Geometric Observer for a Distillation Column - American Chemical

Nov 2, 2005 - The estimation of the product composition profiles for a distillation column is addressed by using an observer based on the differential...
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Ind. Eng. Chem. Res. 2005, 44, 9884-9893

PROCESS DESIGN AND CONTROL Geometric Observer for a Distillation Column: Development and Experimental Testing Stefania Tronci,† Fabrizio Bezzo,‡ Massimiliano Barolo,‡ and Roberto Baratti*,† Dipartimento di Ingegneria Chimica e Materiali, Universita` degli Studi di Cagliari, Piazza D’Armi, I-09123 Cagliari, Italy, and Dipartimento di Principi e Impianti di Ingegneria Chimica, Universita` di Padova, via Marzolo 9, I-35131 Padova, Italy

The estimation of the product composition profiles for a distillation column is addressed by using an observer based on the differential geometry theory. The estimation problem is challenging because of the markedly nonlinear behavior of the column and because of the ill-conditioning of the observability matrix at the operating condition investigated in the paper. The proposed solution is to consider the estimator structure as a degree of freedom in order to improve the estimator robustness without affecting the capability to accurately reconstruct the composition profile. Guidelines are provided to select the observer structure and the location of the measurement sensors. The observer is designed and tested experimentally on a 10-m high pilotplant distillation column separating an ethanol/water system. The results indicate that the observer is able to accurately reconstruct the composition dynamics over a wide set of operating conditions. 1. Introduction The on-line estimation of the states of a nonlinear plant is an important problem in process engineering, with major relevance to control and monitoring applications. Typical examples are distillation columns, where the crucial variables to be estimated are the product compositions. In fact, composition measurements are often not available on-line, either because on-line analyzers are not available themselves or because the delay associated with the composition measurements makes it impossible (or impractical) to use them within a control loop. The solution proposed by several authors is to reconstruct the product composition dynamics by secondary measurements (e.g., temperatures and flows). A pioneering work in this field is that of Joseph and Brosilow1 who used a linear combination of temperature and flow measurements to infer the product composition of a distillation column. Later, Yu and Lyuben2 applied the Wang-Henke algorithm to build a nonlinear estimator, based on temperature measurements obtained through the singular value decomposition analysis of the steady-state gain matrix between tray temperatures and manipulated variables. Mejdell and Skogestad3 developed a static partial-least-squares (PLS) estimator using steady-state temperature data only, and the nonlinearities between temperature and composition were partially counteracted by using logarithmic transforms of these variables. More recently, Kano et al.4 proposed a dynamic PLS model where temperature, flow, heat duty, and pressure data were regressed to * Corresponding author. Tel.: +39070675056. +390706755067. E-mail: [email protected]. † Universita ` degli Studi di Cagliari. ‡ Universita ` di Padova.

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provide product composition estimates; process dynamics was accounted for by including both current and past measurements into the input data set. A different estimator based on a nonlinear wave model is attributable to Shin et al.,5 where the profile position of a binary distillation column was estimated by applying a fullorder observer, according to the Luenberger6 form; the weighting function of the correction term is not constant, but it is adapted by taking into account the difference between the actual temperature and the representative temperature, which is related to the representative slope of the wave model. Lang and Gilles7 proposed a full-order observer where the weighting function of the correction term is selected by physical considerations on the process and is set proportional to the total molar flow. Deza and coworkers8,9 designed a high-gain observer for distillation columns, based on composition measurements. With respect to the previous ones, this estimator has the advantage that it is characterized by exponential convergence. Baratti et al.10,11 showed that the product composition profiles can be estimated for binary and ternary distillation columns using an extended Kalman filter and proved experimentally the validity of their technique; they found that a critical issue in the reliability and robustness of the estimator is the tuning of the model error and measurement error covariance matrices. In the present work, the state estimation problem for a pilot-plant continuous distillation column is solved by using an observer based on the differential geometry theory. The point of departure is the robustly convergent estimator proposed by Alvarez and Lopez,12 which is applicable to either observable or detectable systems and is characterized by a systematic construction-tuning

10.1021/ie048751n CCC: $30.25 © 2005 American Chemical Society Published on Web 11/02/2005

