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Variable structure control applied to chemical processes with inverse response
ISA TRANSACTIONS
1
ISA Transactions 38 (1999) 55±72
Variable structure control applied to chemical processes with inverse response Oscar Camacho *, RubeÂn Rojas1, Winston GarcõÂa2 Grupo de InvestigacioÂn en Nuevas Estrategias de Control Aplicadas (GINECA), Postgrado en AutomatizacioÂn e InstrumentacioÂn, Facultad de IngenierõÂa, Universidad de Los Andes, MeÂrida, 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 nonminimum phase systems oer diculty in applying feedback control. Furthermore, there exist system uncertainties that include modeling errors, unmodeled dynamics and disturbances that become challenging 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 * Corresponding author. Tel: +58-74-402-891; fax: +58-74402-890; e-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected]
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. [1±3]. 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
0019-0578/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0019 -0 578(99)00005 -1
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O. Camacho et al. / ISA Transactions 38 (1999) 55±72
against modeling errors, and the controller performance 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 disturbances. At last, the conclusions are presented.
The control objective is to ensure that the controlled variable be equal to its reference value at all times, meaning that e t and its derivatives must be zero. The problem of tracking a reference value can be reduced to that of keeping S t at zero. Once, S t=0 is reached, it is desired to make
2. Sliding mode control
dS t 0 2 dt (the sliding condition), to guarantee the value of S t at zero. After the sliding surface has been selected, the control law must be designed to satisfy the S t=0 condition. The control law, U t, can be written as follows,
Sliding Mode Control is a technique derived from variable structure control (VSC) which was studied originally by Utkin [5]. This kind of control is particularly appealing for a broad class of systems, due to its ability to deal with nonlinearities, time-variance, as well as uncertainties and disturbances in a direct manner, in the face of modeling imprecisions. In VSC, the control can modify its structure. The design problem consists of selecting the parameters of each structure and de®ning the traveling logic. The ®rst step in SMC is to de®ne a sliding surface, S t=0, along which the 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 the switching function S t has dierent sign. The structure of the control system is intentionally altered as its state crosses the sliding surface in the phase plane in accordance with a prescribed control law. So, the second step is to design the control 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 S t was selected as an integral-dierential equation acting on the tracking error [7]. n t d l e tdt 1 S t dt 0 e t is the tracking error between the reference value (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.
U t UC t UD t
3
where the ®rst additive part, UC t, is continuous and the second one, UD t, is discontinuous. The continuous part is given by UC t f X t; R t
4
where f X t; R t is determined using the equivalent control procedure [5], in accordance with the desired motion of the sliding mode. The discontinuous part, UD t, is nonlinear and represents the switching element of the control law. This part of the controller is discontinuous across the sliding surface. Mainly, UD t is designed based on a relay-like function (i.e. UD t sign S t)), because it allows for changes between the structures with a hypothetical in®nitely 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 physical actuators causing chattering around of the sliding surface [5±7]. 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 relaylike function by a saturation or a sigma function (see Fig. 2) which can be written as follows: UD t KD
S t jS tj
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O. Camacho et al. / ISA Transactions 38 (1999) 55±72
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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).
