Synthesis of nonlinear adaptive controller for a batch distillation

ISA Transactions 46 (2007) 49–57 www.elsevier.com/locate/isatrans

Synthesis of nonlinear adaptive controller for a batch distillation Amiya K. Jana ∗ Department of Chemical Engineering, Birla Institute of Technology and Science-Pilani, Rajasthan-333 031, India Received 6 November 2005; accepted 22 May 2006 Available online 22 January 2007

Abstract A nonlinear adaptive control strategy is proposed for a binary batch distillation column. The hybrid control algorithm comprises a generic model controller (GMC) and a nonlinear adaptive state estimator (ASE). The adaptive observation scheme mainly estimates the imprecisely known parameters based on the available tray temperature measurements. The sensitivity of the proposed estimator is investigated with respect to the effect of initialization error, unmeasured disturbance and uncertainty. Then, a comparative study is carried out between the derived nonlinear GMC–ASE controller and a traditional proportional integral law in terms of set point tracking and disturbance rejection performance. The study also includes the effect of measurement noise and parametric uncertainty on the closed-loop performance. The proposed adaptive control algorithm is shown to be quite promising due to the exponential error convergence capability of the ASE estimator in addition to the high-quality control action provided by the GMC controller. c 2006, ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Generic model controller; Adaptive state estimator; Adaptive control algorithm; Batch distillation

1. Introduction Batch distillation is a well-known separation process that is mostly used in many chemical, biochemical, and food industries for the production of small amounts of products with high added value. A single batch distillation column can produce several products from a multicomponent feed mixture within a single operation, whereas a train of columns is required to separate the same mixture continuously. The flexibility provided by a batch rectifier gives rise to challenging control problems that are basically owing to the nonstationary, nonlinear and finite time duration nature of the underlying dynamics. The batch distillation is inherently an unsteady state process, and in such a situation, there is no normal condition at which the controllers can be tuned. Hence, it is really a challenging task to obtain a satisfactory closed-loop response [1,2]. It is recognized that the conventional proportional integral (PI) and proportional integral derivative (PID) controllers provide poor performance due to the nonstationary nature of batch rectifiers [3]. Frattini-Fileti and Rocha-Pereira [4] ∗ Tel.: +91 01596 245073 215; fax: +91 01596 244183.

E-mail address: amiya [email protected].

applied the gain scheduled PI control strategy on a binary batch distillation. The authors proposed the predictive and adaptive control schemes to confront the time-varying nature of the batch rectifier. Li and Wozny [5] showed that the optimal profiles cannot be tracked with conventional linear controllers. Subsequently, Li and Wozny [6] again showed that a predefined optimal policy can be tracked via an adaptive control law to realize the optimum of multiple-fraction batch distillation. In their procedure, a recursive least square estimation with a variable forgetting factor is used for the online identification and to follow the rapidly changing process dynamics. Barolo and Berto [7] proposed a control strategy that is derived in the framework of nonlinear internal model control [8]. To estimate the distillate composition from the selected tray temperature measurements, the authors used the extended Luenberger observer [1]. The derived control law was then successfully applied on a binary and a ternary batch rectifier. Dechechi et al. [9] have shown that the nonlinear model predictive controller provides good results for overhead composition regulation. In their approach, an extended Luenberger observer is employed for state estimation. Alvarez-Ramirez et al. [10] developed a strategy that comprises a controller (classical PID with antireset windup) and an observer to estimate the modeling error. A drawback of

c 2006, ISA. Published by Elsevier Ltd. All rights reserved. 0019-0578/$ - see front matter doi:10.1016/j.isatra.2006.05.001

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A.K. Jana / ISA Transactions 46 (2007) 49–57

Nomenclature dt Derivative time D Distillate rate Ds Bias of PI controller whose output is D e, e D Error to the controller K C , K C D Gain of PI controller K 1 , K 2 Tuning parameters of the generic model controller Ln Internal liquid flow rate leaving nth tray m Number of manipulated variables mB Holdup in the still pot m˙ B Time derivative of m B mD Holdup in the reflux drum m Dsp Set point value of m D mn Liquid holdup on the nth tray n Number of states/tray index q Number of measurable disturbances R Reflux flow rate Rs Bias of PI controller whose output is R t Time at which the PI controller produces the outputs TD Temperature in the reflux drum u Vector of manipulated inputs VB Vapor boil-up rate Vn Internal vapor flow rate leaving nth tray x Vector of state variables xˆ Vector of estimated states xB Liquid composition in still pot x˙ B Time derivative of x B xD Distillate composition x˙ D Time derivative of x D x Dsp Set point of x D y Output variables yB Composition of boil-up vapor yc Vector of calculated compositions (sensor outputs) ym Vector of measured states ysp Set point of y Greek letters ε τi , τ D θ

