variable structure control applied to chemical processes

7/29/2019 Variable structure control applied to chemical processes

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Variable structure control applied to chemical processes

with inverse response

Oscar Camacho *, Rube n Rojas1, Winston Garca2

Grupo de Investigacion en Nuevas Estrategias de Control Aplicadas (GINECA), Postgrado en Automatizacion e Instrumentacion,

Facultad de Ingeniera, Universidad de Los Andes, Merida, Venezuela

Abstract

This article proposes the use of a sliding mode controller based on a rst-order-plus-deadtime model of the system

for controlling higher-order chemical processes with inverse response. The controller has a simple and xed structure

with a set of tuning equations as a function of the characteristic parameters of the rst order-plus-deadtime model. The

controller performance was judged via computer simulations using linear and nonlinear models of chemical processes

with inverse response.# 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Inverse response processes; Sliding mode control; First-order-plus-deadtime model

1. Introduction

A system is said to be an inverse response or a

non-minimum phase process if at least one of the

zeros of the transfer function is located in the

closed right half plane. It is well known that non-

minimum phase systems oer diculty in applying

feedback control. Furthermore, there exist system

uncertainties that include modeling errors, unmo-

deled dynamics and disturbances that become chal-

lenging control problems in industrial processes.

These uncertainties arise from an imperfect

knowledge of the system, causing degradation of

the control system. Conventional controllers are

not suciently versatile to compensate for all

dynamical complexities of these processes. These

uncertainties create a need for a generalized

methodology for dealing with nonlinear processes

with inverse response. Sliding mode control

(SMC) is appropriate for just such a purpose.

Recently, the sliding mode controller (SMCr)

from a rst-order-plus-deadtime (FOPDT) model

of the actual process has been used to control

chemical processes with high-order-plus-deadtime

transfer functions, by Camacho et al. [13]. This

article extends the previous work and explores the

viability of applying the same SMCr to inverse

response processes, the overall idea is to develop a

general controller that can be used for a broad class

of industrial processes. This article is organized as

follows. Section 2 gives a brief review of SMC.

Section 3 shows the procedure used to obtain the

controller equation and the set of tuning equations,

as rst estimates. Section 4 shows the simulation

studies to establish the robustness of the SMCr

ISA

TRANSACTIONS1

ISA Transactions 38 (1999) 5572

0019-0578/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.P I I : S 0 0 1 9 - 0 5 7 8 ( 99 ) 0 0 0 0 5 - 1

* Corresponding author. Tel: +58-74-402-891; fax: +58-74-

402-890; e-mail: [email protected] E-mail: [email protected] E-mail: [email protected]


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against modeling errors, and the controller per-

formance when it is applied to a nonlinear model

of a reactor with inverse response [4]. This test was

done in presence of noise, time delays and dis-turbances. At last, the conclusions are presented.

2. Sliding mode control

Sliding Mode Control is a technique derived

from variable structure control (VSC) which was

studied originally by Utkin [5]. This kind of con-

trol is particularly appealing for a broad class of

systems, due to its ability to deal with non-

linearities, time-variance, as well as uncertainties

and disturbances in a direct manner, in the face ofmodeling imprecisions. In VSC, the control can

modify its structure. The design problem consists

of selecting the parameters of each structure and

dening the traveling logic. The rst step in SMC

is to dene a sliding surface, St=0, along whichthe process output can slide to nd its desired nal

value. In general, the sliding surface represents the

system behavior during the transient period,

therefore, it must be designed to represent a

desired system dynamics [6]. The sliding surface

divides the phase plane into regions where theswitching function St has dierent sign. Thestructure of the control system is intentionally

altered as its state crosses the sliding surface in the

phase plane in accordance with a prescribed con-

trol law. So, the second step is to design the con-

trol law such that any state outside of the sliding

surface be driven to reach the surface in nite time

and keep on it. Fig. 1 depicts the SMC objective.

