Ind. Eng. Chem. Res. 2002, 41, 3405-3412
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PROCESS DESIGN AND CONTROL Wavelet-Based Regularization of Dynamic Data Reconciliation Mingfang Kong, Bingzhen Chen,* and Xiaorong He Department of Chemical Engineering, Tsinghua University, Beijing, 100084, P. R. China
Dynamic data reconciliation can supply more accurate data for dynamic optimization, dynamic fault diagnosis, and control by means of incorporating process information in some mathematical model. It will be an ill-posed inverse problem if the sensitive input variables are unmeasured; here, the sensitive input variable is defined as the variable that, if it is unmeasured, can only be estimated through the differentiation of other measured variables. In such a case, existing methods cannot obtain correct and usable data effectively. To address the problem, based on the principle of regularization, the wavelets are adopted to construct regular operators. And, a new approach is proposed to determine the optimal scale level corresponding to the optimal approximate operator in which the prior statistical information of the signal is utilized. The algorithm can deal with the estimation of unknown sensitive input variable effectively. The results show that more accurate estimation of the sensitive input variable can be obtained by using the proposed method as compared with the one obtained by using existing collocation methods based on polynomials. 1. Introduction Process measurements from a modern chemical plant typically contain errors unavoidably so that they are not consistent with corresponding conversation laws. Dynamic data reconciliation can estimate unmeasured process quantities, reduce errors, and provide a set of reconciled measurements for dynamic optimization, dynamic fault diagnosis, and control by means of incorporating process information in a dynamic model. Errors in measurements can be classified into two types, namely, random errors and gross errors. In the following text, it is supposed that gross errors have been eliminated by using existing identification methods.1-4 The general model of dynamic data reconciliation is as follows: T -1 min(Y - Ym) Q (Y - Ym) Y
s.t.
dy(t) ) h(y(t), u(t)) dt f(y(t), u(t)) ) 0 c(y(t), u(t)) e 0
(1)
where y represents a measured variable; u represents an unmeasured one; Y) [y(t1), y(t2), ..., y(tN)] is a reconciled value; ti, i ) 1, ..., N are the sampling times; and Ym is a measurement. Among existing methods of dynamic data reconciliation, nonlinear programming is more effective5 and is used widely.6-9 Depending on the type of basis function * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: 86-10-62784572; Fax: 86-1062770304.
adopted, these methods can be classified into following two categories: methods based on polynomials6,7 and methods based on wavelets.8,9 Dynamic data reconciliation usually needs to estimate the unknown sensitive input variables and is typically formulated as dynamic optimization problems restricted by the process model (here, the sensitive input variable is defined as the variable that, if it is unmeasured, can only be estimated through the differentiation of other measured variables). Because the process model includes differential equations, the estimation problem of unknown sensitive input variables is an inverse one and is ill-posed.8,10,11 In such a case, correct and usable estimation results cannot be obtained by using polynomial-based nonlinear programming methods.7 Binder et al. adopted a wavelet-based Petrov-Galerkin method, but the method is not very effective, because it requires that the whole reconciliation problem be solved recursively to determine the optimal scale level of wavelets in order to find a corner point of the L-curve.8 If the problem is complex, the computation used to determine the optimal scale level increases exponentially. At present, regularization is a commonly used method for solving inverse problems. In this paper, according to the principle of regularization and the multiscale property of wavelets, regular operators are constructed based on the wavelets expansion of a differential operator and a new approach is proposed to determine the optimal scale level. Then, differential equations in the model of dynamic data reconciliation are converted into algebraic ones. The paper is organized as follows: first, section 2 introduces the concept of the inverse problem and explains why regularization is needed; then in section 3, the steps of wavelet-based regularization are listed and explained in detail. Finally, in the section 4, examples and comparisons are given.
