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Ind. Eng. Chem. Res. 2002, 41, 4758-4764
PROCESS DESIGN AND CONTROL New Discrete Controller for a Class of Chemical Reactors Jesu ´ s Gonza´ lez,*,† Miguel Angel Barro´ n,‡ and Felipe Vargas-Villamil§ Departamento de Sistemas and Departamento de Materiales, Universidad Auto´ noma Metropolitana-Azcapotzalco, Apartado Postal 75-162, C.P. 07300, Me´ xico, D.F., Me´ xico; and Matema´ ticas Aplicadas y Computacio´ n, Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, C.P. 07330, Me´ xico, D.F., Me´ xico
The temperature regulation problem of a class of continuous stirred tank reactors, where two consecutive exothermic reactions take place is addressed. Time delay control coupled with an exact sliding differentiator is used to design a discrete input-output linearizing controller. This time delay controller with sliding differentiation is robust in the presence of noise in the output measurements and time delays in the actuator response. Conditions for practical stability are given in terms of the system parameters. Numerical simulations are used to show the dynamic performance of this controller. 1. Introduction Continuous stirred tank reactors (CSTRs) are of the utmost importance in the industry. They are widely used in the chemical industry, e.g., in polymerization, petrochemistry, pharmaceuticals, biochemistry, etc. Due to stringent demands in security and efficiency, and to environmental restrictions, there exists the necessity to develop sophisticated modeling techniques and better control strategies in order to improve the performance of these chemical reactors. Although there is a growing interest in the development of control and estimation strategies for highly nonlinear CSTRs, several problems in this area still remain to be solved. For example, reactor global stability is not guaranteed when PIDcontrollers are used and the stability problem is worsened in the presence of state estimation. In general, these controllers are not robust in the presence of disturbances, and their tuning is frequently difficult. Nevertheless, some authors have proposed modified PID controllers to address the above issues; e.g., Jadot et al.1 reported a robust nonlinear PI controller for a class of unstable reaction systems. Estimation and control strategies based on nonlinear models may improve process performance.2 Nevertheless, model-based control strategies have several weaknesses, such as the assumption of a perfect model and the lack of information about the states. Many of the strategies to estimate nonmeasurable states and disturbances for partially known systems are based on observers such as extended Kalman filters,2-5 high-gain observers,6-8 sliding mode observers,9-11 and so on. Recently, an alternative class of estimators for real time determination of uncertainties has been developed.12,13 * To whom all correspondence should be addressed. † Departamento de Sistemas, Universidad Auto´noma Metropolitana-Azcapotzalco. E-mail:
[email protected]. ‡ Departamento de Materiales, Universidad Auto´noma Metropolitana-Azcapotzalco. § Instituto Mexicano del Petro´leo.
In this context, Schuler and Schmidt12 proposed an uncertainty estimator based on calorimetric balances to infer the reaction heats in chemical reactors. Aguilar et al.13 applied this control scheme to regulate the substrate concentration in a continuous bioreactor. One of the most significant weaknesses of this approach, as shown in ref 13, is that the derivative related with the accumulation terms cannot be adequately calculated when there is noise in the measurements. Nonlinear control theory has been extensively used during the past few years to design controllers for CSTRs.5 Linearization of nonlinear systems is related to the cancellation of input-output nonlinearities under the assumption of a perfect knowledge of these nonlinearities. A feedback control scheme designed using this approach guarantees closed-loop stability and output tracking.14 Nevertheless, a perfect knowledge of the system nonlinearities is not easy to get. Therefore, since the nonlinear terms are not fully known, the linearizing controller can lead to a poor performance or even induce instabilities.15 Some approaches based on Taylor series linearization of the reactor model have been employed for the stabilization of chemical reactors. These approaches assume that all the uncertainties remain bounded.16,17 Unfortunately, the controllers designed under these approaches are conservative in nature and may exhibit a poor closed-loop performance. An alternative approach to achieve input-output linearization is time delay control (TDC). In this work, TDC is used to design a controller for the temperature regulation of a continuous stirred tank reactor. An advantage of TDC with respect to the aforementioned controllers is that it requires little a priori knowledge of the system dynamics.18,19 In TDC, the uncertainty is compensated through the measurement and estimation of the rate of change of the control output. Broadly speaking, TDC uses past observations of the control inputs and outputs to directly modify the control actions, rather than to adjust the controller or to identify the
10.1021/ie010062d CCC: $22.00 © 2002 American Chemical Society Published on Web 08/17/2002
