PID Controller Tuning for Stable and Integrating

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Closed-Loop PI/PID Controller Tuning for Stable and Integrating Processes M. Shamsuzzoha Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie401808m • Publication Date (Web): 12 Aug 2013 Downloaded from http://pubs.acs.org on August 18, 2013

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Industrial & Engineering Chemistry Research



Closed-Loop PI/PID Controller Tuning for Stable and Integrating Process with Time Delay Mohammad Shamsuzzoha Department of Chemical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Kingdom of Saudi Arabia

Abstract— The objective of this study is to develop a new online controller tuning method in closed-loop mode. The proposed closed-loop tuning method overcomes the shortcoming of the well-known Ziegler-Nichols (1942) continuous cycling method and it can be an alternative for the same. This is a simple method to obtain the PI/PID setting which gives the acceptable performance and robustness for a broad range of the processes. The method requires closed-loop step setpoint experiment using a proportional only controller with gain Kc0. Based on simulations for a range of first-order with time delay processes, simple correlations have been derived to give PI/PID controller settings. The controller gain (Kc/Kc0) is only a function of the overshoot observed in the setpoint experiment. The controller integral and derivative time (τI and τD) is mainly a function of the time to reach the first peak (tp). The simulation has been conducted for a broad class of stable and integrating processes, and the results are compared with recently published paper of Shamsuzzoha and Skogestad.1 The proposed tuning method gives consistently better performance and robustness for broad class of processes.

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1. Introduction The proportional, integral and derivative (PID) controller is widely used in the process industries due to its simplicity, robustness and wide ranges of applicability in the regulatory control layer. The stable and integrating processes are very common in process industries in flow, level and temperature loop. Based on a survey of more than 11, 000 controllers in the process industries, Desborough and Miller2 reported that more than 97% of the regulatory controllers utilize the PI/PID algorithm. A recent survey of Kano and Ogawa3 shows that the ratio of applications of different type of controller e.g. PI/PID control, conventional advanced control and model predictive control is about 100:10:1. Although the PI/PID controller has only few adjustable parameters, they are difficult to be tuned properly in real processes. One reason is that tedious plant tests are required to obtain improved controller setting. Due to this reason, finding a simple PI/PID tuning approach with a significant performance improvement has been an important research issue for process engineers. Therefore, the objective of this paper is to develop a method that should be simpler with enhanced performance in closed-loop mode. There are variety of controller tuning approach reported in the literature4-21 and among those two are widely used for the controller tuning, one may use open-loop or closed-loop plant tests. Most tuning approaches are based on open-loop plant information; typically the plant’s gain (k), time constant (τ) and time delay (θ). One popular approach is direct synthesis (Seborg et al.4) and the direct synthesis for the disturbance (DS-d) method proposed by Chen and Seborg5, in that they obtained the PI/PID controller parameters by computing the ideal feedback controller which gives a predefined desired closed-loop response. The IMC based PI/PID tuning method has been proposed by Rivera et al.6, Skogestad7 and Shamsuzzoha and Lee8,9 for different type of processes. Although the ideal controller for both the approach are often more complicated than the PI/PID controller for time delayed processes, the controller form can be reduced to either a PI/PID controller or a PID controller cascaded with a low order filter by performing appropriate approximations of the dead time in the process model. The PI/PID tuning method based on both the approaches is simpler in use with significantly improved performance. It is well-known that the PID tuning based on both the methods give very good performance for setpoint changes but sluggish responses to input (load) disturbances for lag-dominant (including integrating) processes with θ/τ8, Fig. 7 shows that the ratio θ/tp varies between 0.25 (for τ/θ=100 with overshoot=0.1) and 0.36 (for τ/θ=8 with overshoot 0.6). It is reasonable to select the average value θ= 0.305tp which is only 15% lower than 0.36 (the worst case). Also note that for the intermediate overshoot of 0.3, the ratio θ/tp varies between 0.30 and 0.32. In summary, the integral time for a lagdominant process is


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τ I2 =2.44t p

(23)

Conclusion: Therefore, the integral time τI is obtained as the minimum of the above two values as recommended in Eq.(11-b):   b t p , 2.44t p  τ I =min  0.645 A (1- b)  

(24)

