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Ind. Eng. Chem. Res. 2011, 50, 1352–1359
PID Controller Design Directly from Plant Data Xin Yang, Bu Xu, and Min-Sen Chiu* Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, 4 Engineering DriVe 4, Singapore 117576
There are two kinds of model-based proportional integral derivative (PID) design methods depending on whether the PID tuning algorithms have a tunable parameter. For the direct model-based PID design methods without resorting to a tunable parameter, the resulting control performance is usually not satisfactory for higher-order process dynamics owing to the inevitable modeling error. On the other end, while better control performance can usually be achieved by the internal model control (IMC)-PID or λ-tuning method, the corresponding optimal tunable parameter is normally determined by trial-and-error procedure from the plant tests or simulation requiring prior information of process dynamics, resulting in an iterative tuning procedure and at the expense of considerable engineering efforts. To alleviate this drawback, a PID design method is proposed to design PID parameters directly using the process data collected from an off-line experiment in this paper. The optimization problem pertaining to the proposed design is derived, and the associated design issues are addressed. Extensive simulation results show that the proposed PID design outperforms the direct model-based PID design and gives better or comparable control performance than the respective best control performances attained by the IMC-PID and Maclaurin-PID designs that have been tuned by trial and error in the simulated closed-loop tests involving an a priori known process model. Therefore, the proposed method is able to retain the simple design procedure of direct model-based PID design methods while achieving comparable or better performance than the IMC-based PID design methods. 1. Introduction Proportional integral derivative (PID) controllers are the most widely used controllers in the chemical process industries. As a result, PID controller design methods have been an active research topic in the past several decades. In particular, various model-based PID design methods are available in the literature, such as the Ziegler-Nichols continuous cycling method,1 direct synthesis method,2,3 internal model control (IMC) method,4-9 and other techniques.10-25 There are two steps in the modelbased PID designs: an empirical model of the process is identified first, which is subsequently used together with a prespecified tuning algorithm to design a PID controller. Generally speaking, there are two kinds of model-based PID design methods depending on whether the PID tuning algorithms have a tunable parameter. Both techniques have advantages and disadvantages. For example, in the absence of a tunable parameter, the advantage of direct model-based PID design methods is the straightforward controller design procedure using a specific chosen PID tuning algorithm and parameters of the assumed lower-order models identified from the plant tests. Although these methods can give good PID design when the underlying process dynamics are reasonably described by the lower-order models, the effectiveness of these methods would degrade for higher-order process dynamics owing to the inevitable modeling error. On the other hand, when the PID tuning algorithms rely on a tunable parameter, better control performance can usually be achieved because the additional adjustable parameter can be tuned to deal with the performance trade-off caused by modeling error. However, the corresponding optimal tunable parameter is normally determined by a trialand-error procedure from the plant tests or simulated closedloop tests requiring prior information of process dynamics, resulting in an iterative tuning procedure and at the expense of considerable engineering efforts. It was reported that the IMC* To whom correspondence should be addressed. Tel.: +65 65162223. Fax: +65 67791936. E-mail:
[email protected].
PID or λ-tuning method is the most widely adopted tuning method among this class of model-based design method.25 To alleviate the aforementioned drawbacks, it is an attractive alternative to design PID controllers directly based on a set of process input and output data without resorting to a process model. Toward this end, several model-free or data-based controller design methods were developed in the literature. The iterative feedback tuning (IFT) method developed by Hjalmarsson et al.26,27 is the earliest direct data-based method for controller tuning. However, IFT may require considerable computational time to obtain a solution with the risk of being a local optimum, not to mention that its initialization is carried out by trial-and-error procedure. Spall and Cristion28 proposed a stochastic approach for adaptive control design using a function approximator like a neural network to execute the action needed from the controller through the minimization of a cost function. However, since a plant model is not available, the gradient of this cost function has to be evaluated by simultaneous perturbation stochastic approximation instead of quadratic methods. Thus, the computational burden of this method is very demanding due to the iterations and the convergence of the trained parameters may not be guaranteed. To overcome this limitation, the virtual reference feedback tuning (VRFT) method and its variants29-32 were proposed to design controller parameters without an iterative design procedure. Kansha et al.33 recently extended the VRFT design method to adaptive PID design for nonlinear systems. However, previous results on the VRFT methods were developed for discrete-time systems. This motivates our research to extend the VRFT design framework to continuous-time systems with the specific aim of designing PID controllers directly from the process data available from open-loop tests. Consequently, the proposed method can design PID controllers without resorting to the availability of a process model and trialand-error procedure necessitated in some model-based PID design methods. The optimization problem pertaining to the
10.1021/ie100784k 2011 American Chemical Society Published on Web 09/29/2010
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011
ˆ (s) ) C(s)(R(s) - Y(s)) U 1 + τDs L(s)-1 Y(s) ) Kc 1 + τIs
(
Figure 1. Reference model.
