Predictive Function Control Based on Multi-model for pH Plant

Copyright © 2001 IFAC IFAC Conference on New Technologies for Computer Control 19-22 November 2001, Hong Kong

PREDICTIVE FUNCTION CONTROL BASED ON MULTI-MODEL FOR pH PLANT Zhi-huan Zhang** , Shu-qing Wang*

The National Lab. of Industrial Control Technology Zhejiang University 310027, P.R.China* The Collage of1nformation Science & Engineering , Ningbo Univ. 315211, P.R.China**

Abstract: A multi-model structure for predictive function control has been presented for a pH neutralisation process. The nonlinear pH process is partitioned into several local operating region. Within each region, a local predictive function controller is constructed based on the corresponding local linear model, their outputs are combined to form a global control action by using their membership functions . i.e. the global predictive function controller (GC-PFC) consists of a set of local model predictive function controller (LM-PFC) by weighting. The GC-PFC has been applied to quality control of a pH non linear process. Realistic simulation shows that GC-PFC is an attractive option for nonlinear MPC.CopyrighP2001 IFAC Key words:

Nonlinear predictive function control, Multi-model, operating regime,

the interpolation function ,

1.

pH nonlinear process.

needs of fast, highly accurate track system. Recently,

INTRODUCTION

PFC-controller begin to the application of slowly Predictive Function Control (PFC) has been presented

chemical industries process, such as batch reactors, pH

in the base of Model Predictive Control (MPC) and

neutralisation process and heat exchangers. Due to

obtained widespread use in the elastic industrial robot

enthalpic

(H.B.Kuntze et al., 1986)a tracking turret(D.Cuadrado

processes, . complete' modelling of enthalpic processes

et al. , 1991)weapon system guidance and control

is very difficult.Therefore, we focus on the use of a

(D.Cuadrado, P.Guerchet and S.Abu Ei Ata Doss, 1991)

relatively simple way to introduce non linear PFC.

rolling mill (J.M.Compas, P. D- ecarreau, G.Lanquetin,

Nonlinear PFC is designed by using multi-model.

J.L.Estival, N.Fulget, R.Martin and 1.Richalet, 1994).

Further,

Even though PFC was originally developed to meet the

simulation of a multi-model of a typical pH process.

227

processes

this

are

approach

well

IS

known

investigated

nonlinear

through

2.

Note that model M; will be dominating model when E~ This means that w;(
MULTI-MODELS

A typical choice for w;(
function .

non linear processes over a wide operating range is the use of a multi-models. Multi-models can briefly be

3.

PREDICTIVE FUNCTION CONTROLLER

described as follow: consider a process described by a

BASED ON MULTI-MODEL (GC-PFC)

state-space model in the following form The global operation of a non linear process is divided

x=f(x,u)

into several local operating regions. Within each local region, a controller is now designed about each of the

y=g(x,U)

local models. The output of each controller is then passed through the interpolation function which effectively generates a window of validity for each of

Let the operating point E be defined by

the individual controllers. The interpolated output are then summed and used to supply the control


define

an

operation

regime

a

commands to the process. The resultant GC-PFC structure is as shown in Fig. 1

and

subset

j nj

by

=0

The global controller output is obtained as following: 11'

This means that the operating set is

LWju; u=~ N

partitioned into disjunct operating regimes. To each operating regime, we allocate a local model (M;).

LW; ;= 1

Hence, we define N local models M t, . .. ,MN • A local model can often be simple since it is only valid in one

and Uj is the membership function and output of the local controller for the ith operating region. w;(y) is

operation regime (and possibly its neighborhood). In

given by

Wi

this work we will choose the local models as linear

w i (y)

state-space models with a structure based on first

=

NMi(y) L,.Mj(y)

principles insight of the process in question.

j=1

The local linear models are combined into a global nonlinear model (M) by a convex combination of weighting functions w I ( ) , . ..

, W N ( )

~Yi

= O.5(y;,!rulX

- Yi.min )

N

M = L,. w ; (
and the i-th regime is characterized by the intervals

LWi(
, y I.max , ] In addition, the parameter [y ,I,rnm'

the smoothness.

228

'J .,

i

determines

I T

...

r-~ Controller

~ Scheduler ~ ModelllPFCI

pH -set

... f--

l./

.

I

""-1 Weigh 1

1

...

I

Model:!IPFC 2

+1

...