An unknown and bounded function in the adaptive estimator Integral time for PI controllers Vector of augmented states

their strategy [3] is that they strongly rely on an accurate model of the batch distillation process. Subsequently, Oisiovici and Cruz [11] proposed an inferential control structure in conjunction with an extended Kalman filter [12]. They applied this control scheme on a high-purity multicomponent batch distillation column. Han and Park [13] have used the quasidynamic estimator [1] to estimate the distillate composition in a closed-loop batch rectifier. In their research work, a nonlinear wave theory [14] has been used to develop the

nonlinear model-based controller. Then, Monroy-Loperena and Alvarez-Ramirez [3] presented a method for the identification and control of a batch distillation process. In their work, the feedback controller is designed in the framework of robust nonlinear control [15] with modeling error compensation techniques. The success of a nonlinear model-based control system greatly depends on the performance as well as robustness of the state estimator. So far, many observation schemes have been designed and applied with good results on the batch distillation processes [1,16–20]. This research interest is towards developing an adaptive control algorithm for a batch distillation column. The control strategy is designed in the framework of generic model control [21] (GMC), combined with an adaptive state estimator [22,23] (ASE). The GMC is a method of choice because it is an efficient strategy [24] to implement the nonlinear process model-based control, and it has been shown to be very successful in laboratory-scale [25] and industrial-based applications [26,27]. Here, the adaptive observation scheme is mainly employed to estimate the badly known parameters from the readily available temperature measurements instead of using the directly measurable product compositions because the composition analyzers provide large delays in the response in addition to high investment and maintenance costs. In brief, two major contributions of the present work are highlighted in the following: (i) It is true that most of the nonlinear observation schemes involve significant design complexity. Actually, the large predictor model and the complex structure of the closedloop observation technique complicate the overall design. To reduce the design complexity, in this study, an adaptive state estimator has been developed for a rigorous batch distillation column. This estimator has simple structure and the predictor model includes only two component continuity equations of the example process. We must note that the proposed ASE approach only estimates the states as per the GMC controller requirements. Like other observation algorithms, it does not compute all the process states. The estimation scheme that is proposed here for a batch distillation is not reported in the scientific literature. (ii) In addition, an adaptive GMC–ASE control law has been synthesized for the batch distillation. The proposed control structure provides high-quality performance mainly due to the exponential error convergence capability of the ASE estimator. To the best of our knowledge, the design technique of the GMC–ASE control strategy for the batch distillation column is a new one. In this study, first the open-loop performance of the proposed adaptive estimator has been inspected under initialization error in the imprecisely known parameters, unmeasured disturbance, and uncertainty. Subsequently, the adaptive controller is implemented on the simulated batch rectifier. The closedloop performance of this control technique has been compared with that of the conventional PI controller. The closed-loop

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A.K. Jana / ISA Transactions 46 (2007) 49–57

by Signal and Lee [29]: K 1(i,i) =

Fig. 1. Block diagram for the adaptive control algorithm.

simulation experiments show the comparative effect of set point change, unmeasured disturbance, modeling uncertainty, and measurement noise. 2. Nonlinear adaptive control algorithm The proposed adaptive control structure consists of the nonlinear generic model controller and an adaptive state estimator. The closed-loop system having different controller elements in addition with a process is given in Fig. 1. Indeed, the poorly known parameters (augmented states) along with the measured states are estimated by the nonlinear observation scheme. Then the estimated parameters and measured states are used to compute the GMC responses. The detailed synthesis of an adaptive controller is presented in generalized form here. 2.1. Generic model control A nonlinear system can be described by the following form of equation: x = f (x, d)θ + g1 (u, x, d),

