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Ind. Eng. Chem. Res. 2000, 39, 2404-2409
Use of the Sequential Loop Closing Method for Iterative Identification of Ill-conditioned Processes Jin Young Choi and Jietae Lee* Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
Thomas F. Edgar Department of Chemical Engineering, University of Texas, Austin, Texas 78712
For some ill-conditioned processes, small element errors due to the classical element-by-element identification cannot be tolerated and control systems based on models with such errors can suffer from poor control performances. To avoid this problem, identification under closed-loop control or iterative identification can be used. However, for higher order ill-conditioned processes, control systems for closed-loop identification are hard to design and iterative identification requires long field experiments. Here, we show that the sequential loop closing method, which is one of the systematic methods to design multiloop control systems, can be used to identify ill-conditioned processes as a part of designing multiloop control systems. The sequential loop closing identification provides control relevant models for design of control systems and can be used to accelerate the convergence of iterative methods. Introduction Processes whose frequency response matrices have large condition numbers are called ill-conditioned processes. A high-purity distillation column is a well-known example of an ill-conditioned process.1 A continuous stirred tank reactor can have ill-conditioned operating regimes as shown by Ogunnaike and Ray.2 A process with heat and material integration can be ill-conditioned when such an integration is tight. The need for highpurity products and for tight integration of heat and material flows is growing because of global economic competition. Inversion of ill-conditioned processes can be very sensitive to modeling errors. For some ill-conditioned processes, small elementwise errors arising in the classical element-by-element identification are amplified in inversion. Hence, explicit and implicit model-based control systems that use inversion can exhibit poor robust performances.1 Uncertainty condition numbers which measure the effects of the model identification errors on stability and performance of inversion-based control systems have been presented by Lee et al.3 Ill-conditioned processes consist of small and large elements corresponding to small and large singular values, respectively. Because the small singular values become large during inversion, control system design requires accurate determination of the small singular values and corresponding singular vectors instead of individual elements.4 However, the classical elementby-element identification which uses perturbation of each input may not excite the small elements, resulting in a poor signal-to-noise ratio and poor identification. Furthermore, small errors in the large elements will hinder accurate identification of the small elements. The small singular values and corresponding singular vectors can be inaccurate. *To whom all correspondence should be addressed. E-mail:
[email protected]. Fax: +82-53-950-6615.
To identify the small singular values and corresponding singular vectors accurately, Kuong and MacGregor5 have proposed a geometric approach in which inputs are transformed with the input singular vectors known a priori so that each singular value can be excited independently. This gives data that are evenly excited, avoiding a poor signal-to-noise ratio for the small singular values. Kuong and MacGregor6 also have presented an iterative method which refines the input singular vectors, which separates the small elements from the large elements. Similar iterative methods which use transformations with SVD decomposition, QR decomposition, and LU decomposition to separate the small elements from the large elements have been presented by Lee et al.7 Instead of using input transformations, closed-loop experiments can be used to excite all of the directions of outputs evenly without the detailed information of input singular vectors.5,8,9 Li and Lee4 have presented an identification method which fits both the process model and its inverse. The inverse of the process model can be obtained by finding the relative gain array10 with closed-loop experiments. The small singular values and corresponding singular vectors are strongly reflected in the inverse of the process model, and they can be identified accurately by fitting the inverse of the process model. The potential disadvantages of the closed-loop identification methods are that control systems for closed-loop identifications are usually hard to design and long closed-loop experiments are required because of the slow responses.1 The sequential loop closing method to design the multiloop control systems11,12 can also be used to identify process models. In the sequential loop closing method, one transfer function between the paired input and output (a single-input single-output transfer function) is identified at a time. Instead of identifying one transfer function at a time, transfer functions for outputs other than the paired output can also be identified without any additional experiments. Here it is shown that this sequential loop closing identification
10.1021/ie9903947 CCC: $19.00 © 2000 American Chemical Society Published on Web 05/12/2000
Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2405
The stability of a control system designed with the h (s) model G h (s) can be guaranteed13 if both G(s) and G are proper and have the same number of unstable poles, H h (s) is stable, and
σ(H h )(jω) < σ-1(E(jw))
Figure 1. (a) Multivariable control system and (b) an Equivalent internal model control system.
method can be used to identify ill-conditioned processes. Models with accurate determinants are obtained, and these models are far better to design control systems than those obtained with classical element-by-element identification methods. Furthermore, the sequential loop closing identification method can be combined with iterative methods with input and output transformations, resulting in more relevant models to design control systems.