There are many options to select the sliding

surface, in our case St was selected as an inte-gral-dierential equation acting on the tracking

error [7].

St d

dt l

nt0

etdt I

et is the tracking error between the referencevalue (set point) and the measured output process,

n is the system order, and l is a tuning parameter

which is selected by the designer. It determines the

performance of the system on the sliding surface.

The control objective is to ensure that the con-

trolled variable be equal to its reference value at

all times, meaning that et and its derivatives must

be zero. The problem of tracking a reference valuecan be reduced to that of keeping St at zero.Once, St=0 is reached, it is desired to make

dSt

dt 0 P

(the sliding condition), to guarantee the value of

St at zero. After the sliding surface has beenselected, the control law must be designed to

satisfy the St=0 condition. The control law,Ut, can be written as follows,

Ut UCt UDt Q

where the rst additive part, UCt, is continuousand the second one, UDt, is discontinuous. Thecontinuous part is given by

UCt fXtY Rt R

where fXtY Rt is determined using theequivalent control procedure [5], in accordance

with the desired motion of the sliding mode.

The discontinuous part, UDt, is nonlinear andrepresents the switching element of the control

law. This part of the controller is discontinuous

across the sliding surface. Mainly, UDt isdesigned based on a relay-like function (i.e.

UDt signSt)), because it allows for chan-ges between the structures with a hypothetical

innitely fast speed. In practice, however, it is

impossible to achieve the high switching control

because of the presence of nite time delays for

control computations or limitations of the physi-

cal actuators causing chattering around of the

sliding surface [57]. The aggressiveness to reach

the sliding surface depends on the control gain (i.e.

), but if the controller is too aggressive it can

collaborate with the chattering. To reduce the

chattering, one approach is to replace the relay-

like function by a saturation or a sigma function

(see Fig. 2) which can be written as follows:

UDt KDSt

jStj S

56 O. Camacho et al. / ISA Transactions 38 (1999) 5572


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where KD is the tuning gain which is responsible

for the reaching mode, normally determined by

the Lyapunov stability criterion, and is a tuning

parameter used to reduce the chattering problem

[7,8]. The last approach was selected to design the

proposed controller. To summarize, the SMCr has

two parts. A discontinuous part, Eq. (5), respon-

sible for guiding the system to the sliding surface,

and a continuous part, Eq. (4), which is responsible

for keeping the controlled variable on the refer-

ence value.

3. SMCr design for inverse response processes

This section shows the design of SMCrs based

on two models of inverse response processes. The

main idea behind this approach is to show that the

Fig. 1. Graphical interpretation of sliding mode control.

Fig. 2. Chattering reduction using a saturation function (a: =0; b: =0.01; c: =0.1; d: =1.0).

O. Camacho et al. / ISA Transactions 38 (1999) 5572 57


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controller obtained based on the non-minimum

phase model of the process, generates an unstable

controller, which creates the need for a dierent

approach to obtain a stable controller.

3.1. SMCr based on a non-minimum phase model

of the process

Fig. 3, shows a typical step response for an

inverse response systems. The simplest approx-

imation for this kind of system can be done using

a rst order model, as follows:

Gs K(1s 1

(s 1T

or

Xs

Us

K(1s 1

(s 1U

where Xs, is the controlled variable and Us isthe controller output. Now, the continuous part of

the controller can be obtained applying the

equivalent control procedure [5].

First, Eq. (7) can be written in dierential equa-

tion form, as follows,

(dXt

dt Xt K Ut (1

dUt

dt

V

then, from Eq. (1), the sliding surface is obtainedfor n=1

St et l

etdt W

equating the rst derivative ofSt to zero (slidingcondition)

dSt

dt

det

dt let 0 IH

and replacing the well known approximation,

[9,10]

det

dt

dXt

dtII

Eq. (10), can be written as follows:

dSt

dt

dXt

dt let IP

then, solving Eq. (8) for the rst derivative of Xt

and adding Eq. (12)

Fig. 3. Typical step response of inverse response systems.



