Autotuning PID Control for Large Time-Delay Processes and Its

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Ind. Eng. Chem. Res. 2002, 41, 4318-4327

Autotuning PID Control for Large Time-Delay Processes and Its Application to Paper Basis Weight Control Wei Tang* and Song-jiao Shi Automation Department, Shanghai Jiaotong University, Shanghai, P.R. China 200030

Meng-xiao Wang Automation Department, Northwest Institute of Light Industry, Shaanxi, P.R. China 712081

The brief arts and crafts of the ordinary fourdrinier are introduced first in this paper. After analysis of the intractable points of paper basis weight (BW) control, an autotuning PID/PI control algorithm is then presented which is capable of controlling large time-delay processes. The algorithm can be described as below. The oscillation amplitude and frequency induced by a modified relay feedback identification method at the point where the Nyquist curve of the process intersects the negative imaginary axis in the S plane are obtained. In terms of the information obtained above, the autotuning PID/PI parameter values are thus tuned according to the specified robust specifications of amplitude and phase margin. The algorithm has many advantages such as simple parameter adjustment, little dependence on the process model, strong robustness, and ease of implementation. It is also very suitable for controlling industrial processes such as the BW loop with large time delay. 1. Introduction Time-delay phenomena often occur in chemical, biological, and metallurgical plants and other process control systems. Time delay is usually one of the main factors that deteriorate the performance of a control system or even cause it to be unstable.1 Compared with a process without time delay, the existence of time delay significantly increases the difficulty to obtain a controller design. Satisfactory system performance cannot be achieved for conventional proportional-integral-derivative (PID) strategies, whose parameters, for example, are tuned by the Ziegler-Nichols (Z-N) method,2 Astrom-Hagglund (A-H) method,3 refined Z-N (RZN) method,4 etc. The Smith predictor5 is a simple but powerful control algorithm for processes with time delay. However, it is sensitive to model mismatch and has poor disturbance rejection capability.6 Some advanced process control algorithms, for example, minimum variance control, adaptive control, etc., can be used for time-delay process control. However, it is necessary to know a lot of accurate information about the real plants in advance. So, satisfactory control results may not be obtained because of modeling error, parameter perturbation, time-varying nonlinearity, or other factors. Relatively speaking, PID control strategies, especially autotuning PID control, still have the dominant position in process control.7,8 Widely used paradigms can also be seen in recent years in the processes with time delay.9 After analysis of the intractable points of paper basis weight (BW) control, an autotuning PID control algorithm is then proposed in this paper. The algorithm has many advantages, for instance, simple parameter adjustment, little dependence on the process model, strong robustness, ease of implementation, and applicability to large time-delay processes. * Corresponding author. E-mail: [email protected]. Tel: +86-21-62932320-88. Fax: +86-21-62932384.

The main contributions of this paper are summarized as follows. The first one is to obtain the dynamic information of large time-delay processes via a modified relay feedback identification method and propose a correction coefficient C90 of the oscillation amplitude valued within 0.81-1.0. The second one is to present a PID design method based on the specifications of amplitude and phase margin that is similar to the A-H method but applicable to large time-delay process control. The last one is to give a successful paradigm of paper BW control by applying the proposed method. Because relatively accurate dynamics of the process with large time delay can be available by the modified relay feedback identification method and the PID/PI controller is tuned in terms of the robust specification of amplitude and phase margin, the proposed method in this paper is strongly robust. It is also worthwhile to mention that the present method is almost independent of the process mathematical model. The paper is organized as below. The brief arts and crafts of the paper-making process and the intractable points of BW control are analyzed in section 2. The acquisition of dynamic characteristics of the BW loop and PID/PI controller design is detailed in sections 3 and 4, respectively. Simulation analysis and real application to BW control are given in sections 5 and 6, respectively. Finally, the conclusion of this paper is drawn in section 7. 2. An Analysis of the Intractable Points of Paper BW Control 2.1. Introduction to Arts and Crafts of the PaperMaking Process. The simplified schematic diagram of an ordinary fourdrinier10 is depicted as in Figure 1. The pulp that contains cellulose fibers and water comes into the paper mill from a pulp-making process. This raw material is blended with broke (paper or stock that has

10.1021/ie0105324 CCC: $22.00 © 2002 American Chemical Society Published on Web 07/25/2002

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4319

Figure 2. Relay feedback based parameter identification.

