Modelling and Simulations C

International Review on

Modelling and Simulations (IREMOS)

PART

C

Contents

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

(continued from Part B) Effect of Crack Faces Opening Mechanisms on the Combined Stress Intensity Factors Under Bending and Torsion Moments by A. E. Ismail, A. K. Ariffin, S. Abdullah, M. J. Ghazali

3232

EMTP-RV Model of Hydraulic Digital Governor by M. Dabro, I. Juriý-Grgiý, R. Luciý

3239

Emission Controlled Security Constrained Unit Commitment Considering Hydro-Thermal Generation Units by M. S. Javadi, A. Meskarbashi, R. Azami, Gh. Hematipour, A. Javadinasab

3243

Development of an Inertial Measurement System for Electronic Stability Control Applications by V. Mahboubi, M. Khoddam, H. Badri, A. Ebrahimi

3251

Selecting Suitable Journal in Digital Libraries with Web Usage Mining by Farzam Yousefi, Sayed Mehran Sharafi, Mohammad Ali Nematbakhsh

3261

Matlab Based Fluid Level Control Using PID and Fuzzy Logic Controllers by Abdurrahman Ünsal, Ahmet Kabul

3273

A New Time-Domain H2-Norm Robust Fault Detection Filter for a Drum Boiler System Through an LMI Approach by L. Khoshnevisan

3279

Fuel Cell and Solar Cell Hybrid System for Electrical Energy Generation by S. Hossain, F. Sharmin, M. Ahmad, I. Daut, M. A. Rashid

3285

Analyzing the Properties of Mobius Capacitors by Eraldo Banovac, Sejid Tesnjak, Darko Pavlovic

3292

Development of the Lightning Location Mapping System Using Fuzzy Logic Technique by A. Che Soh, R. Z. Abdul Rahman, M. Z. A. Ab. Kadir, N. S. Mohd Shif

3301

Gantry Robot Control with an Observer Based on a Subspace Identification with Multiple Steps Data by A. Bouhenna, M. Chenafa, A. Mansouri, A. Valera

3309

Designing Intelligent Advanced Controller for a Class of Large Scale Non-Canonical Nonlinear Systems: Observer-Based Approach by R. Ghasemi, M. B. Menhaj

3317

Adaptive Gain Scheduling Fuzzy Logic PID Controller in Load Frequency Control of Wind Diesel Micro Hydro Isolated Hybrid Power System by R. Dhanalakshmi, S. Palaniswami

3327

Modelling, Simulation and Analysis of a 5-Dof Planar Parallelogram – Link Biped Mechanism by C. Campos, R. Campa, M. Llama, A. Pámanes

3337

(continued)


Computation of Electric Field and Potential Distribution Around a Polluted Porcelain Insulator by Boundary Element Method by K. Krishnamoorthi, S. Chandrasekar, D. Pradhap

3353

Fuzzy Pattern Recognition Based Fault Diagnosis by Rafik Bensaadi, Leïla H. Mouss, Mohamed D. Mouss, Mohamed E. H. Benbouzid

3361

Smart Assisting Device for Partially Paralyzed People During Locomotion-Design and Control by T. S. Sirish, K. S. Sivanandan

3371

Wind-PV-Grid Connected Hybrid Renewable System in Kish Island by Arash Anzalchi, Babak Mozafari

3376

Coupled-Mode Analysis of a T-Branch Waveguide with a Wavelength-Selective Reflection Feedback by K. Fasihi

3383

Prediction of Module Operating Temperatures for Free-Standing (FS) Photovoltaic (PV) System in Malaysia by H. Zainuddin, S. Shaari, A. M. Omar, S. I. Sulaiman, Z. Mahmud, F. Muhamad Darus

3388

A Study on Duffing Oscillator’s Ability on Detecting Disappearance of the Detected Weak Signal by V. Rashtchi, M. Nourazar

3395

Performance Comparison of Dynamic Models of Proton Exchange Membrane and Planar Solid Oxide Fuel Cells Subjected to Load Change by N. A. Zambri, A. Mohamed, H. Shareef