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), responsible for guiding the system to the sliding surface, and a continuous part, Eq. (4), which is responsible
for keeping the controlled variable on the reference 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
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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 approximation for this kind of system can be done using a ®rst order model, as follows: G s
K ÿ 1 s 1 s 1
6
or X s K ÿ 1 s 1 U s s 1
7
where X s, is the controlled variable and U s is the 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 equation form, as follows,
dX t dU t X t K U t ÿ 1 dt dt
8
then, from Eq. (1), the sliding surface is obtained for n=1 S t e t l e tdt 9 equating the ®rst derivative of S t to zero (sliding condition) dS t de t le t 0 dt dt
10
and replacing the well known approximation, [9,10] de t dX t ÿ dt dt
11
Eq. (10), can be written as follows: dS t dX t ÿ le t dt dt
12
then, solving Eq. (8) for the ®rst derivative of X t and adding Eq. (12)
Fig. 3. Typical step response of inverse response systems.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
X t K dU t le t U t ÿ 1 dt
13
and going back to Laplace Transform domain X s K le s ÿ 1 sU s ÿ U s
14
solving for U s, the continuous part of the controller is obtained h i X s le s 15 UC s K 1 s ÿ 1 which represents an unstable controller. Similar results are obtained for higher order non-minimum phase linear model approximations. Therefore, as has been shown, the direct use of the conventional sliding mode control theory, [5±7], to an inverse response process model produces an unstable controller. 3.2. SMCr based on an FOPDT model of the process
G1 s K
eÿt0 s s 1
16
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 subsection. Then, a ®rst order Taylor series approximation was used. It is given by eÿt0 s
1 1 e t0 s t 0 s 1
are low. For example, Barney [11] makes references about the sample frequencies of the most common industrial processes like ¯ow, pressure and temperature (see Table 1). As we can see, the most signi®cant process is ¯ow and its sample frequency 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). Additionally, this kind of process does not have a considerable deadtime. Which means that the product wto is most of the time small as can be observed in the Bode diagram for eÿt0 s (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: X s K U s s 1 t0 s 1
17
It is important to recall that chemical processes are slow which means that the natural frequencies
18
that is t0
In this case, the Smith and Corripio approximation [10] for the process was used. They suggested that a chemical process with inverse response can be approximated by a First-OrderPlus-Deadtime model (FOPDT), as follows:
59
d2 X t dX t X t KU t t0 2 dt dt
19
since Eq. (18) represents a second order system, then from Eq. (1), S t becomes t de t l1 e t lo e tdt 20 S t dt 0 From the sliding condition dS t d2 e t de t l0 e t 0 l1 2 dt dt dt
21
and substituting the de®nition of the error, e t R t ÿ X t, into the ®rst two terms of the above equation ÿ
d2 X t dX t l ÿ l0 e t 0 1 dt2 dt
22
Solving Eq. (19) for the second derivative of X t, adding Eq. (22), and solving for U t, the continuous part of the controller is obtained
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Fig. 4. Inverse response system approximated by an FOPDT model.
Table 1 Sample frequency of most chemical processes Process Flow Pressure Temperature
Sample frequency Maximum process frequency (Hz) (Hz) 1.0 0.2 0.02
0.5 0.1 0.01
t t dX t X t 0 0 ÿ l1 l0 e t UC t K t0 dt t0 23
l0
UC t can be simpli®ed by doing l1
t0 timeÿ1 t0
24
the resulting SMCr is summarized as follows: t0 X t S t l0 e t KD 25 U t K t0 jS tj t dX t l1 e t l0 e t S t sign K ÿ dt 0
Furthermore, it has been shown that this choice of l1 , is the best for the continuous part of the controller [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 given system. 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
26
l21 4
27
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, UD t 0, then U t UC t
U t
t0 X t l0 e t K t0
28
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Fig. 5. Bode diagram for Taylor approximation and eÿt0 s .
this control law must guarantee a stable closed loop response. Replacing it in Eq. (8) and substituting the error de®nition, the following ®rst order ODE is obtained dX t t0 X t X t K l0 R t ÿ X t dt K t0 t0 1 dX t dX t ÿ 1 l0 ÿ K t0 dt dt 30
Summarizing, the previous equation can be written as follows: dX t 1 ÿ 1 l0 t0 X t R t dt t0 l0
31
To be stable, it must satisfy the following condition 1 ÿ 1 l0 t0 50 32 t0 l0
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therefore, if l0 > 0 1 5 1 l0 t0 which implies that 1 timeÿ2 0 < l0 < 1 t0
33
l1 34
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 t0 4 1
35
adding the approximated time constant , and dividing by the product of the time constants, and 1 and the deadtime t0 , in both sides, the following relationship is obtained t0 1 4 1 t0 1 t0
36
and substituting, 37
l1 1 4 1 1 t0
38
then
which implies that if l0 4
l1 1
it will satisfy 1 0 < l0 < 1 t0
t0 timeÿ1 t0
42
l1 l21 ; timeÿ2 0 < l0 < min 1 4
43
0:51 0:76 CO KD K t0
44
TO 0:68 0:12 KKD l1 time
45
where KD and are the tuning parameters for the discontinuous part of the controller [1±3]. The parameters (to ; ; 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
t0 t0
l1
in conclusion, the set of initial tuning parameters will be given by
39
40
Finally, all of the above can be summarized, as follows: l1 l21 ; 41 timeÿ2 0 < l0 < min 1 4
In this section, two examples are used. The ®rst one is a linear second order non-minimum phase system. The idea behind this simulation test was to show the performance of the SMCr against modeling errors, the range of these errors varies between ÿ20 and +20%. The second one is a nonlinear reactor which was used to test the SMCr performance against changes in set point and disturbances 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 G S
1ÿs s 12
46
For the process model, a step change of +10% in set point was introduced at t 1 s, and the parameters of the open loop step response were
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
TO obtained (K 1:00 CO , 1:53s, t0 =1.39 [s]). Using the tuning equations given previously, the initial adjustment of the SMCr parameters were done (l0 0:47, l1 1:37, 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
63
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 were dierent 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 signi®cant modeling errors in time constant and deadtime. In contrast 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.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
signi®cant changes in the settling time of the condition S t=0. Fig. 14 depicts the closed-loop responses obtained for the set point changes when all the modeling parameters (K; ; t0 ) present extreme errors in the same sense. This means that they are increased by 20% (M) or decreased by the
65
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 TO parameters error were simulated (C:K=1.2 CO , 1:22s, TO t0 1:67s; c:K=1.2 CO , =1.22 [s], t0 1:11s; 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.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
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 steadystate error in all cases.