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procedure. These properties make the nonlinear geometric estimator of Alvarez and Lopez12 a valuable alternative when compared to other estimation approaches, such as the following: (i) the extended Kalman filter, which is formally intended for completely observable systems, lacks convergence and a systematic parameter-tuning criteria, and can be very heavy to be calculated on-line when the number of states is large; (ii) the geometric observer,13,14,15 which guarantees convergence with linear output error dynamics but is applicable only to a restricted class of plants; (iii) the high-gain observer,9,16 which guarantees convergence but has a complex tuning procedure; and (iv) the slidingmode approach,17 which guarantees robust stability but requires an elaborate design. The geometric estimator proposed by Alvarez and Lopez12 has been successfully applied to solve the local nonlinear estimation problem of a free radical homopolymer reactor,12,18 to infer the concentration in a catalytic reactor,19 and to monitor the pollutant concentration in an activated sludge process for wastewater treatment.20 Continuous distillation columns are not included in these applications, but they can provide an interesting case study, in particular when they operate within a region where the plant dynamics is markedly nonlinear; in fact, they can potentially take advantage from a robust nonlinear estimation method used in conjunction with a suitable mathematical model of the process. The negative aspect related to the development of a nonlinear geometric observer for distillation columns stems from the coordinate transformation on which the estimator algorithm is based. The geometric approach requires a quasi-linearization of the nonlinear system, which is obtained by applying a Lie derivative transformation to the measured outputs, with a derivative order depending on the dimension of the state vector and on the number of available measured outputs. Thus, for the simple case of the separation of a binary mixture, for which even a single temperature measurement may theoretically suffice to guarantee the system observability,2,21 a full-order observer requires the calculation of the Lie derivatives up to the order of the state variables minus one, consequently making the computational burden very large and weakening the estimator’s robustness. The solution proposed here to solve this problem is based on the partition of the state vector into innovated and noninnovated components, leading to the formulation of a reduced-order observer, which allows one to overcome the problem of the ill-conditioning of the system observability matrix. According to this approach, the estimator structure will be considered as a degree of freedom in the estimator design,22 and criteria for the selection of the estimator structure will be introduced. The results are verified by comparison with the dynamic behavior of a distillation pilot plant that separates an ethanol/water mixture. The selected binary system exhibits a very marked nonideal behavior, providing an interesting challenge for the development of the estimator. Furthermore, the pilot column is overdesigned with respect to the number of trays required for the separation duty; this results in the generation of an almost flat temperature profile within a large section of the column, which makes the system ill-conditioned from the observability point of view.

2. The Plant The stainless steel column has an outer diameter of 300 mm and is equipped with 30 sieve trays with a 200mm tray spacing (total height ∼10 m). The average reflux drum and sump holdups during normal operation are about 15 and 70 kg, respectively. A shell-and-tube water-cooled total condenser provides condensation (with subcooling) of the overhead vapor, while a steamheated vertical thermosiphon reboiler ensures the necessary vapor boilup. The column is operated at the atmospheric pressure and separates an ethanol/water mixture. In all the runs presented in this study, the (subcooled) feed was introduced above tray 8 (tray 1 is the bottom tray, stage 0 corresponds to the bottom sump, and stage 31 corresponds to the reflux drum). Electro-pneumatic valves are mounted on all of the process lines and on the steam line. The column is equipped with several Pt-100 probes to monitor the temperature profile on selected trays and on the bottom sump; namely, the following temperature measurements are available on-line: T0, T4, T8, T12, T18, T22, T26, and T30 (subscripts indicate the stage number). Coriolis meters provide on-line measurements for the reflux, bottom, distillate, and feed mass flows. A vortex meter is used to measure the steam volumetric rate; the steam pressure and temperature are also measured, so that a mass measurement is indirectly available on-line for this stream. Composition measurements are not available on-line, but product samples can be taken manually and analyzed off-line by means of a gas chromatograph. In all the experimental runs considered in this work, the column inventories were controlled by manipulating the distillate and bottom flows (proportional-only level controllers), and the column was operated in open-loop mode by maintaining the steam and reflux flows at the desired values by means of two flow controllers. As mentioned before, the column is actually oversized for the mixture considered in this project; therefore, most of the trays in the rectifying section work at a very high purity, i.e., close to the azeotropic composition. In practice, the composition profile in the column can be thought as the union of two straight lines with different slopes: a high-slope line in a short section of the stripping zone (where the relative volatility is ∼15 at very low ethanol concentrations) and a horizontal line (null-slope) representing the composition profile in the remaining section (where the relative volatility is ∼1 for ethanol mole fractions within 0.5-0.87). The transition zone characterized by a regular temperature gradient is limited to a very few stages. Another important aspect that makes this case study quite demanding from the estimation point of view is the ill-conditioned relationship between temperature and composition for the ethanol/water system. For example, the equilibrium temperature does not change significantly when the ethanol mole fraction in the liquid phase changes from 0.33 to 0.87; in particular, the temperature varies by ∼3 °C within this mole fraction interval, with a ∼0.3 °C variation in the interval from 0.548 to 0.87. This implies that a small error (or noise) in the measured temperature output can lead to a large error in the top composition estimation. On the other hand, at high ethanol dilution on the bottom sump, the temperature is very sensitive to composition changes and measurement noise has a much lower impact on the estimation accuracy.

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3. Geometric Observer The nonlinear geometric observer used to reconstruct the dynamics of the distillate and bottoms compositions is in the form proposed by Alvarez and Lopez12 and Alvarez.18 This observer is based on the property of robust exponential estimability and can be applied to partially observable systems. In the present case study, no unobservable states exist because we assume that two temperature measurements are available;21 yet, their location is to be determined among the ones available in the plant. For the sake of conciseness, only the definition of robust exponential observability and a short description of the geometric algorithm are reported in the following. Let us consider a nonlinear dynamical system in the general form:

x3 ) f(x,u(t),r), x(t0) ) x0, x∈Rn (1)