(1)

y = cx,

where the state x ∈ Rn , the model parameter θ ∈n , the measurable disturbance d ∈ Rq , and the input u ∈ Rm . Moreover, f and g1 are matrices of nonlinear functions. In this study, it is assumed that all states are measurable and c (coefficient matrix) is a unity matrix. From the basic principle of GMC [21], the following control law can be derived [28]: Z t f (x, d)θ + g1 (u, x, d) − K 1 e − K 2 e dt = 0, (2) 0

where e is the error (=ysp − y) to the controller, ysp is the set point of the output, and K 1 and K 2 are diagonal n × n tuning parameter matrices. Eq. (2) shows that the GMC algorithm comprises a dynamic process model, proportional action term and integral action term. If g1 is linear with respect to u, then one can write g1 (u, x, d) = b(x, d)u. Accordingly, Eq. (2) yields   Z t −1 u = [b(x, d)] K1e + K2 e dt − f (x, d)θ . (3) 0

The values of the elements of the tuning parameter matrices can be determined based on the following relationships given

2τ1i τ2i

K 2(i,i) =

1 τ2i2

(4)

where τ1i and τ2i determine the shape and speed of the desired closed-loop trajectory (the reference trajectory), respectively. The reference trajectory gives pseudosecond-order response for a step change in the set point. However, Yamuna and Gangiah [30] confirmed that the earlier relationships could be applied to compute the specified response accurately. Once the values of τ1i and τ2i are obtained, then K 1 and K 2 can be calculated from Eq. (4). This control strategy has several advantages, such as straightforward formulation and simple tuning. Importantly, the relationship between feedforward and feedback control is explicitly accounted for in the GMC law. 2.2. Adaptive state estimation It is assumed that the parameter dynamics in the nonlinear system (Eq. (1)) obey the following first-order equation: θ˙ = g2 (u, x, d) + ε,

(5)

where g2 is a nonlinear function and ε is an unknown function that may depend on x, θ, u, d, noise, and so on. The assumptions that have been made are: ε is an unknown but bounded function and the disturbance d with its time derivative are also bounded. The nonlinear system equations ((1) and (5)) can be expressed in the following condensed form: (6) Z˙ = F(x, d)Z + G(u, x, d) + ε¯ y = CZ hi h i where Z = θx , F(x, d) = 00 f (x,0 d) , G(u, x, d) = h i hi   g1 (u, x, d) ¯ = 0ε and C = In , 0 , with In the n × n g2 (u, x, d) , ε identity matrix. f is an n × n matrix which is differentiable and the corresponding partial derivative is continuous. According to Farza, Busawon and Hammouri [22], the nonlinear adaptive observer can be used to track the vector Z as follows: Z˙ˆ = F(y, d) Zˆ + G(u, y, d) − Γ −1 (y, d)S −1 C T (C Zˆ − y),(7) where hi y (i) Zˆ = θˆ ∈ R2n , θˆ ∈ Rn are the estimated vectors of state and parameter, h respectively. i

(ii) Γ (y, d) =

In 0

0 f (y, d)

.

(iii) S is the unique symmetric positive-definite matrix which satisfies the algebraic Lyapunov equation αS + B T S + B S − C T C = 0, (8) h i 0 In where B = 0 0 and α > 0 is a design parameter [31]. However, the solution of Eq. (8) is given by

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A.K. Jana / ISA Transactions 46 (2007) 49–57

 S =

1 In α  1 In − α2

1 In α2  , 2 In 3 α

−

estimator is Γ

−1

(y, d)S

−1



and consequently, the gain of the



 2α In C = 2 −1 . α f (y, d) T

(9)

It has been shown [22] that under the earlier conditions and when there exist finite real numbers β, λ, with 0 < β ≤ λ such that, for time t ≥ 0,

Table 1 Operating conditions for the batch column System

Ethanol/water

Total feed charge, g mol Feed composition (startup), mol fraction Tray holdup (startup), g mol Heat input to the still pot, kJ/min Distillate flow rate (production phase), g mol/min Distillate composition (steady state), mol fraction