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system parameters. Given that TDC is not a modelbased controller, it can deal with large unpredictable system parameter variations and disturbances. Nevertheless, TDC requires the measurement or the estimation of the output time derivative; however, the current schemes for on-line differentiation exhibit serious drawbacks. One of these drawbacks is the tradeoff between robustness with respect to measurement errors and noisy inputs. One way to design a differentiator is to approximate the time derivative by using the backward finite differences technique. However, this technique leads to large errors in the time derivative when measurements are noisy.13 To overcome this problem, in this work an exact reconstruction of the output time derivative is obtained through the sliding mode technique proposed by Levant.20 This exact value of the output time derivative is coupled with the TDC, and the resulting controller named time delay controller with sliding differentiation (TDCSD) is robust in the presence of noise in the output measurements while also exhibiting good closed-loop performance. The TDC controller deals with model uncertainties while the ESD estimates the output derivatives. This work is organized as follows. In section 2, the chemical reactor to be studied is discussed, and a precise statement of the problem is given. In section 3, a TDC controller and an exact sliding differentiator (ESD) for the chemical reactor are developed, and the resulting TDCSD is presented. In addition, in this section, the TDC and the TDCSD controllers are compared and stability conditions for the TDCSD are established. Also, convergence conditions for the ESD are given. Section 4 illustrates the reactor open-loop and closed-loop dynamic behaviors by means of numerical simulations. The conclusions of this work are given in section 5.
of A, the dimensionless concentration of B, the dimensionless temperature, and the dimensionless time, respectively. The other dimensionless parameters in eqs 1-3 are the chemical time (τch), the reaction ratio (φ), the dimensionless temperature of the cooling jacket (θj), and the Newtonian cooling time (τN). Gray and Scott22 found that the dynamic behavior of the autonomous reactor is highly complex for the following parameter values: τch ) 1.81, θj ) 17.5, and φ ) 0.01. Depending on the value of τN, chaotic behavior or limit cycles with one, two, four, or eight periods can arise. The assumptions imposed on the reactor model limit its applicability to a wider class of chemical reactors, however, those assumptions do not limit the applicability of the proposed control approach. Specifically, our control approach can be applied to a class of SISO dynamical systems with the structure given by eq 4. It is expected that different activation energies and reaction heats will change the dynamic behavior of the considered system, however, the corresponding analysis is out of the scope of this work. 2.2. The Ideal Input)Output Linearizing Controller. The main drawback that arises during the design of a temperature controller for the CSTR described above is the system nonlinearity. Input-output linearization offers a base for nonlinear control system design. A controller derived using this approach provides global closed-loop stability and tracking capabilities23 whenever the system model is completely known. Following Nijmeijer and Van der Schaeft,23 the model described by eqs 1-3 can be rewritten as
dy ) f (y, z) + g(y, z)u dτ
(4)
dz ) Ω(y, z) dτ
2. Background 2.1. System Description. The dynamic behavior of chemical reactors with consecutive exothermic reactions is highly nonlinear. Steady-state multiplicity and periodic or chaotic dynamics may arise.21 The system to be controlled in this work is a CSTR with a cooling jacket where two consecutive reactions occur:
AfBfC
f (y, z) ) -(1 + τN-1)y g(y, z) )
Energy and mass balances are used to obtain the reactor mathematical model. The assumptions imposed on the reactor model are as follows: (i) There is no inflow of reactants B and C. (ii) The activation energy of the two reactions is the same. (iii) Both reactions have the same heat of reaction. (iv) The inflow temperature is equal to the water temperature of the cooling jacket. The dimensionless model of this system is21,22
dR 1 ) 1 - R - R exp(θ) dτ τch
where y ) θ is the system output and u ) θj is the system input; z ) [z1, z2] ) [R,β]T are the internal states; f (y, z) ∈ R , g(y, z) ∈ R , and Ω(y, z) ∈ R 2 are given by
(1)
dβ exp(θ) (R - φβ) - β ) dτ τch
(2)
dθ θj exp(θ) ) (R + φβ) - (1 + τN-1)θ dτ τch
(3)
where R, β, θ, and τ are the dimensionless concentration
[
exp(y) (z1 + φz2) τch
1 1 - z1 z exp(y) τch 1 Ω(y, z) ) exp(y) (z1 - φz2) - z2 τch
]
(5)
Using the aforementioned input-output linearizing technique, the ideal controller for the system represented by eq 4 is given by
u)
1 (-τc-1e - f (y, z)) g(y, z)
(6)
where τc > 0 is the characteristic time, e ) y - yref is the tracking error, and yref is the temperature set point. τc-1 is equivalent to the control gain. An important issue related to the input-output linearization of the system of eq 4 is the stability of the zero dynamics.14 The zero dynamics is the internal dynamics of the system with the output kept identically to the reference value; if deviation variables are used, the output is equal to zero.
