A Modified Smith Predictor with a New Structure for Unstable Processes

process model to guarantee stability. Accordingly, the proposed Smith predictor can predict the dynamics of the actual process much better than previo...
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Ind. Eng. Chem. Res. 1999, 38, 405-411

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PROCESS DESIGN AND CONTROL A Modified Smith Predictor with a New Structure for Unstable Processes Hee Jin Kwak, Su Whan Sung, and In-Beum Lee* Department of Chemical Engineering, Automation Research Center, Pohang University of Science and Technology, San 31 Hyoja Dong, Pohang 790-784, Korea

Jin Yong Park† Department of Environmental Science, Hallym University, 1 Okchun Dong, Chunchon 200-702, Korea

In this paper, we propose a modified Smith predictor with a new structure for unstable processes. The proposed new structure makes it possible to use the unstable model itself as a process model, while the previous modified Smith predictors provided no option to use a mismatched stable process model to guarantee stability. Accordingly, the proposed Smith predictor can predict the dynamics of the actual process much better than previous approaches. It also guarantees good robustness to modeling errors by separating servo and regulatory problems. Moreover, it demonstrates good disturbance rejection performances because the adjustable parameters of the PID controller for the regulatory problem are tuned systematically with an internal feedback loop. 1. Introduction Time delays are common phenomena in many industrial processes, and they cause complications in the control-related problems associated with these processes. Conventional controllers, like the proportionalintegral-derivative (PID) controller, are often so ineffective at controlling such processes that a significant amount of detuning is required to maintain closed-loop stability. To overcome this problem, Smith1 proposed a deadtime compensator called a Smith predictor, and it attracted many researchers and process engineers. While the Smith predictor offers potential improvement in the closed-loop performance over conventional controllers,2 it has a defect pointed out by several researchers3-5 in that the classical Smith predictor cannot stabilize unstable processes. To overcome this limitation, De Paor6 developed an integrated design procedure for a modified Smith predictor and associated controller for unstable processes, such as an exothermic irreversible chemical reactor,7 a nonlinear bioreactor,8 and a continuous emulsion polymerization process.9 In the paper, the author described how he changed the unstable process model to a stable one by replacing the unstable pole with a stable one and how he used the changed stable model as a process model of the Smith predictor. After changing the process model, the author defined an auxiliary system to analyze the stability of * To whom all correspondence should be addressed. Email: [email protected]. Phone: 82-562-279-2274. Fax: 82562-279-3499. † E-mail: [email protected]. Phone: 82-361-2401532. Fax: 82-361-256-3420.

the closed-loop system and used root locus topology to guarantee the asymptotic stability of the system. De Paor and Egan10 extended and partially optimized the method proposed by De Paor.6 Their work results in a very significant extension to the range of the process parameter value  ) λθ, where λ and θ denote an unstable pole and time delay, for which the closed-loop asymptotic stability can be achieved. By such extension and optimization, the obtained control results show better performances than the previous one. These two methods will be reviewed in the next section for comparison with the proposed method. Surely, these two researches contributed to extension of the application of the Smith predictor to unstable processes, but they have some drawbacks. For example, because they use graphical approaches and an optimization technique, they cannot be applied in an on-line manner. Also, their methods are very sensitive to plant/ model mismatches when a high gain controller is used for good set-point tracking, because the characteristic equation includes the modeling error term between the process and the model. This sensitivity has been considered as a typical structural limitation of the Smith predictor under the high controller gain. Together with these drawbacks, another shortcoming of these two previous methods is that they use a stable model instead of an unstable model to guarantee the asymptotic stability of the system. Apparently, the stable and unstable process models show different dynamics. So, their Smith predictor cannot guarantee good prediction, even though it is the most important role of the Smith predictor. In this paper, we propose a modified Smith predictor with a new structure for unstable processes to remove the drawbacks of the previous methods. The proposed