3402

Design and Analysis of a Composite Cone Type Spacer in Gas Insulated Systems Under Various Abnormal Conditions Using Finite Element Method by D. Deepak Chowdary, J. Amarnath

3410

Air Traffic Flow Modelling and Simulations Based on Dynamic Networks by K. Bousson, Tiago M. Domingues

3418

Performance Enhancement in Rolling Processes Using Mixed Sensor-Based and Estimation-Based Control Algorithms by Shahab Amelian, Hamid Reza Koofigar

3425

Coordination of Overcurrent Relays Using New Two Step Algorithms, a Comparative Study Versus Classic Optimization Algorithms by Hadi Hosseinian Yengejeh, Hossein Nasir Aghdam, Hossein Askarian Abyaneh

3430

International Review on Modelling and Simulations (I.RE.MO.S.), Vol. 4, N. 6 December 2011

A Study on Duffing Oscillator’s Ability on Detecting Disappearance of the Detected Weak Signal V. Rashtchi1, M. Nourazar2 Abstract – After the successful use of duffing oscillator in weak signal detection, many researches are done on this subject. Most of these studies only analyze the efficiency of duffing oscillator on detecting the existence of weak signal and they ignore to study duffing oscillator’s ability on detecting the loss of signal that was already detected. Considering this, we are going to analyze this ignored issue. In this paper we introduce two methods to add this ability to duffing oscillator. Copyright © 2011 Praise Worthy PrizeS.r.l. - All rights reserved. Keywords: Weak Signal Detection, Duffing Oscillator, Largest Lyapunov Exponent, Oscillator Resetting

Nomenclature ‫ݔ‬ǡ ‫ݕ‬ ܾ ߛ ߱଴ ߮ ܽ ο߱ ݄ ݀଴

Phase space variables Damping Ratio Driving force amplitude Internal driving force frequency Input signal primary phase difference Input signal amplitude Input signal frequency difference Numerical solution’s step size Distance of two nearby pointsof LLE method

I.

Introduction

Detection and measurement of weak signals in noisy environment is one of the most challenging problems in many applications. Especially in some areas like communication, radar, sonar, fault detection, GPS and industrial measurement, detecting the weak signal plays an important role [1]-[6]. Although some methods have been presented previously for weak signal detection, like Single-layer autocorrelation, multi-layer autocorrelation, discrete wavelet transform [7], [8], [9] and some other methods but immunity to noise was an unachieved goal for decades. After the first introduction of chaotic process by Lorenz in 1963 [10], scientists and engineers pay a particular attention to the applications of chaos. With fast development of chaos theory, application of chaos for detecting weak signal was introduced in 1992. Donald L Birx found that the chaotic phase trajectory has good sensitivity to the weak signal and also it is immune to the noise [11]. Holmes Duffing equation is one of the mostly used chaotic systems for detecting weak signals [12]. Duffing equation is a non-linear second-order differential equation, which is named after G. Duffing, who studied on this equation in 1918. Since 1992, there has been a lot of work done on this application. Manuscript received and revised November 2011, accepted December 2011

3395

Wang proved the validity of this method by using it to detect weak sinusoidal signal in the presence of the strong noise, and obtained measured result with low SNR and he has done a quantitative study on detection and estimation of weak signals by using duffing oscillators [10], [12]. In fault detection, duffing oscillator is used to detect sideband components that appear in the stator current around the fundamental frequency of the supplied current, when dynamic eccentricity fault occurs in an induction motor [13]. In GPS, Weak signal detection by Duffing oscillator is combined with traditional correlation methods and improved GPS receivers [14]. Duffing oscillator improves the abilities of sonar detection and target identification [15]. Since the precision of these applications depend on correct adjustment of threshold value and detecting the state of duffing oscillator, some methods have been suggested to do this job and detect any transition in the state of duffing oscillator. One of these methods for threshold value is Melnikov. But precision of this method does not meet the requirement of the weak signal detection [17]. Other methods are also presented for threshold value that has a good precision, like the one in [17]. Some of the well-known methods for detecting the state of oscillator are: Poincare map, FFT, Correlation Dimension, Lyapunov Exponent, 0-1 test, SALI and CZP [16], [18]-[21]. Most of papers about weak signal detection by chaotic oscillators only pay attention to detect the existence of weak signal and they ignored study of detecting disappearance of signal after existence of it. In this paper we are going to consider this issue and analyze the ability of duffing oscillator to go back to chaotic state after transition to periodic state. In this regard, two methods are suggested to overcome this problem and improve the ability of duffing oscillator to go back to chaotic state after disappearance of signal. For detecting the state of Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

