Intelligent Automation and Soft Computing, Vol. 1X, No. X, pp. 1-3, 2012

Copyright © 20XX, TSI®

Press Printed in the USA. All rights reserved

BFO APPROACH TO IDENTIFY AND CONTROL UNSTABLE

PROCESS USING A DOUBLE FEEDBACK LOOP

V. RAJINIKANTH Department of Electronics and Instrumentation, St.Joseph’s College of Engineering,

IT Highway, Chennai, India.

rajinisjceeie@gmail.com

K. LATHA Department of Instrumentation Engineering, M.I.T. Campus,

Anna University, Chennai, India.

klatha@annauniv.edu

ABSTRACT— System identification is the preliminary practice, widely executed in the field of process

control to develop the mathematical model from experimental data. In this paper, we attempted the

proportional controller based system identification procedure for a class of time delayed unstable systems

using Bacterial Foraging Optimization algorithm. In the proposed work, a double feedback control loop is

considered to identify and control the unstable system. The inner loop with a „P‟ controller is considered

for the system identification procedure and outer loop with the „PI/PID‟ controller is used to control the

system. The proposed method is tested on a class of unstable systems in the presence and absence of

measurement noise. The performance of proposed identification procedure is compared with the classical

identification procedures existing in the literature. The results evident that, the discussed method helps to

accomplish a satisfactory model with considerably reduced model mismatch. The identified model is then

considered for the controller tuning manoeuvre. The method has been tested successfully on many time

delayed unstable processes.

Keywords - Bacterial foraging optimization, CSTR, Double feedback control, Identification, Performance

measure, Unstable process

1. INTRODUCTION

System identification is the preliminary practice, widely performed in process industries to develop mathematical models from experimental data. Derivation of accurate models from nonlinear chemical systems such as Continuous Stirred Tank Reactor (CSTR), biochemical reactor, polymerization reactor, etc., is difficult because of their complex nature. Based on the operating region, these chemical systems may exhibit stable or unstable steady states. System identification procedure followed for stable system is quite uncomplicated and simple open loop test is sufficient to develop an approximated model. For unstable systems, the identification procedure should be performed in closed loop condition. Closed loop step test and relay tuning are the two most popular schemes widely employed in industries to identify the approximated process model for time delayed unstable systems. The practical application of relay based system identification is limited by the delay time to process time constant (θ/τ) ratio. An unstable process under relay feedback produce limit cycle when θ/τ ratio < 0.693 [11]. Moreover the relay based identification cannot provide an appropriate response when the process time of the real time system is very large (Ex: cell growth in bioreactor). Hence, closed loop step test can be used as an alternate for relay based identification.

2 Intelligent Automation and Soft Computing

The closed loop system identification procedure for unstable system is widely addressed in the literature [10-12, 14, 16]. To reduce the computation time and also to increase the identified model accuracy, it is necessary to develop soft computing based system identification technique. In recent years, nature inspired heuristic algorithms such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Bacterial Foraging Optimization (BFO) are widely proposed to find solutions for a variety of engineering problems [1, 4-9, 13].

The heuristic algorithm based system identification procedure is initially discussed by Shin

et.al for stable processes [6]. In their work a real-coded GA based system identification procedure is discussed to identify transfer function model from the step response data. From the recent literature, it is observed that the Bacterial Foraging Optimization (BFO) algorithm have emerged as a powerful tool for finding the solutions for a variety of complex engineering problems [5, 8, 15]. In a classical BFO, the number of parameters to be assigned is large compared to other existing nature inspired algorithms. Hence, in this work we considered the recently discussed enhanced Bacterial Foraging Optimization (BFO) algorithm [15].

In the present work, we combined the „P‟ controller based closed loop system identification

procedure discussed in the book by Padmasree and Chidambaram [12] with the BFO algorithm. A double feedback control structure discussed by Vijayan and Panda [17] is considered for identification and control. The proposed procedure helps to attain an approximated First Order Plus Time Delayed (FOPTD) model for a class of unstable processes. Through a comparative study with the classical system identification techniques existing in the literature, the performance of the identified model is validated. The outer loop with a PI/PID controller is then tuned using the BFO algorithm. The BFO tuned double feedback controller provides better process performances such as smooth reference tracking, reduced peak overshoot, reduced settling time, and error minimization for the unstable processes considered in this paper.