10.1021/ie980515n CCC: $18.00 © 1999 American Chemical Society Published on Web 01/14/1999

406 Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999

B* ) B h 1B2

(2)

Gm*(s) ) k1A/B*

(3)

Gm(s) ) Figure 1. Structure of the original Smith predictor.

method does not need any graphical or optimization techniques. Moreover, it can incorporate the dead time effectively without degrading the closed-loop robustness to modeling errors, and it can use the unstable process model directly while guaranteeing a stable closed-loop system, owing to the new structure. Note that because we use a synthesized controller with the inverse model rather than conventional feedback controllers for servo problems, its control results are superior to those of the previous methods. Especially, if we have a perfect model, we can achieve a perfect set-point tracking result. Together with the synthesized controller for setpoint tracking, we use an additional PID controller including an internal feedback loop implicitly to improve the control performance in rejecting effects of modeling errors and disturbances. Here, the internal feedback loop plays an important role in converting the unstable process to a stable one, and then we can obtain almost optimal tuning parameters of the PID controller very simply compared with the previous tuning methods for unstable processes. The proposed Smith predictor shares the same purpose with the 2-degree-of-freedom IMC controller (TDFIMC).11 That is, both approaches intend to achieve good control results for servo as well as regulatory processes with two independently designed controllers. However, they have an essential difference in structure. This difference will be discussed briefly in section 3. 2. Previous Methods In this section, we will review two previously modified Smith predictors for unstable processes. The aim of this section is to clarify the difference between previous approaches and the proposed Smith predictor. For the first time, De Paor6 proposed an integrated design procedure for the Smith predictor to incorporate unstable processes. The author chose the process model in the following form:

Gp(s) )

k1A exp(-sθ) B1B2

(1)

where A and B1B2 are monic polynomials in s of degree n - l and n, respectively. A and B2 are Hurwitz polynomials, B1 ) s - λ denotes an anti-Hurwitz polynomial, and θ is the time delay of the process. It is noticeable that, instead of the unstable process (Gp(s)), the author used the following modified stable process models (Gm(s) and Gm*(s)) as the models of the classical Smith predictor of Figure 1. In the subsequent section, we will discuss the reason why the author used the stable model even though the actual process is unstable.

k1A exp(-sθ) B*

(4)

Here, B h 1 ) s + λ is a Hurwitz polynomial. After replacing the unstable process model of (1) by the stable ones of (3) and (4), the author designed the controller to stabilize the overall system asymptotically while satisfying some other design criteria. The controller includes the inverse of the modified process model (3) as follows:

Gc(s) )

k2B* k2B h 1B2 ) sH hA sH hA

(5)

Subject to (5), the characteristic equation of the Smith predictor of Figure 1 is given as (6).

h 1B2A{[sH h + k1k2]B1 + k1k2[B h1 ∆De Paor(s) ) B B1] exp(-sθ)} (6) The criterion used for choosing H h and k1k2 is that the characteristic equation has to be as stable as possible, in the sense that its rightmost eigenvalue is as deep as possible in the left half-plane. To design a controller satisfying this stability criterion, root locus topology was used. On the basis of the previous paper,6 de Paor and Egan10 proposed another modified Smith predictor for unstable processes. It is, in a sense, only an extension of the previous paper6 so it shares many aspects with the original research.6 However, this paper has two major differences from the previous paper.6 First, the configuration is changed, as shown in Figure 2. Compared with the previous approaches,6 the summarized changes are that the output of the modified Smith predictor has been moved around the loop through the error detector and an additional cascade gain g is introduced. The first change was made to subtract the modified predictor output from the controller input rather than add it to the process output. Then additional error processing could be performed most simply by means of a static gain element. Second, instead of substitution of an unstable pole for a stable one by conversion of the minus sign to a plus, B h 1 is suggested as follows:

B h 1 ) s + γλ, 0 < γ < 1

(7)

where γ is an arbitrary real number subject to the given inequality range. De Paor6 showed that the asymptotic stability of the system could be achieved if the inequality,  < c, were satisfied. Here c is determined depending on the value of l. On the basis of this result, De Paor and Egan10 proved an interesting and meaningful fact: the adequate choice of γ can make c a larger value than given by the previous paper.6 Here γ was chosen to provide the widest range of stabilizability. After obtaining a modified model of the process, they executed root locus analysis exactly parallel to that of De Paor6 (the only difference is in B h 1). In summary, we reviewed two previous papers about a modified Smith predictor for unstable processes. The main point of their methods is that to guarantee the closed-loop stability a stable model should be used

Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 407

Figure 2. System with a modified Smith predictor proposed by De Paor and Egan.10

where us(t) is the synthesized controller output and it can be calculated from (10) by using differential equation solvers like the Runge-Kutta method and several MATLAB subroutines, and so on. Especially, we can estimate the control output directly for the case of m ) 0. The synthesized servo controller, Gcs(s) compensates the time delay and achieves good control performance, as much as desired, by adjusting the transfer function of the desired trajectory, gd(s). In this research, we use the following equation to obtain the desired trajectory.

yd(s) ) Gd(s) ys(s) ) Figure 3. Structure of the proposed Smith predictor.

instead of the unstable process model. With this simple idea they solved the stability problem of the original Smith predictor. However, in a strict sense, the modified Smith predictors with a modified stable model cannot guarantee the advantage of the Smith predictor completely. That is, because the modified stable model is used in place of the unstable process model, the process dynamics cannot be predicted accurately. In the next section, we will propose a modified Smith predictor with a new structure for unstable processes to solve the stability problem as well as to predict the dynamics of the process more accurately. 3. Modified Smith Predictor with a New Structure The structure of the proposed Smith predictor is shown in Figure 3. Here, yd(s) denotes the desired trajectory and Gcs(s) and Gcd(s) are controllers for servo and regulatory problems, respectively. Consider a general high-order process represented by (8). (n)

(n-1)

(1)

f(y (t),y (t),...,y (t),y(t)) ) g(u(m)(t-θ),u(m-1)(t-θ),...,u(1)(t-θ),u(t-θ)) (8) where the process is strictly proper (i.e., n > m) and y(t) and u(t) are the process and controller outputs, respectively, and y(i)(t) and u(i)(t) are ith derivative of y(t) and u(t), respectively. θ is the time delay of the process. Assuming that the inverse of g( ) is stable, if g( ) has nonminimum phase zeros, the inverse will be unstable. In this case, we should separate the minimum and nonminimum phase and then choose the inverse of g( ) as the inverse of the minimum phase part (Morari and Zafirious11). If the desired trajectory (yd(t)) satisfies the conditions

d(i)yd(t) dt(i)

)

d(i)y(t) , 0eteθ dt(i)

and

i ) 1, 2, ..., n (9)

we can obtain perfect nominal control performance for the servo problem by using the following synthesized controller, Gcs(s). (m-1) (t),...,u(1) g(u(m) s (t),us s (t),us(t)) ) (n-1) f(y(n) (t+θ),...,y(1) d (t+θ),yd d (t+θ),yd(t+θ)) (10)

1 ys(s) (τdes + 1)n

(11)

where ys denotes the set point and τde is an adjustable parameter to specify the control performance. If we choose τde as a smaller value, we will achieve better control performances. Until now, we have discussed the use of a synthesized controller to incorporate the servo problem for unstable processes with a long time delay. Here, it should be noted that the synthesized controller does not use the actual process output, so that the stability problem due to modeling error can be simply avoided. If the model is exact and disturbances are zero, y(t) is exactly the same as yd(t) in Figure 3. However, because disturbances and plant/model mismatches always exist in practice, a feedback controller Gcd(s) is needed to compensate them. We choose the following PID controller as Gcd(s).