10 000.0 0.3/0.7 30.0 5000.0 60.0 0.87

• nth tray

β 2 In ≤ f T (y, d) f (y, d) ≤ λ2 In , the convergence of the nonlinear observer can be guaranteed. It is obvious from Eq. (9) that only a single tuning parameter α is involved in the estimator. When ε = 0, the convergence of the observer error is an exponential one. In the case where ε 6= 0, the asymptotic error can be made arbitrarily small by choosing a sufficiently large value of α. However, a very large value of α may make the observer sensitive to noise. Thus, the choice of α [23] is a compromise between fast convergence and sensitivity to noise. This adaptive observation scheme offers several appealing advantages, such as ease of construction and online implementation, negligible computational load, transparent tuning, and more importantly, guaranteed convergence. 3. The process To apply the adaptive control algorithm, the mathematical model of a batch distillation column is derived and this model will be referred to as ‘the process’. In order to represent realistic operation of actual batch distillation column, a rigorous nonlinear model is required. Such a distillation model may be developed involving dynamic material and energy balance, and algebraic enthalpy equations supported by vapor–liquid equilibrium (VLE) and physical properties. The model of a binary batch distillation process is formulated on the following assumptions: • staged batch distillation column with trays numbered from the bottom and up (total fifteen trays including still pot), • perfect mixing and equilibrium on all trays, • constant stage pressures (atmospheric) and tray efficiencies (vapor-phase Murphree efficiency = 80%), • negligible tray vapor holdups, • total condensation with no subcooling in the condenser, • nonlinear Francis weir formula [32] for tray hydraulics calculations, • variable liquid holdup in each tray (excluding reflux drum during only total reflux condition), and • Wilson activity coefficient model [33] for VLE calculations.

Total continuity: m˙ n = L n+1 + Vn−1 − L n − Vn

(13)

Component continuity: m˙ n x˙n = L n+1 xn+1 + Vn−1 yn−1 − L n xn − Vn yn (14) L V Energy equation: m˙ n H˙ nL = L n+1 Hn+1 + Vn−1 Hn−1 − L n HnL − Vn HnV .

(15)

• Reflux drum (16) Total continuity: m˙ D = VNT − R − D Component continuity: m˙ D x˙ D = VNT y NT − (R + D)x D . (17) Here, the time derivative of the multiplication of two variables, say m and x, is denoted by m˙ x[=d(mx)/dt]. ˙ Due to very fast internal energy dynamics, the vapor and liquid enthalpies are computed using an algebraic form of equations [33]. The energy balance equations, with left-hand sides equal to zero, have been used for the calculation of vapor flow rates. The operating conditions for the example batch distillation column, which deals with the separation of a highly nonideal ethanol/water mixture, are reported in Table 1. 4. Controller synthesis for the batch distillation In the closed-loop simulation experiments, the distillate composition (x D ) of the example rectifier is to be controlled by manipulating the reflux flow rate (R). The model-based adaptive controller and conventional PI are implemented under the same process conditions to investigate their comparative performance. These control structures are synthesized here. 4.1. Adaptive control strategy As stated previously, the adaptive controller is designed in the framework of a generic model controller, incorporating an adaptive state estimator. In order to apply this to the prescribed batch distillation process, the controller elements are required to be derived based on the control theories reported in Section 2. 4.1.1. Generic model controller The component continuity equation for the condenser– accumulator system is

The summary of the modeling equations is as follows: • Reboiler Total continuity: m˙ B = L 1 − VB = −D Component continuity: m˙ B x˙ B = L 1 x1 − VB y B

(10) (11)

Energy equation: m˙ B H˙ BL = Q R + L 1 H1L − VB H BV .

(12)

m˙ D x˙ D = VNT y NT − (R + D)x D .

(18)

Actually, a level controller (PI) is employed to control the holdup in the reflux drum (m D ). The variation of liquid holdup

A.K. Jana / ISA Transactions 46 (2007) 49–57

is so small (±0.5%) that it is reasonable to assume constant m D , and accordingly, the above equation becomes x˙ D =

VNT y NT − (R + D)x D . mD

(19)

Using Eq. (2), the following form of controller equation can be obtained for the process concerned:   Rt VNT y NT − m D K 1 e + K 2 0 e dt R= − D, (20) xD where e = x Dsp − x D . It is obvious from the above controller equation that the component vapor flow rate leaving the top tray (VNT y NT ), a poorly known parameter, is required to estimate for the implementation of GMC. The values of tunable parameters are obtained based on the guidelines given in Section 2 as K 1 = 5.0 and K 2 = 0.0005. 4.1.2. Adaptive state estimator In this study, selected tray temperatures are assumed to be measurable. It is not practically very difficult to measure online even all tray temperatures. Increasing the number of measurements did not lead to significant improvements in the estimator performance [7,34]. However, from a measured tray temperature, the corresponding liquid composition can be computed using a sensor model. The derivation of a sensor model is discussed now. For a binary system ( j = 2), the vapor-phase composition (y) in equilibrium with liquid phase (x) can be expressed as yj = kjx j.