The remaining portion of the paper is organized as follows: Section 2 provides the

description of „P‟ controller based system identification procedure. Section 3 and 4 provide overview of BFO and BFO based system identification procedure respectively. Results and discussions are presented in Section 5. The conclusion of the present work is drawn in Section 6.

2. CLOSED LOOP SYSTEM IDENTIFICATION PROCEDURE

The proportional (P) controller based system identification technique is very simple and the parameter to be adjusted is only the proportional gain „Kp1‟. It is a closed loop test and it can be applicable to the unstable system having the „θ/τ‟ ratio up to 0.8. The previous study reports that, the identification procedure with a stable under damped like process response can minimize the mismatch between the original process and the identified process model [14]. The closed loop structure of the double feedback scheme attempted in this work is shown in Fig. 1 [17].

Where Gp(s) is the unstable system to be identified and controlled, Gc1(s) is the inner loop „P‟

controller considered for identification, and Gc2(s) is the outer loop „PI/ PID‟ controller to stabilize the system (parallel for of PID controller with a derivative filter is considered in Gc2(s)).

Gc1(s)

Y(S)

R(S) - Gc2(s)

Gp(s)

N(S) -

1

2

S

Inner loop

Outer loop

Figure 1. Double feedback loop for Identification and Control of unstable system

BFO Approach to Identify and Control Unstable Process using a Double Feedback Loop 3 During the system identification procedure, the selector switch „S‟ is placed at position „1‟

and the inner loop controller is tuned manually by the operator. The inner loop is used to identify the process model using the step response data. In the proposed work, the identification procedure is performed in the presence and absence of the measurement noise term „N(s).

The key steps in „P‟ controller based system identification using the double feedback control loop is as follows;

Step.1 Place the selector switch ‘S’ at terminal 1. Consider the loop with ‘Gc1’ only, and

Excite the system with an unity step signal ‘R(s)’,

Step.2 Adjust the value of ‘Kp1’ until the closed loop response of the inner loop is similar

to an under damped second order system as in Fig 2.

Step.3 Calculate the values of Yp, Yv, Δt = (Tv-Tp), and Y∞

Step.4 Find the FOPTD model of the system using Equations 1 to 8,

Step.5 Validate the model.

Step.6 Place the selector switch ‘S’ at terminal 2,

Step.7 Consider the identified model to tune the outer loop PID controller

Fig.2 depicts the under damped second order like response of the inner loop for both the noisy and the noise free response for the unstable system. Inorder to get the noisy response, a band-limited white noise with a noise power of 0.2 and a mean of „zero‟ is introduced along with the feedback signal as depicted in Fig 1. The parameters of the process model are identified by considering the following equations discussed in book by Padmasree and Chidambaram [12];

)1K)(1(P k2

1 (1)

1)(K(1)(K)1K(P k2

kk2 (2)

1pk K KK (3)

22 )Vln( /)Vln( (4)

)YY/()YY(V PV (5)

Where: P1, P2, V = variables, ξ = damping ratio, Kk = closed loop gain.

The process parameters can be found by considering the following mathematical relations;

0 50 100 150 200 250 300 350 400 450 500-0.5

0

0.5

1

1.5

2

2.5

3

Time, sec

Ste

p r

esponse w

ith 'P

' contr

olle

r

Step response (Noise)

Step response (Noise free)

Overshoot (Yp)

Openloop delay

Final value (Yinf)

Undershoot (Yv)

Tv - Tp

Figure 2. Closed loop response of the system with ‘P’ controller

4 Intelligent Automation and Soft Computing

Process gain : ))1Y(K/(YK p (6)

Process time constant : π

)P P t(τ 21 / )P P t(τ 21 (7)

Closed loop delay : π P/)P t 2( 21 (8)

The identified FOPDT unstable process model will be =1τs

e K(s)G

θs

p

(9)

A comparative study is executed between the identified FOPDT model with the original

process, and the model developed using the classical system identification procedures existing in

the literature [15] in order to find the model with best fit.

3. BACTERIA FORAGING ALGORITHM

Bacteria Foraging Optimization (BFO) algorithm is a new division of population based optimization technique developed by inspiring the foraging manners of E.coli bacteria. The basic

operation of BFO algorithm has four key steps namely chemo-taxis, swarming, reproduction, and

elimination-dispersal. A detailed description of the above procedures could be found in [4,5,8,15].

The parameters of the BFO algorithms are defined below:

D = the dimension of search space, N = the total number of E.Coli bacteria, Nc= total number

of chemotaxis steps, Ns = swim length during the search, Nre = total number of reproduction steps,

Ned = total number of elimination – dispersal events, Nr = number of reproduced bacteria, Ped =

the probability that each bacterium will be eliminated / dispersed, n = the run length.

