Improved Saturation Relay Test for Systems with Large Dead Time

Improved Saturation Relay Test for Systems with Large Dead Time ...... Yuwana, M.; Seborg, D. E. A new method for on-line controller tuning, AIChE J. ...
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Ind. Eng. Chem. Res. 2005, 44, 2183-2190

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Improved Saturation Relay Test for Systems with Large Dead Time E. Sivakumar, Vivek Sathe, and M. Chidambaram* Department of Chemical Engineering, Indian Institute of Technology, Madras, Chennai 600 036 India

Yu (Yu, C. C. Auto tuning of PID controllers: Relay feedback approach; Springer-Verlag: Berlin, 1999.) introduced the saturation relay feedback test for determining the ultimate gain and ultimate frequency of the system that is required for the design of PI/PID controllers. In the present work, two modifications are proposed to improve the efficiency of the saturation relay feedback test, especially for systems with a large dead time. In the first method, the safety factor is corrected based on the ideal relay feedback test. In the second method, higher order harmonics are included in the analysis of ideal relay. Simulation results are given to show the improvement of the proposed method over that of Yu method for a FOPTD system and a higher order system. The methods proposed by Srinivasan and Chidambaram (Srinivasan K.; Chidambaram, M. Chem. Biochem. Eng. Q. 2003, 18 (3), 249-256.) and Luyben (Luyben, W. L. Ind. Eng. Chem. Res. 2001, 40, 4391-4402.) for identifying FOPTD model parameters are compared. Two example systems are considered for the identification of a FOPTD model. For both the examples, after identification, the IMC tuning method is used for the controller design. The closed loop performances are compared with that of the actual system. Introduction Åstro¨m and Ha¨gglund4 have suggested the use of an ideal relay to generate sustained closed loop oscillations. The ultimate gain and the ultimate frequency can be found by the principal harmonic analysis. PID controllers can then be designed by using the closed loop Ziegler-Nichol’s tuning method. Li et al.5 have pointed out that the error in the estimation of ultimate gain ranges from -18% to +27% (with respect to the theoretical value) in ideal relay feedback test. To reduce the error in the estimation of ultimate gain, Yu1 has suggested the saturation relay feedback test. In this method, the ultimate gain estimated from the ideal relay feedback test is multiplied with a safety factor to get the slope of the saturation relay and the saturation relay feedback test is conducted to get better results for the ultimate gain and frequency. The method proposed by Yu1 gives accurate results for small L/τ ratio (generally L/τ < 2). However, Yu has not reported any result for large L/τ ratio (generally L/τ > 5). In the present study, the error involved in the estimation of ultimate gain for large L/τ ratio is analyzed. To get an improved result, two modifications are proposed mainly focusing on the selection of proper slope for the saturation relay. In the first method, the safety factor is corrected based on the trend of the system output waveform during the ideal relay feedback test. In the second method, higher order harmonics are included in the analysis of the ideal relay, based on the improved conventional relay auto tune method proposed by Srinivasan and Chidambaram.2 Li et al.5 have proposed use of two auto tune relay tests for identification of a first-order plus time delay (FOPTD) transfer function model. The first is a normal one and second with an additional known dead time to obtain ultimate frequency. Then a least-squares method * To whom correspondence should be addressed.E-mail: [email protected]. Tel: +91-44-22578210. Fax: +91-4422570509.

Figure 1. Block diagram for saturation relay feedback system.

is used to determine the unknown parameters. Friman and Waller6 have proposed a method where relay is replaced by construction called a two-channel relay, which consists of two relays connected in parallel, used to identify for small dead time and large dead time. Scali et al.7 have proposed the use of two alternative tests to calculate the required parameters of a completely unknown process. It is an extension of the Li et al.5 method. Luyben3 has pointed out that additional information can be obtained from the relay feed back test, namely the shape of the response. Thyagarajan and Yu8 have proposed a method that uses a conventional ideal relay feedback test and have given a procedure based on the shape information to find parameters for the corresponding model. Srinivasan and Chidambaram2 have proposed a method of identification of FOPTD model parameters using ultimate frequency and corrected ultimate gain and using the input and output data of the process from a relay test. In the present work, the methods proposed by Srinivasan and Chidambaram2 and Luyben3 for identifying FOPTD model parameters are compared. Importance of the Saturation Relay Slope Since the square wave output of the ideal relay results in an abrupt change in the slope of the relay at the zero point, [e(t) ) 0], Yu1 has suggested the saturation relay, which provides a smooth transition of the slope of the relay around the zero point. The saturation relay is characterized by its height and slope (refer to Figure 1). For a feedback loop with a saturation relay, the output of the relay is similar to a sine wave with