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Barron et al.24 proved that the zero dynamics of this system is Lyapunov stable. The nonlinear control law of eq 6 is named ideal because all the terms are assumed to be known. Thus, good performance is obtained when f (y, z) and g(y, z) are known. Nevertheless, when model uncertainties exist, the system cannot be linearized unless these uncertainties are estimated. 3. Controller Design This section shows that it is possible to design an input-output linearizing controller for system 4 without knowing the exact functional form of f (y, z) and g(y, z) nor the exact value of the related parameters. Several advanced control techniques have been developed for uncertain systems. Some of these methods include adaptive control,6 sliding mode control,25 and robust control.26 However, the time delay controller with sliding differentiator (TDCSD), which is developed in the next subsection, has several advantages over traditional control schemes: (i) To implement a TDCSD it is only necessary to know the upper bounds of certain process parameters. These parameters are related to the physical properties of the system, and its upper bounds can be obtained through open-loop numerical experiments. (ii) Despite the fact that the TDCSD is based on the upper bounds of the process parameters, the resulting controller is not a high-gain controller. (iii) The performance of most of the traditional control schemes is not satisfactory when nonmodeled dynamics are present. A frequent nonmodeled dynamics is the actuator one. During the design of the TDCSD, the actuator dynamics is considered. (iv) The noise in the measurements is another factor which affects closed-loop performance. In a TDCSD, the noise is internally smoothed. When the noise is too severe, the control input could be filtered without affecting the performance. 3.1. Time Delay Control. The TDCSD controller proposed in this work is based on a TDC controller and a ESD estimator. In this section a TDC controller is developed. TDC has been described in detail in ref 19. It depends on the direct estimation of a function which represents the uncertainties. Generally speaking, TDC uses past observations of the control inputs and outputs to directly modify the control actions rather than to adjust the controller gains or to identify the system parameters. For these reasons, TDC can deal with large unpredictable system parameter variations and disturbances. Following refs 18 and 19, the TDC controller for the system of eq 4 is given by
uj )
[( ( )
) (( )
dy 1 + gˆ uj-1 + gˆ dτ j-1
dyref - τc-1ej dτ j
)]
(7)
where gˆ is a constant to be determined and uj and uj-1 are the control inputs at time t ) jL and t ) (j - 1)L, respectively. L represents a time delay which, in this work, is equal to the sampling time. When the sampling time tends to zero, the reactor closed-loop dynamics using the above control law is given by
de ) -τc-1e dτ
which is equal to the closed-loop dynamics obtained by means of the ideal control law given by eq 6. Note that if the true value of g is used in the controller given by eq 7, the above dynamics is not obtained because the time delay prevents the exact cancellation of the model nonlinearities. In practice, a zero sampling time is not attainable, therefore, a non zero sampling time situation must be considered. For small sampling times, YoucefToumi and Ito19 proved that if the following condition is satisfied
-1 < 1 - g(yj, zj)gˆ -1 < 1
(8)