Figure 1. Schematic diagram of an ordinary fourdrinier.

been rejected and recovered in the paper mill) and chemicals to form a suspension. This suspension is then refined and diluted into a thick stock with a fiber consistency of 3% by a consistency dilution system. The thick stock is further diluted with white water from a wire pit to form a stock with a fiber consistency of about 0.7%. Here, filler is also added. Later in the paper machine the stock is formed into a sheet, pressed, dried, and reeled up on a reeling drum. 2.2. Intractable Points Analysis of BW Control. Paper BW is one of the important specifications of paper quality. The reason it is rather difficult to control BW well is that a lot of factors have an influence on it.10 The intractable points of BW control are described as below: (a) Large time delay inherent in the BW control loop. There is a very long distance between the BW detection point (at the point where the sheet is reeled up on a reeling drum) and the actuator (on the thick stock pipe of the stuff box outlet), which leads to the occurrence of a large time delay. The time delay varies from 1 to 15 min or larger according to the different paper machine speed. (b) The strong couple feather between BW and paper moisture. Generally, the common way to deal with this problem is to decouple them via decoupling algorithms. However, the complicated mechanism of the papermaking process and the modeling uncertainty often make this strategy ineffective. (c) Parameter perturbation and time-varying nonlinearity inherent in the paper-making process. These factors lead to the accurate mathematical model of the paper-making process available. Hence, some advanced process algorithms such as multivariable decoupling control, minimum variance control, and adaptive control are hard to use, which can reduce the control effects to BW. According to the previous analysis, the crux to improving the BW control performance is, therefore, to find some algorithms that are strongly robust, little dependent on the process model, and suitable for controlling the processes with large time delay. 3. Acquisition of Dynamic Characteristics of the BW Loop The merits of the A-H method3 are that the autotuning process is carried out in a closed loop and the whole system is still in normal operation near the operating point, which can not only keep the system working in good order but also overcome the influence on parameter tuning due to nonlinearity of the real

process. Luyben11 and Yu12 have summarized the advantages of the conventional relay feedback test first proposed by Astrom and Hagglund3 very well. The positive features of it are summed up as five points. However, it is insufficient for the identification of processes with large time delay.11 Recently, Luyben11 has done much work on it and successfully got more information from it so that it can be applicable to large time-delay processes of first order or high order, unstable processes, etc. However, much complicated calculation and data processing are needed, and thus it is not suitable for identification online. An autotuning method with a modified relay feedback test, shown as in Figure 2, is employed here to acquire the dynamic characteristics of the BW loop. Compared with the A-H method, an integral element is introduced following the relay element. For dynamic characteristic identification of the process with large time delay, the introduction of an integral element has two main functions as follows: (a) Improving the approximating ability of the describing function to a nonlinear element. The existence of a large time delay leads to the decrease of the real process’s corner frequency, and the oscillation frequency will also decrease consequently. When dynamic characteristic identification experiments are carried out by the modified autotuning method, the output of the closed-loop system (i.e., the input of the relay element) is notably different from a sinusoid wave, which decreases the description precision of the describing function greatly. By means of the attenuation characteristic of an integral element to high-frequency signals, the high harmonics will be attenuated as far as possible when they pass through the integral element. The ratio of the first harmonic will then increase relatively, which improves the approximation of the describing function to nonlinear elements. Hence, accurate dynamics can be available. (b) Correcting the measured oscillation amplitude. The introduction of an integral element leads to the output of an approximate triangular wave in the closed loop when conducting a relay feedback identification experiment. So, the input of a relay is not a sinusoid signal, which also affects the description precision of the describing function. However, the first-harmonic amplitude of the output triangular wave can be easily figured out because of its special waveform, and the value of the first-harmonic amplitude is independent of the oscillation frequency. The resulting value can be used as a theoretical basis to correct the measured oscillation amplitude, which will be detailed in the last paragraph of this section. The controller design procedure proposed in this paper can be illustrated as follows. The limit cycle information of the BW loop at ω90 in Figure 3 induced by a relay feedback experiment is obtained first. Then the PID/PI parameters are tuned in terms of the specifications of amplitude and phase margin, as well as the corrected limit cycle information.

