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Observer-Based Control of a Continuous Stirred Tank Reactor with Recycle Time-Delay*
Observer-Based Control of a Continuous Stirred Tank Reactor with Recycle Time-Delay M.P. Di Ciccio ∗ M. Bottini ∗∗ P. Pepe ∗ P. U. Foscolo ∗∗ ∗
Department of Electrical and Informaton Engineering,University of L’Aquila, 67040, Poggio di Roio, L’Aquila, Italy; (e-mail: [email protected], [email protected]) ∗∗ Department of Chemical Engineering,University of L’Aquila, 67040, Poggio di Roio, L’Aquila, Italy; (e-mail: [email protected], [email protected]) Abstract: This paper proposes the design of an observer-based control law for a continuous stirred tank reactor with recycle time-delay. Recent results on differential geometry for nonlinear time-delay systems are used. The many performed simulations show the high performance of the proposed control law. Keywords: Chemical industry, Chemical variables control, Temperature control, Nonlinear control systems, Observers, Time delay. 1. INTRODUCTION The control of the operation of chemical reactors has attracted the attention of researchers for a long time. The underlying motivation relies on the fact that industrial chemical reactors are frequently operated at unstable operating conditions, which often corresponds to optimal process performance (K. Adebekun [1996], Aguilar et al. [2001], Alvarez-Ramirez et al. [2000], A.K. Jana [2007], N. Assala et al. [1997], Chang et al. [2004], Chen et al. [2006], Daaou et al. [2008], D. Fissore [2008], Gibon-Fargeot et al. [2000], Hashimoto et al. [2000], Iyer et al. [2000], Kosanovich et al. [1995], Kravaris et al. [1994], Pierri et al. [2006], Prakash et al. [2008], Senthil et al. [2006], M. Soroush [1997],Wright et al. [2006]). Polymerization processes (N. Assala et al. [1997]) and bio-reactors fermentors (M. Soroush [1997]) are important examples of largescale chemical reactors operated at unstable conditions. In addition there exist many process control strategies, in which the information about the internal state of the process is necessary to calculate the control input. Consequently the presence of unknown state variables becomes a difficulty which can be over-come with the inclusion of an appropriate state estimator (Aguilar et al. [2001], Chang et al. [2004], Daaou et al. [2008], D. Fissore [2008], K. Adebekun [1996], Salehi et al. [2000], Iyer et al. [2000], A.K. Jana [2007], Kosanovich et al. [1995], Kravaris et al. [1994], Biagiola et al. [2001], Hashimoto et al. [2000]). To our best knowledge, all these papers do not consider the problem of state estimator in presence of the time-delays in the state variables. Other papers propose methodologies borrowed by time-delay systems nonlinear control theory (W. Wu [1999], P. Pepe [2009], Cao et al. [2000]), but it is assumed that all the state variables are available B This paper is supported by the University of L’Aquila ex 60% and Ph.D. school funds.
for measure, thus a nonlinear observer for state variables estimation is not employed. In the paper Antoniades et al. [1999] an observer-based nonlinear control law for chemical reactors with recycle time-delay is presented. A model with only two state variables (reagent concentration and reactor temperature) is used, and the the dynamics of jacket refrigerant temperature is not taken into account. Also, the manipulated input is the inlet reagent concentration in the fresh feed, which can arise some further complications in the accuracy of the practical implementation of the control law. In this paper we deal with the controller design, by means of partial state measurements, for a continuous stirred tank reactor, where an irreversible, exothermic, liquidphase reaction A → B evolves. The unreacted amount of species A is fedback to the reactor through a recycle loop, in order to minimize reaction wastes. Such recycle is responsible of a non-negligible time-delay due to the fluid transport. The model of the plant is given by three nonlinear delay differential equations (the dynamics of the jacket temperature is considered too). The control input consists of the cooling water flowrate. The controlled output is the reactor temperature. The measured output consists of the reactor and jacket temperatures. Here we do not use the measurement of the reactor concentration, which is expensive to obtain by techniques of gas-chromatography. The reactor concentration is estimated by means of a nonlinear observer for time-delay systems (Germani et al. [2001], Germani et al. [2005] ). The state feedback control law is obtained by means of the method of the exact inputoutput linearization, with delay cancellation, of nonlinear time-delay systems (Germani et al. [2000], Germani et al. [2003]). The observer yields local convergence of the estimation error to zero, as well as, if the state were available, the state feedback control law yields the convergence of the state variables to the operating point. In the spirit of the
separation principle, the controller is here implemented by using the state estimations provided by the observer instead of the unavoidable full state measurements. First order approximated linearization methods are not used here. The control law is actuated by means of a pneumatic valve which regulates the cooling water flow-rate. This actuation process is easier to implement then the ones based on the use of the inlet reactor concentration or of the jacket temperature as manipulated inputs.Many performed computer simulations show the high performance of the proposed observer-based controller. Saturation effects have been taken into account in simulations. The convergence of the state variables to the arbitrarily chosen operating point is always obtained with the many considered system initial conditions (even start-up initial conditions) and initial estimation error, and with different values of the recirculation coefficient. As well, the convergence of the estimated state variables to the true state variables is always obtained in such simulations.
2. THE MODEL OF THE CSTR WITH RECYCLE TIME-DELAY
Here we study a CSTR with jacket cooling in which a firstorder irreversible exothermic reaction takes place (Luyben [2007]): A → B . A separator is considered, which yields a recycle time delay (Fig. 1). If in the reactor out flow an important quantity of not converted reagent is present, it is possible to feedback the unreacted amount of reagent to the reactor, by means of a separator, in order to minimize the reactant waste and to increase the overall conversion of the reaction. The recirculation coefficient, here denoted with Φ, is defined as: Φ=
Effluent flow-rate Total Reactor flow-rate
Therefore Φ = 1 means no recycle, Φ = 0 means total recycle. Denoting with Fr the flow-rate of the recirculating stream and with F the total reactor flow rate, it results that
Fresh feed
T0,Ca0 , F Coolant flow
F (1- ) Ca(t- ) TR(t-
)
Ca t T
FJ , TJ (t)
R
Tcin , FJ
V
R
t
A B
F
Total Reactor Ca(t) flow-rate
TR(t)
Recycle
Effluent
Separator
TR(t),Ca(t), F Fig.1. Schematic CSTR-separator process with recycle
The liquids in the reactor and in the jacket are assumed perfectly mixed, that is, with no radial, axial, or angular gradients in properties (temperature, concentrations). This allows to consider reactor and jacket temperatures and reactant concentrations as space invariant variables. If physical properties are assumed constant (densities and heat capacities), the reactor volume is constant and the jacket volume is constant, the mathematical model of the CSTR is given by the following set of nonlinear functional differential equations: F (Φ Ca0 + (1 − Φ) x1 (t − ∆) − x1 (t)) Vr − E −x1 (t) K0 e Rx2 (t)
x˙ 1 (t) =
F (Φ T0 + (1 − Φ) x2 (t − ∆) − x2 (t)) Vr − E −λ x1 (t) K0 e Rx2 (t) ρ−1 Cp −1
x˙ 2 (t) =
− x˙ 3 (t) =
U Aj (x2 (t) − x3 (t)) Vr ρ Cp u(t) (Tcin − x3 (t)) U Aj (x2 (t) − x3 (t)) + Vj Vj ρj Cj
where: x1 = Ca = reactant concentration ( kmol m3 ) x2 = TR = reactor temperature (K)
def
Φ =
F − Fr ⇒ Fr = F (1 − Φ) F
The presence of this recycle loop introduces the associated recycle loop dead time, ∆. In this situation, the inlet stream to the reactor is characterized by fresh feed with pure A, at flow-rate F Φ, concentration Ca0 , and temperature T0 and the recycle stream is characterized by flow— rate F (1 − Φ), concentration Ca (t − ∆) and temperature TR (t − ∆), where Ca and TR are the reagent concentration and temperature in the reactor, respectively.