Derivative time (τD): Although a significant number of the PID controllers switched off their derivative part but proper use of derivative action can increase stability and improve the closedloop performance. The derivative action is very important for slow moving loops where overshoot is undesirable e.g., temperature loop. The motivation of this section is to develop the approach for inclusion of the derivative action from closed-loop data. In this study the derivative action is recommended for the process having    1 . The addition of the derivation action in that kind of slow process could be useful for the performance and stability improvement. Substitute the value of    I  0.5 into    1 , and after rearrangement the resulting equation is given as

 I  0.5   1

(25)



After simplification it is  I   1.5 and resulting constrain is kK c  1.0 . The corresponding closed-loop condition for the derivative action is given as: A

b

1-b 

(26)

1

Case I: For approximately integrating process (τ>> θ), where integral time is τI =8θ. In the closed-loop the time delay and tp relation is θ= 0.305tp. The derivative time τD1 in Eq. (11c) can be approximated as

 D1 

   0.305t p     0.15t p 2   2 2 2

(27)

Case II: The process with a relatively large delay, for this case integral time τI=(τ+0.5θ) and time delay in closed-loop is θ=0.43tp. For such cases, the derivative action is recommended only if τ/θ ≥ 1. Assuming the case when τ=θ, the τD2 is given from Eq. (11c) as 13 ACS Paragon Plus Environment


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 D2 

 2  2  0.43t p      0.1433t p 2   2   3 3 3

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(28)

Note: The derivative action is only recommended for the processes which have    1. The resulting criteria in the closed-loop to add derivative action is A

b  1. 1-b  

Summary: The derivative action for both the cases i.e., τD1 and τD2 are approximately same and the conservative choice for the selection of τD is recommended as  D  0.14t p

if

A

b 1 1- b 

(29)

5. Selection of Proportional Controller Gain (Kc0) It is mentioned earlier that the proposed method is valid for the overshoot between 0.1 to 0.6 however, an overshoot of around 0.3 is recommended for a better response. Sometimes achieving the P-controller gain (Kc0) via trial and error that gives the overshoot around 0.3 can be time consuming. Therefore, an effective approach to get the value of Kc0 that gives the overshoot around 0.3 is very significant for the proposed method. It is important to note that this procedure requires initial information of the first closed-loop experiment. Let us assume that the first closed-loop test has P-controller gain of Kc01 and resulting overshoot OS1 is achieved that is between 0.1 to 0.60. It is not close to recommended value of overshoot 0.30. Let the target overshoot be OS and the target P-controller gain be Kc0. In the proposed closedloop tuning method the goal is to match the performance with the PID tuning rule. This can be achieved only by maintaining a constant proportional gain Kc, regardless of the overshoot that resulted from the closed-loop setpoint test. Ideally, Kc should be the same as that determined with different overshoots from various closed-loop setpoint tests and the resulting correlation is given as: 1.55  OS1 2  2.159  OS1   1.35 Kc 01  1.55  OS2  2.159  OS  1.35 K c 0    

(30)

The above Eq. (30) gives a general guideline for choosing the P-controller gain for the next closed-loop setpoint test. As it is mentioned earlier, the proposed method is in good agreement


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with the PID setting for the overshoot around 0.3. Therefore, the overshoot in Eq.(30) is set as 0.30, and after simplification the gain for the next closed-loop test is recommended as:





K c 0  1.19 1.45  OS1   2.02  OS1   1.27 K c 01 2

(31) Note: It is not so important to achieve the precise fractional overshoot of 0.3, so few trial is sufficient to get the desire overshoot around 0.3 from above Eq.(31). A high order process given in Example E2 is considered to show the effectiveness of the proposed Eq.(31) for the calculation of the desired overshoot in the step test experiment. First trial: Let us suppose P-controller is applied with initial controller gain Kc01=0.85, and after step test the resulting overshoot comes out to be OS1=0.13. From Eq. (31), resulting controller gain for the next trial would be 1.042. Second trial: similar to first trial, use controller gain of 1.042 in second test and resulting overshoot would be 0.18. Based upon these two new information the controller gain would be 1.182, and corresponding to this controller gain the overshoot will be 0.22. 6. Final Choice of the Controller Settings (Detuning) The proposed method has been derived to match the performance with the PID tuning rule in Eq. (11). It is based on the closed-loop time constant equal to the time delay (τc=θ). In real practice one may want to use less aggressive (detuned) settings (τc>θ), or one may even want to speed up the response (τc1 corresponds less aggressive settings and F1, but in special cases one may select F1 results in more robust controller settings. A standard practice (Shamsuzzoha and Lee8,9; Chen and Seborg5) of using lead-lag set-point filter is recommended to remove the excessive overshoot from the sepoint response in the proposed method if it is required. 8. Analysis The proposed closed-loop method is based on the IMC-PID tuning rule given in Eq. (11). Therefore, it is interesting to compare the results of both the methods and ensure the effectiveness of the proposed closed-loop method. To compare the results of the both the method three typical process model has been considered and those are given below E11

