based on these notations, we can write the criterion

J in the following matrix form

UUYYYYJ

T

c

T

c

∆∆+−−=

λ

]

ˆ

[]

ˆ

[

(13)

where

Y

c

is the vector formed by the future set point

sequence.

The optimal solution is obtained while annulling the

gradient of

J in relation to the vector of the

increment control:

)(][

1

lc

T

N

T

YYQIQQU

u

−+=∆

−

λ

(14)

where I

Nu

is a unity matrix of dimension (N

u

,N

u

) and

al

UHYHY ∆+=

21

.

The GPC is a receding control strategy, only the first

element of the vector

U

∆

is used to compute the

control to be applied to the process.

)1()1()( Ukuku ∆+−=

(15)

3.2 Multi objective GPC

The objectives are often conflicting or competing. A

powerful method for dealing with multiple

objectives is the Pareto optimality concept. Multi

objective problems usually have no unique solution,

but a set of non dominated solutions, known as the

Pareto optimal set (Xin et al. 2004). In the case of

non convex objectives, genetic algorithms are used

to solve the multi objective problems (Colette and

Siarry 2002, Silva and Fleming 2002, Andrès-Toro

et al. 2002). In this work, local models are linear,

consequently, the criterion J

i

is convex in the

controller parameters and it can be efficiently solved

by the weighting sum approach. The weighting

functions method transforms the multi criteria

problem to a single criterion one as follows (Colette

and Siarry 2002).

∑

=

=

n

i

ii

JwJ

1

(16)

where

∑

=

=

n

i

i

w

1

1 and 0≥

i

w . (17)

The weighting sum approach consists to take all

objectives in a single aggregating function. The

modification of the w

i

values that respect the

constraint (17), leads to the Pareto optimal set. Since

the optimization problem is convex, the solutions are

uniformly repatriated on the Pareto surface. In this

work, we have considered the optimal control, the

value from the Pareto set that gives the minimum of

the sum of all objectives.

The following algorithm is used to compute the

optimal sequence of control:

1- Take w

1

=0 and fix the step

∆

w

1

.

2- Choose w

i

, i=2,…,n that verify the relation

(17),

3- Compute

U

J

i

∆∂

∂

using (14) and

U

J

∆∂

∂

.

4- Compute the control sequence

)(kU∆

.

5- Increment

w

1

(w

1

=w

1

+

∆

w

1

), if w

1

<1, return to

step 2.

6- Take

opt

U∆ that gives the minimum of the

sum of all criteria.

7- Compute the control law as follows:

)1()1()(

opt

Ukuku ∆+−=

In this algorithm, the size of the Pareto optimal set

depends on the choice of the step

∆

w

1

. In this work,

we have used

∆

w

1

=0.1, then at each sampling time,

we compute 10 solutions which form the Pareto set.

4 EXPERIMENT RESULTS

The multi objective predictive control scheme based

on the CARIMA model was applied to level control

in a laboratory process. In order to compare the

behavior of the standard GPC and the MOGPC in

presence of non-stationary process, we have fixed

the draining vane of the process described in section

2, in the first position during 300 sampling period

then we turn in the second position. The constraints

imposed on the input signal are as follows:

mnlku /3.2)(0 ≤≤ (18)

A discrete PID regulator can be given by the

following relation (Borne

et al. 1993):

)2()1()21(

)()1()1()(

−+−+−

+++−=

k

T

k

k

T

k

k

T

k

k

T

Kkuku

e

d

e

d

e

d

i

e

P

εε

ε

(19)

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

236

where K

p

is the proportional gain; k

i

is the integral

constant;

k

d

is the derivate constant; T

e

is the

sampling time period; and

ε(k) is the error between

the set point and the output signal. The PID

parameters are computed based on the step response

of the system and the Takahashi method (Borne

et

al.

1993).

The results obtained with the PID controller are

shown in Figure 5. The control signal presents many

fluctuations and the tracking error is not zero. In this

work, we have considered fixed PID controller

parameters. One can ameliorate the closed loop

performances by using an adaptive PID controller to

cope with the process dynamic changes. The results

obtained with the GPC are shown in Figure 6. In this

case, the controller is based on a nominal model

which is obtained by:

))()((

2

1

)(

21

kkk

θθθ

+=

(20)

It’s clear, from this figure, that the nominal model

assures good performances in closed loop. The

evolutions of the output/input and the set point

signals, in the case of the MOGPC, are given in

Figure 7. Obviously, we notice that performances in

terms of the tracking error and the variance of the

control signal are substantially ameliorated. Table 1

gives the variance of the control (

V) for the three

controllers.

()

∑

=

=

N

k

ku

N

V

1

2

)(

1

(21)

where N is the number of data measurement.

The variance obtained with the MOGPC is the

lower, because the control signal obtained with this

controller has few fluctuations compared to those

obtained with the GPC and the PID controllers.

Table 1: Variance of the control

Controller V

PID controller 1.6108

GPC (single criterion) 1.4546

GPC (multi objective strategy) 1.1248

Figure 5: PID controller (k

p

=0.5; k

i

=3; k

d

=0.1)

Figure 6: GPC (single quadratic criterion)

(N

2

= 7; N

u

= 1; 1=

λ

)

MULTI-OBJECTIVE PREDICTIVE CONTROL: APPLICATION FOR AN UNCERTAIN PROCESS

237

Figure 7: MOGPC (multi objective strategy)

(N

2

= 7; N

u

= 1; 1=

λ

)

5 CONCLUSIONS

This paper has presented the multi objective

predictive control. The process is characterized by a

set of CARIMA model. Since considered models are

linear, performance criteria are convex.

Consequently, the weighted sum approach is used to

compute the Pareto optimal set. An application of

the studied strategy to a nonlinear model plant has

been also presented.

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