x3 = TJ = jacket temperature (K) 3
u = flow rate of coolant (control input) ( ms ) K0 = pre-exponential factor (s−1 ) J E = activation energy ( kmol )
R = universal gas constant, 8314 Jkmol−1 K −1 3
F = flowrate of feed and product ( ms ) kg ρ = density of product stream ( m 3)
(1)
3. THE CONTROL LAW IN THE CASE OF FULL STATE AVAILABLE MEASURES
Ca0 = concentration of reactant A in feed ( kmol m3 ) VR = volumetric holdup of liquid in reactor (m3 ) T0 = temperature of feed (K) Cp = heat capacity of product (Jkg −1 K −1 ) J λ = heat of reaction ( kmol )
U = overall heat transfer coefficient (W K −1 m−2 ) Aj = jacket heat transfer area (m2 ) ³ ´ kg ρj = density of coolant m 3
In the case of full state available measures, the control law is obtained through recently developed tools of differential geometry for time-delay systems (Germani et al. [2000], Germani et al. [2003]). Taking into account that the relative degree (of type II) of the system (1), with input u and output y = x2 , is r = 2, the input-output linearizing control law, which drives the output exponentially to the equilibrium, T req (guaranteeing stable zero dynamics, see P. Pepe [2009]), is the following:
Cj = heat capacity of coolant (Jkg −1 K −1 ) Tcin = supply temperature of cooling medium (K) Let χ ∈ [0, 1] be the reaction conversion χ=
Ca0 − Ca Ca0
(2)
Let χeq be the chosen operating point reaction conversion. By (2) it follows that the operating point concentration of reactant in the reactor Caeq is given by Caeq = Ca0 (1 − χeq ) (3)
From equations in (1), setting zero the left-hand side, it follows that the operating point reactor temperature, the operating point jacket temperature, the necessary coolant water flowrate for the chosen operating point are given by: ¶¶−1 µ µ F Φ (−Ca 0 + Ca eq ) R−1 (4) Tr eq = −E ln − Ca0 (1 − χeq )K0 Vr
Tj eq = −
F ρ Cp Φ T0 U Aj
µ µ −F Φ ρ Cp EU −1 Aj −1 ln
FΦχ (1 − χ) K0 Vr λ Ca 0 F Φ χ2 λ Ca 0 F Φ χ − + U Aj (1 − χ) U Aj (1 − χ) µ µ ¶¶−1 FΦχ −E ln R−1 (1 − χ) K0 Vr
ueq
¶¶−1
R−1 (5)
¶ ⎞ µ FΦχ R −λ Ca 0 χ ln − ⎜ µ (−1 + χ) K0 Vr¶ ⎟ ⎜ ⎟ FΦχ =−⎜ (6) ⎟ R ⎠ ⎝ +ρ Cp T0 ln − (−1 + χ) K0 Vr +ρ Cp E ·Φ F Aj U ¶ ⎞−1 µ ⎛ FΦχ R −λ Ca 0 F Φ χ ln − ⎟ ⎜ r µ (−1 + χ) K0 V¶ ⎟ ⎜ FΦχ ⎟ ⎜ R ⎟ ⎜ +Tcin Aj U ln − ·⎜ µ (−1 + χ) K0 Vr ¶ ⎟ ⎟ ⎜ FΦχ ⎟ ⎜ R ⎠ ⎝ +F Φ ρ Cp T0 ln − (−1 + χ) K0 Vr +F Φ ρ Cp E + U Aj E ⎛
·Cj −1 ρj −1
½ F 2 x2 (t) F 2 x2 (t − ∆) F 2 x2 (t − 2 ∆) u(t) = − +2 − 2 Vr Vr 2 V 2 à ¶¶−1 r ! µ µ FΦχ +k01 x2 (t) + E ln − R−1 (−1 + χ) K0 Vr ⎛ ⎞ F (Φ T0 + (1 − Φ) x2 (t − ∆) − x2 (t)) ⎜ ⎟ Vr ⎜ ⎟ − RxE(t) −1 −1 ⎟ 2 +k02 ⎜ −λ x1 (t) K0 e ρ Cp ⎜ ⎟ ⎝ ⎠ U Aj (x2 (t) − x3 (t)) − Vr ρ Cp F 2 x2 (t − 2 ∆) Φ F 2 Φ2 T0 +2 + Vr 2 Vr 2 2 2 F x2 (t − 2 ∆) Φ F 2 Φ x2 (t − ∆) − −2 Vr 2 Vr 2 (7) ³ ´ −1 E +F λ x1 (t − ∆) K0 Vr −1 e Rx2 (t−∆) ρ−1 Cp −1 F U Aj x2 (t − ∆) F U Aj x3 (t − ∆) +2 − Vr 2 ρ Cp V 2ρ C ³ r E p ´−1 −F Φ λ x1 (t − ∆) K0 Vr −1 e Rx2 (t−∆) ρ−1 Cp −1 F Φ U Aj x2 (t − ∆) F Φ U Aj x3 (t − ∆) −2 + Vr 2 ρ Cp V 2 ρ Cp ³ E ´−1r −2 F λ x1 (t) K0 Vr −1 e Rx2 (t) ρ−1 Cp −1 F U Aj x2 (t) F U Aj x3 (t) + Vr 2 ρ Cp V 2ρ C ³ r E p´−1 +λ K0 F Φ Ca0 Vr −1 e Rx2 (t) ρ−1 Cp −1 ³ E ´−1 +F λ x1 (t − ∆) K0 Vr −1 e Rx2 (t) ρ−1 Cp −1 ³ E ´−1 −F Φ λ x1 (t − ∆) K0 Vr −1 e Rx2 (t) ρ−1 Cp −1 ³ E ´−2 −λ K0 2 x1 (t) e Rx2 (t) ρ−1 Cp −1 −2
−2
+λ x1 (t) K0 EF Φ T0 R−1 (x2 (t)) ·ρ−1 Cp −1 Vr −1
³ E ´−1 e Rx2 (t)
Then
³ E ´−1 −2 +λ x1 (t) K0 EF x2 (t − ∆) R−1 (x2 (t)) e Rx2 (t) −1
·ρ
Cp
−1
Vr
−1
−2
−λ x1 (t) K0 EF Φ x2 (t − ∆) R−1 (x2 (t)) ·ρ−1 Cp −1 Vr −1
³ E ´−1 −λ x1 (t) K0 EF R−1 (x2 (t))−1 e Rx2 (t)
³ E ´−1 e Rx2 (t)
∙
¸ 100 y(t) = z(t), 001
and, (taking into account, by model 1, that f2 involves only x1 (t − ∆) = z1 (t − ∆)) for a suitable function ψ, x(t) = ψ(z(t), z(t − ∆)) (15) By this change of variables, the model (1) can be rewritten, taking into account that x1 (t − 2∆) = z1 (t − 2∆), as Ã
! 010 z(t) ˙ = 0 0 0 z(t) (16) 000 Ã !∙ ¸ 0 J1 (z(t), z(t − ∆), z(t − 2∆)) 1 + +J2 (z(t))u(t) 0 Ã ! 0 + 0 J3 (z(t)), 1
·ρ−1 Cp −1 Vr −1
³ E ´−2 −λ2 (x1 (t))2 K0 2 ER−1 (x2 (t))−2 e Rx2 (t) ·ρ−2 Cp −2
³ E ´−1 −λ x1 (t) K0 EU Aj R−1 (x2 (t))−1 e Rx2 (t) ·ρ−2 Cp −2 Vr −1
³ E ´−1 +λ x1 (t) K0 EU Aj x3 (t) R−1 (x2 (t))−2 e Rx2 (t) ·ρ−2 Cp −2 Vr −1 ³ E ´−1 U Aj F Φ T0 + − U A λ x (t) K e Rx2 (t) j 1 0 Vr 2 ρ Cp ·Vr −1 ρ−2 Cp −2
where J1 , J2 , J3 are suitable functions obtained by the model (1) taking into account of the change of variables. In order to perform the estimation of all the variables z(t) (which involve the reactor concentration), by means of only the measured output y(t) (i.e. the reactor and jacket temperatures), the Luenberger-like observer for nonlinear systems with time-delay proposed in (Germani et al. [2001], Germani et al. [2005]) is here applied as follows:
U 2 Aj 2 x2 (t) U 2 Aj 2 x3 (t) U 2 Aj 2 x2 (t) + − Vr ρ Cp Vj ρj Cj Vr 2 ρ2 Cp 2 V 2 ρ2 Cp 2 ¾ rµ ¶−1 2 2 U Aj Tcin U Aj x3 (t) U Aj x3 (t) · + − , Vr ρ Cp Vj ρj Cj Vr ρ Cp Vj Vr ρ Cp Vj −
# 010 zˆ(t) = 0 0 0 zˆ(t) 000 " #∙ ¸ 0 J1 (ˆ z (t), zˆ(t − ∆), zˆ(t − 2∆)) + 1 z (t))u(t) +J2 (ˆ 0 " # 0 z (t)) + k (y(t) − C zˆ(t)) + 0 J3 (ˆ 1 ·
where [ k01 k02 ] ∈ R2 is chosen such that the matrix ∙ ¸ ∙ ¸ 01 0 + [ k01 k02 ] (8) 00 1 is Hurwitz. 4. OBSERVER DESIGN FOR REACTOR CONCENTRATION ESTIMATION
"
with The system (1) can be rewritten in the form: x(t) ˙ = f (x(t), x(t − ∆)) + g(x(t)) · u(t) y = h(x(t)) where: # " x1 (t) x(t) = x2 (t) ; x3 (t) the measured output function h is given by ∙ ¸ x2 (t) h(x(t)) = ; x3 (t)
(14)
k=
(9)
"
k1 0 k2 0 0 k3
(17)
#
such that the matrix (10)
(11)
"
# ∙ ¸ 010 100 0 0 0 −k 001 000
(18)
is Hurwitz. zˆ(t) is the estimation of z(t) and, by the function ψ, we can obtain the estimation of all the variables x(t) (including the un-measured reactor concentration). The hypotheses in Germani et al. [2001] are satisfied, locally, by the underlying CSTR model, and thus local convergence of the estimation error to zero is guaranteed, by suitable choice of the gain vector k.