  s  1 e  s  6 s  1 2 s  12

e s E17 5s  1 100e s E22 100s  1

E17 and E22 are first-order plus delay processes, similar to those used to develop the method. E22 is almost a integrating with delay process. The output responses of the proposed method are similar to the IMC-PID responses which is shown in Fig. 23 and 24. It seems that the response is almost independent of the value of the overshoot in all three cases. The comparison of the proposed and IMC-PID method has been conducted for the high order process (E11), and result is shown in Fig. 25. The model reduction technique (Half rule, Skogestad7) has been utilized to obtain the first order plus delay process and resulting process parameters are obtained as k=1, τ=7 and θ=5. As expected, the output result of the proposed method and approximated IMC-PID is close enough, its agreement with the IMC-PID method is best for the intermediate overshoot (around 0.3). The proposed tuning method is based on the IMC-PID tuning rule given in Eq. (11) whereas the setpoint overshoot method1 is based on the SIMC rule.7 It is important to note that the



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performance of both the proposed and the setpoint overshoot method mainly depends upon their original tuning rule. The performance of the SIMC and IMC-PID has been compared and also shown in Fig. 25 for the high order process plus time delay (E11). The figure clearly shows that the IMC-PID tuning rule gives better performance than SIMC rule. The same observations have been found for the several other processes, though it is not shown. It is assumed that the best controller tuning method results in the best closed-loop output response. However, since both the methods utilize some kind of model reduction techniques to convert the PI/PID controller to the closed-loop method, an approximation error necessarily occurs. On the basis of above observation, it is clear that the proposed method has better performance because of superior performance in its original IMC-PID tuning rule. The proposed method has advantage over other PI/PID tuning method because of its simplicity and consistently better performance and robustness for broad class of the processes. It also has limitation because of step test experiment in setpoint change, which might perturb the process even for a short period of time. Sometimes in the chemical process industries, setpoint step test experiment is not desirable due to several reasons. For example, changing the setpoint of a column temperature loop is not recommended because of off-specification of the products. Due to these reason, occasionally we may have limitation in setpoint step test in chemical process industries. The proposed method is based on the step test in closed-loop with proportional controller (Kc0). Suitable selection of initial controller gain (Kc0) and subsequently number of trials can significantly reduce the time of step test experiment and eventually off-specification in the product. One can stop the closed-loop experiment just after obtaining the information of first peak and valley. The required information (overshoot, tp) can be obtained after first peak and valley and then Eq.(12) can be utilized to obtain parameter b. Along with these lines one can reduce the off-specification of the product during controller tuning. It is not recommended to use large test signal amplitudes because that will cause off-specification of product and/or will excite nonlinearity.


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9. Conclusion A simple approach has been developed for PI/PID controller tuning by the closed-loop setpoint step experiment using a P-controller with gain Kc0. The PI/PID-controller settings are obtained directly from following three data from the setpoint experiment: 

Overshoot, (Δyp - Δy∞) /Δy∞



Time to reach overshoot (first peak), tp



Relative steady state output change, b = Δy∞/Δys.

If one does not want to wait for the system to reach steady state and speedup the closed-loop experiment, it is recommended to use the estimate Δy∞ = 0.45(Δyp + Δyu). The proposed PID tuning method is: K c = K c0 A F

 I  min  0.645 A



 b t p F , 2.44t p F  (1- b) 

 D  0.14t p

b 1 1- b 



if A

where, A=[1.55(overshoot)2 -2.159 (overshoot)+1.35] F is a detuning parameter. F=1 gives the “fast and robust” PI/PID settings corresponding to τc=θ. To detune the response and get more robustness one can selects F>1, but in special cases one may select F

PID Controller Tuning for Stable and Integrating

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