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Adaptive fuzzy approach for H∞ temperature tracking control of continuous stirred tank reactors
ARTICLE IN PRESS
Control Engineering Practice 16 (2008) 1101–1108 www.elsevier.com/locate/conengprac
Adaptive fuzzy approach for HN temperature tracking control of continuous stirred tank reactors Shahin Salehia, Mohammad Shahrokhib, a
Department of Measurement and Automation, Petroleum University of Technology, Tehran, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11365-9465 Azadi Avenue, Tehran, Iran
b
Received 24 June 2007; accepted 22 December 2007 Available online 15 February 2008
Abstract In this paper, an adaptive fuzzy temperature controller is proposed for a class of continuous stirred tank reactors (CSTRs) based on input–output feedback linearization. Since for control implementation concentrations of all species are needed, based on the observability concept, a fuzzy logic system is used to estimate the concentration dependent terms and other unknown system parameters in the control law, using temperature measurements. It has been shown that the HN tracking control performance with a prescribed attenuation level is achieved, by using the proposed controller. Finally the effectiveness of the proposed controller has been demonstrated by applying it to a benchmark chemical reactor through computer simulation. r 2008 Elsevier Ltd. All rights reserved. Keywords: Feedback linearization; HN performance; Adaptive control; Fuzzy logic systems; Chemical reactors
1. Introduction Temperature control of chemical reactors is an important objective for achieving higher product qualities. Complex static and dynamic behaviors, system nonlinearity, unavailability of states, multiplicity and instability of equilibrium points and input constraint have made it a challenging problem. Numerous temperature controllers for continuous stirred tank reactors (CSTRs), based on conventional PID structure and state feedback control technique, have been proposed (Brown, Gonyie, Schwber, & James, 1998; Lee, Coronella, Bhadkamkar, & Seader, 1993; Ratto, 1998). Stabilization of chemical reactors by output feedback with PI controllers has been reviewed in a Ph.D. thesis (Jadot, 1996). Global stability of a reactor with an exothermic reaction using state feedback was proved by Adebekun and Schork (1991) and the same problem, using a state observer, has been considered by Adebekun (1992) and Kosanovich, Piovoso, Rokhlenko, and Guez (1995). Advanced control strategies, like predictive control Corresponding author. Tel.: +98 21 66165419; fax: +98 21 66072679.
E-mail address: [email protected] (M. Shahrokhi). 0967-0661/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2007.12.005
scheme, have been also used for temperature control of CSTRs (Santos, Afonso, Castro, Oliveira, & Biegler, 2001; Sistu & Bequette, 1992; Yu & Gomm, 2003). For instance, some comparisons among these techniques have been made (Sistu & Bequette, 1992). In addition, backstepping and feedback linearization methodologies are applied either in non-adaptive or adaptive forms (Gopaluni, Mizumoto, & Shah, 2003; Haugwitz, Hagander, & Nore´n, 2007; Limqueco & Kantor, 1990; Salehi, Shahrokhi, & Salahshoor, 2006; Wu, 1999; Zhang & Guay, 2001). On the other hand, since fuzzy logic systems are used as universal approximators with arbitrary accuracy for any real continuous function on a compact set (Wang & Mendel, 1992), it has attracted great interests in utilizing heuristic-based approaches to cope with the control problem of nonlinear and ill-conditioned systems (Wang, 1993, 1994, 1996). Experimental evaluation of fuzzy logicbased controllers for controlling chemical processes has been investigated by Fileti, Antunes, Silva, Silveira, and Pereira (2007). A novel nonlinear adaptive controller based on fuzzy logic systems and Fourier Integral has been proposed for temperature control of a CSTR (Huaguang & Cai, 2002). Most of advanced temperature controllers proposed for CSTRs require concentration measurements
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for implementation which are not available in practice. Therefore, designing a model-based controller which does not require composition measurements is highly motivated. The main objective of this paper is to modify our previous control scheme (Salehi et al., 2006) to achieve the HN temperature tracking performance. In that work, temperature control of a CSTR in which a single reaction takes place is discussed. In the present work, a general class of CSTRs in which multiple reactions are taking place is considered. By modifying the adaptive law, the fuzzy minimum approximation error has become independent of control action which results in achieving the HN temperature tracking performance. By taking advantage of the observability concept, a fuzzy controller is designed such that no composition measurement is required. In addition, stability of the closed loop system has been established and boundedness of the internal dynamics has been discussed. The paper is organized as follows. In Section 2, mathematical model of CSTRs is presented. In Section 3, an adaptive fuzzy nonlinear controller is designed by input–output feedback linearization to achieve the HN temperature tracking performance. In Section 4, the effectiveness of the proposed controller has been demonstrated by applying it to a benchmark chemical reactor and a comparison between the proposed controller and a PID whose parameters are tuned by Ziegler–Nicholes technique is made. Finally, conclusion is drawn in Section 5. 2. Mathematical model of CSTRs In this section, a model for a general class of CSTRs is presented. It is assumed that a set of M reactions with N components are taking place in the reactor. The dynamical model of the CSTR is obtained from mass and energy balances and can be written in the following matrix form (Alvarez-Ramirez, 1995): ~ x_ 1 ¼ F r ð~ x1;in ~ x1 Þ=V r þ K st ~ jð~ x1 ; x2 Þ,
(1)
~ x1 ; x2 Þ~ x_ 2 ¼ F r ðx2;in x2 Þ=V r DHð~ jð~ x1 ; x2 Þ þ gr ðx3 x2 Þ, x_ 3 ¼ gj ðx2 x3 Þ þ F j ðx3;in x3 Þ=V j ,
Fig. 1. Schematic diagram of the process.
It is assumed that ~ x1 vector (concentrations of all species) is unavailable, x2 and x3 are measured, and Kst, ~ j, ~ gr and gj are not known. Also, the volumes of reactor DH, and jacket, Vr and Vj, are assumed to be constant. The system equations can be written as ( ~ x_ ¼ f~ð~ xÞ þ ~ gð~ xÞu; (4) y ¼ hð~ xÞ; xÞ ¼ x2 , where ~ x ¼ ½~ xT1 x2 x3 T 2
f1
3
6 7 f 7 f~ð~ xÞ ¼ 6 4 25 f3
2
F r ð~ x1;in ~ x1 Þ=V r þ K st ~ jð~ x1 ; x2 Þ
6
3 7
~ x1 ; x2 Þ~ jð~ x1 ; x2 Þ þ gr ðx3 x2 Þ 7, F ðx x2 Þ=V r DHð~ ¼6 4 r 2;in 5 gj ðx2 x3 Þ
(2)
2
0
3
(3)
6 ~ gð~ xÞ ¼ 4
0
7 5.