! f1 (x(t), x(t − ∆)) (12) f (x(t), x(t − ∆)) = f2 (x(t), x(t − ∆)) , f3 (x(t)) with f1 , f2 , f3 , functions which can be easily extrapolated from the model (1). Let us consider the following change of variables: # Ã ! " 5. OBSERVER-BASED CONTROL LAW x2 (t) z1 (t) (t) (x(t), x(t − ∆)) z f = φ(x(t), x(t−∆)) = z(t) = 2 2 z3 (t) x3 (t) The equations of the observer for the CSTR (1), by the (13) methodology shown in the previous section, is as follows: Ã
·
zˆ1 = zˆ2 (t) + k1 (TR (t) − zˆ1 (t)) ·
zˆ2 = −
U Aj u ˆzˆ3 (t) F 2 zˆ1 (t − ∆) + Vr ρ Cp Vj Vr 2
+k2 (TR (t) − zˆ1 (t)) + F zˆ2V(t−∆) r ³ E ´−1 +K0 U Aj zˆ3 (t) e Rˆz1 (t) ρ−1 Cp −1 Vr −1
³ E ´−1 F U Aj zˆ3 (t) Rˆ z (t) − λ K F Φ Ca ρ−1 Cp −1 Vr −1 0 0 e 1 ρ Cp Vr 2 ³ E ´−1 E F U Aj zˆ1 (t) 2 Rˆ z1 (t−∆) Rˆ z (t) − − F e Φ T Vr −2 0 e 1 ρ Cp Vr 2 ³ E ´−1 U Aj uTcin + + K0 F Φ T0 e Rˆz1 (t) Vr −1 Vr ρ Cp Vj ³ E ´−1 −K0 F zˆ1 (t − ∆) Φ e Rˆz1 (t) Vr −1 +
−
E zˆ2 (t) F zˆ1 (t − ∆) Vr R (ˆ z1 (t))
2
+
F 2 zˆ1 (t) F 2 Φ T0 + Vr 2 Vr 2 ³ E ´−1 U 2 Aj 2 zˆ3 (t) −K0 U Aj zˆ1 (t) e Rˆz1 (t) ρ−1 Cp −1 Vr −1 − Vr ρ Cp Vj ρj Cj ³ ´ −1 E E −F e Rˆz1 (t−∆) Φ U Aj zˆ1 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2 −
E zˆ2 (t) F zˆ1 (t − ∆) Φ 2
Vr R (ˆ z1 (t)) ³ E ´−1 U 2 Aj 2 zˆ1 (t) F zˆ2 (t) + −2 − K0 zˆ2 (t) e Rˆz1 (t) Vr ρ Cp Vj ρj Cj Vr ³ E ´−1 F 2 zˆ1 (t − ∆) Φ −K0 F zˆ1 (t) e Rˆz1 (t) Vr −1 − Vr 2
U Aj zˆ2 (t) E zˆ2 (t) F Φ T0 E zˆ2 (t) F − 2 + V Rˆ Vr ρ Cp r z1 (t) Vr R (ˆ z1 (t)) ³ E ´−1 E E zˆ2 (t) U Aj +F e Rˆz1 (t−∆) zˆ2 (t − ∆) e Rˆz1 (t) Vr −1 + Vr Rˆ z1 (t) ρ Cp ³ E ´−1 E −F e Rˆz1 (t−∆) Φ zˆ2 (t − ∆) e Rˆz1 (t) Vr −1
³ E ´−1 E −F e Rˆz1 (t−∆) U Aj zˆ3 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2
³ E ´−1 E +F e Rˆz1 (t−∆) Φ U Aj zˆ3 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2 ·
zˆ3 =
+k3 (Tj (t) − zˆ3 (t)) Now, by using the relationship between z(t) and x(t) in the previous section, the control law (7) can be rewritten as a function of z(t), z(t − ∆) and z(t − 2∆). By using the estimation of z(t), zˆ(t), given by the observer, the observer based control law is the following:
˜ u(t) = A˜ · B where ˜= B
−
2
+
E (ˆ z2 (t))
2
E
+ 2 F 2 e Rˆz1 (t−∆) zˆ1 (t − 2 ∆)
R (ˆ z1 (t)) ³ E ´−1 Vr −2 Φ e Rˆz1 (t)
³ E ´−1 E −F 2 e Rˆz1 (t−∆) zˆ1 (t − 2 ∆) Φ2 e Rˆz1 (t) Vr −2 ³ E ´−1 E −F 2 e Rˆz1 (t−∆) zˆ1 (t − 2 ∆) e Rˆz1 (t) Vr −2 F zˆ2 (t − ∆) Φ E zˆ2 (t) U Aj zˆ3 (t) − 2 Vr Vr R (ˆ z1 (t)) ρ Cp ³ E ´−1 +K0 F zˆ1 (t − ∆) e Rˆz1 (t) Vr −1 −
³ E ´−1 E +F 2 e Rˆz1 (t−∆) zˆ1 (t − ∆) e Rˆz1 (t) Vr −2
³ E ´−1 E +F e Rˆz1 (t−∆) U Aj zˆ1 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2 ³ E ´−1 E +F 2 e Rˆz1 (t−∆) Φ2 T0 e Rˆz1 (t) Vr −2
³ E ´−1 E −F 2 e Rˆz1 (t−∆) zˆ1 (t − ∆) Φ e Rˆz1 (t) Vr −2
u(t) (Tcin − zˆ3 (t)) (ˆ z1 (t) − zˆ3 (t)) Aj U + Vj Vj ρj Cj
µ
U Aj zˆ3 (t) U Aj Tcin − + Vr ρ Cp Vj Vr ρ Cp Vj
¶−1
and F U Aj zˆ3 (t − ∆) F Φ U Aj zˆ3 (t − ∆) A˜ = − + Vr 2 ρ Cp Vr 2 ρ Cp à ! ¶¶−1 µ µ FΦχ −1 +k01 zˆ1 (t) + E ln − R (−1 + χ) K0 Vr ⎛ ⎞ F (Φ T0 + (1 − Φ) zˆ1 (t − ∆) − zˆ1 (t)) ⎜ ⎟ Vr ⎜ ⎟ zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 − ⎜ ⎟ ⎜ F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ ⎟ ⎜ ⎟ +k02 ⎜ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ⎟ ⎜+ ⎟ ⎜ ⎟ Vr ρ Cp ⎜ ⎟ ⎝ ⎠ (ˆ z1 (t) − zˆ3 (t)) Aj U − Vr ρ Cp ⎞ ⎛ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ −F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ E ´−1 ³ ´−1 E − · e Rˆz1 (t) ρ−1 Cp −1 Vr −2 e Rˆz1 (t−∆) ⎞ ⎛ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ +F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ E ´−1 ³ ´−1 E − ·Φ e Rˆz1 (t) ρ−1 Cp −1 Vr −2 e Rˆz1 (t−∆)
⎞ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ +F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ ´−1 ³ ´−1 E E F 2 zˆ1 (t) − ρ−1 Cp −1 − ·ΦVr −2 e Rˆz1 (t−∆) e Rˆz1 (t−∆) Vr 2 ⎞ ⎛ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ −F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ ´ ´−1 −1 ³ E E − ·Vr −2 e Rˆz1 (t−∆) ρ−1 Cp −1 e Rˆz1 (t−∆) ⎛
F 2 zˆ1 (t − 2 ∆) F 2 zˆ1 (t − ∆) U 2 Aj 2 zˆ1 (t) +2 − 2 Vr Vr 2 Vr 2 ρ2 Cp 2 ³ E ´−1 F U Aj zˆ1 (t) −2 + λ K0 F Φ Ca 0 e Rˆz1 (t) ρ−1 Cp −1 Vr −1 2 Vr ρ Cp −
U 2 Aj 2 zˆ1 (t) F 2 zˆ1 (t − ∆) Φ F 2 zˆ1 (t − 2 ∆) Φ2 − −2 − Vr ρ Cp Vj ρj Cj Vr 2 Vr 2 +2
F 2 zˆ1 (t − 2 ∆) Φ U 2 Aj 2 zˆ3 (t) F 2 Φ2 T0 + + Vr 2 Vr 2 ρ2 Cp 2 Vr 2
³ E ´−2 ³ ´−1 − E · e Rˆz1 (t) ρ−1 Cp −1 e Rˆz1 (t) Vr −1
! zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 − −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EU Aj zˆ3 (t) +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t))−2 e Rˆz1 (t) ρ−2 Cp −2 Ã
F Φ U Aj zˆ1 (t − ∆) Vr 2 ρ Cp ⎞ ⎛ zˆ2 (t) Vr ρ Cp ⎜ −F ρ Cp Φ T0 − F ρ Cp zˆ1 (t − ∆) ⎟ EF Φ T0 −⎝ +F ρ Cp zˆ1 (t − ∆) Φ + F ρ Cp zˆ1 (t) ⎠ +U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t))−2 e Rˆz1 (t) ρ−1 Cp −1 −2
Ã
zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 + −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 − E ·ˆ z1 (t − ∆) e Rˆz1 (t) Vr −2 R−1 −2
· (ˆ z1 (t))
!
