Set-Point Weighted PID Controller Tuning for Time-Delayed Unstable

Aug 6, 2008 - The set-point weighted proportional, integral, and derivative (PID) controller has been shown to be equivalent to an error feedback PID ...
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Ind. Eng. Chem. Res. 2008, 47, 6983–6990

6983

Set-Point Weighted PID Controller Tuning for Time-Delayed Unstable Processes Chan-Cheng Chen,*,† Hsia-Ping Huang,‡ and Horng-Jang Liaw† Department of Occupational Safety and Health, China Medical UniVersity, 404, Taiwan, and Department of Chemical Engineering, National Taiwan UniVersity, 106, Taiwan

The set-point weighted proportional, integral, and derivative (PID) controller has been shown to be equivalent to an error feedback PID controller with a PD controller in the inner loop; and, many nonerror feedback designs have been unified under the frame of tuning the set-point weighted PID controller. A simple method is then proposed to reform an existing PID control system into a set-point weighted PID control system for unstable processes. To reform an existing error feedback PID control system into the set-point weighted PID control system, the proposed method needs neither the information of the process nor the information of the design methods of the original error feedback PID control system. The only information needed to reform the existing PID control system is the values of the tuning parameters of the original error PID control system. Theoretical analysis shows that if the original error feedback PID control system is stable, the stability of the proposed set-point weighted PID control system is guaranteed automatically. Simulation results show that the performance of the proposed set-point weighted PID control system can be effectively enhanced in comparison with that of the original error feedback PID control system; further, this performance of the proposed set-point weighted PID control system is comparable to that of many other nonerror feedback systems reported in the literature. 1. Introduction In process industries, proportional, integral, and derivative (PID) type controllers are extremely popular; in fact, most controllers used in industrial applications have the PID-type structure. Many instrument and process engineers are familiar with PID-type controllers, and techniques for installing, tuning, and using PID-type controllers in process industries have been well-established. The popularity of PID-type controllers can be attributed to their simplicity and satisfactory performance for a wide class of systems. Although several methods, such as the well-known Ziegler and Nichols method, are available for tuning PID controllers to achieve satisfactory performance, there are cases in which these simple tuning rules cannot be applied.1 One such case is the control of an unstable process. It is extremely difficult to synthesize a control system for processes with an unstable pole. These difficulties arise primarily due to the unstable nature of the dynamics for which most design tools cannot be applied. The strongly stabilizing problem is a serious challenge for controlling an unstable process. The parity interlacing property shows that in some cases, right-half-plane (RHP) poles should be introduced in the controller to stabilize an unstable process.13 Many studies in the literature also indicate that a proportional controller can stabilize a time-delayed firstorder unstable process only if the ratio of the time delay to the unstable time constant of the process is less than unity. Even if a PD/PID controller is considered to stabilize a first-order timedelayed unstable process, the stabilizable region of this ratio is still limited to two.4 Therefore, PID controller tuning methods for unstable processes differ from those for stable processes, and many research works were proposed to tune the PID controller for unstable processes.5–7,14,15 A good survey on this subject could be found in a recent book by Sree and Chidambaram.11 * To whom correspondence should be addressed. E-mail address: [email protected]. Tel.: 886-4-222053366 ext. 6210. Fax: 886-4-22076435. † China Medical University. ‡ National Taiwan University.

In addition to the stabilization problem, the achievable performance for controlling an unstable process is significantly different from that of a stable one. Many performance specifications that appear to be very common for stable processes become impractical for a system with unstable poles. For example, with an unstable pole, the overshoot of the system with conventional feedback is bounded from below.3 For the control of such unstable processes, alternative nonerror feedback structures are required to overcome this drawback.3,7–10 Huang and Chen3 have considered a three-element structure, as shown in Figure 1, to enhance the control performance of a closed loop without excessive overshoot. In this case, C1 is a PD-type controller, which is used to stabilize an unstable process, and C2 and C3 are used for the tracking and attenuation problem of the stabilized the inner loop. This three-element structure has been shown to be equivalent to the so-called prefilter form of the two-degrees-of-freedom (2 dof) structure, as shown in Figure 2.3 Jung et al. have proposed a direct synthesis tuning method for unstable first order plus time delay process using this prefilter form structure.7 In their method, the prefilter F(s) is a first order lag and the feedback controller is a PI controller. This prefilter form structure was also adopted by Lee et al. to control unstable processes.8 In their study, the prefilter F(s) is a first order lag for processes with one unstable pole. Park et al. have considered an inner loop, as shown in Figure 3, to minimize the undesirable overshoot.9 In their method, the inner feedback controller kci is a P-type controller, while the controller Gc(s) is an ideal PID controller. In addition to these structures, the set-point weighted PID control algorithm is another possible structure for reducing

Figure 1. Three-element control structure.