10.1021/ie010529k CCC: $22.00 © 2002 American Chemical Society Published on Web 06/13/2002
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2. Inverse Problem For dynamic reconciliation, if the sensitive input variables are unmeasured, then the problem is an inverse one and is ill-posed. The solution will not depend on data continually, which means that minor errors in other variables will result in the solution of a sensitive input variable fluctuating greatly. In the following two subsections, first, the basic concept of the inverse problem will be introduced, then, the ill-posedness of the dynamic reconciliation is analyzed. 2.1. Inverse Problem. Levrentiev defined the inverse problem as follows: the inverse problem of differential equation is to determine its coefficient or item of right-side.11 Most often, the inverse problem does not fulfill Hadamard’s postulates of well-posedness, which makes it difficult to carry out both theoretical analysis and numerical calculation. Any well-posed problem in the sense of Hadamard must possess the following properties: (1) a solution exists for all admissible data; (2) the solution is unique for all admissible data; and (3) the solution depends continuously on the data. If one of the three conditions is not fulfilled, the problem is called an ill-posed one. The third condition, “the solution depends continuously on the data”, means that when all known variables in the problem, such as parameters, initial conditions, or boundary conditions, have small changes, the change of the solution is also very small. This condition is very important to numerical calculation; otherwise, the unavoidable small changes in data such as errors in measurements and rounding errors will result in very great changes of the solution, and the problem is very hard to solve.11 2.2. Analysis of Dynamic Data Reconciliation. For model 1, suppose that the result of variable classification12-14 is T -1 min(Y - Ym) Q (Y - Ym) Y
s.t.
dy1(t) ) g(y(t), u1(t)) dt dy2(t) ) h(y(t), u2(t)) dt f(y(t), u2(t)) ) 0 c(y(t), u2(t)) e 0
(2)
In this situation, the variable u1 is an unknown sensitive input variable and can only be determined by y1. So, this is an inverse problem and is ill-posed in the sense of Hadamard’s definition. For example, the actual signal is y, and suppose its corresponding measurement is yδ ) y + δ sin(nt/δ). If x is estimated by using x ) dy/dt - ay - u, then there will be a great error in the estimation of x because yδ needs to be differentiated. Although |yδ - y|8 ) δ exists, the estimation error of x does not depend continuously on the data because |xδ - x|∞ ) |n cos(nt/δ) + aδ sin(nt/δ)|∞. So, the problem is an ill-posed inverse one, and it is difficult to obtain an accurate result.8 The existence of random errors leads to the result that differentiation does not depend continuously on the data. Because errors are unavoidable, inaccuracy of results is definite. The reason is that the differential operator is unbounded. Then, regularization is needed which uses a
bounded operator to approximate the unbounded one and converts the ill-posed problem to a well-posed one. In fact, regularization is to make a tradeoff between inaccuracy that results from operator approximation and measurement error.10 3. Wavelet-Based Regularization On the basis of the principle of regularization, the construction of regular operators is required to approximate the differential operator. Utilizing the multiscale property of wavelets, the regular operators are constructed based on the wavelet expansion of a differential operator. The degree of approximation changes along with the scale level. To determine the optimal operator, a new approach is proposed. Then, the differential equations in the model of dynamic data reconciliation are converted into algebraic ones, and the SQP approach is adopted to solve the discrete model. The dynamic data reconciliation is computed through the following steps: (1) de-noising the measurements using wavelets;15 (2) constructing regular operators and determining the optimal scale level, namely, determining the optimal approximate differential operator; and (3) solving the reconciliation model by using the SQP method. 