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Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002

N(m)

1 G (j$) ) -1 (j$) p

(2)

where Gp(.) is a controlled object. Substituting eq 1 into eq 2 yields

Gp(j$) ) -

Figure 3. Nyquist curve of the controlled object.

In Figure 2, the negative reciprocal describing function is3

-

1 πm )4d N(m)

(1)

where d is the relay height and m is the amplitude of the relay input (the same as the amplitude of the closedloop output). Now turn the switch in Figure 2 from position b to a; then a stable limit cycle will occur, and we have13

Figure 4. Dynamic responses of Gp1(s).

πm$ 1 j$ ) j 4d N(m)

(3)

Equation 3 means that the oscillation occurs at the point of $90 in Figure 3 and that the oscillation amplitude m is proportional to the relay height d. Therefore, m can be adjusted by d automatically so that the amplitude of the stable limit cycle can be kept within the acceptable limits, which is the most important advantage of the relay feedback based autotuning methods. Generally,3,13 the period of the limit cycle oscillation can easily be determined from the times between zero crossings. The amplitude m can be determined by measuring the peak-to-peak values of the output. These



estimation methods are easy to implement because they are based on counting and comparison only. For large time-delay processes, however, a large number of simulations show that the resulting value of m obtained by the proposed method in this paper often tends to be larger, which must be corrected. Here, let

m ˆ ) mC90

(4)

where C90 is a correction coefficient, whose value will vary slightly for the different plants. It can be valued within 0.81-1.0, where 0.81 is the ratio between the first-harmonic amplitude of the triangular wave and the triangular wave amplitude (the accurate value is 8/π2) and 1.0 corresponds to the triangular wave amplitude. It is suggested to choose C90 ) 0.91 for large time-delay systems and C90 ) 1.0 for little time-delay systems. In fact, the value 0.91 is the relative magnitude of the first-harmonic signal of the closed-loop system output. This result can be obtained via fast Fourier transform (FFT).

4. PID/PI Controller Design The PID controller is given by

(

Gc(s) ) Kp 1 +

)

1 + Tds Ti s

(5)

The design goal is to shift the point $90 in the Nyquist curve of the controlled object to the point of (1/Am)e-j(π-φm) by choosing the PID/PI controller parameters, where Am and φm are specified amplitude margin and phase margin, respectively. Then we have

|Gc(j$90) Gp(j$90)| )

1 Am

∠Gc(j$90) + ∠Gp(j$90) ) -π + φm

(6) (7)

Substituting eqs 3 and 5 into eqs 6 and 7, respectively, gives

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x (

1 + $90Td -

1 $90Ti

)

2

)

4d πm$90AmKp

(8)

1 - $90Td ) cot φm $90Ti

(9)

Ti ) δTd

(10)

exists [1 + (1/$90Ti)2]1/2 ) 4d/πm$90AmKp, where $90Ti ) tan φm. When the previous two equations are solved, the PI controller parameters are

Kp )

Let

where δ is an adjustable parameter, δ ) 1.5-4. From eqs 8-10, the PID controller parameters can be solved, which are

-cot φm + xcot2 φm + 4/δ 4d sin φm , Td ) , Kp ) πm$90Am 2$90 Ti ) δTd (11) For PI controller design, let it be of the form Gc(s) ) Kp[1 + (1/Tis)]. Similarly to the design of PID, there

4d sin φm tan φm , Ti ) πm$90Am $90

(12)

It is noted that the proposed method in this paper is different from the A-H method, though they are similar. As far as PID/PI controller design is concerned, the difference between the two methods is that the point $180 shown in Figure 3 on the Nyquist plot is shifted to the point of 0.5e-j(3π/4) in the A-H method while the point of $90 is shifted to the point of (1/Am)e-j(π-φm) in the proposed method. Here, Am and φm can be specified arbitrarily according to the practical requirements. The slight change seemingly, however, leads to great difference in the performance of large time-delay process control. This will be discussed in detail in section 5.