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Xt

( let

K

(Ut (1

dUt

dt

IQ

and going back to Laplace Transform domainXs

( les

K

((1sUs Us IR

solving for Us, the continuous part of the con-troller is obtained

UCs (

K

Xs(

lesh i

(1s 1IS

which represents an unstable controller. Similar

results are obtained for higher order non-mini-mum phase linear model approximations.

Therefore, as has been shown, the direct use of

the conventional sliding mode control theory, [57],

to an inverse response process model produces an

unstable controller.

3.2. SMCr based on an FOPDT model of the

process

In this case, the Smith and Corripio approx-

imation [10] for the process was used. They sug-gested that a chemical process with inverse

response can be approximated by a First-Order-

Plus-Deadtime model (FOPDT), as follows:

G1s Ket0s

(s 1IT

note that the inverse response time is considered as

the dead time term to see [Fig. 4].

Now, to handle the deadtime, two rst order

approximations can be used, Pade' and Taylor. If

the Pade approximation is used, a right half plane

zero is introduced, which generates an unstable

controller as has been shown in the previous sub-

section.

Then, a rst order Taylor series approximation

was used. It is given by

et0s 1

et0s

1

t0s 1IU

It is important to recall that chemical processes

are slow which means that the natural frequencies

are low. For example, Barney [11] makes refer-

ences about the sample frequencies of the most

common industrial processes like ow, pressure

and temperature (see Table 1). As we can see, themost signicant process is ow and its sample fre-

quency is around 1 Hz, which means that based on

the sampling theorem [12], a ow process has a

frequency less than 0.5 Hz ( 3.14 rad/s). Addition-

ally, this kind of process does not have a con-

siderable deadtime. Which means that the product

wto is most of the time small as can be observed

in the Bode diagram for et0s (a) and Taylor

approximation (b) (see Fig. 5). In summary, for

chemical processes, the use of the Taylor series

approximation is a good approach.

Then, applying Taylor series approximation,Eq. (17) can be written as follows:

Xs

Us

K

(s 1t0s 1IV

that is

t0(d2Xt

dt2 t0 (

dXt

dt Xt KUt IW

since Eq. (18) represents a second order system,then from Eq. (1), St becomes

St det

dt l1et lo

t0

etdt PH

From the sliding condition

dSt

dt

d2et

dt2 l1

det

dt l0et 0 PI

and substituting the denition of the error,

et Rt Xt, into the rst two terms of theabove equation

d2Xt

dt2 l1

dXt

dt

l0et 0 PP

Solving Eq. (19) for the second derivative of Xt,adding Eq. (22), and solving for Ut, the continuouspart of the controller is obtained



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UCt t0(

K

t0 (t0(

l1

dXt

dt

Xt

t0( l0et

!

PQ

UCt can be simplied by doing

l1

t0 (

t0( time

1

PR

the resulting SMCr is summarized as follows:

Ut t0(

K

Xt

t0( l0et

! KD

St

jStj PS

St signK dXt

dt l1et l0

t0

et

!PT

Furthermore, it has been shown that this choice of

l1, is the best for the continuous part of the con-

troller [1]

The function sign(K), in Eq. (26), was included

in the sliding surface equation to guarantee the

appropriate action of the controller for the givensystem. Note that sign(K) only depends on the

static gain of the plant model, therefore it never

switches. Furthermore, for industrial applications,

Eq. (26), can be considered as a PID algorithm [3].

Now, to ensure that the sliding surfaces behaves

as a critical or overdamped system, then

l0 l

21

4PU

besides this, an extra restriction is imposed by the

unstable zero of the non-minimum phase system,

which can be derived as follows:

when the system response has reached the sliding

surface, UDt 0, then

Ut UCt PV

Ut t0(

K

Xt

t0( l0et

!PW

Fig. 4. Inverse response system approximated by an FOPDT model.

