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PROCESS DESIGN AND CONTROL Advanced Proportional-Integral-Derivative Tuning for Integrating and Unstable Processes with Gain and Phase Margin Specifications Ya-Gang Wang* and Wen-Jian Cai School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
Proportional-integral-derivative (PID) control is widely used to control stable processes; however, its application to integrating and unstable processes is less common. In the paper, simple formulas are derived to tune the PID controller for integrating and unstable processes with time delay to meet gain and phase margin specifications. With the proposed PID tuning method, a loop transfer function with a good shape, such as phase margin 60°, gain margin over 3, and the real part close to -0.5 in low frequencies, can be obtained. This guarantees both robustness and performance. Simulation examples are given to show the performance of the method. 1. Introduction In the process control, more than 95% of the control loops are of the proportional-integral-derivative (PID) type.1 The main reason is its relatively simple structure, which can be easily understood and implemented in practice. Over the years, there are many formulas derived to tune the PID controllers for stable processes, such as Ziegler-Nichols, Cohen-Coon, internal model control, integral absolute error optimum (ISE, IAE, and ITAE), and recently proposed tuning methods.2,3 However, it is not easy to control integrating and unstable processes with time delay. Recently, some PID tuning methods for integrating and unstable processes have been proposed.5-22 However, they usually either show poor closed-loop response such as excessive overshoot and large settling time or have complicated formulas. For controller design purposes, many integrating and unstable processes are often approximated by a loworder plus time-delay model, which can be identified by the P control method4 or relay control method.5,6 Because the resulting models are usually imprecise and the parameters of all physical systems vary with the working condition and time, robustness is always a primary concern when analyzing and designing the control systems. In this paper, advanced tuning methods for the PID controller with setpoint weighting1 are proposed for integrating and unstable processes to meet both gain and phase margin specifications, which are the primary measure for control system robustness. The control scheme first adopts the internal loop design strategy proposed by Jacob and Chidambaram7 and developed by Park et al.,5 Sung and Lee,8 and Kwak et al.9 Then, simple and effective PID-type controllers with setpoint weighting are designed based on gain and phase margin specifications. With a good loop transfer function shape, such as that for a phase margin of 60°, * Corresponding author. Tel: +65 790 4220. Fax: +65 790 5471. E-mail:
[email protected].
a gain margin over 3, and the real part close to -0.5 in low frequencies, the control system guarantees both robustness and performance. Simulation examples show that the proposed tuning methods achieve better control performance and robustness compared with other methods. 2. PID Tuning Method 2.1. Controller Design Strategy. Denote controller transfer functions by Gc(s) and integrating or unstable process transfer functions by Gp(s), where Gc(s) is given by
Gc(s) ) Kp +
Ki + Kds s
(1)
We first introduce an inner feedback loop; the block diagram of the controller design strategy is shown in Figure 1. Here, the P controller (Kl) in the inner feedback loop plays an important role in changing the integrating or unstable processes to stable ones. Because e(s) ) r(s) - y(s), the process input u(s) can be written as the following equation:
(
u(s) ) Kp +
)
Ki + Kds [r(s) - y(s)] - Kly(s) s
) (Kp + Kl)
[
] (
)
Kp Ki r(s) - y(s) + + Kds e(s) Kp + Kl s (2)
Introducing b ) Kp/(Kp + Kl) and K′p ) Kp + Kl, we have
u(s) ) K′p[br(s) - y(s)] +
(
)
Ki + Kds e(s) s
(3)
Equation 3 is in the form of a 2 degree of freedom PID controller;1 thus, the block diagram of Figure 1 can be changed into a PID control loop without an inner
10.1021/ie000739h CCC: $22.00 © 2002 American Chemical Society Published on Web 05/15/2002
Ind. Eng. Chem. Res., Vol. 41, No. 12, 2002 2911
described by a first-order plus dead-time transfer function:10
Gp(s) )
K e-Ls Ts - 1
(11)
With the P controller in the inner feedback loop, the internal closed-loop transfer function Gl(s) can be obtained as Figure 1. Block diagram of a two-loop controller.