10.1021/ie049242o CCC: $30.25 © 2005 American Chemical Society Published on Web 02/23/2005

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Table 1. Theoretical Value and Estimations of Ultimate Gain by Relay Tests for the FOPTD Systemsa L/τ

ku

ωu

ideal relay feedback test

Yu method

proposed method I

proposed method II

1 5 10 15 20 25 30

2.262 1.132 1.040 1.019 1.011 1.007 1.005

2.028 0.530 0.286 0.196 0.149 0.12 0.101

2.019 [10.73] 1.282 [13.24] 1.273 [22.41] 1.273 [24.93] 1.273 [25.92] 1.273 [26.40] 1.273 [26.28]

2.231 [1.379] 1.260 [11.33] 1.203 [15.65] 1.203 [18.03] 1.202 [18.96] 1.202 [19.42] 1.202 [19.67]

2.231 [1.379] 1.149 [1.50] 1.126 [8.29] 1.126 [10.51] 1.126 [11.39] 1.126 [11.81] 1.126 [12.05]

2.231 [1.38] 1.128 [0.36] 1.069 [2.74] 1.066 [4.62] 1.066 [5.42] 1.066 [5.83] 1.066 [6.05]

a Safety factor (S ) )1.4 for Yu method;1 S ) 1 for the proposed method I and proposed method II [% error in the estimation of ultimate f f gain].

extreme cuts, based on the height of the relay.1 Hence, the saturation relay feedback test provides more accurate results than that of an ideal relay. To conduct a saturation relay feedback test, a proper slope has to be chosen for the saturation relay. If the slope is too high, the relay output is similar to a square wave, and as a result, the accuracy of the feedback test is lost. As a general rule, the lesser the slope value, the greater is the accuracy of the feedback test. However, if the slope of the saturation relay is less than the ultimate controller gain of the actual system, the feedback test fails to produce sustained oscillations.1 Thus, the success of the feedback test depends mainly on the selection of proper slope for the saturation relay. Proposed Method I. In this method, the slope of the saturation relay is found from the ideal relay feedback test. Since the ultimate controller gain from the ideal relay test is only an approximate value, a safety factor is necessary to ensure that the slope selected for the saturation relay is greater than the ultimate controller gain of the actual system. Yu suggested a safety factor of 1.4 to be multiplied with the approximate ultimate controller gain from the ideal relay feedback test to get the slope of the saturation relay. Yu1 has presented the analysis for only a smaller time delay to time constant ratio (L/τ < 1). In the case of a large L/τ ratio (generally L/τ > 5), the ultimate gain obtained from the ideal relay feedback test is always overestimated.2,5 Hence, the safety factor cannot be more than unity. The safety factor, 1.4, actually increases the slope of the saturation relay unnecessarily and causes additional error in the saturation relay feedback test. Instead, the ultimate gain found from the ideal relay feedback test can be directly assigned as the slope of the saturation relay, without need for any safety factor. The system output waveform during the ideal relay feedback test provides the clue to finding whether the L/τ ratio is smaller or larger. If the system output waveform is triangular, then the L/τ ratio is small.6,2 If the system output waveform is similar to a square waveform, then the L/τ ratio is large.6,2 Hence, the procedure for the proposed method I is as follows: (1) Perform the ideal relay feedback test. (2) Find approximate ultimate controller gain (ku), based on principal harmonics only,

ku ) 4h/(πa)

(1)

where a is the observed amplitude of the system’s output waveform. (3) Observe the trend of system output waveform. (4) If the waveform is triangular, take 1.4 as the safety factor (Sf) as suggested by Yu.1 If the waveform is closer to square waveform, take Sf ) 1.0.

