Znd. Eng. Chem. Res. 1995,34, 4127-4132
4127
Modified Proportional-Integral Derivative (PID) Controller and a New Tuning Method for the PID Controller Su Whan Sung and In-Beum Lee* Department of Chemical Engineering, Pohang University of Science and Technology, S a n 31 Hyoja Dong, Pohang 790-784, Korea
Jitae Leet Department of Chemical Engineering, Kyungpook National University, Taegu 702-701,Korea
The proportional-integral derivative (PID) controller is widely used in industry because of its simplicity and ability to solve the majority of the control problems. However, there are rare tuning rules to tackle general nonlinear processes. Therefore, a systematic tuning rule for the linear or nonlinear process models is proposed in this paper. Basically the control performance of the PID controller has limitations due to the f x e d structure and only three tuning parameters Moreover, the PID controller tuned well for the set point change process cannot work well for the disturbance rejection process. We thus develop a modified PID controller to achieve a good control performance in both the set point tracking and disturbance rejection process.
Introduction The proportional-integral derivative (PID) control algorithm is widely used in the process industries even though many advanced control algorithms have been developed. This is due to its simple structure and robustness to the modeling error. There are many identification methods to obtain the first order plus time delay model or the second order plus time delay model (process reaction curve method, Lee and Sung (1993), Lee (19891, Sung et al. (19941, Chen (19891, Jutan and Rodriguez (19841, Huang et al. (19821, Huang and Huang (19931, Huang and Chou (1994)). Also, there are many tuning rules developed for the first order plus time delay model such as ZN (Ziegler-Nichols), IMC (Internal Model Control), ITAE (the Integral of the Time weighted Absolute value of the Error), Cohen-Coon tuning methods (Ziegler and Nichols (19421, Lopez et al. (19671, Cohen and Coon, Morari and Zafiriou (1989)). The IMC tuning rule shows a superior performance to the other tuning rules for the set point tracking but an inferior performance in the disturbance rejection because this tuning rule used the concept of the IMC control (pole-zero cancellation). Usually, the ITAE-load tuning rule shows a superior performance in the disturbance rejection in comparison with Cohen-Coon, ZN (Miller et al. (1967)), and the IMC tuning rule. Until now, general tuning rules have not been developed for high-order linear and nonlinear models except the ZN tuning rule. The ZN tuning rule uses the ultimate information of the process. Therefore, it cannot systematically consider the concrete control performance to tune the PID parameters. Liaw (1992) proposed a systematic tuning rule for the development of highperformance motor drives, where the tuning parameter is systematically determined so that the motor position tracks the desired trajectory defined by the user. However, his method can be applied t o only the firstorder process and needs a root-finding technique. In recent years, many modified PID controllers are proposed to improved the control performance. For * T o whom all correspondence should be addressed. Email:
[email protected]. FAX: 82-562-279-2699. E-mail:
[email protected]. FAX: 82-053-950+
6615.
example, the proportional gain can be varied corresponding to the magnitude of the error (Luyben (1989)). The control action becomes more aggressive as the process output further deviates from the set point. Cheung and Luyben (1980) have studied a PI controller with proportional and integral gain varied as a function of the error. Although the controller of these types shows better control performance than the linear PID controller, it is difficult to tune these nonlinear controllers. A nonlinear PI-type algorithm proposed by Jutan (1989) uses trends of the past errors or process measurements instead of the derivative term of the error of the linear PID controller. This controller shows better performance than the linear PID controller in terms of sensitivity to measurement noise and ITAE. Wang and Rugh (1987) consider some basic properties between a nonlinear system and its family of linearized systems at constant operating points. Rugh (1987) designs a nonlinear PID controller with parameters that are continuous functions of the manipulated input corresponding to the operating point. The nonlinear PID controller maintains the same response characteristics at various operating points. However, these modified PID controllers have limitations to the nominal control performance. Moreover, even through the tuning parameters are well tuned for the set-point change process, this controller cannot work well for the disturbance rejection process. A systematic tuning rule using a desired trajectory is now proposed. More or less, this method is similar t o Liaw’s (1992) method. Here, the three tuning parameters of the PID controller are determined by leastsquares method so that the response of the process may be close to the desired trajectory as much as possible. Because the desired trajectory is determined by the user, characteristics of the closed-loop response can be predicted and the operating region can be directly considered. Moreover, the tuning procedure can be done with the consideration of the concrete control performance. In addition, a modified PID controller with a time variant bias term is proposed to track the desired trajectory more accurately. Here, we add a time variant bias term to a PID-type controller. This modified PID controller shows a perfect nominal control performance for the set-point change processes. If we use a linear
0888-5885/95/2634-4127$09.oo/o0 1995 American Chemical Society
4128 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
PID controller as the PID-type controller, the bias term is time variant for the time being after the set point is changed and soon becomes constant. Therefore, the advantages of a linear PID controller such as the robustness to the modeling error can be guaranteed. Moreover, instead of a linear PID controller, modified PID controllers (Luyben (1989), Rugh (19871, and Cheung and Luyben (1980)) can be used as a PID-type controller t o guarantee the superiorities of these controllers for the disturbance rejection. The PID part of the modified PID controller is well tuned for the disturbance rejection to obtain a good control performance for both the set-point tracking and regulatory processes.