EF Φ
³ E ´−1 ρ−1 Cp −1 e Rˆz1 (t)
! Ã F Φ U Aj T0 F U Aj zˆ3 (t) U 2 Aj 2 zˆ3 (t) zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 + + 2 2 Vr ρ Cp Vj ρj Cj Vr ρ Cp Vr ρ Cp +U Aj −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ ! Ã +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 ³ ´−1 ³ E ´−1 E − −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EF zˆ1 (t − ∆) − Rˆ −2 −2 −2 z1 (t) ·V ρ C e e Rˆz1 (t) r p +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 F U Aj zˆ1 (t − ∆) − E −2 · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t)) ρ−1 Cp −1 e Rˆz1 (t) +2 Vr 2 ρ Cp !2 Ã zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 − −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ E 6. SIMULATION RESULTS +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−2 ³ E ´−2 Simulations, using MatLab Software Package, have been − E −2 · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t)) ρ−2 Cp −2 e Rˆz1 (t) carried out to verify the effectiveness of the proposed method. The values of the model parameters used in ! Ã zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 simulation are given in Table 1. + −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EU Aj Table 1. Reactor and controller parameters values +F ρ C zˆ (t) + U A zˆ (t) − U A zˆ (t) +
p 1
j 1
j 3
³ ´−1 ³ E ´−1 − E −1 · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t)) ρ−2 Cp −2 e Rˆz1 (t) Ã
zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 +2 F −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 e Rˆz1 (t) ρ−1 Cp −1
!
! zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 + −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EF +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t))−1 e Rˆz1 (t) ρ−1 Cp −1 Ã
+K0
Ã
zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t)
!
K0 = 20.75 × 106 s−1 E = 69.71 × 106 J · kmol−1 ρ = 801 kg · m−3 R = 8314 J · kmol−1 · K −1 ρj = 1000 kg · m−3 Cp = 3137 J · kg −1 · K −1 Cj = 4183 J · kg −1 · K −1 λ = −69.71 × 106 J · kmol−1 U = 851 W · K −1 · m−2 F = 4.377 × 10−3 m3 · s−1 Ca0 = 8.01 kmol · m−3
∆ = 40 s Tcin = 294 K T0 = 294 K Aj = 101 m2 Vr = 102 m3 Vj = 100 m3 k01 = −6 · 10−6 k02 = −5 · 10−3 k1 = 3 · 10−3 k2 = 2 · 10−6 k3 = 5 · 10−2
In all the carried out simulations, a minimum and a maximum value for the input u, specifically the volumetric
flow rate of the refrigerant, is imposed, to take into account of the actuator saturation. It is considered :
0
50000
150000
200000 314
TJ TJ
312
FJ min ≤ u(t) ≤ FJ max
310
312
Φ = 0.3 χ = 0.85
308
300,0
TJ, eq = 312.48 K
0
100
200
300
400
500
310
600 300,0
299,5
299,5
308
299,0
299,0
306
298,5
298,5
304
298,0
298,0
297,5
297,5 600
= 40 s 306
TJ (t=0) = TJ, eq TJ [K]
TJ [K]
with the following acceptable physical values for a hypothetical control valve:
100000
314
TJ (t=0) = 298 304 302 300
0
100
200
300
400
500
302 300
Time [s]
3
FJ min = 0.00678 m /s
298
298
296
296
3
FJ max = 0.08 m /s
0
50000
100000
150000
200000
Time [s]
We performed several sets of simulation runs and all showed the high performance of the method here shown. In the simulation here presented, the process is initially assumed to be at the start up. The Fig. 2 to Fig. 4 show the evolution of the true and estimated state.
Fig. 4. Real and estimated temperature of jacket TJ In Fig. 5 the control signal u(t) is plotted. 0
40000
80000
120000
25000
50000
75000
100000
125000
150000
175000
3
3
= 40 s
-1
0,035
0,030
0,030
0,025
0,025
0,020
0,020
0,015
0,015
0,010
0,010
0,005
0,005
1
1 0
0
1000
2000
3000
4000
3
5000 3
0
0
-3
-3
-6
-6
Ca(t=0) = Caeq Ca(t=0) = 0
Φ = 0.3 χ = 0.85
Caeq = 1.2015 kmol m-3 = 40 s
-3
Ca [kmol m ]
-3
Ca [kmol m ]
3
-1
u [m s ]
2
ueq = 0.0081 m s
0,035
Ca Ca 2
0,040
χ = 0.85
200000
3
200000 0,045
Φ = 0.3
0,040 0
160000
0,045
-1
-9
-9
-12
-12
-15
-15
-18
-18
0
1000
2000
3000
4000
0
-1
0,000 0
40000
80000
120000
160000
0,000 200000
Time [s]
5000
Tempo [s]
-20
-20 0
25000
50000
75000
100000
125000
150000
175000
Fig. 5. Real flowrate of coolant, u = FJ
200000
As can be seen, the estimated state variables converge to the true ones, and, more important, the state variables are driven to the chosen equilibrium point.
Time [s]
Fig. 2. Real and estimated concentration CA
7. CONCLUSION 0
50000
100000
150000
200000 320
320
TR TR
0
1500
3000
4500
6000
298,5
298,5
298,0
298,0
315 Φ = 0.3 χ = 0.85
TR [K]
TR, eq = 319.78 K = 40 s
310
TR [K]
315
297,5
297,5
297,0
297,0
310
TR(t=0) = TR, eq TR(t=0) = 298 305
305 0
1500
3000
4500
6000
Time [s]
300
300
295
295 0
50000
100000
150000
200000
Time [s]
Fig. 3. Real and estimated temperature of reactor TR
In this paper, in the spirit of the separation principle, a nonlinear feedback control law obtained by tools of differential geometry and observer theory for time-delay systems has been built up for a continuous stirred tank reactor with recycle. The reactor and jacket temperature are assumed available for measures, the reactor concentration, as often happens in practice, is not assumed to be available for measures. A recycle time delay is considered. The feedback control law, when all the state variables are available for measures, yields local exponential convergence of the reactor tracking error to zero. The observer guarantees local exponential convergence of the estimation error to zero. Many simulations have been performed, showing the high performance of the observer based nonlinear feedback control law here presented. Further theoretical investigations will concern the problems related to the saturation, which here is taken into account only in the simulations, and the problems related to the uncertainties on the parameters, such as the reactor heat transfer coefficient, by means of
recently developed adaptive algorithms for nonlinear time delay systems (P. Pepe [2004]). REFERENCES K. Adebekun. Output feedback control of a stirred tank reactor. Computers chem. Engng, volume 20, No. 8, pages 1017-1021, 1996. R. Aguilar, R. Martìnez-Guerra and A. Poznyak. PI observers for uncertainty estimation in continuous chemical reactors.IEEE International Conference on Control Applications, México City, México, September 5-7, 2001. J. Alvarez-Ramirez, A. Morales.PI control of continuously stirred tank reactors: stability and performance. Chemical Engineering Science, volume 55, pages 5497-5507, 2000. C. Antoniades, P. D. Christofides.Feedback control of nonlinear differential difference equation systems. Chemical Engineering Science, volume 54, pages 5677-5709, 1999. N. Assala, F. Viel and J. P. Gauthier. Stabilization of polymerzation CSTR under input constraints. Computers chem. Engng, volume 21, No. 5, pages 501-509, 1997. S. I. Biagiola, J. L. Figueroa. A high gain nonlinear observer: application to the control of an unstable nonlinear process.Computer & Chemical Engineering, volume 28, pages 1881-1898, 2004. Y.-Y. Cao and P.M. Frank. Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach. IEEE Transactions On Fuzzy Systems, volume 8, No.2, pages 200-211, April 2000. J.-S. Chang and Kui-Yi Chen. An integrated strategy for early detection of hazardous states in chemical reactors. Chemical Engineering Journal, volume 98, pages 199211, 2004. C.-T. Chen and Shih-Tien Peng. A sliding mode control scheme for non-minimum phase non-linear uncertain input-delay chemical processes. Journal of Process Control, volume 16, pages 37-51, 2006. B. Daaou, A. Mansouri, M. Bouhamida and M. Chenafa.A robust ronlinear observer for state variables estimation in multi-input multi-output chemical reactors. International Journal of Chemical Reactor Engineering, volume 6, Article A86, 2008. D. Fissore. Robust control in presence of parametric uncertainties: observer-based feedback controller design. Chemical Engineering Science, volume 63, pages 18901900, 2008. A. Germani, C. Manes, P. Pepe.Local Asymptotic Stability for Nonlinear State Feedback Delay Systems. Kybernetica, volume 36, No. 1, pages 31-42, 2000. A. Germani, C. Manes, P. Pepe. An asymptotic State Observer for a Class of Nonlinear Delay Systems. Kybernetica, volume 37, No. 4, pages 459-478, 2001. A. Germani, C. Manes, P. Pepe. Input- Output Linearization with Delay Cancellation for Nonlinear Delay Systems: The Problem of the Internal Stability. International Journal of Robust and Nonlinear Control, volume 13, No. 9, pages 909-937, 2003. A. Germani, P. Pepe. A State Observer for a Class of Nonlinear Systems with multiple discrete and Distributed Time Delays.European Journal of Control, volume 11, No. 3, pages 196-205, 2005. A.M. Gibon-Fargeot, F. Celle-Couenne, H. Hammouri. Cascade estimation design for CSTR models.Computers