In the literature there is no apparent guide line to assign parameters for the BFO algorithm. In this work, we considered the BFO algorithm recently discussed by Rajinikanth and Latha [15]. The

chief benefit of this method is, the number of parameters to be assigned is only the values of

bacteria size (N), search dimension (D), performance index (PI), and search boundary (SB).

4. BFO BASED IDENTIFICATION and CONTROL In this paper, we proposed a time domain based Objective Function (OF), to support the BFO

based system identification search. Fig 3 shows the time response for the closed loop system.

Where; T1 = process delay, T2 = rise

time, T3 = peak time (Tp), T4 = valley time

(Tv), T5 = settling time, Tmax = time to

reach Yinf (Y∞), Y= desired process output

(reference signal), Y1 = peak overshoot, Y2

= peak undershoot, Y3 = maximum limit for

Y; Y4 = minimum limit for Y.

The objective function for system

identification with „N(s) = 0‟ can be written

as follows;

maxT

4

4

3

3

1

1

T

4

T

T

23

T

T

12

T

0

1ident YWYWYWWOF

(10)

0 5 10 15 20 25

0

0.5

1

1.5

Time (sec)

Pro

cess r

esponse

Reference Input Process output

T2T1 T4 T5

Y1

Y2

Y3

Y4

TmaxT3

Figure 3. Time domain objective function

BFO Approach to Identify and Control Unstable Process using a Double Feedback Loop 5

Where W1 to W4 are the weighting parameters (W1 = W2 = W3 = W4 =10), which supports the

optimization of the parameters such as delay (θ), over shoot (Yp), under shoot (Yv), the time

difference (Δt = Tv - Tp), and the final steady state value of process response (Y∞).

0 50 100 150 200 250 300 350 400 450 500-0.5

0

0.5

1

1.5

2

2.5

3

Time, sec

Ste

p r

esponse d

ata

for

identification

P5

P2

P4

V2

V1

V3

V4

P3

P1

Y1 Y2

Y4

Y5

V5

Y3

Search boundry for Yp

Delay

Search boundry for Yv

Search boundry for Yinf

Figure 4. Closed loop response with the search boundary for identification

Fig.4 shows the closed loop step response with noise. For noisy data, a predefined boundary

is assigned as depicted in Fig. 4, and the BFO algorithm is allowed to search five best possible

values for peak point, valley point, and final steady state value within the boundary. The average

of the above value is considered to identify the model.

The peak value Yp = [(P1+P2+P3+P4+P5)/5],

The valley value Yv = [(V1+V2+V3+V4+V5)/5], and

The final steady state value Y∞ = [(Y1+Y2+Y3+Y4+Y5)/5].

Figure 5. Block diagram: (a) Identification, (b) Control

A relative analysis between the identified model and the original process is carried to calculate the model mismatch. The identified model shows smaller contradiction with the original

system and the existing models in the literature. Fig.5 depicts the block diagram for BFO based

identification and controller tuning for unstable system. The inner loop with „P‟ controller is

utilised to find the FOPDT model of the process. The identified model is then adopted to tune the

outer loop controller „Gc2‟. The following performance criterion with five parameters, such as ISE,

IAE, Mp, tr and ts are considered in this work to tune the outer loop PI/PID controller.

OFtune = J(Kp, Ki, Kd) = (w1. IAE) + (w2. ISE) + (w3 . Mp) + (w4 . ts) + (w5. tr) (11)

Where w1, w2, ..., w5 are weighting functions used to set the priority of the objective function

(w1 = w2 = 5, w3=10, w4= w5 =7). The performance criterion J(Kp, Ki, Kd) guides the BFO

algorithm to get appropriate values for the PI/PID controller parameters.

Input parameters for BFO

based Identification

( θ, Yp, Tp, Yv, Tv, Y∞ )

Unstable

FOPTD

Model

Gc1(s)

Y(S) R(S)

-

Gp(s)

N(S)

(a)

Gc1(s)

Y(S) R(S)

- Gc2(s)

Gp(s)

N(S) -

BFO based

Controller tuning

Error Kp Ki Kd

Monitoring parameters

Outer loop

Inner loop

(b)

6 Intelligent Automation and Soft Computing

5. Results and Discussions To study the performance of the BFO based identification and control practice, four examples

are considered from the literature. The following simulation study demonstrates the ability of the

proposed method.