67
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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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.
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!B!C
47
2A ! D
48
The process model consists of two mol mass balances: dCA F ÿk1 CA ÿ k3 C2A CA f ÿ CA dt V
49
dCB F k1 CA ÿ k2 CB ÿ CB dt V
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O. Camacho et al. / ISA Transactions 38 (1999) 55±72
Where CA is the euent concentration of component 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 minÿ1, k2 =1.667 minÿ1 and k3 =0.167 Lmolÿ1minÿ1. The concentration of A in the feed stream is given by CAf and equal to 10 molLÿ1. In steady-state the process concentrations present the following values CA =3.0 mol Lÿ1 and CB =1.117 mol Lÿ1. The process is instrumented with a transmitter: TO t ÿ and a valve:
CB t ÿ CBmin CB
51
F Cv Vp
69
52
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 position. The control objective is to regulate CB by manipulating the input ¯ow F. Fig. 19 shows the transmitter output when a step change of ÿ10% in set point was done. The ®gure 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 disturbance of ÿ10% 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 settling 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 identi®ed deadtime, t0 =0.545 min (a: t20 ; b : t0 ; c : 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 transmitter 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.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
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Fig. 22. Transmitter outputs in the presence of a disturbance when dierent transmitter deadtimes, , were introduced.
introduced deadtime was near two times the identi®ed dead time (see Fig. 21). Note that the eective 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 (t 0 41) [10] (a:t 0 1.26; b: t 0 1.70; c: t 0 2.53). Now, in the presence of the disturbance, an inlet concentration change, the transmitter output exhibits the characteristic inverse response with approximately 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 systems 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 out®tted based on PID algorithm, this SMCr seems to be a good alternative to control a myriad of systems. References [1] O. Camacho, A new approach to design and tune sliding mode controllers for chemical processes. Doctoral Dissertation, University of South Florida, Tampa, Florida, August 1996. [2] O. Camacho, C. Smith, Application of sliding mode control to nonlinear chemical processes with variable deadtime, in: Proceedings of 2nd Congress of Colombian Association of Automatics Bucaramanga, Colombia, 1997, 122±128. [3] O. Camacho, C. Smith, E. Chacon, Toward an implementation of sliding mode control to chemical processes in: Proceedings of ISIE'97 Guimaraes, Portugal, 1997, 1101±1105. [4] A. Aoyama, F.J. Doyle III, V. Venkatasubramsinion, Control-ane neural network approach for non-minimum-phase nonlinear process control, Journal Process Control 6 (1) (1995) 17±26. [5] V.I. Utkin, Variable structure systems with sliding modes, Transactions of IEEE on Automatic Control AC(22) (1977) 212±222. [6] J.Y. Hung, Variable structure control: A survey, IEEE Transactions on Industrial Electronics 40(1) (1993) 2±21. [7] J.J. Slotine, W. Li, Applied nonlinear control, PrenticeHall, New Jersey, 1991.
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[8] A.S. Zinober, Variable Structure and Liapunov Control, Spring±Verlag, London, 1994. [9] T.E. Marlin, Process Control E, McGraw-Hill, New York, 1995. [10] C. Smith, A. Corripio, Control Automatico de Procesos, Limusa, Mexico, 1991.
[11] G.C. Barney, Intelligent Instrumentation, Prentice Hall, New York, 1988. [12] K.J. Astrom, B. Wittenmark, Computer-Controlled Systems, Prentice Hall, New Jersey, 1990. [13] L. Povy, Identi®cation des Processus, Bordas, Paris, 1975.