Figure 7. Setpoint tracking responses of Gp1(s).

5. Simulation Examples 5.1. Robustness Simulation. By far the most commonly used model in process control13 is G31(s) ) [K/(Ts + 1)]e-Ls. There is no exception for the papermaking process.14 However, ref 13 emphasizes that G31(s) is not a representative model and that the conclusions drawn based on it may often be misleading when applied to real processes. One reason for this is that the step response of the model is not S-shaped, while most of the nonintegral real processes have S-shaped step responses. The frequently used low-order nonintegral models with S-shaped step responses are G32(s) ) [K/(Ts + 1)2]e-Ls and G4(s) ) [K/(T1s + 1)(T2s + 1)]e-Ls, and T1 * T2. At the same time, G32(s) and G4(s) can be converted to G(s) ) [1/(as2 + bs + c)]e-Ls. Reference 15 points out that the Nyquist plots of model G(s) fit those of the real processes very well within the frequency range that interests engineers. Therefore, on the basis of this model, the PID/PI controller designed according to the specified amplitude and phase margin can guarantee a real process to have almost the same amplitude and phase margin, respectively, as the specified ones. To verify the proposed PID/PI controller’s performance and robustness, some simulation comparisons based on G31(s), G32(s), and G4(s) are done among the A-H method, the Z-N method, the Smith predictor, and the method proposed in this paper. For the sake of convenience, the PID controller in the Smith predictor is tuned by the Z-N method, because, for it, there are

Table 1. Controller Parametersa controlled controller object param Gp1(s) Gp2(s) Gp3(s)

a

Kp Ti Td Kp Ti Td Kp Ti Td

new PID 0.3660 5.2739 1.3185 0.3699 5.8013 1.4503 0.3613 6.3287 1.5822

new PI

A-H

0.3660 0.4587 6.3662 14.3703 3.5926 0.3699 0.4601 7.0028 16.061 4.0152 0.3613 0.4525 7.6394 17.9053 4.4763

ZN-Smith and ZN-PID 0.732 9.35 2.338 0.726 11 2.75 0.69 11.7 2.925

C90 ) 0.91 in the proposed method.

not good analytical tuning methods commonly accepted and extensively used at present except rules of thumb. Take Gp1(s) ) [1/(2s + 1)]e-8s, Gp2(s) ) [1/(s + 1)(2s + 1)]e-8s, and Gp3(s) ) [1/(s + 1)4]e-8s, respectively, as examples for simulation. Let δ ) 4, Am ) 2, and φm ) π/4, which is consistent with that in the A-H method for the convenience of comparison. The setpoint inputs are unit step signals, and the load-disturbance signals are introduced at t ) 100 s with an amplitude of 50% unit step. The values of PID/PI controller parameters are tabulated in Table 1, and the simulation results are shown in Figures 4-6. In the case of parameter perturbation, the time-delay constants (L) and process gains (K) in all models are increased by 30% and 10%, respectively, and all time constants (T, T1, T2) are decreased by 20%, while PID/PI parameters remain invariable for every model.

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From Figures 4-6, such conclusions can be drawn as follows: (a) Although the Z-N method has a fast response speed, the overshoot is large and the response curve is severely oscillatory. So, it is not suitable for large time-delay process control. (b) Although the A-H method is robust, the response speed is slow and oscillation also occurs. So, it is not fit for controlling processes with large time delay either. (c) The Smith predictor has a good performance for large time-delay processes, while its robustness is relatively poor. (d) The PID/PI controller proposed in this paper is not only strongly robust but also capable of controlling processes with a large time delay. At the same time, its response speed is also superior to that of the Smith predictor whose PID parameters are tuned by the Z-N method. Why does the A-H method not have a good performance for large time-delay processes while the proposed one does? It can be illustrated by the two aspects below. One reason is perhaps due to the improvement of identification precision to large time-delay processes. Simulations show that the identification precision of the conventional relay feedback test will decrease with an increase of the time delay. Some measures should be taken to improve it. In this paper an integrator is inserted into the identification loop and a correction coefficient C90 is introduced to improve the identification precision. This can be seen more clearly from section 5.3. The other reason is attributed to the tuning of PID controller parameters. For the A-H method, the tuning expressions3 are Kp ) 4d cos φm/πmAm, Td ) [tan φm +