Example 1: Continuous Stirred Tank Reactor (CSTR) with nonideal mixing considered by Liou

and Yu-Shu has the following transfer function model [3, 12];

s20p e

198.3s

)s133.111(22.2sG

(12)

This system has one unstable pole and a stable zero. The system identification task is

executed using the inner loop with a proportional controller. The controller gain „Kp1‟ is manually

adjusted until the system provides an under damped like response (as discussed in Section 2).

Fig.6 (a) shows the closed loop response for Kp1=2. The BFO algorithm based system

identification procedure is performed for the closed loop response in the presence and the absence

of the measurement noise component „N(s)‟. The process parameters for identification task and the

developed models are presented in Table I (similar procedure is considered for other Examples

considered in this study). A comparative analysis is carried using Nyquist plot to calculate the

model fit with the original process. From Fig.7 (a), the observation is that, the Nyquist plot by

„M1‟ has a best fit with the original process „Gp(s)‟ compared to the model „M2‟ (M1 is the model

developed when N(s) = 0, and M2 is the model developed when N(s) = 0.2).

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

Time, sec

Re

sp

on

se

Step input

Response (Noise)

Response (Noise free)

(a)

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

Time, sec

Refe

rence t

rackin

g r

esponse

Reference input

Response (Noise)

Response (Noise free)

(b)

Figure 6. Closed loop response for Example 1

The BFO algorithm based „PI‟ parameter tuning is executed with the model with the best fit

(M1). Eqn 12 is considered during the controller tuning procedure, and the optimization search convergences at 74th iteration. The optimized values of proportional gain „Kp‟, and integral gain

„Ki‟ are presented in Table II. The model „M1‟ is then replaced by the original process model

„Gp(s)‟ and the reference tracking performance is tested with the BFO tuned „PI‟ controller in the

presence and absence of measurement noise component. From Fig.6 (b), the observation is that,

the proposed method provides a smooth reference tracking response for the CSTR model.

Example 2: An isothermal Continuous Stirred Tank Reactor (CSTR) studied by Liou and Yu-Shu

is represented by the following mathematical model [3, 12];

)1s69.99(

exp3226.3)s(G

s20

p

(13)

The system identification procedure for the above model is performed for the above model as

discussed in Ex.1. The parameters considered for identification and the final models are presented

in Table I. For this system „M2‟ shows a best fit compared to „M1‟ (Fig.7 (b)). The BFO algorithm

based „PID‟ tuning is executed for the outer loop with „M2‟. The optimization search convergences

at 105th iteration and the final controller parameters are given in Table II.

BFO Approach to Identify and Control Unstable Process using a Double Feedback Loop 7

The reference tracking performance for process model „Gp(s)‟ is presented in Fig.8 (a) and the

performance measure values are given in Table II. The BFO tuned PID controller with a double

feedback loop provides a flat reference tracking response for the isothermal CSTR model.

Table I. Closed loop process parameters and identified FOPDT model

Process N(s) Kp1 Process parameters Identified model % Fit

Yp Yv Δt Y∞ K τ D

Ex. 1 0 2.0 1.618 1.037 20.0 1.291 2.2176 29.218 9.4691 86.35

0.2 1.643 0.997 21.7 1.318 2.0694 27.800 10.773 84.91

Ex. 2 0 1.5 1.925 0.903 52.6 1.256 3.2708 92.629 22.701 94.73

0.2 1.967 0.852 52.9 1.248 3.3548 95.732 23.432 96.10

Ex. 3 0 1.2 8.860 4.645 9.40 6.000 1.0000 2.0703 1.6372 83.05

0.2 8.521 4.255 12.1 5.820 1.0061 2.7018 2.2042 80.68

Ex. 4 0 1.5 4.08 2.61 15.5 3.00 1.0000 6.0207 3.6585 84.17

0.2 4.27 2.55 16.8 3.24 0.9643 5.8513 4.2938 82.39

-2 -1.5 -1 -0.5 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Real axis

Ima

gin

ary

axis

Gp(s)

M1

M2

(a)

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Real axis

Ima

gin

ary

axis

Gp(s)

M1

M2

(b)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Real axis

Ima

gin

ary

axis

Gp(s)

M1

M2

(c)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Real axis

Ima

gin

ary

axis

Gp(s)

M1

M2

(d)

Figure 7. Nyquist plots for real process and identified models

8 Intelligent Automation and Soft Computing

Example 3: Let us take an unstable second order process as stated below [2];

1s5.0 12s

1esG

s

p

(14)

The system identification and controller tuning procedure for the above process model is

proposed as discussed in Ex.1 using BFO algorithm. The parameters obtained during the identification study are presented in Table I and Fig. 7 (c) depicts the Nyquist plot. The BFO

algorithm based PID controller tuning converges at 77th iteration. Table II gives the controller

parameters for the outer loop and the overall performance measure for the double feedback loop

for the considered mode. Fig.8 (b), shows the satisfactory reference tracking performance for the

original process model with the BFO tuned PID controller.