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ISA Transactions 38 (1999) 55±72
Variable structure control applied to chemical processes with inverse response Oscar Camacho *, RubeÂn Rojas1, Winston GarcõÂa2 Grupo de InvestigacioÂn en Nuevas Estrategias de Control Aplicadas (GINECA), Postgrado en AutomatizacioÂn e InstrumentacioÂn, Facultad de IngenierõÂa, Universidad de Los Andes, MeÂrida, 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 nonminimum phase systems oer diculty in applying feedback control. Furthermore, there exist system uncertainties that include modeling errors, unmodeled dynamics and disturbances that become challenging 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 * Corresponding author. Tel: +58-74-402-891; fax: +58-74402-890; e-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected]
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. [1±3]. 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
0019-0578/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0019 -0 578(99)00005 -1
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O. Camacho et al. / ISA Transactions 38 (1999) 55±72
against modeling errors, and the controller performance 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 disturbances. At last, the conclusions are presented.
The control objective is to ensure that the controlled variable be equal to its reference value at all times, meaning that e t and its derivatives must be zero. The problem of tracking a reference value can be reduced to that of keeping S t at zero. Once, S t=0 is reached, it is desired to make
2. Sliding mode control
dS t 0 2 dt (the sliding condition), to guarantee the value of S t at zero. After the sliding surface has been selected, the control law must be designed to satisfy the S t=0 condition. The control law, U t, can be written as follows,
Sliding Mode Control is a technique derived from variable structure control (VSC) which was studied originally by Utkin [5]. This kind of control is particularly appealing for a broad class of systems, due to its ability to deal with nonlinearities, time-variance, as well as uncertainties and disturbances in a direct manner, in the face of modeling imprecisions. In VSC, the control can modify its structure. The design problem consists of selecting the parameters of each structure and de®ning the traveling logic. The ®rst step in SMC is to de®ne a sliding surface, S t=0, along which the 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 the switching function S t has dierent sign. The structure of the control system is intentionally altered as its state crosses the sliding surface in the phase plane in accordance with a prescribed control law. So, the second step is to design the control 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 S t was selected as an integral-dierential equation acting on the tracking error [7]. n t d l e tdt 1 S t dt 0 e t is the tracking error between the reference value (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.
U t UC t UD t
3
where the ®rst additive part, UC t, is continuous and the second one, UD t, is discontinuous. The continuous part is given by UC t f X t; R t
4
where f X t; R t is determined using the equivalent control procedure [5], in accordance with the desired motion of the sliding mode. The discontinuous part, UD t, is nonlinear and represents the switching element of the control law. This part of the controller is discontinuous across the sliding surface. Mainly, UD t is designed based on a relay-like function (i.e. UD t sign S t)), because it allows for changes between the structures with a hypothetical in®nitely 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 physical actuators causing chattering around of the sliding surface [5±7]. 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 relaylike function by a saturation or a sigma function (see Fig. 2) which can be written as follows: UD t KD
S t jS tj
5
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
57
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).
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), responsible for guiding the system to the sliding surface, and a continuous part, Eq. (4), which is responsible
for keeping the controlled variable on the reference 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
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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 approximation for this kind of system can be done using a ®rst order model, as follows: G s
K ÿ 1 s 1 s 1
6
or X s K ÿ 1 s 1 U s s 1
7
where X s, is the controlled variable and U s is the 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 equation form, as follows,
dX t dU t X t K U t ÿ 1 dt dt
8
then, from Eq. (1), the sliding surface is obtained for n=1 S t e t l e tdt 9 equating the ®rst derivative of S t to zero (sliding condition) dS t de t le t 0 dt dt
10
and replacing the well known approximation, [9,10] de t dX t ÿ dt dt
11
Eq. (10), can be written as follows: dS t dX t ÿ le t dt dt
12
then, solving Eq. (8) for the ®rst derivative of X t and adding Eq. (12)
Fig. 3. Typical step response of inverse response systems.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
X t K dU t le t U t ÿ 1 dt
13
and going back to Laplace Transform domain X s K le s ÿ 1 sU s ÿ U s
14
solving for U s, the continuous part of the controller is obtained h i X s le s 15 UC s K 1 s ÿ 1 which represents an unstable controller. Similar results are obtained for higher order non-minimum phase linear model approximations. Therefore, as has been shown, the direct use of the conventional sliding mode control theory, [5±7], to an inverse response process model produces an unstable controller. 3.2. SMCr based on an FOPDT model of the process
G1 s K
eÿt0 s s 1