(3)
where Kc denotes the proportional gain, τI is the integral time, and τD is the derivative time. Substituting s ) jω into eq 3, the next equation lays the foundation for the proposed PID design:
Figure 2. Feedback system.
proposed design is derived, and the associated design issues are addressed. A wide range of process models is used to evaluate the proposed PID design and benchmark model-based PID design methods. For direct model-based PID design methods, the Connell-PID algorithm23 is considered, whereas the IMC-PID6,36 and Maclaurin-PID8 designs are chosen as the benchmark model-based PID design methods having the tunable parameter. Extensive simulation results show that the proposed PID design not only outperforms the Connell-PID design, but also in most examples gives better control performance than the best control performance achieved by the IMC-PID design that has been tuned by trial and error in the simulated closedloop tests involving an a priori known process model, which is not required for the proposed PID design. On the other hand, the proposed method gives comparable performance to the best control performance achieved by the Maclaurin-PID design that has been tuned by the same procedure of IMC-PID design. Therefore, the proposed method is able to retain the simple design procedure of direct model-based PID design methods while achieving comparable or better performance than the IMCbased PID design methods. Consequently, the proposed design is an attractive alternative to the model-based PID design methods and can thus be applied to design the initial PID parameters with good control performance.
31,34
Similar to the discrete-time VRFT design framework, the proposed direct PID controller design approximately solves a model-reference problem in continuous time as depicted in Figure 1, where the output Y(s) is related to the set point R(s) through the reference model by L(s) R(s) 1 + L(s)
(1)
where L(s) is the desired loop transfer function. The controller design problem is to assign a reference model in terms of L(s) that describes the desired servo response of a feedback control system consisting of a linear time-invariant plant G(s) and a PID controller C(s) as shown in Figure 2. Assuming that G(s) is unknown and only a set of process input and output data, u(t) and y(t), are available from the open-loop test, the design goal of the proposed method is to obtain PID parameters such that the corresponding feedback control system in Figure 2 behaves as closely as possible to the prespecified reference model. To this end, given the available output signal y(t), the reference signal in Figure 1 is obtained from eq 1 as -1
R(s) ) (1 + L(s) )Y(s)
ˆ (jω) ) Ω(jω) -Ω(jω) j Ω(jω) jω W U ω
(4)
Ω(jω) ) L(jω)-1 Y(jω)
(5)
[
]
where
[
W ) Kc
Kc KcτD τI
]
T
(6)
It is noted that, even though the process model is not known, when the process is subject to the measured input signal u(t), it generates y(t), i.e., the corresponding output signal available for the proposed PID design. Therefore, a good controller generates u(t) or its Laplace transform U(s) for that matter when the error signal is given by R(s) - Y(s). Since U(s) is known, the controller design problem is equivalent to minimizing the ˆ (jω) given in eq 4 and U(jω) in a frequency difference between U range [0 ωmax] formulated as follows: ˆ | 2 ) |Φ - ΨW| 2 min min J ) |Φ - Φ
(2)
where Y(s) is the Laplace transform of y(t) and R(s) is called the virtual reference signal because it does not actually exist. As Y(s) is considered to be the desired output of the feedback system when the reference signal is specified by R(s), the corresponding controller’s output is calculated as
(7)
W
θ
where θ is a parameter of the desired loop transfer function L(s) to be discussed in the ensuing development, and Φ ) [U(jω0) U(jω1) · · · U(jωn-1) ]T
[
ˆ (jω0) U ˆ (jω1) · · · U ˆ (jωn-1) ]T ˆ ) [U Φ
2. Proposed PID Controller Design
Y(s) )
)
1353
-Ω(jω0) j ω0 -Ω(jω1) Ω(jω1) j ω1 Ψ) l l -Ω(jωn-1) Ω(jωn-1) j ωn-1 Ω(jω0)
Ω(jω0) jω0 Ω(jω1) jω1 l Ω(jωn-1) jωn-1
(8)
]
(9)
(10)
