Performance Analysis of Controllers Designed For Delayed Industrial Process

_______________________________________________________________________________________________

7

Performance Analysis of Controllers Designed For Delayed Industrial Process

Priyanka Singh Assistant Professor

EEE Department, Amity University, Greater Noida

[email protected],

Sunil Singh Assistant Professor,

EE Department ,College of Technology, GBPUAT Pantnagar

[email protected],

Abstract

:

In a process industry, many processes can be modelled by integral plus dead time model. In present analysis, dead time is approximated by using Taylor’s, Pade first order & Pade second order approximation techniques for an integral plus dead time model. Controller has been designed using various tuning techniques for the dead time approximated models and the performance of these controllers is compared for their set point tracking capability

.

Keywords:integral plus dead time model, tuning techniques, set point tracking capability, process industries

__________________________________________________*****_________________________________________________

I. INTRODUCTION

The control of the process is the main objective of the process industries. The process without delay is easy to analyze but in real time optimization the processes are with time delay. This delay is associated with plant due to transportation lag or the location of the sensors [1].

It is to be noted that most real-time systems have time-delay associated with them. Time delay can originate due to one or more of the following reasons [2]:

1. Measurement of system variables

2. Physical properties of the equipment used in the system 3. Signal transmission (transport delay)

Integral processes are generally found and usually described by an integrator plus dead time (IPDT) transfer function which is defined as [3]

G s = K^e−sθ_s (1) Where K is gain of the process

Ɵ is the delay

The control of integrating systems has been investigated widely in the last years. Integrating systems with time delay are found in the modelling of liquid level systems, liquid storage tanks, boilers, batch chemical reactors and the bottom level control of a distillation column [4]. Time delay alters the dynamics of the controller. The model is simple to identify as it contains only two parameters gain and delay.

Although, the simplicity in identification, it is very difficult to analyze these processes. Since most of industrial processes are stable, many efficient design methods were

developed only for stable processes in the past.

Recent work begins to pay respect to the control problem of integrating processes. Though a great number of controllers have been devised in the past decades, the most widely used controller is still of PID type. In industries 95% of the controllers are of PID type [5], but this is also a fact that

they are poorly tuned. PID controller is the special case of the PI controller. PI controllers are suitable for integrating processes. These controllers are tuned to control the process performance. The tuning of the PI controller is method of computing the two control parameters proportional gain and integral time. There are a number of parameters that generally process control systems aim to control. These comprise of the rise time which is the time required for the controlled parameters to go from 10 to 90% of the final desired values .Settling time, the time required for the transient’s damped oscillations to reach and stay within ±2%

of the steady-state value and the maximum overshoot, the maximum amount that the controlled variables overshoot the desired value. To control the process, the main aim of the controller is to obtain fast response with good stability.

II. INTEGRAL PLUS DEAD TIME MODEL

The integral plus dead time model is generally encountered in the process industries. Many tuning techniques are for first order plus dead time model but there are relatively less tuning rules for integral plus dead time model. The common examples of these processes are distillation column, chemical reactor and level control of the boiler steam drum.

Various tuning techniques for tuning PI controller have been proposed for better control system response on desirable control objectives [7]. O'Dwyer et al. [8] provided a survey of tuning rules for continuous-time PI and PID control of time-delayed single-input, single-output (SISO) processes.

The ability of PI and PID controllers to balance many practical processes has led to their broad acceptance in industrial applications. Then requirement to choose two or three controller parameters is most conveniently done using tuning rules. The selected tuning techniques for comparisons are as follows

1.Chidambaram & Sree Tuning Technique 2.Rotach Tuning Technique

_______________________________________________________________________________________________

8 3.Ziegler & Nichols Tuning Technique: The tuning relations

reported by Ziegler and Nichols [9] were determined empirically to provide closed-loop Z-N.

