PID controller tuning for integrating processes

ISA Transactions 49 (2010) 70–78

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PID controller tuning for integrating processes Ahmad Ali a,∗ , Somanath Majhi b a

Electrical Engineering Department, Indian Institute of Technology, Patna 800013, Bihar, India

b

Electronics & Communication Engineering Department, Indian Institute of Technology, Guwahati 781039, Assam, India

article

info

Article history: Received 23 September 2008 Received in revised form 29 June 2009 Accepted 10 September 2009 Available online 25 September 2009 Keywords: Integrating process Nyquist curve PID controller

abstract Minimizing the integral squared error (ISE) criterion to get the optimal controller parameters results in a PD controller for integrating processes. The PD controller gives good servo response but fails to reject the load disturbances. In this paper, it is shown that satisfactory closed loop performances for a class of integrating processes are obtained if the ISE criterion is minimized with the constraint that the slope of the Nyquist curve has a specified value at the gain crossover frequency. Guidelines are provided for selecting the gain crossover frequency and the slope of the Nyquist curve. The proposed method is compared with some of the existing methods to control integrating plant transfer functions and in the examples taken it always gave better results for the load disturbance rejection whilst maintaining satisfactory setpoint response. For ease of use, analytical expressions correlating the controller parameters to plant model parameters are also given. © 2009 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction The ability of proportional-integral-derivative (PID) controllers to meet most of the control objectives has led to their widespread acceptance in the control industry. A comprehensive summary of the PID tuning methods reported in the literature is given in [1]. It can be concluded from [1] that the number of tuning rules are much more for stable overdamped processes as compared to integrating processes. Several PI/PID tuning methods for integrating processes have been proposed in the literature [2–9]. Chien and Fruehauf [2] have proposed an internal model control (IMC) approach to select tuning constants of a PI controller for a process transfer function consisting of a pure integrator and a dead time. Tyreus and Luyben [3] have shown that the IMC based PI controller can lead to poor control performance if the closed loop time constant is not chosen properly and have proposed simple tuning rules based on the classical frequency response method to achieve maximum closed loop log modulus of 2 dB. Kookos et al. [5] have obtained the parameters of the PI controller by gain-phase margin method (KPGM) and by minimizing weighted ISE. Tuning rules for PI controller based on the maximum peak resonance specification is proposed in [6]. However, the above methods do not give settings for PID controllers. The method proposed in [3] is extended for designing PID controllers in [4]. The desired control signal trajectory is used as a performance specification to design the PID controller in [8].

∗

Corresponding author. Fax: +91 612 2277383. E-mail address: [email protected] (A. Ali).

Chidambaram and Sree [9] have obtained the parameters of a PI, PID and PD controller by matching the coefficients of corresponding powers of s in the numerator and that in the denominator of the closed loop transfer function for a servo problem. The method proposed in [9] is further improved by Sree and Chidambaram [10] to avoid the oscillations in the system output for perturbation in the plant delay. Skogestad [11] has used the IMC framework to derive PI/PID tuning rules (SIMC) for various class of integrating processes. However, none of the above reported works give the settings for PID controller for integrating plus time delay (IPTD), integrating plus first order plus time delay (IFOPTD) and double integrating plus time delay (DIPTD) process models, respectively. One of the methods of obtaining the controller parameters is by minimizing an integral criterion. The integral of squared error (ISE) has often been used for control system design since the integral can be evaluated analytically in the frequency domain. Although the ISE method provides a good way to obtain optimal PID controller settings, it weights all errors equally independent of time and hence results in a response with a relatively small overshoot but a long settling time. For integrating processes, the ISE criterion results in a PD controller which gives good servo response but fails to reject the load disturbances. Recently, Karimi et al. [12] have used the Bode’s integrals to approximate the derivatives of amplitude and phase of a system with respect to frequency at a given frequency and these derivatives are then used to adjust the slope of the Nyquist curve at the gain crossover frequency to obtain the parameters of a PID controller using the modified Ziegler–Nichols method. However, the selection of gain crossover frequency and the slope of the Nyquist curve has not been discussed in [12]. In this work, it is shown that minimizing the ISE criterion so that the slope