x1 2
ðx3;in x3 Þ=V j
3. Adaptive fuzzy controller design In this section, a nonlinear adaptive fuzzy controller, based on input–output linearization, has been designed for a class of CSTRs whose dynamic models are given by Eq. (4). The basic controller structure is proposed by Chang (2001) and Chen, Lee, and Chang (1996). In these papers, two adaptive fuzzy controllers, using linguist rules, for a class of nonlinear system have been introduced. Using their strategy and input–output linearization technique, a
ARTICLE IN PRESS S. Salehi, M. Shahrokhi / Control Engineering Practice 16 (2008) 1101–1108
modified adaptive controller has been proposed for temperature tracking of CSTRs given by Eq. (4). In the first step of design, the system of Eq. (4) is transformed into the normal form (Isidori, 1995; Khalil, 1996; Slotine & Li, 1991). Under this transformation, Eq. (4) can be written as 8 _ ~ > c¼~ qð~ c; ~ xÞ; > > > < x_ ¼ x ; 1 2 (5) _ > x2 ¼ að~ xÞ þ bð~ xÞu; > > > : y ¼ x1 ; where ~ x ¼ ðx1 ; x2 Þ 2 <2 is the external state vector, ~ c2 n2 < is the internal state vector, a and b are given by following equations: ~ jÞ=q~ xÞ ¼ ðf 1 þ f 2 ÞqðDH~ x1 að~ xÞ ¼ L2f~hð~ ðF r =V r þ gr Þf 2 þ gr f 3 ,
the approximator. First, fuzzy logic systems are universal approximators with arbitrary accuracy for any given continuous function (Wang & Mendel, 1992). Second, fuzzy logic systems are constructed from fuzzy IF-THEN rules using some specific fuzzy inference, fuzzification, and defuzzification strategies, hence, linguistic information from human experts (in the form of the fuzzy IF-THEN rules) can be directly incorporated into the approximator design. að~ xÞ is a function of available (temperature measurements) and unavailable states (concentrations). It is assumed that the system is observable through reactor temperature measurements. By taking into account the observability concept, it can be concluded that the species concentrations, ~ x1 , can be estimated from the system output (x2). Consequently, að~ xÞ can be approximated using temperature measurements (x2, x3). If estimate of að~ xÞ is denoted by a^ , then the estimator can be constructed as T
ya ~ a^ ðx2 ; x3 Þ ¼ ~ Bðx2 ; x3 Þ,
xÞ ¼ gr ðx3;in x3 Þ=V j . bð~ xÞ ¼ L~g Lf~hð~ As can be seen, the relative degree of the system is two. e ¼ ðe1 ; e2 ÞT ¼ ðe; e_ÞT , where yr is Defining e ¼ yyr and ~ the desired output, the tracking error dynamic can be expressed as ~ xÞ þ bð~ ~ e_ ¼ Ao~ e þ bðað~ xÞu y€ r Þ, where 0 Ao ¼ 0
1 0
;
(6)
0 b~ ¼ . 1
~ ¼ ðk2 ; k1 Þ 2 <12 such that A Ao Choosing matrix K ~ ~ bK is Hurwitz, Eq. (6) becomes: ~ K~ ~e þ að~ ~ e_ ¼ A~ e_ þ bð xÞ þ bð~ xÞu y€ r Þ.
(7)
If the system mathematical model is exactly known and there is no model uncertainty, by applying the following nonlinear control law: ~e að~ xÞÞ=bð~ xÞ, u ¼ ðy€ r K~
(8)
the error dynamic becomes ~ e_ ¼ A~ e.
(9)
This error dynamic equation is asymptotical stable because A is Hurwitz. It should be noted that for implementation of control law (8), concentrations of all species in the reactor and system parameters are needed. But in practice, concentrations are not measured because compositionmeasuring devices are expensive and include lag. Usually, exact kinetic model is not available and there are uncertainties in the system parameters. To make the problem more realistic, it is assumed that all of the required ~ gr and gj are not available. data such as Kst, ~ j, DH, One way to solve the above problem is to estimate að~ xÞ and bð~ xÞ by appropriate estimators. In this work, a fuzzy approximator is used to estimate að~ xÞ. There are two main reasons for using the fuzzy logic as basic building blocks of
1103
(10)
T T where ~ ya ¼ ð~ y1;a ; . . . ; ~ ym;a ÞT is a fuzzy parameter vector, and
~ BT1 ðx2 ; x3 Þ; . . . ;~ BTm ðx2 ; x3 ÞÞT is the fuzzy regresBðx2 ; x3 Þ ¼ ð~ sive vector with the regressor ~ Bi ðx2 ; x3 Þ and m is the number of fuzzy rule bases (Chen et al., 1996; Wang, 1997). However, if a fuzzy approximator is used for estimating bð~ xÞ, then the minimum approximation error depends on control input and therefore, the HN tracking performance cannot be achieved (Basar & Berhard, 1990; Doyle, Glover, Khargonekar, & Francis, 1989; Francis, 1987; Kang, Lee, & Park, 1998; Stoorvogel, 1992; Van der Schaft, 1992). To solve this problem, bð~ xÞ is approximated differently. Considering the functionality of bðx3 Þ ¼ gr ðx3;in x3 Þ=V j , the following approximator is chosen: ^ 3 Þ ¼ y1;b þ y2;b x3 . bðx
(11)
By selecting the above approximator, the minimum approximation error does not depend on the control input which will be shown later. By substituting estimates of a and b in Eq. (8) and adding additional term v, the following control law is obtained: ^ 3 Þ, ~e a^ ðx2 ; x3 Þ þ vÞ=bðx u ¼ ðy€ r K~
(12)
where v, to be defined later, is considered to achieve the HN tracking performance. By substituting Eq. (12) into Eq. (7), the following closed loop error dynamic is obtained: ~ xÞ a^ ðx2 ; x3 Þ þ ðbðx3 Þ bðx ^ 3 ÞÞu þ vÞ. ~ e_ ¼ A~ e þ bðað~ Let us define the optimal parameter estimates ~ ya as " # ~ y ¼ arg min sup k^aðx2 ; x3 Þ að~ xÞk , a
~ ya 2Oa
(13)
(14)
~ x2Ox
where Oa and Ox denote a set of suitable bounds on ~ ya and ~ x. By using Eq. (14), the minimum approximation error is
ARTICLE IN PRESS S. Salehi, M. Shahrokhi / Control Engineering Practice 16 (2008) 1101–1108
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defined as w ¼ að~ xÞ
4. Simulation ya Þ. a^ ðx2 ; x3 j~
(15)
As can be observed, w does not depend on the control input. Using the above equation in Eq. (13), the closed loop error dynamic can be rewritten as ~ aðx2 ; x3 j~ ~ e_ ¼ A~ e þ bð^ ya Þ a^ ðx2 ; x3 j~ ya Þ ^ 3 ÞÞu þ v þ wÞ. þ ðbðx3 Þ bðx
(16)
½y1;b ; y2;b T , ~ yb
Defining ~ yb ¼ ¼ ½gr x3;in ; 1T =V j and using Eqs. (10) and (11), Eq. (16) becomes T ~T ~~ ~ e_ ¼ A~ e þ bð y~ a ~ (17) Zðx3 Þu þ v þ wÞ, Bðx2 ; x3 Þ þ y~ b ~ ~ ~ ya ~ ya , y~ b ¼ ~ yb ~ yb and ~ Zðx3 Þ ¼ ½1; x3 T . where y~ a ¼ ~
Theorem. Consider the control law (12) with
H2 O=Hþ
H2 O=Hþ
C5 H6 ! C5 H7 OH ! C5 H8 ðOHÞ2 ; 2C5 H6 !C10 H12 :
1 T v ¼ b~ P~ e, q
(18)
T _ ~ ya ¼ ga b~ P~ e~ Bðx2 ; x3 Þ,
(19)
T _ ~ yb ¼ gb b~ P~ e~ Zðx3 Þu,
(20)
where ga, gb and q are positive constants and P ¼ PT40 is the solution of the following Riccati equation: T T 2 1 PA þ AT P þ Q Pb~b~ P þ 2 Pb~b~ P ¼ 0, q r
The process considered for simulation is the one used by other researchers (Chen, Kremling, & Allgo¨wer, 1995; Gopaluni et al., 2003; Klatt & Engell, 1998). It involves production of cyclopentenol (B) from cyclopentadiene (A) by acid-catalyzed electrophilic addition of water in dilute solution. Because of the strong reactivity of the reactant A and product, dicyclopentadiene (V) is produced by Diels–Alder reaction as a side product, and cyclopentanediol (C) is generated as a consecutive product through the addition of another water molecule. The complete reaction scheme is
(21)
where Q ¼ QT40. If w 2 L2 , then the HN tracking performance is achieved with a prescribed attenuation level r for the closed loop error dynamic given by Eq. (17).