and Chemical Engineering, volume 24, pages 2355-2366, 2000. Y. Hashiimoto, K. Matsumoto, T. Matsuo, T. Hamaguchi, A. Yoneya, Y. Togari.Anti-saturation discrete-time nonlinear controllers and observers. Computers and Chemical Engineering, volume 24, pages 815-821, 2000. N. M. Iyer, A. Farell. Indirect schemes for adaptive linearizing control of a continuous stirred tank reactor . IEEE International Conference on Control Applications, 1994. A. K. Jana. Nonlinear state estimation and generic model control of a continuous stirred tank reactor.International Journal of Chemical Reactor Engineering, volume 5, Article A42, 2007. K.A. Kosanovich, M.J. Piovoso, V. Rokhlenko and A. Guez. Nonlinear adaptive control with parameter estimation of a CSTR. J. Proc. Cont., volume 5, No. 3, pages 137-148, 1995. C. Kravaris, P. Daoutidis and R.A. Wright. Output feedback control of nonminimum-phase nonlinear processes. Chemical Engineering Science, volume 49,No. 13, pages 2107-2122, 1994. W.L. Luyben Chemical Reactor Design. Wiley, 2007. P. Pepe Adaptive Output Tracking for a Class of Nonlinear Time Delay Systems International Journal of adaptive control and signal processing, vol. 18, N. 6, pp. 489-503, (2004). P. Pepe A Robust State Feedback Control Law for a Continuous Stirred Tank Reactor with recycle. in J.J. Loiseau, W.Michiels, S.-I. Niculescu, R.Sipahi (Eds), Topics in Time Delay Systems Analysis, Algorithms, and Control, LNCIS, Springer Verlag, (2009), to appear. F. Pierri, F. Caccavale, M. Iamarino, V. Tufano. A controller-observer scheme for adaptive control of chemical batch reactors.Proceedings of the 2006 IEEE American Control Conference , Minneapolis, Minnesota, USA, June 14-16, 2006. J. Prakash, R. Senthil. Design of observer based nonlinear model predictive controller for a continuous stirred tank reactor. Journal of Process Control, volume 18, pages 504-514, 2008. R. Senthil, K. Janarthanan and J. Prakash.Nonlinear state estimation using fuzzy kalman filter. Ind. Eng. Chem. Res., volume 45, No. 25, pages 8678-8688, 2006. S. Salehi, M. Shahrokhi. Two observer-based nonlinear control approaches for temperature of a class of continuous stirred tank reactors.Chemical Engineering Science, volume 63, Issue 2, pages 395-403, January 2008. M. Soroush. Nonlinear state-observer design with application to reactors. Chemical Engineering Science, volume 52,No. 3, pages 387-404, 1997. R.A. Wright, C. Kravaris. Two-degree-of-freedom output feedback controllers for discrete-time nonlinear systems. Chemical Engineering Science, volume 61, pages 46764688, 2006. W. Wu. Systematic design of nonlinear controllers for a nonisothermal CSTR with delay in the recycle stream. Journal of the Chinese institute of chemical engineers, 30 (1), 11-21 Jan 1999. W. Wu. Finite difference-based output feedback control for a class of nonlinear systems with sampling time-delay. Journal of Process Control, available at ScienceDirect, 2009.
Department of Electrical and Informaton Engineering,University of L’Aquila, 67040, Poggio di Roio, L’Aquila, Italy; (e-mail: [email protected], [email protected]) ∗∗ Department of Chemical Engineering,University of L’Aquila, 67040, Poggio di Roio, L’Aquila, Italy; (e-mail: [email protected], [email protected]) Abstract: This paper proposes the design of an observer-based control law for a continuous stirred tank reactor with recycle time-delay. Recent results on differential geometry for nonlinear time-delay systems are used. The many performed simulations show the high performance of the proposed control law. Keywords: Chemical industry, Chemical variables control, Temperature control, Nonlinear control systems, Observers, Time delay. 1. INTRODUCTION The control of the operation of chemical reactors has attracted the attention of researchers for a long time. The underlying motivation relies on the fact that industrial chemical reactors are frequently operated at unstable operating conditions, which often corresponds to optimal process performance (K. Adebekun [1996], Aguilar et al. [2001], Alvarez-Ramirez et al. [2000], A.K. Jana [2007], N. Assala et al. [1997], Chang et al. [2004], Chen et al. [2006], Daaou et al. [2008], D. Fissore [2008], Gibon-Fargeot et al. [2000], Hashimoto et al. [2000], Iyer et al. [2000], Kosanovich et al. [1995], Kravaris et al. [1994], Pierri et al. [2006], Prakash et al. [2008], Senthil et al. [2006], M. Soroush [1997],Wright et al. [2006]). Polymerization processes (N. Assala et al. [1997]) and bio-reactors fermentors (M. Soroush [1997]) are important examples of largescale chemical reactors operated at unstable conditions. In addition there exist many process control strategies, in which the information about the internal state of the process is necessary to calculate the control input. Consequently the presence of unknown state variables becomes a difficulty which can be over-come with the inclusion of an appropriate state estimator (Aguilar et al. [2001], Chang et al. [2004], Daaou et al. [2008], D. Fissore [2008], K. Adebekun [1996], Salehi et al. [2000], Iyer et al. [2000], A.K. Jana [2007], Kosanovich et al. [1995], Kravaris et al. [1994], Biagiola et al. [2001], Hashimoto et al. [2000]). To our best knowledge, all these papers do not consider the problem of state estimator in presence of the time-delays in the state variables. Other papers propose methodologies borrowed by time-delay systems nonlinear control theory (W. Wu [1999], P. Pepe [2009], Cao et al. [2000]), but it is assumed that all the state variables are available B This paper is supported by the University of L’Aquila ex 60% and Ph.D. school funds.
for measure, thus a nonlinear observer for state variables estimation is not employed. In the paper Antoniades et al. [1999] an observer-based nonlinear control law for chemical reactors with recycle time-delay is presented. A model with only two state variables (reagent concentration and reactor temperature) is used, and the the dynamics of jacket refrigerant temperature is not taken into account. Also, the manipulated input is the inlet reagent concentration in the fresh feed, which can arise some further complications in the accuracy of the practical implementation of the control law. In this paper we deal with the controller design, by means of partial state measurements, for a continuous stirred tank reactor, where an irreversible, exothermic, liquidphase reaction A → B evolves. The unreacted amount of species A is fedback to the reactor through a recycle loop, in order to minimize reaction wastes. Such recycle is responsible of a non-negligible time-delay due to the fluid transport. The model of the plant is given by three nonlinear delay differential equations (the dynamics of the jacket temperature is considered too). The control input consists of the cooling water flowrate. The controlled output is the reactor temperature. The measured output consists of the reactor and jacket temperatures. Here we do not use the measurement of the reactor concentration, which is expensive to obtain by techniques of gas-chromatography. The reactor concentration is estimated by means of a nonlinear observer for time-delay systems (Germani et al. [2001], Germani et al. [2005] ). The state feedback control law is obtained by means of the method of the exact inputoutput linearization, with delay cancellation, of nonlinear time-delay systems (Germani et al. [2000], Germani et al. [2003]). The observer yields local convergence of the estimation error to zero, as well as, if the state were available, the state feedback control law yields the convergence of the state variables to the operating point. In the spirit of the
separation principle, the controller is here implemented by using the state estimations provided by the observer instead of the unavoidable full state measurements. First order approximated linearization methods are not used here. The control law is actuated by means of a pneumatic valve which regulates the cooling water flow-rate. This actuation process is easier to implement then the ones based on the use of the inlet reactor concentration or of the jacket temperature as manipulated inputs.Many performed computer simulations show the high performance of the proposed observer-based controller. Saturation effects have been taken into account in simulations. The convergence of the state variables to the arbitrarily chosen operating point is always obtained with the many considered system initial conditions (even start-up initial conditions) and initial estimation error, and with different values of the recirculation coefficient. As well, the convergence of the estimated state variables to the true state variables is always obtained in such simulations.
2. THE MODEL OF THE CSTR WITH RECYCLE TIME-DELAY
Here we study a CSTR with jacket cooling in which a firstorder irreversible exothermic reaction takes place (Luyben [2007]): A → B . A separator is considered, which yields a recycle time delay (Fig. 1). If in the reactor out flow an important quantity of not converted reagent is present, it is possible to feedback the unreacted amount of reagent to the reactor, by means of a separator, in order to minimize the reactant waste and to increase the overall conversion of the reaction. The recirculation coefficient, here denoted with Φ, is defined as: Φ=
Effluent flow-rate Total Reactor flow-rate
Therefore Φ = 1 means no recycle, Φ = 0 means total recycle. Denoting with Fr the flow-rate of the recirculating stream and with F the total reactor flow rate, it results that
Fresh feed
T0,Ca0 , F Coolant flow
F (1- ) Ca(t- ) TR(t-
)
Ca t T
FJ , TJ (t)
R
Tcin , FJ
V
R
t
A B
F
Total Reactor Ca(t) flow-rate
TR(t)
Recycle
Effluent
Separator
TR(t),Ca(t), F Fig.1. Schematic CSTR-separator process with recycle
The liquids in the reactor and in the jacket are assumed perfectly mixed, that is, with no radial, axial, or angular gradients in properties (temperature, concentrations). This allows to consider reactor and jacket temperatures and reactant concentrations as space invariant variables. If physical properties are assumed constant (densities and heat capacities), the reactor volume is constant and the jacket volume is constant, the mathematical model of the CSTR is given by the following set of nonlinear functional differential equations: F (Φ Ca0 + (1 − Φ) x1 (t − ∆) − x1 (t)) Vr − E −x1 (t) K0 e Rx2 (t)
x˙ 1 (t) =
F (Φ T0 + (1 − Φ) x2 (t − ∆) − x2 (t)) Vr − E −λ x1 (t) K0 e Rx2 (t) ρ−1 Cp −1
x˙ 2 (t) =
− x˙ 3 (t) =
U Aj (x2 (t) − x3 (t)) Vr ρ Cp u(t) (Tcin − x3 (t)) U Aj (x2 (t) − x3 (t)) + Vj Vj ρj Cj
where: x1 = Ca = reactant concentration ( kmol m3 ) x2 = TR = reactor temperature (K)
def
Φ =
F − Fr ⇒ Fr = F (1 − Φ) F
The presence of this recycle loop introduces the associated recycle loop dead time, ∆. In this situation, the inlet stream to the reactor is characterized by fresh feed with pure A, at flow-rate F Φ, concentration Ca0 , and temperature T0 and the recycle stream is characterized by flow— rate F (1 − Φ), concentration Ca (t − ∆) and temperature TR (t − ∆), where Ca and TR are the reagent concentration and temperature in the reactor, respectively.