Gl(s) )
feedback loop, where K′p, Ki, Kd, and setpoint weighting b are PID settings. 2.2. Integrating Process. For controller design purposes, we adopt the following simple integrating model:
Gp(s) 1 + KlGp(s)
)
Ke-Ls Ts - 1 + KKle-Ls
(12)
Again using Taylor series expansion (6), eq 12 is written as
(4)
Ke-Ls 0.5KKlL2s2 + (T - KKlL)s + KKl - 1 (13)
With the P controller in the inner feedback loop, the internal closed-loop transfer function Gl(s) can be obtained as
Because the characteristic equation of G′p(s) should have negative poles to be stable, the following condition must be satisfied from the Routh-Hurwitz stability criterion:
Gp(s) )
Gl(s) )
K e-Ls s(Ts + 1)
Gp(s) 1 + KlGp(s)
)
Ke-Ls Ts + s + KKle-Ls 2
Kmin )
(5)
Using a Taylor series expansion, the time-delay term in the denominator of eq 5 can be approximated by
e-Ls = 1 - Ls + 0.5L2s2
Gl(s) = G′p(s) )
(6)
Kl )
Gl(s) = G′p(s) )
Here, G′p(s) denotes the second-order plus time-delay model obtained from the Taylor series expansion method. Because the characteristic equation of G′p(s) should have negative poles to be stable, the following condition must be satisfied from the Routh-Hurwitz stability criterion:
Kl < 1/KL
(8)
For optimum disturbance rejection, the P controller gain was derived by Sung and Lee8
Kl ) 0.2/KL
(9)
which satisfies the stability criterion (8). If we choose eq 9 as a design value of the P controller gain in the inner feedback loop, then eq 7 is given by
G′p(s) )
e-Ls T + 0.1L 2 0.8 0.2 s + s+ K K KL
(10)
2.3. Unstable Process. For controller design purposes, many of the unstable processes are adequately
(14)
The above expression indicates that a condition T/L > 1 for unstable processes should be satisfied. That means that the proposed method is suitable for unstable processes with small time delays. For the optimum gain margin, the P controller gain was derived by De Paor and O’Malley11
Substituting eq 6 into eq 5, Gl(s) can be expressed by
Ke-Ls (7) (T + 0.5KKlL2)s2 + (1 - KKlL)s + KKl
1 T < Kl < ) Kmax K LK
1 K
xTL
(15)
which happens to be Kl ) xKminKmax. Hence, it satisfies the stability criterion (14). We choose expression (15) as a design value of the P controller gain in the inner feedback loop, and then eq 13 is written as
G′p(s) )
(
e-Ls
)
L 1 1 0.5 xTL s2 + (T - xTL)s + K K K
(x )
T -1 L (16)
Because the integrating and unstable processes are stabilized with the P controller in the inner feedback loop, we can have a peace of mind when designing the PID controller for the stabilized processes. Notice that the stabilized processes (10) and (16) are all of secondorder plus dead-time process structure. For convenience of the outer loop controller design, we express both eqs 10 and 16 as
e-Ls as2 + bs + c
G′p(s) )
(17)
2.4. Controller Design. The PID controller transfer function (1) can also be written as
(
Gc(s) ) k
)
As2 + Bs + C s
(18)
where A ) Kd/k, B ) Kp/k, and C ) Ki/k. We choose the
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controller zeros to be equal to the poles of model G′p(s), i.e., A ) a, B ) b, and C ) c. Hence
G′p(s) Gc(s) ) ke-sL/s
(19)
where k is determined based on gain and phase margin specifications. Denoting the gain and phase margins as Am and Φm and their crossover frequencies as ωg and ωp, respectively, we have
arg[Gc(jωg)G′p(jωg)] ) -π
(20)
( )
Figure 2. Process responses for example 1.