4. Skogested Tuning Technique

Skogestad S. et al. [10] presented analytic rules for PID controller tuning that are symbolic and still resulted in good closed loop behaviour. Instead of deriving separate rules, it presented a single tuning rules for FOPDT and Second Order Plus Dead Time (SOPDT)

5. Shinskey Tuning Technique: In this technique the controller parameters are represented in terms of the 6. Hazebroek & Vanderwarden Tuning Technique

III. METHODOLOGY

In the present analysis the integral plus dead time model is selected for the analysis process. The selected process transfer function is of water level system .The selected transfer is given by equation (2)[11]

𝐺𝑝(𝑠) =^{.00449 𝑒}_𝑠 ^−10𝑠 (2) Where

Gain of the process K= .00449 Delay of the process Ɵ= 10 secs

The delay associated with the process is approximated with the different selected approximation techniques. The delay is approximated by Taylor’s approximation the transfer function of the process is given by equation (3)

𝐺𝑝(𝑠) =_{𝑠(1+10𝑠)}^.00449 (3)

The delay is approximated by Pade I approximation the process transfer function is given by equation (4)

𝐺𝑝 𝑠 =.00449 (1−5𝑠)

(1+5𝑠) (4) The delay is approximated by Pade second order approximation the process transfer function is given by equation (5)

𝐺𝑝(𝑠) =^{.00449 𝑠2} − .02694 𝑠+.005388 𝑠³ + 0.6𝑠²+0.12𝑠 (5)

The controllers are designed for the approximated process to control the dynamics of the process. The most dominant PI controller architecture is the ideal PI controller as given in equation (6)

𝐺𝑐(𝑠) = 𝐾𝑐(1 +_𝑠𝑇¹

𝑖 ) (6)

Controller designed from the controller parameters of the selected tuning techniques [12] are as follows

1.Chidambaram & Sree controller The tuning formulas is given as

K_c=^1.1111

𝐾𝜃 (7)

T_i=4.5 𝜃 (8) The transfer function of the controller is given by equation (9)

𝐺𝑐 𝑠 =1113 .3𝑠+24.74

45𝑠 (9) 2. Rotach controller

The tuning formula is given by

K_{c =} ^0.75_{K θ} (10)

Ti = 2.41 𝜃 (11)

The transfer function of the controller designed by the Rotach tuning technique is given by equation (12)

𝐺𝑐 𝑠 =402.47𝑠+16.70

24.1𝑠 (12) 3. Ziegler & Nichols controller

K_{c =} ^0.9

𝐾 𝜃 (13)

Ti = 3.33 𝜃 (14)

The controller designed by the tuning parameters of the Ziegler & Nichols tuning technique is given by equation (15)

𝐺𝑐 𝑠 =667.33𝑠+20.04

33.3𝑠 (15) 4. Skogested controller

The tuning formula for the Skogested tuning technique is given by

K_c = ^0.49

Kθ (16)

Ti = 3.77 𝜃 (17)

The transfer function of the controller designed by Skogested tuning technique is given by equation (18)

𝐺𝑐 𝑠 =411.30𝑠+10.91

37.7𝑠 (18)

5. Shinskey controller

The tuning parameters is given by K_{c =} ^0.9259

Kθ (19)

Ti = 4 θ (20)

_______________________________________________________________________________________________

9 The transfer function of the controller designed by tuning

parameter of the Shinskey tuning technique is given by equation (21)

𝐺𝑐 𝑠 =848.90𝑠+10.91

37.7𝑠 (21) 6. Hazebroek & Vanderwarden controller

The tuning parameters are given as K_{c =} ^1.5

K θ (22) Ti= 5.56 θ (23)

The controller transfer function designed by the Hazebroek

& Vanderwarden is given by equation (24) 𝐺𝑐 𝑠 =1857 .04𝑠+33.40

55.6𝑠 (24) The controllers designed is subjected to the step input and the performance of the control system is analyzed on the basis of the steady state and transient characteristics i.e. rise time, peak time ,settling time and the maximum peak overshoot.