x3 = TJ = jacket temperature (K) 3
u = flow rate of coolant (control input) ( ms ) K0 = pre-exponential factor (s−1 ) J E = activation energy ( kmol )
R = universal gas constant, 8314 Jkmol−1 K −1 3
F = flowrate of feed and product ( ms ) kg ρ = density of product stream ( m 3)
(1)
3. THE CONTROL LAW IN THE CASE OF FULL STATE AVAILABLE MEASURES
Ca0 = concentration of reactant A in feed ( kmol m3 ) VR = volumetric holdup of liquid in reactor (m3 ) T0 = temperature of feed (K) Cp = heat capacity of product (Jkg −1 K −1 ) J λ = heat of reaction ( kmol )
U = overall heat transfer coefficient (W K −1 m−2 ) Aj = jacket heat transfer area (m2 ) ³ ´ kg ρj = density of coolant m 3
In the case of full state available measures, the control law is obtained through recently developed tools of differential geometry for time-delay systems (Germani et al. [2000], Germani et al. [2003]). Taking into account that the relative degree (of type II) of the system (1), with input u and output y = x2 , is r = 2, the input-output linearizing control law, which drives the output exponentially to the equilibrium, T req (guaranteeing stable zero dynamics, see P. Pepe [2009]), is the following:
Cj = heat capacity of coolant (Jkg −1 K −1 ) Tcin = supply temperature of cooling medium (K) Let χ ∈ [0, 1] be the reaction conversion χ=
Ca0 − Ca Ca0
(2)
Let χeq be the chosen operating point reaction conversion. By (2) it follows that the operating point concentration of reactant in the reactor Caeq is given by Caeq = Ca0 (1 − χeq ) (3)
From equations in (1), setting zero the left-hand side, it follows that the operating point reactor temperature, the operating point jacket temperature, the necessary coolant water flowrate for the chosen operating point are given by: ¶¶−1 µ µ F Φ (−Ca 0 + Ca eq ) R−1 (4) Tr eq = −E ln − Ca0 (1 − χeq )K0 Vr
Tj eq = −
F ρ Cp Φ T0 U Aj
µ µ −F Φ ρ Cp EU −1 Aj −1 ln
FΦχ (1 − χ) K0 Vr λ Ca 0 F Φ χ2 λ Ca 0 F Φ χ − + U Aj (1 − χ) U Aj (1 − χ) µ µ ¶¶−1 FΦχ −E ln R−1 (1 − χ) K0 Vr
ueq
¶¶−1
R−1 (5)
¶ ⎞ µ FΦχ R −λ Ca 0 χ ln − ⎜ µ (−1 + χ) K0 Vr¶ ⎟ ⎜ ⎟ FΦχ =−⎜ (6) ⎟ R ⎠ ⎝ +ρ Cp T0 ln − (−1 + χ) K0 Vr +ρ Cp E ·Φ F Aj U ¶ ⎞−1 µ ⎛ FΦχ R −λ Ca 0 F Φ χ ln − ⎟ ⎜ r µ (−1 + χ) K0 V¶ ⎟ ⎜ FΦχ ⎟ ⎜ R ⎟ ⎜ +Tcin Aj U ln − ·⎜ µ (−1 + χ) K0 Vr ¶ ⎟ ⎟ ⎜ FΦχ ⎟ ⎜ R ⎠ ⎝ +F Φ ρ Cp T0 ln − (−1 + χ) K0 Vr +F Φ ρ Cp E + U Aj E ⎛
·Cj −1 ρj −1
½ F 2 x2 (t) F 2 x2 (t − ∆) F 2 x2 (t − 2 ∆) u(t) = − +2 − 2 Vr Vr 2 V 2 à ¶¶−1 r ! µ µ FΦχ +k01 x2 (t) + E ln − R−1 (−1 + χ) K0 Vr ⎛ ⎞ F (Φ T0 + (1 − Φ) x2 (t − ∆) − x2 (t)) ⎜ ⎟ Vr ⎜ ⎟ − RxE(t) −1 −1 ⎟ 2 +k02 ⎜ −λ x1 (t) K0 e ρ Cp ⎜ ⎟ ⎝ ⎠ U Aj (x2 (t) − x3 (t)) − Vr ρ Cp F 2 x2 (t − 2 ∆) Φ F 2 Φ2 T0 +2 + Vr 2 Vr 2 2 2 F x2 (t − 2 ∆) Φ F 2 Φ x2 (t − ∆) − −2 Vr 2 Vr 2 (7) ³ ´ −1 E +F λ x1 (t − ∆) K0 Vr −1 e Rx2 (t−∆) ρ−1 Cp −1 F U Aj x2 (t − ∆) F U Aj x3 (t − ∆) +2 − Vr 2 ρ Cp V 2ρ C ³ r E p ´−1 −F Φ λ x1 (t − ∆) K0 Vr −1 e Rx2 (t−∆) ρ−1 Cp −1 F Φ U Aj x2 (t − ∆) F Φ U Aj x3 (t − ∆) −2 + Vr 2 ρ Cp V 2 ρ Cp ³ E ´−1r −2 F λ x1 (t) K0 Vr −1 e Rx2 (t) ρ−1 Cp −1 F U Aj x2 (t) F U Aj x3 (t) + Vr 2 ρ Cp V 2ρ C ³ r E p´−1 +λ K0 F Φ Ca0 Vr −1 e Rx2 (t) ρ−1 Cp −1 ³ E ´−1 +F λ x1 (t − ∆) K0 Vr −1 e Rx2 (t) ρ−1 Cp −1 ³ E ´−1 −F Φ λ x1 (t − ∆) K0 Vr −1 e Rx2 (t) ρ−1 Cp −1 ³ E ´−2 −λ K0 2 x1 (t) e Rx2 (t) ρ−1 Cp −1 −2
−2
+λ x1 (t) K0 EF Φ T0 R−1 (x2 (t)) ·ρ−1 Cp −1 Vr −1
³ E ´−1 e Rx2 (t)
Then
³ E ´−1 −2 +λ x1 (t) K0 EF x2 (t − ∆) R−1 (x2 (t)) e Rx2 (t) −1
·ρ
Cp
−1
Vr
−1
−2
−λ x1 (t) K0 EF Φ x2 (t − ∆) R−1 (x2 (t)) ·ρ−1 Cp −1 Vr −1
³ E ´−1 −λ x1 (t) K0 EF R−1 (x2 (t))−1 e Rx2 (t)
³ E ´−1 e Rx2 (t)
∙
¸ 100 y(t) = z(t), 001
and, (taking into account, by model 1, that f2 involves only x1 (t − ∆) = z1 (t − ∆)) for a suitable function ψ, x(t) = ψ(z(t), z(t − ∆)) (15) By this change of variables, the model (1) can be rewritten, taking into account that x1 (t − 2∆) = z1 (t − 2∆), as Ã
! 010 z(t) ˙ = 0 0 0 z(t) (16) 000 Ã !∙ ¸ 0 J1 (z(t), z(t − ∆), z(t − 2∆)) 1 + +J2 (z(t))u(t) 0 Ã ! 0 + 0 J3 (z(t)), 1
·ρ−1 Cp −1 Vr −1
³ E ´−2 −λ2 (x1 (t))2 K0 2 ER−1 (x2 (t))−2 e Rx2 (t) ·ρ−2 Cp −2
³ E ´−1 −λ x1 (t) K0 EU Aj R−1 (x2 (t))−1 e Rx2 (t) ·ρ−2 Cp −2 Vr −1
³ E ´−1 +λ x1 (t) K0 EU Aj x3 (t) R−1 (x2 (t))−2 e Rx2 (t) ·ρ−2 Cp −2 Vr −1 ³ E ´−1 U Aj F Φ T0 + − U A λ x (t) K e Rx2 (t) j 1 0 Vr 2 ρ Cp ·Vr −1 ρ−2 Cp −2
where J1 , J2 , J3 are suitable functions obtained by the model (1) taking into account of the change of variables. In order to perform the estimation of all the variables z(t) (which involve the reactor concentration), by means of only the measured output y(t) (i.e. the reactor and jacket temperatures), the Luenberger-like observer for nonlinear systems with time-delay proposed in (Germani et al. [2001], Germani et al. [2005]) is here applied as follows:
U 2 Aj 2 x2 (t) U 2 Aj 2 x3 (t) U 2 Aj 2 x2 (t) + − Vr ρ Cp Vj ρj Cj Vr 2 ρ2 Cp 2 V 2 ρ2 Cp 2 ¾ rµ ¶−1 2 2 U Aj Tcin U Aj x3 (t) U Aj x3 (t) · + − , Vr ρ Cp Vj ρj Cj Vr ρ Cp Vj Vr ρ Cp Vj −
# 010 zˆ(t) = 0 0 0 zˆ(t) 000 " #∙ ¸ 0 J1 (ˆ z (t), zˆ(t − ∆), zˆ(t − 2∆)) + 1 z (t))u(t) +J2 (ˆ 0 " # 0 z (t)) + k (y(t) − C zˆ(t)) + 0 J3 (ˆ 1 ·
where [ k01 k02 ] ∈ R2 is chosen such that the matrix ∙ ¸ ∙ ¸ 01 0 + [ k01 k02 ] (8) 00 1 is Hurwitz. 4. OBSERVER DESIGN FOR REACTOR CONCENTRATION ESTIMATION
"
with The system (1) can be rewritten in the form: x(t) ˙ = f (x(t), x(t − ∆)) + g(x(t)) · u(t) y = h(x(t)) where: # " x1 (t) x(t) = x2 (t) ; x3 (t) the measured output function h is given by ∙ ¸ x2 (t) h(x(t)) = ; x3 (t)
(14)
k=
(9)
"
k1 0 k2 0 0 k3
(17)
#
such that the matrix (10)
(11)
"
# ∙ ¸ 010 100 0 0 0 −k 001 000
(18)
is Hurwitz. zˆ(t) is the estimation of z(t) and, by the function ψ, we can obtain the estimation of all the variables x(t) (including the un-measured reactor concentration). The hypotheses in Germani et al. [2001] are satisfied, locally, by the underlying CSTR model, and thus local convergence of the estimation error to zero is guaranteed, by suitable choice of the gain vector k.