[

ud(t) ) kc (yd(t) - y(t)) +

∫0t(yd(t) - y(t)) dt +

1 τi

τd

]

d(yd(t) - y(t)) (12) dt

Here, the tuning procedure of the PID controller will be discussed in the next section. Before jumping into the discussion of the PID design, let us investigate what merits the proposed structure of Figure 3 has for the stability and the control performances compared with the previous approaches. The original Smith predictor of Figure 1 has the following characteristic equation:

∆Smith ) 1.0 + Gc(Gp - Gm + Gm*)

(13)

Usually, Gc(s) is a high gain controller because it is designed on the basis of the delay free process model of Gm*(s) to achieve a good set-point tracking performance. As shown in (13), because the plant/model mismatch (Gp - Gm) is fed to the controller, this mismatch is apt to be amplified severely by the high gain of Gc(s). Occasionally, this mismatch results in an unstable closedloop system. So, there are limitations in increasing the controller gain. This fact means that the servo control performance and the robustness are in a trade-off relation. Especially, in the case of an unstable process, this mismatch causes more serious problems. If there is any infinitesimal plant/model mismatch, then Gp Gm + Gm* has two unstable poles of the process and the model so that any Gc(s) cannot stabilize this system. As an ad hoc method, De Paor6 replaced Gm(s) and Gm*(s) by stable models to reduce the number of

408 Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999

Figure 4. Two-degree-of-freedom IMC controller.

Figure 5. Control structure with an internal feedback loop.

unstable poles. However, this change means that the main role of the Smith predictor is discarded for stability. It is notable that we are able to predict the process dynamics for the servo problem based on the process model, but it is impossible to predict the dynamic behavior of unpredictable disturbances. Therefore, conceptually the original Smith predictor cannot contribute to improving disturbance rejection performances. Also, note that the feedback loop to reject disturbances degrades the robustness, as mentioned above. In other words, the strategy to reject disturbance of the original Smith predictor degrades the achievable set-point tracking performance as well as the robustness without any profitable aspects. On the other hand, our approach can solve this dilemma partially by separating the servo and regulatory processes. The proposed method of Figure 3 has the following characteristic equation:

ing errors, similar to the original and previously modified Smith predictors. In this section, we proposed a modified Smith predictor with a new structure. Through this section, the following advantages of the proposed method have been discussed clearly. The proposed method compensates the time delay efficiently without degrading the robustness, because it treats the servo and regulatory problems independently. In particular, the proposed method shows perfect tracking for the servo problem if the model is exact. Moreover, since it uses the unstable process model directly as a process model, the original prediction function of the Smith predictor can be secured, unlike the previous approaches.6,10

∆proposed(s) ) 1 + Gcd(s) Gp(s)

(14)

Here, notice that Gcs(s) does not appear in this equation so we can infer that the servo control performance does not affect the closed-loop stability totally. Therefore, we can increase the gain of Gcs(s) without worrying about the stability problem. Equivalently, we can achieve very good set-point tracking performances without degrading the robustness. Moreover, since Gp(s) has one unstable pole, Gcd(s) can stabilize the system without any modification only if the given process is stabilizable. Note that the original Smith predictor cannot stabilize the system without modifications, as in previous approaches. Many researchers8,9,10,12 have presented the stabilizability conditions for unstable processes. For details, refer to the papers. On the basis of the discussion given up to now, we can conclude that the main characteristic of the proposed Smith predictor is that good control performances for both the set-point tracking and the disturbance rejection can be achieved by separating the two problems structurally. As mentioned in the Introduction section, the proposed control strategy has the same purpose with the TDF-IMC of Figure 4. Here Gcs and Gcd are controllers designed for the set-point tracking and the disturbance rejection, respectively. The effects of the set point (ys) and disturbance (d) on the process output (y) are described by (15).

y)

GcsGp

ys +

1 + Gcd(Gp - Gm)

(1 - GcdGm)Gp

d

1 + Gcd(Gp - Gm)

(15)