(5) Select the slope of the saturation relay (k), from the approximate ku found by ideal relay feedback test.

slope,

k ) Sfku

(2)

(6) Perform the saturation relay feedback test. (7) Find the ultimate controller gain and ultimate frequency, based on principal harmonics only,

ku ) (2h/π)[sin-1(a1/a)/a1 + (a2 - a12)0.5/a2]

(3)

where, a1 ) h/k from saturation relay,

ωu ) 2π/Pu

(4)

where Pu is the period of oscillation of the system’s output waveform. Though the method involves two tests, the improvement in the accuracy of the estimation of ultimate controller gain and ultimate frequency really pays for the additional amount of work. Hence, the error caused by the improper selection of a slope value is avoided in the proposed method I. The safety factor, which is chosen by observing the system output waveform in the ideal relay feedback test, prevents selecting an unnecessary high slope value for the saturation relay. Simulations are carried out for a FOPTD model, with kp ) 1, τ ) 1, and L ranging from 1, 5, and 10, to 30. Theoretical values of ultimate gain and ultimate frequency are calculated and tabulated in Table 1. Estimations of the ultimate gain based on the ideal relay feedback test by the Yu1 method and the proposed method I are also tabulated in Table 1. Proposed Method II. Though the error in the estimation of the ultimate controller gain is reduced significantly by selecting a proper safety factor, still there is a possibility of reducing it further. The actual cause for the error in the ultimate gain estimation is the misinterpretation of a waveform as a sine waveform, during the analysis of the ideal relay. Hence, the error can be considerably reduced, only if higher order harmonics are included during the analysis of the ideal relay test. Srinivasan and Chidambaram2 have proposed a method to include higher order harmonics to improve the ideal relay feedback test. However, in that method, the number of higher order harmonics that has to be included varies with L/τ ratio and it has to be carefully chosen in order to get the accurate results. However, in the proposed method, only three harmonic components are used to characterize the output of the ideal relay. This will increase the accuracy of the ideal relay feedback test. Further, the selection of the slope for the saturation relay based on an ideal relay feedback test improves. As a net result, this will definitely help in

Ind. Eng. Chem. Res., Vol. 44, No. 7, 2005 2185 Table 2. PI Controller Settings for FOPTD System with Kp ) 1, τ ) 1, and L ) 10a' parameter

theoretical value

Yu method

proposed method I

proposed method II

Luyben method

ku ωu kc τI ISE ISEb

1.0402 0.2863 0.47 [0.35] 18.29 [6.0] 18.49 [12.61] 15.21 [11.78]

1.203 0.2909 0.54 [0.45] 17.99 [6.68] 16.26 [11.85] 14.42 [11.93]

1.1264 0.2904 0.51 [0.41] 18.03 [6.41] 17.12 [12.08] 14.77 [11.74]

1.0687 0.2934 0.48 [0.36] 17.85 [6.02] 17.75 [12.52] 15.09 [11.76]

1.2732 0.2939 0.57 [0.43] 17.83 [5.85] 15.6 [11.80] 14.3 [12.27]

a k ) 0.45 k ; τ ) 2π/(1.2ω ); S ) 1.4 for Yu method; S ) 1 for the proposed methods. [The PI controller parameters and the ISE c u I u f f values by using IMC method are reported in the brackets]. b ISE for FOPTD system when 25% perturbation in kp.

reducing the error in the estimation of ultimate controller gain. Moreover, there is no confusion in selecting the appropriate number of higher order harmonics that has to be included, as the ideal relay feedback test is the initial test and the results of this test is further improved by saturation relay feedback test. Hence, the procedure for the proposed method II is as follows: (1) Perform the ideal relay feedback test. (2) Observe the trend of the system output waveform. (3) If the waveform is triangular, (a) find approximate ultimate controller gain (ku), based on principal harmonics only and (b) take 1.4 as the safety factor (Sf) as suggested by Yu.1 (4) If the waveform is similar to square waveform, (a) find improved ultimate controller gain (ku), based on higher order harmonics (n ) 1, 3, 5) only [3].

ku ) 4h/(πaC)

(5)

where

aC ) y(t*)/(1 - 1/3 + 1/5); t* ) π/(2ωu)

(6)

(b) take 1 as the safety factor (Sf) for conducting the saturation relay test. (5) Using the above selected safety factor and ku, calculate the slope of the saturation relay (k):

slope,

k ) Sfku

(7)

