Wavelet-Based Linearization Method in Nonlinear Systems

Transcription

1 Wavelet-Based Linearization Method in Nonlinear Sstems Xiaomeng Ma A Thesis In The Department of Mechanical and Industrial Engineering Presented in Partial Fulfillment of the Requirements for the Degree of Master of Applied Science (Mechanical Engineering) at Concordia Universit Montreal, Quebec, Canada Jul, Xiaomeng Ma,

2 This is to certif that the thesis prepared Concordia universit School of graduate studies B: Xiaomeng,Ma Entitled: Wavelet-based Linearization Method in Nonlinear Sstems And submitted in partial fulfillment of the requirement for the degree of Master of Applied science(mechanical engineering) Complies with the regulations of the universit and meets the accepted standards with respct to originalit and qualit. Signed b the final eamining committee: Dr. AK.W,Ahmed Dr. A.Agunduz Dr. N.Bouguila Dr. W.F.ie Chair Eaminer Eaminer Supervisor Approved b Chair of department or graduate program director Dean of Facult Date

3 ABSTRACT The mathematical theories for linear sstems provide a unified framewor for understanding all possible linear sstem behaviors but no correspondingl general theoretical ones eist for treating non-linear problems. Due to the inherent nonlinearities of almost phsical sstems, the subect of nonlinear control is an important area of automatic control. Among man methods to deal with a nonlinear sstem, the most commonl used one is to linearize it which allows taing advantage of mature linear sstem control techniques to control the nonlinear sstems approimatel. This thesis introduces a wavelet-based linearization method to estimate the nonlinear sstem response based on the traditional equivalent linearization technique. The mechanism b which the signal is decomposed and reconstructed using the wavelet transform is investigated. The properties and characteristics of some famous harmonic wavelet transforms are analzed. Since the wavelet analsis can capture temporal variations in the energ and frequenc content, a nonlinear sstem can be approimated as a time dependent linear sstem b combining the wavelet analsis technique with well-nown traditional equivalent linearization method. The nonlinear sstems including Duffing oscillator and bilinear hsteresis sstem are used as eamples to verif the effectiveness of the proposed wavelet-based linearization method. Results from this stud demonstrate that the wavelet linearization approach is more accurate and promising tool compared with traditional equivalent linearization methods in nonlinear control sstems in the real world. iii

4 ACKNOWLEDGEMENTS I would lie to epress m sincere gratitude to m academic supervisor and thesis adviser Professor Dr. Wen-Fang, Xie for her guidance, trust and encouragement throughout m stud in Concordia Universit. It is impossible to complete this wor without her continuous and invaluable help and support. Heartfelt thans go to m friends and m classmates who gave me great support in ever stage at m stud. In particular, I would lie to than Wang Tao, Han Xuan, and Zhao ZhongYu for their help in m stud and m thesis. I would lie to epress m thans to m best friend Julie for her alwas being around whenever I needed. I also lie to than to the facult at Industrial and Mechanical Engineering Department for providing great help and thoughtful services. And the financial support that I received from Concordia Universit is gratefull acnowledged. Finall, I would lie to than to m parents and m husband for everthing the did for me. iv

5 TABLE OF CONTENTS LIST OF FIGURES... VIII LIST OF TABLES... X LIST OF SYMBOLS, ABBREVIATIONS AND NOMENCLATURE... XI INTRODUCTION.... MOTIVATION.... OBJECTIVES OF THE THESIS....3 THESIS OUTLINE... 3 LITERATURE REVIEW TIME FREQUENCY ANALYSIS Fourier Transform Short Time Fourier Transform Wigner-Ville Distribution Wavelets Transform.... LINEARIZATION TECHNIQUES Lapunov s Linearization... 9 v

6 .. Feedbac Linearization Method Describing Function Method Statistical Linearization Method CONCLUSION WAVELET TRANSFORM SIGNAL DECOMPOSITION AND RECONSTRUCTION USING THE WAVELETS WAVELET FUNCTION DESIGN Dilation Equations Dilation Wavelets Wavelet Famil Introduction HARMONIC WAVELET DISCRETE HARMONIC WAVELET TRANSFORM CONCLUSION WAVELET-BASED LINEARIZATION FOR DUFFING OSCILLATOR SYSTEM DUFFING OSCILLATOR SYSTEM MODEL MULTI-FREQUENCY LINEARIZATION FOR STATIONARY INPUTS WAVELET-BASED LINEARIZATION FOR DUFFING OSCILLATOR vi

7 4.4 CONCLUSION WAVELET-BASED LINEARIZATION FOR BILINEAR HYSTERETIC SYSTEMS BILINEAR HYSTERETIC SYSTEM MODEL STATISTICAL LINEARIZATION FOR BILINEAR HYSTERETIC SYSTEM INPUT - OUTPUT RELATIONSHIP FOR A LINEAR SYSTEM WAVELET-BASED EQUIVALENT LINEARIZATION LINEARIZATION RESULTS CONCLUSION CONCLUSIONS CONTRIBUTION OF THE THESIS FUTURE WORK... REFERENCES... vii

8 LIST OF FIGURES Figure. Time-frequenc plane partition b using STFT... 9 Figure. Partition of the time-scale plane due to the wavelet transform... 3 Figure. 3 A nonlinear sstem... 9 Figure. 4 A linearization sstem... 3 Figure. 5 Detection of limit ccle... 3 Figure 3. Two-level wavelet decomposition using high-pass and low-pass filters Figure 3. The wavelet decomposition tree Figure 3. 3 Two-level wavelet reconstruction using high-pass and low-pass filters. 39 Figure 3. 4 Construction of the scaling function... 4 Figure 3. 5 Construction of D4 scaling function b iteration from a bo function over the interval = to Figure 3. 6 Scaling function for D4 wavelet calculated for 88 points using the method described above Figure 3. 7 D4 wavelet calculated b iteration Figure 3. 8 D4 wavelet according to (.8) from scaling function in Figure Figure 3. 9 Haar wavelet with unit amplitude plotted for level -,,, Figure 3. Morlet and Meican hat Wavelet forms Figure 3. Magnitudes of the Fourier transforms of harmonic wavelets of different level... 5 viii

9 Figure 3. Real part of the harmonic wavelet defined b (3.4)... 5 Figure 3. 3 Imaginar part of the harmonic wavelet defined b (3.4)... 5 Figure 3. 4 Real part of the harmonic scaling function defined b (3.8) Figure 3. 5 Imaginar part of the harmonic scaling function defined b (3.8) Figure 3. 6 FFT algorithm to compute the harmonic wavelet transform: (a) for a sequence of 6 real elements;[46]... 6 Figure 3. 7 FFT algorithm to compute the harmonic wavelet transform: (b) for a sequence of 6 comple elements[46]... 6 Figure 4. SDOF mass-spring-damper sstem[ 54] Figure 4. Equivalent linearization for stationar input... 7 Figure 4. 3 Wavelet-based linearization solution Figure 5. The bilinear sstem model... 8 Figure 5. The bilinear hsteresis force h characteristic tan... 8 Figure 5. 3 Elasto-plastic hsteresis h... 8 Figure 5. 4 The bilinear hsteresis loop Figure 5. 5 The elaso-plastic hsteresis loop Figure 5. 6 Numerical solution of bilinear hsteretic sstem to stationar ecitation Figure 5. 7 Wavelet-based linearization method comparing with numerical solution i

