Popular PID Tuning Rules for Stable

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Ind. Eng. Chem. Res. 2008, 47, 344-368

A Comparative Study of Recent/Popular PID Tuning Rules for Stable, First-Order Plus Dead Time, Single-Input Single-Output Processes M. G. Lin, S. Lakshminarayanan, and G. P. Rangaiah* Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, 4 Engineering DriVe, Singapore 117576

Numerous PID tuning rules exist, and new rules continue to be proposed in the literature. The primary objective of this article is to evaluate comprehensively and compare several recent or popular PID tuning methods based on the integral of absolute error in the controlled variable, total variation of the manipulated variable and maximum sensitivity. Several tuning rules such as internal model control, direct synthesis, and optimizationbased methods were considered and evaluated for a range of first-order plus dead time process dynamics subjected to set-point changes and/or load disturbances via simulation. On the basis of the results, recommendations were then provided for selecting the optimal tuning rule for industrial processes according to the desired control objective (servo versus regulatory) and the process dead time to time constant ratio. 1 Introduction In spite of all of the advances in process control over the past several decades, the proportional, integral and derivative (PID) controller remains the most-dominant form of feedback controller in use today. The transparency of the PID control mechanism, the availability of a large number of reliable and cost-effective commercial PID modules, and their widespread acceptance by operators are among the reasons for its success. Koivo and Tanttu1 estimated that there are perhaps only 5-10% of control loops that cannot be controlled by single-input singleoutput (SISO) PID controllers. According to Desborough and Miller,2 a survey of more than 11 000 controllers in the refining, chemicals, and pulp and paper industries showed that 97% of regulatory controllers had the PID structure. Because the PID controller finds widespread use in the process industries, considerable amounts of research efforts have also been directed toward finding the optimal values for the PID controller’s three parameters: the proportional gain (Kc), the integral time (τi), and the derivative time (τd). In a striking statistic, O’Dwyer3 reported that 293 out of 408 sources of tuning rules have been recorded since 1992, reflecting the upsurge of interest in the PID controller in the past decade. Despite these research efforts, surveys conducted on the state of the art of industrial control practices reported sobering results. For instance, Van Overschee and De Moor4 reported that 80% of PID controllers are badly tuned. About 30% of the PID controllers operate in manual mode. Among the PID controllers that are in auto mode, 25% use default factory settings, implying that they have never been tuned. This and other surveys show that the determination of PID controller parameters is a vexing problem in many industrial applications. The first-order plus dead time (FOPDT) model is a good approximation for many chemical processes that exhibit a monotonic response to step inputs. Experimental data from processes that exhibit higher order and even slightly nonlinear behavior are often fitted with FOPDT models to facilitate controller design.5 Whereas our analysis can include any linear transfer function model (e.g., higher-order models and processes with numerator dynamics) and corresponding controller tuning * To whom correspondence should be addressed. E-mail: chegpr@ nus.edu.sg. Fax: 65-67791936.

Figure 1. Block diagram of a feedback control system.

Figure 2. Internal model control (IMC) structure.

Figure 3. Rearrangement of the IMC structure.

methods, we restrict the treatment to FOPDT models in this article because of the above reasons. There are also many industrially proven techniques now available for deriving empirical models from plant data. The main contribution of this study is that it facilitates the determination of the best tuning rule to compute optimal PID controller settings for stable SISO FOPDT processes under different scenarios. Depending on the objective of the control system (set-point tracking or disturbance rejection) and the different kinds of disturbances (deterministic/stochastic) it is likely to experience, our study aids the identification of appropriate PID tuning parameters. Our work is also fairly comprehensive in the sense that it includes the control of lagdominant, balanced, and delay-dominant stable SISO FOPDT systems. We also consider the severity of manipulated variable action and robustness in addition to output performance.

10.1021/ie0704546 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/09/2008

Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 345 Table 1. Summary of Selected PID Tuning Rules tuning rule Nichols7

basis

proportional gain (Kc)

Ziegler and Rivera et al.9

E IMC

1.2τ/Kθ

Lee et al.10

IMC

τi/K(λ + θ)

Skogestad11 Chen and Seborg12

IMC DS

τ/K(λ + θ)

integral time (τi)

θ τ+ 2 θ K λ+ 2

(

)

(

)(

)

2

(

Sree et al.13 Huang and Jeng14

DS CO

2θ τ + θ/2

θ/2 τθ/(2τ + θ)

τ + θ2/2(λ + θ)

θ2 θ 13τi 2(λ + θ)

min[τ,4(λ + θ)]

0

(

θ θ 3λ + 2 2 3 2 2λ - 3λ θ 1 K 3 θ2 θ 2λ + 3λ2θ + 3λ + 2 2 2τθ +

derivative time (τd)

)(

)

θ θ 3λ + 2 2 3 2 2λ - 3λ θ (2τ + θ)θ

2τθ +

)

λ g 0.1τ λ g 0.8θ

[

2

2

] λ)θ regulatory

τθ θ 3λ + 2 2 2(τ + θ)λ3

3λ2τθ +

(

remarks

(

)

)(

)

θ2 θ 3λ + 2 2 3 2 2λ - 3λ θ

2τθ +

1/K(τ/θ + 0.5) (0.36θ + 0.76τ)/Kθ

τ + θ/2 τ + 0.47θ

0.5θ(τ + 0.1667θ)/(τ + 0.5θ) 0.47τθ/(0.47θ + τ)

(0.8194τ + 0.2773θ)/Kτ0.0262θ0.9738

1.0297τ + 0.3484θ

θ

aθ

series settings ko ) 0.65,a ) 0.4

R6τθ/(θ + R7τ)

regulatory MS ) 1.4

Huang et al.15

CO

koτ/Kθ

τ

Åstro¨m and Ha¨gglund16

CO

(R1θ + R2τ)/Kθ

θ

Syrcos and Kookos17

CO

1 τ 0.31 + 0.6 K θ

(

(

)