10.1021/ie800001m CCC: $40.75  2008 American Chemical Society Published on Web 08/06/2008

6984 Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008

Figure 2. Prefilter form control structure and its equivalence to the threeelement control structure. Figure 4. Equivalent control structure of a set-point weighted PID controller.

in the book by Astrom and Hagglund,1 can be represented by the following transfer function:

(

Gc(s) ) Kc′ 1 +

Figure 3. Inner loop feedback control structure.

the overshoot. Prashanti and Chidambaram have extended some weighted PID controller tuning methods for stable processes to the case of unstable processes and compared their performance.10 Their method focused on reforming an error feedback PID control system into the set-point weighted PID control system. However, their method must be combined with the method of original error feedback PID control design. Thus, different error feedback PID control design must have different guides to tune the parameters in a set-point weighted PID controller in their method. Moreover, their method also needs the process information to determine the set-point weighted parameter. All the abovementioned 2 dof methods reduce the excessive overshoot caused by an unstable pole in the error feedback structure; however, they have different degrees of success. In fact, nonerror feedback PID controllers are less popular than conventional PID controllers. However, in many commercial distributed control systems, the set-point weighted PID algorithm is provided. Thus, this type of PID controller is the most easily accessible nonerror feedback structure to process engineers. In general, it is difficult to model an open-loop unstable process and it must usually be stabilized in the first stage. Therefore, a tuning method that can begin with a conventional PID controller in the first stage to achieve higher performance is required. The advantage of this approach is that it avoids the use of tedious and cost-intensive modeling and identification procedures. This study is organized as follows. First, different PID control algorithms and the set-point weighted PID control algorithm are reviewed and discussed in section 2. A low frequency approximation of the design proposed by Huang and Chen3 is discussed in section 3 to establish the relationship between the settings of a set-point weighted PID controller and those of an error feedback PID controller. In section 4, simulation examples are given to demonstrate the performance of the proposed method. Finally, section 5 concludes this study. 2. PID Algorithms and the Set-Point Weighted PID Algorithm

(

1 TR′

∫ e(τ) dτ + T t

0

D′

de(t) dt

)

(2)

A slightly different form, which is known as the series form PID control algorithm, is extensively used in commercial controllers. PID controllers of this type are described by the following transfer function.

(

Gc(s) ) Kc′′ 1 +

)

1 (1 + sTD′′) sTR′′

(3)

Both the ISA and series algorithms operate on an error signal. Therefore, such control systems are known as feedback on error systems (or unity feedback systems). However, a more flexible structure is also provided in commercial controllers, which treat the set-point and the process output separately. A PID controller of this type, known as a set-point weighted PID controller, is given by

(

u(t) ) Kc ep(t) +

1 TR

∫ e(τ) dτ + T t

D

0

ded(t) dt

)

(4)

ep(t) ) βysp - y(t) eD(t) ) γysp - y(t) e(t) ) ysp - y(t) The parameters β and γ are introduced to shape the errors in the proportional and derivative terms, respectively. To further understand the function of the set-point weighted PID controller, we can rewrite eq 4 as follows:

{

u(t) ) Kc βe(t) - (1 - β)y(t) + (1 - γ)TD

}

} {

1 TR

∫ e(τ) dτ + γT t

D

0

Kc dy(t) ) Kcβe(t) + dt TR

{

de(t) dt

∫ e(τ) dτ + t

0

}

de(t) dy(t) - Kc(1 - β)y(t) + Kc(1 - γ)TD ≡ dt dt t dy(t) de(t) K1e(t) + K2 0 e(τ) dτ + K3 - K4y(t) + K5 dt dt

KcγTD

{

} {

∫

}

where

The well-known “ideal form” of the PID algorithm can be described as u(t) ) Kc′ e(t) +

)

1 + sTD′ sTR′

(1)

where u is the control variable and e is the control error (e ) y - ysp). This form, which is also known as the ISA algorithm

K1 ) Kcβ

(5)