10 LIST OF TABLES Table. Comparison with four methods in signal analsis 7

11 List of Smbols, Abbreviations and Nomenclature Smbol Definition FFT IFFT STFT WVD DWT LTI SDOF Fast Fourier Transform Inverse Fast Fourier Transform Short-Time Fourier Transform Wigner-Ville Distribution Discrete Wavelet Transform Linear Time Invariant Single Degree Of Freedom i

12 INTRODUCTION. Motivation In engineering applications, the use of a linear model for the sstem under consideration leads to fairl simple and often useful results when the sstem behaves linearl and ecitation is stationar. The evaluation of the sstem response is not difficult. However, no real sstem is eactl linear. In mechanical and structural sstems nonlinearities can arise in various forms and usuall becomes progressivel more significant as the amplitude of motion increases to ecite nonlinear behavior requiring techniques which go beond linear approach. The mathematical theories for linear sstems provide a unified framewor for understanding all possible linear sstem behaviors but no correspondingl general theoretical ones eist for treating non-linear problems. Due to the inherent nonlinearities of almost phsical sstems, the subect of nonlinear control is an important area of automatic control. Among man methods to deal with a nonlinear sstem, the most commonl used one is to linearize it which allows taing advantage of mature linear sstem control techniques to control the nonlinear sstems approimatel. This dissertation introduces a wavelet-based method to linearize nonlinear sstems to its equivalent linearization sstems described b their wavelet coefficients. Time-frequenc localization is the main motivation for using wavelet theor to approimate the nonlinear

13 response as represented in this thesis. In this dissertation, the harmonic wavelet function is chosen as the basis function in the linearization procedure due to its eplicit form in mathematical epression and its non-overlapping characteristic in the frequenc domain which can be interchanged from the scale to frequenc band. This thesis also demonstrates that the harmonic wavelet function is used for its computational simplicit where there is a simple algorithm for harmonic wavelet analsis which uses the Fast Fourier Transform (FFT) and which is generall faster than the dilation wavelet analsis algorithm. The mechanism b which the signal is decomposed and reconstructed using the wavelet transform is investigated. The properties and characteristics of some famous harmonic wavelet transforms are analzed. In order to investigate the validit of this proposed method, the Duffing oscillator sstem and the bilinear hsteresis sstem are used as the eamples to illustrate that the wavelet-based linearization method is a quite promising tools to find the equivalent linear parameters of the nonlinear sstems in terms of time-frequenc localization.. Obectives of the thesis In this research, the main goal is to develop wavelet-based linearization method to linearize nonlinear dnamic sstems. The obectives are listed as follows:

14 To investigate the mechanism b which the signal is decomposed and reconstructed using the wavelet transform and to analze the properties and characteristics of some famous harmonic wavelet transforms, To implement a multi-frequenc linearization and wavelet-based linearization methods for Duffing oscillator sstem when the sstem is subect to multiple harmonic inputs, To develop a wavelet-based linearization method for the nonlinear sstem- bilinear hsteresis sstem and to develop numerical algorithm to find the equivalent linear parameters..3 Thesis Outline The thesis is organized as follows. Chapter introduces the motivation, obective and contributions of the thesis. The thesis outline is given as well. Chapter reviews time frequenc signal analsis and linearization technique as the foundation of this dissertation. Chapter 3 focuses on the wavelet transform including its mechanism b which a signal is decomposed and reconstructed using wavelet basis function, and how to obtain the wavelet function based on dilation equation method. Also some famous wavelet families are displaed as some eamples to illustrate the signal decomposition at different levels and time locations. Finall, the harmonic wavelets and their significant properties are 3

15 discussed. As the most important features, these characteristics pla ver critical roles in the wavelet-based linearization method. Chapter 4 gives the detailed discussion about the statistical linearization procedure for a nonlinear sstem including the traditional equivalent method and the multi-frequenc linearization. The linearization methods are applied to Duffing oscillator sstem. The wavelet-based linearization approach is implemented. The effectiveness of the proposed method is verified b numerical method. Chapter 5 addresses the wavelet-based linearization method with application in the bilinear hsteresis sstem and the comparison with statistical linearization method. The simulation results demonstrate that the wavelet-based linearization is a promising and applicable technique in linearization of bilinear hsteresis sstems. Chapter 6 concludes the thesis b reviewing the results. In addition, it points out the future research wor. 4

16 LITERATURE REVIEW In this chapter, we will review the methods related to the time-frequenc analsis of signal processing. The properties and limitations of each method will be given. Also, various inds of linearization techniques in different engineering areas will be reviewed.. Time Frequenc Analsis The organization of this section is as follows: Section.. starts with Fourier transform and its properties as the basic of signal analsis and then etends to Time-Frequenc analsis. In Section.., the short- time Fourier transform (STFT) and the Wigner-Ville(WV) transform are emphasized and their properties and limitations are discussed. Net, Section..3 introduces the concept of wavelets transform, its advantages and definitions as the foundation of this thesis... Fourier Transform Most engineers are familiar with the idea of frequenc analsis b which a periodic function can be broen down into its harmonic components. And such a periodic function ma be snthesized b adding together its harmonic components, namel, the Fourier series of function f(t) defined on the interval p, p is given b:[7] 5

17 ( ) f t = a + a cos n n t + bn sin n p p n= t (.) where: a a b n n p = f ( p t)dt (.) (.) p p = f ( t) n p cos p tdt (.3) p p = f ( t) n p sin p tdt (.4) p In order to obtain a representation for a non-periodic function defined for all real t, it seems desirable to tae the limits as p, t leads to the formulation of the famous Fourier integral theorem. Mathematicall, this is continuous version of the completeness propert of Fourier series. Phsicall, form (.) can be resolved into an infinite number of harmonic components with continuousl varing frequenc ( i amplitude f t e d ), and, called Fourier transform whereas the ordinar Fourier series represents a resolution of a given function into an infinite but discrete set of harmonic components. The Fourier transform originated from the Fourier integral theorem is believed to be one of the most remarable discoveries in the mathematical sciences and engineering applications including the analsis of stationar signals and real-time signal processing which mae an effective use of the Fourier transform in time and frequenc domains. The success of Fourier transform is due to the fact that under some conditions, the function f(t) 6

18 can be reconstructed b the Fourier inversion formula that defines: f t it it f e f d f e F, (.5) Thus, the Fourier transform theor has been ver useful for analzing harmonic signal. On the other hand, in spite of some remarable successes, Fourier transform analsis seems to be inadequate for studing non-stationar signal that the spectral content of the signal changes with time [5]. First, the Fourier transform of a signal does not contain an local information in the sense that reflects the change of wave number with space. Second, the Fourier transform method can be used to investigate problems either in time domain or in frequenc domain, but not simultaneousl in both domains. There are probabl the maor weanesses of the Fourier transform analsis. It is necessar to define a single transform of time and frequenc that can be used to describe the energ densit of a signal simultaneousl in both time and frequenc domains. Such a single transform would give complete time and frequenc information of signal... Short Time Fourier Transform It is well-nown that the Fourier transform analsis is a ver effective tool for studing stationar signal. However, signals are, in general, non-stationar, and hence cannot be analzed completel b Fourier transform. Therefore, a complete analsis of non-stationar signal requires both time-frequenc. 7