(

τ 0.777 + 0.45

Balestrino et al.6 compare three PID tuning rules (derived in 1967, 1981, and 1993) in terms of normalized performance indices for controlling FOPDT models. They consider the normalized integral of absolute error (IAE), the normalized integral of squared error (ISE), and also the variability of control action (normalized). However, they do not consider noise and disturbance effects as we do here. In our work, the following 10 PID tuning rules were selected after a review of the recent and/or popular tuning rules. The Ziegler and Nichols7 tuning method was included for comparison purposes because it was recently shown to provide the best compromise between robustness and performance.8 Because of the popularity of model-based PID tuning methods, a comprehensive study was conducted on tuning methods proposed by Rivera et al.,9 Lee et al.,10 Skogestad,11 and Chen and Seborg12 as they reduce the complexity of the PID tuning problem to the specification of a single adjustable parameter, λ. Recently proposed tuning methods by Sree et al.,13 Huang and Jeng,14 Huang et al.,15 Åstro¨m and Ha¨gglund,16 and Syrcos and Kookos17 were then included for comparison to encompass the latest developments in PID control. 2. Background 2.1. Feedback Control Loop. Figure 1 shows a block diagram of a SISO feedback control system where ysp is the set point, y is the controlled output, u is the controller output/ manipulated input, and e is the error ysp - y. The load disturbance d enters at the process input, and the feedback signal is corrupted by random measurement noise n at the process output. Gp(s) represents the combined dynamics of the stable process, the transmitters, and the final control element and is assumed to be well approximated by a FOPDT transfer function -θs

Gp(s) )

Ke τs + 1

)

R3θ + R4τ θ + R5τ

(1)

θ τ

)

servo θ/τ < 1/3

0.4575τ + 0.0302θ (1.0297τ + 0.3484θ)

[

τ 0.44 - 0.56

θ/τ g 1/3

regulatory θ/τ ∈ [2,5]

(θτ) ] 2.2

Gc(s) is the PID controller transfer function and may be represented as follows:

(

Gcideal(s) ) Kc 1 +

)

1 + τds τis

(2)

Known as the ideal or parallel controller algorithm, eq 2 is unfortunately physically unrealizable in practice because no device could be constructed that truly differentiates an input signal.5 Even if it can be realized, a parallel PID controller has the potential problem of giving a very unsteady high-frequency control signal due to noise in the measurement - such noise is very often present. Consequently, over the years, several approximations to the derivative term have been developed by the vendors. In this article, the PID controller was implemented in the popular parallel with filtered deriVatiVe form:

(

Gc,parallel ) Kc,parallel 1 +

1 τi,parallels

+

τd,parallels Rτd,parallels + 1

)

(3)

In this equation, the derivative term includes a parameter R, known as the derivative filter parameter, which reduces the sensitivity of control calculations to high-frequency measurement noise. Most commercial PID modules have a preset value of R ) 0.10. Another popular implementation of PID control is the series with filtered deriVatiVe form:

(

Gc,series ) Kc,series 1 +

)(

τd,seriess + 1 τi,seriess Rτd,seriess + 1 1

)

(4)

The relation between controller parameters for the parallel form (eq 3) and the series form (eq 4) can be obtained through a term by term matching (for a ) 0) of eqs 3 and 4. This yields eqs 5-7:

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Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008

( ) ( ) ( )

Kcparallel ) Kcseries 1 + τiparallel ) τiseries 1 + τdparallel ) τdseries

1+

τdseries τiseries

τdseries τiseries

1 τdseries

(5)

(6) (7)

τiseries

In the literature, the tuning rules are specified for a particular form of the PID controller - ideal (parallel) form, series (also known as cascade or interacting) form etc. Equations such as 5-7 are useful in converting one form to the other when feasible. Of the methods considered here (also listed in Table 1), only Skogestad11 and Huang et al.15 use the series form; others derive tuning relations are based on the ideal form of the PID controller. In the case of Skogestad,11 we use only the PI controller, and hence there is no difference between the alternate forms of the controller. 2.2. Controller Performance Metrics. Understandably, there are several measures for control-loop performance. Integral error criteria such as IAE, ITAE, and ISE are typically considered (Åstro¨m and Ha¨gglund18) and are quite popular.5 For example, the IAE in the controlled variable,

IAE )

∫0∞ |e(t)|dt

(8)

Whereas the ISE criterion penalizes large errors (resulting in the most-aggressive settings) and the ITAE criterion penalizes persistent errors (resulting in the most-conservative settings), the IAE criterion tends to produce moderate settings that are between those for the ISE and ITAE criteria. Hence, in this study, IAE criterion was selected as one of the metrics for controller performance. As another measure of the controller performance, the required control effort was computed by calculating the total variation (TV) of the manipulated input, u ∞

TV )

|u(i + 1) - ui| ∑ i)0

(9)

The TV of u(t) is the sum of all of the control moves, both up and down. Thus, it is also a good measure of the smoothness of the manipulated input signal. It is desired that TV be as small as possible. For a control system to remain stable and function effectively, it should not be unduly sensitive to small changes in the process or to the inaccuracies in the process model. A control system that satisfies this requirement is said to be robust. On the basis of the block diagram in Figure 1, the sensitivity function S can be defined as

S(jω) )

1 1 + GcGp(jω)

(10)

The amplitude ratio of S(jω) provides a measure of sensitivity of the closed-loop control system to changes in the process. Consequently, the maximum sensitivity MS, defined as the maximum value of |S(jω)| for all of the frequencies, serves as a very useful metric for robustness analysis:

MS ) max |S(jω)| ω

(11)

The Nyquist stability criterion states that a closed-loop system will be stable if the Nyquist plot of the open-loop transfer function GcGp(s) does not encircle the critical point (-1,0). Geometrically, the maximum sensitivity MS is the inverse of the shortest distance from the Nyquist plot of GcGp(s) to the critical point (-1,0). Thus, as MS decreases, the system becomes more robust. For satisfactory robustness, MS should be in the range of 1.2 to 2.5 3. PID Tuning Rules The selected 10 tuning rules as applied to FOPDT processes are briefly described in this section. They can be classified according to the basis for their development, that is, empirical, internal model control, direct synthesis, or constrained optimization, although overlaps exist. The formulas for the selected tuning rules are summarized in Table 1. 3.1. Empirical Methods (E). The tuning method proposed by Ziegler and Nichols7 is based on the open-loop process step response. Because FOPDT processes have a maximum slope of K/τ at t ) θ for a unit-step input change, these same rules can be used to derive a set of PID tuning parameters for a FOPDT process, which gives roughly quarter-decay response.5 3.2 Internal Model Control (IMC). The IMC method is based on an assumed process model, G ˜ p(s) and leads to an analytical expression for the controller settings related directly to the model parameters. The IMC controller G/c (s) has the advantage of taking into consideration model uncertainties and allowing the tradeoffs between system performance and robustness to be considered in a more systematic fashion via the specification of a single adjustable parameter, λ:

G/c (s) )

1 G ˜ p-(s)(λs + 1)r

(12)

where G ˜ p-(s) contains all of the stable left-half plane zeros, r is a positive integer selected such that G/c (s) is proper, and λ denotes the desired closed-loop time constant. Rearranging the IMC system in Figure 2 as in Figure 3, it can be shown that the standard feedback controller Gc(s) in Figure 3 and the IMC controller G/c (s) in Figure 2 are related by

Gc(s) )

G/c (s) 1 - G/c G ˜ p(s)

(13)

Assuming that G ˜ p(s) is a FOPDT transfer function, Rivera et al.9 obtained a set of parallel PID controller settings after introducing a first-order Pade´ approximation for the time-delay term. For sufficient robustness, λ g 0.8θ and λ g 0.1τ are recommended. Expanding Gc(s) in the Maclaurin series in s results in a controller, which contains the proportional, integral, and derivative terms, in addition to an infinite number of higherorder derivative terms. Because it is impossible to fully implement such a controller, Lee et al.10 approximated the parallel PID controller by retaining only the first three terms. For FOPDT processes, Skogestad11 does not recommend the use of derivative action. Hence, by introducing a first-order Taylor series approximation for the time-delay term, a set of parallel PI tuning relations is obtained. It is also well-known that for lag-dominant processes, the choice of τi ) τ results in a long settling time for load disturbances entering at the process

Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 347

Figure 4. Various perturbations studied.

input. For this reason, Skogestad11 requires that τi be computed as the lesser of τ and 4(λ + θ). To achieve a fast response with good robustness, λ ) θ is recommended. 3.3. Direct Synthesis (DS). In the DS method, the controller design is based on an assumed process model and a desired closed-loop transfer function. As such, it is very similar to the IMC approach and can produce identical controllers if the design parameters are specified in a consistent manner. The two selected DS methods are outlined below. Unlike the usual DS methods where the PID controller settings are based on the specification of a closed-loop transfer function for set-point changes, Chen and Seborg12 developed a modified DS method based on disturbance rejection. Consider Figure 1 where it is assumed that the disturbance dynamics is equal to the process dynamics. Thus, in the following development, Gd(s) ) Gp(s); as well, the assumed disturbance dynamics G ˜ d(s) is taken to be equal to the assumed process dynamics G ˜ p(s). Because the closed-loop transfer function for load disturbance (y/d) is not known a priori and the actual process Gp(s) may not be exactly known, it is practical to replace them with the desired closed-loop transfer function for disturbance rejection, (y/d)desired and an assumed process model, G ˜ p(s), respectively. This gives the design equation

Gc(s) )

G ˜ d(s) (y/d)desiredG ˜ p(s)

-

1 G ˜ p(s)

(14)

As stated before, G ˜ p(s) ) G ˜ d(s). Assuming that G ˜ p(s) is well described by a FOPDT model and approximating the time-delay

term by a first-order Pade´ approximation, a set of parallel PID controller settings is obtained. It is apparent from Table 1 that Kc, τi, and τd of Chen and Seborg12 can become negative for large values of λ. Thus, care should be taken in specifying the desired closed-loop time constant λ. Unlike Chen and Seborg,12 the DS method proposed by Sree et al.13 is based on servo responses and does not have an adjustable parameter, λ. The closed-loop transfer function relating set point ysp to the controlled output y is given by

GcGp(s) y ) ) [K1q + K2 + K3q2]e0.5q × ysp 1 + GcGp(s)

[

]

e-q (15) (s + 1)qe0.5q + (K1q + K2 + K3q2)e-0.5q

where K1 ) KcK, K2 ) K1τ/τi, K3 ) K1(τd/τ), ) θ/τ, and q ) sτ. Removing the e-q term in the numerator from further analysis (because it will only shift the corresponding time axis), a first-order Taylor series expansion is introduced to approximate e0.5q and e-0.5q in the denominator such that the order of the numerator becomes equivalent to that of the denominator. Because the objective of the controller is to achieve (y/ysp)desired ) 1, coefficients of powers of q in the numerator are equated with the corresponding coefficients in the denominator to solve for K1, K2, and K3, resulting in a set of parallel PID controller settings. 3.4. Constrained Optimization (CO). Tuning methods developed using constrained optimization are mostly based on

348


Figure 5. Effect of the tuning parameter on the performance of controller tuning methods for a unit-step change in the set point: (a) θ/τ ) 0.1, (b) θ/τ ) 1.0, and (c) θ/τ ) 2.0.

With reference to Figure 1 for a FOPDT process, the openloop transfer function GcGp(s) is

GcGp(s) ) GcidealG ˜ p(s) )

KcK(τiτds2 + τis + 1)e-θs τis(τs + 1)

)

p1(p2s2 + p3s + 1)e-θs s(τs + 1)

(16)

where p1 ) KcK/τi, p2 ) τiτd and p3 ) τi. The optimization problem can then be formulated as a search for the optimal parameters, p1, p2 and p3, such that IAE is minimized. Huang and Jeng14 solved the optimization problem for various values of θ/τ and derived two sets of parallel PID controller relations; one for θ/τ < 1/3 and another for θ/τ g 1/3. For FOPDT processes, Huang and Jeng19 found that the best achievable GcGp(s) such that IAE is minimized, and this can be approximated as

GcGp(s) ≈ Figure 6. Legend for Figures 5, 7-25, and 27-32.

the minimization of certain integral of error criterion subject to desired constraints. A unique advantage of this design approach is that the tuning methods developed are obtained via direct optimization of the selected objective. Such a desired characteristic is not shared by many other classical design methods. In addition, the constrained optimization approach also serves to complement classical model-based design methods, as demonstrated by Huang et al.15 and Åstro¨m and Ha¨gglund.16

ko (1 + aθs) -θs e θ (1 + µθs)s

(17)

where the denominator transfer function (1 + µθs) is implemented to make the PID controller realizable; ko and a denote the model parameters returned from the minimization search. On the basis of the above findings, Huang et al.15 further developed correlations between ko and a with gain and phase margins and proposed a model-based design method that results in a series with a filtered derivative PID controller with two adjustable parameters ko and a, which are closely related to robustness and performance of the closed-loop control system. The default values of ko ) 0.65 and a ) 0.4 for a gain margin of 2.7 and a phase margin of 65° are recommended.