y ) h(x,r), y∈Rm

where x are the states, u are the inputs, r are the model parameters, y are the measured outputs, n is the number of states, and m is the number of measured outputs. According to Alvarez and Lopez,12 the motion x(t) is robustly exponentially observable if there are m integers (observability indexes) k1, k2, ..., km (k1 + k2 + ... + km ) k ) n, ki > 0) such that, in some neighborhood I about [x(t), u(t), r], the following conditions are verified: (i) the map Φ(x,u,r) ) [h1, Lfh1, ..., Lkf 1-1h1, ..., hm, Lfhm, ..., Lkf m-1hm]T, continuously differentiable with respect to x and r and with dim(Φ) ) (n × 1), is robustly invertible with respect to x (Rx-invertible); (ii) the map φ(x,u,r) ) [Lkf 1h1, ..., Lkf mhm]T, with dim(φ) ) (m × 1), is Lipschitz -continuous (L-continuous), meaning that for each neighborhood I and for (x′,u′,r′)∈ I, some constants C1, C2, and C3 exist such that |φ(x′,u′,r′) - φ(x,u,r)| e C1|x′ - x| + C2|u′ - u| + C3|r′ - r|. Condition (i) of robust invertibility means that there is an inverse map Φ-1, such that Φ-1[Φ(x,u,r),u,r] ) x(t) for t g 0, and that Φ and Φ-1 are both L-continuous. In such a case, the following observer can be constructed:

xˆ3 ) f(xˆ ,u(t),r) + Φx-1K0[y - yˆ ] (2)

y ) h(xˆ ,r)

where Φx is the Jacobian of the map Φ with respect to the states of the system, with dim(Φx) ) (n × n), and K0 is a block diagonal matrix whose entries are the constant gains of the observer:

[ ] [] [] [ ]

B1 0 K0 ) l 0

0 B2 l 0

l 0 l 0 , l l l Bm

k11 k21 , B2 ) B1 ) l kk11

k12 k22 , ..., Bm ) l kk22

k1m k2m l kkm,m

(3)

The value of the gains is set such that the reference

linear, noninteractive, and pole-assignable error dynamics is stable.12 4. Estimation Problem The first step for the development of the estimator (eq 2) requires developing the dynamical model of the distillation system, i.e., to determine the form of the function f(x,u,r). Since the estimator should be suited for on-line applications, the use of a simple model is preferred. The simplified model used in the present paper is the one proposed by Baratti et al.10 This model considers only the mass balance equations for one of the components of the binary system, while the tray energy balances are neglected; the reboiler is modeled as an additional tray. The dynamics of the molar tray holdups is neglected, too, and tray holdup values are updated after an input variation by considering steady-state correlations. Therefore, the internal vapor and liquid molar flows are assumed to be constant in each section of the column between two successive variations of the manipulated inputs. Perfect top and bottom level control is also assumed. Vapor-liquid equilibria (VLE) on each tray are calculated by using the NRTL (nonrandom twoliquid) model for the estimation of activity coefficients and assuming an ideal vapor phase. The parameter values are reported in Gmehling and Onken.23 The composition of the vapor phase leaving the generic tray is calculated from the corresponding equilibrium value by using the Murphree efficiency equation. The efficiency was kept constant ()0.54) for all trays and was used as a tuning parameter to calibrate the steady-state model. Finally, a linear pressure profile was assumed along the column. The resulting mathematical model is constituted by 32 ordinary differential equations describing the dynamics of the liquid composition on each stage. With the equilibrium temperatures being implicitly dependent on the states x, the model can be written in the form:

x3 ) f(x,u(t),r), x∈R32 H(x,y) ) 0, y∈R2

(4)

In particular, indicating with Ts the temperature measurement in the stripping section and with Tr the temperature measurement in the rectifying section, y is defined as

y ) [Ts, Tr]T

(5)

where the locations of Ts and Tr need to be specified in order to design the observer. Note that more than two measurements could be used in principle; however, we prefer to mimic a real situation, where more than two measurements are not always available. Building a full-order geometric observer would imply that the sum of the observability indexes must be equal to 32, which is the order of the system. With y being an order-two vector, and associating the observability index k1 to Ts and k2 to Tr, the following condition for a fullorder observer must, therefore, be satisfied:

k1 + k2 ) 32

(6)

In particular, considering that the feed entering tray 8 divides the column into a rectifying section (23 stages) and a stripping section (9 stages), a possible estimator

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structure may be {k1, k2} ) {9, 23}; this also sets the order of the Lie derivatives in the Φ map, from which the observer gain is derived, and the following map should, therefore, be constructed,

Φ ) [h1, Lfh1, L2f h1, ..., Lkf 1-1h1, h2, Lfh2, ..., Lkf 2-1h2]T (7) where the highest order of the directional derivative of temperature Tr (i.e., Lkf 2-1h2), for such a structure selection, is equal to 22, and the dimension of Φ is equal to (k1 + k2). This high-order structure can be detrimental for the robustness of the estimator, making the application of a full-order geometric observer impractical. Actually, because the use of time derivatives with an order >2 should be avoided, it can be easily concluded that the development of an observer based on the differential geometry theory for a distillation column is a very challenging task, given the fact that the number of the measured variables is usually much smaller than the number of states to be estimated. An alternative approach can be taken by considering a peculiar property of the geometric observer proposed by Alvarez and Lopez,12 which is the possibility to correct only a subset of the observable states, that is, to set an estimator degree κ such that