(21)

The vapor–liquid equilibrium ratio has the general form kj =

γ j p sj pt

,

(22)

where γ , p s and pt represent the liquid-phase activity coefficient, vapor pressure and total pressure, respectively. Since p s solely depends on the temperature according to the Antoine equation, it is easy to calculate the liquid compositions knowing the measured tray temperatures. The activity coefficients are determined using the Wilson equation. For the prescribed batch column, the temperatures in the reflux drum, still pot and in the bottom tray are assumed to be measurable. Accordingly, the composition of the liquids in the reflux drum (x D ), still pot (x B ) and bottom tray (x1 ) may be computed from the sensor model following the steps described earlier. Note that these calculated compositions based on the readily available temperatures would be considered as measured compositions in what follows. The generic model controller only needs the information on VNT y NT . In this study, another imprecisely known parameter, the vapor–liquid equilibrium constant in the still pot (k B ), along with VNT y NT , have been estimated based on the measured x D , x B and x1 . Such attempts have been made in order to provide a better test scenario for the proposed procedure. It is worth mentioning that although x D and x B (measured states)

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are directly obtained from the sensor model, these compositions are also estimated in the ASE to compute the residual (xˆ − x). The predictor model, which is required to design an adaptive state estimator, consists of only two balance equations around the reflux-condenser system and still pot in addition to two extra state equations having no dynamics. The mathematical representation of the predictor model is VNT y NT − R x D − Dx D mD L 1 (x1 − x B ) − VB (y B − x B ) x˙ B = mB ˙ VNT y˙ NT = 0.0 k˙ B = 0.0. x˙ D =

(23)

Since the component continuities around the reflux-accumulator system and the still pot have no common states, the adaptive state estimator may be partitioned into two subsystems. The final structures of the state estimator can be obtained in matrix form combining Eqs. (7), (9) and (22): Subsystem 1:    " #    −Rx D − Dx D 1 x˙ˆ D x ˆ 0 D  + = mD mD  ˆ VNT yˆ NT V˙ˆ NT y˙ˆ NT 0 0 0   2α1 − 2 [xˆ D − x D ]. (24) α2 m D Subsystem 2:    " #  −VB x B   L 1 (x1 − x B ) + VB x B x˙ˆ B x ˆ 0 B  = mB  ˆ +  mB kB k˙ˆ B 0 0 0   2α3 (25) −  −α42 m B  [xˆ B − x B ]. VB x B In the earlier estimator structures, α1 , α2 , α3 and α4 are the tuning parameters. The values of these parameters are obtained based on the guidelines suggested by Farza et al. [23] as α1 = 200.0, α2 = 50.0, α3 = 30.0 and α4 = 60.0. It was mentioned earlier that the two augmented states (VNT y NT and k B ) are estimated based on the three measured states (x D , x B and x1 ). Therefore, it is ensured in the present case that the number of parameters to be adapted is not greater than the number of measurements used [23,35]. 4.2. PI controller A single-loop PI controller, which is used to control the nonlinear batch distillation process, is written as   Z 1 t (26) R = Rs + K c e + e dt . τi 0 The tuning of the PI controller parameters has been performed using integral square error ISE performance criteria and the optimal values of these parameters are: K c = −9997.50 and τi = 0.992. For both the adaptive controller and traditional PI, the manipulated variable is bounded as R ≤ 225.0 g mol/min.

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Fig. 3. Performance of the state estimator at nominal condition. Fig. 2. Startup profile of the process.