16
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 subsection. Then, a ®rst order Taylor series approximation was used. It is given by eÿt0 s
1 1 e t0 s t 0 s 1
are low. For example, Barney [11] makes references about the sample frequencies of the most common industrial processes like ¯ow, pressure and temperature (see Table 1). As we can see, the most signi®cant process is ¯ow and its sample frequency 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). Additionally, this kind of process does not have a considerable deadtime. Which means that the product wto is most of the time small as can be observed in the Bode diagram for eÿt0 s (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: X s K U s s 1 t0 s 1
17
It is important to recall that chemical processes are slow which means that the natural frequencies
18
that is t0
In this case, the Smith and Corripio approximation [10] for the process was used. They suggested that a chemical process with inverse response can be approximated by a First-OrderPlus-Deadtime model (FOPDT), as follows:
59
d2 X t dX t X t KU t t0 2 dt dt
19
since Eq. (18) represents a second order system, then from Eq. (1), S t becomes t de t l1 e t lo e tdt 20 S t dt 0 From the sliding condition dS t d2 e t de t l0 e t 0 l1 2 dt dt dt
21
and substituting the de®nition of the error, e t R t ÿ X t, into the ®rst two terms of the above equation ÿ
d2 X t dX t l ÿ l0 e t 0 1 dt2 dt
22
Solving Eq. (19) for the second derivative of X t, adding Eq. (22), and solving for U t, the continuous part of the controller is obtained
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Fig. 4. Inverse response system approximated by an FOPDT model.
Table 1 Sample frequency of most chemical processes Process Flow Pressure Temperature
Sample frequency Maximum process frequency (Hz) (Hz) 1.0 0.2 0.02
0.5 0.1 0.01
t t dX t X t 0 0 ÿ l1 l0 e t UC t K t0 dt t0 23
l0
UC t can be simpli®ed by doing l1
t0 timeÿ1 t0
24
the resulting SMCr is summarized as follows: t0 X t S t l0 e t KD 25 U t K t0 jS tj t dX t l1 e t l0 e t S t sign K ÿ dt 0
Furthermore, it has been shown that this choice of l1 , is the best for the continuous part of the controller [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 given system. 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
26
l21 4
27
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, UD t 0, then U t UC t
U t
t0 X t l0 e t K t0
28
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O. Camacho et al. / ISA Transactions 38 (1999) 55±72
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Fig. 5. Bode diagram for Taylor approximation and eÿt0 s .
this control law must guarantee a stable closed loop response. Replacing it in Eq. (8) and substituting the error de®nition, the following ®rst order ODE is obtained dX t t0 X t X t K l0 R t ÿ X t dt K t0 t0 1 dX t dX t ÿ 1 l0 ÿ K t0 dt dt 30
Summarizing, the previous equation can be written as follows: dX t 1 ÿ 1 l0 t0 X t R t dt t0 l0
31
To be stable, it must satisfy the following condition 1 ÿ 1 l0 t0 50 32 t0 l0
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therefore, if l0 > 0 1 5 1 l0 t0 which implies that 1 timeÿ2 0 < l0 < 1 t0
33
l1 34
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 t0 4 1
35
adding the approximated time constant , and dividing by the product of the time constants, and 1 and the deadtime t0 , in both sides, the following relationship is obtained t0 1 4 1 t0 1 t0
36
and substituting, 37
l1 1 4 1 1 t0
38
then
which implies that if l0 4
l1 1
it will satisfy 1 0 < l0 < 1 t0
t0 timeÿ1 t0
42
l1 l21 ; timeÿ2 0 < l0 < min 1 4
43
0:51 0:76 CO KD K t0
44
TO 0:68 0:12 KKD l1 time
45
where KD and are the tuning parameters for the discontinuous part of the controller [1±3]. The parameters (to ; ; 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
t0 t0
l1
in conclusion, the set of initial tuning parameters will be given by
39
40
Finally, all of the above can be summarized, as follows: l1 l21 ; 41 timeÿ2 0 < l0 < min 1 4
In this section, two examples are used. The ®rst one is a linear second order non-minimum phase system. The idea behind this simulation test was to show the performance of the SMCr against modeling errors, the range of these errors varies between ÿ20 and +20%. The second one is a nonlinear reactor which was used to test the SMCr performance against changes in set point and disturbances 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 G S
1ÿs s 12
46
For the process model, a step change of +10% in set point was introduced at t 1 s, and the parameters of the open loop step response were
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
TO obtained (K 1:00 CO , 1:53s, t0 =1.39 [s]). Using the tuning equations given previously, the initial adjustment of the SMCr parameters were done (l0 0:47, l1 1:37, 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
63
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 were dierent 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 signi®cant modeling errors in time constant and deadtime. In contrast 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.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
signi®cant changes in the settling time of the condition S t=0. Fig. 14 depicts the closed-loop responses obtained for the set point changes when all the modeling parameters (K; ; t0 ) present extreme errors in the same sense. This means that they are increased by 20% (M) or decreased by the
65
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 TO parameters error were simulated (C:K=1.2 CO , 1:22s, TO t0 1:67s; c:K=1.2 CO , =1.22 [s], t0 1:11s; 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.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
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 steadystate error in all cases.