where frequency responses of U(jωi) and Y(jωi) at various frequencies ωi ) iωmax/(n - 1) are obtained using the discrete Fourier transform (DFT) of process input and output data collected from a pulse input test in this paper. Our simulation experience shows that the proposed design method is insensitive to the different pulse signals used in the open-loop tests. Finally, the frequency ωmax is the upper bound of the frequency range for the DFT computation. Discussion concerning the specification of ωmax will be provided in section 2.1. After some algebra, eq 7 is recast as11 ˜ -Ψ ˜ W| 2 min min J ) |Φ θ
[
(11)
W
]
˜ ) Re(Φ) ; Φ Im(Φ)
[
˜ ) Re(Ψ) Ψ Im(Ψ)
]
(12)
where Re(A) and Im(A) denote the real matrix (or vector) with elements being the real and imaginary parts of a complex matrix (or vector) A, respectively.
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Figure 4. Input and output data used for the proposed PID design. Figure 3. Solution of ω ˆ b for eq 16.
For a given θ, eq 11 is solved by the least-squares method as ˜ TΨ ˜ )-1Ψ ˜ TΦ ˜ W*(θ) ) (Ψ
(13)
In the proposed design, by searching the smallest J(θ) in the specified range of θ, say J(θ*), its corresponding solution W*(θ*) is used to obtain the PID parameters. 2.1. Specification of L(s) and ωmax. In this paper, the following desired loop transfer function L(s) is considered.16 L(s) )
0.76(1 + 0.47θs) -θs e θs
(14)
It is clear from eq 14 that the minimization problem in eq 11 is implicitly dependent on θ. As θ is related to the apparent time delay of the process, a reasonable choice for the prespecified range of θ should cover the apparent time delay of the process, which is available from open-loop experiment or a priori process knowledge. Next, the frequency range specified for the DFT in eq 11 is addressed. When ωmax is fixed, we can select a sufficiently large n to represent process frequency response with better resolution. On the other end, because ωmax is closely related to the controller design, it is logical to specify ωmax as the bandwidth frequency of the feedback system, ωb, which is the frequency where the amplitude ratio of the complementary sensitivity function H(s) ) [G(s)C(s)]/[1 + G(s)C(s)] crosses 1/√2 from above.35 However, since process model and controller are unknown, the information of ωb is not available prior to controller design. The only way to evaluate |H(jω)| ) 1/√2 is by the reference model. Thus, for each θ, ωb is obtained by solving
|
L(s) 1 + L(s)
|
1 √2 Define sˆ ) θs and let sˆ ) jω ˆ , eq 15 reduces to )
s)jωb
+ 0.47sˆ) | sˆ +0.76(1 0.76(1 + 0.47sˆ)e | -sˆ
) sˆ)jω ˆb
1 √2
(15)
ω ˆb 2.94 ) θ θ
3. Examples and Results 3.1. Linear Processes. Consider a high-order process given by G1(s) )
(16)
(17)
e-8s (2s + 1)3(s + 1)2
To proceed with the proposed PID design, a pulse input is used to generate the process data u(t) and y(t) as shown in Figure 4. With the specified range of θ ) [6 15], the optimal solutions are obtained by θ* ) 11.87 and W* )[0.63 0.065 1.74]. Consequently, the PID controller obtained by the proposed design is given by
(
C1(s) ) 0.63 1 +
1 + 2.77s 9.63s
)
(18)
For comparison purposes, three model-based PID controllers are also designed based on the following first-order plus time delay (FOPDT) and second-order plus time delay (SOPDT) models identified from a step test. 1 e-12.63s 3.73s + 1
(19)
1 e-10.97s 6.64s2 + 5.20s + 1
(20)
M1(s) )
The solution ω ˆ b for eq 16 is obtained numerically as illustrated in Figure 3, where ω ˆ b ) 2.94. Consequently ωb )
In summary, the implementation of the proposed PID controller design method is described as follows: Step 1. Given the process data u(t) and y(t) obtained from an open-loop test and a prespecified range of θ. Step 2. The minimization problem given in eq 11 is solved by iteration and for each chosen θ, the corresponding ωb is calculated by eq 17 and set ωmax ) ωb. Step 3. The frequency responses of U and Y are calculated by applying the DFT to the process data collected in step 1. Step 4. Repeat step 2 and step 3 and obtain the smallest J(θ) in the specified range of θ, while the corresponding W*(θ*) is used to calculate the PID parameters.