Figure 7. Responses of controller tunings with similar IAE values for the set-point change and θ/τ ) 0.1: (a) Rivera et al.9 (IAE ) 2.0) and (b) Skogestad11 (IAE ) 3.0).

Figure 8. Performance of controller tuning methods for a unit-step change in the set point: θ/τ ) 0.1.

Panagopoulos et al.20 proposed a tuning method based on the optimization of load disturbance rejection with constraints on robustness to model uncertainties. For processes well described by a FOPDT model, Åstro¨m and Ha¨gglund16 derived a set of

generalized parallel PID controller settings by varying the specified design parameter MS. For satisfactory performance with excellent robustness, MS ) 1.4 is recommended. For this method, the parameters of the ideal parallel PID algorithm are

350


Figure 9. Performance of controller tuning methods for a unit-step change in the set point and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 0.1.

Figure 10. Performance of controller tuning methods for a unit-step change in the set point: θ/τ ) 1.

provided in terms of parameters R1 to R7 (Table 1). The Ri values for various values of maximum sensitivity (MS) are tabulated by Åstro¨m and Ha¨gglund.16 For example, if MS ) 1.4 is desired, they are given by: R1 ) 0.168, R2 ) 0.460, R3 ) 0.363, R4 ) 0.871, R5 ) 0.111, R6 ) 1.70, and R7 ) 4.37.

Syrcos and Kookos17 formulated a nonlinear programming problem for a PID controller design solely in the time domain. With no approximations introduced and solved using direct optimization for a number of delay-dominant FOPDT processes with θ/τ ∈ [2,5], subjected to a unit-step load disturbance d, a


Figure 11. Performance of controller tuning methods for a unit-step change in the set point and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 1.

Figure 12. Performance of controller tuning methods for a unit-step change in the set point: θ/τ ) 2.

set of parallel PID controller settings for delay-dominant processes were derived. 4. Simulation and Comparison Strategy For this study, the following stable FOPDT process model was used:

Gp(s) )

1 e-θs 10s + 1

(18)

Furthermore, the PID controller was implemented in the widely popular parallel with filtered deriVatiVe form with the derivative filter parameter specified as R ) 0.1:

(

Gcparallel(s) ) Kc 1 +

τds 1 + τis τd s+1 10

)

(19)

To provide a holistic comparison of the various tuning methods, the closed-loop system is subjected to both set-point

352


Figure 13. Performance of controller tuning methods for a unit-step change in the set point and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 2. Table 2. Summary of Recommended Tuning Rules in the Order of Preference for Servo Control lag-dominant (θ/τ ) 0.1)

delay-significant (θ/τ ) 1 and 2)

step

1. Huang and Jeng14 2. Huang et al.15 3. Lee et al.10

1. Huang and Jeng14 2. Lee et al.10

step (noisy)

1. Skogestad11 2. Huang et al.15 3. Lee et al.10

1. Skogestad11 2. Huang and Jeng14 3. Lee et al.10

inputs

changes and load disturbances: (a) for the servo problem, the set point ysp is modeled as a unit-step input in the absence of measurement noise, n, and is then repeated in the presence of measurement noise with mean, nj ) 0, and variance, σ2n ) 0.052; and (b) for the regulatory problem, the load disturbance d is first modeled as a unit-step input, followed by a random input with mean dh ) 0 and variance σ2d ) 1; the latter is also known as a stochastic load disturbance. Similar to the servo problem, the regulatory problem is first studied in the absence of measurement noise n and then repeated in the presence of measurement noise with mean nj ) 0 and variance σ2n ) 0.052 (Figure 4). All of the simulations carried out in this study can be categorized into two distinct types. Type 1. To explore the limits of model-based tuning methods with a single adjustable parameter λ, the recommended values for λ in Section 3 were not utilized. Instead, λ was adjusted incrementally from the minimum value of λ that gives closedloop system stability. On the other hand, the upper limit for λ is of secondary importance because the closed-loop system tends toward better robustness as λ increases. The same approach is applied the tuning method of Åstro¨m and Ha¨gglund,16 where the adjustable parameter is MS. To cover the various FOPDT processes, the methodology was repeated for three different θ/τ ratios: θ/τ ) 0.1 is representative of a lag-dominant process,

θ/τ ) 1.0 is representative of a balanced process, and θ/τ ) 2.0 is representative of a delay-dominant process. Type 2. For FOPDT processes, an important factor that affects the performance and robustness of the closed-loop system is the θ/τ ratio. Termed as the controllability index by Deshpande and Ash,21 the θ/τ ratio can be interpreted to represent the real or apparent time delay of the closed-loop system.22 To explore the effects of the θ/τ ratio on the various tuning methods, the time delay θ was varied from 1 to 20 such that θ/τ ∈ [0.1,2]. To facilitate the simulations and comparisons described above, an integrated MATLAB/SIMULINK program was developed; this code can be made available upon request. For the ease of crossreferencing, all of the simulations were carried out for 200 units of time using the Dormand-Prince (4,5) variable step algorithm as the ordinary differential equation solver. 5. Results and Discussion Using the comparison strategy described in Section 4, the results are presented separately for the two problems: servo control and regulatory control. Nonetheless, before these two problems are addressed, it is useful to discuss the curves observed for the model-based tuning methods as a result of varying λ. 5.1. The Nature of Model-Based Tuning Methods. Figure 5 illustrates the variation in the performance of the selected controller tuning methods having λ or MS as an adjustable parameter, for the servo control of three representative processes: lag-dominant (θ/τ ) 0.1), balanced (θ/τ ) 1), and delaydominant (θ/τ ) 2). For the model-based methods with a single adjustable parameter λ, the marker x denotes λ f 0 and the marker O denotes the recommended value for λ, if any. The legend for all of the plots in this article, unless otherwise specified, is as shown in Figure 6.


