Modified Smith Predictors for Integrating Processes - American

The proposed Smith predictor can accelerate the servo tracking response as much as we want without destabilizing the closed-loop response by separatin...
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Ind. Eng. Chem. Res. 2001, 40, 1500-1506

PROCESS DESIGN AND CONTROL Modified Smith Predictors for Integrating Processes: Comparisons and Proposition Hee Jin Kwak, Su Whan Sung, and In-Beum Lee* Department of Chemical Engineering, Automatic Research Center, Pohang University of Science and Technology, San 31 Hyoja Dong, Pohang, 790-784, Korea

We summarize and systematically analyze the strong and weak points of previous modified Smith predictors for integrating processes, and we improve previous approaches by introducing an enhanced Smith predictor. The proposed Smith predictor can accelerate the servo tracking response as much as we want without destabilizing the closed-loop response by separating the servo and regulatory control problems. Its structure is very compact so that it can realize the good control performances of the previous approaches while retaining simplicity. Moreover, its design strategy can incorporate much more general integrating processes than previous approaches. 1. Introduction predictor1

The Smith is a common and effective deadtime compensator. It guarantees superior control performances for open-loop stable processes with a large time delay if an accurate model is available compared to conventional controllers, like the proportionalintegral-derivative (PID) controller. However, many researchers have pointed out that the Smith predictor cannot be used with its original structure to control processes with an integral mode because a constant load disturbance results in steady-state error.2,3 To overcome this problem, many modified Smith predictors with different structures have been proposed.3-8 The modified Smith predictors proposed until now have been classified into two groups according to the manner in which the modeling error signal originated from the plant/model mismatch or disturbance is treated. The first group3-5 feeds the modeling error signal to the main servo controller so that the servo and regulatory problems cannot be separated. On the other hand, the second group6-8 feeds the modeling error to a regulatory controller designed to reject only modeling error, so the two control problems can be separated. In this paper, for convenience of explanation, we name the first group3-5 Smith predictors with dependent structure [hereafter, dependent Smith predictors (DSPs)] and the second6-8 Smith predictors with independent structure [hereafter, independent Smith predictors (ISPs)]. In this paper, we review the two types of Smith predictors and discuss their strong and weak points. Also, we propose another control structure to overcome weak points of the previous approaches and preserve their strong points. Because the proposed Smith predictor is a kind of the ISP, it can achieve good servo control performances without degrading the closed-loop robust* To whom all correspondence should be addressed. E-mail: [email protected]. Phone: 82-562-279-2274. Fax: 82-562-279-3499.

ness to modeling errors. The robustness property of the ISPs will be explained in detail in section 3. We use a synthesized controller with an inverse model as the servo controller, from which the process output is exactly the same as the given set point if we have an exact process model. Together with the servo controller, we use a regulatory PID controller to improve disturbance rejection performances. Note that the proposed structure is simpler while achieving the same or better performances than previous models. Moreover, the design strategy of the proposed Smith predictor can be applied to more general integrating processes, whereas the previous approaches can incorporate only the pure integrator plus time delay model. This paper is organized as follows: In section 2, we review the modified Smith predictors that have previously been proposed and comment briefly on their features. In addition, we introduce an enhanced Smith predictor and propose a systematic design procedure for the regulatory controller. Section 3 consists of an analysis of the previous approaches and the proposed Smith predictor to clarify advantages and disadvantages of each Smith predictor. Several simulation results will be given to confirm the analyzed results. Finally, conclusions are given in section 4. 2. Previous Approaches and Proposed Smith Predictor 2.1. Dependent Smith Predictors (DSPs). Watanabe and Ito3 proposed the first DSP with the structure given in Figure 1. They used the following G1(s)

G1(s) )

1 1 ‚ 1 + θs s

(1)

where θ is the time delay of the process. The main weak point of their approach is that zero steady-state error

10.1021/ie0004996 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/17/2001

Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001 1501

Figure 1. Modified Smith predictor proposed by Watanabe and Ito.3

Figure 3. Modified Smith predictor proposed by Astro¨m et al.6

Figure 2. Modified Smith predictor proposed by Matausˇek and Mici.4,5

Figure 4. Modified Smith predictor proposed by Tian and Gao.8

can be achieved only when the time delay of the process is exactly known. Otherwise, there is a steady-state error. Moreover, simulation studies have shown that with a PI controller, the set-point and disturbance response are either very oscillatory or highly damped when the process has a large time delay. Matausˇek and Mici4,5 pointed out these disadvantages and tried to overcome them. Matausˇek and Mici4,5 proposed a modified Smith predictor that has an additional controller F(s) to remove the load disturbance effectively with the main servo controller R(s) ) Kr, as shown in Figure 2. The only difference between two approaches4,5 is found in the structure of the additional controller, F(s). Matausˇek and Mici used a proportional (P) controller for F(s) mainly to achieve zero steady-state error in the first paper4 and replaced the P controller by the following lead-lag compensator in their later paper5 for faster disturbance rejection.