M2(s) )
The first model-based PID design is the Connell-PID23 algorithm, which gives the best control performance among its class23 (i.e., without a tunable parameter) for almost all the processes considered in this paper. The other two model-based
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Table 1. PID Controllers Obtained by Various Design Methods for G1(s) PID parameters design methods
Kc
τI
τD
τF
IAE
proposed design Connell-PID IMC-PID (FOPDT) with λ ) 5.2 IMC-PID (SOPDT) with λ ) 0.04 Maclaurin-PID (FOPDT) with λ ) 3 Maclaurin-PID (SOPDT) with λ ) 2.17
0.63 0.47 0.56
9.63 21.05 10.05
2.77 5.05 2.35
1.84
18.72 37.67 21.35
0.24
5.20
1.28
0.02
23.87
0.56
8.84
2.67
18.70
0.58
8.83
2.75
18.77
PID design methods are the IMC-PID6,36 and Maclaurin-PID8 algorithms having λ as a tunable parameter. As the resulting control performances obtained by these two design methods vary with the selected value of λ, the adjustable parameter λ is tuned by trial and error via simulation studies involving an a priori known process model G1(s). The optimal λ is thus determined by giving the best control performance as measured by the integral absolute error (IAE) index. The design equations used for these model-based PID controllers are given in the Appendix for ease of reference. Table 1 summarizes the model-based PID controllers designed for G1(s) together with the optimal values of λ, if applicable. Figure 5 compares the servo performances of the proposed PID design C1(s) and various model-based PID controllers. It is clear that the proposed design gives better control performance than the Connell-PID and IMC-PID designs, leading to the respective reductions of IAE by 50.31%, 12.32% (IMC-PID using the FOPDT model), and 21.58% (IMC-PID using the SOPDT model). In contrast, the proposed design is marginally better than the Maclaurin-PID designs. It is worthwhile pointing out again that both IMC-PID and Maclaurin-PID designs require either a reasonably accurate process model to be used in the closed-loop simulation studies or the availability of online closed-loop tests conducted in the actual plant in order to determine their respective optimal values of λ by trial and error, which would demand considerable engineering efforts. In comparison, the proposed PID design obtains PID parameters in a direct manner by using the process data collected from an off-line experiment. Next, the sensitivity of the proposed PID design method to the effect of process noise is investigated. Figure 6 shows the
Figure 6. Input and output data used for the proposed PID design (with noise).
process input and output data under (5% Gaussian white noises, which are now used in the proposed PID design. By using the same prespecified range of θ as before, the resulting PID controller is given by
(
C2(s) ) 0.62 1 +
)
(21)
Compared with the setting of PID controller obtained under noise-free condition, i.e. C1(s), no dramatic change in the PID parameters is seen, which illustrates the robustness of the proposed direct PID design method. Besides the process G1(s) studied above, the proposed PID design method is applied to other processes exhibiting a wide range of process dynamics,17,18 such as dynamics with long time delay (G8), zero dynamics (G7), underdamped dynamics (G5, G6), and higher-order dynamics (G2, G9), which are given in the following: G2(s) ) G3(s) )
G5(s) )
1 (s + 1)20
1 e-3s 2 (s + 10s + 1)(s + 1) 2
G4(s) )
1 e-1.5s (s + 1)5
1 e-2.2s (4s2 + 2.8s + 1)(s + 1)2
G6(s) )
1 e-2s (9s + 2.4s + 1)(s + 1) 2
G7(s) )
1.5s + 1 e-s (2s + 1)(s + 1)3
G8(s) )
1 e-20s 20s + 1
G9(s) )
Figure 5. Set-point responses of the proposed controller C1(s) and modelbased PID designs using FOPDT (top) and SOPDT (bottom) models.