Figure 14. Performance of controller tuning methods for a unit-step load disturbance: θ/τ ) 0.1.

Figure 15. Performance of controller tuning methods for a unit-step load disturbance and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 0.1.

The curves shown in Figure 5 have the following implications. (i) For each tuning method with a single adjustable parameter, there exists an optimum value of λ for which a minimum IAE value can be achieved. However, this does not mean that the minimum achievable IAE value for each tuning method is the same. Hence, if satisfactory output performance cannot be

achieved by a particular tuning method, an alternative tuning may be tried because a more aggressive setting (smaller λ) will only cause the output performance to deteriorate. (ii) Further, as illustrated in Figures 5 and 7, it is possible to achieve a desired level of output performance with two different values of λ, such that both settings give a stable closed-loop

354


Figure 16. Performance of controller tuning methods for stochastic disturbances (dh ) 0, σ2d ) 1): θ/τ ) 0.1.

Figure 17. Performance of controller tuning methods for stochastic disturbances (dh ) 0, σ2d ) 1) and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 0.1.

response with similar IAE values. Consequently, the selection of an appropriate adjustable parameter should be based on an output performance related metric (i.e., IAE, ISE) and at least one non-output performance related metric, which could be a control effort metric (i.e., TV) or a robustness metric (i.e., MS).

(iii) Whereas it is safe to conclude that increasing λ will lead to a more-robust control system, the conclusion that overly decreasing λ will lead to an unsatisfactory performance is subjective. Figure 7 shows that decreasing λ will lead to a faster and more oscillatory response; because the controller gain Kc


Figure 18. Performance of controller tuning methods for a unit-step load disturbance: θ/τ ) 1.

Figure 19. Performance of controller tuning methods for a unit-step load disturbance and corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 1.

increases as λ decreases, this translates to greater control action and, consequently, a greater amount of control effort required. If rapid responses are desired, λ can be decreased at the expense of robustness and greater control effort. 5.2. Servo Control. Because TV and MS are plotted on the y axis against IAE on the x axis and minimal values of IAE, TV,

and MS are desirable properties, the optimal tuning rule/s can be located in the lower-left portion of the individual plots. For the tuning method of Huang et al.,15 default values of ko ) 0.65 and a ) 0.4 were used in this study. 5.2.1. Servo Control of Lag-Dominant FOPDT Processes. From Figure 8, it is apparent that Rivera et al.,9 Lee et al.,10

356


Figure 20. Performance of controller tuning methods for stochastic disturbances (dh ) 0, σ2d ) 1): θ/τ )1.

Figure 21. Performance of controller tuning methods for stochastic disturbances (dh ) 0, σ2d ) 1) and corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 1.

Huang and Jeng,14 and Huang et al.15 all provide excellent setpoint responses with sufficient robustness for the control of lagdominant processes in the absence of measurement noise. This is expected for Huang and Jeng14 because the tuning method is based on the global minimization of IAE for a unit-step change

in set point. For more-robust tuning, Huang et al.15 or Lee et al.10 with larger λ values can be considered as alternatives. If the control effort required is a priority in the selection of an optimal tuning method, Skogestad11 can be considered at the expense of achievable maximum output performance.


Figure 22. Performance of controller tuning methods for a unit-step load disturbance: θ/τ ) 2.

Figure 23. Performance of controller tuning methods for a unit-step load disturbance and corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 2.

However, this is discouraged, given that the minimum IAE / ≈ 2.5) is about 1.8 times achievable for Skogestad11 (JSkogestad / ≈ 1.4) (Figure 8). that of Huang and Jeng14 (JHuang&Jeng Because the use of derivative action improves the responsivenessof the control system, thus improving the minimum achiev-

able IAE, more-strenuous control action would be required and hence a greater TV compared to a PI controller. Performance presented in Figure 9 resulted when random measurement noise n with mean nj ) 0 and variance σ2n ) 0.052 was introduced into the process output. In terms of optimal

358


Figure 24. Performance of controller tuning methods for stochastic disturbances (dh ) 0, σ2d ) 1): θ/τ ) 2.

Figure 25. Performance of controller tuning methods for stochastic disturbances (dh ) 0, σ2d ) 1) and corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 2.

output performance with minimum control effort and maximum robustness, it is evident that Skogestad11 is the superior tuning method for the servo control of noisy lag-dominant processes, / where the difference between JSkogestad (∼10.5) and / JLee(∼10.1) is only about 4%, but the control effort required by Lee et al.10 is about 18 times that required by Skogestad.11 The contrast of results between Figures 8 and 9 shows that, for the servo control of lag-dominant processes, derivative action

should not be implemented if the system is corrupted by substantial measurement noise. 5.2.2. Servo Control of Balanced and Delay-Dominant FOPDT Processes. For the servo control of balanced and delaydominant processes, collectively known as delay-significant processes, similar conclusions can be drawn for both the noiseless and noisy cases. From Figures 10-13, it is apparent that the method of Huang and Jeng14 provides the best servo


Figure 26. Schematic of a typical industrial fired heater. Table 3. Summary of Recommended Tuning Rules in the Order of Preference for Regulatory Control inputs step step (noisy)

lag-dominant (θ/τ ) 0.1)

delay-dominant (θ/τ ) 2)

1. Åstro¨m and Ha¨gglund16

1. Åstro¨m and Ha¨gglund16

1. Lee et al.10

1. Skogestad11 2. Åstro¨m and Ha¨gglund16

1. Skogestad11 2. Åstro¨m and Ha¨gglund16

1. Skogestad11 2. Lee et al.10

1. Rivera et al.9

1. Åstro¨m and Ha¨gglund16 2. Lee et al.10

1. Rivera et al.9

1. Skogestad11 2. Åstro¨m and Ha¨gglund16 3. Lee et al.10

1. Skogestad11 2. Rivera et al.9

random 2. Huang et al.15 random (noisy)

balanced (θ/τ ) 1)

1. Skogestad11 2. Rivera et al.9 3. Huang et al.15

Table 4. Optimal Tuning Rules Identified for the Fired Heater servo control

step step (noisy)

Huang and Jeng14 Skogestad11

regulatory control

step step (noisy) random random (noisy)