The set point tracking capability of the designed controller is analyzed. The controller with the best set point tracking capability is decided on the control system dynamics.

IV. RESULT & DISCUSSIONS

The comparison of the responses of the Chidambaram and Sree tuning technique is shown in the figure 1. In the figure 1 ,a, a1 and a2 are the response of the Taylors, Pade I and Pade II order approximations respectively.

Fig.1 Comparison of Taylors, Pade I and Pade second order approximation for Chidambaram & Sree tuning technique On comparing these responses it is concluded that the rise time and peak time is better for the Pade first order approximation of the time delay and settling time and peak overshoot is better for Taylor’s approximation of the time delay as depicted from Table 2

Figure 2 shows the comparison of the approximation of delay for the Rotach tuning technique. In figure 2, b ,b1 and b2 shows the response of Taylor’s ,Pade I and Pade II order approximation respectively .

Fig.2 Comparison of Taylors, Pade I and Pade second order approximation for Rotach tuning technique

On comparison on the basis of the transient and steady state characteristics it is concluded that rise time , peak time and settling time is better for Pade first order approximation .Peak overshoot is better for Taylor approximation as depicted by table 3.

Figure 3 shows the comparison of the different selected time delay approximation. In the figure 3,c,c1,c2 shows the response of Taylor’s, Pade first order and Pade second order approximation of time delay respectively.

Fig.3 Comparison of Taylors, Pade I and Pade second order approximation for Ziegler & Nichols tuning technique On the basis of dynamics it is concluded that rise time, peak time and settling time is better for Pade first order approximation. Peak overshoot is better for Taylor approximation of time delay as depicted from table 4.

In figure 4 the comparison for set point tracking capability of the controller tuned by Skogested tuning technique is shown .In the figure 4,d,d1 and d2 shows the behaviour of

0 50 100 150 200 250 300 350 400

-0.5 0 0.5 1 1.5 2 2.5

Step Response

Time (sec)

Amplitude

a a1 a2

0 50 100 150 200 250 300

-0.5 0 0.5 1 1.5 2

Step Response

Time (sec)

Amplitude

b b1 b2

0 50 100 150 200 250 300

-0.5 0 0.5 1 1.5 2

Step Response

Time (sec)

Amplitude

c c1 c2

_______________________________________________________________________________________________

10 the Taylor’s, Pade first order and Pade second order

approximation respectively.

Fig.4 Comparison of Taylors, Pade I and Pade second order approximation for Skogested tuning technique Comparing on the basis of the transient and steady state analysis it is concluded that rise time, peak time and settling time is better for Pade first order approximation .Maximum peak overshoot is better for Taylor’s approximation as depicted from table 4.

Figure 5 shows the comparison of the responses of the Shinskey tuning technique. In the figure 5,e,e1,e2 shows the response of the Taylor’s ,Pade I and Pade second order approximation of the time delay respectively.

On comparison it is concluded that rise time, peak time and settling time is better for Pade first order approximation of the time delay. The Taylor’s approximation shows the better result for the peak overshoot.

Fig.5 Comparison of Taylors, Pade I and Pade second order approximation for Shinskey tuning technique Figure 6 show the comparison of time delay for the set point tracking capability for Hazebroek & Vanderwarden tuning technique. In the figure 6 ,f,f1 and f2 shows the Taylor’s , Pade first order and Pade second order approximation of the time delay respectively.

Fig.6 Comparison of Taylors, PadeI and Pade second order approximation for Hazebroek and Vanderwarden tuning

technique

On comparing the set point tracking capability of the Taylor’s , Pade first order and Pade second order approximation the Pade first order gives unrealizable response. The rise time and peak time is better for Pade first order approximation. The maximum peak overshoot and the settling time is better for the Taylor’s approximation of the time delay.

fig.7. Comparison of all tuning techniques for Taylor approximation

Figure 7 shows the comparison of all tuning techniques for the Taylor approximation of time delay of the process model . In the figure 7, a,b.,c,d,e and f shows the response of Chidambaram & Sree, Rotach , Ziegler & Nichols , Skogested ,Shinskey and Hazebroek &Vanderwarden respectively.