V. Rashtchi, M. Nourazar

duffing oscillator in our simulations we use largest lyapunov exponent. This paper is organized in four sections as follows: In section II, the principles of Weak signal detection by duffing oscillator will be described. In section III calculation of the largest Lyapunov exponent will be explained. And the last section IV is about the results of our simulations.

II.

Principles of Weak Signal Detection Based on Duffing Oscillator

The basic form of Holmes Duffing equation can be considered as Eq. (1) [8]: ‫ ݔ‬ƍƍ ሺ‫ݐ‬ሻ ൅ ܾ‫ ݔ‬ƍ ሺ‫ݐ‬ሻ െ ‫ݔ‬ሺ‫ݐ‬ሻ ൅ ‫ ݔ‬ଷ ሺ‫ݐ‬ሻ ൌ ߛ ȉ ሺ‫ݐ‬ሻ

The transition from the chaotic state to the periodic state is the basis for detecting weak signals by means of chaotic oscillators. In all methods considered in this paper, transition from chaotic state to periodic state, displays the existence of a signal with specific frequency. That specific frequency is same as the internal frequency of oscillator and this is the main advantage of weak signal detection by chaotic oscillator. Being sensitive only to signals with frequency equal to the internal frequency of oscillator and immunity to noise, make this method a good choice for detecting weak signals. Figs. 1 show the state of the duffing oscillator in the chaotic and periodic states.

(1)

In Eq. (1), ߛǤ ሺ‫ݐ‬ሻ and ܾ‫ ݔ‬ƍ ሺ‫ݐ‬ሻ are respectively driving force and damping to remove energy conservation from the system. Since this equation is normalized in frequency, some frequency transformation must be done to be useable in other frequencies. If we take‫ ݐ‬ൌ ߱଴ ߬, duffing equation can be written as Eq. (2) [12]: ܾ ƍ ͳ ƍƍ ‫ ݔ‬ሺ߱଴ ߬ሻ ൅ ‫ ݔ‬ሺ߱଴ ߬ሻ െ ‫ݔ‬ሺ߱଴ ߬ሻ ൅ ‫ ݔ‬ଷ ሺ߱଴ ߬ሻ ൌ (2) ߱଴ ߱଴ଶ ൌ ߛ ȉ ሺ߱଴ ߬ሻ

(a)

If we add our input signal (to be detected signal) to this equation, we get Eq. (3): ͳ ƍƍ ܾ ƍ ሺ߱଴ ߬ሻ ൅ ‫ ݔ‬ሺ߱଴ ߬ሻ െ ‫ݔ‬ሺ߱଴ ߬ሻ ൅ ‫ ݔ‬ଷ ሺ߱଴ ߬ሻ ൌ ଶ‫ݔ‬ (3) ߱଴ ߱଴ ൌ ߛ ȉ ሺ߱଴ ߬ሻ ൅ ‫ݐݑ݌݊ܫ‬ where‫ݐݑ݌݊ܫ‬ሺ‫ݐ‬ሻ will be in the form of Eq. (4): ‫ ݐݑ݌݊ܫ‬ൌ ȉ ሺሺ߱଴ ൅ ο߱ሻ߬ ൅ ߮ሻ ൅ ݊‫݁ݏ݅݋‬ሺ߬ሻ