4.3. Level controller A conventional PI controller is used to control the holdup in reflux drum by manipulating the distillate flow rate. The controller equation is   Z t 1 (27) e D dt , D = Ds + K C D e D + τD 0 where e D = m Dsp − m D . In this study, in addition to the constant distillate composition (using an adaptive/PI controller) the overhead flow rate is also kept almost constant using this level controller. The controller parameters are selected accordingly as K C D = 0.000001 and τ D = 10 000.00. 5. Results and discussion The performance of the adaptive state estimator and adaptive control algorithm is evaluated by applying them to the separation of an ethanol/water system in a batch rectifier. First the column is brought to the steady state by considering the conventional total reflux startup procedure. Then both the controller and estimator are switched on, and the controllers (adaptive and PI) as well as estimator parameters are tuned. Note that immediately after the withdrawal of overhead product is started, the estimator outputs may be somewhat inaccurate and those outputs may lead to the production of very aggressive controller responses. This happens because the distillate composition is moved from the steady state value to the set point value and this results in severe spiking of the reflux rate. Under total reflux startup conditions, the batch column takes almost 140 min to reach the steady state, as shown in Fig. 2. All the performance evaluation plots are given only for the production phase of the batch rectifier. However, first the tracking performance of the proposed observation scheme is tested on the open-loop system. Subsequently, the performance of the adaptive controller is compared with that of the conventional PI.

Fig. 4. Performance of the state estimator under initialization error in the following two parameters (bVN T y N T , K B c = [118.12, 2.395]; bVN T y N est , T K Best c = [80, 1]).

5.1. Observer performance As previously mentioned, only two balance equations (the component continuities around the condenser–accumulator system and the still pot) constituted the state predictor causing significant structural mismatch between the reduced-order model and the process. Despite this discrepancy, the nonlinear estimator shows promising error convergence ability in Fig. 3 at the nominal condition. At the beginning of the production phase, the estimated parameters deviate from their actual values due to the sudden distillate withdrawal. Fig. 4 depicts the result of a sensitivity test on the estimator performance under large initialization error in VNT y NT and k B . It is practically true that the model parameters are not precisely known. However, in this situation, the closed-loop observation approach adequately estimates the states with sufficiently fast convergence of the estimation error towards zero. The effects of unmeasured disturbance in heat input to the still pot and of uncertain tray efficiency have been demonstrated in Figs. 5 and 6, respectively. In order to provide a realistic test scenario for the proposed observer, two consecutive step

A.K. Jana / ISA Transactions 46 (2007) 49–57

Fig. 5. Performance of the state estimator under unmeasured disturbance in heat input to the still pot (changed from 5.0 × 103 to 4.0 × 103 kJ/min at time = 165 min and then from 4.0×103 to 5.0×103 kJ/min at time = 175 min).

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Fig. 8. Comparative servo performance of the adaptive controller and PI with set point step changes in distillate composition (changed from 0.87 to 0.85 and then from 0.85 to 0.83).

Fig. 6. Performance of the state estimator under uncertain tray efficiency (changed from 80% to 50% at time = 165 min and then from 50% to 80% at time = 175 min). Fig. 9. Comparative performance of the adaptive controller and PI under unmeasured disturbance in heat input to the still pot (changed from 5.0 × 103 to 4.0 × 103 kJ/min at time = 160.3 min and then from 4.0 × 103 to 5.0 × 103 kJ/min at time = 165 min).

Fig. 7. Performance of the state estimator under unknown column top pressure (changed from 760 to 780 mm/Hg at time = 165 min and then from 780 to 760 mm/Hg at time = 175 min).

changes with large magnitudes have been considered in both cases. It is observed from the figures that the top and bottom compositions are negligibly affected by the step changes. The reason may be that the effect of step changes is insignificant compared to that of distillate withdrawal. However, excellent agreement has been achieved between the estimated and true process values even for large step changes in the presence of structural discrepancy. Fig. 7 shows the effects of unknown pressure at the top of the column on the ASE performance. Here, the top pressure has been changed twice. The open-loop simulation results show satisfactory tracking performance of the ASE estimator. This simulation experiment again confirms excellent estimation error convergence ability of the proposed observation scheme.

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Fig. 10. Comparative performance of the adaptive controller and PI under uncertain tray efficiency (changed from 80% to 30% at time = 160.3 min and then from 30% to 80% at time = 165 min).