(6) as in the proposed method 1 and (7) as in the proposed method 1 Simulations are carried out for a FOPTD model, with kp ) 1, τ ) 1, and L ranging from 1, 5, and 10 to 30. The simulation results show that the proposed method II has the best accuracy in the estimation of ultimate controller gain (refer to Table 1). Case Study 1 Consider a FOPTD system with kp ) 1, τ ) 1, and L ) 10. The controller is designed based on ultimate gain and frequency of the identified model. The designed PI controller is simulated on the actual system. The closed loop responses of the controllers by the Yu1 method and the present two methods are evaluated on the actual system. The comparison shows that the proposed method II gives a closed loop response closer to that of the actual system. Hence, the proposed method II is recommended. The servo response is found to be closer to that of the actual system (refer to Figure 2). The controller is designed by Ziegler-Nichols method based on ultimate gain and frequency of the identified model. The designed PI controller is simulated on the actual system. The performance of the controller designed on the identified

Figure 2. Comparison of closed loop responses for the FOPTD system (with L/τ ) 10) using the controllers from various methods.

model should be closer to that designed on the actual system. Luyben3 has suggested for L/τ large values (L/τ ) 10), the IMC tuning method gives better results than the Ziegler-Nichols tuning method. For the present example system, IMC tuning method is also used. The controller parameters using the IMC method are given in Table 2. As shown by Luyben,3 we found in the present study that the IMC also gives an improved performance over that of the Ziegler-Nichols for systems with a large delay. After designing the controller by the Yu1 method, proposed method I, proposed method II, and the Luyben3 method, the controllers are evaluated for robustness. A perturbation of 25% in process gain is given. The simulation result shows that all the methods are robust (refer to Table 2). The proposed method II gives results close to that of the actual system. Effect of Noise The effect of noise is also considered. It is observed that, in the presence of noise, the ultimate gain and frequency (and hence the controller settings) are not affected for all three methods. An example system with kp ) τ ) 1.0 and L ) 10 (i.e., L/τ ) 10) is considered. The noise with zero mean Gaussian distribution and a standard deviation of 0.5% is added to the output of the system. The corrupted signal is used in the relay identification test. The amplitude and the period of osculation are identified by taking average at the peak locations. ku and ωu are calculated using the Yu1 method, modified method I, and modified method II. These calculated values of ku, a, and pu are given in Table 3. The ultimate frequency is 0.2938 for all three

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Figure 3. Process output without and with measurement noise. Table 3. Identified ku and pu while Using Noisy Data parameters

Yu method

proposed method I

proposed method II

a pu ωu ku

0.999 21.4443 0.293 1.2034

1.0 21.4443 0.293 1.1263

0.999 21.443 0.293 1.0478

Table 4. PI Controller Settings for SOPTD System with Kp ) 1, τ1 ) 2, τ2 ) 4, and L ) 40a' parameter

theoretical value

Yu method

proposed method I

proposed method II

ku ωu kc τI ISE

1.0465 0.0685 0.4709 76.4378 75.6721

1.203 0.0694 0.5414 75.4465 67.3882

1.1265 0.0692 0.5069 75.6646 70.8979

1.0687 0.0698 0.4809 75.0142 73.5425

a k ) 0.45 K ; τ ) 2π/(1.2ω ); S )1.4 for Yu method; S )1 for c u I u f f the proposed method II.

methods. Figure 3 shows the system output response with and without noise. After introducing the noise, it is observed that the initial dynamics and response are shifted on the time axis. While calculating ku and pu the initial dynamics is not used. Hence, in the presence of noise, the ultimate gain and frequency (and hence the controller settings) are not affected for all the three methods. Case Study 2 Consider a SOPTD system with unit process gain, τ1 ) 2, τ2 ) 4, and L ) 40. The above-described procedure for the FOPTD system is followed for this higher order system (refer to Table 4). The ideal relay test shows a waveform close to that of a square form. Hence, a scaling factor of 1 is considered for the proposed methods. Again, the comparison shows that the proposed method II follows the actual behavior of the system very closely (refer to Figure 4 and Table 4). From these simulation results, it is clear that the modified methods are more accurate than the conventional saturation relay feedback test.

Figure 4. Comparison of closed loop response for SOPTD system (with kp ) 1, τ1 ) 2, τ2 ) 4, and L ) 40) using the controllers from various methods.