The reactor inflow contains only the reactant A. Constant density and an ideal residence time distribution within the reactor are assumed. The dynamics of the reactor can be descried by the set of Eqs. (1)–(3). x11 and x12 represent components of A and B, respectively, their corresponding vector is ~ xT1 ¼ ½x11 x12 , and the inlet vector is T ~ are as ~ x1;in ¼ ½x11;in x12;in . For this system Kst, ~ j and DH follows: 1 0 1 K st ¼ , 1 1 0 2
k10 x11 eE 1 =x2
3
6 7 ~ jð~ x1 ; x2 Þ ¼ 4 k20 x12 eE 2 =x2 5, k30 x211 eE 3 =x2
Proof is given in Appendix A. Remark 1. The Riccati equation (Eq. (21)) has a positive semi-definite solution if and only if 2r2 Xq (Chen et al., 1996). Remark 2. In order to constraint ~ ya ; ~ yb to some predefined sets ðOa ; Ob Þ, the projection algorithm can be used to modify adaptation laws (19) and (20) (Goodwin & Mayne, 1987; Luenberger, 1984; Narendra & Annaswamy, 1989). It should be noted that as r decreases, q is decreased resulting in a higher control action. On the other hand, in all physical applications, actuators are constrained and therefore the attenuation level is limited in practice. Finally in order to complete the stability analysis of the closed loop system under control law of Eq. (12), the stability of internal states, ~ c, must be checked. For CSTRs it has been shown that all concentration trajectories ~ x1 ðt; ~ x1;o Þ, ~ x1;o 2
~¼ DH
1 BC ½DH AB DH AD R DH R R , rp cp
where ki0’s are reaction constant rates, Ei’s are activation energies, rp and cp are density and heat capacity of the reactor content. Values for the physical and kinetic parameters are given in Table 1. By constructing the observability matrix, it can be shown that the system is observable if the reactor temperature is measured. k1 and k2 are set to 50 and 120, respectively. By choosing Q ¼ I, 2r2 ¼ q and solving the Riccati equation (Eq. (21)), P is obtained as 0:2184 0:5 P¼ . 0:5 1:21 The fuzzy rule bases of Eq. (10) are designed as follows: Rði;jÞ : if x2 is F j2 and x3 is F i3 then a^ is yi;j for 1pip10 1pjp4.
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Ziegler–Nicholes technique, controller parameters are determined (kc ¼ 1:5, tI ¼ 18 and tD ¼ 4:5). In the simulation, first performance of the proposed controller for temperature transition from 408 to 407.14 K is examined. Fig. 2 depicts the reactor temperature and the corresponding control action for r ¼ 0.05 and 0.5. The result for the PID controller is also shown in Fig. 2. The integral squares of error for the above values of r and the PID controller are tabulated in Table 2. As can be seen, by decreasing the attenuation level, the integral squares of error is decreased which indicates that HN tracking performance has been achieved for the proposed controller. Moreover, from Fig. 2 and Table 2, it is found that by decreasing the attenuation level, a better performance can be achieved compared to the PID controller. In the next simulation, temperature tracking is tested and the desired trajectory is assumed to vary as yr ¼ 407:17 þ 5 sinðt=500Þ. The closed loop responses and the corresponding control actions for the proposed and PID controllers are shown in Fig. 3. Their corresponding integral squares of error are given in Table 2. Again, it is observed that a better performance, compared to PID
Table 1 Reactor parameters Symbol
Value
k10 k20 k30 E1 E2 E3 DH AB R DH BC R DH AD R rp cP Vr Vj Fr x11,in x12,in x2,in x3,in gr gj
3.575 108/s 3.575 108/s 1.256 106 mol/s 9758.3 K 9758.3 K 8560 K 4.20 kJ/mol 11.0 kJ/mol 41.85 kJ/mol 0.9342 kg/l 3.01 kJ/(kg K) 10 l 5l 0.0522 l/s 5.01 mol/l 0 403 K 363 K 85.6347 104/s 2.408 102/s
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The following memberships have been selected: 8 > < 1=ð1 þ expð0:4 ðx 350ÞÞÞ; 2 mF i2 ðxÞ ¼ expððx ð10 ði 1Þ þ 350ÞÞ =50Þ; > : 1=ð1 þ expð0:4 ðx 440ÞÞÞ;
i ¼ 1; 2pip9; i ¼ 10;
8 1=ð1 þ expð0:16 ðx 350ÞÞÞ; > < 2 mF i3 ðxÞ ¼ expððx ð25 ði 1Þ þ 350ÞÞ =50Þ; > : 1=ð1 þ expð0:16 ðx 440ÞÞÞ;
i ¼ 1; 2pip3; i ¼ 4:
Using the above memberships and the fuzzy rule bases, the fuzzy approximator of Eq. (10) is constructed and can be written as P10 Q4 i¼1 j yi;j mF i2 ðx2 ÞmF j3 ðx3 Þ a^ ðx2 ; x3 Þ ¼ P10 Q4 , i¼1 j mF i2 ðx2 ÞmF j ðx3 Þ 3
~ ya and ~ B are ~ ya ¼ ðy1;1 ; . . . ; y1;4 ; y2;1 ; . . . ; y2;4 ; . . . ; y10;4 ÞT , T
~ Bðx2 ; x3 Þ ¼ ðB1;1 ; . . . ; B1;4 ; B2;1 ; . . . ; B2;4 ; . . . ; B10;4 Þ where mF i ðx2 ÞmF j ðx3 Þ 3 Bi;j ðx2 ; x3 Þ ¼ P10 Q2 4 . i ðx2 Þm j ðx3 Þ m i¼1 j F 2 F 3
The initial conditions for the CSTR are selected to be ~ xT1 ð0Þ ¼ ½2 0:5, x2 ð0Þ ¼ 408, x3 ð0Þ ¼ 393 and the manipulated variable is subject to the following constraints: 0pup0:5 l=s. To compare the performance of the proposed controller with a PID controller, system has been modeled by a linear dynamic and using the
Fig. 2. (a) Reactor temperature transient responses from an initial condition to the desired value: r ¼ 0.05 (solid line), r ¼ 0.5 (dashed line), PID (dotted line), set-point (dash-dot line) and (b) corresponding control actions.