there exists a positive integer M such that for all j > M, the input-output linearization of the chemical reactor described by eq 4 using the TDC given by eq 7 is guaranteed. This means that for times greater than τM ) LM, the closed-loop reactor behaviors using the control laws given by eqs 6 and 7 are identical. In addition, given that g(y, z) depends on the reactor parameters, the above condition can be easily satisfied from the physical knowledge of the reactor. In the ideal situation, when g is equal to gˆ , inequality 8 is satisfied; however, this does not guarantee that the input-output linearization is obtained, because, in addition, the sampling time must be small enough. The implementation of the control law described by eq 7 requires the measurement or the estimation of the time derivative of the control output. However, until now the synthesis of on-line differentiators has been a difficult task. A tradeoff between robustness in the presence of measurement errors and robustness in the presence of noisy inputs arises when on-line differentiators are designed. One way to design a differentiator is to approximate the time derivative by means of backward finite differences, as will be discussed later. In those cases where the measurements are noisy, this approach cannot be implemented if the sampling time is too small. In the next subsection, a robust approach to obtain an estimate of the time derivative is presented. 3.2. Time Delay Control with Sliding Differentiation. TDC requires the estimation of the control output time derivative. However, linear differentiators with constant coefficients may achieve asymptotically exact differentiation for a small class of input signals.20 In this work, a robust exact differentiation is proposed using the sliding mode approach developed by Levant.20 The main feature of the exact sliding differentiator (ESD) is that the maximum error of the derivative is proportional to the square root of the input noise magnitude after a finite-time transient process. According to Levant,20 his approach can be applied if the following two conditions are satisfied: (i) the Lipschitz’s constant of the input signal derivative must be bounded by a given constant C, and (ii) the input signal noise must be a bounded function of time. For the problem considered here, both conditions are fulfillled. For the first one, it is well-known that mathematical functions describing CSTR dynamics are Lipchitz functions.5,27 To show this, let us define y(τ) ) θ(τ) as the input signal of the ESD; then, from eqs 1-3. one can see that there exists a Lipschitz’s constant C for the control output dynamics dy/dτ ) f (y, z) + g(y, z)u such that 0 < C < ∞. The above constant C can be obtained by means of open-loop experiments. The second condition is trivially satisfied given that the considered measurement noise is white in nature.
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Following Levant,20 an estimate dˆ (τ) of dy(τ)/dτ can be calculated in this way
dˆ (τ) ) ξ - µ|ζ - y(τ)1/2 sign(ζ - y(τ))
(9)
where ζ and ξ are internal variables whose values are obtained by solving the following system
dζ ) dˆ (τ) dτ
(10)
dξ ) - sign(ζ - y(τ)) dτ
In eqs 9 and 10, and µ are positive constants to be defined below, and sign(•) is the sign function, which is defined as follows
{
The proof of the convergence of the ESD can be found in Levant.20 The sufficient conditions for the convergence of dˆ (τ) to dy(τ)/dτ are
( -+ CC)
To implement the ESD, it can be assumed that the initial values of the internal variable, ζ(0), and the control output, y(0), are equal. The initial value of dˆ (0) can be set to zero. Last, using the TDC for system 4 and the ESD given by eqs 9 and 10, the TDCSD for the chemical reactor is expressed as
uj ) uj-1 +
[
1 -dˆ j-1 + gˆ
(( )
dyref - τc-1ej dτ j
)]
ref
dy )0 dτ
(12)
Considering the TDC controller given by eq 7 and supposing that the derivative of the tracking error with respect to time is approximated using finite differences, i.e.