Table 1

Sample frequency of most chemical processes

Process Sample frequency

(Hz)

Maximum process frequency

(Hz)

Flow 1.0 0.5

Pressure 0.2 0.1

Temperature 0.02 0.01



7/18

this control law must guarantee a stable closed

loop response. Replacing it in Eq. (8) and sub-

stituting the error denition, the following rst

order ODE is obtained

(dXt

dt Xt K

t0(

K

Xt

t0( l0Rt Xt

!

(1t0(

K

1

(t0

dXt

dt

l0

dXt

dt

!

QH

Summarizing, the previous equation can be writ-

ten as follows:

dXt

dt

( (1 (1l0t0(

t0(l0

! Xt Rt QI

To be stable, it must satisfy the following condi-

tion

( (1 (1l0t0(

t0(l0

!50 QP

Fig. 5. Bode diagram for Taylor approximation and et0 s.



8/18

therefore, ifl0 b 0

( (15(1l0t0( QQ

which implies that

0 ` l0 `( (1

(1(t0

!time2 QR

On the other hand, the deadtime approximation,

to, of the inverse response is less than the unstable

zero time constant, (1, as can be observed in Fig. 6.

From the graphic

t04(1 QS

adding the approximated time constant (, and

dividing by the product of the time constants, (

and (1 and the deadtime t0, in both sides, the fol-

lowing relationship is obtained

( t0

((1t04

( (1

((1t0QT

and substituting,

l1

( t0

(t0 QU

then

l1

(14

( (1

(1(t0QV

which implies that if

l04l1

(1QW

it will satisfy

0 ` l0 `( (1

(1(t0

!RH

Finally, all of the above can be summarized, as

follows:

0 ` l0 ` minl1

(1Yl

21

4

!time2 RI

in conclusion, the set of initial tuning parameters

will be given by

l1

t0 (

t0( time1

RP

0 ` l0 ` minl1

(1Yl

21

4

!time2 RQ

KD 0X51

K

(

t0

!0X76CO RR

0X68 0X12KKDl1

TO

time !

RS

where KD and are the tuning parameters for the

discontinuous part of the controller [13]. The

parameters (toY (Y (1 and K), needed to calculate

the initial tuning of the controller, are obtained

from the open loop step response [10,13].

4. Simulation models

In this section, two examples are used. The rstone is a linear second order non-minimum phase

system. The idea behind this simulation test was to

show the performance of the SMCr against mod-

eling errors, the range of these errors varies

between 20 and +20%. The second one is anonlinear reactor which was used to test the SMCr

performance against changes in set point and dis-

turbances in presence of noise.

4.1. SMCr robustness to modeling errors

To test the SMCr robustness against modeling

errors, the following nonminimum phase linear

model of a process was used

GS 1 s

s 12RT

For the process model, a step change of +10% in

set point was introduced at t 1 s, and the para-meters of the open loop step response were



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obtained (K 1X00 TOCO

, ( 1X53s, t0=1.39 [s]).

Using the tuning equations given previously, the

initial adjustment of the SMCr parameters were

done (l

0 0X47,l

1 1X37, KD=0.55,

=0.77).Fig. 7 shows the closed-loop response obtained

for the set point change when the designed SMCr

with the proposed initial adjustment was applied.

It is clear from this that the proposed controller

works properly for this kind of system. Then using

the same plant, the initial adjustment was done

simulating modeling errors in the static gain of

20% (see Fig. 8). Although the overshoots weredierent when the static gain was changed, the

same settling times were obtained. Note that when

the static gain was changed, the sliding surface did

not go to zero, but rather to a dierent constant

Fig. 6. Relationship between estimated deadtime, t0 t0, and the unstable zero time constant tao1=(1.

Fig. 7. System step response when the SMCr was applied.