Even though the TDF-IMC and the proposed Smith predictor are designed for the same purpose, there is a difference in the implementing methods. As given in (15), because the former uses the process model explicitly, its characteristic equation has the modeling error term Gp - Gm. So, the structure of Figure 4 cannot stabilize unstable processes when there are any model-

4. Tuning of the PID Controller for Regulatory Problems In this section, we propose a tuning rule for the PID controller (Gcd(s)) of Figure 3. The function of this controller is to reject the effects of disturbances and modeling errors. If we want to consider only the stability of the system, we can tune the controller by using the tuning rule proposed by De Paor and O’Malley,13 because it secures the stability of the system. However, as the tuning rule of De Paor and O’Malley13 uses only two-frequency information, it frequently gives poor control performances. So, we suggest a more efficient tuning procedure for the PID controller, considering much more frequency information. It is worth noticing that although acceptable tuning rules for unstable processes are rare, many excellent PID tuning rules have been proposed for stable processes. On the basis of this fact, we would utilize the tuning rules for stable processes to tune the unstable process more efficiently. Consider the control system with an internal feedback loop of Figure 5. Here, Gc(s) is the conventional PID controller to control the overall process (the shaded part). If ki can stabilize the unstable process Gp(s), then the overall process becomes a stable system. Therefore, Gc(s) can be tuned by any tuning rule for stable processes, as we intended, only if ki is tuned appropriately. Here, it is interesting that the control system including the internal feedback loop of Figure 5 can be changed to the conventional control system which does not have the internal feedback loop explicitly for the regulatory problem as follows. The process input of Figure 5 for the regulatory process (ys ) 0) is

[

unet(t) ) kc (-y(t)) +

∫0t(-y(t)) dt + τd

1 τi

]

d(-y(t)) dt ki(-y(t)) (16)

and by simple manipulation, (16) can be rewritten as

Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 409

(17) of the conventional PID controller with parameters of (18)-(20).

[

unet(t) ) kc* (-y(t)) +

∫0t(-y(t)) dt + τd*

1 τi*

]

d(-y(t)) dt (17)

kc* ) kc + ki

(18)

τi* ) τikc*/kc

(19)

τd* ) τdkc/kc*

(20)

Here, kc*, τi*, and τd* are the proportional gain and integral and derivative time constants of the conventional PID controller without the internal feedback loop. That means we can design the control system of Figure 5 efficiently using tuning rules for stable processes, and we can convert the PID controller plus the internal feedback loop of Figure 5 to the PID controller (Gcd(s)) of Figure 3 using (18)-(20). Note that the internal feedback loop is only used implicitly to estimate the parameters of the conventional PID controller of Figure 3, so that we can use a standard PID controller as Gcd(s) like (17). The procedure for determining kc, τi, τd and ki is summarized as follows: First of all, ki of Figure 5 is estimated by the tuning rule proposed by De Paor and O’Malley.13

ki )

1

x|Gp(jωu)||Gp(0)|

x|Gp(0)| GM ) x|Gp(jωu)|

(21)

Goverall(s) )

1 + kiGm(s)

=

5. Simulation Results For the purpose of comparison, we adopt an unstable process (24) from De Paor and Egan.10

Gp(s) )

(22)

km exp(-θms)

) τm2s2 + 2τmζms + 1 Greduced(s) (23)

where km, θm, τm, and ζm denote the static gain, time delay, time constant, and damping factor of the reduced model, respectively. Now, we tune the outer PID controller by using the second-order plus time delay tuning rule.15 The tuning rule was developed by fitting the optimal data sets obtained from the optimization with the integral of the time-weighted absolute value of the error (ITAE) as the objective function. Although it is composed of only several algebraic equations without

e-0.5s s-1

(24)

By the proposed method, the controller output for the servo problem is found as follows for the desired trajectory (26):

us(t) )

where ωu denotes the ultimate frequency of the unstable process. Here, (21) guarantees the optimal gain margin (for details, refer to De Paor and O’Malley13), and GM of (22) denotes the gain margin when ki is determined by (21). Once we determine the ki value, the overall process is defined. In this step, the overall process should be reduced to the second- or first-order plus time delay model, because most tuning rules for the PID controller are based on such low order models. We reduce the overall process to the second-order plus time delay model like (23) using the reduction method proposed by Sung and Lee.14 Here, the reduced second-order plus time delay model can represent various dynamics better than the first-order plus time delay model, and this reduction method is simple and efficient.