3.1. Wavelet De-Noising. First, wavelet de-noising can eliminate the abnormal data in measurements.16 Second, wavelet de-noising can utilize the temporally redundant information of measurements to reduce random errors.15 So, it can improve the accuracy of reconciliation as compared with other methods where only a dynamic model is used, namely, only spatial redundancy is utilized. On the other hand, accurate reconciled results of measured variables will be benefit to the estimation of unknown variables. On the other hand, it is obvious that wavelet denoising does not utilize the spatially redundant information of measurements so that it cannot obtain results consistent with constraints and cannot estimate unknown variables. So, the data reconciliation is needed along with the wavelet de-noising. In the proposed method, both kinds of spatial and temporal redundancy are used, so the corresponding reconciled results will be better. Roughly speaking, the filtered signal can be taken as data measured through more accurate instruments. In section 4.4, a computation example without wavelet de-noising will be given to show that the estimation will deteriorate without denoising. 3.2. Representation of the Differential Operator in Bases of Wavelets. To use the regularization method to solve the inverse problem, a series of approximate operators need to be constructed and an optimal one needs to be selected. First, the differential operator is expanded by using wavelets. The regular operator can be constructed at different scale levels, and then operator series can be obtained. The compact supported orthogonal wavelet is
ψjk(x) ) 2-j/2 ψ(2-jx - k), j, k ∈ Z
(3)
The twin-scale relations are
φ(x) ) x2
L-1
∑ hkφ(2x - k)
k)0
(4)
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ψ(x) ) x2
L-1
∑ gkφ(2x - k) k)0 m ) 0, 1, ..., M - 1
T ≈ Tm ) PmTPm : L2(R) f Vm
dt
(10)
∫φ(x - i) dφ(x) ) 2∫∑hmφ(2x - 2i - m) d ∑hjφ(2x - j)
ri )
(6)
m
)2
(7)
j
∑ ∑hmhj ∫φ(2x - 2i - m) dφ(2x - j) m j ∑ ∑hmhjr2i-j+m m j
where Pm is the projection operator from L2(R) to Vm.11 The approximate operator is different as the selected scale varies. Then,
T(f) )
∑ hkφ(2x - k)
then, we can obtain
Let the differential operator be T and scale level be m, then T is approximated by using operator Tm
df
L-1 k)0
The vanishing moment of the wavelet is M, namely,
∫-∞∞ψ(x)xm dx ) 0,
φ(x) ) x2
(5)
)2
(11)
For a one-order differential operator, there exists
≈ Tk(f) ) PkTPk(f) )
∑i ∑j 〈 f, φk,i〉
〈
〉
dφk,i , φk, j φk, j (8) dt
In the given additive formula, the first part 〈 f, φk,i〉 can be computed through wavelet transform of f; if the inner product of the scale function and its derivative is known, the coefficient of φk, j can be calculated, and then, through the inverse wavelet transformation, the approximate derivative Tk(f) can be obtained. It is obvious that the computation of the inner product of the scale function and its derivative is important to the expansion of the operator. The detailed computation is illustrated in following subsections. In actual computation, suppose that the original signal and its derivative all belong to space V0. Because f is a discrete signal, it requires that the calculated result Tk(f) should be discrete too. Thus, the dynamic model of dynamic data reconciliation is discretized, and the differential equations are converted into algebraic ones. The computation steps are listed as follows: (1) decomposing the original signal to the scale level k and the coefficient Fk,i ) 〈 f, φk,i〉 can be obtained; (2) calculating the inner product ri ) ∫φ(x - i) dφ(x) ) 〈dφ/ dt, φ0,i〉; (3) calculating the convolution of R ) [ri] and Fk, Gk,j ) ∑〈 f, φk,i〉〈dφk,i/dt, φk,j〉 is obtained; and (4) processing the inverse wavelet transform for signal Gk and the approximation is obtained. 3.2.1. Computation of Inner Product. It is obvious that the computation of the inner product of the scale function and its derivative is important to the expansion of the operator.
Let ri )
,φ ∫φ(x - i) dφ(x) ) 〈dφ dt 0,i〉
Through variable substitution, the computation of the inner product of the scale function and its derivative can be converted into the calculation of ri, namely,
〈
〉
dφk,i , φk, j ) dt
∫φk,j dφk,i
) 2-k
,φ 〈dφ 〉 dt
) 2-krj-i Using a twin-scale equation
0, j-i
(9)
∑i iri ) -1
(12)
Solving eqs 11 and 12 simultaneously can obtain ri.11,17,18 Then, the approximate operator can be achieved. 3.2.2. Computation of Approximation Coefficients of Derivative Signal. After obtaining the inner product of the scale function and its derivative, we can calculate the coefficients of the derivative signal.