Criteria whether models are relevant to design control systems are very important in the selection of identification methods. For some ill-conditioned processes, the accuracy of each element of the process transfer function matrix does not guarantee relevance for designing control systems. Hence, the accuracy of each element cannot be used to determine whether models are adequate or not and the classical element-by-element identification methods which provide accurate elements cannot be used to identify ill-conditioned processes. Consider the multivariable control system in Figure 1a. The transfer function matrix between the disturbances and the outputs is
Y(s) ) [I + G(s) C(s)]-1D(s)
(1)
When a process model G(s) is used for designing the control system, it becomes
Y(s)) [I + (I + E(s))-1G h (s)]-1D(s) ≈ [I + (I - E(s))G h (s) C(s)]-1D(s) h (s)]-1D(s) ) [I + G h (s) C(s)]-1[I - E(s) H
(2)
where
h (s) - G(s))G(s)-1 (3) E(s) ) G h (s) G(s)-1 - I ) (G H h (s) ) [I + G h (s) C(s)]-1G h (s) C(s)
for any angular frequency ω, where σ(‚) means the largest singular value. This stability condition was derived by transforming the multivariable control system to the internal model control framework14 as shown in Figure 1b. Identification methods that provide models with small σ(E(jω)) are required for robust stability. For example, because σ(H h (jω)) ≈ 1 for frequencies up to the bandwidth, models with σ(E(jω)) < 1 should be used. The sign consistency of the relative gain array (RGA) can be used for another simple criterion. When multiloop control systems are designed, the RGA at the steady state is often used to screen unworkable pairings. A pairing which has negative diagonal RGA should not be used because it has an integrity problem when integral action is included in the controller for offsetfree operations.15,16 That is, if the sign of RGA is not correct, multiloop control systems with integral action can be unstable when some loops are in an open loop. The RGA is given by
Λ(G) ) G-ToG ) {adj(G)ToG}/det(G)
Criteria for Control Relevant Models
(4)
If a norm of E(jω)H h (jω) is small for a wide range of frequencies, the performance of the control system would not be much different from what is expected at the design stage. Hence, the maximum singular value of E(jω), σ(E(jω)), can be used as a criterion to check the model relevance which is independent of the controller. When models with small σ(E(jω)) are used, performances of control systems are guaranteed.
(5)
(6)
where o means the element-by-element product. For some ill-conditioned processes, elements of the RGA are very large and such processes can be singular for small model identification errors.3,17 This means that det(G) is very sensitive to model identification errors and it is very important to find models with accurate determinants. Hence, without an accurate determinant of G, we cannot expect sign consistency of RGA and models relevant for control system designs. Here, the sequential loop closing identification is shown to provide models with accurate determinants. Identification with the Sequential Loop Closing Method In the sequential loop closing method,11,12 the controller for the first pair of input and output is designed and this loop is then closed. The second loop is designed for the second pair with the first loop closed. In this manner, loops are designed sequentially up to the final loop. Each controller can be designed systematically with a single-input single-output (SISO) tuning method. For such multiloop controllers, SISO automatic tuning methods can be applied.18,19 Shen and Yu19 have shown that transfer functions identified by the autotuning sequential loop closing method are relevant for control system designs. However, they identified one transfer function at each step in the sequential loop closing design and can suffer from poor control performances when pairings and design sequences are not appropriate. Instead of identification of one transfer function between the paired input and output at each step, transfer functions for other outputs in addition to the paired output can also be identified. In this way, the whole multiple-input multiple output (MIMO) transfer function matrix can be obtained while individual loops are being designed. The number of identification experiments is the same as those in the autotuning sequential
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Figure 3. Input and output transformations for the iterative identification.