1 + 2.94s 9.73s
1 (10s + 1)8
The PID controllers designed by the proposed method and their IAEs are summarized in Table 2, while those of the modelbased PID designs are given in Tables 2 and 3. The control
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Table 2. Proposed PID and Connell-PID Designs for G2(s)-G9(s) proposed design
Connell-PID
process
Kc
τI
τD
IAE
Kc
τI
τD
IAE
G2 G3 G4 G5 G6 G7 G8 G9
0.62 2.11 0.89 0.73 0.56 1.40 1.13 0.93
11.93 12.07 4.29 4.29 2.70 3.22 29.41 52.89
3.56 2.46 1.65 2.01 4.29 1.26 6.39 21.85
23.72 9.12 6.77 8.23 9.39 3.98 33.21 83.35
0.42 3.04 0.76 0.54 0.46 1.27 1.6 0.77
27.28 8.68 7.61 9.28 8.97 4.32 33.33 94.30
6.55 2.08 1.83 2.23 2.15 1.04 8 22.63
51.60 10.28 9.71 15.10 17.43 3.49 39.97 116.83
Table 3. IMC-PID and Maclaurin-PID Designs for G2(s)-G9(s) IMC-PID process
model
Kc
G2
FOPDT SOPDT FOPDT SOPDT FOPDT SOPDT FOPDT SOPDT FOPDT SOPDT FOPDT SOPDT FOPDT SOPDT FOPDT SOPDT
0.56 0.21 2.38 1.12 0.97 0.43 0.58 0.41 0.38 0.41 1.27 0.95 1.24 0.57 0.98 0.44
G3 G4 G5 G6 G7 G8 G9
τI
τD
Maclaurin-PID τF
12.46 2.81 2.20 6.05 1.48 0.02 12.50 2.06 0.02 10.44 0.52 0.02 4.47 1.12 0.02 3.08 0.75 0.02 4.66 1.12 0.86 3.21 1.46 0.02 4.22 0.98 1.37 2.46 3.68 0.02 3.35 0.80 0.02 2.98 0.72 0.02 30 6.67 1.74 21.37 1.29 0.02 55.46 13.86 0.02 38.38 9.40 0.02
IAE
Kc
27.01 30.96 8.91 10.60 7.45 8.05 9.70 9.18 11.89 10.10 3.89 4.19 37.66 41.12 89.37 98.21
0.55 0.56 2.40 2.18 0.86 0.85 0.62 0.64 0.30 0.53 1.29 1.63 1.21 1.22 0.94 0.90
τI
τD
IAE
11.01 3.40 23.62 10.99 3.45 23.70 12.50 2.24 8.88 12.31 2.06 8.97 4.29 1.35 6.82 4.24 1.36 6.83 4.17 1.27 8.55 4.10 1.67 8.29 2.98 0.58 10.90 2.95 3.29 9.65 3.35 0.96 3.87 3.49 1.05 3.77 28.51 6.52 34.24 28.51 6.50 34.28 54.74 18.07 82.99 53.54 17.48 83.24
performances of the proposed PID design for these processes are compared with the model-based controller designs in Figures 7 and 8, respectively. It can be seen that the proposed controller design outperforms the Connell-PID design except for G7, and gives better performance than those of IMC-PID designs for processes G2, G4, G5, G6, G8, and G9. In contrast, for these six processes, the proposed design gives comparable or slightly better performances than those of Maclaurin-PID designs. For G3 and G7, the proposed design is slightly inferior to the IMCPID design using the FOPDT model and Maclaurin-PID design, but it gives slightly better performance than the IMC-PID design using the SOPDT model. Again, the PID parameters of the IMCPID and Maclaurin-PID designs have been obtained by optimizing the tunable parameter λ in the simulated closed-loop tests for set-point change involving the actual process models G2-G9, which is not required for the proposed design method. 3.2. Nonlinear Process. Consider an isothermal free-radical polymerization of methyl methacrylate (MMA) that takes place in a jacket CSTR using azobisisobutyronitrile (AIBN) as initiator and toluene as solvent. The control objective is to regulate the product number-average molecular weight (NAMW) by manipulating the flow rate of the initiator (FI). The dynamics of the reactor can be described by the following equations:37 F(Cmin - Cm) dCm ) -(kp + kfm)CmP0 + dt V
FD1 dD1 ) Mm(kp + kfm)CmP0 dt V y)
D1 D0
where P0 )
(
2f kICI kTd + kTc
)
0.5