I

~r

H~

""-1 Weigh 2

1

pHplant

I~

J~

...r=

~r

L

""-1 ModelNIPFC N

Fig. I

4.

~

I

1 WeighN

I

Predictive Function Controller Based On Multi-Model

LOCAL PREDICTIVE FUNCTION

parameters. The choice of the basis functions

CONTROL

defines the

input profile

predetermined Consider the following local ARMAX model n~

behavior

and can (smooth

signal,

controlling non linear systems.

Ym(k)= L(-aJYm(k-i)+ Lb;u(k-i)

as

The cost function to be minimized is:

representing the process behavior. The prediction is

J = L[y(k + j) - w(k + j)]2

obtained

adding

i=1

an

between

the

p

autocompensation

calculated as a nmction of the model

~bserved

and

differences

past

j=1

term output.

where w(k+j) is usually a first order approach to the known reference:

w(k + j) = c(k + j) - a k (c(k) - y(k»

y(k + j / k) = Ym(k + j) + e(k + j) The future control signal is stmctured as a linear combination of the basis function Bi , that are chosen according to the nature of the process and

In order to smooth the control signal, a quadratic factor of the form A[.6.u(k)f can be added to the cost function.

the reference:

u(k + j) =

for

instance). This can result in an advantage when

flh

;=1

assure a

I.!l; (k)B; (j) ;=1

In the case of SISO processes without constraints, the control law can be obtained as follows. First, the output is decomposed into free

Normally these functions are polynomial type: steps,

output and forced one, and the structurization of the

ramps or parabolas, as the majority of references

control signal is employed to give:

can be expressed as combinations of these functions . With this strategy a complex input profile can be specified using a small number of unknown

Y1(k + j) is the system free response, Yf (k + j)

229

is the system forced response.

Minimizing J with respect to the coefficients Now the coefficients of the vector /J. are

Now the cost function can be written as:

computed and the control signal, taking into

p

J = L[y(k + j) - w(k + j)f

accolmt the receding horizon strategy, is given by:

j=J

u(k) =

p

=L [~

T

YB

-

d(k + j) f

I,~i (k)Bi (0) i=1

j=1

5.

Where:

APPLICATION TO A pH NEUTRALISATION PLANT

The neutralisation of pH represents a highly non-linear process, and offers a suitable case study for the demonstration and evaluation of GC-PFC techniques. Fig.2 shows a schematic

d(k + j) = c(k + j) - ~j (c(k) - y(k»

of the plant.

- YJ (k + j) - e(k + j)

Fig.2 Schematic of pH neutralisation plant The process consists of weak concentration acid, base and buffer streams being continuously mixed within a

10

reaction vessel whose effluent pH is measured at a distance from the plant, introducing a time delay. The

8

aim is control the pH value of the OUtpllt stream by varying the inlet base flow re te (q2) . The outlet flow

4

rate (q4) is dependent upon the fluid height (h) within

5

10

15

20

25

30

the vessel and the position of the outlet valve, which is

q2

set manually. Ramping the base flow rate while recording the outlet stream pH value produced an estimate of the process titration curve, as shown in Fig.3.

Fig.3 Titration curve for pH process which the gain is nearly constant. Also, within each region, linear identification experiments

For this plant, the static gain between the base flow rate

suggested

and the pH (the slope of the titration curve) varies

sufficient to describe the dynamic process

considerably as these variables change. It can be seen

behavior. The local second-order, linear models

that three regions in

with three were constructed to give:

that a

second-order

model

K e-1os G (s) - _ _-:..1_ _I - (300S + 1)(5S + 1)

230

was

the pH neutralisation process. Fig.4 shows the results obtained from a series of step changes in pH from 4.0 to 10.0.

K e-1os G (5) - _ _ 2 - (300S + 1)(5S + 1)

..::....2- - -

K e-10s G (5)- _ _ 3 - (300S + 1)(5S + 1) ..::....3_ _ _

This GC-PFC strategy was test in simulation on 10r---~--~----.----r----.---~---'----~---,

8.5 :r:: Co

7

4L-~~--~----~--~----~--~--~----~---J

o

50

100

150

200

250

300

350

400

450

Time (min)

Fig. 4 Set-point tracking for local model based PFC and GC-PFC controller In this case, the GC-PFC controller shows a better performance for large operating regions.

H.B.Kuntze,

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Ata

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