Am|Gc(jωg)G′p(jωg)| ) 1
(21)
|Gc(jωp)G′p(jωp)| ) 1
(22)
Φm ) π + arg[Gc(jωp)G′p(jωp)]
(23)
Inserting expression (19) into eqs 20-23 gives
ωgL ) π/2
(24)
Am ) ωg/k
(25)
k ) ωp
(26)
Φm ) π/2 - ωpL
(27)
From eqs 25 and 26, we get
Amωp ) ωg
(28)
Multiply both sides of eq 28 by L, and considering eqs 24 and 27, we obtain
Φm )
(
)
π 1 12 Am
(29)
Equation 29 gives the constraints of the gain and phase margins for eq 19. Typical values of the gain and phase margins range from 2 to 5 and from 30° to 60°, respectively.1 If we assign Am ) 3, then Φm ) 60°. From eqs 24 and 25, we have
k)
π π ) 2AmL 6L
( )
(30)
Hence, PID settings for integrating and unstable processes are given, respectively, as
2π + 3 K′p ) 15KL π Ki ) 30KL2 π(T + 0.1L) Kd ) 6KL 2π b) 2π + 3
(31)
and
( x) (x )
x
1 T π T + K L 6K L T π -1 Ki ) 6KL L π xTL Kd ) 12K L 1T b) 6 L 1+ -1 π T
K′p )
T L
(32)
x
(
)x
While designing controllers, we can ignore the inner feedback loop and directly design PID controllers for integrating or unstable time delay processes by eqs 31 and 32. The proposed tuning formulas are very simple and straightforward. Moreover, simulation examples given in next section will demonstrate that the proposed control system can achieve better performance and robustness than other tuning methods. 3. Simulation Examples In the following, we will give comparisons between the proposed PID tuning method and other PID tuning methods, such as Poulin’s method,12 Tan’s method,13 and Ho’s method.14 In Poulin’s method, we choose the PID tuning formula because the PI tuning formula gives a worse performance for integrating processes. These methods are the most suitable candidates for comparison because they are the best and latest methods among existing tuning methods. Example 1. Consider the integrating process transfer function used by Poulin and Pomerleau12 and Tan et al.13
Gp(s) )
e-0.2s s(s + 1)
The proposed method yields the PID control settings as K′p ) 3.0944, Ki ) 2.6180, Kd ) 2.6704, and b ) 0.6768. The control performance of the proposed method is compared with Poulin’s and Tan’s PID tuning methods for integrating processes. Figure 2 shows the closed-loop process output responses for these three designs with a unit-step setpoint change occurring at t ) 0 s and a unit-step disturbance occurring at t ) 30 s (proposed method, solid line; Poulin’s method, dotted line; Tan’s method, dashed line).
Ind. Eng. Chem. Res., Vol. 41, No. 12, 2002 2913
Figure 3. Process responses for example 2.
Figure 4. Process responses for example 3.
The proposed method shows a better control performance for both setpoint change and load disturbance. We notice that Tan’s method also gives a good disturbance attenuation in this example even though its response for setpoint change is poor. Thus, let us consider another example to compare the proposed tuning method with Tan’s tuning method again. Example 2. Consider an integrating process with a large dead time
Gp(s) )
e-2s s(s + 1)
The proposed method yields the PID control settings as K′p ) 0.3094, Ki ) 0.0262, Kd ) 0.3142, and b ) 0.6768. The control performance of the proposed method is compared with Tan’s PID tuning method for integrating processes. Figure 3 shows the closed-loop process output responses for these two designs with a unit-step setpoint change occurring at t ) 0 s and a unit-step load disturbance occurring at t ) 80 s. The proposed method also shows a better control performance for both the setpoint change and load disturbance (proposed method, solid line; Tan’s method, dotted line). Example 3. Consider an unstable process transfer function
Gp(s) )
e-0.2s s-1
The proposed method yields the PID control settings as K′p ) 3.6833, Ki ) 3.2360, Kd ) 0.1171, and b ) 0.3929. The control performance of the proposed method is compared with the result of Ho’s method, as shown as Figure 4 (proposed, solid line; Ho’s, dotted line), where a unit-step load disturbance is added at t ) 5 s. The
Figure 5. Process responses for example 4.
Figure 6. Process responses for example 5.
proposed method shows a better control performance for both setpoint change and step input disturbance. To verify the feasibility and robustness of the proposed tuning method, let us reconsider examples 1 and 3, where we change the poles and the structure of the processes. Example 4. The process transfer function of example 1 is changed into
Gp(s) )
e-0.2s s(0.1s + 1)(s + 1.2)
while we still use the original PID settings. The control performance of the proposed method is compared with the results of Poulin’s and Tan’s PID methods, as shown in Figure 5, where a unit-step load disturbance is added at t ) 30 s (proposed method, solid line; Poulin’s method, dotted line; Tan’s method, dashed line). The proposed method shows better robustness. Example 5. The process transfer function of example 3 is changed into
Gp(s) )
e-0.2s (0.01s + 1)(s - 1.5)
while we still use the original PID settings. The control performance of the proposed method is compared with the results of Ho’s method, as shown in Figure 6, where a unit-step load disturbance is added at t ) 10 s (proposed, solid line; Ho’s method, dotted line). The proposed method shows good robustness. With the proposed method for designing PID controller, the Nyquist curves of the loop transfer functions for examples 1-3 are shown in Figure 7 (example 1, solid line; example 2, dotted line; example 3, dashed line). They are similar, and all have the desired phase margin of 60°.
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Ind. Eng. Chem. Res., Vol. 41, No. 12, 2002
Figure 7. Nyquist plots of the loop transfer functions.