(4)

Since there is no absolute analytical solution for duffing equation, numerical solution would be our choice. For solving this equation by Runge-Kutta 4th order algorithm, the state equation form is needed that can be obtained from Eq. (5): ‫ ݔ‬ሺ‫ݐ‬ሻ ൌ ߱Ͳ ‫ݕ‬ሺ‫ݐ‬ሻ ቐ‫ݕ‬ƍ ሺ‫ݐ‬ሻ ൌ ߱Ͳ ሾ‫ݔ‬ሺ‫ݐ‬ሻ െ ‫ ͵ݔ‬ሺ‫ݐ‬ሻ െ ܾ‫ݕ‬ሺ‫ݐ‬ሻ ൅ ߛ ሺ߱‫ݐ‬ሻ ൅ (5) ƍ

൅‫ݐݑ݌݊ܫ‬ሿ

The mentioned duffing equation, like most nonlinear dynamic systems, has four states [12]: 1. the chaotic motion, 2. the quasi periodic motion, 3. the periodic motion, and 4. the fixed point.


(b) Figs. 1. States of duffing oscillator (a) Periodic state. (b) Chaotic state

To increase the sensitivity to signals with low amplitude, the oscillator should work in critical state. Critical state means chaotic state but on the verge of transition to periodic state. In the critical state a small signal is enough to force the oscillator to change its state to periodic. To calibrate oscillator for working in critical state, we fixed ܾ ൌ ͲǤͷ and gradually increased ߛ until the oscillator enter the large periodic state. Last ߛ that forces the oscillator to work in chaotic state before entering to International Review on Modelling and Simulations, Vol. 4, N. 6

3396


periodic state will be considered as ߛ஼ . Another parameter that should be considered before solving the Eq. (5) by Runge-Kutta is the step of Runge-Kutta algorithm (݄). With different steps, different threshold values obtained for critical state. As mentioned in [16], ݄ is limited to the range of: ͳ ͳ ൑݄൏ ͷ݂ ͳͲ݂

(6)

This limitation is for ensuring the stability and efficiency of Runge-Kutta algorithm. The recursive equations that we use for 4th order Runge-Kutta are as Eqs. (7), (8) and (9):

where:

and:

݄ ‫݇ ۓ‬ଵ ൌ ݂ሺܻ௡ ሻ ʹ ۖ ݄ ۖ ݈ଵ ൌ ݃ሺ‫ݐ‬௡ ǡ ܺ௡ ǡ ܻ௡ ሻ ʹ ۖ ۖ ݇ ൌ ݄ ݂ሺܻ ൅ ݈ ሻ ௡ ଵ ۖ ଶ ʹ ۖ ݄ ݄ ݈ଶ ൌ ݃ ൬‫ݐ‬௡ ൅ ǡ ܺ௡ ൅ ݇ଵ ǡ ܻ௡ ൅ ݈ଵ ൰ ʹ ʹ ‫ ݇ ۔‬ൌ ݄݂ሺܻ ൅ ݈ ሻ ଷ ௡ ଶ ۖ ݄ ۖ ݈ଷ ൌ ݄݃ ൬‫ݐ‬௡ ൅ ǡ ܺ௡ ൅ ݇ଶ ǡ ܻ௡ ൅ ݈ଶ ൰ ʹ ۖ ݄ ۖ ݇ ൌ ݂ሺܻ ൅ ݈ ሻ ௡ ଷ ۖ ସ ʹ ݄ ۖ ሻ ‫݈ ە‬ସ ൌ ʹ ݃ሺ‫ݐ‬௡ ǡ ܺ௡ ൅ ݇ଷ ǡ ܻ௡ ൅ ݈ଷ

(7)

‫ ܭ‬ൌ ሺ݇ଵ ൅ ʹ݇ଶ ൅ ݇ଷ ൅ ݇ସ ሻȀ͵

(8)

‫ ܮ‬ൌ ሺ݈ଵ ൅ ʹ݈ଶ ൅ ݈ଷ ൅ ݈ସ ሻȀ͵

(9)