Gm(s)

any complicated numerical technique, the control results obtained by this tuning rule are almost the same as those of the optimal tuning results. After estimating all of the tuning parameters of Figure 5, we determine the adjustable parameters of the equivalent conventional PID controller Gcd(s) of Figure 3 using (18)-(20). In this section, we proposed a tuning rule for the PID controller by using the internal feedback loop implicitly to improve the disturbance rejection performance compared with previous unstable tuning rules. Note that because we use the internal feedback loop only implicitly, its usage does not add any complexity to the structure of the proposed method. Because the secondorder plus time delay tuning rule is almost optimal and the model reduction provides sufficiently good accuracy, the PID controller designed by the proposed tuning strategy gives almost optimal performance for a step input disturbance rejection process. We confirmed these aspects through extensive simulations.

d(yd(t + 0.5)) - yd(t + 0.5) dt

(25)

e-0.5s 0.5s + 1

(26)

Gd(s) )

The internal feedback loop gain is estimated by (21).

ki ) 1.5927

(27)

Next, with the model reduction method, the following reduced second-order plus time delay model is obtained.

Goverall(s) )

Gm(s)

= 1 + 1.5927Gm(s) 1.6872 exp(-0.3417s)

(0.6255)2s2 + 2(0.6255)(0.3948)s + 1

) Greduced(s) (28)

Then, the tuning parameters of the PID controller (Gc(s)) of Figure 5 are decided as follows:

kc ) 1.0073, τi ) 0.6906, τd ) 0.6384

(29)

With the tuning results of (27) and (29), we finally determine the parameters of the PID controller of the proposed control structure of Figure 3 for the regulatory problem, Gcd(s), using (18)-(20).

(

Gcd(s) ) 2.6 1.0 +

1.0 + 0.2473s 1.7825s

)

(30)

We compared the proposed method with the method of De Paor and Egan10 of Figure 2 with the following

410 Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999

Figure 6. Control results of two Smith predictors with the nominal process model: (a) set point tracking; (b) disturbance rejection.

design specifications. These specifications are given in the paper of De Paor and Egan.10

g ) 1.5999 Gc(s) )

(31)

14.245(s + 0.3) s

(32)

e-0.5s s + 0.3

(33)

Gm(s) )

Control results of two methods are compared in Figure 6 with a perfect model. Clearly, the proposed method shows superior control performances for both servo and regulatory problems. Here, if we choose a smaller time constant of the desired trajectory, we can achieve a much better set-point tracking performance. For comparison of the robustness of the two methods, we simulated many cases with various levels of uncertainties. Figure 7 shows typical control results for setpoint tracking with parameter uncertainties ((10%) of the time delay. These results confirm that the proposed method is much more robust than the previous method, and it also provides better control performances under the parameter uncertainty because it treats the servo and regulatory problems independently. 6. Conclusions In this paper, we proposed a modified Smith predictor with a new structure for unstable processes. Previously

Figure 7. Control results of two Smith predictors for set-point tracking with (a) +10% and (b) -10% error in time delay of the process.