Table 2 Integral squares of error for different cases Case
r ¼ 0.05
r ¼ 0.5
PID
Initial temperature transient response in regulation Temperature tracking
14
1100
258
33
1960
465
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Fig. 3. (a) Reactor temperature tracking responses: r ¼ 0.05 (solid line), r ¼ 0.5 (dashed line), PID (dotted line), set-point (dash-dot line) and (b) corresponding control actions.
Fig. 4. Reactor temperature transient response for 5% increase in the feed temperature applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
controller, is achieved for the proposed controller by decreasing the tuning parameter r. To demonstrate the performance of the proposed controller for load rejection, the feed specifications are changed and the closed loop responses are shown in Figs. 4–6 for r=0.05. The results indicate that the proposed controller has a good load rejection performance. Finally, to show the effectiveness of the proposed controller against model uncertainties, 15% decrease in
Fig. 5. Reactor temperature transient response for 50% increase in the feed concentration applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
Fig. 6. Reactor temperature transient response for 15% increase in the feed flow rate applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
heat transfer coefficient and reaction rate constants are applied to the system at t=50 s for r=0.05 and the results are illustrated in Figs. 7 and 8, respectively. As can be seen the controller is robust against model uncertainties as well. 5. Conclusion In this work, an adaptive controller is proposed for temperature control of a class of CSTRs by means of input–output feedback linearization technique. Since all
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has a better performance compared to a PID controller tuned by Ziegler–Nicholes technique. Appendix A. Proof of Theorem Consider the following Lyapunov function: T T 1 T 1 ~ 1 ~ ~ ~ y~ a y~ a þ y~ b y~ b . eþ e P~ V¼ ~ 2 2ga 2gb
(A.1)
The time derivative of V is 1 _T 1 T _ 1 ~_~ T~~ 1 ~_ T~ e P~ V_ ¼ ~ eþ ~ e þ ya ya þ y~ b y~ b . e P~ 2 2 ga gb
(A.2)
Using Eq. (17) in Eq. (A.2) yields T 1 T T ~T ~T ~T ~T V_ ¼ ½~ e A P~ e þ y~ a ~ e þ y~ b ~ eu þ vb~ P~ e Zb P~ Bb P~ 2 T ~BT~ ~ZT~ y~ a þ ~ y~ b u þ wb~ P~ e þ~ eT PA~ e þ~ eT Pb~ eT Pb~
Fig. 7. Reactor temperature transient response for 15% decrease in the heat transfer coefficient applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) (b) corresponding control action.
_ T~ _ T~ 1~ ~ þ~ ~ þ 1~ y~ a y~ a þ y~ b y~ b . eT Pbw þ~ eT Pbv ga gb
(A.3)
Using Eq. (18) and rearranging Eq. (A.3) yields T 1 T 2 V_ ¼ ~ e e ðPA þ AT P Pb~b~ PÞ~ 2 q _T ~ 1 ~BT þ ~ y~ a Þy~ a þ ðga~ eT Pb~ ga _T ~ T 1 ~ZT u þ ~ y~ b Þy~ b þ wb~ P~ þ ðgb~ e. eT Pb~ gb
(A.4)
Using the adaptive laws (19) and (20) and Eq. (21), V_ can be rewritten as
Fig. 8. Reactor temperature transient response for 15% decrease in the reaction rate constants applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
states are required for control implementation and it is assumed that concentrations are not available, a fuzzy estimator is designed to estimate concentration dependant term in control law using temperature measurements. The overall stability of the closed loop is established using the Lyapunov stability theorem. It has been shown that by using the proposed controller, HN temperature tracking performance has been achieved. The effectiveness of the proposed scheme has been demonstrated through computer simulation and it is shown that the proposed controller
T 1 T 1 T ~~T V_ ¼ ~ e 2~ e þ wb~ P~ e e Q~ e Pbb P~ 2 2r 2 1 T 1 1 ~T b P~ ¼ ~ e e rw e Q~ 2 2 r 1 2 2 1 T 1 þ r w p ~ e þ r2 w2 . e Q~ 2 2 2 Integrating the inequality (A.5) yields Z Z 1 T T 1 2 T 2 ~ e dtpV ð0Þ V ðTÞ þ r w dt. e Q~ 2 0 2 0
Since V ðTÞX0, as T ! 1, it results in Z Z 1 T T 1 2 T 2 ~ e dtpV ð0Þ þ r w dt, e Q~ 2 0 2 0
(A.5)
(A.6)
(A.7)
i.e. the HN tracking performance is achieved for the closed loop error dynamic (17) (Basar & Berhard, 1990; Doyle et al., 1989; Francis, 1987; Stoorvogel, 1992; Van der Schaft, 1992). References Adebekun, A. K. (1992). The robust global stabilization of a stirred tank reactor. AIChE Journal, 38(5), 651–659.