K2 ) Kc/TR

(6)

K3 ) KcγTD

(7)

K4 ) Kc(1 - β)

(8)

K5 ) Kc(1 - γ)TD

(9)

Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008 6985

Thus, as shown in Figure 4, we can conclude that the setpoint weighted PID algorithm is equivalent to an error feedback PID controller with a PD controller in the inner loop. For convenience in discussion, we label the set-point weighted PID controller functions as a PID-PD controller in this study. We use hyphens to separate the control functions in the main control loop and the inner control loop. It should also be noted that there is a one-to-one correspondence between the parameter sets (Kc, TR, TD, β, γ) and (K1, K2, K3, K4, K5). Hence, if the parameter set (K1, K2, K3, K4, K5) is given, we can obtain the corresponding parameter set of the set-point weighted PID controller by using the following equations: Kc ) K1 + K4

(10)

β ) K1/(K1 + K4)

(11)

TR ) (K1 + K4)/K2

(12)

TD ) (K3 + K5)/(K1 + K4)

(13)

γ ) K3/(K3 + K5)

(14)

Theoretically, Kc, TR, TD, β, and γ could be tuned independently; however, the process engineers usually tune the setpoint weighted PID controller in two stages. They generally prefer the generic use of a PID controller and provide the parameters β and γ, thereby making a small contribution to the reduction of P and D actions, respectively. Hence, it appears important to determine the effective controller function when we change the values of β and γ. First, we consider four extreme cases. If we set β ) γ ) 1, then we obtain K4 ) K5 ) 0. Thus, the control function of the set-point weighted PID controller will be the same as that of the error feedback PID controller. When β ) 1 and γ ) 0, we obtain K3 ) K4 ) 0, and the set-point weighted controller functions as a PI-D controller. This case is very common in process industrial applications. For example, this algorithm can be found in the Honeywell TDC 3000 system as equation B of control functions.2 If we set β ) 0 and γ ) 0, we will obtain K1 ) K3 ) 0, and the set-point weighted PID controller will function as an I-PD controller. In fact, this is the so-called I-PD algorithm used in some commercial controllers. When β ) 0 and γ ) 1, we obtain K1 ) K5 ) 0, and the set-point weighted PID controller functions as an ID-P controller. It is noteworthy that if we fix the values of (Kc, TR, TD), then the values of β and γ have no effect on the response to the load disturbance or measurement noise; however, the response to set-point changes depends on the values of β and γ being used. In some process applications, γ may be selected as 0 to avoid abrupt changes in the controller output during the set-point changes.1 As mentioned earlier, γ is usually fixed to 0 or 1 in process applications, and only β is considered to shape the response. Here, we discuss the problem for tuning β with γ ) 1 or γ ) 0. When γ ) 1 and β ∈ (0, 1) (i.e., K5 ) 0), the set-point weighted PID controller functions as a PID-P controller and its control function is the same as the structure shown in Figure 3. Thus, it could be concluded that the method proposed by Park et al. is the same as the tuning of the set-point weighted PID controller with γ ) 1. When γ ) 0 and β ∈ (0, 1), the set-point weighted PID algorithm works as a PI-PD controller. It will be shown later that the design of Huang and Chen3 could also be considered for tuning the set-point weighted PID controller with γ ) 0. The control functions of the set-point weighted PID controller for different weighting parameters are summarized in Table 1.

Table 1. Control Functions of the Set-Point Weighted PID Controller for Different Values of β and γ β

γ

control function

1 1 0 0 β ∈ (0, 1) β ∈ (0, 1) β ∈ (0, 1)

1 0 1 0 0 1 γ ∈ (0,1)

PID PI-D I-PD ID-P PI-PD PID-P PID-PD

3. Tuning Set-Point Weighted PID Controllers with Existing Error Feedback PID Controller Settings In the study of Huang and Chen,3 the three-element control structure shown in Figure 1 was proposed to reduce the overshoot caused by an unstable pole. The controller C1, known as the stabilizing controller in their study, was a PD-type controller and was considered to stabilize the unstable process. After the stabilization of the unstable process by C1, the controllers C2 and C3 were used for the tracking and attenuation of the stabilized inner loop, respectively. For the stabilized inner loop, they used the internal model control (IMC) design method to design C2 and C3. If the IMC filters for tracking and attenuating are selected to be the same in their study, the controller C3, which is used for disturbance attenuation, will be equal to unity. It is evident that C2 must include an integrator to meet the requirements of the zero steady-state error. In other words, C2 can be approximated as 1/ TR′′s in the low frequency range. Since C1 is a PD-type controller, i.e., C1 ) Kc′′(1 + TD′′s), the following relation holds, referring to Figure 2, in the low frequency range (note that C3 ) 1): u(s) ) C1C2r - C1(1 + C2C3)y ) C1C2(r - y) - C1y 1 ≈ Kc′′(1 + TD′′s) e - Kc′′(1 + TD′′s)y TR′′s