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A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

of the Nyquist curve attains a specified value at the gain crossover frequency gives satisfactory closed loop performance for a class of integrating processes. The paper is organized as follows. Section 2 presents the problem formulation: process model, PID structure and the optimization problem. The expressions for the PID parameters in terms of plant model parameters are presented in Section 3. Simulation results are discussed in Section 4 followed by conclusions in Section 5. 2. Problem formulation In this section, the controller equation is presented as well as the assumed process model structure and the optimization problem that is posed in order to tune the PID controller. 2.1. PID controller The general parallel form of a PID controller is given by Gc (s) =

U (s) E (s)

 = Kc 1 +

Td s Tf s + 1

+

1



Ti s

(1)

where Kc , Ti and Td are the controller parameters to be obtained for the satisfactory closed loop performance of the system. Tf = α Td and usually, a small value of the derivative filter constant (α ) is considered in the literature. In this work α is set at 0.10. Further, e(t ) and u(t ) are the input and output signals of the PID controller, respectively. 2.2. Process model For controller design purposes, the commonly encountered integrating processes are adequately described by low order transfer function models such as integrating plus time delay (IPTD), Gp (s) =

Kp e

(2)

s

Kp e−θ s s(Ts + 1)

Gp (s) =

Kp e

s2

.

(6) 2 where φm is the required phase margin and ng the slope of the open loop gain at the crossover frequency. The open loop gain should have a slope of about −1 around the crossover frequency, with preferably steeper slopes before and after the crossover and hence ng is assumed as −1 [15]. For the considered integrating process models, Eq. (6) reduces to

θ ωg ≤

π

− φm . (7) 2 ◦ ◦ Typical values of φm range from 30 to 60 [16]. Therefore, in this work three values of φm , i.e. 30◦ , 45◦ and 60◦ are considered and the corresponding values of θ ωg (considering the equality sign in (7)) are 1.05, 0.78 and 0.52, respectively. 2.4. Loop slope adjustment In this subsection, a relationship between Ti and Td is obtained to achieve a user specified slope at any given frequency. The slope of the Nyquist curve of the loop transfer function L(s) = Gp (s)Gc (s) at any frequency ωo is equal to the phase of the derivative of L(jω) at ωo . dL(jω)

dGc (jω) dGp (jω) = Gp (jω) + Gc (jω) . (8) dω dω dω Eq. (8) gives the derivative of the loop transfer function with respect to ω. Also, we have ln Gp (jω) = ln Gp (jω) + j6 Gp (jω).





(9)

Differentiating the above equation, we get

dω

(3)

dGc (jω) dω

( = Gp (jω)

d ln Gp (jω)





dω

+j

d6 Gp (jω) dω

) .

(10)

  1 = jKc Td + 2 . ω Ti

(11)

Substituting (1), (10) and (11) in (8), we get (4)

dL(jω) dω

2.3. Selection of gain crossover frequency

( 

(5)

   1 + 1 + j T ω − d ω2 Ti ω Ti !) d ln Gp (jω) d6 Gp (jω) +j . (12) dω dω

= Gp (jω)Kc j Td +

×

The gain crossover frequency is related to the rise time and thus to the bandwidth of the closed loop system and hence, can be used as a measure of system performance [13]. Also, the slope of the Nyquist curve at ωg gives a measure of the minimum distance of the Nyquist plot from the critical point (−1, 0) which indicates the system robustness. Hence, we can say that the slope of the Nyquist curve at gain crossover frequency drastically affects the performance and robustness of the closed loop system. In this subsection, an upper bound on the gain crossover frequency (ωg ) is obtained for the above mentioned class of integrating processes using the Åström’s inequality proposed in [14]. The best achievable control performance for a process is limited by its non-minimum phase nature and to figure out this achievable performance, the process model is factorized into Gp (s) = Gmp (s)Gnmp (s)

