for (1.3)
In the equation above, denotes the n th order output frequency response of the system. This expression reveals
how the non-linear mechanisms operate on the input signal to produce output response in frequency domain.
(1.4)
is the definition of the nth-order Generalized Frequency Response Function (GFRF)
(Billings and Yusof 1996, Boaghe et al . 2000, 2002) , and
denotes the integration of over the n-dimensional hyper plane Equation (1.3) extends the well-known linear relationship (1.1) to the non-linear case.
From equation (1.1), it shows that the possible output frequencies are exactly the same as the frequencies in the input for
linear system. For equation (1.2), the situation of non-linear system becomes more complicated than that in linear case,
that is, the relationship between input and output frequencies is generally given by
(1.5)
Where represents the non-negative frequency ranges of the system output, and denotes the non-negative frequency
ranges produced by the n th-order system non-linearity.
In 1997, Lang and Billings derived an explicit expression for the output frequency range of non-linear systems when
subjected to a general input with a spectrum given by
when
0 otherwise
where b>a The result obtained is
(1.7)
means to take the integer part
for k=0,…….,
In (1.7) p* could be taken as 1,2,…, , the specific value of which depends on the system nonlinearities. If the
system GFRFs for i=1,…,q-1, and , then p*=q. This is the first analytical description
for the output frequencies of non-linear systems, which is an extension of the relationship between the input and output
frequencies of linear systems in equation (1.1) to the non-linear case.
The theory above shows the basic properties of the relationship between the input and output frequencies. However, more
details about the relocation of the frequency domain response taking place of a non-linear system can not be acquired.
Therefore, a new frequency domain analyse method of non-linear system which is known as
NOFRFs is introduced in the next section.
Charpter 2 Non-linear output frequency response functions (NOFRFs)
2.1 Some basic concept
First consider two different systems in time domain Static linear system
y(t) =ku(t)
Static non-linear system
y (t)=ku(t)
The frequency domain representations of the systems above are
Y(jw)= kU(jw)
and
Y(jw)= kU(jw)
respectively. U(jw) represents the Fourier transform of u(t) and it can be written
as
(2.1)
The equation (2.1) should be taken as an expression of a second order non-linear system.
In this case, a more generalized expression can be deduced
Equation (2.2) is the Fourier transform of the system input u(t) raised to the n th power. From the concepts above, the
expression of output frequency response of a stable time invariant linear system described in (1) can be rewritten as
Y(jw)=G(jw)U(jw)
Where Y(jw)= Y (jw), G(jw)= G (jw), U(jw)= U (jw)
Under the definition that Un(jw)is the natural extension of U(jw) to the n th order homogeneous non-linear situation, equation (3) can be rewritten as
= G(jw)U (jw)
under the condition
The concept of NOFRFs is defined as
which was introduced by Lang and Billings [1]
One of the important properties of NOFRFs is
Y (jw)= Y (jw) = G(jw)U (jw)
2.2 Evaluation of G(jw)
In this section, a direct evaluation of G(jw)will be educed from the definition in (2.5).
From equation (2.6), it is known that
Y (jw) = G(jw)U (jw)=
=
in which and are the real and imaginary part of , and are the real and imaginary part of . Therefore
in which and
Consider the case of where u(t)=u(t) where is a constant and u(t) is the input signal.
The equation above can be written as
in which and are the real and imaginary part of , and are the real and imaginary part of G *.
Stimulate the system with input signals
where
From NOFRFs, the output frequency response under input can be expressed as below.
where
and
Using a least square based approach, the expression of G * can be written as
(2.7)
=
The requirement of this algorithm for the determination of the non-linear output frequency response is experimental or
simulation data of the system under different input excitations. In the following chapter, this algorithm will be applied
to an example non-linear system to evaluate its NOFRFs.
Charpter 3 NOFRFs based analysis for a mechanical oscillator
3.1 Introduction of the non-linear system
In this section, the procedure of quantitatively calculates the NOFRFs to investigate the generation of new frequency
components of output signal in non-linear system will be applied to a non-linear system.
The system which can be described by the non-linear system is shown below
Figure 1. A mechanical oscillator
the differential equation of this system can be described as
(3.1)
where u(t) and y(t) are the input and output of the system. and are the system parameters. In this case, assuming m =1, c =20, , , , to study the behaviour of the non-linear system in frequency domain when subject to a specific input using the analysis procedure in §1.3
3.2 Analysis of system output frequency response
The specific input is given by
(3.2)
t = -511 0.005 sec, …, 512 0.005 sec
Having been given the input signal, the output can be obtained in MATLAB by solving ordinary difference equation.
MATLAB uses Runge-Kutta-Fehlberg method solving ordinary difference equations (ODE). The result is calculated
within limited sampling point. In this case, the MATLAB function ode45 will be used to calculate the time domain output
signal which is shown in Figure 3 (b). The code is shown in Appendix A. Figure 4 shows the spectra of input and
corresponding output signal using Fast Fourier Transform from the time domain input and output signals, the MATLAB
code is shown in Appendix B. From Figure 4, it is very clear that the output spectra have richer frequency components
than those of input signal. Which indicates that some of the input energy is transferred by the system from the input
frequency band (30,55) Hz to a lower frequency range (0,30) Hz. This phenomenon is a very significant property of non-
linear systems. However, further study on the properties of this non-linear system will depends on the investigation of
non-linear output frequency response corresponding to input (3.2).
const constant 0
x= y output 1000
u input 2000
e noise 3000
subsystem =up to 9 subsystems are allowed = number from 1 to 9 times 100
lag number from 1 to 99
The maximum power that a term can be raised to is 4.