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Kosanovich, K. A., Piovoso, M. J., Rokhlenko, V., & Guez, A. (1995). Nonlinear adaptive control with parameter estimation of a CSTR. Journal of Process Control, 5, 133–148. Lee, S. Y., Coronella, C. J., Bhadkamkar, A. S., & Seader, J. D. (1993). Modelling and temperature regulation of a thermally coupled reactor system via internal model control strategy. Industrial & Engineering Chemistry Research, 32, 3029–3036. Limqueco, L. C., & Kantor, J. C. (1990). Nonlinear output feedback control of and exothermic reactor. Computers Chemical Engineering, 14(4/5), 427–437. Luenberger, D. G. (1984). Linear and nonlinear programming. Reading, MA: Addison-Wesley. Narendra, K. S., & Annaswamy, A. M. (1989). Stable adaptive systems. Englewood Cliffs, NJ: Prentince-Hall. Ratto, M. (1998). Theoretical approach to the analysis of PI-controlled CSTR’s with noise. Computers & Chemical Engineering, 22, 1581–1593. Salehi, S., Shahrokhi, M., & Salahshoor, K. (2006). Adaptive fuzzy output feedback control of an unstable reactor. In Proceeding of IEEE international conference on control application (pp. 1374–1379), Munich, Germany. Santos, L. O., Afonso, P. A. F. N. A., Castro, J. A. A. M., Oliveira, N. M. C., & Biegler, L. T. (2001). On-line implementation of nonlinear MPC: An experimental case study. Control Engineering Practice, 9(8), 847–857. Sistu, P. B., & Bequette, B. W. (1992). A comparison of nonlinear control techniques for continuous stirred tank reactors. Chemical Engineering Science, 47(9–11), 2553–2558. Slotine, E., & Li, W. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Stoorvogel, A. (1992). The HN control problems: A state space approach. Englewood Cliffs, NJ: Prentice-Hall. Van der Schaft, A. J. (1992). L2-gain analysis of nonlinear systems and nonlinear state feedback HN control. IEEE Transactions on Automatic Control, 37, 770–784. Wang, L. X. (1993). Stable adaptive fuzzy control of nonlinear system. IEEE Transactions on Fuzzy Systems, 1(2), 146–155. Wang, L. X. (1994). Adaptive fuzzy systems and control: Design and stability. Englewood Cliffs, NJ: Prentice-Hall. Wang, L. X. (1996). Stable adaptive controllers with application to inverted pendulum tracking. IEEE Transactions on Systems Man and Cybernetics Part B, 26, 677–691. Wang, L. X. (1997). A course of fuzzy system and control. New York: Prentice-Hall. Wang, L. X., & Mendel, J. M. (1992). Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Transactions on Neural Networks, 3, 807–814. Wu, W. (1999). Adaptive nonlinear control of non-minimum phase processes. Chemical Engineering Science, 54, 3815–3829. Yu, D. L., & Gomm, J. B. (2003). Implementation of neural network predictive control to a multivariable chemical reactor. Control Engineering Practice, 11(11), 1315–1323. Zhang, T., & Guay, M. (2001). Adaptive nonlinear control of continuously stirred tank reactor system, In Proceedings of the American control conference (pp. 1274–1277).
Control Engineering Practice 16 (2008) 1101–1108 www.elsevier.com/locate/conengprac
Adaptive fuzzy approach for HN temperature tracking control of continuous stirred tank reactors Shahin Salehia, Mohammad Shahrokhib, a
Department of Measurement and Automation, Petroleum University of Technology, Tehran, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11365-9465 Azadi Avenue, Tehran, Iran
b
Received 24 June 2007; accepted 22 December 2007 Available online 15 February 2008
Abstract In this paper, an adaptive fuzzy temperature controller is proposed for a class of continuous stirred tank reactors (CSTRs) based on input–output feedback linearization. Since for control implementation concentrations of all species are needed, based on the observability concept, a fuzzy logic system is used to estimate the concentration dependent terms and other unknown system parameters in the control law, using temperature measurements. It has been shown that the HN tracking control performance with a prescribed attenuation level is achieved, by using the proposed controller. Finally the effectiveness of the proposed controller has been demonstrated by applying it to a benchmark chemical reactor through computer simulation. r 2008 Elsevier Ltd. All rights reserved. Keywords: Feedback linearization; HN performance; Adaptive control; Fuzzy logic systems; Chemical reactors
1. Introduction Temperature control of chemical reactors is an important objective for achieving higher product qualities. Complex static and dynamic behaviors, system nonlinearity, unavailability of states, multiplicity and instability of equilibrium points and input constraint have made it a challenging problem. Numerous temperature controllers for continuous stirred tank reactors (CSTRs), based on conventional PID structure and state feedback control technique, have been proposed (Brown, Gonyie, Schwber, & James, 1998; Lee, Coronella, Bhadkamkar, & Seader, 1993; Ratto, 1998). Stabilization of chemical reactors by output feedback with PI controllers has been reviewed in a Ph.D. thesis (Jadot, 1996). Global stability of a reactor with an exothermic reaction using state feedback was proved by Adebekun and Schork (1991) and the same problem, using a state observer, has been considered by Adebekun (1992) and Kosanovich, Piovoso, Rokhlenko, and Guez (1995). Advanced control strategies, like predictive control Corresponding author. Tel.: +98 21 66165419; fax: +98 21 66072679.
E-mail address: [email protected] (M. Shahrokhi). 0967-0661/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2007.12.005
scheme, have been also used for temperature control of CSTRs (Santos, Afonso, Castro, Oliveira, & Biegler, 2001; Sistu & Bequette, 1992; Yu & Gomm, 2003). For instance, some comparisons among these techniques have been made (Sistu & Bequette, 1992). In addition, backstepping and feedback linearization methodologies are applied either in non-adaptive or adaptive forms (Gopaluni, Mizumoto, & Shah, 2003; Haugwitz, Hagander, & Nore´n, 2007; Limqueco & Kantor, 1990; Salehi, Shahrokhi, & Salahshoor, 2006; Wu, 1999; Zhang & Guay, 2001). On the other hand, since fuzzy logic systems are used as universal approximators with arbitrary accuracy for any real continuous function on a compact set (Wang & Mendel, 1992), it has attracted great interests in utilizing heuristic-based approaches to cope with the control problem of nonlinear and ill-conditioned systems (Wang, 1993, 1994, 1996). Experimental evaluation of fuzzy logicbased controllers for controlling chemical processes has been investigated by Fileti, Antunes, Silva, Silveira, and Pereira (2007). A novel nonlinear adaptive controller based on fuzzy logic systems and Fourier Integral has been proposed for temperature control of a CSTR (Huaguang & Cai, 2002). Most of advanced temperature controllers proposed for CSTRs require concentration measurements
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for implementation which are not available in practice. Therefore, designing a model-based controller which does not require composition measurements is highly motivated. The main objective of this paper is to modify our previous control scheme (Salehi et al., 2006) to achieve the HN temperature tracking performance. In that work, temperature control of a CSTR in which a single reaction takes place is discussed. In the present work, a general class of CSTRs in which multiple reactions are taking place is considered. By modifying the adaptive law, the fuzzy minimum approximation error has become independent of control action which results in achieving the HN temperature tracking performance. By taking advantage of the observability concept, a fuzzy controller is designed such that no composition measurement is required. In addition, stability of the closed loop system has been established and boundedness of the internal dynamics has been discussed. The paper is organized as follows. In Section 2, mathematical model of CSTRs is presented. In Section 3, an adaptive fuzzy nonlinear controller is designed by input–output feedback linearization to achieve the HN temperature tracking performance. In Section 4, the effectiveness of the proposed controller has been demonstrated by applying it to a benchmark chemical reactor and a comparison between the proposed controller and a PID whose parameters are tuned by Ziegler–Nicholes technique is made. Finally, conclusion is drawn in Section 5. 2. Mathematical model of CSTRs In this section, a model for a general class of CSTRs is presented. It is assumed that a set of M reactions with N components are taking place in the reactor. The dynamical model of the CSTR is obtained from mass and energy balances and can be written in the following matrix form (Alvarez-Ramirez, 1995): ~ x_ 1 ¼ F r ð~ x1;in ~ x1 Þ=V r þ K st ~ jð~ x1 ; x2 Þ,
(1)
~ x1 ; x2 Þ~ x_ 2 ¼ F r ðx2;in x2 Þ=V r DHð~ jð~ x1 ; x2 Þ þ gr ðx3 x2 Þ, x_ 3 ¼ gj ðx2 x3 Þ þ F j ðx3;in x3 Þ=V j ,
Fig. 1. Schematic diagram of the process.