κ ) (κ1 + κ2) < k ) n

(8)

where κ1 and κ2 represent the estimation indexes.22 This can be accomplished by partitioning the estimated states into an innovated (“closed-loop”) component (xI) and a noninnovated (“open-loop”) one (xII). This reduces the order of the Lie derivatives to be calculated (because these derivatives only refer to subset xI), which in turn increases the estimator robustness and reduces the algebraic computational burden. In this case, according to the geometric approach,12 the state vector x(t) is robustly estimable if (i) rank Φx(xI,xII,u,κ1,κ2) ) nI and (ii) the noninnovated state motion xII(t) is stable, where now Φx is the Jacobian with respect to the innovated states xI of the map Φ (eq 7) and nI ) (κ1+κ2) < n. By proceeding in this way, the following inferential system is obtained:

xˆ3 I ) fI(xˆ I,xˆ II,u(t),r) + Φx-1K0[y - yˆ ] xˆ3 II ) fII(xˆ I,xˆ II,u(t),r)

(9)

H(xˆ I,yˆ ) ) 0 where xˆ I ∈ RnI, nI ) κ1 + κ2 , n, the coordinate transform map Φ has dimension (nI × 1), and Φx has dimension nI × nI. This approach sets forth an additional problem in the design of the observer, which is to properly select xI. To this purpose, one would like to keep the dimension of the innovated state vector as small as possible, to avoid the use of high-order Lie derivatives in the Φ map (because they are detrimental to the estimator robustness), without decreasing the capability of the estimator to reconstruct the actual system dynamics. A further issue related to the design of the observer should be mentioned, i.e., the location of the temperature measurements. This issue may be regarded as an optimal sensor location problem for a distributed-sensor system, and several studies on this topic have been

published in the past years. A common approach is to define an optimality criterion based on an observability index, as happens in the work of Waldraff et al.,24 where the observability matrix, the observability Gramian, and the Popov-Belevitch-Hautus rank test were considered for locating sensors at optimal positions in a tubular reactor. A different technique is that of Vande Wouwer et al.,25 who proposed a simple test of independence between the sensor responses, using the Gram determinant as a measure to single out the optimal sensor locations. Unfortunately, these “formal” techniques are mostly useful for the design of full-order observers, whereas in the present application, we are looking for a reduced-order observer that guarantees a good estimation accuracy with as small as possible a number of innovated states. As will be further clarified in the next section, we are looking for temperature sensor locations (i) where the relationship between temperature and product composition is “strong”, (ii) which ensure that the “open-loop” dynamics is well-conditioned, and (iii) for which the propagation error among the noninnovated states is minimized. In this framework, the following general guidelines are proposed for the location of measurement sensors: (i) the condition number of the Φx matrix should be minimized; (ii) the dynamical open-loop system xˆ II should be well-conditioned from the integration point of view; and (iii) the propagation of the estimation error in the open-loop dynamics should be minimized. 5. Selection of the Observer Structure The selection of the observer structure is carried out by selecting two design parameters: the location of the two temperature measurement sensors (which implicitly assigns the values of two of the innovated states because of the VLE relationships) and the value of the estimation indexes κ1 and κ2 (which sets the dimension of the xI vector). The minimum dimension of xI is nI ) m ) 2. However, it is worth noting that the design of an estimator with an order >2 can be easily accomplished by including into the innovated state vector the states appearing in the material balance equations of those states that already belong to xI. This means that if r and s are the selected temperature measurement locations in the rectifying and stripping sections, then, respectively, xr and xs (where x is the ethanol mole fraction in the liquid phase) will be the first two innovated states. Now, consider, for example, the dynamic material balance equation on tray s: besides xs∈ xI, this equation also contains states xs+1 and xs-1 (the latter one through the vapor-phase composition ys-1); therefore, either xs+1 or xs-1 can be also included into xI, with Φx still being nonsingular. If xs-1 is selected, the observer order can be further increased by adding xs+1 or xs-2 to the innovated state vector; by proceeding in this way, the existence of Φx-1 is always guaranteed. A similar argument is used for the rectifying section. By using this approach, the information acquired on a tray through a measured output is propagated stage by stage along the column by applying successive Lie derivatives to that output, which ensures proper state innovation. The effects of the observer structure on the robustness and accuracy of the estimator can be evidenced by observing the condition number c(Φx) of the Φx matrix, the condition number of the Jacobian matrix of the openloop system dynamics, and the propagation of the

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Figure 1. (a) Evolution of the experimental temperatures: Ti, i ) 0, 4, 8, 12, 18, 22, 26, and 30 for Run 1; (b) Evolution of the ethanol composition in the residue (solid diamond) and in the distillate (solid circle) for Run 1.

Figure 2. Condition number c(Φx) for the following structures: S2a (dotted line), S4a (dashed line), and S6a (solid line).