j
ej - ej-1 L
(13)
1 gˆ τc-1
[( ) 1+
]
then, the following expression for the TDC is obtained
uj ) u j j-1 -
[( ) (
)
]
τc τc ej - ej-1 L L
(14)
A more familiar expression can be obtained by adding
]
τc 2τc τc 1 1 + e 1 + e + e j j-1 L L L j-2 gˆ τc-1 (15)
where u j j-1 ) uj-1 + (uj-1 - uj-2). On the other hand, a digital PD controller in its velocity form is expressed as28
[(
) (
)
]
τD 2τD τD ej - 1 + ej-1 + ej-2 L L L (16)
where Kp and τD are the proportional constant and the derivative time, respectively. Then, by comparison of eqs 15 and 16, it is clear that the TDC is similar to a digital PD controller. The following remarks can be made about the TDC given by eq 15: (i) For a first order-like system, its response will exhibit steady-state offset. (ii) The propagation of the noise of the control output measurements into the control input is inversely proportional to the sampling time. Therefore, as L f 0, the system becomes unstable when the control output measurements are noisy. Let us apply the previous analysis to the TDCSD. Integrating the system given by eq 10 and substituting ξ into eq 9, then the following discrete version of the ESD is obtained j-1
dˆ j = -L
γi - µ|e˜ j|1/2 γj ∑ i)1
(17)
where e˜ j ) ζj - yj is the estimation error and γj ) sign(e˜ j). Using the above estimate of the output time derivative, the TDCSD becomes j-1 1 uj ) uj-1 + [L γi + µ| e˜ j|1/2 γj - τc-1ej] (18) gˆ i)1
∑
By adding and subtracting the term uj-1 on the righthand side of eq 18, a more familiar expression can be obtained
j j-1 uj ) u
1 [e - ej-1] + ψj gˆ τc j
(19)
where u j j-1 ) uj-1 + (uj-1 - uj-2) and ψj is defined as
1 ψj ) [Lγj-1 + µ(|e˜ j|1/2 γj - |e˜ j-1|1/2 γj-1)] gˆ
hence
uj ) uj-1 -
[( )
τc τc 1 1+ ej-1 - ej-2 -1 L L gˆ τc
(11)
where dˆ (τ), ζ(τ), and ξ(τ) are defined in eqs 9 and 10. Below, a discussion on the advantages of the TDCSD is given. 3.3. Comparison between TDC and TDCSD. For the sake of simplicity, only the regulation problem will be considered in order to compare the stability properties of the TDC and the TDSCD. For this case
(dτde) =
uj-1 ) uj-2 -
uj ) uj-1 - Kp 1 +
+1 if ω > 0 sign(ω) ) 0 if ω ) 0 -1 if ω < 0
> C and µ2 g 4C
and subtracting the term uj-1 on the right-hand side of eq 14 and considering that
(20)
Now, a digital proportional controller P in its velocity form is derived from the eq 16 considering the derivative time to be null:
uj ) uj-1 - Kp[ej - ej-1]
(21)
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Figure 2. Open-loop behavior of the ESD. Figure 1. Chemical reactor open-loop dynamic behavior for different values of τN.
By comparing eqs 19 and 21, one can see that the TDCSD proposed in this work is similar to the sum of a digital P controller with an uncertainty compensation term. The following remarks may be made about the TDCSD given by eq 19: (i) The system response will exhibit steady-state offset. However, the offset can be made arbitrarily small using a small enough closed-loop characteristic time. (ii) The noise propagation of the control output measurements into the control input is not a function of the sampling time. (iii) The noise propagation is bounded. 4. Application Example To illustrate the performance of the ESD and the behavior of the closed-loop reactor, numerical simulations were carried out using the following values for the chemical reactor parameters: parameter
value
τch θj φ
1.81 17.5 0.01
For this reactor, Gray and Scott22 found that depending on the value of τN, chaotic behavior or limit cycles with one, two, four, or eight periods may arise. To test the robustness of the proposed control scheme in the presence of different qualitative open-loop behaviors in the simulations shown in Figures 1-4, the following values for τN were used: τN ) 0.123 for τ ∈ [0, 20]; τN ) 0.13723 for τ ∈ [20, 40]; τN ) 0.14 for τ ∈ [40, 60]. The change of the value of τN could be viewed as external disturbances to the system. The open-loop dynamic behavior of the reactor is shown in Figure 1. This figure shows that the behavior of the dimensionless states strongly depends on τN. Figure 2 shows the performance of the open-loop sliding
differentiator. In this figure, the real and the estimated dimensionless temperature time derivatives are shown. The values for µ and were chosen in order to obtain the best fitting between the real and the estimated time derivatives. As can be observed, the estimated value of the dimensionless temperature time derivative approximates fairly well the real derivative. Figure 2 exhibits two of the main properties of the ESD.20 The first one is that the correspondence between the real and the estimated values of the input signals is very good even for high-frequency variations. The second one is that no lag is introduced into the estimated time derivative. To illustrate the performance of the TDCSD controller, two sets of numerical simulations were carried out. In the first one, the regulation control problem was addressed, see Figures 3 and 4, while in the second one, the tracking control problem was studied, see Figures 5 and 6. In the regulation problem the objective is to maintain the control output at the reference value, i.e., yref ) θref ) 1.6261. Gray and Scott22 found that for u ) θj ) 17.5 and τN ) 0.14, the reactor exhibits one unstable equilibrium point located at Re ) z1,e ) 0.2634, βe ) z2,e ) 0.7166, ye ) θe ) 1.6261. Figure 3 shows the closed-loop performance for gˆ -1 ) 1.0. This value satisfies the condition given by eq 8. Values of gˆ considerably smaller than 1.0 may be employed. However, with this choice some undesirable effects may arise, such as the peaking phenomena. As can be observed in this figure, the control output, i.e., the reactor dimensionless temperature, is maintained fairly close to the reference value. However, the sliding diferentiator induces high-frequency chattering. It is well-known that due to the discontinuity present in the sign function, high-frequency chattering is generated.10 To filter the control input signal, several corrective actions may be taken. In this work, the sign function was substituted, as is usually done in sliding mode control, by this approximating function
sign(ω) =
ω |ω| +δ
where δ is a small enough constant.