19 In order to get information about local frequenc spectrum, a lot of research wor have been carried out to cut the signal to segments with a short data windows centered at time t,[7] and spectral coefficients are computed for this short length of data. The window is then moved to a new position and the calculation is repeated. This transform is called the short time Fourier transform (STFT). Based on such a procedure, Dennis Gabor [4] first introduced the windowed Fourier transform (the Gabor transform) to measure localized frequenc component of signal and used a Gaussian distribution function as window function. The main idea was to use a time-localization window function t b g a for etracting local information from the Fourier transform of a signal where the parameter a measures the width of the window and the parameter b is used to translate the window in order to cover the whole time domain. The propert of the Gabor transform provides the local aspect of the Fourier with time resolution equal to the size of the window.[4] g t ep i gt, (.6) As the window function b translating, the Gabor transform of f with the respect to g, is denoted b ~ f g i t, f gt e d f, g (.7) t, And the inversion formula for the Gabor transform is given b Equation (.6). ~ The Gabor transform f g t, of a given signal f(t) depends on both time and frequenc, 8

20 thus the Gabor transform completel describes the spectral signal varing with time and changes an one dimensional signal into a comple value function of two real parameters: time and frequenc in the two dimensional time-frequenc space. The short time Fourier Transform (STFT) remains the time information but has strong time-frequenc resolution limitations. The problem with STFT can be demonstrated b Heisenberg`s Uncertaint Principle [38], [7]. According to this principle, it is impossible to now the eact time -frequenc representation of a signal, but one can obtain the frequenc bands eisting in a time interval which is a resolution problem and once the window is chosen, the frequenc and time resolution are fied for all frequencies and all times.[77] Figure. Time-frequenc plane partition b using STFT [ 69]..3 Wigner-Ville Distribution Wigner-Ville Distribution (WVD) is another popular tool for time-frequenc analsis. It can also be thought of as a short time Fourier transform (STFT) where windowing 9

21 function is time-scaled, time-reversed cop of the original signal. It is the Fourier transform of the signal s autocorrelation function with respect to the dela variable. It has much better resolution than STFT.[7] In 948, Ville [65] proposed the Wigner Distribution of a function or a signal f t in the form: W f, (.8) iw t f t f t e d For analsis of the time-frequenc structure of non-stationar signal, where f t is the comple conugate of f t, this time-frequenc representation of a signal t f is nown as the Wigner-Ville Distribution [68] which is one of the fundamental methods which have been developed over the ears for the time- frequenc signal analsis. In the view of its remarable mathematical structure and properties, the Wigner-Ville Distribution is now well-recognized as an effective method for the time-frequenc analsis of non-stationar signals and non-stationar random process based on the generalizing relationship between the power spectrum and autocorrelation function. Suppose that it is possible to calculate an instantaneous correlation function:, t E t t R (.9) At time t, the operator E denotes the average of a statistical ensemble which is assumed

22 to be available for the analsis. The Fourier transform of R,t gives: S R, tep i d (.) S is called the spectral densit of the process and is a function of angular frequenc. It provides information on how the mean square of the ensemble is distributed over frequenc and time. But in practice, it is never possible to compute ensemble average correlation function of R,t. Instead, the ensemble averaging operation is replaced b: t *, t t ep i d (.),t is called the Wigner-Ville Distribution of and * denotes comple conugate. From theoretical and application points of view, the Wigner-Ville Distribution [53] plas a central role and has several important structures and properties. First, it provides a high-resolution representation in time and in frequenc for some non-stationar signals. Second, it has the special propert of satisfing the time and frequenc analsis in terms of the instantaneous power in time and energ spectrum in frequenc. In spite of these desired features, its disadvantage is that although the integral is center at time t, it covers an infinite range of and thus depends on the characteristic of far awa from the local t. Therefore, it does not describe the trul local behavior of at time t, this is the fundamental uncertaint principle since

23 time-dependent spectral high resolution cannot be obtained simultaneousl in time and frequenc. Also it often possesses severe cross-terms or interference terms between different time-frequenc regions leading to undesirable properties...4 Wavelets Transform In order to overcome some of the inherent weanesses of the short time Fourier transform and the Wigner-Ville Distribution, there has been considerable recent interest in looing for a general time-frequenc distribution analsis as a mathematical method for time-frequenc signal. Wavelet Transform as the new concept can be viewed as a snthesis of various ideas originating from different disciplines including mathematic, phsics and engineering [7]. The pioneer research wor in wavelet transform can be found in [8], [4], [9]. Morlet (984)[8] first introduced the idea of wavelet as a famil of function constructed from translation and dilation of a single function called mother wavelet t defined b: t b t a, b a ; a, b R, a (.) a where a is called a scaling parameter which measures the degree of compression or scale, and b is a translation parameter which determines the time location of the wavelet.

24 If a <, the wavelet (.) is the compressed version (smaller support in time domain) of mother wavelet and corresponds mainl to higher frequencies. On the other hand, where a >, t has a bigger time width than t a,b and corresponds to lower frequencies. Thus wavelets have time-width adapted to their frequencies. This is the main reason for the success of wavelet in signal processing and time-frequenc signal analsis. It ma be noted that the resolution of wavelets at different scales varies in time and frequenc domains as governed b the Heisenberg uncertaint principle[7][38], that is, at large scale, the resolution is coarse in time domain and fine in the frequenc domain. As the scale a decrease, the resolution in time domain becomes fine whereas the resolution in frequenc domain becomes coarse. Figure. Partition of the time-scale plane due to the wavelet transform [ 69] A more etensive stud has been carried out b Grossman, [9] and the wavelet transform of function f t is defined b: 3

25 W a, b a, b f a, b f, f t t dt (.3) where t plas the same role as the ernel it a,b ep in the Fourier transform. However, unlie the Fourier transform, the continuous wavelet transform is not a single transform, but an transform obtained in this wa. The inverse wavelet transform can be defined so that f C f t can be reconstructed b means of the formula: t b (.4) a a t W f a, b dadb where parameter C satisfies the so-called admissibilit condition; ˆ C d < (.5) where parameter ˆ is the Fourier transform of the mother wavelet t. In practical applications involving fast numerical algorithms, the continuous wavelet can be computed at discrete grid points. To do this, a general function a,b t can be defined b replacing a with a m and maing m m t a a t m, n nb, a, b with nb a m b, where n m, are integers (.6) The discrete wavelet transform of f m, n f, f t t dt f t is defined as the W m, n m, n (.7) 4

26 If the wavelets form an orthogonal basis and complete, its discrete wavelet transform m, n f, f W m, n f t f, m, n m, n t f m, n f t can be reconstructed from b means of the formula (.8) provided that wavelets form an orthogonal basis. For some ver special choices of and a, b, the m, n constitute an orthogonal basis for L. In fact, if a and b, then there eists a function with good time-frequenc localization properties: m m t t n m, n (.9) It forms an orthogonal basis. These different orthogonal basis functions have been founded to be ver useful in application to speech processing [73], image processing [76], computer vision and so on [66]. Generall speaing, comparing with the short time Fourier transform and the Wigner-Ville Distribution, wavelet transform has more advantages in time-frequenc signal analsis due to its localization in time and frequenc domains and fleible resolution to a certain etent. Meanwhile, wavelet allow comple information such as music, speech, images and patterns to be decomposed into elementar forms at different positions and scales that are generated from a single function called mother wavelet b 5

27 translation and dilation operations. This information is subsequentl reconstructed with higher precision. Table. summarizes the comparison of four methods in signal analsis. 6