݂ሺܻ௡ ሻ ൌ ܻ߱௡

x when the equations of a chaotic system are known; and x when only experimental data are available from observation (or the equations are too complicated for calculation). The first way enjoys better precision and is also more easily calculable than the second one [22]. Since the equation for weak signal detection by the duffing oscillator is known, the first way will be chosen. Since chaotic oscillators are highly sensitive to initial conditions, the trajectories of the first two values which are close enough will separate exponentially as time increases, which can be described by LE [17], [23]. The objective is to follow two nearby orbits and to calculate their average logarithmic rate of separation [22]. Whenever these two orbits get too far apart, one of the orbits has to be moved back to the vicinity of the other along the line of separation. The procedure for calculating the LLE is as follows [22]: 1- Iterate until the oscillator begins working in the critical state and it has passed its transient response; 2- Choose an arbitrary point a0 from the phase trajectory; 3- Choose or imagine point b0 as a nearby point separated by݀଴ (a suitable value for݀଴ is about 1000 times larger than the precision of the floating point number being used); 4- Iterate both points and calculate the new distance of the two selected points in steps 2 and 3 (a1, b1): ݀ଵ ൌ ඥሺ‫ݔ‬௔ଵ െ ‫ݔ‬௕ଵ ሻଶ ൅ ሺ‫ݕ‬௔ଵ െ ‫ݕ‬௕ଵ ሻଶ 5- Calculate ‫ ܦܮ‬ൌ ݈‫݃݋‬

ௗభ ௗబ

(13)

.

6- Readjust the second point so its separation is ݀଴ in the same direction as ݀ଵ :

(10)

݀଴ ሺ‫ ݔ‬െ ‫ݔ‬௔଴ ሻ ݀ଵ ௕଴ ݀଴ ൌ ‫ݕ‬௔ଵ ൅ ሺ‫ݕ‬௕଴ െ ‫ݕ‬௔଴ ሻ ݀ଵ

‫ݔۓ‬௕଴ ൌ ‫ݔ‬௔ଵ ൅

݃ሺ‫ݐ‬௡ ǡ ܺ௡ ǡ ܻ௡ ሻ ൌ ߱ሾܺ௡ െ ܺ௡ ଷ െ ܾܻ௡ ൅ ൅ߛ ሺ߱‫ݐ‬௡ ሻ ൅ ‫ݐݑ݌݊ܫ‬ሺ‫ݐ‬௡ ሻሿ

(11)

ܺ௡ାଵୀ ܺ௡ ൅ ‫ܭ‬ ൝ ܻ௡ାଵୀ ܻ௡ ൅ ‫ܮ‬ ‫ݐ‬௡ାଵୀ ‫ݐ‬௡ ൅ ݄

(12)

‫ݕ۔‬ ‫ ە‬௕଴

7- Repeat step 4-6 and the LLE will be the average of ld.

IV.

௛

Because of ‫ݐ‬௡ ൅ in Runge-Kutta fourth order ଶ algorithm, ‫ ݐݑ݌݊ܫ‬signal should be sampled with period ௛ of . ଶ

III. Largest Lyapunov Exponent The LLE is the most common test in chaos. In this test a positive largest lyapunov exponent indicates the chaotic state and, consequently, the negative LLE points to the periodic state. LLE calculations can be carried out in two ways: Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

(14)

Results of the Simulations

The chaotic system of our simulation is based on Homes Duffing equation with state equations mentioned in Eq. (5). The internal driving force frequency (߱଴ ) is set to ͷͲ‫ ݖܪ‬and we consider the fixed value of ܾ ൌ െͲǤͷ for damping force. The numerical solution that we use is Runge-Kutta 4th order that was mentioned in Eq. (7)-(12). Since for different values of ݄, different values of ߛ will make the oscillator work in critical state, the proper value for ݄ is the largest number which for the smaller ones, the value of ߛ௖௥௜௧௜௖௔௟ does not differ specially.

