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ISA Transactions 48 (2009) 10–15

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Research note

PI/PID controller design based on IMC and percentage overshoot specification to controller setpoint change Ahmad Ali ∗ , Somanath Majhi Department of Electronics and Communication Engineering, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India

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Article history: Received 30 May 2008 Received in revised form 20 August 2008 Accepted 15 September 2008 Available online 9 October 2008 Keywords: Internal model control Maximum sensitivity PID Controller Robustness

a b s t r a c t In this work, the normalized Internal Model Control (IMC) filter time constant is designed to achieve a specified value of the maximum sensitivity for stable first and second order plus time delay process models, respectively. Since a particular value of the maximum sensitivity results in an almost constant percentage overshoot to controller setpoint change, an empirical relationship between the normalized IMC filter time constant and percentage overshoot is presented. The main advantage of the proposed method is that only a user-defined overshoot is required to design a PI/PID controller. Simulation examples are given to demonstrate the value of the proposed method. © 2008 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction The ability of proportional-integral (PI) and proportionalintegral-derivative (PID) controllers to meet most of the control objectives has led to their widespread acceptance in the control industry. A number of methods for tuning PI/PID controller proposed in the literature have been compiled in [1]. The Internal Model Control (IMC)-based approach for controller design proposed in [2] has gained widespread acceptance in the control industry because a clear tradeoff between the closed loop performance and robustness is achieved by means of a single tuning parameter namely the IMC filter time constant (λ). Several methods of obtaining λ have been proposed in the literature [3–5]. The IMC filter parameter is obtained by minimizing a weighted function of integral square error (ISE) and the maximum of the complementary sensitivity function in [3]. Chen et al. [4] have proposed PI/PID tuning formula for stable first and second order plus time delay (FOPTD and SOPTD) process models based on the IMC principle to achieve a +2 dB maximum closed loop amplitude ratio. Skogestad [5] has proposed the SIMC tuning rules for various process models by equating the IMC filter time constant to the process model’s equivalent time delay. The controller is usually designed based on an approximate model of the actual plant. Also, the parameters of the physical system vary with operating conditions and time and hence, it is

∗

Corresponding author. Tel.: +91 361 2582510; fax: +91 361 2582542. E-mail addresses: [email protected] (A. Ali), [email protected] (S. Majhi).

essential to design a control system that shows robust performance in the case of the aforementioned situations. Gain and phase margins are two well known measures of robustness and simple analytical formulas to tune PI/PID controller for stable/unstable FOPTD and SOPTD plant models to meet user defined gain and phase margins have been proposed in [6,7], respectively. However, the gain and phase margin specifications give poor results for processes with unusual frequency response curve and may fail to give reasonable bounds on the sensitivity functions. The maximum sensitivity (Ms ) is defined as the inverse of the shortest distance from the Nyquist curve of the open loop transfer function to the critical point (−1, 0). Ms measures the closeness of the Nyquist plot from the critical point at all frequencies and not just the two frequencies as associated with gain and phase margins, and hence can therefore serve as a better measure of system robustness. In a recent article, Skogestad [8] reports that there are two approaches for controller design: tight control and smooth control. The fastest possible control with acceptable robustness is achieved by tight control whereas smooth control results in the slowest possible control subject to achieval of acceptable disturbance rejection. In this work, the normalized IMC filter time constant is designed to achieve a particular value of Ms that results in smooth and tight control for stable FOPTD and SOPTD process models, respectively. Also, it is observed that the percentage overshoot remains constant for a particular value of Ms and hence analytical expression relating the IMC filter constant to percentage overshoot is also obtained. The paper is organized as follows. The design method for the PI/PID controller is presented in Section 2. Simulation results are given in Section 3 followed by the conclusions in Section 4.

0019-0578/$ – see front matter © 2008 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2008.09.002

A. Ali, S. Majhi / ISA Transactions 48 (2009) 10–15 Table 1 Controller parameters for various process models Gp (s)

Kp

K e−θ s τ s+ 1 K e−θ s τ 2 s2 +2ξ τ s+1

1 τ K λ+θ 1 2ξ τ K λ+θ

Table 2 Ms∗ , Ms , percentage overshoots and λn Ti

= =

1 τ θ K λn +1 1 2ξ τ θ K λn +1

Td

τ 2ξ τ

τ 2ξ

2. Controller design Process dynamics are often approximated by low order transfer function models for ease in controller design. The dynamics of a large number of industrial processes can be represented by FOPTD and SOPTD transfer function models of the forms: K e−θ s

Gp (s) =

K e−θ s

τ 2 s2 + 2ξ τ s + 1

.