π

ω is given by

double integrating plus time delay (DIPTD) −θ s

arg Gnmp (jωg ) ≥ −π + φm − ng

The derivative of the controller (neglecting Tf ) with respect to

integrating plus first order plus time delay (IFOPTD) and Gp (s) =

where Gmp (s) is the minimum phase part and Gnmp (s) the nonminimum phase part, with |Gnmp (jω)| = 1. Recently, Åström [14] has shown that the gain crossover frequency satisfies the following inequality

dGp (jω)

−θ s

71

1



The slope of the Nyquist curve at ωo denoted by ψ is given by

ψ = ϕo + arctan



1 + Ti Td ωo2 + sp (ωo )ωo Ti + (Ti Td ωo2 − 1)sa (ωo )



sa (ωo )ωo Ti − (Ti Td ωo2 − 1)sp (ωo )

(13) where

ϕo = 6 Gp (jωo )

d ln Gp (jω)



sa (ωo ) = ωo

sp (ωo ) = ωo

dω



ωo

d6 Gp (jω) dω

. ωo

Rearranging (13), we get the equation given in Box I.

(14)

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A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

Table 1 Performance summary for load disturbance rejection.

ωg

ψ = 55◦

0.52 0.78 1.05

Td =

ψ = 60◦

ψ = 65◦

ymax

ts (s)

ISE

ymax

ts (s)

ISE

ymax

ts (s)

ISE

1.49 1.19 1.13

27.04 16.27 10.58

1.35 1.30 1.50

1.54 1.19 1.14

20.81 12.12 8.04

1.35 1.28 1.46

1.56 1.20 1.15

25.50 14.60 9.42

1.34 1.26 1.42

sa (ωo ) − 1 + sp (ωo ) tan(ψ − ϕo ) − Ti ωo (sp (ωo ) − sa (ωo ) tan(ψ − ϕo ))

ωo2 Ti (1 + sa (ωo ) + sp (ωo ) tan(ψ − ϕo )) Box I. Table 4 Controller parameters for DIPTD processes.

Table 2 Controller parameters for IPTD processes.

θ

Kp Kc

Ti

Td

θ

Kp Kc

Ti

Td

0.5 1 2 3 4

2.05 1.03 0.51 0.34 0.26

1.59 3.17 6.35 9.52 12.70

0.24 0.49 0.98 1.47 1.96

0.5 1 2 3 4

0.67 0.17 0.04 0.02 0.01

4.25 8.51 17.02 25.53 34.05

1.43 2.87 5.73 8.60 11.46

3.2. IFOPTD process model

Table 3 Controller parameters for IFOPTD processes.

θ

Kp Kc

Ti

Td

ψ◦

0.5 1 2 3 4

2.37 0.88 0.39 0.25 0.19

3.01 5.13 11.21 16.57 22.50

0.58 0.98 1.60 2.22 2.80

45 60 70 70 70

For IFOPTD plant models, ωg is assumed as 0.78/θ and the values of ψ that give satisfactory closed loop performance as well as the corresponding controller parameters are given in Table 3. Analytical expressions correlating the controller parameters to plant model parameters obtained using the curve fitting toolbox are:

2.5. Problem formulation

Kp Kc T = 0.78

Using the condition that |L(jωg )| = 1, we obtain the following equation for IPTD process model

Ti

Kp Kc = q

ω

2 g Ti

(1 − ωg2 Ti Td )2 + ωg2 Ti2

.

(15)

The proposed method of obtaining the controller parameters can be therefore summarized as: Given any IPTD plant model find Ti such that the ISE is minimum subject to the equation given in Box I and (15), where ωo = ωg and ωg and ψ are user specified values. On similar lines, the controller parameters are obtained for IFOPTD and DIPTD process models respectively.

T Td T

 −1.53 θ T

+ 0.10

 1.02 θ

= 5.44

T

= 0.96

 0.77 θ T

.