Gk, j )
∑i 〈f, φk,i〉
Gk, j )
∑i 〈f, φk,i〉
then
)
〈
dφk,i , φk, j dt
〉
〈
dφk,i , φk, j dt
〉
∑i Fk,i2-krj-i
(13)
(14)
The approximation coefficients Gk of a derivative signal can be calculated by the convolution of R ) [ri] and Fk. Then, through inverse wavelet transform, the derivative signal can be obtained. 3.3. Determination of the Optimal Scale. It is inferred from the error formula of regularization that an optimal operator exists for a certain inverse problem.10 The multiscale property of wavelets makes it possible to approximate the signal at different scales, so the approximate operator series can be obtained. Because different scale levels correspond to different approximate operators, the selection of an optimal operator is equivalent to the determination of the optimal scale level. Because the deviation δ between the measurement and the actual signal is unknown, it is impossible to utilize the error formula of regularization to determine the optimal operator directly. A new approach is proposed in order to determine the optimal scale level in which the prior statistical information of the signal is used. Let Tn be approximate operator and yδ is the measurement signal, then ynδ t KTnyδ is an estimation of the true signal. The purpose of regularization is to obtain a more accurate derivative of the signal yδ, and then the optimal approximate operator should make ynδ close to the actual signal. Suppose that the expected value and standard variance of random errors contained
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in measurements are 0 and σ, respectively, then the following model can be used to determine the optimal scale level. The objective function contains two parts. The first part represents that the standard variance of ynδ - yδ should be close to σ as far as possible. The second part means that the expected value of ynδ - yδ should be close to zero. All of these mean that ynδ - yδ should have the same statistical properties as random error.
min k
(s -σ σ)
2
+ yj2
s.t. ynδ ) KTkyδ yj ) s)
1
(yn,iδ - yiδ) ∑ N i
x
1
∑i (yn,iδ - yj)2
N-1
(15)
In this model, the scale k corresponds to the approximate operator Tk. The optimal operator can be obtained by solving the model. Unlike the previous method8 which needs to solve the whole reconciliation model repeatedly for determining the optimal scale, this approach only needs to process the single signal. 3.4. Solution of Dynamic Data Reconciliation. Because the optimal approximate operator Tm of differential operator is obtained, the original reconciliation problem is converted into T -1 min(Y - Yf) Q (Y - Yf) Y
s.t. Tm(y(t)) ) h(y(t), u(t)) f(y(t), u(t)) ) 0 c(y(t), u(t)) e 0
(16)
It should be noted that only the differential operator associated with unknown sensitive input variables is required to be approximated based on the principle of regularization. The reason is that only the estimation of unknown sensitive input variables is ill-posed and that the reconciliation of other variables is not ill-posed. Because Yf is a signal after filtering, the variance covariance matrix needs to be determined. After filtering, Yf ) Y + δ, where Y is an unknown true signal, δ is random error, and its distribution is unknown. Assume that δi is uncorrelated. Then, using Stein’s unbiased estimation formula,19 E |Yf - Y| can be estimated15,19 and the standard variance of Yf can be obtained. Model 16 is solved by using the SQP approach. 4. Examples Consider a simple nonlinear dynamic systemsCSTR with a first-order exothermic reaction.6 The model can be written as follows:
dA q ) (A0 - A) - kA dt V -∆Hr UAR dT q kA ) (T0 - T) + (T - Tc) (17) dt V Fcp FcpV
k ) k0 exp
( ) -EA T
(18)
where A and T are tank concentration and tank temperature, respectively, and A0 and T0 are feed concentration and feed temperature, respectively. q is feed flow rate, V is tank volume, k is an Arrhenius rate expression, EA is activation energy, AR is heat transfer area, F is density, cp is heat capacity, ∆Hr is heat of reaction, U is a heat transfer coefficient, and Tc is coolant temperature. In the following calculation cases, concentrations and temperatures were scaled by using a nominal reference concentration Ar and a nominal reference temperature Tr. Detailed information about the model can be found in Liebman’s paper.6 Data are generated by simulation, and random errors are added. The compact supported orthogonal wavelet, named Daubechies Wavelet, is adopted and the vanished moment of the selected wavelet is 2. And, in following cases, the feed temperature T0 is unmeasured. Because feed temperature T0 can only be determined through dT/dt, the problem is ill-posed. In the following sections, the problem is solved by using finite element collocation based on polynomials (FECP),7 orthogonal collocation on finite elements (OCFE),6 and the proposed wavelet method. 