( )

(15)

Equation 15 can be rewritten as:

(

) (

TD′′ dy(t) 1 t e(t) + e(τ) dτ - Kc′′ y(t) + TD′′ ) TR′′ TD′′ 0 dt TR′′ + TD′′ TD′′ t 1 r(t) - y(t) + e(τ) dτ Kc′′ TR′′ TR′′ + TD′′ TR′′ + TD′′ 0 TD′′ TR′′ dy(t) (16) TR′′ + TD′′ dt

u(t) ) Kc′′

∫

(

)

∫

)

We now define ep(t) ≡

TD′′ r(t) - y(t) TR′′ + TD′′

ed(t) ≡ 0 · r(t) - y(t) By comparing with the set-point weighted PID control law, i.e., eq 4, we have Kc ) Kc′′

TR′′ + TD′′ TR′′

(17)

TR ) TR′′ + TD′′

(18)

TD )

TD′′TR′′ TR′′ + TD′′

(19)

β)

TD′′ TR′′ + TD′′

(20)

γ)0

(21)

6986 Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008

Thus far, we have shown that Huang and Chen’s work could be considered, in the low frequency range, as a set-point weighted PID controller tuning method. However, their results could also be explained on the basis of a series PID-type controller Gc(s) and a prefliter, F(s), in the low frequency range. The transfer functions of Gc(s) and F(s), referring to Figure 2, are given below:

(

Gc(s) ) C1(1 + C2C3) ≈ Kc′′(1 + TD′′s) 1 + F(s) ) C2/(1 + C2C3) ≈

1 TR′′s + 1

1 TR′′s

)

(22) (23)

If the dynamics of F(s) is further neglected, we can obtain the settings of a series PID controller in the error feedback structure. The stability of the control system does not change when F(s) becomes unity. On the basis of this approach, we find that the settings of the series PID controller are closely related to those of the weighted PID controller. Set-point weighted PID controller tuning could be considered as the low frequency approximation of Huang and Chen’s work, and the tuning of the error feedback PID controller is a further simplified case of set-point weighted PID controller tuning with F(s) ) 1. As discussed in the literature, a different IMC filter structure or filter constant would result in a different control system in the three-element structure. Hence, different error feedback PID tuning methods proposed in the literature could be conceptually considered as the low frequency approximation of a special design in the three-element system with F(s) ) 1. The corresponding set-point weighted PID settings given by eq 17–21 could be considered to introduce a prefilter in the error feedback PID control structure. Since we do not change the characteristic equation, the new set-point weighted PID control system will be stable if the original error feedback system is stable. According to the three-element structure design procedure, the set-point weighted PID control system is an approximation of the original design in the low frequency range, while the error feedback PID control system is an approximation of the set-point weighted PID control system with F(s) ) 1. Thus, the performance of the set-point weighted PID control system will be higher than that of the error feedback control system. This will be shown by simulation examples. In the literature, most settings for error feedback PID tuning are provided in the ideal form. Therefore, eq 17–21 could be rewritten in terms of the ideal form PID settings as follows: Kc ) Kc′

(24)

TR ) TR′

(25)

TD ) TD′

(26)

1 - √(TR′ - 4TD′)/TR′ (27) 2 γ)0 (28) Hence, if the ideal form PID controller settings are given, the set-point weighted PID controller is tuned as follows. First, the controller gain, reset time, and derivative time must be the same as those of the original controller. Second, β must be tuned according to eq 27 and γ must be set to 0. The proposed method coincides with the practice of process engineers for tuning the set-point weighted PID controller. β)

4. Simulation Examples Three examples are used to demonstrate the advantage of the proposed method. All simulations are performed by using

Figure 5. Servo response for example 1 using the present method with HC99 method for controller design: (solid) HC99 + this study, (dash) HC99.