Figure 2. System when some loops are closed.
loop closing method18,19 and the classical element-byelement identification method. Here, we show that this identification method, even when applied to ill-conditioned processes, provides models with accurate determinants. Assume that identification of the principal m × m subsystem yields an accurate determinant and has been closed with a stable control system with integral action as shown in Figure 2. Identification is performed with perturbing inputs disconnected. Because the first m loops are closed, we have
such models do not guarantee small σ(E(jω)). Iterative identification methods6,7 can be used to obtain models with smaller σ(E(jω)). In iterative identification methods, input and output transformations are used as in Figure 3. The transformed process
G ˜ k ) YkGXk
(11)
is identified. Identified models can have errors repre˜ ko∆. Then the kth approximate model sented as G ˜ k + G becomes
˜ k + G ˜ ko∆)Xk-1 G h k) Yk-1(G ) G + Yk-1(G ˜ ko∆)Xk-1
(12)
-1
u1 ≡ Q3(s) u2 ) -G1 (s) H1(s) G3(s) u2 y2 ≡ Q4(s) u2 ) (G4(s) + G2(s) Q3(s))u2
and
(7)
where H1(s) ) (I + G1(s) C1(s))-1G1(s) C1(s). The new transfer functions, Q3(s) and Q4(s), are identified, and they will have errors, in the frequency domain,
Q h 3(jω) ) Q3(jω) + (ω) Q3(jω)o∆3(jω) Q h 4(jω) ) Q4(jω) + (ω) Q4(jω)o∆4(jω)
(8)
where ω is the angular frequency and (ω) ∆(jω) is an element-by-element model identification error with (ω) > 0 and |∆(jω)| ) 1. Hence, we have a transfer function matrix
G h )
[
G h 1 -H h 1-1G h 1Q h3 G h2 Q h4 - G h 2Q h3
]
(9)
h 1(jω) For the frequency range of H h 1(jω) ) (I + G h (jω) C1(jω) ≈ 1, C1(jω))-1G
h4 - G h2 Q h3 + G h2 G h 1-1G h 1Q h 3) det(G h ) ≈ det(G h 1) det(Q ) det(G h 1) det(Q h 4) ) det(G h 1) det(Q4) det(I + (Q4o∆4)Q4-1)
(10)
If Q4 is well-conditioned, the above determinant is accurate. We can apply this procedure to the process sequentially. At each step in the sequential identification, Q4 can be a scalar and hence well-conditioned, yielding a model with an accurate determinant. Usually a multiloop control system designed by the sequential loop closing method has a bandwidth that is sufficiently large for application of this identification method. When a larger bandwidth is required, a decoupling control system at each step instead of the multiloop control system can be used. Iterative Methods with Input/Output Transformations While sequential loop closing identification provides transfer function models with accurate determinants,
σ(E)) σ(G h kG-1 -I) ˜ ko∆)G ˜ k-1Yk) ) σ(Yk-1(G
(13)
e γ(G ˜ k) κ(Yk) σ(∆) where κ(Yk) ) σ(Yk) σ(Yk-1) and γ(G ˜ k) ) maxσ(∆))1 σ((G ˜ ko∆)Gk-1) (definition and computational methods of the uncertainty condition number γ(‚) can be found in Chen et al.20 and Lee et al.3 If Yk is well-conditioned and the uncertainty condition number of the transformed process, γ(G ˜ k), is small, then σ(E) is small. Because σ(E) is small for diagonal matrices or some tridiagonal matrices, iterative identification methods transform process transfer functions to the diagonal or tridiagonal matrices with well-conditioned input and output transformation matrices of Xk and Yk. Specifically, the SVD method21 uses Xk+1 ) Vk and Yk+1 ) WkT, where matrices Wk and Vk are the left and right singular vector matrices of the kth estimates G h k, respectively. This transformation will diagonalize the process. In the QR method, the kth identified model is decomposed as G h k ) PkRkQkT, where Pk is a permutation matrix, Qk is an orthogonal matrix, and Rk is an upper triangular matrix. The transformation of Xk+1 ) Qk and Yk+1 ) PkT will transform the process to the triangular form. In the LU method, the kth estimates are decomposed as
G h k ) PkLkUk
(14)
where Lk and Uk are lower and upper triangular matrices, respectively, and the input and output transformations
XK+1 ) Uk-1 YK+1 ) PkT
(15)
are used to triangularize the process. Convergences of these iterative methods have been proved for a small relative identification error at each step. The SVD and QR methods use transformations with orthogonal matrices, and such transformations are not
Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2407
Figure 4. Artificial recycle system.
available in general for the dynamic processes. On the other hand, the LU method uses analytical transformations and can be easily extended to the dynamic case.