F(s) )

Ko(Tds + 1) Tf s + 1

(2)

Simulation results given in the second paper5 show superior disturbance rejection performances compared with their previous approach.4 However, because their design procedure for F(s) is based on the phase margin criterion, their approach cannot guarantee optimality in the time-domain performance criterion, and the design strategy can be applied to only the pure integrator plus time delay model. Also, it is notable that the modeling error is magnified by the servo controller. Therefore, there is a limitation in speeding up the servo control performance by increasing the servo controller gain. This will be discussed again later. 2.2. Independent Smith Predictors (ISPs). Astro¨m et al.6 proposed a modified Smith predictor that decouples the set-point response from the load response, as shown in Figure 3. By separating the two control problems, they make it possible to optimize set-point response and disturbance rejection independently. However, they did not recommend any systematic tuning method for many parameters of M(s). Moreover, they considered only the restricted integrating process G(s) ) e-θs/s. Zhang and Sun7 pointed out the drawbacks of Astro¨m’s approach and tried to provide solutions for them. They extended the previous approach to the more

general (but still restrictive) integrating process G(s) ) e-θs/τs and tried to provide a simple tuning rule for the controller. They used a proportional controller with gain λ1 as the servo controller R(s) and proposed the following disturbance estimator M(s)

M(s) ) M0(s) )

sM0(s) 1 - sM0(s)Gm(s)

βmsm + Λ + β1s +β0 (λ2s + 1)n

(3)

(4)

where both βi (i ) 0, 1, Λ, m) and λ2 are constants. They used two tuning parameters, λ1 and λ2, to optimize the set-point and disturbance responses, respectively. However, they rely on a general guideline to tune these two parameters rather than a systematic approach. Tian and Gao8 proposed another ISP for integrating processes with time delays to improve disturbance rejection performances. This approach has the same control structure as Astro¨m’s Smith predictor except that it uses a local proportional feedback controller for prestabilizing as shown in Figure 4. Here, the use of the additional controller makes the local closed-loop transfer function relate the disturbance d to the process output y as follows

(Kp/s) exp(-θs) Y(s) ) D(s) 1 + (kiKp/s) exp(-θs)

(5)

Therefore, the following Go(s) is needed to eliminate the effect of ki on set-point tracking:

Go(s) ) 1 +

kiKp* exp(-θ*s) s

(6)

As shown in Figure 4, by adding the internal feedback loop, this approach has four controllers, and its structure becomes unnecessarily complex. However, simulation results given in the paper show just slightly better control performances than the Matausˇek and Mici4 scheme despite the added structural complexity. Like other previous approaches, their tuning strategy can be applied to only pure integrator plus time delay models. 2.3. Proposed Smith Predictor. We recommend the structure given in Figure 5 as the modified Smith

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Figure 5. Proposed Smith predictor.

predictor for integrating processes. Here, yd(s) denotes the desired trajectory and Gcs(s) and Gcd(s) are the controllers for the servo and regulatory problems, respectively. It should be noted that, in this structure, the servo and regulatory problems are separated. Because of the separated structure, we can accelerate the set-point tracking as much as we want, regardless of the closed-loop stability. The structure of the proposed Smith predictor was originally proposed to solve problems related to Smith predictors for unstable processes.9 We extend it to integrating processes in this paper and compare with previous approaches. Zhang et al.10 also used a similar concept to control a stable first-order process with time delay more effectively. Note that, because the proposed Smith predictor uses a synthesized controller with the inverse process model as the servo controller and the regulatory controller designed only for disturbance rejection performance, it has the potential capability of achieving maximum servo tracking performance. Consider a general high-order integrating process represented by

f[y(n)(t), y(n-1)(t), Λ, y(1)(t)] ) g[u(m)(t-θ), u(m-1)(t-θ), Λ, u(1)(t-θ), u(t-θ)] (7) where the process is strictly proper (i.e., n > m); y(t) and u(t) are the process and controller outputs, respectively; and y(i)(t) and u(i)(t) are ith derivatives of y(t) and u(t), respectively. θ is the time delay of the process. Also, assume 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 phases 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 following conditions