(tan2 φm + 4/δ)1/2]/2$180), and Ti ) δTd. Compare them with eq 11, and we can find that the A-H method tends to give a too big Ti and Td for processes with a large time delay, which leads to a sluggish response with inverse response phenomenon in the starting part. These can be seen from Table 1 and Figures 4-6. As a matter of fact, the A-H method was not proposed, aiming at large time-delay processes. Zhuang and Atherton16 and Luyben17 have pointed out that it is not suitable for large time-delay process control. Many simulations done by us demonstrated that it performs very well for processes without large time delay. 5.2. Setpoint Tracking Simulation. It is wellknown that internal model control (IMC)18 is an alternative method applicable to time-delay process control with perfect performance and robustness. Now design controllers according to the conventional IMC design procedure and compare its performance with that of the proposed method. The IMC controllers can be denoted -1 as CI(s) ) Gp(s) F(s) and F(s) ) Kf/(τs + 1)n, where Gp-(s) is the minimum part of the controlled object model and F(s) is a filter. Kf is determined from that F(0) Gp+(0) ) 1, where Gp+(s) denotes the nonminimum part of the controlled object model. The multiplier n must be chosen such that CI(s) is proper. Therefore, τ in F(s) is the only parameter to tune, which has a great influence on the close-loop system response. For the sake of comparison of the setpoint tracking, here τ is tuned in this way such that the closed-loop system has



approximately the same response speed as that regulated by the proposed PID controller. The response speed here is evaluated by the specification of rising time. With respect to the controlled objects of Gp1(s), Gp2(s), and Gp3(s), the corresponding IMC controllers are CI1(s) ) (2s + 1)/(5.5s + 1), CI2(s) ) (s + 1)(2s + 1)/ (4.1s + 1)2, and CI3(s) ) (s + 1)4/(2.6s + 1)4, respectively. The simulation results are shown in Figures 7-9. In the case of parameter perturbation, L and K in all models are increased by 50% and T, T1, and T2 are decreased by 50%, while controller parameters remain invariable for every model. It is shown from Figures 7-9 that both IMC structure and the proposed method have good setpoint tracking and robustness and that the proposed method performs better in the case of parameter perturbation. Luyben discussed an improved IMC structure17 based on the FOPDT model developed by Morari and Zafiriou.19 The IMC controller is of the form of PID, for example, CI(s) ) Kp[1 + 1/Tis + Tds][1/(Tfs + 1)], in which PID parameters are tuned as follows: For the PI controller, λ ) max (1.7L, 0.2T), KpK ) (2T + L)/2λ and Ti ) T + L/2. For the PID controller, λ ) max (0.25L, 0.2T), KpK ) (2T + L)/[2(λ + L)], Ti ) T + L/2, Td ) TL/(2T + L), and Tf ) λL/[2(λ + L)]. Take simulation model Gp1(s) ) [1/(2s + 1)]e-8s as an example for comparisons. The corresponding PI/PID parameters are as follows: For the PI controller (IMC-PI), λ ) 13.6, Kp ) 0.4412, and Ti ) 6.0. For the PID controller (IMCPID), λ ) 2.0, Kp ) 0.6, Ti ) 6.0, Td ) 1.3333, and Tf )