International Review on Modelling and Simulations, Vol. 4, N. 6

3397


So by practice ݄ is set to 0.0002 and for this ݄, oscillator works in critical state when ߛ௖ ൌ ͲǤͺʹͷͺ. External input signal (Eq. (4)) is a cosine function with a fixed frequency equal to ͷͲ‫ ݖܪ‬same as the internal frequency of oscillator. White Gaussian noise also is added to this cosine function. Amplitude of cosine function vary depend on kinds of simulations. The main goal of our simulations is to remove the signal after detecting it and analyze the duffing oscillator’s ability to detect disappearance of signal. In other words, study the ability of duffing oscillator in going back to critical chaotic state after transition to periodic state. To analyze this, the input signal is applied in some periods and omitted in some periods. We divided the time of simulation which is 72 second to six equal parts and input signal exists only in three parts (second and fourth and sixth parts). Fig. 2 is showing the input signal during these six parts. In Figs. 3, for better study, outputs of duffing oscillator are shown in these six parts separately. And Fig. 4 shows the LLE calculation result during the simulation time.

2- The LLE, because of its averaging, need some time to detect existence of signal. This defines the necessary minimum time of signal existence. 3- The oscillator need some time to finish its transient state because of unknown initial values and be ready to work in critical state. The first idea to overcome the problem is to reset the oscillator every few second. To avoid the change of phase shift between input signal and internal frequency of oscillator on each reset, only phase space variables should be reset to their initial values.

(a)

(b)

(c)

(d)

(e)

(f)

TABLE I SIX DIFFERENT PARTS OF THE INPUT SIGNAL USED IN THE SIMULATION OF THE DUFFING OSCILLATOR Case

Input signal Amplitude

Time

A B C D E F

0.0000 0.0001 0.0000 0.0001 0.0000 0.0001

0-12 second 12-24 second 24-36 second 36-48 second 48-60 second 60-72 second

Figs. 3. States of duffing oscillator’s output for cases mentioned in Table I

Fig. 2. Input signal during the simulation time

There are three obvious points in Figs. 3 and Fig. 4: 1- The duffing oscillator has a problem in detecting the loss of input signal and couldn’t go back to its chaotic state after transition to periodic state.


Fig. 4. Largest lyapunov exponent of the oscillator output during the simulation of different input signals (threshold value=0.005)


3398


But before resetting the oscillator, some problems should be considered. After loading the phase space variables with their unknown initial values, some time is needed to oscillator work in critical state and finish its transient state. Transient state of oscillator during this time can cause wrong state detection. So this transient time should be as short as possible. Reducing this time can be achieved by choosing known values for initial values. The best value is a sample of phase space variables while oscillator is working in critical state. Another problem that should be considered in resetting the duffing oscillator is the time between each reset. This should be chosen by considering the minimum time that the signal must exist to be detected. Our simulation shows that when the amplitude of weak signal is increased the minimum time is decreased and vice versa. So for a known minimum value of amplitude, this value can be adjusted by practice. Time between each reset should be larger than the mentioned minimum time. The result of duffing oscillator’s output state with resetting method is shown in Figs. 5 with two different times between each reset.

(a)

Another method to improve the ability of duffing oscillator in detecting the loss of input signal is to repeat the simulation with larger amplitude for input signal according to Table II and also two different ߛ௖ : (a) ߛ௖ ൌ ͲǤͺʹͷͺ (critical state) and (b) ߛ௖ ൌ ͲǤͺʹͷͲ (chaotic state). As given in Figs. 6, the duffing oscillator is able to could go back to chaotic state only when it is in the chaotic state before applying the weak signal. This simulation shows that when the oscillator works in critical state, it can only detect the existence of weak signal by transition to periodic state. TABLE II SIX DIFFERENT PARTS OF THE INPUT SIGNAL USED IN THE SIMULATION OF THE DUFFING OSCILLATOR WITH LARGER AMPLITUDE Case A B C D E F