4.1. Results Obtained by Using FECP Method. If the feed temperature is unmeasured, it needs to utilize the differentiation of tank temperature to estimate it. In such a case, the reconciliation is an ill-posed inverse problem. First, the reconciled results obtained by using FECP method7 are shown in Figure 1. This method has no assumption about data. In this figure, the red line represents reconciled values, the blue line represents measurement, and the black line represents true values. The mean deviation of results of this method is shown in Table 1. The bound of feed temperature T0 is [0, 10]. The results obtained by using the FECP method fluctuate tempestuously within this range, and they are unusable. The fluctuation results from the fact that there are random errors in tank temperature and that the errors cannot be avoided. After differentiation, the error is enlarged. Unless the true values of tank temperature were known, the accurate estimation could not be obtained. 4.2. Results Obtained by Using OCFE Method. Liebman et al. adopted OCFE to solve the problem of dynamic data reconciliation.6 However, it should be noted that the method assumes that, in the moving window, the feed temperature, and feed concentration are all constant. Usually, this assumption does not accord with a practical situation, and in fact, its effect is similar to a filter. On the other hand, if there is no such assumption, this method cannot estimate the unknown feed temperature because all data at the inner collocation points are unknown. In this example, the length of moving window is 10. The results are shown in Figure 2 and Table 1. The method cannot obtain a good estimation of feed temperature either. 4.3. Results Obtained by Using Wavelet-Based Method. In the proposed method, there is no assumption similar to the one in the OCFE method. The results are shown in Figure 3 and Table 1. The compact supported orthogonal wavelet, named Daubechies Wavelet, is used and its vanishing moment selected is 2. As shown in Figure 3, the estimation of feed temperature obtained through the wavelet-based method are
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Figure 1. Results of the FECP method. Table 1. Comparison of Two Methodsa mean deviation (MD) methods
tank concentration
tank temperature
feed concentration
feed temperature
FECP OCFE wavelet no de-noising
0.070 795 2 0.055 155 7 0.037 348 8 0.101 630 5
0.597 843 0 0.419 321 0 0.282 861 9 0.780 527 0
0.396 003 4 0.223 357 6 0.255 202 9 0.839 139 6
81.799 544 6.764 113 6 1.023 605 6 2.829 725 1
a Note: MD ) (∑|X - X |)/Nσ, where X and X are reconciled and simulated values, respectively, σ is the corresponding standard r s r s variance, and N is the sampling count.
mostly within the range of (3 × 0.05, and the results are usable. Nevertheless, the estimation still has some deviation in the sense of a general concept of accuracy, which is due to the fact that the problem is ill-posed. It is known from the error formula of regularization (eq A4, Appendix) that an optimal approximate operator exists when the errors of measurement are fixed,10 (i.e., the errors between reconciled and actual values have a lower limit and cannot be infinitesimal). That means that it is impossible to obtain very accurate results. 4.4. Results Obtained by Using Wavelet-Based Method Without De-Noising. Except that wavelet denoising is not used, the method used in this section is the same as the method in section 4.3. The results are shown in Figure 4 and Table 1. The computation shows that reconciled results will deteriorate if there is no wavelet de-noising. 5. Discussion In dynamic data reconciliation, if the sensitive input variable is unknown, then the problem is an inverse problem and is ill-posed in the sense of Hadamard’s sense. Namely, the solution does not depend on the data continuously. This is because there are errors in data
and the differential operator is unbounded. The unboundedness of the differential operator leads to error enlargement in estimation. It is impossible to eliminate errors in data completely, so this fluctuation cannot be avoided completely. For the ill-posed reconciliation problem, the FECP and OCFE methods failed and cannot obtain satisfactory results. The proposed regularization based on wavelets uses a bounded operator to approximate the unbounded one, so the first item of errors formula (eq A4), which represents the influence of errors on estimation of unknown sensitive input variable, is reduced. In fact, regularization is to make a tradeoff between inaccuracy that results from operator approximation and error. So, more accurate estimation can be obtained by using the proposed method. From Table 1, the reconciled results of feed concentration obtained by using the OCFE method have less mean deviation than other methods, and the estimation of feed temperature has less mean deviation than the one by using the FECP method. This performance is mainly due to the assumption of the OCFE method that in the moving window all feed concentrations are constant. It can be expected that, under the assumption,
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Figure 2. Results of the OCFE method.