where τ0, τ1, and τ2 are time constants of the preheater, reactor 1, and reactor 2, respectively. As heat integration becomes tight, the process becomes ill-conditioned. Simulations are carried out for the sets (k01, k11, k12, k21, k22) ) (0.6, 2.6, 0.02, 0.6, 0.02) and (τ0, τ1, τ2) ) (1, 1, 1). Frequency responses are obtained by injecting pseudo random binary sequence (PRBS) signals and fitting the outputs with a Laguerre-type transfer function and the least-squares method.22,23 The PRBS signal is applied first to the input u1, and coefficients of the Laguerretype transfer function
(s - p)i-1
m
g(s) )
Proposed Method methods,7
the usual eleIn iterative identification ment-by-element identification is used for identification of each G ˜ k. Here, the sequential loop closing identification method is applied to identify G ˜ k, reducing the iteration number considerably. That is, the sequential loop closing identification will reduce the total experimental time for control relevant identification of illconditioned processes. The identification procedure is summarized as follows: Step 1. Identify the process with the sequential loop closing identification method. Step 2. Determine the order of input and output variables for the LU decomposition and the sequence of the loop closing identification. Step 3. Apply input and output transformations and identify the process model with the sequential loop closing identification method. Step 4. Repeat steps 2 and 3 until the transformed process is not much different from a triangular matrix. Here, the LU decomposition method is used because it can be easily extended to the dynamic case. The LU decomposition method designs a one-way decoupler iteratively7 because the process is transformed to a triangular one with a triangular transformation matrix. The performances of the LU decomposition method and the loop closing identification method are dependent on the order of input and output variables. Because errors of transfer functions identified early can be transferred to identification of the subsequent transfer functions, one criterion for the sequence of loop closing identification is that a well-conditioned principal subsystem be identified first. Also the order of input and output variables is chosen for the uncertainty condition number of the final G ˜ k to be small. Example 1. Consider the recycle process shown in Figure 4, where temperatures T1 and T2 are the controlled variables and P1 and P2 are the manipulated variables. Its dynamics can be described as
[ ]
T1 T2 )
[
-
1
k21 1 τ2s+1 )
k01k11 (τ0s+1)(τ1s+1)
] [ ][ ] -1
k12 0 τ1s+1 0
P1 k22 P2 τ2s+1
1 (τ0s+1)(τ1s+1)(τ2s+1)-k01k11k21 k12(τ0s+1)(τ2s+1) k01k11k22 P1 k12k21(τ0s+1) k22(τ0s+1)(τ1s+1) P2
[
][ ]
aix2p ∑ i)1
(s + p)i
are identified with the least-squares method. The number of terms m is chosen as 6, and the time scale parameter, p, is set to 0.06. Random noises of (0.025 are added to the outputs. Transfer functions of g11(s) and g21(s) are obtained in this way. Based on the identified transfer function g11(s), the first PI controller is designed as c1(s) ) 50(1 + 1/40s). To obtain g12(s) and g22(s), a PRBS signal is applied to the second input u2, as shown in Figure 5. When the least-squares method is applied to the outputs in Figure 5, Laguerre-type transfer functions of g12(s) and g22(s) are identified. First-order plus time delay (FOPTD) models are also extracted by fitting the frequency responses of the Laguerre models. That is, model parameters of
f(s) )
ke-θs τs+1
are calculated which will approximate the frequency response of g(s) as follows:
k ) g(0)
(x
τ ) avg
(
θ ) avg
)
(k/|g(jω)|)2 - 1 ω
)
-∠g(jω) - tan-1(τω) ω
where avg(‚) means the average for the frequency range of [0.0001 0.1]. The FOPTD model obtained with the usual element-by-element identification is
[
0.3112e-2.749s 0.4871e-3.519s 39.47s+1 42.07s+1 G h EBE(s) ) 0.1898e-1.812s 0.3059e-2.496s 42.46s+1 39.68s+1
]
and that with the sequential loop closing identification is
[
0.3112e-2.749s 0.4863e-4.552s 39.47s+1 42.07s+1 G h SEQ(s) ) 0.1898e-1.812s 0.3166e-1.231s 42.46s+1 42.46s+1
]
Figure 6 shows plots of σ(E(jω)) for both models. The sequential loop closing identification gives a model with small σ(E(jω)) of less than 1. Because σ(E(jω)) is small
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The steady-state gain obtained by the element-byelement identification for 2% relative errors is
G h EBE(0) )
[
86.04 -88.13 110.36 -107.41
]
where the diagonal elements are decreased by 2% and the off-diagonal elements are increased by 2%, which is the worst case in the sense that σ(E) is maximized. After the first loop is closed with a control system having integral action, the steady-state gains for the second input u2 are
Q3(0) ) -g12(0)/g11(0) ) 0.98405 Q4(0) ) g22(0) - g21(0) g12(0)/g11(0) ) -3.1253
Figure 5. Typical responses in the sequential loop closing identification tests for example 1 (the first loop is closed, and the second input u2 is perturbed).