(2)

Let the forms of the PI and PID controllers be Gc (s) = Kp Gc (s) = Kp

 1+

1

1 sTi

 + sTd

(4)

(5)

zs

where z = λ + θ . Consequently, the sensitivity function S (s) becomes 1

zs + e−θ s

.

(6)

Substituting the delay term by a 1/1 Padé approximation, the sensitivity function can be written in the frequency domain as S (jω) =

−θ ω2 z + j2ωz . −ω θ z + 2 + jω(2z − θ ) 2

(7)

Squaring the magnitude of both sides of (7), we get 1 y2

=

θ 2 z 2 ω4 + 4z 2 ω2 (2 − z θ ω2 )2 + ω2 (2z − θ )2 1 M∗

Smooth control Tight control

Ms

λn

1.38 1.71

1.47 0.74

Table 4 Proposed PI controller parameters Kp

Ti

0.40τ Kθ 0.57τ Kθ

τ τ

Table 5 Proposed PID controller parameters

Smooth control Tight control

Kp

Ti

Td

0.81ξ τ Kθ 1.15ξ τ Kθ

2ξ τ

2ξ

2ξ τ

τ τ

2ξ

Nyquist plots given in [9]. Therefore, the condition that ω2 should have repeated roots gives (10)

Dividing the above equation by θ , (10) becomes

e−θ s

=

Table 3 Ms and λn for smooth and tight control

2

(λ + θ )s

1 + L(s) zs

3.5568 2.1384 1.4458 1.0375 0.7691 0.5796

4c 2 λ2 + λ(8c 2 θ + 4θ c − 8θ ) + 4c θ 2 (1 + c ) − 7θ 2 = 0.

e−θ s

S (s) =

λn

0 0 0.17 3.48 8.79 14.51

(3)

L(s) = Gp (s)Gc (s)

=

Overshoot (%)

1.2 1.3 1.4 1.6 1.7 1.8

Tight control

where Kp , Ti and Td are the proportional gain, the integral time constant and the derivative time constant, respectively. Table 1 shows the IMC-based PI/PID controller settings for the FOPTD and SOPTD process models, where λn = λ/θ . The loop transfer function L(s) using (1) and (3) or (2) and (4) becomes

=

Ms

1.2 1.3 1.4 1.5 1.6 1.7

Smooth control



sTi

 1+

Ms∗

(1)

τs + 1

Gp (s) =

11

4c 2 λ2n + λn (8c 2 + 4c − 8) − 7 + 4c + 4c 2 = 0.

(11)

The solution of (11) gives

√

λn =

2 + 2 1 − c − c − 2c 2 2c 2

.

(12)

The λn obtained by solving (11) for various values of c and the corresponding percentage overshoots are given in Table 2. Ms is the true value of the maximum sensitivity corresponding to the obtained set of controller parameters and is calculated using the robust control toolbox of MATLAB. The difference between Ms∗ and Ms is because of the 1/1 Padé approximation of the delay term. An important point to be observed is that the overshoot remains constant for a particular value of Ms for a wide range of FOPTD and SOPTD processes, respectively. Analytical expressions correlating Ms and λn to percentage overshoot (ov ) obtained using the curve fitting toolbox are Ms = 32.79 × 10−5 × (ov)3 − 89.04 × 10−4 × (ov)2

(8)

1 kS (jw)k ∞

where y = = is the inverse of the user specified s maximum sensitivity. Simplification of (8) gives

ω4 θ 2 z 2 c 2 + ω2 (4z 2 c 2 + 8θ z − θ 2 ) − 4 = 0 (9) p where c = 1 − y2 . For y to be the minimum distance of the Nyquist curve from the critical point, the Nyquist curve of the loop transfer function should touch the circle with centre (−1, 0) and radius y at only one frequency as is evident from the various

+ 88.75 × 10−3 × (ov) + 1.385 λn = −47.89 × 10−5 × (ov)3 + 0.0144 × (ov)2 − 0.1698 × (ov) + 1.474.

(13)

(14)

The normalized IMC filter parameter and the corresponding maximum sensitivity for smooth and tight control are obtained by calculating the respective values for 0 and 10 percent overshoot and are given in Table 3. The proposed tuning rules for stable FOPTD and SOPTD process models are given in Tables 4 and 5, respectively.