MATLAB 6.5 and SIMULINK 5.0. The options of the SIMLINK solver are set at ODE5 and fixed-step with step size ) 0.001. All settings of the error feedback PID controller provided in the simulation examples are based on those of the ideal PID controller. Example 1. The first-order delayed unstable process (FODUP) with the following transfer function is considered: 4e-2s 4s - 1 Many studies have proposed different settings of the error feedback PID controller for this process. Here, we use the PID settings of Huang and Chen5 as Kc ) 0.565, TR ) 12.276, and TD ) 0.608; those of Sree et al.12 as Kc ) 0.548, TR ) 11.117, and TD ) 1.024; and those of Visioli14 as Kc ) 0.624, TR ) 11.551, and TD ) 1.161. The corresponding values of the setpoint weighted parameter β are then calculated using eq 27 as 0.052, 0.103, and 0.101 for these three sets of settings, respectively. Figures 5–7 show the servo responses without and with the set-point weighting for the aforementioned three PID control systems. It could be found from these figures that the proposed set-point weighted PID tuning method significantly reduces the overshoot of the system in comparison with that of the original error feedback PID control system. Park, Sung, and Lee9 have developed a control system with an inner loop feedback structure, as shown in Figure 3, to reduce the overshoot for this process, and their results have been proved to be equivalent to those of a set-point weighted PID controller with Kc ) 0.418, TR ) 11.587, TD ) 0.699, β ) 0.163, and γ ) 1. Jung, Song, and Hyun7 have also suggested a control system with a PI controler and a prefilter to control this FODUP. Their tuning parameters for this FOPUD are the following: Kc ) 0.384, TR ) 30.301, and a prefilter of 1/(30.301s + 1). The servo responses of the proposed three set-point weighting systems and those of aforementioned two works are compared in Figure 8. It turns out from this figure that the proposed three set-point weighting systems, which are reformed from different PID control systems, all give better performance than those of aforementioned two nonerror feedback works. For convience, the controller settings for all abovementioned system are summarized in Table 2. Gp(s) )

Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008 6987

Figure 6. Servo response for example 1 using the present method with the Visioli method for controller design: (solid) Visioli + this study, (dash) Visioli.

Figure 8. Servo responses of different 2 dof systems for example 1: (thick solid) SSC + this study, (thin solid) Visioli + this work, (dash) HC99 + this work, (dash dot) JSH, (dot) PSL. Table 2. Tuning Parameters in Example 1 method

Kc

TR

TD

HC99 SSC Visioli HC99 + this work SSC + this work Visioli + this work PSL JSH

0.565 0.548 0.624 0.565 0.548 0.624 0.418 0.384

12.276 11.117 11.551 12.276 11.117 11.551 11.587 30.301

0.608 1.024 1.161 0.608 1.024 1.161 0.699 0.000

β

γ

0.052 0.103 0.113 0.163

0.00 0.00 0.00 1.00

prefilter

1/(30.301s + 1)

Table 3. Performance Comparison of Example 1 for Different Methods method

ISE

IAE

ymax

HC99 SSC Visioli HC99 + this work SSC+this work Visioli + this work SSC+this work PSL JSH

13.47 10.10 8.28 4.76 4.43 4.36 4.43 5.48 8.01

11.71 11.87 10.87 6.21 6.28 6.14 6.28 10.10 10.61

2.78 2.32 2.30 1.00 1.09 1.05 1.09 1.42 1.00

Figure 7. Servo response for example 1 using the present method with the SSC method for controller design: (solid) SSC + this study, (dash) SSC.

Example 2. The second order delayed unstable process (SODUP) with the following transfer function is considered:

To make a quantitative comparison, the performance of aforementioned systems are summarized in Table 3. The performance indices considered in Table 3 include the ISE (integral square error), IAE (integral absolute error), and maximum response. As shown in Table 3, after introducing the set-point weighting, the maximum response decreases from 2.30 to 1.05 for Visioli’s settings, from 2.32 to 1.09 for Sree et al.’s settings, and from 2.78 to 1.00 for Huang and Chen’s settings. Table 3 also shows that the proposed set-point weighting method effectively reduces the values of the ISE and IAE as compared with the original error feedback system. When compared with the two nonerror feedback methods proposed the by Park et al. and Sree et al., the proposed tuning parameter sets all provide higher performance in all performance indices. With these facts, the proposed method could effectively reform an error feedback PID control system into a set-point weighting PID control system in this example.