Table 1. Operating Condition of the Experimental Runs run

1

2

3

feed flowrate (kg/h) feed ethanol mole fraction feed temperature (K) steam flowrate (kg/h) reflux flowrate (kg/h)

142.6 0.335 299.0 110.0 140.0

146. 0.335 304.9 106.0 138.0

131.0 0.335 304.9 100.0 135.4

estimation error along the noninnovated dynamics. The first index depends on the order of the Lie derivatives (increasing the Lie derivative order, the condition number of the Φx matrix is expected to increase as well) and on the location of the temperature sensors (the entries of Φx depend on the derivative of the measured outputs with respect to the states). The second index provides information on the integration of the differential equations describing the open-loop dynamics; an ill-conditioned system is more difficult to integrate and can increase the estimation error and should, therefore, be avoided. The third index is related to the distribution of the noninnovated state variables along the column and to the requirement to have a low-order estimator. Using a temperature sensor located far from the column ends may increase the estimation error, because the measurement will be propagated through those noninnovated states that are located between the innovated one and the outlet composition; the estimation error can be reduced by either augmenting the number of the innovated states or by locating the sensor closer to the column ends. A further issue related to the selection of the observer structure should be mentioned. Application of the Lie algebra to a complex set of relationships such as those describing the VLE of the ethanol/water mixture implies very tedious computations, which may well lead to errors if a computer algebra system is not used. However, even when such a tool is used, the drawback is the creation of burdensome equations that make the computation time increase. Hence, minimization of the derivative order is important with respect to this issue as well. In the following, the three indexes described above will be evaluated, with the purpose of selecting the observer structure that leads to a robust and efficient inferential system. The analysis will be carried out with respect to the results obtained in experimental Run 1, during which the column was moved from the initial steady state by applying a step decrease to the steam flow rate (from 110 to 90 kg/h at ∼16 min). The main operating conditions for this run are indicated in Table 1, while parts a and b of Figure 1 show, respectively, the transients of the tray temperature profiles and of

Figure 3. Condition number c(Φx) for the following structures: (a) S2a ) [(1,1),(x0,x18)] (solid line), S2b ) [(1,1),(x0,x22)] (dashed line), and S2c ) [(1,1),(x0,x26)] (dotted line); (b) S2d ) [(1,1),(x4,x18)] (solid line), S2e ) [(1,1),(x4,x22)] (dashed line), and S2f ) [(1,1),(x4,x26)] (dotted line).

the product composition profiles (note that the distillate composition is close to the azeotrope). The following notation will be used to define the different estimator structures,

SnIi ) [(κ1,κ2),(xI(1),xI(2),...)]

(10)

where SnIi indicates the structure (S) of dimension nI ) κ1+κ2 and i is a letter (i ) a, b, c, ...) that will be used to differentiate reduced-order observers having the same order nI, but with different components for vector xI ) [xI(1), xI(2), ...]T. In the following, for the sake of conciseness, only cases with κ1 ) κ2 will be reported, while it was verified that assuming κ1 * κ2 does not cause major changes in the results. 5.1. Analysis of the Condition Number of the Φx Matrix. The first measure we consider is the condition number of the Φx matrix. Calculations were carried out by integrating the model at the operating conditions of Run 1. In Figure 2, the condition number c(Φx) is reported for different orders of the observer, when the measured output vector is y ) [T0, T18]T. For the sake of brevity, among the different structures for temperature sensor locations that were explored, only the results concerning the following ones are shown:

S2a ) [(1,1),(x0,x18)]

(11a)

S4a ) [(2,2),(x0,x1,x17,x18)]

(11b)

S6a ) [(3,3),(x0,x1,x2,x17,x18,x19)]

(11c)

The time profiles of c(Φx) show that an increase of the number of the innovated states (from S2a to S6a)

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causes a marked increase of the condition number because of the presence of higher-order derivatives in the Φx map. Although not shown in Figure 2, the same increasing trend with the dimension of the innovated state vector was obtained by considering different sensor locations (i.e., different components in the xI vector). The value of the condition number of Φx for a given dimension of xI is affected by the location of the innovated states along the column. This behavior is reported in Figure 3 for structures with nI ) 2 only, but a similar trend was shown for higher-order observers as well. Note that when nI ) 2, Φx is the following 2 × 2 matrix:

matrix (fII)xII of the noninnovated dynamics;22 this condition number will be denoted by CN in the following. Figure 4 shows the plot of CN for test Run 1 and different partitioning of the state vector. The open-loop system obtained for different choices of the innovated states is indicated with SIInIi, where nI represents the dimension of the vector xI that generates the noninnovated structure SII and i represents a given selection of the xI components (i ) a, b, c, ...). Parts a and b of Figures 4 refer to the case of two innovated states (with different locations of Ts and Tr); namely, they represent the condition number of the following noninnovated state dynamics:

∂Ts 0 ∂xs Φx ) ∂Tr 0 ∂xr

SII2a ) [x1,x2,...,x17,x19,...,x31]

(13a)

SII2b ) [x1,x2,...,x21,x23,...,x31]

(13b)

SII2c ) [x1,x2,...,x25,x27,...,x31]

(13c)

SII2d ) [x0,...,x3,x5,...,x17,x19,...,x31]

(13d)

SII2e ) [x0,...,x3,x5,...,x21,x23,...,x31]

(13e)

SII2f ) [x0,...,x3,x5,...,x25,x27,...,x31]

(13f)

[ ]

(12)