Case Studies 3 and 4 Consider a third-order transfer function model with large time delay considered by Luyben3

G(s) ) 0.125 exp(-10s)/(s + 1)3

(8)

Using the proposed method II, the ultimate gain and ultimate frequency are calculated. The method suggested by Srinivasan and Chidambaram2 for identifying the FOPTD model parameters is applied, and the results are compared with that proposed by Luyben,3 A FOPTD model is also analytically derived using the amplitude criterion and phase angle criterion for the actual system and keeping the same process gain (kp). The identified FOPTD model parameters, for the Srinivasan and Chidambaram2 and the Luyben3 method, match well with that of the actual system. After the identification step, the IMC tuning method is used for the PI controller design. The details are given in Table 5. The closed loop response on the actual system is evaluated The servo and regulatory responses are shown in Figures 5 and 6, and the ISE values are given

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Figure 5. Comparison of closed loop servo response using PI controller for case study 3.

Figure 6. Comparison of closed loop regulatory response using PI controller for case study 3.

in Table 5. The controller by the proposed method II gives a response close to that based on the actual system. Another example reported by Lee et al.9 for the coupled tanks system with L/τ ) 5.48 is also considered.

G(s) ) 1.09 exp(-8.5s)/(1.55s + 1)

Table 5. PI Controller Settings Using IMC Method for Case Studies 3 and 4 case study 3

(9)

The actual process parameters and identified parameters using the Luyben3 method and proposed method II are given in Table 5. The identified parameters using proposed method II follows that of the actual system closely. The PI controllers are designed by the IMC method, and the performance on the actual system is evaluated. The servo and regulatory responses are given in Figures 7 and 8, and the ISE values are given Table 5. The controller by the proposed method II gives a response close to that based on the actual system. For case studies 3 and 4, the response shows that the performance of the proposed method II gives close to that of actual system.

4

parameters

actuala

Luyben method

proposed method II

kp τ L λ kc τI ISE kp τ L λ kc τI ISE

0.125 1.78 11.26 19.14 3.097 7.41 14.36 1.09 1.55 8.50 14.45 0.3682 5.8 10.93

0.121 2.98 9.95 16.915 3.8867 7.955 13.58 0.9993 1.8639 7.9023 13.433 0.433 5.815 10.45

0.125 1.85 11.18 19.00 3.132 7.44 14.32 1.0893 1.5438 8.5024 14.455 0.368 5.795 10.93

a Actual FOPTD identified using phase angle criteria and gain criteria (i.e., making kp, ku, and pu of the original model and reduced model same).

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Figure 7. Comparison of closed loop servo response using PI controller for case study 4.

Figure 8. Comparison of closed loop regulatory response using PI controller for case study 4.

Non-relay Identification Methods Now we will consider some of the nonrelay methods used for closed loop identification. Apart from the Ziegler-Nichols continuous cycling method and the Cohen-Coon open loop reaction curve method, there are at least four methods available in the literature for closed loop identification of stable first-order plus time delay transfer models. Yuwana and Seborg10 proposed a simple method that uses the closed loop step response using P controller and the Pade approximation of the dead time element to evaluate the parameters of a firstorder process model. This method does not force the system to marginal stability, and because of the closed loop identification, the model parameters are less sensitive to disturbance and measurement noise occurring during the identification stage. Jutan and Rodriguez11 modified the Yuwana and Seborg10 algorithm by using a higher order approximation for the process model delay. They used a least-squares method to optimize the

parameters in their approximation, which also provided for an analytical solution to the closed loop dynamics. Lee12 matched the dominant poles of the closed loop system with a second-order process response in order to determine the process parameters. This modification enables the method to be used for the process with large dead times. Chen13 has modified the Yuwana and Seborg10 method by determining the process ultimate frequency of oscillation and the controller gain directly from the closed loop system and then used these data to identify a delay plus first-order model. Let us consider a case study 4 example (FOPTD system) with kp ) 1.09, L ) 8.5, and τ ) 1.55. For the present problem using kc ) 0.5, a closed loop response for a unit step change in the set point is shown in the Figure 9. From the response, the values of yp1, yp2, ym1, and ∆t are 0.5427, 0.4012, 0.2548, and 10.3820, respectively. Using these values, the identified model parameter kp, L, and τ are calculated from the relevant

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Figure 9. Typical closed loop under damped response for a step change in the set point using kc ) 0.5.