Input signal Amplitude 0.0000 0.0016 0.0000 0.0016 0.0000 0.0016

Time 0-12 second 12-24 second 24-36 second 36-48 second 48-60 second 60-72 second

(a)

(b)

(b)

Fig. 5. LLE of duffing oscillator after applying the resetting method for two different times between each reset: (a) reset every 18 second. (b) reset every 2 second (threshold value=0.005)

Figs. 6. LLE of duffing oscillator after using large amplitude with two different ߛ௖ : (a) ߛ௖ ൌ ͲǤͺʹͷͺ (critical state) and (b) ߛ௖ ൌ ͲǤͺʹͷͲ (chaotic state). (Threshold value=0.005)



3399


But when the signal is disappeared, since the oscillator is working in periodic state, not critical state, it can only approach to the verge of critical state (its initial state) but still in periodic state. When the oscillator is adjusted to work in chaotic state, although it can’t detect signals with very small amplitude but it is possible to detect signals with larger amplitude. When the signal is disappeared, the amplitude of the omitted signal is large enough to change the oscillator’s state from periodic state to its initial chaotic state. Although in this method signals with very small amplitudes can’t be detected but we are free of resetting problems.

V.

Conclusion

References

[2]

[3]

[4]

[5]

[6]

[7]

[9]

[10]

[11]

[12]

[13]

In a brief review, the ability of duffing oscillator on detecting the disappearance of the weak signal that is already detected by oscillator is studied in this paper. The original method of weak signal detection by duffing oscillator in which the oscillator is adjusted to work in critical sate, was unable to change its state to chaotic state when the detected signal was removed. In this paper two methods are introduced to add the mentioned ability to duffing oscillator. Resetting the oscillator every few seconds was the first method. Although this method has some consideration mostly about the time between each reset but shows acceptable results for detecting the disappearance of the detected weak signal. The second method was adjusting the duffing oscillator to work in chaotic state instead of critical chaotic state mentioned in the section II. By adjusting the duffing oscillator with smaller values of ߛ௖ that makes it to work in chaotic state, sensitivity of oscillator to weak signals is decreased but instead, transition from periodic state to chaotic state is possible when the detected signal is removed.

[1]

[8]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

C. Li, Study of Weak Signal Detection Based on Second FFT and Chaotic Oscillator. Nature and Science, vol. 3(Issue 2):59-64, 2005. A. Jalilvand, H. kargar, H. Fotoohabadi, High Impedance Fault Detection Using Duffing Oscillator and FIR Filter, International Review of Electrical Engineering (IREE), vol. 5 (Issue3):1255 – 1265, May 2010. V. Rashtchi, R. Aghmashe, M. Ojaghi, Application of Duffing Oscillators to Dynamic Eccentricity Fault Detection in Squirrel Cage Induction Motors, International Review of Electrical Engineering (IREE), vol. 6 (Issue 3):1196-1203, June 2011. J. Wang, J. Zhou, B. Peng, Weak Signal Detection Method Based on Duffing Oscillator, Kybernetes, vol. 38 (Issue 10): 1662-1668, 2009. P. Huang, Y. Pi, Z. Zhao, Weak GPS Signal Acquisition Algorithm Based on Chaotic Oscillator, EURASIP Journal on Advances in Signal Processing, vol. 2009:1-6, January 2009. C. Li, L. Qu, Applications of Chaotic Oscillator in Machinery Fault Diagnosis, Mechanical Systems and Signal Processing, vol. 21 (Issue 1):257-269, January 2007. Z. Zheng, F. Zhang, Comparative study on weak signal detection algorithms. E-Business and E-Government (ICEE): 3825–3828, Guangzhou, China, May 2010.