Table II. BFO tuned controller parameters and its performance appraisal

Process Outer loop PI/PID values Performance measure

Kp Ki Kd Mp tr ts ISE IAE

Ex. 1 0.0952 0.0217 0.0000 0.062 78.15 191.9 31.90 41.09

Ex. 2 0.1496 0.0211 3.7014 0.011 115.5 210.2 27.15 38.77

Ex. 3 0.1309 0.0587 0.3914 0.025 4.137 35.48 1.860 3.192

Ex. 4 0.1378 0.0469 0.6638 0.000 36.50 36.50 4.093 7.112

Example 4: Let us consider an unstable third order process with one unstable pole [2,10,16].

1s21s5.0 15s

1esG

s5.0

p

(15)

Majhi derived a second order model, GM1 (s) = 1.001 e-0.939s / (10.354s2 + 2.932s -1) using the

relay feedback based identification procedure [10]. Liu and Gao developed a second order model,

GM2 (s) = 1.0001 e-0.9422s / ((4.9996s-1) (2.07s+1)), and a FOPDT model, GM3 (s) = 1.0001 e-3.2821s /

(5.7663s-1), using the relay feedback test [15]. The second order model GM1 and GM2 shows a

better fit with the original process Gp(s) (≈ 97% fit). The FOPDT model GM3 (s) shows a model fit

of 91.47% with the real process given in Eqn.15. The FOPDT model developed by the proposed method is tabulated in Table I. The model „M1‟ is very close with the FOPTD model „GM3 (s)‟

developed by Liu and Gao [10].

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

Time, sec

Re

sp

on

se

Reference input

Process output

(a)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

Time, sec

Re

sp

on

se

Reference input

Process output

(b)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Time; sec

Re

sp

on

se

Reference input

Gp1(t)

Gp2(t)

(c)

Figure 8. Reference tracking response of unstable systems with BFO tuned PID controller

BFO Approach to Identify and Control Unstable Process using a Double Feedback Loop 9

The BFO algorithm based „PID‟ tuning for „M1‟ convergences at 68th iteration and the

optimized controller parameters and the closed loop performance of the system are presented in

Table II. Form Fig.8(c), it is noted that, the double feedback control loop provides a smooth

reference tracking response with almost zero overshoot with satisfactory values for tr, ts, ISE, and

IAE .

6. Conclusions

In this paper, BFO based system identification and control procedure is discussed for a class of unstable processes using a double feedback control loop. A novel time domain based objective function is developed to guide the BFO based system identification search. The inner loop with proportional controller enables us to develop the first order plus time delayed model for the unstable processes. The identification analysis is tested on a varied class of unstable process models in the presence and absence of measurement noise. The proposed identification procedure shows a robust performance on the process data with measurement noise. The simulation result illustrates that, the discussed method helps to attain a satisfactory process model with considerably reduced model mismatch. The PI/PID controller in the outer loop is used to control the unstable system. The BFO tuned double feedback controller provides better process performances such as smooth reference tracking, reduced peak overshoot, reduced settling time, and error minimization for the unstable processes considered in this study.

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10 Intelligent Automation and Soft Computing

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ABOUT THE AUTHORS

V. Rajinikanth received B.E. in Instrumentation and Control Engineering from the University of Madras (1999) and M.E. in Process Control and Instrumentation from

Annamalai University (2002). Currently he is working as an Assistant Professor in the Department of Electronics and Instrumentation Engineering in St.Joseph‟s College of Engineering, Chennai. His research interests are Unstable system identification and control, and Soft computing.

Dr. K. Latha received Ph.D in Electrical Engineering from the Anna University,

Chennai in 2006. Currently she is working as an Associate Professor in the Department of Instrumentation Engineering, Madras Institute of Technology Campus, Anna University, Chennai, India. Her research interests are Process modeling, Dynamic system modeling and control, and Soft computing.