28 Table. Comparison of Four Methods in Signal Analsis Signal Analsis Basis Function Dimension Resolution Fourier Transform(FT) stationar signal triangle function onl one, either in the time domain or in the frequenc domains Global function without an local information in time and frequenc domain. Short time Fourier transform(stf T) non-stationar signal time -localization window function(a Gaussian distribution function) two, both time and frequenc domains Local information in time and frequenc but their resolutions are fied and no an eact amount because of uncertaint principle. Wiger-Ville Distribution( WVD) non-stationar signal autocorrelatio n function two, both time and frequenc domaisn Better than STFT but similar with STFT in the uncertaint principle Wavelet Transform(WV) non-stationar signal wavelet functions which are constructed from translation and dilation mother wavelet two, both time and frequenc domains More fleible resolution and location in time and frequenc due to the wavelets have time-width adapted to their frequenc 7

29 . Linearization Techniques In man practical applications, if one wants to predict the sstem performance accuratel, the sstem nonlinearit must be taen into account. As it is mentioned previousl, the importance of studing nonlinear sstems in automatic control area lies in: No model of real sstem is trul linear even if the operating range of a control sstem is small to stud its linear approimation b a linearized method. Nonlinear equations cannot be solved analticall and the tpical analsis tools lie Laplace and Fourier transform are not suitable for analzing the nonlinear sstems. As a result, nonlinear sstems ma demonstrate comple effects, such as limit ccles, bifurcations and even chaos which cannot be anticipated. In order to design an accurate and desirable controller to meet specified control performance, we need some powerful techniques to analze such sstem s behaviour which must be predicted and properl compensated for. Since the linear control is a mature subect with a plent of powerful methods and great achievements in the industrial and engineering applications, it is natural to adopt linear control techniques to analze nonlinear problems. In the following sections, some tpical linearization methods are introduced and, first the 8

30 Lapunov s linearization is reviewed in Section... Section.. introduces the method of feedbac linearization. The describing function method is discussed in the Section..3 and the statistical linearization method is introduced in the last section... Lapunov s Linearization The main idea of the Lapunov s linearization technique [3] is to approimate a nonlinear sstem b a linear one that is around the equilibrium point, and do hope the behaviour of the solutions of the linear sstem will be the same as the nonlinear one. We stud Lapunov linearization for autonomous sstem, and for non-autonomous sstems that the parameters var with time, we etend the research of the linearization method for non-autonomous nonlinear sstem. Consider an autonomous sstem of second order differential equation has the form: d dt d dt f, g, (.) The constant solution to this sstem is called equilibrium point that satisfies: f,, g, (.), If the sstem is linear with constant coefficients, we can obtain the solution of this sstem. However, for the nonlinear sstem, no method is developed for deriving a general 9

31 solution to this equation. To deal with a nonlinear problem, we must use an approimation method which is called linearization approach. As, is an equilibrium point, we tr to find the linear sstem when when, is close to, that is, to approimate f, and g,,, is close to. This is to calculate the functions f, and g,, b the Talor epansion approimation at a point,. f g f f, f,,, g g, (.), g,, B replacing f, and near,, we obtain: g, in (.) with their linear approimations d dt f f f,,, d dt g g g,,, (.3) Since, is an equilibrium point, d dt g, f., f f,,

32 d dt g g,, (.4) This is a linear sstem and its coefficients matri is: f g,,,, f g (.5) This matri is called Jacobian matri of sstem at the point,, and the eigenvalues of the Jacobian matri tell the information about the stabilit of the linear sstem and that of original nonlinear sstem. If the eigenvalues are negative or comple with negative real part, then the equilibrium point is stable, i. e., all solutions are convergent at the equilibrium point or if the eigenvalues are comple, then the solution will spiral around the equilibrium point. Hence, the linearized sstem is stable and so is the original nonlinear sstem. If the eigenvalues are positive or comple with positive real part, then the equilibrium point is unstable, i.e., all solutions will move awa from the equilibrium point or if eigenvalues are comple, then the solutions will spiral awa from equilibrium point. The linearized sstem is unstable and so is the nonlinear one. If the eigenvalues are real number with different sign (one positive and one

33 negative), then the equilibrium point is a saddle. Both of two sstems are unstable. If the eigenvalues are comple with real part equal to zero, then the equilibrium point is center point, and all solutions form ellipses around the equilibrium point in center or at least, there is one on the imaginar ais and another is comple with negative part. The equilibrium point ma be stable or unstable and hence one cannot draw an conclusion about the stabilit of the nonlinear sstem. Note that the stabilit characteristics of linear sstems are determined b their equilibrium point, therefore, if the nonlinear sstem is linearized, the local behaviour of this nonlinear sstem can be approimated b its linear replacing one. The Lapunov linearization method for non-autonomous nonlinear sstem involves more conditions and the powerful autonomous theorem does not appl to non-autonomous nonlinear sstem. However, there still eist some rules about instabilit for non-autonomous nonlinear sstem from that of its linear approimation as long as its linear approimation is time invariant. In summar, Lapunov s linearization method both for autonomous and non-autonomous nonlinear sstem provide the linearization method to analze the nonlinear sstem stabilit issue which is critical and essential in nonlinear control sstem design.

34 .. Feedbac Linearization Method Feedbac linearization is a common approach [6] used in nonlinear control sstems. The idea of this method is to transform original sstem model into equivalent linear one which is changed to the state variables and a suitable input instead. This method is different from the Jacobian linearization mentioned previousl, which is approimated b its linearized sstem, actuall, feedbac linearization method transforms the eact state variables in some algebraic forms so that the nonlinear sstem is changed into a linear one and linear control techniques can be applied. As one of the most important control techniques, the feedbac linearization is applied successfull in the control of high performance aircraft, industrial robots and so on. This part provides a procedure of feedbac linearization including the basic concepts of this method and mathematical tools to transform nonlinear components into linear sstems. First, the basic concept of feedbac linearization is to cancel the nonlinearities in a nonlinear sstem so that the closed-loop dnamics is in a linear form [37]. The nonlinear sstem has the forms: 3

35 f X bx u n (.6) where u is the scalar control input, is the scalar output, and X n T,,..., is the state vector, and f X and X (.6) can be written as the form: b are nonlinear function of the states. The Equation d dt... n n f... n X bx u (.7) Thus the sstem is epressed in the controllabilit canonical form, and the input u is designed as: u v f b, (.8) b We cancel the nonlinearities and obtain the simple input output relationship in the form that is: n v, (.9) We can design the control law that is: n v c c... c n (.3) With the parameters n n c chosen so that the polnomial p cn p... c has all its i roots strictl in the left-half comple plane, the nonlinear sstem leads to the eponentiall stable dnamics. n n cn... c (.3) 4

36 If the control of the tracing problem is considered, the control law can be: n n v c c e (.3) d e ce... n where et t t, t d d is a desired output. Second, some mathematical tool related to feedbac linearization technique is introduced. Lie derivatives: Consider nonlinear sstem: f gu h (.33) where n R is the state vector, P m u R is the vector of inputs, and R is the vector of outputs. d dh dh dh dh f gu (.34) d dt d d d Defining the lie derivative h along f as dh L f h f, (.35) d And similarl, the lie derivative h along g as dh L g h g (.36) d (.37) Thus, L h L hu f g The goal of the feedbac linearization is to find a direct and simple relationship between the sstem output and the control input u b differentiating the output until the input u 5

37 6 is epressed in the same equation. Thus we obtain: h L h L h z z z T z n f f n n (.38) u h L L h L z z h L z z h L z n g n f n f f f 3... (.39) We assumed the relative degree of sstem is n, the lie derivative of the form h L L i f g for i= n- are all zeros. And the original X coordinate is transformed into the Z coordinate so that the new linearized sstem with z variables is obtained as well as the relationship between input and output. Therefore, the control law is that: ( ) ( ) ( ) v h L h L L u n f n f g + = (.4) Also the input-output relationship from v to z is that v z z z z z n... 3 (.4) Choosing the suitable C in Cz v with standard linear sstem methodolog, we get the linear sstem with the linearizing state z and linearizing control law v.