(16)

3.3. DIPTD process model The values of ωg and ψ are assumed as 0.52/θ and 15◦ and the proposed controller settings are: Kc = 0.17/(Kp θ 2 ), Ti = 8.51θ and Td = 2.87θ (Table 4). 3.4. PI controller

3. Tuning formulas 3.1. IPTD process model The values of ωg and ψ are selected based on the extensive simulation studies. To illustrate the method adopted for selecting ωg and ψ , an IPTD process model is considered with both Kp and θ assumed as unity. The settling time and the maximum value of the system output (ymax ) for various values of ωg and ψ for a unit step change in the load disturbance at the system input are given in Table 1. It can be observed from Table 1 that the minimum value of ymax is obtained for ωg = 1.05 and hence, ωg is selected as 1.05. Also, the least settling time is obtained for ψ = 60◦ and hence, ωg and ψ are taken as 1.05/θ and 60◦ , respectively. The controller parameters obtained for various values of theta are given in Table 2. The recommended PID controller settings are: Kc = 1.03/(θ Kp ), Ti = 3.17θ and Td = 0.49θ .

Substituting Td = 0 in the equation given in Box I and rearranging, we get the following equation for the integral time constant of a PI controller Ti =

sa (ωo ) − 1 + sp (ωo ) tan(ψ − ϕo ) . ωo (sp (ωo ) − sa (ωo ) tan(ψ − ϕo ))

(17)

Once Ti is obtained, Kc can be calculated using the following equations for IPTD and IFOPTD process models, respectively. Kc Kp = q

ωg2 Ti 1 + ωg2 Ti2

q ωg2 Ti 1 + ωg2 T 2 Kc Kp = q . 1 + ωg2 Ti2

(18)

(19)

A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

Fig. 1. Responses for perfect model for Example 1: (a) proposed, (b) Sree and Chidambaram [10] and (c) ISE.

Fig. 2. Responses for a perturbation of +20% in process time delay for Example 1: (a) proposed, (b) Sree and Chidambaram [10] and (c) ISE.

Fig. 3. Responses for a perturbation of +20% in process gain for Example 1: (a) proposed, (b) Sree and Chidambaram [10] and (c) ISE.

73

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Table 5 Performance summary. Servo ISE

Regulatory ymax

ts (s)

ISE 1.26

ymax

ts (s)

0.34 – 0.36

48.23 – 66.39

Example 1

Proposed ISE Sree and Chidambaram [10]

8.78 6.82 9.07

1.70 1.22 1.53

49.50 36.25 64.76

Example 2

Proposed ISE SIMC

6.07 5.32 7.82

1.38 1.17 1.28

53.71 30.92 78.09

– 1204.07

5.47 – 7.76

62.59 – 107.20

Example 3

Proposed SIMC

5.57 3.93

2.03 1.84

29.33 23.32

235.13 607.15

7.15 8.32

22.45 52.06

Example 4

Proposed SIMC KPGM [5]

16.22 14.45 15.78

1.38 1.28 1.30

95.72 143.52 126.15

252.51 308.07 339.99

2.92 2.90 3.06

97.40 192.90 145.32

– 1.45 397.64

(2) IFOPTD process model (a) ωg = 0.35/θ (b) ψ = 50◦ (c) TKc Kp = 0.37(θ /T )−1.18 (d) Ti /T = 3.85(θ /T )1.35 + 5.85. The proposed tuning rules are valid for θ and θ /T as large as 10. However, as the PI/PID controller gives sluggish closed loop responses for delay dominated plants, the Smith predictor should be used for integrating plant models with large time delay. 4. Simulation study

Fig. 4. Nyquist plots for Example 2: (a) proposed and (b) SIMC.

Based on the extensive simulation studies, the following parameters are recommended for the integrating process models. (1) IPTD process model (a) ωg = 0.52/θ (b) ψ = 50◦ (c) Kc Kp = 0.48/θ (d) Ti = 5.24θ

In this section, the results of simulations of four examples to illustrate the value of the proposed tuning rules are given. The results are compared with Sree and Chidambaram’s method for IPTD process model and with Skogestad’s tuning rules (SIMC) for IFOPTD and DIPTD process models. The output control performance is measured by computing ISE values and by determining the settling time (ts ) and peak value of the system output (ymax ) for a unit step change in both the setpoint and the load disturbance at the plant input. For a fair comparison, the tuning constants for the parallel form of PID are given for SIMC tuning rules in this work. 4.1. Example 1 −6s

An IPTD plant model with transfer function 0.0506 e s is considered. The method proposed in [10] gives the PID controller settings as Kc = 2.95, Ti = 15 and Td = 3 whereas the proposed

Fig. 5. Responses for perfect model for Example 2: (a) proposed, (b) SIMC and (c) ISE.