0.8. The simulation results are shown in Figure 10. In the case of parameter perturbation, L and K are increased by 30% and T is decreased by 30%, while controller parameters remain invariable. Conclusions similar to those above can be drawn from Figure 10. In addition, IMC-PID and IMC-PI give faster responses than the proposed method in the nominal case, while a big overshoot occurs for IMC-PID. In the perturbed case, the proposed method demonstrates better robustness. 5.3. Identification Precision Simulation. It is said, in section 4, that the proposed method has better identification precision to large time-delay processes than the A-H method. Here we will verify this by simulation. For the class of model identification method based on relay feedback, the oscillation frequency of the close-loop system depends on the controlled object completely, and it has nothing to do with the relay characteristic. So, an accurate oscillation frequency can be available no matter what method is employed, for example, the A-H method or the proposed method. However, many simulations indicate that the oscillation amplitude is sensitive to the difference of the relay feedback identification methods, which can be reflected by the dynamic gain of the controlled object. Theoretically, the dynamic gain Kth of a process Gp(s) at frequency $ can be calculated by Kth ) |Gp(j$)|. For G31(s), for example, Kth ) K/(1 + $2T2)1/2. From eqs 1-4 and ref 3, we have K90id ) πC90m90$90/4d90 for the proposed method and K180id ) πm180/4d180 for the A-H method, where K90id and K180id are dynamic gains

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Figure 10. Setpoint tracking responses of Gp1(s) under the control of IMC-PID/PI and the proposed PID/PI. Table 2. Theoretical and Identified Dynamic Gains of the Controlled Objectsa proposed method

A-H method

object

oscillation frequency

theoretical gain

identified gain

relative error (%)

oscillation frequency

theoretical gain

identified gain

relative error (%)

Gp1(s) Gp2(s) Gp3(s)

0.1571 0.1428 0.1309

0.9540 0.9519 0.9666

0.9659 0.9558 0.9785

1.24 0.41 1.23

0.3360 0.3006 0.2697

0.8300 0.8207 0.8690

0.7707 0.7684 0.7814

7.14 6.38 10.09

a

C90 ) 0.91 in the proposed method.

of the controlled objects obtained by relay feedback identification methods; m90 and m180 are oscillation amplitudes at frequencies $90 and $180 shown in Figure 3, respectively; and d90 and d180 are the corresponding relay amplitudes. With respect to Gp1(s), Gp2(s), and Gp3(s), the corresponding theoretical and identified dynamic gains are tabulated in Table 2. Such conclusions can be drawn from Table 2 as follows. (a) Compared with the A-H method, oscillation will occur at relatively low frequency for the proposed method and thus the dynamic gain is relatively big. (b) The identification precision of the proposed method to large time-delay processes is better than that of the A-H method, which can be drawn from the relative errors in Table 2. This is one of the reasons that the proposed method has a better performance than that of the A-H method. 6. Real Applications The autotuning PID/PI algorithm proposed in this paper has been imbedded in a distributed control system

series (DCS) of the paper-making process named BMDS. BMDS is a kind of standard DCS used for paper-making process control, which is developed by Microcomputer Application Institute of Northwest Institute of Light Industry in China. It mainly consists of four subsystems: a stock consistency control subsystem, an aircushioned headbox control subsystem, a drying section temperature control subsystem, and a paper BW and moisture control subsystem. This BMDS series has been developed nearly 20 years and has been in operation for several decades in paper mills throughout China. It is still in development and improvement. Figure 11 shows the BWMD (machine direction) curve within 1 h in normal conditions in Bositeng Lake Pulp and Paper Co. Ltd. of Sinkiang Autonomous District in China. The paper machine produces 52 g/m2 news-used paper in long terms, and the quality specification of BW fluctuation range is -3 to +2 g/m2. After the BDMS is in operation, the fluctuation range of BW is restricted to within -1 to +1 g/m2, as shown in Figure 11. So, the

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4327 CI(.) ) internal model control (IMC) controller F(.) ) filter of IMC Kf, τ, n ) parameters of the IMC filter (gain, time constant, and multiplier of the filter, respectively)

Literature Cited

Figure 11. BWMD curve in normal conditions.