38 In summar, feedbac linearization method is based on the idea of transforming nonlinear sstem into a linear one b using state feedbac with input-state linearization or input-output linearization method which can be used for both stabilization and tracing control problems and has been applied in man practical nonlinear controlling sstems to achieve successful results. However, this linearization method needs the eact nonlinear function in the sstem dnamics which is usuall difficult to obtain in the real practice. In addition, the controller design based on the linearized model will cause infinit control action if L g L n f h in (.4) becomes ver small...3 Describing Function Method Basicall, the describing function method is the statistical etension of [4] linearization technique in the electrical engineering literature which is an approimation procedure for analzing certain nonlinear control problems that were originall developed b [] and used as a tool in control engineering. Subsequent developments of this method in this field have described in [56] and [5]. The describing function method is an approimation procedure for analzing certain nonlinear control problems. It is based on replacing each nonlinear element with a quasi-linear descriptor which is the approimation of nonlinear sstem b a linear 7

39 transfer function whose gain is determined b the input amplitude while a true linear sstem transfer function does not depend on that amplitude. And describing function can also be treated as an etended version of the frequenc response method to predict nonlinear behavior, especiall, for limit ccle prediction in nonlinear sstems. The describing function is a powerful tool to discover the eistence of limit ccles and determine their stabilit. [6] Before discussing the propert of describing function, some assumptions must be given to satisf some conditions: There is onl a single nonlinear component. The nonlinear component is the time invariant. Corresponding to a sinusoidal input sint t in the output t, onl the fundamental component has been considered. This assumption implies that higher frequenc harmonics can be neglected, and the approimated linear one has low-pass properties. The nonlinearit is odd that means the relationship with input and output of nonlinear element is smmetric about origin so that the static term in the Fourier epansion of output can be ignored. 8

40 A. Basic Definitions Consider a sinusoidal input Asin t to nonlinear element. where A is amplitude and is frequenc, as show in Figure.3. Figure. 3 A nonlinear sstem The output (t) is alwas a periodic function that is etended b Fourier series epressed as: a t a cosn t b sinnt n n (.4) n where the Fourier coefficients a, b are determined b n n a t d t (.43) a n b n t cosnt d t (.44) t sinnt d t (.45) Due to the fourth assumption impling that a, the third assumption implies that onl the fundamental component t is considered: t t a t b sin t M sin t cos (.46) 9

41 where M A, a b and A, arctan. And also this output can be written as : t t t Me b a e. a b Since the concept of frequenc response function is the frequenc ratio of the sinusoidal input and the sinusoidal output of sstem, we define the describing function of nonlinear element to be comple ratio of the fundamental component of nonlinear element b input sinusoidal, that is: t Me M NA, e b a (.47) t Ae A A Thus, the describing function [64] can be treated as the approimation of linear one with a frequenc response function N A,, which is considered as an etension of the concept of frequenc response function. However, the describing function of nonlinear element differs from linear sstem frequenc response function in that it depends on the input amplitude A. B. Describing Function Analsis Nonlinear Sstem As we now how to get the describing function for the nonlinear element, we use this tool to predict the limit ccles that is based on the linear approach of the famous Nquist criterion in control sstem to investigate the stabilit. Consider a self-sustained oscillation of amplitude A and frequenc in the sstem of 3

42 Figure.4. Figure. 4 A linearization sstem The variables in the loop must satisf the following relations: w N A, G w Therefore, G NA So, NA, G,. And. (.48) G and it can be written as: NA, (.49) The amplitude A and frequenc of the limit ccles must satisf (.49), if the above equation has no solution and the nonlinear sstem has no limit ccles. However, it is difficult to get the solution b this analtical approach, especiall, for higher order sstem, thus the graphical wa is usuall used. The idea is to plot both sides of (.49) in comple plane and find the intersection points of two curves. And also according to Nquist criterion, we can determine the stabilit of limit ccles as shown in Figure.5. 3

43 Figure. 5 Detection of limit ccle Generall speaing, the describing function method which is an etension of the frequenc response method of linear control can be used as the approimate analsis to predict some important characteristics of nonlinear sstem including sstems with hard nonlinearities such as, saturation, baclash and hsteresis []. Particularl, it is the main tool to predict the limit ccles in the nonlinear sstem in the graphical nature of frequenc domain [5]...4 Statistical Linearization Method Among those various possible approaches which are available, the statistical method or equivalent linearization method has proven to be ver useful approimate technique in structure dnamics and earthquae engineering [6]. Methods of predicting the random vibration response of mechanical and structural sstem 3

44 to random eternal forces has grown rapidl. Since the ecitation and response of such sstem are alwas non-periodic, highl irregular and no repeatabilit, linear method is not suitable to appl, and new techniques are required. However, in engineering application, because the linear sstem is fairl simple and there a lot of resources to obtain the useful and desired solution, the sstems can be modeled in terms of the linear differential equations of motion. Although no real sstem is eactl linear, in mechanical and structural sstems, non-linearit can arise in various forms and usuall become more significant as the amplitude of vibration increase. In order to predict the response of this ind of sstem or to get an approimation solution of non-linear equation, the statistical method or equivalent linearization method is applied to estimate the accurate equivalent linear parameters. The statistical equivalent linearization method is based on the idea that the non-linear sstem is replaced b an equivalent linear equation b minimizing the difference between the two sstems in some appropriate sense. This technique has been used for deterministic non-linear problems for man ears, for instance, referred to as describing function method [4], and the adaptation of approach to deal with stochastic problem was first developed b [] to use as a tool in control engineering And subsequent developments of this field have been described b[56],[5],[57]. This method nown as 33

45 statistical linearization or equivalent linearization was proposed b [4], [5]as a wa to solve nonlinear vibration problems, furthermore, the statistical method is applied and served as the fundamental concept[58],[59] to propose the novel wavelet-based linearization method.[8],[9] In summar, this part reviews some tpical linearization methods and their applications in the nonlinear sstem control design and analsis. Comparing with their properties and limits, it provides fundamental bacground nowledge for the proposed linearization method that is based on the statistical linearization approach. The details about this method will be discussed in the Chapter 4 & 5 b using some tpical nonlinear sstems as the eamples..3 Conclusion In this chapter, comprehensive review and introduction on time-frequenc analsis of signal processing and linearization techniques are given. The properties and limitations of each method have been analzed. 34

46 3 WAVELET TRANSFORM This chapter presents the mechanism b which the signal is decomposed and reconstructed using the wavelet transform and introduces some famous wavelet basis functions. The properties and characteristics of the harmonic wavelet transform are discussed. The discrete harmonic wavelet transform will be also described. 3. Signal Decomposition and Reconstruction Using the Wavelets The discrete wavelet scheme is used for this presentation due to its simplicit and ease of implementation. In discrete wavelet transform (DWT), a time-scale representation of a digital signal is obtained using digital filtering techniques [44]. At each of filtering process, a signal is decomposed in high scale-low frequenc components which are called approimations (A) and in low scale-high frequenc components which are called details (D). This signal is passed through a series of low- pass filters to analze the low frequencies and pass through a series of high- pass filters to analze the high frequencies. During the decomposition process, twice the amount of the original signal data is generated at each stage. To avoid this result, the data has to been down-sampled b a factor of that is shown as: 35