! f1 (x(t), x(t − ∆)) (12) f (x(t), x(t − ∆)) = f2 (x(t), x(t − ∆)) , f3 (x(t)) with f1 , f2 , f3 , functions which can be easily extrapolated from the model (1). Let us consider the following change of variables: # Ã ! " 5. OBSERVER-BASED CONTROL LAW x2 (t) z1 (t) (t) (x(t), x(t − ∆)) z f = φ(x(t), x(t−∆)) = z(t) = 2 2 z3 (t) x3 (t) The equations of the observer for the CSTR (1), by the (13) methodology shown in the previous section, is as follows: Ã
·
zˆ1 = zˆ2 (t) + k1 (TR (t) − zˆ1 (t)) ·
zˆ2 = −
U Aj u ˆzˆ3 (t) F 2 zˆ1 (t − ∆) + Vr ρ Cp Vj Vr 2
+k2 (TR (t) − zˆ1 (t)) + F zˆ2V(t−∆) r ³ E ´−1 +K0 U Aj zˆ3 (t) e Rˆz1 (t) ρ−1 Cp −1 Vr −1
³ E ´−1 F U Aj zˆ3 (t) Rˆ z (t) − λ K F Φ Ca ρ−1 Cp −1 Vr −1 0 0 e 1 ρ Cp Vr 2 ³ E ´−1 E F U Aj zˆ1 (t) 2 Rˆ z1 (t−∆) Rˆ z (t) − − F e Φ T Vr −2 0 e 1 ρ Cp Vr 2 ³ E ´−1 U Aj uTcin + + K0 F Φ T0 e Rˆz1 (t) Vr −1 Vr ρ Cp Vj ³ E ´−1 −K0 F zˆ1 (t − ∆) Φ e Rˆz1 (t) Vr −1 +
−
E zˆ2 (t) F zˆ1 (t − ∆) Vr R (ˆ z1 (t))
2
+
F 2 zˆ1 (t) F 2 Φ T0 + Vr 2 Vr 2 ³ E ´−1 U 2 Aj 2 zˆ3 (t) −K0 U Aj zˆ1 (t) e Rˆz1 (t) ρ−1 Cp −1 Vr −1 − Vr ρ Cp Vj ρj Cj ³ ´ −1 E E −F e Rˆz1 (t−∆) Φ U Aj zˆ1 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2 −
E zˆ2 (t) F zˆ1 (t − ∆) Φ 2
Vr R (ˆ z1 (t)) ³ E ´−1 U 2 Aj 2 zˆ1 (t) F zˆ2 (t) + −2 − K0 zˆ2 (t) e Rˆz1 (t) Vr ρ Cp Vj ρj Cj Vr ³ E ´−1 F 2 zˆ1 (t − ∆) Φ −K0 F zˆ1 (t) e Rˆz1 (t) Vr −1 − Vr 2
U Aj zˆ2 (t) E zˆ2 (t) F Φ T0 E zˆ2 (t) F − 2 + V Rˆ Vr ρ Cp r z1 (t) Vr R (ˆ z1 (t)) ³ E ´−1 E E zˆ2 (t) U Aj +F e Rˆz1 (t−∆) zˆ2 (t − ∆) e Rˆz1 (t) Vr −1 + Vr Rˆ z1 (t) ρ Cp ³ E ´−1 E −F e Rˆz1 (t−∆) Φ zˆ2 (t − ∆) e Rˆz1 (t) Vr −1
³ E ´−1 E −F e Rˆz1 (t−∆) U Aj zˆ3 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2
³ E ´−1 E +F e Rˆz1 (t−∆) Φ U Aj zˆ3 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2 ·
zˆ3 =
+k3 (Tj (t) − zˆ3 (t)) Now, by using the relationship between z(t) and x(t) in the previous section, the control law (7) can be rewritten as a function of z(t), z(t − ∆) and z(t − 2∆). By using the estimation of z(t), zˆ(t), given by the observer, the observer based control law is the following:
˜ u(t) = A˜ · B where ˜= B
−
2
+
E (ˆ z2 (t))
2
E
+ 2 F 2 e Rˆz1 (t−∆) zˆ1 (t − 2 ∆)
R (ˆ z1 (t)) ³ E ´−1 Vr −2 Φ e Rˆz1 (t)
³ E ´−1 E −F 2 e Rˆz1 (t−∆) zˆ1 (t − 2 ∆) Φ2 e Rˆz1 (t) Vr −2 ³ E ´−1 E −F 2 e Rˆz1 (t−∆) zˆ1 (t − 2 ∆) e Rˆz1 (t) Vr −2 F zˆ2 (t − ∆) Φ E zˆ2 (t) U Aj zˆ3 (t) − 2 Vr Vr R (ˆ z1 (t)) ρ Cp ³ E ´−1 +K0 F zˆ1 (t − ∆) e Rˆz1 (t) Vr −1 −
³ E ´−1 E +F 2 e Rˆz1 (t−∆) zˆ1 (t − ∆) e Rˆz1 (t) Vr −2
³ E ´−1 E +F e Rˆz1 (t−∆) U Aj zˆ1 (t − ∆) e Rˆz1 (t) ρ−1 Cp −1 Vr −2 ³ E ´−1 E +F 2 e Rˆz1 (t−∆) Φ2 T0 e Rˆz1 (t) Vr −2
³ E ´−1 E −F 2 e Rˆz1 (t−∆) zˆ1 (t − ∆) Φ e Rˆz1 (t) Vr −2
u(t) (Tcin − zˆ3 (t)) (ˆ z1 (t) − zˆ3 (t)) Aj U + Vj Vj ρj Cj
µ
U Aj zˆ3 (t) U Aj Tcin − + Vr ρ Cp Vj Vr ρ Cp Vj
¶−1
and F U Aj zˆ3 (t − ∆) F Φ U Aj zˆ3 (t − ∆) A˜ = − + Vr 2 ρ Cp Vr 2 ρ Cp à ! ¶¶−1 µ µ FΦχ −1 +k01 zˆ1 (t) + E ln − R (−1 + χ) K0 Vr ⎛ ⎞ F (Φ T0 + (1 − Φ) zˆ1 (t − ∆) − zˆ1 (t)) ⎜ ⎟ Vr ⎜ ⎟ zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 − ⎜ ⎟ ⎜ F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ ⎟ ⎜ ⎟ +k02 ⎜ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ⎟ ⎜+ ⎟ ⎜ ⎟ Vr ρ Cp ⎜ ⎟ ⎝ ⎠ (ˆ z1 (t) − zˆ3 (t)) Aj U − Vr ρ Cp ⎞ ⎛ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ −F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ E ´−1 ³ ´−1 E − · e Rˆz1 (t) ρ−1 Cp −1 Vr −2 e Rˆz1 (t−∆) ⎞ ⎛ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ +F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ E ´−1 ³ ´−1 E − ·Φ e Rˆz1 (t) ρ−1 Cp −1 Vr −2 e Rˆz1 (t−∆)
⎞ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ +F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ ´−1 ³ ´−1 E E F 2 zˆ1 (t) − ρ−1 Cp −1 − ·ΦVr −2 e Rˆz1 (t−∆) e Rˆz1 (t−∆) Vr 2 ⎞ ⎛ zˆ2 (t − ∆) Vr ρ Cp − F ρ Cp Φ T0 ⎜ −F ρ Cp zˆ1 (t − 2 ∆) + F ρ Cp zˆ1 (t − 2 ∆) Φ ⎟ −F ⎝ ⎠ +F ρ Cp zˆ1 (t − ∆) + U Aj zˆ1 (t − ∆) −U Aj zˆ3 (t − ∆) ³ ´ ´−1 −1 ³ E E − ·Vr −2 e Rˆz1 (t−∆) ρ−1 Cp −1 e Rˆz1 (t−∆) ⎛
F 2 zˆ1 (t − 2 ∆) F 2 zˆ1 (t − ∆) U 2 Aj 2 zˆ1 (t) +2 − 2 Vr Vr 2 Vr 2 ρ2 Cp 2 ³ E ´−1 F U Aj zˆ1 (t) −2 + λ K0 F Φ Ca 0 e Rˆz1 (t) ρ−1 Cp −1 Vr −1 2 Vr ρ Cp −
U 2 Aj 2 zˆ1 (t) F 2 zˆ1 (t − ∆) Φ F 2 zˆ1 (t − 2 ∆) Φ2 − −2 − Vr ρ Cp Vj ρj Cj Vr 2 Vr 2 +2
F 2 zˆ1 (t − 2 ∆) Φ U 2 Aj 2 zˆ3 (t) F 2 Φ2 T0 + + Vr 2 Vr 2 ρ2 Cp 2 Vr 2
³ E ´−2 ³ ´−1 − E · e Rˆz1 (t) ρ−1 Cp −1 e Rˆz1 (t) Vr −1
! zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 − −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EU Aj zˆ3 (t) +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t))−2 e Rˆz1 (t) ρ−2 Cp −2 Ã
F Φ U Aj zˆ1 (t − ∆) Vr 2 ρ Cp ⎞ ⎛ zˆ2 (t) Vr ρ Cp ⎜ −F ρ Cp Φ T0 − F ρ Cp zˆ1 (t − ∆) ⎟ EF Φ T0 −⎝ +F ρ Cp zˆ1 (t − ∆) Φ + F ρ Cp zˆ1 (t) ⎠ +U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t))−2 e Rˆz1 (t) ρ−1 Cp −1 −2
Ã
zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 + −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 − E ·ˆ z1 (t − ∆) e Rˆz1 (t) Vr −2 R−1 −2
· (ˆ z1 (t))
!