In the identification of Q3(0) and Q4(0), a 2% relative error is added to Q3(0) and subtracted from Q4(0), respectively. We have
gj 12(0) ) -1.02Q3(0) gj 11(0) ) -86.36 gj 22(0) ) 0.98Q4(0) - gj 21(0) gj 12(0)/gj 11(0) ) -113.84 Hence
G h SEQ(0) )
Figure 6. Plots of σ(E(jω)) for models by the element-by-element identification and the sequential loop closing identification.
enough, iterative identification is not applied to this example. However, the element-by-element identification gives a model with σ(E(jω)) greater than 1 at low frequencies. Because σ(H h (jω)) ≈ 1 at low frequencies for control systems with integral action, the stability of the control system designed with the model of element-byelement identification may not be guaranteed. Example 2. Consider the following high-purity distillation column:24
[
86.4(12.1s+1) 87.8 194s+1 (194s+1)(15s+1) G(s) ) 109.6(17.3s+1) 108.2 194s+1 (194s+1)(15s+1)
]
The RGA of this process is very large at low frequencies. Control relevant models at low frequencies are to be found. Ignoring the fast dynamic pole (15s + 1), we identify models of the form
G h (s) )
[
1 k11 k12 τs+1 k21 k22
]
Because the time constant τ does not affect σ(E(jω)) much, we concentrate on finding the steady-state gain matrix.
[
86.04 -86.36 110.36 -113.84
]
The RGAs for the process and two models are λ11(G(0)) h EBE(0)) ) -19, and λ11(G h SEQ(0)) ) 37. The ) 35, λ11(G sign of the RGA of G h EBE(0) is different from that of the process. When a multiloop control system is designed, G h EBE(s) suggests off-diagonal pairing and it has an integrity problem15,16 when integral action is used in the control system. In contrast, an accurate RGA having a consistent sign is obtained with G h SEQ(s). Because σ(E(0)) are greater than 1, we apply the iterative identification method of LU decomposition.7 Before applying the input and output transformations with LU decomposition, we change the order of the input and output variables because it provides a smaller uncertainty condition number γ(G ˜ k). Identification results are shown in Figure 7 for worst case errors at each iteration. Worst case errors are such that the absolute values for the diagonal elements are decreased and those of off-diagonal elements are increased. For 5% relative errors, the iterative identification method with the sequential loop closing identification provides a model with σ(E(0)) smaller than 1 in iteration 2, while that with the element-by-element identification achieves this condition in iteration 3. We can also see that improvement by the sequential loop closing identification is more significant as the identification error is increased. Conclusion The sequential loop closing identification method is applied to ill-conditioned processes and is shown to provide transfer function matrices whose determinants are accurate. For some ill-conditioned processes, elements of the RGA can be very large and such processes become singular for small model errors. That is, determinants of the process transfer function matrices are very sensitive to model identification errors. Models with accurate determinants via the sequential loop closing identification method will be more relevant to
Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2409
Figure 7. Identification results for example 2 and ∆ ) [-1, 1; 1, -1] (solid line, iterative identification with the sequential loop closing identification; dashed line, iterative identification with the element-by-element identification).