47 That means a discrete time signal f(t) consists of omitting ever other value and we can thin of a sstem whose input is f(t) and whose output is g(t)=f(t). The same calculation holds for the detail coefficients. The filter process is demonstrated in Figure 3. Figure 3. Two-level wavelet decomposition using high-pass and low-pass filters. The decomposition process can be iterated with successive approimations being decomposed in turn [4] so that one signal is broen down into man lower resolution components. This is called wavelet decomposition tree which is shown in Figure 3. 36

48 Figure 3. The wavelet decomposition tree The decomposition lasts until the detail coefficients consist of repetitions of a single sample. To snthesize the signal, the inverse process of up sampling and filtering is followed b using the low (L ) and high (H ) pass reconstruction filters. Up sampling of discrete time signal f(t) consists of inserting zeros between the values. We can thin about a sstem with input f(t) and output g(t)=f(t/) for even values of t, and g(t)= for odd values of t shown as: The design of the low and high pass filter for the decomposition and reconstruction process is ver important because it determines the shape of the wavelet function used in analsis.[6],[],[44],[74],[75] In fact, the filter design and wavelet function choice 37

49 develop at the same time. The filters used in the decomposition and reconstruction process are different in that the must be satisfied some conditions:[] Condition : H L L H (3.) where H and L are Fourier transform of high-pass filter h and low-pass filter l, also H and L are conugate to H and L and the same as h and l to h and l. It is defined b, H And L n hne (3.) n n gne (3.3) n Condition : H L L H (3. 4) In the signal decomposed or reconstructed process, the function is involved into successive two filters L (low frequenc) and H(high frequenc) or their conugate filter L and H that so-called quadric mirror filters. 38

50 Figure 3. 3 Two-level wavelet reconstruction using high-pass and low-pass filters. There are infinite numbers of possibilit to choose the analzing wavelet. However, onl a small group of these meets the conditions that are necessar if the wavelets are given an accurate decomposition and reconstruction and also be orthogonal to each other[45]. This is etremel critical and efficient for computing wavelet transform which does for wavelet analsis the same as what FFT (fast Fourier transform) does for Fourier analsis. 3. Wavelet Function Design As mentioned previousl, the wavelet function design is based on accurac and calculating efficienc. There are two classes of orthogonal wavelets-- Dilation wavelet and harmonic wavelet. First we consider the theor of dilation wavelets and later discuss harmonic wavelet. The successful development of dilation wavelet in mathematical communit has been led 39

51 b Daubechies[9], [], [] who invented so-called dilation schemes and for first time gave the method to calculate the dilation wavelet coefficients. 3.. Dilation Equations To dilate is to spread out so that dilation means epansion [6]. The basis function φ () is a dilated horizontall version φ () that has the same height but is stretched over twice the horizontal scale of, where is a independent variable that ma represent time or position depending on the applications. In dilation equation, φ () is epressed as a finite series of terms, each of which involves φ (). Each of these φ () terms is positioned at different places on the horizontal ais b maing the translation (-) instead of, where is an integer(positive or negative). The dilation equation has a form: c c c c 3 (3.5) 3 where c s are numerical constants. It is impossible to solve (3.5) directl to find function φ () ecept for ver simple case. But indirectl setting up an iterative algorithm in which each new approimation approimation b the scheme: c c c c 3 3 is calculated from the previous (3.6) 4

52 The iterative process continues until becomes the same as. Consider starting from a bo function: ; =, elsewhere. After one iteration the bo function over interval = to has epanded into a staircase function over the interval = to also shown in the Figure 3.4 Figure 3. 4 Construction of the scaling function Each contribution to is shown separatel and then four contributions are shown added together. A particular set of the coefficients c, c, c, c3 have been obtained as 4

53 follows: c 3 / 4; c 3 3 / 4; c 3 3 / 4; 3 / 4; c (3.7) 3 This is the orthogonal D4 wavelet. The D stands for Daubechies. The function coming from a unit bo (unit height and length) is called scaling function and a corresponding wavelet function will be obtained from it. When the iterative process continues, it has been seen that the first iteration produces the four coefficients c, c, c, c3, at =,.5,,.5. At the second process, each coefficient produces another four coefficients. For eample, c at = produces four new coefficients which are c, cc, c c, cc3, at =,.5,.5,.75; and c at =.5 contributes to four new coefficients c c, c, cc, cc3 at =.5,.75,,.5; and so on. After second iteration, all the resulting ordinates corresponding to the coefficients are obtained. Figure 3.5 displas the first eight iterations of the development of the D4 scaling function. Figure 3.6 shows scaling function for D4 wavelet calculated for 88 points. 4

54 43 X Figure 3. 5 Construction of D4 scaling function b iteration from a bo function over the interval = to From this process we find the matri scheme M M c c c c c c c c c c c c c c c c c c c c

55 where r M denotes a matri of order r r r r in which each column has a sub-matri of coefficients c, c, c, c3 positioned two places below the sub-matri to left. The number of points increases in the sequence and after eight iterations it reaches points and each of them spaced / =/56. Although the iteration method is not an efficient wa, it is simple to understand and easil to be programmed. Figure 3. 6 Scaling function for D4 wavelet calculated for 88 points 44

56 3.. Dilation Wavelets The wavelet function W() is derived from scaling function b taing differences. For the four coefficient scaling function defined b (3.5) the dilation wavelet function is w c c c c 3 3 (3.8) The same coefficients are used as scaling function but in reverse order and with alternate terms having their signs changing from plus to minus. The results of maing the calculation (3.8) for D4 scaling function in Figure 3.6 are shown in Figure 3.8 and the iteration of development of the D4 wavelet is shown in Figure 3.7. So far using the scaling function to construct orthogonal wavelet function which has been developed b Daubeches[9] [], [] who first computed the values of required number N of coefficients c, c,... c N in order to achieve accurate level epansion for signal decomposition and reconstruction 45

57 Figure 3. 7 D4 wavelet calculated b iteration Figure 3. 8 D4 wavelet according to (.8) from scaling function in Figure

58 3..3 Wavelet Famil Introduction There is a lot of wavelet functions which can be used as the mother wavelet function to achieve a signal s wavelet transform. Among those choices we introduce some ver commonl used wavelets in engineering applications. [38] First, it is the Haar wavelet which is the simplest orthogonal basis function. Figure 3.9 shows how a signal is decomposed in levels. Figure 3. 9 Haar wavelet with unit amplitude plotted for level -,,, w The Haar wavelet function is: w for / for/ (3. 9) 47

59 The time or position is plotted along the horizontal ais and wavelet function w is to which are located at different position along ais. There are two half length wavelets represented b ( ) w and w and four-quarter length wavelets represented b 4, w4, w4, w4 3, w as shown in Figure 3.9. The scale and position can be obtained from its argument. For eample, w is the same as w ecept that it is compressed into half the horizontal length and it begins at =/ instead of =. The wavelet s level is determined b how man wavelets fit into the unit interval = to. At level there is wavelet lie the third top view of Figure3.9. And there are wavelets which fit between to and at level, there are 4 wavelets which fit between to lie the top view Figure 3.9 and so on. The decomposition of a signal or function f() is the same as its Fourier transform in which the sequence length N of signal being analzed determines how man separate frequencies can be represented. In the wavelet transform, the sequence length determines how man wavelet levels there are with different scales and positions in which the sequence of N= n points consist of n+ levels (scales) running from - to n- and n position. 48