10/18

value to correct the steady-state error (see Fig. 9).

This shows that the integral action of the SMCr

works properly to avoid steady-state errors in

these conditions.Figs. 10 and 12 show the closed-loop responses

obtained for the set point change when modeling

errors of 20% in the time constant, (, and

deadtime, t0, were simulated. In these cases, the

system outputs show slightly dierent transient

responses with the same settling times showing

that the SMCr action is robust against signicant

modeling errors in time constant and deadtime. Incontrast with the static gain modeling errors case,

the sliding surface outputs went to zero for all the

cases (see Figs. 11 and 13). Note that the sliding

surface outputs show the respective delay without

Fig. 8. System step responses when (20%) modeling errors in static gain, K, were introduced.

Fig. 9. Sliding surface outputs to a set point change when (20%) modeling errors in static gain, K, were introduced.



11/18

signicant changes in the settling time of the con-

dition St=0. Fig. 14 depicts the closed-loopresponses obtained for the set point changes when

all the modeling parameters (KY (Y t0

) present

extreme errors in the same sense. This means that

they are increased by 20% (M) or decreased by the

same value (m) and their outputs are compared with

the nominal output (N) The system outputs show

similar responses to those obtained in the previous

cases. In the same sense, the sliding surface output

shows a behavior similar to that of Fig. 9, due to the

static gain modeling errors (see Fig. 15).

Fig. 10. System responses to a set point change when (20%) modeling errors in time constant, (tao(), were introduced.

Fig. 11. Sliding surface outputs toa set point change when (20%) modeling errors in time constant, (tao(), were introduced.



12/18

Looking at the worst case, as far as modeling

errors are concerned, the dierent error combinations

of the three model parameters were done. The critical

cases were obtained when the static gain, K, was 20%

above and the time constant, (, was 20% below of the

nominal value. Fig. 16 shows the close-loop responses

for the set point changes when the critical parameters

error were simulated (C:K=1.2 TOCO

, ( 1X22s,

t0 1X67s; c:K=1.2TOCO

, (=1.22 [s], t0 1X11s;

N: Nominal values). Even though the critical output

responses show large overshoot and underdamped

behavior, they reach steady-state in a reasonable

Fig. 12. System responses to a set point change when (20%) modeling errors in deadtime, (t0t0), were introduced.

Fig. 13. Sliding surface outputs to a point change when (20%) modeling errors in time constant, (t0t0), were introduced.



13/18

time. Note that the sliding surface output shows a

similar behavior as before (see Fig. 17).

In summary the SMCr is shown to be robust

against modeling errors, guaranteeing zero steady-

state error in all cases.

4.2. SMCr performance when it is applied to a

nonlinear model

To test the controller behavior against set point

changes, the presence of disturbances and noise

Fig. 14. System step responses against extreme (M,m) modeling errors.

Fig. 15. Sliding surface outputs to a set point change when extreme (M,m) modeling errors were introduced.



14/18

the Van de Vusse non linear model was used [4]. A

schematic drawing of the Van de Vusse reactor is

shown in Fig. 18. The isothermal series/parallel

reactions which take place in the reactor are:

A 3 B 3 C RU

2A 3 D RV

The process model consists of two mol mass

balances:

dCA

dt k1CA k3C

2A

F

VCAf CA RW

dCB

dt k1CA k2CB

F

VCB SH

Fig. 16. System step responses against critical (C,c) modeling errors.

Fig. 17. Sliding surface outputs to a set point change when critical (C,c) modeling errors were introduced.



15/18

Where CA is the euent concentration of compo-

nent A, CB is the euent concentration of B, F is

the input ow and V is the reactor volume. The

operating values for this study are

k1=0.833 min1, k2=1.667 min

1 and k3=0.167

Lmol1min1. The concentration of A in the feed

stream is given by CAf and equal to 10 molL1. In

steady-state the process concentrations present the

following values CA=3.0 mol L1 and CB=1.117

mol L1.