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

)

d(i)y(t) dt(i)

0 e t e θ and i ) 1, 2, Λ, n (8)

we can obtain perfect nominal control performance for the servo problem by using the synthesized controller Gcs(s)

g[us(m)(t), Λ, us(1)(t), us(t)] ) f[yd(n)(t+θ), yd (n-1)(t+θ), Λ, yd(1)(t+θ)] (9) where us(t) is the synthesized controller output, which can be calculated from eq 9 by using differential equation solvers such as the Runge-Kutta12 method or several MATLAB13 subroutines. In particular, we can estimate the control output directly for the case of m ) 0. The synthesized servo controller Gcs(s) compensates for the time delay and achieves good control performance, as much as desired, by adjusting the transfer function of the desired trajectory, Gd(s).

Figure 6. Control structure with an internal feedback loop.

In this research, we use the following equation to obtain the desired trajectory

yd(s) ) Gd(s) ys(s) )

e-θ*s ys(s) (τdes + 1)n

(10)

where ys denotes the set point, τde is an adjustable parameter for specifying the control performance, θ* is the time delay of the process model, and n is the process order. If we choose τde as a smaller value, we will achieve better control performances. In this paper, we recommend that τde be set equal to 0.5Te, where Te is the sum of the time constants of the process model. Until now, we have discussed a synthesized controller for the servo problem. Here, it should be noted that the synthesized controller does not use the actual process output so that the stability problem due to modeling errors can simply be avoided. Also, if the model is exact and disturbances are zero, y(t) is exactly the same as yd(t). However, because disturbances and plant/model mismatches always exist in practice, a feedback controller Gcd(s) should be used to compensate for them. We choose the following ideal PID controller as Gcd(s):

[

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

1 τi

∫0t (yd(t) - y(t)) dt + τd

]

d(yd(t) - y(t)) (11) dt

Here, if we want to guarantee only the stability of the system, we can tune the controller by using the tuning rule proposed by Ziegler and Nichols14 (ZN tuning rule), because it secures the stability of the system. However, because the ZN tuning rule uses only one-frequency information, it frequently gives poor control performances. Thus, we suggest a more efficient tuning procedure for the PID controller considering much more frequency information. Notice that, although acceptable tuning rules for integrating processes are rare, many excellent PID tuning rules have been proposed for stable processes. On the basis of this fact, we propose a tuning rule for the integrating process that utilizes tuning rules for stable processes. This approach was also used in a previous Smith predictor for unstable processes.9 Consider the control system with an internal feedback loop of Figure 6. Here, Gc(s) is a conventional PID controller to control the overall process (the part boxed by dotted line). If ki stabilizes the integrating process Gp(s), then the overall process becomes a stable one. Therefore, Gc(s) can be tuned by any tuning rule for stable processes, as we intended. It is notable that the conventional PID controller with the internal feedback controller of Figure 6 is equivalent to the conventional PID controller without the internal feedback loop. To verify this, refer to the following. The process input of Figure 6 for the regulatory process (ys ) 0) is

Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001 1503

[

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

1 τi

∫0t (-y(t))dt + τd

]

d(-y(t)) dt ki[y(t)] (12)

and by simple manipulation, eq 12 can be rewritten as eq 13 of the conventional PID controller with parameters of eqs 14-16.

[

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

1 τi*

∫0t (-y(t)) dt + τd*

kc* ) kc + ki

]

d(-y(t)) dt (13) (14)

τi * )

τikc* kc

(15)

τd* )

τdkc kc *

(16)

Here, kc*, τi*, and τd* are, respectively, the proportional gain and integral and derivative times of the conventional PID controller without the internal feedback loop. This means that we can design the control system of Figure 6 efficiently using tuning rules for stable processes and that we can convert the PID controller plus the internal feedback loop of Figure 6 to the PID controller [Gcd(s)] of Figure 5 using eqs 14-16. In this step, it is important to recognize that the internal feedback loop is only used implicitly to estimate the parameters of the conventional PID controller of Figure 5. Thus, we can use a standard PID controller as in eq 13 as Gcd(s). The procedure for determining kc, τi, τd, and ki is summarized as follows. First, ki of Figure 6 is obtained by the modified Ziegler-Nichols tuning rule for integrating processes proposed by Kwak et al.15

ki )

kcu 4

(17)

where kcu denotes the ultimate gain of the integrating process. After determining 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 as in eq 18 using the reduction method proposed by Sung and Lee.16 Here, the reduced secondorder 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.