Figure 3. Results of method based on wavelets.
increasing the length of moving window will obtain more accurate results because, in this case, the feed temper-
ature is constant. However, usually it is not the general case. If the feed temperature is not constant, a good
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Figure 4. Results of method based on wavelets without de-noising.
estimation cannot be obtained by using this method. So, wavelet-based regularization is required and is more general. 6. Conclusions Because differential equations exist in the model of dynamic data reconciliation, if unknown sensitive input variables exist, then the problem will be an ill-posed inverse one. Correct and usable estimation results can be obtained by using existing polynomial-based methods. Here, according to the regularization theory, regular operators are constructed based on the wavelet expansion of a differential operator and a new approach is proposed to determine the optimal scale level where the prior statistical information is used. Compared with the other wavelet-based method, the proposed approach simplified the selection of the optimal scale level. It does not need to solve the whole reconciliation problem and only needs to process the single signal. The computation results show that more accurate estimation results can be obtained by using the proposed method than those that are obtained by using methods based on polynomials. Acknowledgment This work is supported by the National Natural Science Foundation of China, Grant No. 29910761863, and the National High-tech Research & Development Plan (863 plan fund, No. 863-511-945) that are gratefully acknowledged. Appendixes A.1. Regularization.10 At present, regularization is proved to be an effective method to solve the inverse
problem. It converts an ill-posed problem into a wellposed one through operator approximation, and the approximate solution is obtained. The inverse problem can be expressed as the following operator equation:
Kx ) y x ) Ty
(A1)
where y is known, x needs to be estimated, K is a linear compact operator, and T is the inverse of K. In dynamic data reconciliation, K is an integral operator, and T is a differential operator. The operator T is unbounded, so stability is not guaranteed. A regularization strategy is a family of linear and bounded operators
TR : Y f X, R > 0
(A2)
such that
lim TRKx ) x Rf0
for all x ∈ X
(A3)
(i.e., the operators TRK converge pointwise to the identity). R is called the regularization parameter. From the errors formula of regularization, an optimal approximate operator TR exists. And, the accuracy of results has a lower limit. A.2. Errors of Regularization.10 Now, let y ∈ K(x) be the exact right-hand side of Kx ) y and yδ ∈ Y be the measured data with |y - yδ| ) δ. We define xR,δ t TRyδ as an approximation of the solution x of y ) Kx. Then, the error splits into two parts by the following obvious application of the triangle inequality:
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|xR,δ - x| ) |TRyδ - x| e |TRyδ - TRy| + |TRy - x| ) |TR(yδ - y)| + |TRKx - x| e δ|TR| + |TRKx - x|
(A4)
The first term on the right-hand side describes the error in the data multiplied by the “condition number” |TR| of the regularized problem. This term tends to infinity as R tends to zero. The second term denotes the approximation error at the exact right-hand side y ) Kx. This term tends to zero as R tends to zero. If there is very small difference between TR and T, then the estimation error is not small because the first error item is very large. On the other hand, if operator TR is not a good approximation of T, then the estimation error is also great because the second item is primary as compared with the