5.2. Comparative closed-loop performance of adaptive controller and PI In the servo performance study, the effects of two consecutive set point step changes in distillate composition have been presented in Fig. 8. It is obvious that the proposed adaptive controller ensures fast and smooth response, and the PI controller is clearly outperformed by the adaptive controller. Moreover, in terms of reflux rate profile, the

nonlinear controller provides better response than PI because the PI outputs hit the specified upper bound of the manipulated input. The unmeasured disturbance rejection capability of both the controllers is studied by implementing two consecutive step changes in heat input to the still pot. The comparative closed-loop process responses are analyzed in Fig. 9. In this simulation experiment, the adaptive control scheme confirms better performance as compared to the conventional PI. In practice, the efficiency of every tray in a distillation column is different and may vary with time. It is practically infeasible to have the proper knowledge of tray efficiencies. Fig. 10 depicts the comparative effect of uncertain tray efficiency on the controller responses. From the figure, it is obvious that the nonlinear control strategy performs better than the PI controller. Fig. 11 shows the simulation results when both the controllers are implemented on the batch rectifier with a set point step change in distillate composition under measurement noise. It is considered that an additive Gaussian noise with zero mean corrupts the measurement of temperature in the reflux drum. The figure confirms that the proposed adaptive controller provides better noise rejection, whereas the produced PI control input appears chattering. 6. Conclusions An adaptive control strategy is proposed for the control of constant composition operations of a batch distillation column.

Fig. 11. Comparative servo performance of the adaptive controller and PI under measurement noise in TD (amp TD = ±0.05).

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The strategy comprises a nonlinear generic model controller and a closed-loop adaptive state estimator. Simple structure, easy tuning and promising performance make the adaptive controller attractive for online use. Structural and parametric mismatches were considered between the actual process and its model in order to provide a realistic test scenario for the proposed strategy. In this study, the open-loop performance of the nonlinear adaptive estimator was tested. Despite the structural discrepancy, unmeasured disturbance and uncertainty, the observation scheme provided fast converge of the estimation error towards zero. Moreover, the adaptive control strategy confirmed its superiority over the conventional PI. References [1] Quintero-Marmol E, Luyben WL, Georgakis C. Application of an extended Luenberger observer to the control of multicomponent batch distillation. Ind Eng Chem Res 1991;30:1870–80. [2] Kim YH. Optimal design and operation of multi-product batch distillation column using dynamic models. Chem Eng Process 1999;38:61–72. [3] Monroy-Loperena R, Alvarez-Ramirez J. A note on the identification and control of batch distillation columns. Chem Eng Sci 2003;58:4729–37. [4] Frattini-Fileti AM, Rocha-Pereira JAF. The development and experimental testing of two adaptive control strategies for batch distillation. In: Darton R, editor. Distillation and absorption ‘97’. IChemE, Rugby. UK; 1997. p. 249. [5] Li P, Wozny G. Optimization of multiple-fraction batch distillation with detailed dynamic process model. In: Darton R, editor. Distillation and Absorption ‘97’. IChemE, Rugby. UK; 1997. p. 289. [6] Li P, Wozny G. Tracking the predefined optimal policies for multiplefraction batch distillation by using adaptive control. Comp Chem Eng 2001;25:97–107. [7] Barolo M, Berto F. Composition control in batch distillation: Binary and multicomponent mixtures. Ind Eng Chem Res 1998;37:4689–98. [8] Henson MA, Seborg DE. An internal model control strategy for nonlinear systems. AICHE J 1991;37:1065–81. [9] Dechechi EC, Luz Jr LFL, Assis AJ, Maciel MRW, Maciel Filho R. Interactive supervision of batch distillation with advanced control capabilities. Comp Chem Eng 1998;22:S867–70. [10] Alvarez-Ramirez J, Monroy-Loperena R, Cervantes I, Morales A. A novel proportional-integral-derivative control configuration with application to the control of batch distillation. Ind Eng Chem Res 2000;39:432–40. [11] Oisioyici RM, Cruz SL. Inferential control of high-purity multicomponent batch distillation columns using an extended Kalman filter. Ind Eng Chem Res 2001;40:2628–39. [12] Baratti R, Bertucco A, Da Rold A, Morbidelli M. Development of a composition estimator for binary distillation columns. Application to a pilot plant. Chem Eng Sci 1995;50:1541–50. [13] Han M, Park S. Profile position control of batch distillation based on a nonlinear wave model. Ind Eng Chem Res 2001;40:4111–20. [14] Han M, Park S. Control of high-purity distillation column using a nonlinear wave theory. AICHE J 1993;39:787–96. [15] Alvarez-Ramirez J. Adaptive control of feedback linearizable systems: A modeling error compensation approach. Internat J Robust Nonlinear Control 1999;9:361–77.

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