Goverall(s) )

Gm(s) 1 + kiGm(s)

=

km 2 2

exp(-θms)

) τm s + 2τmζms + 1 Greduced(s) (18)

In eq 18, 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.17 Because this 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)17 as the objective function, despite

its simplicity, 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 6, we determine the adjustable parameters of the equivalent conventional PID controller Gcd(s) of Figure 5 using eqs 14-16. As discussed before, Tian and Gao8 also recommend use of the internal feedback loop controller to improve disturbance rejection performance. However, because they used it explicitly, they have no choice but to make control structure too complex to eliminate its effect from the servo control performance. In contrast, we use the internal feedback loop only implicitly, so its usage does not add any complexity to the structure of the proposed method. Also, note that the proposed tuning strategy can be applied to general integrating processes unlike the previous approaches. 3. Comparison Works In this section, we compare previous Smith predictors with the proposed approach by analyzing them from the theoretical and practical points of view and confirming the analyzed results through a simulation study. 3.1. Robustness. DSPs have a limitation in improving servo control performance maintaining the system stability. This limitation can be understand easily by considering their characteristic equation. All DSPs of Figures 1 and 2 have the following term in their characteristic equation

∆DSPs ) 1.0 + R(s)(Gp - Gm + Gm*) + Λ

(19)

where R(s) is a servo controller. As shown in eq 19, the plant/model mismatch (Gp Gm) is multiplied by the servo controller R(s). In general, R(s) is a high-gain controller because it is designed on the basis of the delay-free process model Gm*(s) to achieve good set-point tracking performance. Therefore, the amplified mismatches can be large enough to make the overall system unstable. In other words, to guarantee the stability of the system, we have no choice but to use a low-gain servo controller despite sacrificing servo control performances. On the other hand, ISPs, including the proposed approach, do not have the amplified mismatch term because they do not feed the mismatch term to the servo controller. For example, consider the characteristic equation of the proposed approach

∆proposed(s) ) 1 + Gcd(s) Gp(s)

(20)

Equation 20 shows that the magnitude of the model mismatch has no relation to the servo system stability because they do not appear in ∆proposed(s). Therefore, if we tune Gcd(s) only to guarantee the stability, we can improve the set-point control performance as much as we want, regardless of the modeling error. This separation is a big advantage of ISPs compared with DSPs. 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 improvements in disturbance rejection performances. Also, note that the feedback loop for rejecting disturbances degrades the model robustness, as mentioned above. In other words, the strategy for rejecting distur-

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Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001 Table 2. ITAE Indexesa Gp1(s)

MM5 ZS7 TG8 proposed

Gp2(s)

nominal case

20% error in delay

nominal case

20% error in delay

23.6 (57.0) 23.6 (105.6) 23.6 (84.4) 15.9 (68.6)

228.7 (798.9) 84.9 (113.3) 41.0 (88.7) 44.4 (68.3)

246.3 (1015.7) 289.7 (2650.7) 233.8 (1182.4) 33.5 (396.7)

221.01 (889.6) UAb 216.3 (1103.1) 83.0 (400.6)

a For set-point ITAE (load ITAE). b UA ) unacceptable with oscillatory responses.

Figure 7. Control results of the previous and proposed Smith predictors for Gp1(s). Dashed-dotted line: MM. Dotted line: ZS. Dashed-dotted-dotted line: TG. Solid line: Proposed (a) nominal case, (b) 20% error in time delay. Table 1. Controller Parameters of the Previous and Proposed Smith Predictors Matausˇek and Mici5

Zhang and Sun7

approach

Kr

Ko

Td

Tf

λ1

λ2

Gp1(s) Gp2(s)

0.6 0.2

0.1448 0.0639

2 4.528

0.2 0.4528

0.6 0.2

4 5.66

Tian and Gao8 approach Kr Gp1(s) Gp2(s)

ki

Kc

proposed Td

τde

Kc

τI

τd

0.6 0.1 0.1758 2.5 0.5 0.2561 10.6419 2.1872 0.2 0.0442 0.0777 5.66 1.0 0.1695 20.0567 6.3400

bances 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 following simulation results will confirm the conclusion of the above discussion. We adopted the process in eq 21 from the previous papers.5,7,8