Two considerations can be drawn from the analysis of Figure 3. The first one is that, whatever the location of Ts, moving Tr closer to the top worsens the conditioning of Φx. This can be ascribed to the fact that, when the liquid composition becomes closer to the azeotropic one, very small changes in the temperature imply large changes in the liquid composition. The second remark coming from Figure 3 is that, whatever the location of Tr, when Ts is located on tray 4 (Figure 3b), the condition number decreases more quickly than it happens when Ts is located at the bottom of the column (Figure 3a). The reason is related to the slower dynamics of the bottom sump and to the influence of the state values on the equilibrium temperature. On tray 4, the ethanol concentration after the disturbance suddenly steps to a high value (see the temperature profiles in Figure 1), and therefore, the value of the derivative of Ts ) T4 with respect to the state is comparable to the one calculated with reference to Tr. However, when Ts is located in the bottom sump, the composition values therein imply a much higher ∂T/∂x derivative (see Section 2), and the condition number of Φx is, therefore, higher. Furthermore, the dynamics of the bottom is the slowest one in the column, and this fact explains the small decreasing rate of the condition number when Ts ) T0. To summarize, the analysis of c(Φx) suggests that the observer order should be 2, with Tr located on tray 18 and Ts located on tray 4 (structure S2d in Figure 3). 5.2. Analysis of the Condition Number of the Jacobian Matrix of Noninnovated States. The drawback of the S2d structure is that the two innovated states [x4, x18]T are located quite far from the top and the bottom of the column; hence, the error on the noninnovated states [x0, ..., x3, x5, ..., x17, ..., x19, ..., x31]T may affect the estimation performance of the observer. In fact, [x4, x18]T must be propagated through the noninnovated states to get the desired estimates for x0 and x31. Furthermore, it should be noted that innovating only a part of the state vector through the geometric algorithm changes the dynamics of the system to be integrated. This fact can generate an open-loop system that is numerically ill-conditioned, making its integration more difficult and increasing the risk of numerical errors, which will be summed to the modeling ones with a related loss of estimation performance. This issue suggests that the observer structure should be selected by also controlling the condition number of the Jacobian

Parts c and d of Figure 4 report the result when 6 states are corrected, and the following 26 states are not innovated:

SII6a ) [x3,x4...,x16,x20,...,x31]

(14a)

SII6b ) [x3,x4,...,x20,x24,...,x31]

(14b)

SII6c ) [x3,x4,...,x24,x28,...,x31]

(14c)

SII6d ) [x0,...,x2,x6,...,x16,x20,...,x31]

(14d)

SII6e ) [x0,...,x2,x6,...,x20,x24,...,x31]

(14e)

SII6f ) [x0,...,x2,x6,...,x24,x28,...,x31]

(14f)

Although the difference between the CN values is not excessive, the results show that the most ill-conditioned open-loop system is the one with 30 noninnovated states (which corresponds to xI∈ R2). Generally speaking, the lower the number of states that are operated in openloop mode, the easier the integration of those states. In all the structures considered, the condition number starts increasing as soon as the top composition reaches a high purity and keeps decreasing when the bottom composition starts changing (see Figure 1). However, changes are less marked when Tr is located at tray 18 (SII2a, SII2d, SII6a, and SII6d). In this case, CN does not change significantly during the whole run: it is significantly lower than the CNs of most structures until t ≈ 30 min and slightly exceeds the value of CN of all the other structures afterward, that is, when the ethanol concentration becomes high on every stage of the column. Since the structures with Tr ) T18 have almost the same final value of the condition number, regardless of the location of Ts, this trend of CN seems to depend on the value of the ∂T/∂x term in the rectifying section rather than on the location of Ts. Transferring states of the upper stages from the noninnovated into the innovated state vector makes the dynamics of the former

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Figure 5. (a) Ethanol composition in the distillate: experimental (solid circle), model (dashed line), S2d ) [(1,1),(x4,x18)] (solid line), S4d ) [(2,2),(x3,x4,x17,x18)] (dashed-dotted line), and S6d ) [(3,3),(x3,x4,x5,x17,x18,x19)] (dotted line); (b) Ethanol composition in the residue: experimental (solid circle), model (dashed line), S2d (solid line), S4d (dashed-dotted line), and S6d (dotted line).

Figure 4. Condition number CN for the structures: (a) (solid line),SII2b (dashed line), and SII2c (dotted line); (b) (solid line), SII2e (dashed line), and SII2f (dotted line); (c) (solid line), SII6b (dashed line), and SII6c (dotted line); (d) (solid line), SII6e (dashed line), and SII6f (dotted line).

SII2a SII2d SII6a SII6d

less ill-conditioned. This fact is confirmed by the diminishing trend of the condition number when the observer order is increased (i.e., the noninnovated vector dimension decreases). Furthermore, the peak value of CN is slightly delayed by moving Ts from tray 4 to the bottom, and this depends on the different dynamics between the two stages, as previously evidenced in the analysis of c(Φx). Therefore, as far as the noninnovated system dynamics is concerned, the analysis of CN suggests locating Tr on tray 18, in such a way as to avoid peaks in the value of CN, while no unequivocal indication is obtained for the location of Ts. With regards to the order of the observer, CN suggests the use of a sixth-order observer. 5.3. Analysis of the Estimation Error. The third issue considered for the selection of the estimator structure concerns the accuracy of the observer, i.e., the minimization of the error between the estimated compositions and the real ones. In fact, it is quite obvious that, when a reduced-order observer is used, the error of the noninnovated states can make the system diverge. So far, the criteria related to the analysis of c(Φx) and of CN have both indicated tray 18 as the candidate for the temperature measurement location in the rectifying section. As for the location of Ts and for the order of the observer, the indications are somewhat conflicting. Although engineering judgment would actually suggest associating Ts to the slowest mode of the column (i.e., to the bottom sump), we will nevertheless start our analysis with Ts ) T4, as suggested by the analysis of c(Φx). Figure 5 shows the estimation results with secondorder, fourth-order, and sixth-order observers with y ) [T4 , T18]T for Run 1. The distillate composition profile is reconstructed quite well by all the observers (Figure 5a). This is mainly due to the fact that the change in distillate composition is not very marked, and the “plain” model itself does a good job in tracking this change; therefore, in this case, the correction term of eq 9 is almost useless. On the other hand, the estimation