60 49 w a a w a w a w a w a w a w a w a a f (3.) There are some other wavelet famil functions such as Mecian hat wavelet whose analtical epression is:[] 4 3 e (3.) And Morlet wavelet is defined as: t e e where 75. (3.) Both basis wavelet functions shown in Figure 3. are applied in different signals analsis and other engineering applications. Therefore mother wavelets choice should tae into account the details and particular application so that the wavelet transform can represent function characteristics effectivel. Figure 3. Morlet and Meican hat Wavelet forms

61 3.3 Harmonic Wavelet In this section harmonic wavelet is introduced and its properties and characteristics are discussed as the fundamental concept to construct the rest of this thesis. As the number N of wavelet coefficients is increased, the wavelet s Fourier transform becomes compact. Onl the values of N up to are included in the program. However the number of wavelet coefficients for N> is required to be computed and it turns out that the spectrum of a wavelet with N coefficients are more lie bo-lie as N increases. This fact lead Newland [46] to see a wavelet w whose spectrum is eactl lie a bo so that the magnitude of Fourier transform w is zero ecept for an octave band of frequencies. A simple wa to define w is shown in Figure 3.. Figure 3. Magnitudes of the Fourier transforms of harmonic wavelets of different level 5

62 All the Fourier transforms are identicall zero for, and for level zero the definition of w is : w for 4 elsewhere B calculating the inverse Fourier transform of w (3. 3), we obtain the corresponding comple wavelet as: i4 i e e w (3.4) i whose real and imaginar parts are shown in Figure 3. and Figure 3.3 Figure 3. Real part of the harmonic wavelet (3.4) 5

63 Figure 3. 3 Imaginar part of the harmonic wavelet (3.4) For the general comple wavelet, at level and translated b step of size define: i w elsewhere e for 4 (3. 5) we B calculating the inverse Fourier transform, this gives: w e i4 i e i (3.6) where = to, = to. For level - with the frequenc band in Figure 3. the wavelet is defined as 5

64 w e elsewhere i for (3. 7) The scaling function is defined as: e i i (3. 8) = to, the real and imaginar parts of are shown in Figure 3.4 and Figure 3.5. Figure 3. 4 Real part of the harmonic scaling function (3.8) 53

65 Figure 3. 5 Imaginar part of the harmonic scaling function (3.8) The idea to choose the wavelet and scaling function is that the form the orthogonal set and have the properties: w * r w sd w for all,, s;, r (3.9) r w sd for all,, r, s;, r (3.) Ecept for the case when r=, and s=. when 54 w d Also sd for all, s (3. ) * sd for all, s (3. ) Ecept s=, when d And finall, sd w for all,, s; ( ) (3. 3)

66 * sd w for all,, s; ( ) (3. 4) The proof of these results depends on the properties of their Fourier transform and derives from two results from Fourier transform theor of w() and v() respectivel, then one has [48] w v d W V d (3. 5) * * d W V d w v (3.6) From Equations(3.5),(3.6),(3.7), the wavelets of different of levels have Fourier transform that occup different frequenc band while for the wavelets in the same frequenc band, each wavelet is onl orthogonal to the comple conugate of another wavelet. The result of all this is that the w defined b (3.6) together with defined b (3.8) construct an orthogonal famil of wavelet that offer an alternative to wavelet famil from dilation equation. The differences of two wavelet functions are that Harmonic wavelet can be described b a simple analtical formula which are compact in frequenc domain and can be described b a comple function so that there are two real wavelets for each, pair. While Dilation wavelets can not be epressed in functional form. The are compact in -domain and there is one real wavelet for each, pair. Harmonic wavelet has an advantage that is computation simplicit which uses FFT (fast 55

67 Fourier transform) to harmonic wavelet analsis to a signal or function. 3.4 Discrete Harmonic Wavelet Transform Because of comple wavelets, two amplitude coefficients have to be defined as follows:[46] * w d a, f (3.7) a~, f d (3.8) w If f() is a real function, a ~ is the comple conugate of, a,. However, if f() is comple ~ and a, are different. a, Similarl, a * d, f (3.9) a ~ f d (3.3), In terms of these coefficients, the wavelet epansion of a general function f() for which f d (3.3) is given b: f ~ ~ *,, a, w a, w * a a (3.3) Now we tr to find out an algorithm to compute the coefficients ~ a,, a,, a,, a ~, in this epansion. As an eample, consider a, in (3.7). First substitute for 56

68 * w in terms of its Fourier transform where from (3.5) w * 4 i i e e d (3.33) So that a, in (.7) becomes a i i de df e d (3.34), 4 * Since the Fourier transform of w is zero outside w 4 and second term integral over is ust the Fourier transform of f() multiplied b so that a 4 i F e d (3.35), Now we replace integral b summation to get the discrete form of the wavelet transform. In this case, we step samples in increment of in frequenc band so that there steps in level. Then F F is replaced b the discrete coefficient F when s F s. s And the integral in (3.35) becomes the summation a F s s e i s (3.36) Because cancels with from (3.7) and i e for all integer we obtain the formula: is a F e s s = to (3.37) This is the inverse discrete Fourier transform for the sequence of frequenc coefficients, s= to F s. 57

69 To compute the wavelet amplitude coefficients b a discrete algorithm, we begin b representing f() b the discrete sequence f(r), r= to N-(where N is power of ). Net FFT is used to compute the set of comple frequenc coefficients F(p), p= to N-, then octave blocs of F(p) are processed b IFFT to generate the amplitudes of the harmonic wavelet epansion of f(r) with the coefficients: is a~ s F se = to (3.38) Because the discrete Fourier transform does not have negative indices but instead of F s b F that becomes: N S a ~ _ s F N s e is (3.39) As an eample, for N=6, Figure 3.6 shows how the algorithm wors in diagram. For (a) f(r) is a real function and ~ are comple conugates of a. It is not necessar to a calculate these coefficients. And in Figure 3.7 for (b) f(r) is comple as shown. And the computation of a and First, for a consider i i e e d a N in algorithm involves special cases. (3.4) i i i e f e df e * a, d d (3.4) Because is zero ecept for when it is, the integral is covered b a single of. If the value of F( ) is replaced b F, 58 then

70 F 4F, a Second, for a N F,, a~, F. Hence, a a ~, a, F., the initial FFT operation which transform f(r) to F(p) preserves the N N mean square so that p F p f N r r holds. Each subsequent IFFT operation on octave bands of F p for p<n/ or its comple conugate on octave band for p>n/ similarl preserves the mean square in octave bands so that = + = = a = F + holds. Hence one has a. N F N The Parseval s theor in form is: a N N a a a N N N r r f (3.4) The summation is made over all levels = to n-, the highest level n- being the maimum that can reach with initial sequence of length n. For eample an original sequence length of 4 6 can provide harmonic wavelet amplitudes at level -,,, and n-=, comparing with the levels that obtained from the dilation wavelet, the levels from -,,, and n-=3. Since for each pair of,, the dilation method has onl one real dilation wavelet whereas harmonic wavelet has two real ones that are even and odd which are given b the real and imaginar parts of the comple harmonic wavelet [49]. The algorithm illustrated in Figure 3.6 is a fast wa to compute the discrete harmonic wavelet transform and for the inverse transform, the same algorithm wors in reverse 59