1 Gp1(s) ) e-5s s

(21)

The parameters of the controllers of the previous and proposed approaches are given in Table 1. The param-

eters of the previous approaches are adopted from those given in the papers. A unit step is introduced at time t ) 0, and a load disturbance d ) -0.1 is introduced at time t ) 70, as in the previous papers.5,7,8 Figure 7a illustrates the control results when the model is exactly the same as the actual process. The control results for the set point of the previous approaches are slow compared with the proposed approach, but all approaches show acceptable performances. However, if the servo controller gain is increased to accelerate the servo tracking and there are parameter errors, each approach shows different behavior. For example, we apply the above controllers with the increased gain (servo controller gain)1.0) to the process G(s) ) e-4s/s (20% time delay error). The control results are given in Figure 7b and Table 2. As shown in Figure 7b, DSP [Matausˇek and Mici (MM)] approach shows an unstable response. On the other hand, ISPs [the Zhang and Sun (ZS), Tian and Gao (TG), and proposed approaches] show a stable response. This means that there is no way to accelerate the set-point response guaranteeing the stability with DSPs when there is model mismatch and that ISPs are more robust than DSPs because of their structure. 3.2. Model Availability. It is worth noticing that all previous approaches can be applied to only pure integrator plus time delay processes. Therefore, the integrating processes with additional dynamics should be reduced to the pure integrator plus time delay process. However, this reduction is reasonable only when the process has only fast additional dynamics, and it can cause serious problems for unmodeled slow dynamics. In particular, because Zhang and Sun7 derived the estimator M(s) for only the pure integrator plus time delay model and did not sufficiently discuss expansion to high-order processes, it is sensitive to structural mismatch of the process model. To show the effect of the unmodeled dynamics, consider the integrating process in eq 22 that has the additional dynamics.

Gp2(s) )

1 e-5s s(2s + 1)(5s + 1)

(22)

As in Matausˇek and Mici,4 the above high-order process is reduced to the pure integrator plus time delay model as follows. Controller parameters are summarized in Table 1.

1 Gm(s) ) e-11.32s s

(23)

Here, the proportional gain Kr of the Matausˇek and Mici5 approach cannot be decided by one step. Therefore, it is decided by a trial-and-error procedure considering the trade-off between performance and stability. Be-

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4. Conclusions

Figure 8. Control results of the previous and proposed Smith predictors for Gp2(s). Dashed-dotted line: MM. Dotted line: ZS. Dashed-dotted-dotted line: TG. Solid line: Proposed.

cause no guideline was recommended for the proportional gain of the other approaches,7,8 they are chosen to have the same value of Kr. A unit step set point is introduced at time t ) 0, and a load disturbance d ) -0.1 is introduced at time t ) 70. The control results for this simulation are given in Figure 8 and Table 2. As shown in Figure 8, the existence of the slow dynamics that were ignored in the reduction step degrades the control performances of the previous approaches for both set-point tracking and disturbance rejection cases. In particular, the control results of Zhang and Sun’s approach are unacceptable, and in this case, the process output diverges. Other simulations also show that Zhang and Sun’s approach is most sensitive to structural mismatch of the process model among the previous approaches. Contrarily, our approach with a PID controller shows a better, stable response. That is, the proposed method can incorporate more general model structures unlike the previous approaches. 3.3. Design Difficulty. Except for two strong points discussed in the previous subsections, the proposed Smith predictor has another merit that the design rule for the controller to reject load disturbance, Gcd(s), is very simple; can guarantee good control performances in the time domain because it is based on time-domain optimization; and can be applied to general high-order integrating processes. On the other hand, both Matausˇek and Mici4,5 and Tian and Gao8 recommended the design procedures for F(s), the internal feedback-loop controller gain ki and Gc(s) on the basis of the phase-margin criterion. However, it is too complex to extend their design methods to more general integrating processes. Also, because these design procedures are based on the phase-margin criterion, their approach cannot guarantee optimality for the time-domain performance criterion. Zhang and Sun7 used λ2 as a design parameter to specify control performances of M(s). They recommended the tuning rule for λ2 as 0.5θ-1.5θ based on a general guideline rather than a systematic tuning rule. Thus, users have difficulty in deciding the value of λ2 within the recommended bound, and the general guideline might not be systematic from the time-domain point of view even though the control performances are highly dependent on the λ2 values. Moreover, it has also limitations in applying to general integrating processes.