EF Φ
³ E ´−1 ρ−1 Cp −1 e Rˆz1 (t)
! Ã F Φ U Aj T0 F U Aj zˆ3 (t) U 2 Aj 2 zˆ3 (t) zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 + + 2 2 Vr ρ Cp Vj ρj Cj Vr ρ Cp Vr ρ Cp +U Aj −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ ! Ã +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 ³ ´−1 ³ E ´−1 E − −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EF zˆ1 (t − ∆) − Rˆ −2 −2 −2 z1 (t) ·V ρ C e e Rˆz1 (t) r p +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 F U Aj zˆ1 (t − ∆) − E −2 · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t)) ρ−1 Cp −1 e Rˆz1 (t) +2 Vr 2 ρ Cp !2 Ã zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 − −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ E 6. SIMULATION RESULTS +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−2 ³ E ´−2 Simulations, using MatLab Software Package, have been − E −2 · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t)) ρ−2 Cp −2 e Rˆz1 (t) carried out to verify the effectiveness of the proposed method. The values of the model parameters used in ! Ã zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 simulation are given in Table 1. + −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EU Aj Table 1. Reactor and controller parameters values +F ρ C zˆ (t) + U A zˆ (t) − U A zˆ (t) +
p 1
j 1
j 3
³ ´−1 ³ E ´−1 − E −1 · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t)) ρ−2 Cp −2 e Rˆz1 (t) Ã
zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 +2 F −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 e Rˆz1 (t) ρ−1 Cp −1
!
! zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 + −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ EF +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t) ³ ´−1 ³ E ´−1 − E · e Rˆz1 (t) Vr −2 R−1 (ˆ z1 (t))−1 e Rˆz1 (t) ρ−1 Cp −1 Ã
+K0
Ã
zˆ2 (t) Vr ρ Cp − F ρ Cp Φ T0 −F ρ Cp zˆ1 (t − ∆) + F ρ Cp zˆ1 (t − ∆) Φ +F ρ Cp zˆ1 (t) + U Aj zˆ1 (t) − U Aj zˆ3 (t)
!
K0 = 20.75 × 106 s−1 E = 69.71 × 106 J · kmol−1 ρ = 801 kg · m−3 R = 8314 J · kmol−1 · K −1 ρj = 1000 kg · m−3 Cp = 3137 J · kg −1 · K −1 Cj = 4183 J · kg −1 · K −1 λ = −69.71 × 106 J · kmol−1 U = 851 W · K −1 · m−2 F = 4.377 × 10−3 m3 · s−1 Ca0 = 8.01 kmol · m−3
∆ = 40 s Tcin = 294 K T0 = 294 K Aj = 101 m2 Vr = 102 m3 Vj = 100 m3 k01 = −6 · 10−6 k02 = −5 · 10−3 k1 = 3 · 10−3 k2 = 2 · 10−6 k3 = 5 · 10−2
In all the carried out simulations, a minimum and a maximum value for the input u, specifically the volumetric
flow rate of the refrigerant, is imposed, to take into account of the actuator saturation. It is considered :
0
50000
150000
200000 314
TJ TJ
312
FJ min ≤ u(t) ≤ FJ max
310
312
Φ = 0.3 χ = 0.85
308
300,0
TJ, eq = 312.48 K
0
100
200
300
400
500
310
600 300,0
299,5
299,5
308
299,0
299,0
306
298,5
298,5
304
298,0
298,0
297,5
297,5 600
= 40 s 306
TJ (t=0) = TJ, eq TJ [K]
TJ [K]
with the following acceptable physical values for a hypothetical control valve:
100000
314
TJ (t=0) = 298 304 302 300
0
100
200
300
400
500
302 300
Time [s]
3
FJ min = 0.00678 m /s
298
298
296
296
3
FJ max = 0.08 m /s
0
50000
100000
150000
200000
Time [s]
We performed several sets of simulation runs and all showed the high performance of the method here shown. In the simulation here presented, the process is initially assumed to be at the start up. The Fig. 2 to Fig. 4 show the evolution of the true and estimated state.
Fig. 4. Real and estimated temperature of jacket TJ In Fig. 5 the control signal u(t) is plotted. 0
40000
80000
120000
25000
50000
75000
100000
125000
150000
175000
3
3
= 40 s
-1
0,035
0,030
0,030
0,025
0,025
0,020
0,020
0,015
0,015
0,010
0,010
0,005
0,005
1
1 0
0
1000
2000
3000
4000
3
5000 3
0
0
-3
-3
-6
-6
Ca(t=0) = Caeq Ca(t=0) = 0
Φ = 0.3 χ = 0.85
Caeq = 1.2015 kmol m-3 = 40 s
-3
Ca [kmol m ]
-3
Ca [kmol m ]
3
-1
u [m s ]
2
ueq = 0.0081 m s
0,035
Ca Ca 2
0,040
χ = 0.85
200000
3
200000 0,045
Φ = 0.3
0,040 0
160000
0,045
-1
-9
-9
-12
-12
-15
-15
-18
-18
0
1000
2000
3000
4000
0
-1
0,000 0
40000
80000
120000
160000
0,000 200000
Time [s]
5000
Tempo [s]
-20
-20 0
25000
50000
75000
100000
125000
150000
175000
Fig. 5. Real flowrate of coolant, u = FJ
200000
As can be seen, the estimated state variables converge to the true ones, and, more important, the state variables are driven to the chosen equilibrium point.
Time [s]
Fig. 2. Real and estimated concentration CA
7. CONCLUSION 0
50000
100000
150000
200000 320
320
TR TR
0
1500
3000
4500
6000
298,5
298,5
298,0
298,0
315 Φ = 0.3 χ = 0.85
TR [K]
TR, eq = 319.78 K = 40 s
310
TR [K]
315
297,5
297,5
297,0
297,0
310
TR(t=0) = TR, eq TR(t=0) = 298 305
305 0
1500
3000
4500
6000
Time [s]
300
300
295
295 0
50000
100000
150000
200000
Time [s]
Fig. 3. Real and estimated temperature of reactor TR
In this paper, in the spirit of the separation principle, a nonlinear feedback control law obtained by tools of differential geometry and observer theory for time-delay systems has been built up for a continuous stirred tank reactor with recycle. The reactor and jacket temperature are assumed available for measures, the reactor concentration, as often happens in practice, is not assumed to be available for measures. A recycle time delay is considered. The feedback control law, when all the state variables are available for measures, yields local exponential convergence of the reactor tracking error to zero. The observer guarantees local exponential convergence of the estimation error to zero. Many simulations have been performed, showing the high performance of the observer based nonlinear feedback control law here presented. Further theoretical investigations will concern the problems related to the saturation, which here is taken into account only in the simulations, and the problems related to the uncertainties on the parameters, such as the reactor heat transfer coefficient, by means of
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