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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]
7/29/2019 Variable structure control applied to chemical processes
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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.
58 O. Camacho et al. / ISA Transactions 38 (1999) 5572
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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
O. Camacho et al. / ISA Transactions 38 (1999) 5572 59
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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
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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.
O. Camacho et al. / ISA Transactions 38 (1999) 5572 61
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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
62 O. Camacho et al. / ISA Transactions 38 (1999) 5572
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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.
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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.
64 O. Camacho et al. / ISA Transactions 38 (1999) 5572
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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.
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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.
66 O. Camacho et al. / ISA Transactions 38 (1999) 5572
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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.
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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.
68 O. Camacho et al. / ISA Transactions 38 (1999) 5572
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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.
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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.
70 O. Camacho et al. / ISA Transactions 38 (1999) 5572
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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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Fig. 22. Transmitter outputs in the presence of a disturbance when dierent transmitter deadtimes, , were introduced.
O. Camacho et al. / ISA Transactions 38 (1999) 5572 71
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