The model parameters and steady-state operation condition used are the same as those reported in the literature.33,37 To proceed with the proposed PID design, multiple step changes around the nominal value of process input FI are introduced. Both process input and output data are corrupted by 5% Gaussian white noise as shown in Figure 9. Using this data, the PID controller obtained by the proposed design is as follows:
(
C3(s) ) -5.26 1 +
1 + 0.0078s 0.17s
)
(23)
For comparison purposes, Table 4 summarizes the optimal IMC-PID and Maclaurin-PID designs based on the following FOPDT and SOPDT models:
FICIin - FCI dCI ) -kICI + dt V dD0 FD0 ) (0.5kTc + kTd)P02 + kfmCmP0 dt V
Figure 7. Set-point responses of the proposed PID design and model-based PID designs using FOPDT models.
-0.44 -0.058s e 0.14s + 1
(24)
-0.43 e-0.049s 0.0047s2 + 0.16s + 1
(25)
M1(s) )
(22) M2(s) )
Figures 10 and 11 compare the servo performance of the proposed PID controller with the model-based PID controllers. It can be seen that the proposed PID controller provides
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Table 4. PID Controllers Designed by Various Methods for Polymerization Reactor PID parameters
design methods proposed design IMC-PID (FOPDT) with λ ) 0.04 IMC-PID (SOPDT) with λ ) 0.03 Maclaurin-PID (FOPDT) with λ ) 0.04 Maclaurin-PID (SOPDT) with λ ) 0.01
Kc
τI
τD
IAE
τF
+10% -10% set-point set-point change change
-5.26 0.17 0.008 -3.95 0.17 0.024 0.012
392.8 441.2
396.5 479.7
-2.79 0.16 0.030 0.012
500.6
562.3
-3.67 0.16 0.015
421.9
459.6
-5.72 0.17 0.042
372.3
409.6
Maclaurin-PID designs are tuned by trial and error in the simulated closed-loop tests involving eq 22 as the actual process model. In contrast, the proposed design method obtains PID controller C3(s) based on the process data shown in Figure 9 exclusively in a straightforward manner without resorting to the availability of actual process model. 4. Conclusions
Figure 8. Set-point responses of the proposed PID design and model-based PID designs using SOPDT models.
In this paper, a systematic one-step procedure is developed to design a PID controller directly based on the process data collected from an open-loop experiment. This is in sharp contrast to the IMC-based PID design methods that require trial-anderror procedure to determine the optimal tunable parameter λ from simulation studies requiring prior information of process dynamics or plant tests, which demand considerable engineering efforts. Extensive simulation results show that the proposed PID design provides better or comparable control performance compared to the three model-based PID design methods, i.e., Connell-PID, IMC-PID, and Maclaurin-PID designs, considered in this paper. Therefore, the proposed design is able to retain the simple design procedure of direct model-based PID design methods while achieving comparable or better performance than the IMC-based PID design methods. Consequently, the proposed design is an attractive alternative to the model-based PID design methods. Lastly, it is noted that the proposed PID design is
Figure 9. Input and output data used for the proposed PID design (polymerization reactor).
comparable control performance to the Maclaurin-PID design using the SOPDT model, while it gives better performance than those obtained by the IMC-PID design and the Maclaurin-PID design using the FOPDT model. Lastly, both IMC-PID and
Figure 10. Set-point responses of the proposed PID design and IMC-PID design using FOPDT (left) and SOPDT (right) models.