modified Smith predictors for unstable processes only focus on the stability of the system, and to guarantee the stability, they abandon the original and the most important feature of the Smith predictor. On the other hand, the proposed Smith predictor guarantees the stability of the system, as well as a good prediction capability, owing to the new structure. By using the proposed Smith predictor, we could achieve better control performances for both the set-point tracking and the disturbance rejection cases compared with those of the previous method. Moreover, it shows acceptable robustness and control performances under plant/model mismatches. Acknowledgment This research was supported by The Hallym Academy of Sciences, Hallym University, Korea. Nomenclature A, B2, B h 1 ) Hurwitz polynomials B1, B* ) anti-Hurwitz polynomial and the denominator of the modified stable model d(t) ) disturbance Gp(s), Gm(s), Gm*(s), Gd(s), Goverall(s), Greduced(s) ) transfer function of the actual process, process model, delay free process model, desired trajectory, overall process, and reduced model, respectively

Ind. Eng. Chem. Res., Vol. 38, No. 2, 1999 411 Gc(s), Gcd(s), Gcs(s) ) transfer function of the conventional PID controller and controllers designed for the disturbance rejection and set-point tracking, respectively g ) cascade gain kc, ki, km ) controller gain, internal feedback loop gain, and static gain of the model u(t), u(i)(t), unet(t), us(t) ) controller output and its ith derivative, net process input, and synthesized controller output y(t), y(i)(t), ys(t), yd(t) ) process output and its ith derivative, set point, and desired trajectory Greek Symbols , c ) process parameter value and its maximum value for which the asymptotic stability of the system can be achieved γ ) arbitrary real number λ ) unstable pole θ, θm ) time delay of the process and the model, respectively τi, τd ) integral and derivative time of the PID controller, respectively τde, τm ) time constant of the desired trajectory and the model, respectively ωu ) ultimate frequency of an unstable process ζm ) damping factor of the model ∆De Paor(s), ∆Smith(s), ∆proposed(s) ) characteristic equations of the modified Smith predictor proposed by De Paor,6 the original Smith predictor, and the proposed Smith predictor, respectively

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(3) Furukawa, F.; Shimemura, E. Predictive control for systems with time delay. Int. J. Control 1983, 37, 399. (4) Watanabe, K.; Ito, M. Process-model control for linear systems with delay. IEEE Trans. Autom. Control 1981, 26, 1261. (5) Gawthrop, P. J. Some interpretations of the self-tuning controller. Proc. Inst. Electr. Eng. 1977, 124, 889. (6) De Paor, A. M. A modified Smith predictor and controller for unstable processes with time delay. Int. J. Control 1985, 41, 1025. (7) Luyben, W. L. External versus Internal Open-Loop Unstable Processes. Ind. Eng. Chem. Res. 1998, 37, 2713. (8) Kavdia, M.; Chidambaram, M. On-line Controller tuning for Unstable Systems. Comput. Chem. Eng. 1996, 20, 301. (9) Semino, D. Automatic Tuning of PID Controllers for Unstable Processes. Proc. IFAC Adv. Control Chem. Process. 1994, 321. (10) De Paor, A. M.; Egan, R. P. K. Extension and partial optimization of a modified Smith predictor and controller for unstable processes with time delay. Int. J. Control 1989, 50, 1315. (11) Morari, M.; Zafirious, E. Robust Process Control; PrenticeHall: Englewood Cliffs, NJ, 1989. (12) Poulin, E Ä .; Pomerleau, A. PID tuning for integrating and unstable processes. IEE Proc. Control Theory Appl. 1996, 143, 429. (13) De Paor, A. M.; O’Malley, M. J. Controllers of ZieglerNichols type for unstable process with time delay. Int. J. Control 1989, 49, 1273. (14) Sung, S. W.; Lee, I. Limitations and Countermeasures of PID Controllers. Ind. Eng. Chem. Res. 1996, 35, 2596. (15) Sung, S. W.; O, J.; Lee, J.; Yi, S.; Lee, I. Automatic Tuning of PID Controller using Second-Order Plus Time Delay Model. J. Chem. Eng. Jpn. 1996, 29, 990.

Received for review August 3, 1998 Revised manuscript received November 24, 1998 Accepted November 29, 1998 IE980515N