product quality was improved greatly and notable economical benefits were produced. 7. Conclusion The autotuning PID/PI algorithm proposed in this paper is simple for parameter adjustment and little dependent on the process model. The PID/PI controller parameters can be autotuned by obtaining the limit cycle information at the point where the Nyquist plot of the process intersects the negative imaginary axis in the S plane only. The algorithm is strongly robust and easy to implement. It is also very suitable for controlling industrial processes such as the BW loop with large time delay. Acknowledgment The paper was supported by two National Key Projects in the Ninth Five-Year Plan of China. The corresponding project numbers are 97-619-02-03 and 97-619-0204. Nomenclature N(.) ) describing function of a relay element d ) height of the relay Gp.(.) ) controlled objects Gp-(.) ) minimum part of the controlled object Gp+(.) ) nonminimum part of the controlled object K, T, T1, T2, a, b, c, L ) parameters of controlled objects (gain, dominant time constant, time delay, etc.) m ) oscillation amplitude before correction m ˆ ) oscillation amplitude after correction C90 ) correction coefficient of the oscillation amplitude $ ) oscillation frequency $90, $180 ) oscillation frequency at the point of 90° and 180°, respectively, in the S plane Gc(.) ) PID controllers Kp, Ti, Td, Tf ) PID parameters (proportional gain and integral, derivative, and filtering time constants, respectively) λ ) closed-loop time constant

(1) Malek-Zavarei, M.; Jamshidi, M. Time-delay System Analysis Optimization and Applications; North-Holland Systems & Control Series: New York, 1987; Vol. 9. (2) Ziegler, J. G.; Nichos, N. B. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759. (3) Astrom, K. J.; Hagglund, T. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica 1984, 20 (5), 645. (4) Hang, C. C.; Astrom, K. J.; Ho, W. K. Refinements of the Ziegler-Nichols tuning formula. IEE Proc. D 1991, 138 (2), 111. (5) Smith, O. J. M. Closer control of loops with dead time. Chem. Eng. Prog. 1957, 53, 217. (6) Hung, H. P.; Chen, C. L.; Chao, Y. C.; Chen, P. L. A modified Smith predictor with an approximate inverse of dead-time. AIChE J. 1990, 36, 1025. (7) Yamamoto, S.; Hashimoto, I. Present status and future needs: The view from Japanese industry. Proceedings of the Fourth International Conference on Chemical Process Control, TX, 1991; p 1. (8) Bialkowski, W. L. Dreams versus reality: A view from both sides of the gap. Pulp Pap. Can. 1993, 94 (11), 19. (9) Tao, Y. H.; Yin, Y. X.; Ge, L. S. Novel PID Control and Its Applications; Mechanical Industry Press: Beijing, 2000 (in Chinese). (10) Qian, C. M.; Liu, H. B. Measurement and Control of Pulp and Paper-Making Processes; Light Industry Press of China: Beijing, 1991 (in Chinese). (11) Luyben, W. L. Getting more information from relayfeedback test. Ind. Eng. Chem. Res. 2001, 40, 4391. (12) Yu, C. C. Autotuning of PID Controllers; Springer-Verlag: London, 1999. (13) Astrom, K. J.; Hagglund, T. PID Controllers: Theory, Design and Tuning, 2nd ed.; Instrument Society of America: Research Triangle Park, NC, 1995. (14) Mo, F. C. Microcomputer applications to pulp and paper making processes. Chin. Pulp Pap. 1984, 5, 54 (in Chinese). (15) Wang, Q. G.; Lee, T. H.; Fung, H. W.; Bi, Q.; Zhang, Y. PID tuning for improved performance. IEEE Trans. Control Syst. Technol. 1999, 7 (7), 457. (16) Zhuang, M.; Atherton, D. P. Automatic tuning of optimum PID controllers. IEE Proc. D 1993, 140 (3), 216. (17) Luyben, W. L. Effect of derivative algorithm and tuning selection on the PID control of dead-time processes. Ind. Eng. Chem. Res. 2001, 40, 3605. (18) Garcia, C. E.; Morari, M. Internal model control,1 a unifying review and some new results. Ind. Eng. Chem. Process Des. Dev. 1982, 21 (2), 308. (19) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: New York, 1989.

Received for review June 25, 2001 Revised manuscript received April 26, 2002 Accepted May 22, 2002 IE0105324

Autotuning PID Control for Large Time-Delay Processes and Its

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