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Figure 3. Closed-loop dynamic behavior of the TDCSD for the regulation control problem.
Figure 4. Closed-loop behavior for the regulation control problem when the control input was smoothed.
In addition to the above approximating function, a low-pass filter was used. Such a filter represents the actuator dynamics because the low-pass filter induces a lag in the actuator response. Figure 4 shows the closed-loop behavior for the regulation problem using the same conditions as those employed in Figure 3, but the approximating function of the sign function and the low-pass filter are simultaneously employed. A small overshoot of the dimensionless temperature is observed at τ ) 20 due to the change of τN; however, broadly speaking, the closed-loop performance is satisfactory. Figure 5 presents the simulation results for a tracking problem in which the reference value of the dimension-
Figure 5. Closed-loop dynamic behavior of the TDCSD for the tracking control problem.
Figure 6. Closed-loop behavior for the tracking control problem when the control output measurements are noisy.
less temperature changes from yref ) θref ) 1.6261 for τ ∈ [0, 30] to yref ) θref ) 1.75 for τ ∈ [30, 60]. In this figure, the value of τN was kept constant at 0.13723 during the course of the simulation. For the above reference values, the closed-loop dynamics of the system corresponds to a first order response. This means that the system given by eq 4 is exactly input-output linearized by the TDCSD. Furthermore, Figure 5 shows the dynamic behavior of the internal states z ) [z1, z2] ) [R, β]T, which resembles a first-order response. Finally, the closed-loop performance for the tracking problem in the presence of noisy temperature measurements is depicted in Figure 6. The standard deviation of the white noise was considered to be 0.1. The
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amplitude of the noise is approximately 5% of the reference value. Figure 6 shows that the propagation of the high frequency control output noise into the control input is not too severe. In this manner, the closed-loop performance remains satisfactory despite the extremely noisy nature of the temperature measurements. 5. Conclusions A robust controller to regulate the temperature of a class of continuous stirred tank reactors (CSTR) was developed. The controller is able to regulate the reactor temperature despite complex open-loop dynamics and the presence of model uncertainties. In addition, the controller is robust against noisy temperature measurements. The proposed controller does not depend on an exact knowledge of the reactor model. Literature Cited (1) Jadot, F.; Viel, F.; Bastin, G. Robust nonlinear PI control for a class of unstable reaction systems. Proc. Third Eur. Cont. Conf. 1995, 1870. (2) Kurtz, M. J.; Henson, M. A. State and disturbance estimation for nonlinear systems affine in the unmeasured variables. Comput. Chem. Eng. 1998, 22, 1441. (3) Farza, M.; Busawon, K.; Hammouri, H. Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors. Automatica 1998, 34, 301. (4) Farza, M.; Hammouri, H.; Jalfut, C.; Lieto, J. State observation of a nonlinear system: application to (bio)chemical processes. AIChE J. 1998, 45, 93. (5) Elicabe, G. E.; Ozdeger, E.; Georgakis, C.; Cordeiro, C. Online estimation of reaction rates in semicontinuous reactors. Ind. Eng. Chem. Res. 1995, 34, 1219. (6) Bastin, G.; Dochain, D. On-line Estimation and Adaptive Control for Bioreactors; Elsevier: Amsterdam, 1990. (7) Oliveira, R.; Ferreira, E. C.; Oliveira, F.; Feyo de Azevedo, S. A study on the convergence of observed-based kinetics estimators in stirred tank bioreactors. J. Proc. Control 1996, 6, 367. (8) Tornambe´ A.; Valigi, P. A decentralized controller for the robust stabilization of a class of MIMO dynamical systems. J. Dyn. Syst., Meas., Control 1994, 116, 293 (9) Yi, K.; Hedrick, K. Observer-based identification of nonlinear systems parameters. J. Dyn. Syst., Meas., Control 1995, 117, 175. (10) Slotine, J. J. E.; Hedrick, J. K.; Misawa, E. A. On sliding observers for nonlinear systems. J. Dyn. Syst., Meas., Control 1987, 109, 245.