A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

75

Fig. 6. Responses for a perturbation of +20% in process time delay for Example 2: (a) proposed, (b) SIMC and (c) ISE.

Fig. 7. Responses for a perturbation of +20% in process gain for Example 2: (a) proposed, (b) SIMC and (c) ISE. Table 6 ISE values under parameter uncertainty. Method

+20%θ

+20%Kp Servo

Regulatory

Servo

Regulatory

11.15 7.20 9.59

1.37 – 1.51

13.33 8.40 13.07

–

Example 1

Proposed ISE Sree and Chidambaram [10]

Example 2

Proposed ISE SIMC

6.73 5.69 7.41

406.56 – 1222.41

7.83 6.45 9.20

435.52 – 1279.60

Example 3

Proposed SIMC

5.74 4.27

219 590.90

9.87 5.68

280.47 627.37

Example 4

Proposed SIMC KPGM [5]

15.26 13.66 14.64

258.70 313.33 345.50

19.46 16.99 18.34

278.30 327.56 363.90

PID parameters are Kc = 3.39, Ti = 19.02 and Td = 2.94, respectively. Minimizing the ISE criterion results in a PD controller with Kc = 2.60 and Td = 3.25. Fig. 1 shows the servo and regulatory responses for a unit step change in both the setpoint and load disturbance. The performance indices are given in Table 5. The PD controller gives the best servo response but results in a significant offset for the regulatory problem. It is evident from Fig. 1 and

1.60 1.85

Table 5 that the proposed method gives improved load disturbance rejection as compared to Sree and Chidambaram’s method. The robustness of the proposed controller is studied by assuming a 20% change in the steady state gain and the plant delay and the corresponding step responses are shown in Figs. 2 and 3, respectively. As the PD controller fails to reject the load disturbances, the corresponding robust responses for the regulatory problem are not

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A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

Fig. 8. Responses for perfect model for Example 3: (a) proposed and (b) SIMC.

Fig. 9. Responses for a perturbation of +20% in process time delay for Example 3: (a) proposed and (b) SIMC.

shown. Table 6 shows the ISE values under parameter uncertainty for a servo problem and separately for a regulatory problem. Both the proposed and Sree and Chidambaram’s methods are robust, but it can be concluded from the ISE values given in Table 6 that the present method performs better for regulatory problems. 4.2. Example 2 −4s

The IFOPTD plant model considered in [10] is Gp (s) = s(es+1) . The controller settings proposed in [10] gives an unstable response and hence, is not considered in this work. The PID parameters by the present method are Kc = 0.19, Ti = 22.37 and Td = 2.79. The controller parameters obtained by minimizing ISE are Kc = 0.17 and Td = 2.95 whereas SIMC method gives Kc = 0.13, Ti = 33 and Td = 0.97. Figs. 4 and 5 show the Nyquist plots and the corresponding servo and the regulatory responses. Once again, the PD controller obtained by minimizing ISE gives the best servo response but fails to reject the load disturbances. It can be observed from Fig. 5 and Table 5 that the proposed method outperforms SIMC for both servo and regulatory problems. For servo problem, even though the overshoot is more but the settling time is less as compared to SIMC. To show the robust performance

of the proposed design method, a 20% change in the plant delay and the steady state gain is considered and the corresponding step responses are shown in Figs. 6 and 7, respectively. Also, the ISE values under uncertainty in the model parameters are given in Table 6. A Comparison of ISE values reveals that the proposed method is more robust as compared to SIMC for both servo and regulatory problems. 4.3. Example 3 −s