e-s (0.5s + 1)(2s - 1) For this SODUP, the PID settings of Huang and Chen5 are Kc ) 1.792, TR ) 12.425, and TD ) 0.480. The value of β is then calculated using eq 27 as 0.040. In Prashanti and Chidambaram’s study,10 the PID controller settings for this SODUP are Kc ) 1.586, TR ) 12.00, and TD ) 0.479 and the weighting parameters are β ) 0.22 and γ ) 2.35. If we use their error feedback PID controller settings and apply eqs 27 and 28, we will obtain β ) 0.042 and γ ) 0.00. Lee et al.8 have also propose an IMC based the design for SODUP. With IMC filter constant λ ) 2, their settings for this SODUP are the following: Kc ) 1.949, TR ) 12.063, TD ) 0.826, and a prefilter of 1/(11.119s + 1). For convience, the tuning parameters for the abovementioned systems are summarized in Table 4. Figures 9 and 10 show the responses of the proposed setpoint weighted PID tuning methods and those of their original Gp(s) )

6988 Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008 Table 4. Tuning Parameters in Example 2 method

Kc

TR

HC99 HC99 + this work PC without weighting PC with weighting PC + this work LLP

1.792 1.792 1.586 1.586 1.586 1.949

12.425 12.425 12.000 12.000 12.000 12.063

TD

β

γ

prefilter

0.480 0.480 0.040 0.000 0.479 0.479 0.220 2.350 0.479 0.042 0.000 0.826 1/(11.199s + 1)

Figure 11. Servo responses of different 2 dof systems for example 2: (thick solid) PC + this work, (thin solid) HC99 + this work, (dash) LLP, (dash dot) PC with weighting. Table 5. Performance Comparison of Example 2 for Different Methods

Figure 9. Servo response for example 2 using the present method with the HC99 method for controller design: (solid) HC99 + this study, (dash) HC99.

method

ISE

IAE

ymax

HC99 PC without weighting PC with weighting HC99 + this work PC + this work LLP

20.60 29.00 3.26 3.59 3.67 3.49

15.02 18.85 5.81 5.12 5.42 5.02

3.56 3.77 1.47 1.02 1.18 1.00

To make a quantitative comparison, the performance of aforementioned systems are summarized in Table 5. It could be found that the maximum response decreases from 3.56 to 1.02 for Huang and Chen’s settings and from 3.77 to 1.18 for Prashanti and Chidambaram’s settings. It could be found that although the ISE performance of Prashanti and Chidambaram’s method and Lee et al.’s method are a little higher than those of the proposed two set-point weighted PID controllers, the difference appears insignificant in compariosn with the ISE performance of the original error feedback PID control systems. Thus, it seems reasonable to conclude that both these two setpoint weighting PID control systems, which are reformed from different PID control systems, give comparable performance as aforementioned two nonerror feedback works for this SODUP example. Example 3. A high order delayed unstable process with the following transfer function is considered: e-0.5s (37) (5s - 1)(0.5s + 1)(2s + 1) For this process, the PID settings of Huang and Chen3 are Kc ) 6.186, TR ) 7.170, and TD ) 1.472. From eq 27, we could obtain β ) 0.289. Lee et al.8 have considered the prefilter form structure to reduce the overshoot for this process. They have provided the PID controller tuning parameters as Kc ) 7.144, TR ) 6.684, and TD ) 1.655 with a prefilter of the first order lag having a time constant of 4.276. If we employ their PID controller settings and then apply eq 27, we will obtain β ) 0.289. The tuning parameters for these systems are summarized in Table 6. Figures 12 and 13 show the responses of the proposed setpoint weighted PID tuning method and those of their original Gp(s) )

Figure 10. Servo response for example 2 using the present method with the PC method for controller design: (solid) PC + this study, (dash) PC.

PID systems. As shown in these figures, both the proposed setpoint weighted PID tuning methods effectively reduce the overshoot in comparison with that of the original PID systems. In Figure 11, the servo responses of these two set-point weighting systems are compared with those of two abovementioned nonerror feedback works by Prashanti and Chidambaram, and Lee et al. It turns out from this figure that these two setpoint weighting systems, which are reformed from different PID control systems, give comparable performance as compare with aforementioned two nonerror feedback works.

Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008 6989 Table 6. Tuning Parameters in Example 3 method

Kc

TR

HC97a LLP without filter HC97a + this work LLP. + this work LLP with filter

6.186 7.144 6.186 7.144 7.144

7.170 6.684 7.170 6.684 6.684

TD

β

γ

prefilter

1.472 1.655 1.472 0.289 0.000 1.655 0.451 0.000 1.655 1/(4.276s + 1)

Figure 14. Servo responses of different 2 dof systems for example 3: (thick solid) LLC + this work, (thin solid) HC97a + this work, (dash) LLP with filter. Table 7. Performance Comparison of Example 3 for Different Methods

Figure 12. Servo response for example 3 using the present method with the HC97a method for controller design: (solid) HC97a + this study, (dash) HC97a.

method

ISE

IAE

ymax

HC97a LLP without filter HC97a + this work LLP + this work LLP with filter

3.70 3.03 2.84 2.34 2.49

5.56 5.18 3.97 3.22 3.52

1.85 1.71 1.01 1.07 1.03

nonerror feedback work by Lee et al. To make a quantitative comparison, the performance of aforementioned systems are summarized in Table 7. It could be found that the maximum response decreases from 1.85 to 1.01 for Huang and Chen’s settings and from 1.71 to 1.07 for Lee et al.’s settings without prefilter. It could also be seen from table that the difference in ISE and IAE between Lee et al.’s method and the proposed two methods appears to be insignificant. Thus, it seems reasonable to conclude that both the proposed two set-point weighting PID control systems, which are reformed from different PID control systems, give comparable performance as that of Lee et al.’s nonerror feedback method for this example. 5. Conclusion

Figure 13. Servo response for example 3 using the present method with the LLP method for controller design: (solid) LLP + this study, (dash) LLP.

PID control systems. As shown in these figures, both the proposed set-point weighted PID tuning methods effectively reduce the overshoot in comparison with that of the original PID control systems. In Figure 14, the servo responses of the proposed two set-point weighting systems are compared with those of the nonerror feedback work by Lee et al. It turns out from this figure that both the proposed two set-point weighting systems, which are reformed from different PID control systems, give comparable performance as compare with the reporetd

Studies on tuning PID controllers for unstable processes have been actively conducted in recent years. The excessive overshoot problem in error feedback systems is discussed in the literature and many nonerror feedback structures were proposed to overcome it. As the control function for the set-point weighted PID controller was shown to be equivalent to a PID-PD controller in this study, it was found that many existing nonerror feedback designs could be unified under the frame of tuning set-point weighted PID controller. As earlier mentioned, the process engineers usually tune the set-point weighted PID controller in two stages. They generally prefer the generic use of a PID controller and provide the parameters β and γ, thereby making a small contribution to the reduction of P and D actions, respectively. However, most of the studies focused on designing a nonerror feedback structure instead of reforming the existing error feedback system to a nonerror feedback one. In this study, the relationship between the existing error feedback PID controller settings and the set-

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point weighted PID controller settings is presented. The proposed method only needs the information of PID controller settings of original error feedback PID control system and is independent of the design method of the original PID system. Moreover, the proposed method does not need any process information to tune the set-point weighting parameters of β and γ. The stability of the derived set-point weighted PID control system will be automatically guaranteed by the proposed method if the original error feedback PID control system is stable. The proposed simple rule originates from a low frequency approximation of Huang and Chen’s work. Many error feedback PID settings in the literature, including those of Huang and Chen, Sree et al., Visioli, Prashanti and Chidambaram, and Lee et al., are explored with different examples to evaluate whether the proposed method satisfactorily enhances the performance for different error feedback PID tuning methods. As shown in the simulation examples, the proposed set-point weighted PID settings, in all cases, effectively reduce the overshoot, ISE, and IAE in comparison with those of the original error feedback PID settings. Many nonerror feedback systems reported in the literature, which include the works by Park et al., Jung et al., and Lee et al., are considered in the different examples to compare their performance with that of the proposed set-point weighted PID system. It was found from simuation examples that the propoesed set-point weighted control systems reformed from different original error feedback PID control systems all give comparable performance to those nonerror feedback systems.

Acknowledgment The authors would like to thank the China Medical University of the ROC for supporting this study financially under grant no. CMU95-178.

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ReceiVed for reView January 1, 2008 ReVised manuscript receiVed June 12, 2008 Accepted June 16, 2008 IE800001M