It is assumed that ~ x1 vector (concentrations of all species) is unavailable, x2 and x3 are measured, and Kst, ~ j, ~ gr and gj are not known. Also, the volumes of reactor DH, and jacket, Vr and Vj, are assumed to be constant. The system equations can be written as ( ~ x_ ¼ f~ð~ xÞ þ ~ gð~ xÞu; (4) y ¼ hð~ xÞ; xÞ ¼ x2 , where ~ x ¼ ½~ xT1 x2 x3 T 2
f1
3
6 7 f 7 f~ð~ xÞ ¼ 6 4 25 f3
2
F r ð~ x1;in ~ x1 Þ=V r þ K st ~ jð~ x1 ; x2 Þ
6
3 7
~ x1 ; x2 Þ~ jð~ x1 ; x2 Þ þ gr ðx3 x2 Þ 7, F ðx x2 Þ=V r DHð~ ¼6 4 r 2;in 5 gj ðx2 x3 Þ
(2)
2
0
3
(3)
6 ~ gð~ xÞ ¼ 4
0
7 5.
x1 2
ðx3;in x3 Þ=V j
3. Adaptive fuzzy controller design In this section, a nonlinear adaptive fuzzy controller, based on input–output linearization, has been designed for a class of CSTRs whose dynamic models are given by Eq. (4). The basic controller structure is proposed by Chang (2001) and Chen, Lee, and Chang (1996). In these papers, two adaptive fuzzy controllers, using linguist rules, for a class of nonlinear system have been introduced. Using their strategy and input–output linearization technique, a
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modified adaptive controller has been proposed for temperature tracking of CSTRs given by Eq. (4). In the first step of design, the system of Eq. (4) is transformed into the normal form (Isidori, 1995; Khalil, 1996; Slotine & Li, 1991). Under this transformation, Eq. (4) can be written as 8 _ ~ > c¼~ qð~ c; ~ xÞ; > > > < x_ ¼ x ; 1 2 (5) _ > x2 ¼ að~ xÞ þ bð~ xÞu; > > > : y ¼ x1 ; where ~ x ¼ ðx1 ; x2 Þ 2 <2 is the external state vector, ~ c2 n2 < is the internal state vector, a and b are given by following equations: ~ jÞ=q~ xÞ ¼ ðf 1 þ f 2 ÞqðDH~ x1 að~ xÞ ¼ L2f~hð~ ðF r =V r þ gr Þf 2 þ gr f 3 ,
the approximator. First, fuzzy logic systems are universal approximators with arbitrary accuracy for any given continuous function (Wang & Mendel, 1992). Second, fuzzy logic systems are constructed from fuzzy IF-THEN rules using some specific fuzzy inference, fuzzification, and defuzzification strategies, hence, linguistic information from human experts (in the form of the fuzzy IF-THEN rules) can be directly incorporated into the approximator design. að~ xÞ is a function of available (temperature measurements) and unavailable states (concentrations). It is assumed that the system is observable through reactor temperature measurements. By taking into account the observability concept, it can be concluded that the species concentrations, ~ x1 , can be estimated from the system output (x2). Consequently, að~ xÞ can be approximated using temperature measurements (x2, x3). If estimate of að~ xÞ is denoted by a^ , then the estimator can be constructed as T
ya ~ a^ ðx2 ; x3 Þ ¼ ~ Bðx2 ; x3 Þ,
xÞ ¼ gr ðx3;in x3 Þ=V j . bð~ xÞ ¼ L~g Lf~hð~ As can be seen, the relative degree of the system is two. e ¼ ðe1 ; e2 ÞT ¼ ðe; e_ÞT , where yr is Defining e ¼ yyr and ~ the desired output, the tracking error dynamic can be expressed as ~ xÞ þ bð~ ~ e_ ¼ Ao~ e þ bðað~ xÞu y€ r Þ, where 0 Ao ¼ 0
1 0
;
(6)
0 b~ ¼ . 1
~ ¼ ðk2 ; k1 Þ 2 <12 such that A Ao Choosing matrix K ~ ~ bK is Hurwitz, Eq. (6) becomes: ~ K~ ~e þ að~ ~ e_ ¼ A~ e_ þ bð xÞ þ bð~ xÞu y€ r Þ.
(7)
If the system mathematical model is exactly known and there is no model uncertainty, by applying the following nonlinear control law: ~e að~ xÞÞ=bð~ xÞ, u ¼ ðy€ r K~
(8)
the error dynamic becomes ~ e_ ¼ A~ e.
(9)
This error dynamic equation is asymptotical stable because A is Hurwitz. It should be noted that for implementation of control law (8), concentrations of all species in the reactor and system parameters are needed. But in practice, concentrations are not measured because compositionmeasuring devices are expensive and include lag. Usually, exact kinetic model is not available and there are uncertainties in the system parameters. To make the problem more realistic, it is assumed that all of the required ~ gr and gj are not available. data such as Kst, ~ j, DH, One way to solve the above problem is to estimate að~ xÞ and bð~ xÞ by appropriate estimators. In this work, a fuzzy approximator is used to estimate að~ xÞ. There are two main reasons for using the fuzzy logic as basic building blocks of
1103
(10)
T T where ~ ya ¼ ð~ y1;a ; . . . ; ~ ym;a ÞT is a fuzzy parameter vector, and
~ BT1 ðx2 ; x3 Þ; . . . ;~ BTm ðx2 ; x3 ÞÞT is the fuzzy regresBðx2 ; x3 Þ ¼ ð~ sive vector with the regressor ~ Bi ðx2 ; x3 Þ and m is the number of fuzzy rule bases (Chen et al., 1996; Wang, 1997). However, if a fuzzy approximator is used for estimating bð~ xÞ, then the minimum approximation error depends on control input and therefore, the HN tracking performance cannot be achieved (Basar & Berhard, 1990; Doyle, Glover, Khargonekar, & Francis, 1989; Francis, 1987; Kang, Lee, & Park, 1998; Stoorvogel, 1992; Van der Schaft, 1992). To solve this problem, bð~ xÞ is approximated differently. Considering the functionality of bðx3 Þ ¼ gr ðx3;in x3 Þ=V j , the following approximator is chosen: ^ 3 Þ ¼ y1;b þ y2;b x3 . bðx