71 starting at the bottom of diagrams in Figure 3.7. In summar, the wavelet determined from the dilation equation is a sequential algorithm in the sense that successive wavelets levels are calculated in sequence. In contrast, the FFT algorithm for harmonic wavelets are parallel method that FFT is onl used once to get full sequence and then in the second time in octave bloc all of which are computed simultaneousl. Therefore, for application which is involved ver long sequence or complicate signal processing, harmonic wavelet transform algorithm is a ver significant time-saving method. 6

72 Figure 3. 6 FFT algorithm to compute the harmonic wavelet transform: (a) for a sequence of 6 real elements;[46] 6

73 Figure 3. 7 FFT algorithm to compute the harmonic wavelet transform: (b) for a sequence of 6 comple elements [46] 6

74 3.5 Conclusion In this chapter, the mechanism of a signal decomposition and reconstruction is presented based on the wavelet filter processing. The properties and characteristics of some tpical harmonic wavelet transform are discussed with emphasis on two tpes of wavelet basis functions i.e. dilation wavelet function and harmonic wavelet function. Comparing these two wavelet functions, the conclusion is drawn that the harmonic wavelet function is more suitable to be the basis function in the wavelet-based linearization method which will be introduced in Chapters 4 & 5. 63

75 4 WAVELET-BASED LINEARIZATION FOR DUFFING OSCILLATOR SYSTEM In this chapter, we will first introduce one ind of nonlinear sstem-duffing oscillator sstem with non-linear stiffness. A multi-frequenc linearization and wavelet-based linearization methods are implemented and compared for Duffing oscillator sstem when the sstem is subect to multiple harmonic inputs. 4. Duffing Oscillator Sstem Model To illustrate the procedure of equivalent linearization theor, first we consider the following oscillator with non-linear stiffness [54], as shown in Figure 4.: Figure 4. SDOF mass-spring-damper sstem [ 54] The ordinar differential equation of motion can be written as: t c t g Ft m (4.) where m is the mass, c is the viscous damping coefficient, F(t) is the eternal ecitation, 64

76 and (t) is the displacement response of sstem. Dividing the equation b m, the equation becomes: t t ( t) h( ) f (4.) where is the damping parameter, h is the non-linear restoring force that could depend on displacement, and f t is a zero mean stationar random ecitation. We can alwas find a wa to decompose the non-linear restoring force to one linear component plus a non-linear component, that is: h( ) n H (4.3) where is the non-linear factor that presents the tpe and degree of non-linearit in the sstem, and is the un-damped natural frequenc for linear sstem. n The idea of linearization is replacing the equation (4.) b the following linear sstem: t f (4.4) eq eq where is the equivalent linear damping coefficient per unit mass, eq is the eq equivalent linear stiffness coefficient per unit mass, and eq n, to find an eq epression for is to minimize the epected value of the mean square error which is: eq h (4.5) eq The wa that minimizes the epected value of mean square error, is shown: E eq (4.6) 65

77 B substituting (4.5) into equation (4.6), and carring out the differentiation, the following equations for eq is shown as: eq E h E h (4.7) E where is the standard deviation of (t). B substituting the epression for h, given b (4.3), into equation(4.7), the result is: E H eq n (4.8) This epression shows, ver clearl, how the non-linear component of the stiffness element influences the value of. In equation (4.8), the eact evaluation of eq H E requires a nowledge of the first order densit function of the response process (t) which is unnown [54]. However the displacement (t) is assumed to be Gaussian and Equation (4.7) becomes: eq h dh E n E d (4.9) For eample, the Duffing oscillator is used to illustrate this procedure b which the non-linear restoring force is written as: 3 h n (4.) In this case, eq n eq can be epressed: 3E (4.) 66

78 And b defining: A E (4.) Then assume the densit function of (t) to be the Gaussian form: f ep (4.3) To combine (4.) and (4.3) together, one has A E f d (4.4) And b performing integration the equation (4.4) with the Gamma function, the result is obtained: 3 A (4.5) The net step is to evaluate and the procedure is to use the frequenc domain. The input-output formula S is the spectral densit for f t and f S is the spectral densit for output t. Use the well-nown relationship in the linear vibration : S S (4.6) where f is the frequenc response function given as: (4.7) eq i eqeq Once S is determined, is found from the equation: S d (4.8) 67

79 The equations (4.6), (4.7), and (4.8) ield a relationship between eq and. Another independent relationship between two quantities can be found from Equations (4.) (4.4), (4.3) and (4.5). Hence, two algebraic relationship are obtained for two unnowns, and, and the formulas for eq eq and epression are formed. B iterative process the equivalent coefficient finall is computed. The procedure can be described as follows:. Assign an initial value of equivalent coefficient eq.. Use equation (4.6), (4.7), (4.8) to get. 3. Solve equation (4.) and (4.4) for the new eq. 4. Repeat step and step 3 until the new eq converge to the old eq within a pre-defined threshold. 4. Multi-Frequenc Linearization for Stationar Inputs In this section, we will discuss the multi-frequenc-linearization method based on the equivalent linearization procedure and use Duffing sstem as an eample. For this ind of sstem, the non-linear factor onl depends on the displacement so that the nonlinearit restoring force can be decomposed b linear one term and another 68

80 non-linear one that mentioned in previous part. That is: h( ) h ( ) (4.9) nl The equivalent method is to replace the original non-linear sstem given b: t h f (4.) with the linear sstem given b t nl f (4.) where nl is the equivalent coefficient which needs to be found out. Following the method proposed b Caughe [3][4], it is assumed that the damping is ver small and f t is a wide-band process. Under these circumstances the solution of equation (4.) is the narrow-band process and one can write ( t) A( t)cos eqt ( t) (4.) t A( t)sin t ( t) (4.3) eq eq where (4.4) eq nl When is small, A t and t will be slowl varing function of time over one ccle of response in period of T. Thus the epected value in the equation (4.7) eq can be treated as average value over one ccle. On minimizing the mean square of error one ccle of response, h nl is obtained as: d eq (4.5) d 69

81 7 For Duffing oscillator, the non-linear term is given b 3 = = ) ( h nl nl (4.6) And equation (4.) becomes: t f 3 (4.7) And the equivalent stiffness is obtained according to equation (4.5) T T nl d d 4 (4.8) From the linear vibration theor, the input signal is a sum of sinusoids: t F t F t F t f N N sin... sin sin (4.9) An then the output will be: N N X N t t X t X t sin... sin sin (4.3) where i i i i F X and tan i i i (4.3) Substituting t from (4.3) into (4.8), the additional stiffness due to the nonlinearit found for each frequenc of ecitation is calculated as 3, X X X X X X X N nl 3, X X X X X X X N nl

82 3 X X X 3 nl, N XN... (4.3) 4 X N X N X N To obtain the solution for this equation, we need the iterative scheme. Starting with nl, i and X i i for i=:n (4.33) i F nl, i i The coefficient nl is computed b equation (4.3) until it is converged and the result for above sstem onl produces the same frequenc as the ecitation one. It has been shown that (Stoer 965) [63] when the harmonic solution is nown, the response of the Duffing oscillator to a single frequenc can be epressed b 3 5 X i X i t) X i sin it sin3 it sin5 it (4.34) i ( 4 i i The whole response can be calculated b summing up all the response components as t N i i t (4.35) The eample is given to displa this procedure. The sstem is: 3 f t, t 5sin4t sin8 t 5sint 5sin t,.5, 6, and f 6 (4.36) The input and output process are shown as in Figure 4.. 7

83 Displacement (m) Ecitation Force (N) Time (Second) Time (Second) Figure 4. Equivalent linearization for stationar input 7