(11) Wang, G. B.; Peng, S. S.; Huang, H. P. A sliding observer for nonlinear process control. Chem. Eng. Sci. 1997, 52, 787. (12) Schuler, H.; Schmidt, C. U. Calorimetric-state estimators for chemical reactor diagnosis and control: review of methods and applications. Chem. Eng. Sci. 1992, 47, 899. (13) Aguilar, R.; Alvarez, J.; Gonza´lez, J.; Barro´n, M. A strategy to regulate continuous fermentation processes with unknown reaction rates. J. Chem. Technol. Biotechnol. 1996, 69, 357. (14) Isidori, A. Nonlinear Control Systems, 3rd ed.; SpringerVerlag: London, 1995. (15) Esfandari, F.; Khalil, H. K. Output feedback stabilization of fully linearizable systems. Int. J. Control 1992, 56, 1007. (16) Barmish, B. R.; Corless, M.; Leitman, G. A. A new class of stabilizing controllers for uncertain dynamical systems. SIAM J. Control Optim. 1983, 21, 246. (17) Kravaris, C.; Palanki, S. Robust nonlinear state feedback under unstructured uncertainty. AIChE J. 1988, 7, 1119. (18) Youcef-Toumi, K.; Ito O. A time delay controller for systems with unknown dynamics. J. Dyn. Syst., Meas., Control 1990, 112, 133. (19) Youcef-Toumi, K.; Ito O. Input-output linearization using time delay control. J. Dyn. Syst., Meas., Control 1992, 114, 10. (20) Levant, A. Robust exact differentiation via sliding mode techniques. Automatica 1998, 34, 379. (21) Jorgensen, D.; Aris, R. On the dynamics of stirred tank with consecutive reactions. Chem. Eng. Sci. 1983, 38, 45. (22) Gray, P.; Scott, S. Chemical Oscillations and Instabilities; Clarendon Press: Oxford, England, 1990. (23) Nijmeijer, H.; Van der Schaeft, A. J. Nonlinear Control; Springer-Verlag: London, 1993. (24) Barro´n, M.; Gonza´lez, J.; Aguilar R.; Arce-Medina, E. Successful bounded control for a chemical reactor with nonlinear oscillatory consecutive reactions. Chem. Eng. J. 1997, 66, 27. (25) Slotine, J. J. E.; Sastry, S. S. Tracking control of nonlinear systems using sliding surfaces with applications to robot manipulators. Int. J. Control 1983, 38, 465. (26) Gonza´lez, J.; Aguilar, R.; Alvarez-Ramırez, J.; Barro´n, M. Nonlinear regulation for a continuous bioreactor via numerical uncertainty observer. Chem. Eng. J. 1998, 69, 191. (27) Alvarez-Ramirez, J.; Femat, R.; Gonzalez-Trejo, J. Robust control of a class of uncertain first-order systems with least prior knowledge. Chem. Eng. Sci. 1998, 53, 2701. (28) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling and Control; Oxford University Press: New York, 1994.
Received for review January 22, 2001 Revised manuscript received April 26, 2002 Accepted June 25, 2002 IE010062D