first item. So, one optimal approximate operator exists. Nomenclature A ) tank concentration A0 ) feed concentration AR ) heat transfer area cp ) heat capacity EA ) activation energy gk ) reconstruction high-pass filter hk ) reconstruction low-pass filter Pm, Pk ) projection operator q ) feed flow rate Q ) variance covariance matrix T ) tank temperature T0 ) feed temperature T ) differential operator Tm ) approximate differential operator Tc ) coolant temperature u ) unmeasured variables U ) heat transfer coefficient V ) tank volume yδ ) measurement signal ynδ ) estimation of true signal y ) measured variables Y ) reconciled results Ym ) measurements φ ) scale function ψ ) wavelet function σ ) standard variance of measurements ∆Hr ) heat of reaction
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(2) Albuquerque J. S.; Biegler, L. T. Data Reconciliation and Gross-Error Detection for Dynamic Systems. AIChE J. 1996, 42, 2841-2856. (3) Chen J.; Romagnoli, J. A. A Strategy for Simultaneous Dynamic Data Reconciliation and Outlier Detection. Comput. Chem. Eng. 1998, 22, 559-562. (4) Taiwen, J.; Bingzhen, C.; Xiaorong, H. Industrial application of wavelet transform to the on-line prediction of side draw qualities of crude unit. Comput. Chem. Eng. 2000, 24, 507-512. (5) Jang S. S.; Joseph, B.; Mukal, H. Comparison of Two Approaches to On-line Parameters and State Estimation of Nonlinear Systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 809-814. (6) Liebman, M. J.; Edgar, T. F.; Lasdon, L. S. Efficient Data Reconciliation and Estimation for Dynamic Processes Using Nonlinear Programming Techniques. Comput. Chem. Eng. 1992, 16 (10/11), 963-986. (7) Kong, M.; Chen, B.; Li, B. An integral approach to dynamic data rectification. Comput. Chem. Eng. 2000, 24 (2/7), 749-753. (8) Binder, T.; Blank, L.; Dahmen, W.; Marquardt, W. Regularization of Dynamic Data Reconciliation Problems by Projection; Technical Report LPT-1999-21; Lehrstuhl fu¨r Prozeaˆtechnik: RWTH Aachen, Germany, 1999. (9) Binder, T.; Blank, L.; Dahmen, W.; Marquardt, W. Towards multiscale dynamic data reconciliation. In Nonlinear Model Based Process Control; Berber R., Kravaris C., Eds.; NATO-ASI Series; Kluwer: Dordrecht, The Netherlands, 1998; pp 623-665. (10) Kirsch, A. An introduction to the mathematical theory of inverse problems; Springer-Verlag: New York, 1996. (11) Li, S.; Liu, J. The Basic of Wavelet Transformation and Inverse Problem. Geological Publishing House: Beijing, China, 1994. (12) Crowe, C. M.; Garcia Campos, Y. A.; Hrymak, A. Reconciliation of Process Flow Rates by Matrix Projection. Part I: Linear Case. AIChE J. 1983, 29 (8), 881-888. (13) Crowe, C. M. Reconciliation of process flow rates by matrix projection. Part II: The Nonlinear Case. AIChE J. 1986, 32 (4), 616-623. (14) Bagajewicz, M. J.; Jiang, Q. Integral Approach to Plant Linear Dynamic Reconciliation. AIChE J. 1997, 43 (10), 25462558. (15) Donoho, D. L.; Johnstone, I. M. Adapting to Unknown Smoothness via Wavelet Shrinkage. J. Am. Stat. Assoc. 1995, 90 (432), 1200-1224. (16) Jiang, T.; Chen, B.; He, X. Industrial application of Wavelet Transform to the on-line prediction of side draw qualities of crude unit. Comput. Chem. Eng. 2000, 24 (2/7), 507-512. (17) Dahmen, W.; Micchelli, C. A. Using the Refinement Equation for Evaluating Integrals of Wavelets. SIAM J. Numer. Anal. 1993, 30 (2), 507-537. (18) Beylkin, G. On the Representation of Operators in Bases of Compactly Supported Wavelets. SIAM J. Numer. Anal. 1992, 6 (6), 1716-1740. (19) Stein, C. M. Estimation of the Mean of a Multivariate Normal Distribution. Ann. Stat. 1981, 9 (6), 1135-1151.
Received for review June 13, 2001 Revised manuscript received January 31, 2002 Accepted April 24, 2002 IE010529K