Consider the DIPTD plant transfer function of Gp (s) = es2 . The PID controller settings by the present method are Kc = 0.17, Ti = 8.51 and Td = 2.87 whereas Skogestad’s method gives Kc = 0.125, Ti = 16 and Td = 4. The parameters of the PD controller obtained by minimizing ISE are Kc = 0.07 and Td = 5.98. As the PD controller fails to reject the load disturbances, the corresponding step response plots are not shown. Fig. 8 shows the comparisons of the proposed method with that of SIMC for servo and regulatory responses. From Fig. 8 and Table 5, it can be observed that the proposed method performs better for load disturbance rejection whereas the SIMC gives improved servo response. Also, in process control applications, disturbance

A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

77

Fig. 10. Responses for a perturbation of +20% in process gain for Example 3: (a) proposed and (b) SIMC.

rate is altered [16]. The ISE values given in Table 6 for 20% perturbation in Kp and θ shows that the proposed method is more robust for regulatory problem (Figs. 9 and 10). From the simulation results, it can be observed that the proposed method gives improved load disturbance rejection performances as compared to the recently reported methods, thereby illustrating the usefulness of the proposed tuning formulas. However, the PID controller gives large overshoot for servo response for integrating processes which is not acceptable in the control industry. This problem can be resolved by introducing a second order setpoint filter with the following transfer function: Fsp =

1 s2 Ti Td + sTi + 1

.

(20)

Fig. 11 shows the improved servo response for Example 3, when a setpoint filter given by (20) is used along with the PID controller. It can be observed that the setpoint filter drastically improves the servo response by reducing the overshoot and settling time. Fig. 11. Improved servo response for Example 3.

rejection is much more important than setpoint tracking. This is because setpoint changes are required only when the production

4.4. Example 4 Distillation is the most widely used separation technique in the chemical process industries for the separation of fluid

Fig. 12. Responses for perfect model for Example 4: (a) proposed, (b) SIMC and (c) KPGM.

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A. Ali, S. Majhi / ISA Transactions 49 (2010) 70–78

Fig. 15. Control variables for filter.

e−2s s

: (a) with setpoint filter and (b) without setpoint

Fig. 13. Nyquist plots for Example 4: (a) proposed, (b) SIMC and (c) KPGM.

SIMC

Proposed

KPGM [5]

the setpoint filter results in smooth control signal with no initial kick. This is evident from the plots of the control variables shown in Figs. 14 and 15, respectively. The same holds true for IFOPTD and DIPTD plant models.

0.34 59.20

0.32 38.78

0.30 49.36

5. Conclusion

Table 7 PI parameters for Example 4.

Kc Ti

In this paper, tuning formulas for a class of integrating processes are proposed by minimizing the ISE criterion with the constraint that the slope of the Nyquist curve has a user specified value at gain crossover frequency. Good nominal and robust control performances are achieved with the designed controller. Simulation examples show that the proposed method gives improved load disturbance rejection performances as compared to some of the recently reported methods in the literature. Furthermore, it is shown that the proposed design method gives satisfactory closed loop performance for perturbations in the process parameters. References

Fig. 14. Control variables for filter.

e−.5s s

: (a) with setpoint filter and (b) without setpoint

mixtures. The distillation column separates a small amount of a low boiling material from the final product. The bottom level of the distillation column is controlled by adjusting the steam flow rate. The process model for the level control system is given by Gp (s) = 0.2e−7.4s /s [2]. The controller parameters obtained by various methods are given in Table 7. The step responses are shown in Fig. 12 and the corresponding Nyquist plots are given in Fig. 13. It can be observed that the proposed method outperforms SIMC and KPGM for both setpoint tracking and load disturbance rejection with the least settling time. Furthermore, as the parameters of the physical system vary with operating conditions and time, 20% change in the steady state gain and the plant delay is assumed and the corresponding ISE values are given in Table 6. As is evident from the ISE values, the proposed method gives more robust performances for load disturbance rejection as compared to the other reported approaches for the assumed parameter perturbations. The performance of the PID controller is also evaluated by considering the maximum controller output and it is observed that

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