Åstro¨m and Ha¨gglund16 Skogestad11 Rivera et al.9 Skogestad11

responses for processes with significant apparent time delays (θ/τ g 1), regardless of the effects of measurement noise. For a more-robust system, Lee et al.10 can be considered as it leads to relatively lower values of TV and MS at the expense of slightly higher IAE values. Although the use of derivative action improves the servo response in terms of the minimum IAE achievable for delaysignificant processes, its use for noisy processes is not necessarily warranted. From Figure 13, the minimum achievable IAE / by Skogestad11 (JSkogestad ≈ 45) is about 30% more than that / ≈ 32), whereas the control by Huang and Jeng14 (JHuang&Jeng effort required by the latter is about 25 times more than that required by the former. Hence, for noisy delay-significant processes, it is recommended that Skogestad11 be used first to determine whether satisfactory output performance can be attained. If not, Huang and Jeng14 or Lee et al.10 can be considered to improve the output performance. The inferior minimum IAE achievable using a Skogestad11 PI tuning algorithm compared to PID tuning algorithms is indicative of the benefits of derivative action on systems with significant apparent time delays. It is also noteworthy that although Sree et al.13 was designed for set-point tracking whereas Chen and Seborg12 was designed for disturbance rejection; the latter outperforms the former in the servo control of delay-significant processes. Recommended tuning rules for servo control are summarized in Table 2.

5.3. Regulatory Control. In this section, the performance of the various controller tuning methods for regulatory control is first evaluated for a unit-step load disturbance first in the absence and later in the presence of measurement noise. Because random load disturbances are frequently encountered in the process industries, the load disturbance input is subsequently remodeled as a stochastic input with a mean dh ) 0 and a variance σ2d ) 1, and the evaluation was repeated. 5.3.1. Regulatory Control of Lag-Dominant FOPDT Processes. From Figure 14, it is apparent that for sufficiently robust controller tuning (MS < 2), Åstro¨m and Ha¨gglund16 followed by Chen and Seborg,12 are the superior choices for the regulatory control of lag-dominant processes in the absence of measurement noise. This can be expected because both tuning methods were designed for disturbance rejection. Consequently, one would also expect Åstro¨m and Ha¨gglund16 and Chen and Seborg12 to give favorable results when applied to the regulatory control of balanced and delay-dominant processes. Although Ziegler and Nichols7 delivers the best achievable output performance, it suffers from poor robustness (MS > 4) and thus, is not highly recommended. If the use of Ziegler and Nichols7 is necessary, the process model has to be sufficiently accurate and invariant in nature. When random measurement noise n with mean nj ) 0 and variance σ2n ) 0.052 was introduced into the lag-dominant process output; the performance of various controllers is as shown in Figure 15. Evidently, Skogestad11 exhibits superior output performance with a minimum control effort required and good robustness (MS ≈ 2) for the regulatory control of noisy lag-dominant processes. A comparative study of Rivera et al.,9 Lee et al.,10 Chen and Seborg,12 and Skogestad11 for the regulatory control of FOPDT processes has also been conducted by Foley et al.23 Although the input disturbance was modeled as a random walk and different performance metrics were used, the relative performances of the four tuning rules were essentially similar to the results obtained in Figures 14 and 15. For the regulatory control of stochastically disturbed lagdominant processes not corrupted by measurement noise, Åstro¨m and Ha¨gglund16 and Chen and Seborg12 cease to be the optimal tuning rules (Figure 16). Instead, with a maximum output performance corresponding to the minimum IAE achievable and robustness of MS ≈ 1.8, Rivera et al.9 is recommended for the regulatory control of stochastically disturbed lagdominant processes in the absence of measurement noise. Alternatives such as Huang et al.15 and Huang and Jeng14 can also be considered. Figure 17 shows that Skogestad11 continues to be the preferred tuning rule for the control of noisy lag-dominant processes despite the random nature of the load input. Nonetheless, if the control effort required is not an important consideration, Lee et al.10 can be considered as the alternative tuning rule, though the improvement in output performance is only about 4%MS < 2 / (IAE Skoges t ad ≈ 8.8, JLee ≈ 8.5), whereas the increase in control effort required is about 10 times. Similar conclusions that have been previously drawn for the servo problem verify the disincentive in the use of derivative action for noisy lagdominant processes. 5.3.2. Regulatory Control of Balanced FOPDT Processes. For the regulatory control of noise-free balanced processes, Chen and Seborg12 exhibits superior performance in dealing with both step and stochastic load disturbances (Figures 18 and 20). This was expected because Chen and Seborg12 was designed specifically for the purpose of disturbance rejection. Nevertheless, at optimal IAE values, the tuning rule is mostly nonrobust, that

360


Figure 27. Performance of controller tuning methods for a unit-step change in the set point: θ/τ ) 0.262.

Figure 28. Performance of controller tuning methods for a unit-step change in the set point and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 0.262.

is, when J* ≈ 10.3, MS ≈ 3.0 in Figure 18 and when J* ≈ 7.2, MS ≈ 2.4 in Figure 20. Hence, Åstro¨m and Ha¨gglund16 should be considered as the default tuning rule for a necessarily robust controller (MS < 2), exhibiting relative superior output performance. For the regulatory control of noisy balanced processes (Figures 19 and 21), as the apparent time-delay increases, Skogestad11 is no longer the superior tuning rule because of

the deteriorating minimum IAE achievable. Åstro¨m and Ha¨gglund16 can be considered as the alternative tuning rule for robust and improved output performance if the control effort required is of secondary importance. 5.3.3. Regulatory Control of Delay-Dominant FOPDT Processes. For the regulatory control of noise-free delaydominant processes subjected to unit-step load disturbances


Figure 29. Performance of controller tuning methods for a unit-step load disturbance: θ/τ ) 0.262.

Figure 30. Performance of controller tuning methods for a unit-step load disturbance and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 0.262.

(Figure 22), Lee et al.10 exhibits both superior performance and robustness, whereas for the regulatory control of noise-free delay-dominant processes subjected to stochastic load disturbances (Figure 24), Ziegler and Nichols7 outperforms all of the recent tuning methods in terms of maximum achievable IAE

but is insufficiently robust (MS ≈ 2.5). For a sufficiently robust controller tuning, Rivera et al.9 should be considered instead. Similar to previous conclusions on lag-dominant and balanced noisy processes, it is recommended that Skogestad11 be used first to determine whether satisfactory output performance can

362


Figure 31. Performance of controller tuning methods for a stochastic disturbance (dh ) 0, σ2d ) 1): θ/τ ) 0.262.