On comparing the set point tracking capability of different tuning techniques it is concluded that the rise time, peak time, peak overshoot as well as settling time is better for Hazebroek & Vanderwarden tuning technique. This tuning technique shows better transient as well as steady state behaviour for Taylor approximation of time delay of the process model.

0 50 100 150 200 250 300

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Step Response

Time (sec)

Amplitude

d d1 d2

0 50 100 150 200 250

-0.5 0 0.5 1 1.5 2

Step Response

Time (sec)

Amplitude

e e1 e2

0 50 100 150 200 250 300 350 400

-0.5 0 0.5 1 1.5 2 2.5

Step Response

Time (sec)

Amplitude

f f1

0 50 100 150 200 250 300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Step Response

Time (sec)

Amplitude

a b c d e f

_______________________________________________________________________________________________

11 Figure 8 comparing Pade first order approximation for all

tuning technique

Figure 8 shows the comparison of all tuning techniques for the Taylor approximation of time delay of the process model . In the figure 8 a1,b1,c1,d1,e1 and f1 shows the response of Chidambaram & Sree,Rotach , Ziegler & Nichols , Skogested, Shinskey and Hazebroek & Vander warden respectively for Pade first order approximation of time delay in process model .

The rise time and peak time is better for Hazebroek and Vander warden tuning technique .The settling time of the response is better for the Shinskey tuning technique. The peak overshoot is better for the Skogested tuning technique for Pade first order approximation of time delay of the process model.

figure 9 Comparison of Pade first orderI approximation of all tuning technique

Figure 9 shows the comparison of all tuning techniques for the Taylor approximation of time delay of the process model . In the figure 8 a2,b2,c2,d2,e2 and f2 shows the response of Chidambaram & Sree,Rotach , Ziegler & Nichols , Skogested ,Shinskey and Hazebroek &Vanderwarden respectively for Pade first order approximation of time delay in process model .

The rise time and peak time is better for Chidambaram &

Sree tuning technique .The settling time of the response is better for the Skogested tuning technique. The peak overshoot is better for the Skogested tuning technique for the Pade first orderI approximation of time delay of the process model.

V. CONCLUSION

Different controllers are designed for the different tuning techniques. Controller performance is evaluated for the step input subjected to the process model. The controller shows the different behaviour for the different controller tuning techniques. The controller is evaluated for the set point tracking capability. The basis of the evaluation is the transient and steady state characteristics.

Different conclusion is derived on the basis of the results obtained. From the complete analysis it is concluded the Hazebroek & Vanderwarden tuning technique shows the better transient as well as the steady state response for the Taylor approximation of delay in the selected integral plus dead time model.

The better peak overshoot can be achieved by the Taylor approximation of time delay. As from the analysis it is concluded that the peak overshoot of the integral plus dead time model is better for Taylor approximation of time delay .The Taylor approximation shows better result for peak overshoot for different tuning techniques.

The Hazebroek & Vanderwarden tuning technique is better for Pade first order approximation. The transient behaviour is better for the Pade first order approximation of time delay for Hazebroek & Vanderwarden tuning technique. The response of the controller settles faster in the Shinskey tuning technique for Pade first order approximation. Peak overshoot is better for Skogested tuning technique for Pade first order approximation of time delay.

From the analysis it is concluded that for Pade second order approximation of time delay of the process model the transient behaviour of the response to the step input is better for the Chidambaram & Sree tuning technique. The response settles faster in the Skogested tuning technique. The response settles fast as compared to all other tuning techniques.