that is,
Theoretical Development Let us develop a tuning rule of a linear PID controller for the set-point tracking process. Assume the process model and the PID controller are described as follows.
where y ( t ) and u ( t ) denote the process output and controller output and b and y W ) represent the bias term and the ith derivative of the process output. If the desired trajectory satisfies the following conditions, then the perfect nominal control performance can be achieved.
Therefore, we can estimate the tuning parameters of the PID controller by solving (10). Here, we can reduce the computation load by considering the final steady state. When the final steady state is obtained under the perfect tracking, the following equation should be satisfied.
Therefore, using this condition, we can estimate more easily the tuning parameters of the PID controller from the following equations.
where yd(t) denotes the desired trajectory. Assume that the initial state is a steady state and shows zero offset; then (3) and (4) become the following simple forms:
y d ( t )= y ( t ) = constant, 0
It
I8
(6)
For some desired trajectory determined by the user, condition (4) may not be satisfied exactly because the tuning parameters are constant. Therefore, we propose a least-squares method to obtain the three tuning parameters satisfying the condition (4) as much as possible. That is,
It should be noted that this least-squares problem reduces to a one-dimensional problem when the PI controller is used. The desired trajectory can be determined in various ways. For example, the following desired trajectory can be used for the step set-point change.
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 4129 1.6
*...- ..*
1.o
where z determines the desired control performance. A similar procedure can be done to find the tuning parameters for the disturbance rejection. Consider the following process which has an input disturbance.
0.6 0.4
0.2
-
0.0
0
dis(t) = 0.0, t I0.0
I
I
1
2 time
3
I 4
(19)
For a given dis(t), the condition (3) cannot be satisfied unless we solve the differential equation (18) from t = 0 to t = 8. That is, we cannot consider the initial state at t = 8 when we design a desired trajectory without solving the differential equation. However, any wrong initial derivative value of the desired trajectory does not result in a series problem. Similarly to the set-point case, we can obtain the tuning parameters from a leastsquares method. fk
a
1
b
= fi(")( tk+8),y'n-1'(tk+8)y'n-2)(tk+8),. ..,
y'"(tk+8)J(tk+8)) - b - diS(tk) (20)
We can also reduce the computation load by using the final steady-state condition. (23)
For example, the following desired trajectory can be determined for a step input disturbance. (yd
- d)(s) = a(-yl(s)
d =y ( 0 ) where
21, 22,
recommend the relation of 22 = ~1/2.5based on the analysis of several simulation results. In addition, we propose a modified PID controller to obtain the perfect nominal control performance for the set-point change process and a good control performance for the regulatory process. In general, PID controllers including a nonlinear PID controller cannot guarantee good control performances for both set-point change and disturbance rejection. However, our modified PID controller can solve this problem. The proposed controller is composed of two parts (PID-type and bias parts) as follows:
+ y2(s))
(24)
and a are decided by the user. We
where PID(y(t)) is the output of such a PID controller as linear Luyben's (1989) or Cheung and Luyben's (1980) PID controllers. It is assumed that the desired trajectory satisfies (31, and the tuning parameters of the PID-type controller are well tuned for the disturbance rejection to achieve an effective disturbance rejection. The bias term b(t) is then obtained from the following equation t o perform the perfect nominal control performance for the set-point change.
4130 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 1.6
b ( t )= fCyi;l)(t+e)yi;l-l)(t+e)y~-')(t+e), ...,
1.4
y',l'(t+e)yd(t+e))- PID(yd(t)) (26) If the desired trajectory satisfies the condition (3) and b(t) is obtained from (261, perfect nominal tracking is possible. When the model has some errors, the PID controller can reject the effects of the modeling error. That is, b(t) and PID-type controller become an open loop control part and a feedback control part, respectively. If any desired trajectory has a constant steadystate value, the bias term becomes a constant value after several time units. Therefore, our proposed PID controller returns to the PID controller of a certain type so that the superiorities of a PID-type controller for the disturbance rejection process can be guaranteed.