The process is instrumented with a transmitter:

TOt CBt CBmin

CBSI

and a valve:

F CvVp SP

Where TO is the transmitter output [%], is

transmitter deadtime [min], CBmin

is the minimum

concentration limit, CB is the transmitter span,

Cv is the valve coecient and Vp is the valve posi-

tion. The control objective is to regulate CB by

manipulating the input ow F.

Fig. 19 shows the transmitter output when a

step change of 10% in set point was done. Thegure depicts an inverse response characteristic

with a smooth behavior, a small overshoot and

zero steady-state error as was predicted by the

robustness test. In the presence of a step dis-

turbance of 10% in the inlet concentration, CAf,

Fig. 18. Van de Vusse reactor.

Fig. 19. Transmitter output to a set point step change in presence of noise.



16/18

the system response was smooth with a short set-

tling time and zero steady-state error (see Fig. 20).

In spite of the controller not being derived for

non-minimum phase systems with deadtime, based

on the robustness shown against modeling errors,

the same test was done against changes in dead

time. This test was done when the transmitter

deadtime varies without readjustment of the

SMCr.

The transmitter outputs for set point and inlet

concentration changes are shown in Figs. 21 and

22. In these two cases the transmitter deadtime, ,

was changed as fractions of the identied dead-

time, t0=0.545 min (a: t02

Y X t0Y X 2t0). Although the SMCr was tuned without the

dead time in the transmitter, it worked properly

for a broad range of dead time values. The trans-

mitter output became marginally stable when the

Fig. 20. Transmitter output to an inlet concentration change in presence of noise.

Fig. 21. Transmitter outputs to a set point step change when dierent transmitter deadtimes, , were introduced.



17/18

introduced deadtime was near two times the iden-

tied dead time (see Fig. 21). Note that the eec-

tive dead time, the adding eects of the transmitter

deadtime and the inverse response time, are larger

than the system time constant, (=0.645 min,

which exceeds the controllability relationship

(t0(41) [10] (a:

t0( 1.26; b:

t0( 1.70; c:

t0( 2.53). Now, in

the presence of the disturbance, an inlet con-

centration change, the transmitter output exhibits

the characteristic inverse response with approxi-

mately the same settling time for all the cases.

5. Conclusions

This paper showed by simulations that the

Sliding Mode Controller developed from an

FOPDT model works well for inverse response

systems. The obtained responses showed that the

proposed controller has the potential of being

used to control more complex or nonlinear sys-

tems with inverse response and deadtime, such as

distillation columns, reactors among others. The

robustness of the controller against modeling

errors, disturbance and presence of noise was

clearly shown. Given that the controller presents a

xed structure which allows implementation of the

same algorithm for minimum and non-minimum

phase systems, its implementation in DCS's (Digital

Control Systems) is very simple and can be outtted

based on PID algorithm, this SMCr seems to be a

good alternative to control a myriad of systems.

References

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mode controllers for chemical processes. Doctoral Dis-

sertation, University of South Florida, Tampa, Florida,

August 1996.

[2] O. Camacho, C. Smith, Application of sliding mode con-

trol to nonlinear chemical processes with variable dead-

time, in: Proceedings of 2nd Congress of Colombian

Association of Automatics Bucaramanga, Colombia,

1997, 122128.

[3] O. Camacho, C. Smith, E. Chacon, Toward an imple-

mentation of sliding mode control to chemical processes

in: Proceedings of ISIE'97 Guimaraes, Portugal, 1997,

11011105.

[4] A. Aoyama, F.J. Doyle III, V. Venkatasubramsinion,

Control-ane neural network approach for non-mini-

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Fig. 22. Transmitter outputs in the presence of a disturbance when dierent transmitter deadtimes, , were introduced.



18/18

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variable structure control applied to chemical processes

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