Figure 6. (a) Ethanol composition in the distillate: experimental (solid circle), S2d ) [(1,1),(x0,x18)] (solid line), S2f ) [(1,1),(x0, x26)](dashed line); (b) Ethanol composition in the residue: experimental (solid circle), S2d (solid line), S2f (dashed line).

performance on the bottom composition profile is much worse (Figure 5b), and increasing the order of the observer from 2 to 6 does not result in any significant improvement. The observers shown in Figure 5 were tuned in such a way as to eliminate the error between the measured temperature on tray 4 and the estimated one; therefore, Figure 5b suggests that the correction of x4 (innovated state) amplifies the error on x0 (noninnovated state). When the observer is tuned less aggressively, the response of the estimated system gets closer to that of the model; hence, no improvement in composition estimation results. Conversely, the estimation results are significantly improved when Ts is moved to the bottom of the column, as shown in Figure 6, where the distillate and bottoms compositions inferred using a second-order estimator with xI ) [x0,x18]T are compared to the experimental data of Run 1. Note that the resulting structure is able to reproduce the experimental data accurately, even if the dimension of the innovated state vector is the smallest one. Therefore, a second-order observer with temperature measurement sensors placed on the bottom sump and on tray 18 appears to be able to robustly converge to the actual system for test Run 1. It should however be remarked at this point that the current selection for the location of Tr may be biased by the conditions at which Run 1 was carried out. In fact, as was noted before, during this run the distillate composition is quite close to the azeotropic one and does not change substantially; therefore, the contribution of the correction term in eq 9 is negligible for state xˆ r, which means that a different selection for Tr can possibly provide equally good estimation results on the x31 profile. This is confirmed by the dashed curve in Figure 6b, which was obtained by moving Tr

Ind. Eng. Chem. Res., Vol. 44, No. 26, 2005 9891 Table 2. Step Change Program of Run 2 time (min)

reflux flowrate (kg/h)

steam flowrate (kg/h)

0.0 8.0 20.5 26.5

138.0 191.0 191.0 137.0

106.0 106.0 115.0 115.0

from tray 18 to tray 26. Generally speaking, a correction on tray 18 has a lower effect on the estimated distillate composition than a correction on tray 22 or tray 26. 5.4. Final Design for the Observer. Summing up all the considerations drawn so far, the following estimator structure was finally designed:

nI ) 2, κ1 ) 1, κ2 ) 1

Figure 7. (a) Ethanol composition in the distillate: experimental (solid circle), model (dotted line), S2b ) [(1,1),(x0,x22)] (solid line); (b) Ethanol composition in the residue: experimental (solid circle), model (dotted line), estimator S2b ) [(1,1),(x0,x22)] (solid line).

xI ) [x0,x22]T xII ) [x1,...,x21,x23,...,x31]T y ) [T0,T22]T

(15)

The choice of x22 as an innovated state is the best compromise between the information obtained from the condition numbers c(Φx) and CN (which both indicate state x18 as the preferred one) and the third guideline (which suggests to innovate state x26 to minimize the propagation of the error along the noninnovated dynamics). It is worth noting that the selection of x22, instead of x18, does not cause an excessive increase of the condition numbers (see Figures 3a and 4a). The selection of a second-order observer implies the following gain matrix K0,

K0 )

[

]

k11 0 k11 ) ω11 k 0 22 k22 ) ω22

(16)

where ω11 and ω22 are the characteristic frequencies of the system, which can be selected by considering the dynamics of the column (stage 0 and 22) and imposing a faster response for the estimator. This task was accomplished by carrying out proper step tests to evaluate the characteristic time of the bottom and the upper states of the column, respectively, and then using these values as the first guess in a trial-and-error procedure that led to the best compromise between efficiency and robustness of the estimator. The selected values are ω11 ) 0.03 min-1 and ω22 ) 0.1 min-1. 6. Performance of the Estimator The structure of the estimator designed in the previous section was evaluated by considering two additional experimental tests. In the first experiment (Run 2), the estimator was applied for the reconstruction of the product composition dynamics when both the reflux and the steam flows were subjected to step changes as summarized in Table 2 (the nominal steady state operating conditions are reported in Table 1). Results are shown in parts a and b of Figure 7, where the ethanol mass fractions in the distillate and bottoms, respectively, are compared to the experimental compositions; the profiles calculated by the model are also reported, to show how the nonlinear geometric observer improves the model prediction. The results show that the estimator is able to reconstruct the bottom composition quite accurately; as far as the distillate composition