Figure 32. Performance of controller tuning methods for a stochastic disturbance (dh ) 0, σ2d ) 1) and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052): θ/τ ) 0.262.

be attained before considering Lee et al.10 for unit-step load disturbances or Rivera et al.9 for stochastic load disturbances (Figures 23 and 25). The relative performances of Rivera et al.,9 Lee et al.,10 Chen and Seborg,12 and Skogestad11 in Figures

22 and 23 were also similar to the results obtained in the comparative study conducted by Foley et al.23 Recommended tuning rules for regulatory control under various situations are summarized in Table 3. A comparison of


Figure 33. Legend for the subsequent figures in this section (Figures 34-40).

Figure 34. Robustness of controller tuning methods as θ/τ increases.

PI tuning rules conducted by Foley et al.24 indicates thatSkogestad11 is a well-rounded tuning rule for the control of lagdominant systems. 5.4. Case Study: Fired Heater. This section serves to verify the validity of recommendations in Tables 2 and 3 by using them to predict the optimal tuning rules for the feedback control of a typical industrial process: a fired heater shown in Figure 26 where the temperature of the feed outlet Tout is the controlled output variable and the fuel flow rate FFuel is the manipulate-

dinlet variable. The following transfer function has been reported by Marlin25 for this process

Gp(s) )

Tout(s) FFuel(s)

)

0.1e-1.1s 4.2s + 1

(20)

With the θ/τ known, the optimal tuning rules can be identified using the recommended tuning rules for the servo and regulatory controls listed in Tables 2 and 3. Because θ/τ ) 0.262 and is

364


Figure 35. Performance of controller tuning methods for a unit-step change in the set point as θ/τ increases.

Figure 36. Performance of controller tuning methods for a unit-step load disturbance as θ/τ increases.

,1, the optimal tuning rule with respect to each of the six perturbation types is identified by considering the recommended tuning rules for the lag-dominant processes, and these are summarized in Table 4. Figures 27-32 illustrate the actual simulation results obtained as outlined in Section 4. The optimal tuning rules identified

based on recommendations were congruent with the results obtained. Because the identified optimal tuning rules turn out to be the best based on the above comparisons, the determination of an optimal PID tuning rule for the control of a FOPDT process can be achieved using Tables 2 and 3 and the θ/τ ratio of the process.


Figure 37. Performance of controller tuning methods for a stochastic load disturbance (dh ) 0, σ2d ) 1) as θ/τ increases.

Figure 38. Performance of controller tuning methods for a unit-step change in the set point and output, corrupted by measurement noise (nj ) 0, σ2n ) 0.052), as θ/τ increases.

5.5. Effects of the Controllability Index. To investigate the effects of the controllability index (i.e., the ratio of the apparent time delay to the time constant in the FOPDT model of a process), the IAE, TV, and MS for the various tuning methods were plotted as a function of θ/τ for each of the six different types of external perturbations considered in this study. The

main objective is to generalize the trends of IAE, TV, and MS for the respective tuning rules as θ/τ increases. Because MS is calculated directly from the closed-loop transfer functions and is independent of the process inputs, one MS versus θ/τ plot is sufficient. For the tuning methods proposed by Rivera et al.,9 Skogestad,11 and Åstro¨m and Ha¨gglund,16 the values of the

366


Figure 39. Performance of controller tuning methods for a unit-step load disturbance and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052), as θ/τ increases.

Figure 40. Performance of controller tuning methods for a stochastic load disturbance (dh ) 0, σ2d ) 1) and output corrupted by measurement noise (nj ) 0, σ2n ) 0.052), as θ/τ increases.

adjustable parameter recommended by the respective authors were used. On the other hand, Lee et al.10 and Chen and Seborg12 were excluded in this study because of the lack of such recommended values. The legend for all of the plots in this section is as shown in Figure 33.

From Figure 34, the robustness of the various tuning methods, with the exception of Ziegler and Nichols,7 generally remains constant as θ/τ increases. The decreasing MS trend for Ziegler and Nichols7 can be attributed to the smaller proportional gain and larger integral time as θ/τ increases. Hence, the control


system tends to be less oscillatory and more stable as θ/τ increases. Nevertheless, Ziegler and Nichols7 tuning remains insufficiently robust with MS > 2. With reference to Figures 35-40, nearly identical trends of the effects of the controllability index on the performance of controllers tuned by different tuning rules are observed for all of the six different types of external perturbations considered. For FOPDT processes with θ/τ < 0.2, Skogestad11 is comparable in output performance to the rest of the PID tuning methods but requires significantly less control effort. For noisy systems, the control effort required is even less. Again, this illustrates the disincentive in the use of derivative action for the control of lag-dominant and/or noisy processes. Consequently, Skogestad10 should be considered as the default tuning rule for such processes. As θ/τ increases, Skogestad11 is increasingly inferior in output performance compared to the rest of the PID tuning methods (with the exception of Ziegler and Nichols7). Furthermore, the control effort required using PID tuning algorithms diminishes as θ/τ increases. This relatively enhanced performance of PID control systems is illustrative of the benefits of implementing derivative action on delay-significant processes. In general, the IAE is observed to be increasing as θ/τ increases, whereas the TV decreases as θ/τ increases. More than just illustrating the tradeoff that exists between the output performance and the control effort required, it also indicates the increasing difficulty of controlling processes with larger apparent time delays. 6. Conclusions This article will conclude with specific recommendations on optimal tuning rules for controlling industrial process control loops that are affected by set-point changes, load disturbances, and the corruption of feedback signals by measurement noise. For the general servo control problem, if the system is substantially noise-free, it is recommended that Huang and Jeng14 be used for rapid servo responses with satisfactory robust stability. If a more-robust setting is desired, Lee et al.10 is a favorable alternative to consider. On the other hand, for a noisy system, Skogestad11 should first be considered to determine whether satisfactory output performance can be attained, before considering Huang and Jeng14 or Lee et al.10 For the general regulatory problem, if the system is substantially noise-free, Åstro¨m and Ha¨gglund16 is recommended for the control of lagdominant and balanced processes, whereas Lee et al.10 is recommended for the control of delay-dominant processes. On the other hand, for noisy systems, Skogestad11 should first be considered to determine whether satisfactory output performance could be attained, before considering Åstro¨m and Ha¨gglund16 or Lee et al.10 It is to be noted that we consider only the PI controller for Skogestad’s tuning, whereas all of the other considered controllers are the PID type. Thus, it is no surprise that Skogestad’s tuning appears attractive for noisy systems and that it results in less input usage compared to PID controllers. Nomenclature a ) correlation parameter in Huang et al.15 d ) load disturbance DS ) direct synthesis e ) error FOPDT ) first-order plus dead time Gc(s) ) feedback controller transfer function G/c (s) ) IMC controller transfer function