Because we introduced the Taylor series expansion method in designing PID controllers, this produced model error especially in high frequencies, resulting in a bigger value of the gain margin than 3. Anyway, we obtain a loop transfer function with a good shape, like phase margin 60° and the real part close to -0.5 in low frequencies for an adequate integral effort,1 which guarantees both robustness and performance. 4. Conclusions In this paper, simple PID tuning formulas have been derived for the integrating and unstable processes with time delay. We adopted an additional inner feedback loop design technique and a designing controller based on both gain and phase margin specifications and finally obtained simple tuning formulas for PID controllers with setpoint weighting which can overcome the structural limitation of a typical PID controller for integrating and unstable processes. With the proposed PID tuning method, we can obtain a loop transfer function with a good shape, like phase margin 60°, gain margin over 3, and the real part close to -0.5 in low frequencies, which guarantee both robustness and performance. Simulation results have been given to show the performance that can be achieved. Literature Cited (1) Astrom, K. J.; Hagglund, T. PID Controllers: Theory, Design, and Tuning; Instrument Society of America: Research Triangle Park, NC, 1995. (2) Wang, Y.-G.; Shao, H.-H. PID autotuner based on gain and phase margin specifications. Ind. Eng. Chem. Res. 1999, 38, 3007. (3) Wang, Y.-G.; Shao, H.-H. Optimal tuning for PI controller. Automatica 2000, 36, 147.
(4) Yuwana, M.; Seborg, D. E. A new method for on-line controller tuning. AIChE J. 1982, 28, 434. (5) Park, J. H.; Sung, S. W.; Lee, I. B. An enhanced PID control strategy for unstable process. Automatica 1998, 34, 751. (6) Majhi, S.; Atherton, D. P. Autotuning and controller design for unstable time delay processes. Proceedings of the 1998 International Conference on Control, Swansea, U.K., Sept 1998; p 769. (7) Jacob, E. F.; Chidambaram, M. Design of controllers for unstable first-order plus time delay systems. Comput. Chem. Eng. 1996, 20, 579. (8) Sung, S. W.; Lee, I. Limitations and Countermeasures of PID Controllers. Ind. Eng. Chem. Res. 1996, 35, 2596. (9) Kwak, H. J.; Sung, S. W.; Lee, I. On-line Process Identification and Autotuning for Unstable Processes. Chem. Eng. Res. Des. 2000, 78, Part A, 549. (10) Kavdia, M.; Chidambaram, M. On-Line Controller Tuning for Unstable Systems. Comput. Chem. Eng. 1996, 20, 301. (11) De Paor, A. M.; O’Malley, M. Controllers of Ziegler-Nichols type for unstable process with time delay. Int. J. Control 1989, 49, 1273. (12) Poulin, E.; Pomerleau, A. PID tuning for integrating and unstable processes. IEE Proc.sControl Theory Appl. 1996, 143, 429. (13) Tan, W.; Liu, J.; Tam, P. K. S. PID tuning based on loopshaping H∞ control. IEE Proc.sControl Theory Appl. 1998, 145, 485. (14) Ho, W. K.; Xu, W. PID tuning for unstable processes based on gain and phase-margin specifications. IEE Proc.sControl Theory Appl. 1998, 145, 392. (15) Chidambaram. M. Control of Unstable Systems: A Review. J. Energy, Heat Mass Transfer 1997, 19, 49. (16) Kwak, H. J.; Sung, S. W.; Lee, I.-B. On-line process identification and autotuning for integrating processes. Ind. Eng. Chem. Res. 1997, 36, 5329. (17) Kwak, H. J.; Sung, S. W.; Lee, I. A Modified Smith Predictor for Unstable Processes with a New Structure. Ind. Eng. Chem. Res. 1999, 38, 405. (18) Lee, Y.; Lee, J.; Park, S. PID controller tuning for integrating and unstable processes with time delay. Chem. Eng. Sci. 2000, 55, 3481. (19) Luyben, W. L. Tuning Proportional-Integral-Derivative Controllers for Integrator/Deadtime Processes. Ind. Eng. Chem. Res. 1996, 35, 3480. (20) Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/dead time processes. Ind. Eng. Chem. Process Des Dev. 1992, 31, 2625. (21) Venkatashankar, V.; Chidambaram, M. Design of P and PI controllers for unstable first-order plus dead time delay systems. Int. J. Control. 1994, 60, 137. (22) Wang, L.; Cluett, W. R. Tuning PID controllers for integrating processes. IEE Proc.sControl Theory Appl. 1997, 144, 385.
Received for review August 9, 2000 Revised manuscript received September 4, 2001 Accepted March 13, 2002 IE000739H