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Kc )
τI , K(λ + θ)
τI ) τ +
θ2 , 2(λ + θ) θ2 θ τD ) 12(λ + θ) 3τI
(
)
(A5)
Likewise, for processes approximated by the following SOPDT model: M)
K e-θs τ s + 2ςτs + 1
(A6)
2 2
The IMC-PID design is given by36 Kc )
τI , K(λ + 2θ)
τI ) 2ςτ,
τD )
τ2 , 2ςτ
τF )
λθ λ + 2θ (A7)
The Malcaurin-PID design obtains PID parameters as follows:8 Kc )
τI , K(2λ + θ)
τI ) 2ςτ -
2λ2 - θ2 , 2(2λ + θ) τ2 -
τD ) τI - 2ςτ +
Figure 11. Set-point responses of the proposed PID design and MaclaurinPID design using FOPDT (left) and SOPDT (right) models.
developed primarily for servo response as the reference model is based on the L(s) that is obtained from servo response consideration. Therefore, when the regulatory response is of concern, a new expression of L(s) is required which warrants further investigation. Appendix: Connell-PID, IMC-PID, and Maclaurin-PID Designs In this paper, the Connell-PID, IMC-PID, and Maclaurin-PID designs are the benchmark model-based PID designs in relation to the proposed PID design. The controller parameters for the first method are based on the FOPDT model only, while the latter two design methods are based on the FOPDT and SOPDT models. For processes approximated by a FOPDT model: M(s) )
K e-θs τs + 1
(A1)
The Connell-PID controller is given by23 Kc )
1.6τ , Kθ
τI ) 1.6667θ,
τD ) 0.4θ
(A2)
The formulation of an IMC-PID controller augmented with a low-pass filter is as follows:6
(
C(s) ) Kc 1 +
Kc )
2τ + θ , 2K(λ + θ)
τI ) τ +
)
1 1 + τDs τIs τFs + 1 θ , 2
τθ , 2τ + θ λθ τF ) 2(λ + θ)
(A3)
τD )
(A4)
where λ is the tuning parameter. In addition, the following gives Maclaurin-PID design based on the FOPDT model:8
θ3 6(2λ + θ) τI
(A8)
Literature Cited (1) Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759–768. (2) Chen, D.; Seborg, D. E. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 2002, 41, 4807– 4822. (3) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley & Sons: New York, 2004. (4) Garcia, C. E.; Morari, M. Internal model control. 1. A unifying review and some new results. Ind. Eng. Chem. Process Des. DeV. 1982, 21, 308– 323. (5) Rivera, D. E.; Morari, M.; Skogestad, S. Internal model control. 4. PID controller design. Ind. Eng. Chem. Process Des. DeV. 1986, 25, 252– 265. (6) Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. (7) Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve controller performance. Chem. Eng. Prog. 1990, 86, 33–41. (8) Lee, Y.; Park, S.; Lee, M.; Brosilow, C. PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE J. 1998, 44, 106– 115. (9) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291–309. (10) Åstro¨m, K. J.; Ha¨gglund, T. PID Controllers: Theory, Design and Tuning; Instrument Society of America: Research Triangle Park, NC, 1995. (11) Wang, Q. G.; Hang, C. C.; Bi, Q. A frequency domain controller design method. Chem. Eng. Res. Des. 1997, 75, 64–72. (12) Tan, K. K.; Wang, Q. G.; Hang, C. C. AdVances in PID Control; Springer-Verlag: New York, 1999. (13) Yu, C. C. Autotuning of PID Controllers: Relay Feedback Approach; Springer: New York, 1999. (14) Sung, S. W.; Lee, I. B. An improved algorithm for automatic tuning of PID controllers. Chem. Eng. Sci. 2000, 55, 1883–1891. (15) Sung, S. W.; Lee, T. Y.; Park, S. Optimal PID controller tuning method for single-input/single-output processes. AIChE J. 2002, 48, 1358– 1361. (16) Huang, H. P.; Jeng, J. C. Monitoring and assessment of control performance for single loop systems. Ind. Eng. Chem. Res. 2002, 41, 1297– 1309. (17) Huang, H. P.; Jeng, J. C. Identification for monitoring and autotuning of PID Controllers. J. Chem. Eng. Jpn. 2003, 36, 284–296. (18) Huang, H. P.; Jeng, J. C.; Luo, K. Y. Auto-tune system using singlerun relay feedback test and model-based controller design. J. Process Control 2005, 15, 713–727. (19) Toscano, R. A simple robust PI/PID controller design via numerical optimization approach. J. Process Control 2005, 15, 81–88. (20) Vilanova, R. IMC-based robust PID design: tuning guidelines and automatic tuning. J. Process Control 2008, 18, 61–70.