design control systems than models obtained by the usual element-by-element identification methods. Iterative identification methods which transform processes to diagonal or triangular forms iteratively with input and output transformation matrices can be used to refine models to be more relevant to design stable control systems. The number of iterations is shown to be reduced when the sequential loop closing identification method is incorporated with the iterative identification method based on LU decomposition. Acknowledgment J.Y.C. and J.L. gratefully acknowledge the financial support of Korea Science and Engineering Foundation under Grant 98-0502-05-01-3. Literature Cited (1) Skogestad, S.; Morari, M. Implication of Large RGAElements on Control Performance. Ind. Eng. Chem. Res. 1987, 104, 2323. (2) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling, and Control; Oxford University Press: Oxford, U.K., 1994. (3) Lee, J.; Cho, W.; Edgar, T. F. Effects of Diagonal Input Uncertainties and Element Uncertainties in Ill-Conditioned Processes. Ind. Eng. Chem. Res. 1998, 37, 1009-1017. (4) Li, W.; Lee, J. H. Control-Relevant Identification of IllConditioned Multivariable Systems: Estimation of Gain Directionality. Comput. Chem. Eng. 1996, 20, 1023. (5) Kuong, C. W.; MacGregor, J. F. Design of Identification Experiments for Robust Control. A Geometric Approach for Binary Processes. Ind. Eng. Chem. Res. 1993, 32, 1658.
(6) Kuong, C. W.; MacGregor, J. F. Identification for Robust Multivariable Control; the Design of Experiments. Automatica 1994, 30, 1541-1554. (7) Lee, J.; Cho, W.; Edgar, T. F. Iterative Identification Methods for Ill-Conditioned Processes. Ind. Eng. Chem. Res. 1998, 37, 1018-1023. (8) Andersen, H. W.; Kummel, M.; Jorgensen, S. B. Dynamics and Identification of a Binary Distillation Column. Chem. Eng. Sci. 1989, 44, 2571. (9) Jacobsen, E. W.; Skogestad, S. Inconsistencies in Dynamic Models for Ill-Conditioned Plants: Application to Low-Order Models of Distillation Column. Ind. Eng. Chem. Res. 1994, 33, 631. (10) Bristol, E. H. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Automatic Control 1966, AC-11, 133. (11) Mayne, D. Q. The Design of Linear Multivariable Systems. Automatica 1973, 9, 201. (12) Chiu, M. S.; Arkun, Y. A Methodology for Sequential Design of Robust Decentralized Control Systems. Automatica 1992, 28, 997. (13) Grosdidier, P.; Morari, M. Interaction Measures for Systems Under Decentralized Control. Automatica 1986, 22, 309. (14) Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Ind. Eng. Chem., Process Des. Dev. 1982, 21, 308. (15) Campo, P. J.; Morari, M. Achievable Closed-Loop Properties of Systems Under Decentralized Control: Conditions Involving the Steady-State Gain. IEEE Trans. Automatic Control 1994, AC39, 932. (16) Lee, J.; Edgar, T. F. Conditions for Decentralized Integral Controllability. J. Process Control 1999, accepted for publication. (17) Hovd, M.; Skogestad, S. Simple Frequency-dependent Tools for Control System Analysis, Structure Selection and Design. Automatica 1992, 28, 989-996. (18) Loh, A. P.; Hang, C. C.; Quek, C. K.; Vasnani, V. U. Autotuning of Multiloop Proportional-Integral Controllers Using Relay Feedback. Ind. Eng. Chem. Res. 1993, 32, 1102. (19) Shen, S. H.; Yu, C. C. Use of Relay-Feedback Test for Automatic Tuning of Multivariable Systems. AIChE J. 1994, 40, 627. (20) Chen, J.; Freudenberg, J. S.; Nett, C. N. The Role of the Condition Number and the Relative Gain Array in Robustness Analysis. Automatica 1994, 30, 1029-1035. (21) Horn, R. A.; Johnson, C. R. Matrix Analysis; Cambridge University Press: New York, 1985. (22) Zervos, C.; Belanger, P. R.; Dumont, D. A. On PID Controller Tuning Using Orthogonal Series Identification. Automatica 1988, 24, 165-175. (23) Park, H. I.; Sung, S. W.; Lee, I. B.; Lee, J. On-line Process Identification Using the Laguerre Series for Automatic Tuning of the Proportional-Integral-Derivative Controller. Ind. Eng. Chem. Res. 1997, 36, 101-111. (24) Skogestad, S.; Lundstrom, P. µ-optimal LV-control of Distillation Column. Comput. Chem. Eng. 1990, 14, 401-413.
Received for review June 2, 1999 Revised manuscript received December 6, 1999 Accepted March 15, 2000 IE9903947