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A. Ali, S. Majhi / ISA Transactions 48 (2009) 10–15

Fig. 1. Step responses for G1 (s), G2 (s), G3 (s) and G4 (s): (–) smooth control, (. . . ) tight control, (- -) SIMC.

A. Ali, S. Majhi / ISA Transactions 48 (2009) 10–15

13

Fig. 2. Step responses for G5 (s) and G6 (s): (–) smooth control, (. . . ) tight control, (- -) SIMC, (—·—) Wang et al.

2.1. Limitations of IMC-based controller design As the IMC approach is based on pole zero cancellation, methods which incorporate IMC design principles result in good set point responses. However, the IMC settings result in a long settling time for the load disturbances for lag dominant processes which is not desirable in the control industry. A good trade off between disturbance response and robustness is achieved by selecting Ti = 8θ for lag dominant processes [5]. As suggested in [5], the integral time constant is therefore taken as Ti = min(T , 8θ ) for the PI controller and Ti = min(2ξ τ , 8θ ) for the PID controller. Even though the modified integral settings degrades the setpoint performance, it is observed that the proposed controller settings gives satisfactory performance for both setpoint tracking and load disturbance rejection for lag dominant processes. Extensive simulation results have shown that the closed loop performance of second order processes with θ ξ /τ less than 0.6 can be improved by the following settings of the proportional gain of the PID controller. θξ 0.2 ξ τ if 0.01 < < 0.1 Kp = K θ λn + 1 τ Kp =

1.2

ξτ θξ if 0.1 ≤ < 0.6. K θ λn + 1 τ

(15)

3. Simulation study The proposed tuning scheme is applied to FOPTD process models with various ratios of dead time to time constant and

SOPTD processes to demonstrate the effectiveness of the proposed tuning method. The performance of the proposed controller is compared with the IMC-based approach proposed in [5]. The results are also compared with Wang et al.’s [10] method for SOPTD process models. The PID controller is implemented in the widely used parallel form given by Gc (s) = Kp G1 (s) =

 1+

1 sTi

+

sTd 0.1sTd + 1

 (16)

1

e− s s+1 1 G2 (s) = e−5s s+1 100 G3 (s) = e− s 100s + 1 G4 (s) =

3 100s + 1

e−10s .

(17)

The FOPTD process models considered in [11] are given by (17). The time constant and the plant delay are the same for G1 (s) whereas G2 (s) represents a delay dominated plant. G3 (s) is a lag dominant process with θ /τ ratio of 0.01 and G4 (s) represents the transfer function of an important viscosity loop in a polymerization process. Fig. 1 shows the plant output for a unit change in the step input and a unit step change in the load disturbance. The controller tuned for smooth control gives plant output with no overshoot for G1 (s) and G2 (s) whereas settings for tight control yield faster and oscillatory output. The SIMC gives a response which is in between

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A. Ali, S. Majhi / ISA Transactions 48 (2009) 10–15

Fig. 3. Step responses for G5 (s): smooth control for nominal (–) and perturbed (. . . ) process model, Chen et al. for nominal (- -) and perturbed (—·—) process model.

Fig. 4. Step responses for G7 (s) for zero percent overshoot: (–) proposed, (. . . ) Mnif.

these two outputs as is evident from Fig. 1. Unlike the SIMC, the proposed method can be used to obtain a family of responses starting from the smooth response to tight control. As G3 (s) and G4 (s) represent lag dominant processes, the IMC controller settings result in poor load disturbance rejection and hence the integral time constant suggested in [5] is considered. It can be observed from Fig. 1 that this degrades the system setpoint performance and the overshoot is not zero for G3 (s) and G4 (s). However, the proposed controller gives satisfactory performance for both the setpoint tracking and disturbance rejection even for these lag dominant processes, respectively. It can be thus concluded that the proposed tuning method gives satisfactory performance for a wide range of stable FOPTD process models. The SOPTD plant models considered are given by the following transfer functions: G5 (s) = G6 (s) = Gˆ5 (s) =

1

(s + 1)(s + 5)2 1

( + 0.4s + 1) s2

e−0.5s

(18)

e−0.1s

(19)

1 7.

724s2

+ 32.317s + 25.220

e−0.606s .