[23]

A. Sadoughi, M. Jafarboland, S. Tashakkor, A Practical Bearing Fault Diagnosing System Based on Vibration Power Signal Autocorrelation, International Review of Electrical Engineering (IREE), vol. 5 (Issue 1):148-154, February 2010. H. Akbari, J. Milimonfared, H. Meshgin-Kelk, Axial Static Eccentricity Detection in Induction Machines by Wavelet Technique, International Review of Electrical Engineering (IREE), vol. 5 (Issue 3):936-943, June 2010. G. Wang, S. He, A Quantitative Study on Detection and Estimation of Weak Signals by Using Chaotic Duffing Oscillators, IEEE Transactions on Circuits and Systems, vol. 50 (Issue 7):945-953, July 2003. Ch. Wei, M. Chen, W. Cheng, Z. Zhe.Summary on weak signal detection methods based on chaos theory. Electronic Measurement & Instruments (ICEMI), pp. 1-430-1-435, Beijing, China, Aug. 2009. G. Wang, D. Chen, J. Lin, The application of Chaotic Oscillators to Weak Signal Detection. IEEE Transaction on Industrial Electronics, vol. 46(Issue 2):440-444, April 1999. Rashtchi V, Aghmashe R, Ojaghi M. Application of Duffing Oscillators to Dynamic Eccentricity Fault Detection in Squirrel Cage Induction Motors. International Review of Electrical Engineering (IREE), Volume 6, Issue 3, 1196-1203, Part A, Jun 2011. Q. Honglei, S. Xingli, J. Tian. Weak GPS signal detect algorithm based on duffing chaos system. Signal Processing (ICSP), pp. 2501–2508, Beijing, China, Oct. 2010. H. Wu, W. Liu, J. Lou. Application of chaos in sonar detection. Mechanic Automation and Control Engineering (MACE), pp. 4774–4777, Hohhot, China, July 2011. Y. Fu, D. Wu, L. Zhang, X. Li, A Circular Zone Partition Method for Identifying Duffing Oscillator State Transition and its Application to BPSK Signal Demodulation. Science China Information Sciences, vol. 54(Issue 6):1274-1282, June 2011. D. Liu, H. Ren, L. Song, H. Li. Weak signal detection based on chaotic oscillator. The 40th IEEE Industry Applications Conference, Vol. 3, pp. 2054-2058, Hong Kong, Oct. 2005. S. Chandra, A Tutorial and Diagnostic Tool for Chaotic Oscillators and Time Series, Computers & Graphics, vol. 21 (Issue 2):253-262, March-April 1997. A. Bedri, E. Akin, Tools for Detecting Chaos, Journal of the Institute of Science & Technology ofSakarya University, vol. 9 (Issue 1):60-66, 2005. F. Freistetter, Fractal Dimensions as Chaos Indicators, Celestial Mechanics and Dynamical Astronomy, vol. 78 (Issue 1-4):211225, 2000. G. Gottwald, I. Melbourne, On the Implementation of the 0–1 Test for Chaos, SIAM Journal on Applied Dynamical Systems, vol.2009 (Issue 1):129-145, June 2009. J. Sprott, Chaos and Time-Series Analysis (Oxford University Press 2003). M. Rosenstein, J. Collins, C. Luca. A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets. Physica D, vol. 65 (Issue 1-2):117-134, May1993.

Authors’ information 1,2

Electrical Engineering Department, Zanjan University, Zanjan, Iran,

Vahid Rashtchi was born in 1967 in Zanjan, Iran. He received his B.Sc. in electrical engineering from Tabriz University in 1991 and his M.Sc. and Ph.D. in electrical engineering from Amirkabir University of Technology in 1993 and 2001, respectively. He is currently associate professor at Zanjan University, working on power electronics and applications of artificial intelligent systems since 2001. Up to now, he has had several outstanding positions such as "Head of the Engineering Faculty of Zanjan University" and "Research Vice Chancellor of Zanjan University". E-mail: [email protected]


3400


Mohsen Nourazar received his B.Sc. degree in communication engineering and the M.Sc. degrees in electrical engineering from Zanjan University, Zanjan, Iran, in 2009 and 2011, respectively. His research interests include applications of artificial intelligent systems and hardware implementation based on FPGA. E-mail: [email protected]



3401