(11)
By selecting the above approximator, the minimum approximation error does not depend on the control input which will be shown later. By substituting estimates of a and b in Eq. (8) and adding additional term v, the following control law is obtained: ^ 3 Þ, ~e a^ ðx2 ; x3 Þ þ vÞ=bðx u ¼ ðy€ r K~
(12)
where v, to be defined later, is considered to achieve the HN tracking performance. By substituting Eq. (12) into Eq. (7), the following closed loop error dynamic is obtained: ~ xÞ a^ ðx2 ; x3 Þ þ ðbðx3 Þ bðx ^ 3 ÞÞu þ vÞ. ~ e_ ¼ A~ e þ bðað~ Let us define the optimal parameter estimates ~ ya as " # ~ y ¼ arg min sup k^aðx2 ; x3 Þ að~ xÞk , a
~ ya 2Oa
(13)
(14)
~ x2Ox
where Oa and Ox denote a set of suitable bounds on ~ ya and ~ x. By using Eq. (14), the minimum approximation error is
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defined as w ¼ að~ xÞ
4. Simulation ya Þ. a^ ðx2 ; x3 j~
(15)
As can be observed, w does not depend on the control input. Using the above equation in Eq. (13), the closed loop error dynamic can be rewritten as ~ aðx2 ; x3 j~ ~ e_ ¼ A~ e þ bð^ ya Þ a^ ðx2 ; x3 j~ ya Þ ^ 3 ÞÞu þ v þ wÞ. þ ðbðx3 Þ bðx
(16)
½y1;b ; y2;b T , ~ yb
Defining ~ yb ¼ ¼ ½gr x3;in ; 1T =V j and using Eqs. (10) and (11), Eq. (16) becomes T ~T ~~ ~ e_ ¼ A~ e þ bð y~ a ~ (17) Zðx3 Þu þ v þ wÞ, Bðx2 ; x3 Þ þ y~ b ~ ~ ~ ya ~ ya , y~ b ¼ ~ yb ~ yb and ~ Zðx3 Þ ¼ ½1; x3 T . where y~ a ¼ ~
Theorem. Consider the control law (12) with
H2 O=Hþ
H2 O=Hþ
C5 H6 ! C5 H7 OH ! C5 H8 ðOHÞ2 ; 2C5 H6 !C10 H12 :
1 T v ¼ b~ P~ e, q
(18)
T _ ~ ya ¼ ga b~ P~ e~ Bðx2 ; x3 Þ,
(19)
T _ ~ yb ¼ gb b~ P~ e~ Zðx3 Þu,
(20)
where ga, gb and q are positive constants and P ¼ PT40 is the solution of the following Riccati equation: T T 2 1 PA þ AT P þ Q Pb~b~ P þ 2 Pb~b~ P ¼ 0, q r
The process considered for simulation is the one used by other researchers (Chen, Kremling, & Allgo¨wer, 1995; Gopaluni et al., 2003; Klatt & Engell, 1998). It involves production of cyclopentenol (B) from cyclopentadiene (A) by acid-catalyzed electrophilic addition of water in dilute solution. Because of the strong reactivity of the reactant A and product, dicyclopentadiene (V) is produced by Diels–Alder reaction as a side product, and cyclopentanediol (C) is generated as a consecutive product through the addition of another water molecule. The complete reaction scheme is
(21)
where Q ¼ QT40. If w 2 L2 , then the HN tracking performance is achieved with a prescribed attenuation level r for the closed loop error dynamic given by Eq. (17).
The reactor inflow contains only the reactant A. Constant density and an ideal residence time distribution within the reactor are assumed. The dynamics of the reactor can be descried by the set of Eqs. (1)–(3). x11 and x12 represent components of A and B, respectively, their corresponding vector is ~ xT1 ¼ ½x11 x12 , and the inlet vector is T ~ are as ~ x1;in ¼ ½x11;in x12;in . For this system Kst, ~ j and DH follows: 1 0 1 K st ¼ , 1 1 0 2
k10 x11 eE 1 =x2
3
6 7 ~ jð~ x1 ; x2 Þ ¼ 4 k20 x12 eE 2 =x2 5, k30 x211 eE 3 =x2
Proof is given in Appendix A. Remark 1. The Riccati equation (Eq. (21)) has a positive semi-definite solution if and only if 2r2 Xq (Chen et al., 1996). Remark 2. In order to constraint ~ ya ; ~ yb to some predefined sets ðOa ; Ob Þ, the projection algorithm can be used to modify adaptation laws (19) and (20) (Goodwin & Mayne, 1987; Luenberger, 1984; Narendra & Annaswamy, 1989). It should be noted that as r decreases, q is decreased resulting in a higher control action. On the other hand, in all physical applications, actuators are constrained and therefore the attenuation level is limited in practice. Finally in order to complete the stability analysis of the closed loop system under control law of Eq. (12), the stability of internal states, ~ c, must be checked. For CSTRs it has been shown that all concentration trajectories ~ x1 ðt; ~ x1;o Þ, ~ x1;o 2
~¼ DH
1 BC ½DH AB DH AD R DH R R , rp cp
where ki0’s are reaction constant rates, Ei’s are activation energies, rp and cp are density and heat capacity of the reactor content. Values for the physical and kinetic parameters are given in Table 1. By constructing the observability matrix, it can be shown that the system is observable if the reactor temperature is measured. k1 and k2 are set to 50 and 120, respectively. By choosing Q ¼ I, 2r2 ¼ q and solving the Riccati equation (Eq. (21)), P is obtained as 0:2184 0:5 P¼ . 0:5 1:21 The fuzzy rule bases of Eq. (10) are designed as follows: Rði;jÞ : if x2 is F j2 and x3 is F i3 then a^ is yi;j for 1pip10 1pjp4.