Simulation Study
....'...... -
Response with modeling error Response without modeling error
0.6 0.4
0.2
"."
iI 0
I
1
I
I
I
2
4
6
8
10
time
Figure 2. Control results of the modified PID controller in Example 1.
When (24) is chosen as the desired trajectory, the following equations can be obtained for a step input disturbance rejection from (20)-(23).
Example 1. Consider the following first order plus time delay process controlled by a linear PI controller. (27) If we choose (16) as the desired trajectory for a step setpoint change, the following equations can be obtained. yd(t) = ~ ( 0+)(y, - y(O))(l - exp(-(t - e)/z)), t L
e z(t)= -a(-tl exp(-t - 8/zl)+ z2 exp(-t - 8/zJ
i(-)
+ e)
= (y, - y(o))(r
(31)
+ tl - t z ) (37)
&klXk' k
From (12)-( E), we can find the following relation.
(40)
xXklxk1 k
Here, the final time t, should be chosen so large that the desired trajectory can almost reach to a steady state after this time, and we recommend a large number of t k . Simulation results and data are shown in Figure l a and Table 1. The ZN tuning relation shows an aggressive control action for the set-point change process.
Simulation results are shown in Figure lb, and the data for this simulation are shown in Table 1. From the ZN tuning results in Figure 1, we can realize that, if the linear PI controller is well tuned for the disturbance rejection, then this controller shows a large overshoot for the set-point change process. To overcome this problem, the following modified PI controller can be designed as mentioned above.
where yd(t),p ( t ) ,and z(t)are the same as (28)-(30). From Figure 2, the modified PI controller shows good control
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 4131
loo 90
80
11
'.-'-'
.- -
75
Conventional PID Rugh's( 1987) Method Promed Method Desired Trajectory ]
I-
.-..
.......I..'
/". .i . - :., I
.......... Responses with modeling error
Responseswithout modeling error
h
20
0
40
60
80
100
0
20
65 60
Method
..'..,..,. Proposed Method
Desired Trajectory
3
I
I
I
I
I
1
0
20
40
60
80
100
20
"
0
0
time
Figure 4. Tuning results of several methods at y = 70 for the disturbance rejection in Example 2.
performance in both set-point change and disturbance rejection processes. From comparison of the two responses, we conclude that the linear PI controller compensates for the modeling error. Example 2. Consider a series of 3 identical, interacting tanks (Rugh(1987)).
(43)
To apply the proposed models, hl(t) should be represented in terms of y ( t ) . That is,
.
1 100
40 7
1
- \'.','.,
1
.
100
- - - Conventional PID
..-'-.Rugh's(l987)
,.-. + .'. ,/'
80
time
Figure 3. Tuning results of several methods for the set-point change from y = 50 to 70 in Example 2.
75
60
40
tlme
.
I
I
I
1
20
40
60
80
tlme
Figure 5. Control result of the modified PID controller for the set-point change from y= 50 to 70 and the disturbance rejection at y = 70 in Example 2. Table 2. Data for Simulation in Example 2 set-point change t = 2.5
fromy = 15 to 35: k, = 0.522, tl= 38.318, Zd = 5.461 fromy = 50 to 70: k, = 1.071, tl= 123.954, td = 7.215 fromy = 80 to 100: k, = 1.484, t1= 210.856, t d = 7.741 disturbance rejection a = 0.5, t1 = 2.5, 52 = ~112.5 magnitude of step input disturbance: 2 a t y = 35: k, = 6.028, t,= 6.782, td = 3.176 at y = 70: k, = 6.028, t,= 6.782, td = 3.176 at y = 100: k, = 6.610, r1 = 7.436, t d = 5.258 modified PID controller T = 2.5 time of step input disturbance: t = 50 fromy = 15 to 35: 12, = 6.028, tl= 6.782, 5d = 3.176 fromy = 50 to 70: K, = 6.028, t,= 6.782, td = 3.176 fromy = 80 to 100: k, = 6.610, tl= 7.436, rd = 5.258 process data: A = 0.5, c = 1.0 modeling error: A = 0.45, c = 1.0
We chose the desired trajectory for the step set-point change of (16) and (17) as follows.
The desired trajectory to tune the linear PID controller for the disturbance rejection is also determined as follows.
i+
+ 0.5{
a 1
If we define the desired trajectory, we can design the modified PID controller and tune the linear PID controller using (44). Note that the numerical derivative can be used to obtain the desired dhl(t)/dt for simplicity.