Figure 8. Ethanol composition in the distillate: experimental (solid circle), model (dotted line), S2b ) [(1,1),(x0,x22)] (solid line).

is concerned, it seems that the model performs somewhat better than the estimator (however, note that the change in the distillate composition is fairly small during this run, and the estimation error is within the experimental one). The mismatch between the experimental and the reconstructed distillate composition profiles is mainly related to the VLE correlation, which is probably not accurate when the system operates close to the azeotropic point. As was discussed in Section 2, at these operating conditions small errors in the temperature estimation imply high composition changes. The geometric algorithm cannot eliminate this small mismatch between the measured and estimated temperatures, because it does not take into account the uncertainness of the VLE model parameters; hence, no correction action on the distillate composition is possible. The conditions of the last experimental run (Run 3) were selected in order to test the estimator performance when the input is changed in opposite directions, and the distillate composition is driven very far away from the region where the observer was designed. In this experiment, the steam flowrate was first increased from 100 to 120 kg/h at t ≈ 9 min. After ∼20 min, the steam flow was reverted back to the original value, and the purity of distillate changed accordingly. In Figure 8 the estimated ethanol mass fraction in the distillate is compared to the experimental data and to the model prediction. The dynamic response of the system is not reconstructed accurately by the model, which shows a dynamics significantly slower than that of the real system. Conversely, the correction algorithm markedly improves the model prediction. Therefore, the estimator appears to be sufficiently robust to counteract the

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Figure 9. Temperature profiles along the column: experimental (solid circle), S2b (solid line), S4b (dashed line), S6b (dotted line) (a) at the initial steady state; (b) after 5 min; (c) after 15 min; and (d) after 28 min.

system nonlinearities. An analysis of the bottom composition profile is not reported, because at the conditions of Run 3, the light component is not present in the bottom. 7. Further Remarks The selection of a reduced-order estimator with a minimum number of innovated states leads to good results in the reconstruction of the product composition dynamics, ensures the robustness of the system, and does not involve algebraic complexity due to high-order Lie derivatives. On the other hand, the more states are corrected, the more accurately the estimated temperature profile is expected to reproduce the experimental one. This issue can be noticed in Figure 9, where the internal temperature profiles obtained for different observer structures (corresponding to two, four, and six innovated states) are compared to the experimental profile of Run 1 at different time instants, with temperature sensors located on tray 0 and tray 22. The results show that the higher-order observers are indeed able to correct the internal temperatures faster than the second-order observer. 8. Conclusions This paper considered the development of a composition estimator based on the differential geometry theory for a pilot-plant binary distillation column. Two issues make the estimation problem challenging: the markedly nonlinear static and dynamic behavior of the plant and the order of the full-state estimator. The former issue is mainly related to the plant overdesign and highly nonlinear VLE behavior of the ethanol/water system, while the latter one is due to the intrinsic structure of the observer. In fact, even if the system can be made observable by the availability of two temperature measurements only, using a full-order estimator is impractical because of the large difference between the dimension of the state vector and that of the measured

outputs. Such a large difference implies a high order of the Lie derivatives used in the quasi-linearization of the system. The problem was solved by considering the estimator structure as a degree of freedom. A reduced-order observer was developed, where only a low-order subset of the full state vector is innovated. The design of the observer structure was accomplished by the analysis of three indexes: the condition number of the Jacobian of the transformation map (on which the correction term of the geometric algorithm depends), the condition number of the open-loop dynamics, and the propagation of the error along the noninnovated states. These indexes were calculated for a reference run, and they provided a deep understanding of the reconstruction of the state dynamics. The drawback of the proposed approach was the somehow contradictory information on the observer structure design provided by the three indexes; thus, the final structure was a compromise among all the recommended solutions. The estimator was evaluated by considering additional experimental tests. The results obtained show that the performance of the observer is good even when the column is operated far away from the conditions upon which the observer was designed. With respect to a more conventional extended Kalman filter design, the geometric design does not require the integration of the Riccati equations and, therefore, is much easier to be used on-line; furthermore, the number of tuning parameters is much lower. On the other hand, the main drawback is a complex design procedure, mainly due to the calculation of the Lie derivatives, the order of which depends on the dimension of the observable state vector. Acknowledgment This work was carried out in the framework of the MIUR PRIN 2005 Project “Methodologies for monitoring and integrated management in the process industry”. List of Symbols c(A) ) condition number of the matrix A CN ) condition number of the Jacobian matrix of the noninnovated states h(x) ) explicit function relating the outputs y to the states x H(x y) ) implicit function relating the outputs y to the states x ki ) observability index associated to the measured outputs κi ) estimation index associated to the measured outputs kij ) entries of the matrix K0 K0 ) constant matrix of the observer gains Lnf hi ) nth Lie derivative of the scalar field hi along the vector field f nc ) number of components nI ) dimension of the innovated-state vector S ) estimator structure Tj ) temperature of tray jth x ) state vector y ) vector of the measured outputs Φ ) coordinate transform map ω ) characteristic frequency Subscripts I ) innovated states II ) noninnovated states s ) stripping section r ) rectifying section

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Received for review December 23, 2004 Revised manuscript received July 29, 2005 Accepted September 21, 2005 IE048751N