In this paper, we classified modified Smith predictors for integrating processes into two groups, namely, dependent Smith predictors (DSPs) and independent Smith predictors (ISPs), based on the manner of treating modeling error signal and analyzed characteristics of their approaches. In addition, we proposed another Smith predictor with an independent structure for integrating processes and a systematic design procedure for controllers. We compare works using the previous and proposed approaches to clarify their strong/weak points and perform several simulations to confirm these results. Through the comparison, we can summarize the advantages of the proposed Smith predictor as follows: (1) Because of the independent structure, the servo controller design is free from the stability problem, unlike the DSPs. (2) The proposed design strategy can be applied to more general integrating processes compared with all previous modified Smith predictors. (3) The design rule for the disturbance rejection controller is simple and can guarantee good control performances in the time domain. Simulation results show that the proposed Smith predictor gives better set-point tracking performance and faster load disturbance rejection regardless of whether there are plant/model mismatches. Acknowledgment This work was supported by the Brain Korea 21 project. Nomenclature d(t), dˆ (t) ) disturbance and estimated disturbance, respectively F(s) ) controller for disturbance rejection in Matausˇek and Mici’s5 approach Gp(s), Gm(s), Gm*(s), Gd(s), Goverall(s), Greduced(s) ) transfer functions of the actual process, process model, delay free process model, desired trajectory, overall process, and reduced model, respectively G0(s), Gc(s), Gcd(s), Gcs(s) ) transfer functions of the controller used by Tian and Gao,8 conventional PID controller, and controllers designed for the disturbance rejection and set-point tracking, respectively M(s) ) disturbance estimator Ko, Kr ) gains of the F(s) and servo controller, respectively Kp, Kp* ) gains of the process and process model, respectively kc, ki, km ) controller, internal feedback loop, and static gains of the model, respectively kcu ) ultimate gain of the process R(s) )servo controller Td, Tf, Te ) time constants of the lead and lag terms of F(s), respectively, and sum of the time constants of the process model, respectively u(t), u(i)(t), unet(t), us(t), ud(t) ) controller output and its ith derivative, net process input, synthesized controller output, and regulatory controller output, respectively y*(t), y(t), y(i)(t), ys(t), yd(t) ) estimated and actual process output and its ith derivative, set point, and desired process output trajectory, respectively Greek Symbols βi ) constant λ1, λ2 ) servo controller gain and time constant of M(s) in Zhang and Sun’s7 approach

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θ, θ*, θm ) time delay of the process, process model, and SOPTD model, respectively τi, τd ) integral and derivative time of the PID controller, respectively τde, τ, τm ) time constants of the desired trajectory, integral model, and model, respectively ζm ) damping factor of the model ∆DSPs(s), ∆proposed(s) ) characteristic equations of the DSPs and the proposed Smith predictor, respectively

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(8) Tian, Y.; Gao, F. Control of Integrator Processes with Dominant Time Delay. Ind. Eng. Chem. Res. 1999, 39, 2979. (9) Kwak, H. J.; Sung, S. W.; Lee, I.; Park, J. Y. A Modified Smith Predictor with a New Structure for Unstable Processes. Ind. Eng. Chem. Res. 1999, 39, 405. (10) Zhang, W. D.; Sun, Y. X.; Xu, X. Two Degee-of-Freedom Smith Predictor for Processes with Time Delay. Automatica 1999, 34, 1279. (11) Morari, M.; Zafirious, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (12) Burden, R. L.; Faires, J. D. Numerical Analysis; PWS Publishing Company: Boston, MA, 1993. (13) Saadat, H. Computational aids in control systems using MATLAB; McGraw-Hill: New York, 1993. (14) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers, Trans. ASME 1942, 64, 759. (15) Kwak, H. J.; Sung, S. W.; Lee, I. On-Line Process Identification and Autotuning for Integrating Processes. Ind. Eng. Chem. Res. 1997, 36, 5329. (16) Sung, S. W.; Lee, I. Limitations and Countermeasures of PID Controllers. Ind. Eng. Chem. Res. 1996, 35, 2596. (17) 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 May 18, 2000 Revised manuscript received September 11, 2000 Accepted January 2, 2001 IE0004996