(20)

G5 (s) represents a higher order non-oscillatory process and the approximate second order plus time delay model is given by (20). The second example is an oscillatory plant with ξ = 0.2. The plant outputs and the corresponding control signals are shown in Fig. 2. The proposed smooth controller gives zero percentage

overshoot and satisfactory load disturbance rejection for G5 (s). The controller tuned for tight control gives a faster response with less overshoot for G5 (s) as compared to Wang et al.’s PID as is evident from Fig. 2. The load disturbance response is the same for all the three methods. As the θ ξ /τ ratio for G6 (s) is 0.02, the modified proportional gain suggested in Section 2 is used. Satisfactory performance is achieved for the oscillatory plant G6 (s) by the proposed method whereas Wang et al.’s PID gives large overshoot and long settling time which is not desirable in the control industry. The plant model G5 (s) is considered again and the results are compared with Chen et al.’s [4] method to illustrate the robustness of the proposed tuning method. The controller settings corresponding to gain and phase margins of 3.14 and 61.4◦ (as these are the best settings reported in [4] from robustness point of view) are considered in this work. The step responses for the nominal plant model and an assumed 20% uncertainty in both the steady state gain and the delay are shown in Fig. 3. It can be concluded from Fig. 3 that the robustness of the proposed method towards assumed parameter perturbations are quite satisfactory. Finally, the thermoelectric device TB-127-1.4-1.2 (Kyrotherm) which is used in biomedical Thermocycler for PCR analysis and does not tolerate any overshoot in the output response is considered. A simplified model of the device is given by G7 (s) =

7 100s + 1

e−16s .

(21)

For a fair comparison, the results are compared with Mnif’s method [12] because both the proposed and Mnif’s methods

A. Ali, S. Majhi / ISA Transactions 48 (2009) 10–15

15

Fig. 5. Step responses for G7 (s) for 5% overshoot: (–) proposed, (. . . ) Mnif.

gives the controller parameters such that the response of the compensated system has overshoot close to a prescribed value. The controller parameters proposed in [12] are Kp = 0.31 and Ti = 100 whereas the proposed method gives Kp = 0.36 and Ti = 100. The performances of the closed loop system is evaluated using these controller settings by giving a unit step input in the set point and a negative step input of 0.50 at t = 300 s. It can be observed from the system outputs shown in Fig. 4 that the proposed method results in superior performance for both the setpoint and disturbance rejection. The step responses for 5% overshoot specification are shown in Fig. 5. Exactly 5% overshoot is obtained by the proposed method whereas Mnif’s method gives 7% overshoot, thereby proving the accuracy of the proposed tuning formulas. 4. Conclusion The sufficiency of the maximum sensitivity as a tuning parameter for controller design is illustrated in this paper and guidelines are provided regarding the selection of this tuning parameter for smooth and tight control for stable FOPTD and SOPTD process models, respectively. Furthermore, an empirical relationship between the percentage overshoot and the normalized IMC filter parameter is presented to facilitate controller tuning corresponding to any

user defined value of the percent overshoot. The usefulness of the proposed method is illustrated by several simulation examples. References [1] O’Dwyer A. Handbook of PI and PID controller tuning rules. 2nd edn. Imperial college press; 2006. [2] Rivera DE, Morari M, Skogestad S. Internal model control. 4. PID controller design. Ind Eng Chem Process Des Dev 1986;25:252–65. [3] Liu K, Shimizu T, Inagaki M, Ohkawa A. New tuning method for IMC controller. Chem Eng Japan 1998;31(3):320–4. [4] Chen C, Huang HP, Hsieh CT. Tuning of PI/PID controllers based on specification on maximum closed loop amplitude ratio. Chem Eng Japan 1999;32(6):783–8. [5] Skogestad S. Simple analytic rules for model reduction and PID controller tuning. J Process Control 2003;13:291–309. [6] Ho WK, Hang CC, Cao L. Tuning of PID controllers based on gain and phasemargin specifications. Automatica 1995;31(3):497–502. [7] Ho WK, Xu W. PID tuning for unstable processes based on gain and phase margin specifications. IEE Control Theory Appl 1998;145(5):392–6. [8] Skogestad S. Tuning for smooth PID control with acceptable disturbance rejection. Ind Eng Chem Res 2006;45:7817–22. [9] Åström KJ, Panagopoulos H, Hägglund T. Design of PI controllers based on nonconvex optimization. Automatica 1998;34(5):585–601. [10] Wang QG, Lee TH, Fung HW, Bi Q, Zhang Y. PID tuning for improved performance. IEEE Trans Control Syst Technol 1999;7(4):457–65. [11] Shamsuzzoha M, Lee M. An enhanced performance PID filter controller for first order time delay processes. Chem Eng Japan 2007;40(6):501–10. [12] Mnif F. New tuning rules of PI-like controllers with transient performances for monotonic time-delay systems. ISA Trans 2008;47(4):401–6.