67
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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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.
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!B!C
47
2A ! D
48
The process model consists of two mol mass balances: dCA F ÿk1 CA ÿ k3 C2A CA f ÿ CA dt V
49
dCB F k1 CA ÿ k2 CB ÿ CB dt V
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O. Camacho et al. / ISA Transactions 38 (1999) 55±72
Where CA is the euent concentration of component 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 minÿ1, k2 =1.667 minÿ1 and k3 =0.167 Lmolÿ1minÿ1. The concentration of A in the feed stream is given by CAf and equal to 10 molLÿ1. In steady-state the process concentrations present the following values CA =3.0 mol Lÿ1 and CB =1.117 mol Lÿ1. The process is instrumented with a transmitter: TO t ÿ and a valve:
CB t ÿ CBmin CB
51
F Cv Vp
69
52
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 position. The control objective is to regulate CB by manipulating the input ¯ow F. Fig. 19 shows the transmitter output when a step change of ÿ10% in set point was done. The ®gure 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 disturbance of ÿ10% 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 settling 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 identi®ed deadtime, t0 =0.545 min (a: t20 ; b : t0 ; c : 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 transmitter 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.
O. Camacho et al. / ISA Transactions 38 (1999) 55±72
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Fig. 22. Transmitter outputs in the presence of a disturbance when dierent transmitter deadtimes, , were introduced.
introduced deadtime was near two times the identi®ed dead time (see Fig. 21). Note that the eective 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 (t 0 41) [10] (a:t 0 1.26; b: t 0 1.70; c: t 0 2.53). Now, in the presence of the disturbance, an inlet concentration change, the transmitter output exhibits the characteristic inverse response with approximately 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 systems 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 out®tted based on PID algorithm, this SMCr seems to be a good alternative to control a myriad of systems. References [1] O. Camacho, A new approach to design and tune sliding mode controllers for chemical processes. Doctoral Dissertation, University of South Florida, Tampa, Florida, August 1996. [2] O. Camacho, C. Smith, Application of sliding mode control to nonlinear chemical processes with variable deadtime, in: Proceedings of 2nd Congress of Colombian Association of Automatics Bucaramanga, Colombia, 1997, 122±128. [3] O. Camacho, C. Smith, E. Chacon, Toward an implementation of sliding mode control to chemical processes in: Proceedings of ISIE'97 Guimaraes, Portugal, 1997, 1101±1105. [4] A. Aoyama, F.J. Doyle III, V. Venkatasubramsinion, Control-ane neural network approach for non-minimum-phase nonlinear process control, Journal Process Control 6 (1) (1995) 17±26. [5] V.I. Utkin, Variable structure systems with sliding modes, Transactions of IEEE on Automatic Control AC(22) (1977) 212±222. [6] J.Y. Hung, Variable structure control: A survey, IEEE Transactions on Industrial Electronics 40(1) (1993) 2±21. [7] J.J. Slotine, W. Li, Applied nonlinear control, PrenticeHall, New Jersey, 1991.
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[8] A.S. Zinober, Variable Structure and Liapunov Control, Spring±Verlag, London, 1994. [9] T.E. Marlin, Process Control E, McGraw-Hill, New York, 1995. [10] C. Smith, A. Corripio, Control Automatico de Procesos, Limusa, Mexico, 1991.
[11] G.C. Barney, Intelligent Instrumentation, Prentice Hall, New York, 1988. [12] K.J. Astrom, B. Wittenmark, Computer-Controlled Systems, Prentice Hall, New Jersey, 1990. [13] L. Povy, Identi®cation des Processus, Bordas, Paris, 1975.