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Ziegler–Nicholes technique, controller parameters are determined (kc ¼ 1:5, tI ¼ 18 and tD ¼ 4:5). In the simulation, first performance of the proposed controller for temperature transition from 408 to 407.14 K is examined. Fig. 2 depicts the reactor temperature and the corresponding control action for r ¼ 0.05 and 0.5. The result for the PID controller is also shown in Fig. 2. The integral squares of error for the above values of r and the PID controller are tabulated in Table 2. As can be seen, by decreasing the attenuation level, the integral squares of error is decreased which indicates that HN tracking performance has been achieved for the proposed controller. Moreover, from Fig. 2 and Table 2, it is found that by decreasing the attenuation level, a better performance can be achieved compared to the PID controller. In the next simulation, temperature tracking is tested and the desired trajectory is assumed to vary as yr ¼ 407:17 þ 5 sinðt=500Þ. The closed loop responses and the corresponding control actions for the proposed and PID controllers are shown in Fig. 3. Their corresponding integral squares of error are given in Table 2. Again, it is observed that a better performance, compared to PID
Table 1 Reactor parameters Symbol
Value
k10 k20 k30 E1 E2 E3 DH AB R DH BC R DH AD R rp cP Vr Vj Fr x11,in x12,in x2,in x3,in gr gj
3.575 108/s 3.575 108/s 1.256 106 mol/s 9758.3 K 9758.3 K 8560 K 4.20 kJ/mol 11.0 kJ/mol 41.85 kJ/mol 0.9342 kg/l 3.01 kJ/(kg K) 10 l 5l 0.0522 l/s 5.01 mol/l 0 403 K 363 K 85.6347 104/s 2.408 102/s
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The following memberships have been selected: 8 > < 1=ð1 þ expð0:4 ðx 350ÞÞÞ; 2 mF i2 ðxÞ ¼ expððx ð10 ði 1Þ þ 350ÞÞ =50Þ; > : 1=ð1 þ expð0:4 ðx 440ÞÞÞ;
i ¼ 1; 2pip9; i ¼ 10;
8 1=ð1 þ expð0:16 ðx 350ÞÞÞ; > < 2 mF i3 ðxÞ ¼ expððx ð25 ði 1Þ þ 350ÞÞ =50Þ; > : 1=ð1 þ expð0:16 ðx 440ÞÞÞ;
i ¼ 1; 2pip3; i ¼ 4:
Using the above memberships and the fuzzy rule bases, the fuzzy approximator of Eq. (10) is constructed and can be written as P10 Q4 i¼1 j yi;j mF i2 ðx2 ÞmF j3 ðx3 Þ a^ ðx2 ; x3 Þ ¼ P10 Q4 , i¼1 j mF i2 ðx2 ÞmF j ðx3 Þ 3
~ ya and ~ B are ~ ya ¼ ðy1;1 ; . . . ; y1;4 ; y2;1 ; . . . ; y2;4 ; . . . ; y10;4 ÞT , T
~ Bðx2 ; x3 Þ ¼ ðB1;1 ; . . . ; B1;4 ; B2;1 ; . . . ; B2;4 ; . . . ; B10;4 Þ where mF i ðx2 ÞmF j ðx3 Þ 3 Bi;j ðx2 ; x3 Þ ¼ P10 Q2 4 . i ðx2 Þm j ðx3 Þ m i¼1 j F 2 F 3
The initial conditions for the CSTR are selected to be ~ xT1 ð0Þ ¼ ½2 0:5, x2 ð0Þ ¼ 408, x3 ð0Þ ¼ 393 and the manipulated variable is subject to the following constraints: 0pup0:5 l=s. To compare the performance of the proposed controller with a PID controller, system has been modeled by a linear dynamic and using the
Fig. 2. (a) Reactor temperature transient responses from an initial condition to the desired value: r ¼ 0.05 (solid line), r ¼ 0.5 (dashed line), PID (dotted line), set-point (dash-dot line) and (b) corresponding control actions.
Table 2 Integral squares of error for different cases Case
r ¼ 0.05
r ¼ 0.5
PID
Initial temperature transient response in regulation Temperature tracking
14
1100
258
33
1960
465
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Fig. 3. (a) Reactor temperature tracking responses: r ¼ 0.05 (solid line), r ¼ 0.5 (dashed line), PID (dotted line), set-point (dash-dot line) and (b) corresponding control actions.
Fig. 4. Reactor temperature transient response for 5% increase in the feed temperature applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
controller, is achieved for the proposed controller by decreasing the tuning parameter r. To demonstrate the performance of the proposed controller for load rejection, the feed specifications are changed and the closed loop responses are shown in Figs. 4–6 for r=0.05. The results indicate that the proposed controller has a good load rejection performance. Finally, to show the effectiveness of the proposed controller against model uncertainties, 15% decrease in
Fig. 5. Reactor temperature transient response for 50% increase in the feed concentration applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
Fig. 6. Reactor temperature transient response for 15% increase in the feed flow rate applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
heat transfer coefficient and reaction rate constants are applied to the system at t=50 s for r=0.05 and the results are illustrated in Figs. 7 and 8, respectively. As can be seen the controller is robust against model uncertainties as well. 5. Conclusion In this work, an adaptive controller is proposed for temperature control of a class of CSTRs by means of input–output feedback linearization technique. Since all
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has a better performance compared to a PID controller tuned by Ziegler–Nicholes technique. Appendix A. Proof of Theorem Consider the following Lyapunov function: T T 1 T 1 ~ 1 ~ ~ ~ y~ a y~ a þ y~ b y~ b . eþ e P~ V¼ ~ 2 2ga 2gb
(A.1)
The time derivative of V is 1 _T 1 T _ 1 ~_~ T~~ 1 ~_ T~ e P~ V_ ¼ ~ eþ ~ e þ ya ya þ y~ b y~ b . e P~ 2 2 ga gb
(A.2)
Using Eq. (17) in Eq. (A.2) yields T 1 T T ~T ~T ~T ~T V_ ¼ ½~ e A P~ e þ y~ a ~ e þ y~ b ~ eu þ vb~ P~ e Zb P~ Bb P~ 2 T ~BT~ ~ZT~ y~ a þ ~ y~ b u þ wb~ P~ e þ~ eT PA~ e þ~ eT Pb~ eT Pb~
Fig. 7. Reactor temperature transient response for 15% decrease in the heat transfer coefficient applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) (b) corresponding control action.
_ T~ _ T~ 1~ ~ þ~ ~ þ 1~ y~ a y~ a þ y~ b y~ b . eT Pbw þ~ eT Pbv ga gb
(A.3)
Using Eq. (18) and rearranging Eq. (A.3) yields T 1 T 2 V_ ¼ ~ e e ðPA þ AT P Pb~b~ PÞ~ 2 q _T ~ 1 ~BT þ ~ y~ a Þy~ a þ ðga~ eT Pb~ ga _T ~ T 1 ~ZT u þ ~ y~ b Þy~ b þ wb~ P~ þ ðgb~ e. eT Pb~ gb
(A.4)
Using the adaptive laws (19) and (20) and Eq. (21), V_ can be rewritten as
Fig. 8. Reactor temperature transient response for 15% decrease in the reaction rate constants applied at t ¼ 50 s: r ¼ 0.05 (solid line), set-point (dash-dot line) and (b) corresponding control action.
states are required for control implementation and it is assumed that concentrations are not available, a fuzzy estimator is designed to estimate concentration dependant term in control law using temperature measurements. The overall stability of the closed loop is established using the Lyapunov stability theorem. It has been shown that by using the proposed controller, HN temperature tracking performance has been achieved. The effectiveness of the proposed scheme has been demonstrated through computer simulation and it is shown that the proposed controller
T 1 T 1 T ~~T V_ ¼ ~ e 2~ e þ wb~ P~ e e Q~ e Pbb P~ 2 2r 2 1 T 1 1 ~T b P~ ¼ ~ e e rw e Q~ 2 2 r 1 2 2 1 T 1 þ r w p ~ e þ r2 w2 . e Q~ 2 2 2 Integrating the inequality (A.5) yields Z Z 1 T T 1 2 T 2 ~ e dtpV ð0Þ V ðTÞ þ r w dt. e Q~ 2 0 2 0
Since V ðTÞX0, as T ! 1, it results in Z Z 1 T T 1 2 T 2 ~ e dtpV ð0Þ þ r w dt, e Q~ 2 0 2 0
(A.5)
(A.6)
(A.7)
i.e. the HN tracking performance is achieved for the closed loop error dynamic (17) (Basar & Berhard, 1990; Doyle et al., 1989; Francis, 1987; Stoorvogel, 1992; Van der Schaft, 1992). References Adebekun, A. K. (1992). The robust global stabilization of a stirred tank reactor. AIChE Journal, 38(5), 651–659.
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