Figure 10. Comparison of closed loop servo response using the PI controllers from various methods for case study 4. Table 6. Comparison of Identified Model Parameters, the Designed PI Controller Parameters Using IMC Method and IAE Values of the Closed Loop System for Case Study 4 Example params actual kp τ L λ kc τI IAEa a

Lee

Chen

Yuwana- Jutanproposed Seborg Rodriguez method II

1.09 1.085 1.085 1.085 1.55 2.68 4.15 4.1861 8.50 7.18 7.14 7.694 14.45 12.21 12.13 13.079 0.3682 0.473 0.586 0.565 5.8 6.27 7.717 8.032 14.45 13.89 15.72 15.09

1.085 1.47 11.78 20.03 0.338 7.363 18.86

1.0893 1.5438 8.5024 14.455 0.368 5.795 14.44

IAE value calculated up to 100 s with a sampling time 0.01 s.

equations and the results are given in Table 6. Based on identified model parameters, the PI controller is designed using the IMC tuning formulas proposed Luyben3 (refer to Table 6). The designed controllers are simulated on the actual system. As stated earlier, the

performance of the controller designed on the identified model should be closer to that designed on the actual model. The servo response comparison shows that the proposed method II gives a response closer to that of the actual system (refer to Figure 10 for response comparison). Hence, the proposed method II, using saturation relay is recommended. Out of the nonrelay methods, the Lee12 method gives better results. Conclusions The conventional saturation relay method proposed by Yu1 yields about 15-19% error in the estimation of ultimate gain for systems with large L/τ ratios (L/τ > 5). To reduce the error in the estimation of ultimate gain, two modified methods are proposed, focusing on the selection of a proper slope for the saturation relay. When these methods are applied to systems with large L/τ ratios, the corresponding error is about 8-12% and 2-6%, respectively. Case studies for a FOPTD system

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and a higher order system show that the proposed methods provide a better performance than the conventional saturation relay method. The Srinivasan and Chidambaram2 method and the Luyben3 method are used to identify FOPTD model parameters. Based on the IMC method and also on the original system, PI controllers are designed for the identified model. The performance of the proposed method is also compared with nonrelay closed loop methods. The closed loop performances on the actual system show that the Srinivasan and Chidambaram2 method gives performances close to that of the actual system. Nomenclature a ) observed amplitude aC ) corrected amplitude as per eq 6 G(s) ) process open loop transfer function kc ) controller gain kp ) process steady-state gain ku ) ultimate gain IMC ) internal model tuning L ) dead time Sf ) safety factor as in eq 2 τ ) process open loop time constant τI ) controller integral time pu ) ultimate period ωu ) ultimate frequency

Literature Cited (1) Yu, C. C. Auto tuning of PID controllers: Relay feedback approach; Springer-Verlag: Berlin, 1999.

(2) Srinivasan K.; Chidambaram, M. An improved auto-tune identification method, Chem. Biochem. Eng. Q. 2003, 18 (3), 249256. (3) Luyben, W. L. Getting more information from relay-feedback tests, Ind. Eng. Chem. Res. 2001, 40, 4391-4402. (4) Astrom, K. J.; Hagglund, T. Automatic tuning of simple regulators with specification on phase and amplitude margin, Automatica 1984, 20, 645-651. (5) Li, W.; Eskinat, E.; Luyben, W. L. An improved auto-tune identification method, Ind. Eng. Chem. Res. 1991, 30, 1530-1541. (6) Friman, M.; Waller, K. V. A two-channel relay for autotuning, Ind. Eng. Chem. Res. 1997, 36 (7), 2662-2671. (7) Scali, C.; Marchetti, G.; Semino, D. Relay with additional delay for identification and autotuning of completely unknown process, Ind. Eng. Chem. Res. 1999, 38 (5), 1987-1997. (8) Thyagarajan, T.; Yu, C. C. Improved auto tuning using shape factor from relay feedback, Ind. Eng. Chem. Res. 2003, 42, 4425-4440. (9) Lee, T. H.; Wang, Q. G.; Tan, K. K. Robust smith-predictor controller for uncertain delay systems, AIChE J. 1996, 42 (4), 1033-1040. (10) Yuwana, M.; Seborg, D. E. A new method for on-line controller tuning, AIChE J. 1982, 28, 434-440. (11) Jutan, A.; Rodrigue, E. S. Extensions of a new method of on-line controller tuning, Can. J. Chem. Eng. 1984, 62, 802-807. (12) Lee, J. On-line controller tuning from a single closed loop test, AIChE J. 1989, 35, 329-331. (13) Chen C. I. A simple method of on-line identification and controller tuning, AIChE J. 1989, 35, 2037-203.

Received for review August 19, 2004 Revised manuscript received January 13, 2005 Accepted January 14, 2005 IE049242O