Gd(s) ) disturbance transfer function (assumed equal to Gp(s) in this work) Gp(s) ) process transfer function Gp-(s) ) portion of Gp(s) with left half plane zeros IAE ) integral of the absolute error IMC ) internal model control ISE ) integral of the squared error ITAE ) integral of the time weighted absolute error J* ) minimum IAE ko ) correlation parameter in Huang et al.15 K ) process steady-state gain Kc ) proportional gain of controller MS ) maximum sensitivity CO ) constrained optimization PI ) proportional and integral PID ) proportional, integral and derivative s ) Laplace transform variable S(jω) ) sensitivity function SISO ) single input single output TV ) total variation of manipulated input u u ) manipulated input y ) controlled output ysp ) set point θ ) process dead time or time delay λ ) tuning parameter in IMC and DS methods τ ) process time constant τi ) integral time τd ) derivative time R ) derivative filter parameter Ri ) correlation parameter in Åstro¨m and Ha¨gglund16 ω ) angular frequency µ ) filter parameter in Huang et al.15 Literature Cited (1) Koivo, H. N.; Tanttu, J. T. Tuning of PID Controllers: Survey of SISO and MIMO Techniques, Proceedings of the IFAC Intelligent Tuning and AdaptiVe Control Symposium (Singapore); 1991, 758. (2) Desborough, L.; Miller, R. Increasing Customer Value of Industrial Control Performance Monitoring - Honeywell’s Experience, Sixth International Conference on Chemical Process Control. AIChE Symp. Ser. 2002, 326 (98). (3) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules; Imperial College Press: London, 2003. (4) Van Overschee, P.; De Moor, B. RaPID: The End of Heuristic PID Tuning. Preprints of Proceedings of PID ’00: IFAC Workshop on Digital Control (Terrassa, Spain); 2000, 687-692. (5) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, Second Edition; John Wiley and Sons: New York, 2004. (6) Balestrino, A.; Landi, A.; Medaglia, M; Satler, M. Performance Indices and Tuning in Process Control. Proceedings of the 14th Mediterrenean Conference on Control and Automation (Ancona, Italy); 2006. (7) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 64, 759. (8) Tan, W.; Liu, J.; Chen, T.; Marquez, H. J. Comparison of Some Well-Known PID Tuning Formulas; Comput. Chem. Eng. 2006, 30, 14161423. (9) Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Control: 4. PID Controller Design. Ind. Eng. Chem. Res. Process Des. DeV. 1986, 25, 252. (10) Lee, Y.; Lee, M.; Park, S.; Brosilow, C. PID Controller Tuning for Desired Closed-Loop Responses for SI/SO Systems. AIChE J. 1998, 44 (1), 106. (11) Skogestad, S. Simple Analytic Rules for Model Reduction and PID Controller Tuning. J. Process Control 2003, 13 (4), 291-309. (12) Chen, D.; Seborg, D. E. PI/PID Controller Design Based on Direct Synthesis and Disturbance Rejection. Ind. Eng. Chem. Res. 2002, 41, 4807.

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(13) Sree, R. P.; Srinivas, M. N.; Chidambaram, M. A Simple Method of Tuning PID Controllers for Stable and Unstable FOPDT Systems. Comput. Chem. Eng. 2004, 28, 2201. (14) Huang, H. P.; Jeng, J. C. Identification for Monitoring and AutoTuning of PID Controllers. J. Chem. Eng. Japan. 2003, 36, 284. (15) Huang, H. P.; Jeng, J. C.; Luo, K. Y. Auto-Tune Using SingleRun Relay Feedback Test and Model-Based Controller Design. J. Process Control 2005, 15, 713. (16) Åstro¨m, K. J.; Ha¨gglund, T. Revisiting the Ziegler-Nichols Step Response Method for PID Control. J. Process Control 2004, 14, 635650. (17) Syrcos, G.; Kookos, I. K. PID Controller Tuning Using Mathematical Programming. Chem. Eng. Process. 2005, 44, 41. (18) Åstro¨m, K. J.; Ha¨gglund, T. The Future of PID Control. Control Engineering Practice 2001, 9, 1163-1175. (19) Huang, H. P.; Jeng, J. C. Monitoring and Assessment of Control Performance for Single Loop Systems. Ind. Eng. Chem. Res. 2002, 41, 1297.

(20) Panagopoulos H.; Åstro¨m, K. J.; Ha¨gglund T. Design of PID Controllers Based on Constrained Optimization. IEE Proceedings: Control Theory and Applications; 2002, 149 (1), 32-40. (21) Deshpande, P. B.; Ash, R. H. Elements of Computer Process Control with AdVanced Control Applications; Instrument Society of America, Research Triangle Park: NC, 1981. (22) Åstro¨m, K. J.; Hang, C. C.; Persson, P.; Ho, W. K. Towards Intelligent PID Control. Automatica 1992, 28 (1), 1-9. (23) Foley, M. W.; Julien, R. H.; Copeland, B. R. A Comparison of PID Controller Tuning Methods. Canadian J. Chem. Eng. 2005, 83, 712. (24) Foley, M. W.; Ramharack, N. R.; Copeland, B. R. Comparison of PI Controller Tuning Methods. Ind. Eng. Chem. Res. 2005, 44, 6741. (25) Marlin T. E. Process Control: Designing Processes and Control Systems for Dynamic Performance, Second Edition; McGraw-Hill Higher Education: New York, 2000.

ReceiVed for reView March 28, 2007 ReVised manuscript receiVed September 21, 2007 Accepted September 28, 2007 IE0704546

Popular PID Tuning Rules for Stable

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