Z2
k}')
exp( -
k)
The data for the simulation are shown in Table 2. Tuning results for the step set-point change from y s = 50 to 70 and disturbance rejection at y s = 70 are shown in Figures 3 and 4. In this simulation, the conventional PID controller is tuned by the ZN tuning rule based on
4132 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
the operating pointy = 49 (Rugh (1987)). For various operating regions, similar control performances are obtained. The proposed tuning rule shows better tuning results than the previous tuning rules. The modified PID controller shows good control results in both servo and regulatory problems and a perfect nominal control performance for the servo problem as shown in Figure 5. Here the step input disturbance of magnitude 2 is entered a t t = 50. Good control performances for the set-point change and disturbance rejection are achieved for various operating regions.
Conclusions We proposed a more general tuning rule using a desired trajectory to systematically tune the PID controller for various process models. In addition, a modified PID controller using a time variant bias term is proposed to achieve high-quality control performance for both set point change and disturbance rejection processes. The proposed tuning method can directly consider wide operating region for the nonlinear process and desired control performance systematically. It can achieve a perfect nominal control performance and shows a good control performance in both servo and regulatory problems for simulated processes.
Nomenclature A = parameter of process a = parameter of the desired trajectory for disturbance rejection b = bias term of controller or initial controller output c = parameter of process d = initial value of process output dis = disturbance h = liquid height of tank i(t)= integral part of PID controller under perfect tracking condition k, = proportional gain of PID controller p(t) = proportional part of PID controller under perfect tracking condition t = time u(t) = controller output y ( t ) = process output Yd(t) = desired trajectory ys = set point
n = number of t k p = process s = set-point value
Literature Cited Chen, C. A Simple Method for On-Line Identification and Controller Tuning. AlChE J . 1989,35,2037. Cheung, T. F.; Luyben, W. L.’Nonlinear and Nonconventional Liquid Level Controllers. Ind. Eng. Chem. Fundam. 1980,19, 93. Cohen, G. H.; Coon, G. A. Theoretical Considerations of Retard Control. Taylor Instrument Companies’ Bulletin TDS-1OA102. Huang, C.; Chou, C. Estimation of the Underdamped Second-Order Parameters from the System Transient. Ind. Eng. Chem. Res. 1994,33,174. Huang, C.; Huang, M. Estimation of the Second-Order Parameters from the Process Transient by Simple Calculation. Ind. Eng. Chem. Res. 1993,32,228. Huang, C.; William, C.; Clements, J . Parameter Estimation of the Second-Order-Plus-Dead-Time Model. Ind. Eng. Chem. Process Des. Dev. 1982,21,601. Jutan, A. A Nonlinear PI(D) Controller. AIChE J . 1989,67,458. Jutan, A,; Rodriquez, E. S. Extension of a New Method for OnLine Controller Tuning. Can. J . Chem. Eng. 1984,62,802. Lee, J. On-Line PID Controller Tuning from a Single, Closed-Loop Test. AIChE J . 1989,35,329. Lee, J.; Sung, S. W. Comparison of Two Identification Methods for PID Controller Tuning. AIChE J . 1993,39,695. Liaw, C. M. Design of a Two-Degree-of-Freedom Controller for Motor Drives. IEEE Trans. Autom. Control 1992,37(81, 1215. Lopez, J. A.; Miller, C. L.; Smith, C. L.; Murrill, P. W. Controller Tuning Relations Based on Integral Performance Criteria. Instrum. Technol. 1967,14 (121, 57. Luyben, W. L. Process Modeling, Simulation and Control For Chemical Engineers, 2nd ed.; McGraw-Hill: New York, 1989. Miller, J. A,; Lopez, A. M.; Smith, C. L.; Murrill, P. W. A Comparison of Controller Tuning Techniques. Control Eng. 1967 14 (121, 72. Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. Rugh, W. J . Design of Nonlinear PID Controllers. AIChE J . 1987, 33,1738. Sung, S. W.; Park, H. I.; Lee, I. B.; Yang, D. R. On-Line Process Identification and Autotuning Using P-Controller. First Asian Control Conf. 1994,1 , 411. Wang, J . L.; Rugh, W. J. Parameterized Linear Systems and Linearization Families for Nonlinear Systems. IEEE Trans. Circuits Syst. 1987,34 (6), 650. Ziegler, J. G.; Nichols, N. B. Optimum Setting for Automatic Controllers. Transl. ASME 1942,64, 759.
Received for review March 1, 1995 Revised manuscript received July 14, 1995 Accepted July 28, 1995@
Greek Symbols e = time delay z = time constant of desired trajectory z, = integral tiime of PID controller t d = derivative time of PID controller
Subscripts d = desired value
IE9501510 ~~
@
Abstract published in Advance A C S Abstracts, September
15, 1995.