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 (21) Shamsuzzoha, M.; Lee, M. Design of advanced PID controller for enhanced disturbance rejection of second-order processes with time delay. AIChE J. 2008, 54, 1526–1536. (22) Panda, R. C. Synthesis of PID tuning rule using the desired closedloop response. Ind. Eng. Chem. Res. 2008, 47, 8684–8692. (23) O’Dwyer, A. Handbook of PI and PID controller tuning rules; Imperial College Press: London, 2006. (24) Ang, K. H.; Chong, G.; Li, Y. PID control system analysis, design, and technology. IEEE Trans. Control Syst. Technol. 2005, 13, 559–576. (25) Li, Y.; Ang, K. H.; Chong, G. C. Y. Patents, software and hardware for PID controlsAn overview and analysis of the current art. IEEE Control Syst. Mag. 2006, 26, 42–54. (26) Hjalmarsson, H.; Gunnarsson, S.; Gevers, M. A Convergent iterative restricted complexity control design scheme. Proceedings of 33rd IEEE Conference on Decision and Control; IEEE: New York, NY, 1994; pp 1735-1740. (27) Hjalmarsson, H.; Gevers, M.; Gunnarsson, S.; Lequin, O. Iterative feedback tuning: theory and applications. IEEE Control Syst. Mag. 1998, 18, 26–41. (28) Spall, J. C.; Cristion, J. A. Model-free control of nonlinear stochastic systems with discrete-time measurements. IEEE Trans. Autom. Control 1998, 43, 1198–1210. (29) Guardabassi, G. O.; Savaresi, S. M. Approximate feedback linearization of discrete time nonlinear systems using virtual input direct design. Syst. Control Lett. 1997, 32, 63–74. (30) Savaresi, S. M.; Guardabassi, G. O. Approximate I/O feedback linearization of discrete time nonlinear system via virtual input direct design. Automatica 1998, 34, 715–722.
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(31) Campi, M. C.; Lecchini, A.; Savaresi, S. M. Virtual reference feedback tuning (VRFT): a new direct approach to the design of feedback controllers. Proceedings of 39th IEEE Conference on Decision and Control; IEEE: Piscataway, NJ, 2000; pp 623-629. (32) Guardabassi, G. O.; Savaresi, S. M. Virtual reference direct design method: an off-line approach to data-based control system design. IEEE Trans. Autom. Control 2000, 45, 954–959. (33) Kansha, Y.; Hashimoto, Y.; Chiu, M. S. New results on VRFT design of PID controller. Chem. Eng. Res. Des. 2008, 86, 925–931. (34) Campi, M. C.; Lecchini, A.; Savaresi, S. M. Virtual reference feedback tuning: a direct approach for the design of feedback controllers. Automatica 2002, 38, 1337–1346. (35) Skogestad, S.; Postlethwaite, I. MultiVariable Feedback Control: Analysis and Design; Wiley: New York, 1996. (36) Zhang, W.; Xi, Y.; Yang, G.; Xu, X. Design PID controllers for desired time-domain or frequency-domain response. ISA Trans. 2002, 41, 511–520. (37) Doyle, F. J.; Ogunnaike, B. A.; Pearson, R. K. Nonlinear modelbased control using second-order volterra models. Automatica 1995, 31, 697–714.
ReceiVed for reView April 1, 2010 ReVised manuscript receiVed September 1, 2010 Accepted September 7, 2010 IE100784K