From the analysis it is concluded that the Skogested tuning technique gives better values for peak overshoot for all approximation of time delay of the process model.

REFERENCES

[1] A. O‟Dwyer, “PID compensation of time delayed processes 1998-2002: a survey”, in Proc. American Control Conf., Denver, Colorado, USA, pp. 1494-1499, 2003 [2] M.M. Zavarei, and M. Jamshidi, “Time-Delay Systems:

Analysis, Optimization and Applications”, Elsevier Science Publisher, Amsterdam, The Netherlands, 1987, Chapter 1, pp. 1-15.

[3] Ahmad Ali and Somanath Majhi, “PI Controller Tuning For Integrating Processes With Time Delay”, XXXII national systems conference, NSC 2008, December 17-19, 2008.

[4] K. J. Astrom, and T. Haggland, “PID controllers:

Theory, Design and Tuning”, Instr Soc. America, North Carolina. 1995.

[5] C. C. Yu, Auto-tuning of PID Controller. Berlin: Springer, 1990.

[6] C. V. Nageswara Rao , A. Seshagiri Rao, and R. Padma Sree “Design of PID Controllers for Pure Integrator Systems with Time Delay” International Journal of Applied Science and Engineering 2011. 9, 4: 241-260

0 50 100 150 200 250 300 350 400

-0.5 0 0.5 1 1.5 2 2.5

Step Response

Time (sec)

Amplitude

a1 b1 c1 d1 e1 f1

0 50 100 150 200 250 300 350 400

-0.5 0 0.5 1 1.5 2 2.5

Step Response

Time (sec)

Amplitude

a2 b2 c2 d2 e2

_______________________________________________________________________________________________

12 [7] M. Chidambaram, “Design of PI controllers for

integrator/dead time processes,” Hung. J. Ind. Chem., vol.

22, pp. 37-43, 1994.

[8] O'Dwyer, Aidan, “PI and PID controller tuning rules:an overview and personal perspective”, Irish Signals and Systems Conference, 2006, pp. 161 – 166.

[9] Dale E Seborg, Thomas F Edgar, Duncan A Mellichamp,

“Process Dynamics and Control”, pp.318, Second Edition, Wiley, 2007

[10] Skogestad S., simple analytic rules for model reduction and PID controller tuning”, journal of process control, 13, pp.

291-309, 2003.

[11] J.Sell Nancy, ”Process control fundamentals for the pulp and paper industry”, Tappi press, Atlanta, USA, pp. 164, 1995

[12] W Hu , G Xiao and X Li , “An analytical method for PI controller tuning with specified gain and phase margins for integral plus time delay processes”, ISA Trans. 2011 Apr;50(2):268-76. doi: 10.1016/j.isatra.2011.01.001. Epub 2011 Feb 1.

Table 1 Characteristics of Closed Loop Response for Different Selected Methods

Tuning technique Approximation Tr(sec) Tp(sec) Ts(sec) Mp(%)

Chidambaram &

Sree

Taylor 12.9 34.1 120 44

Pade first order 6.2 27 133 82.6

Pade second order 6.91 30.4 257 100

Rotach Taylor 14.8 42.4 195 60.4

Pade first order 8.63 36.3 183 85.4

Pade second order 8.91 37.2 221 95.8

Zeiglar& Nichols Taylor 14.1 39.1 138 50.3

Pade first order 7.56 31.9 128 78

Pade second order 8.01 33.7 185 91.5

Skogested Taylor 21.2 58.2 168 45.2

Pade first order 14.8 51.3 150 48.9

Pade second order 14.2 50.4 149 51.2

Shinskey Taylor 14 36.4 131 45.6

Pade first order 7.29 30.4 101 73.7

Pade second order 7.81 32.8 175 88.2

Hazebroek &

Vanderwarden

Taylor 11 27.4 103 42.8

Pade first order 4.37 23.1 259 116

Pade second order - - - -