Using the code convention described above, the NARMAX models can be represented as the matrix showing in Appendix F.
Given an input signal, the corresponding output response can be generated using MATLAB in SUN workstation.
The signal (3.1) is chosen as the input signal to the three models
The output responses corresponding to input signal (4.1) of the three given non-linear
systems are generated in laboratory using Sun workstation, and the code is shown in
Appendix G. The input and corresponding output signal of the three studied systems
are showing below:
(1.7)
means to take the integer part
for k=0,…….,
From (4.1), it is easy to find out that the corresponding frequency range of G(jw) n=1,2,3, is (30,55) Hz,(60,100) Hz and
(50,800) Hz respectively.
First, considering the first order homogeneous non-linear system, the first order transform functions over a frequency
range (30, 55) Hz of each model are generated and shown in figure 18 in the form of spectrum.
Figure 19. The first-order non-linear output transfer functions of three models
From Figure 19, it can be clearly observed that the non-linear output transfer function of the first model has the minimal
gain comparing with other two models while the third model holds a maximum gain.
The following non-linear output transfer functions belong to the second order homogenous non-linear system and the
NOFRFs over a frequency range of (60,100) Hz are showing below:
Figure 20. The second-order non-linear output transfer functions of three models
Obviously, the NOFRFs that are shown in Figure 20 share the same trend with Figure 19. of model 1 has a maximum value around 1.5 in frequency range (60,100) Hz while the other two models hold a maximum value of 0.03 and 400 respectively.
Then, the non-linear output transfer functions of the third order non-linear
components of these three models in frequency range (5,80) Hz are evaluated and
shown in Figure 21:
Figure 21. The third-order non-linear output transfer functions of three models
Again, the same phenomenon is presented in Figure 21 that is exactly the same with previous two homogeneous non-
linear systems.
For more details of the comparison of these three models, the integrals of each G(jw) over the investigated frequency
ranges are evaluated and which are shown in the table 3. From the table below, the different between each model is very
evident.
Table 3 compare of integral of for each model
4.4 Conclusion
In this chapter, the concept of NOFRFs is applied to a real mechanical structure andthe output transfer function up to
fourth order homogeneous non-linear systems are evaluated which are shown in figure 18, figure 19 and figure 20. From
above paragraphs, it is able to compare non-linear output transfer function of each model up
to third order in specified frequency range. The result shows a direct relationship between the faults of non-linear system
and the size of output transfer functions. From Table 3, it is even more evident that model 3 always holds the largest
magnitude of G(jw) than other models in each order. It is also known that the first model is a faultless system while
model 3 has the most significant fault. The study in different order of G(jw) and various frequency ranges all indicate
that the more faulty systems are, the larger the magnitudes of output transfer function. This conclusion indicates that the
algorithm of NOFRFs can be used in fault detection of non-linear systems to identify the size of faults of non-linear
systems.
Charpter 5 Conclusion
In this paper, a methodology for fault detection in non-linear systems based on the concept of NOFRF (Non-linear Output
Frequency Response Function) has been introduced, and been applied to two different types of non-linear models.
In chapter 2, the first model is given in ODE (Ordinary Differential Equation) form. With a given input signal in time
domain, the corresponding output response is directly generated in MATLAB using function ode45. The NOFRFs are
then calculated using the approach introduced in Chapter 1 and the validity of this approach also been validated.
The second system is given in NARMAX model in chapter 3. A set of codes are written to obtain its output corresponding
to given input in SUN workstation. In this case, three Aluminium plates with different type of fault are introduced in
NARMAX model form. By studying and comparing the NOFRFs of each plate in certain frequency band, it is easy to
discover the relationship between the types of fault and size of . In this way, it can be found that NOFRFs-based
approach is an effective method in the application of fault detection.
BIBLIOGRAPHY
Banks, Stephen P, Mathematical theories of nonlinear systems. London: Prentice-
Hall International , 1988Cook, P. A. Nonlinear dynamical systems, 2 ed. New York; nd
London: Prentic Hall , c1994
Coleman, Matthew P, An introduction to partial differential equations with
MATLAB/Matthew P. Coleman. London: Chapman & Hall/CRC , c2005
H. Zhang, S. A. Billings, Analysing nonlinear systems in the frequency domain, I the
transfer function, Mechanical Systems and Signal Processing 7 (1993) 531-550
H. Zhang, S. A. Billings, Analysing nonlinear systems in the frequency domain, :
the phase response, Mechanical Systems and Signal Processing 7 (1993) 531-550
Ingle, Vinay K. Digital signal processing using MATLAB. Pacific Grove, Calif.
London: Brooks/Cole Pub , 2000
Kharab, Abdelwahab. An introduction to numerical methods: aMATLAB approach.
Boca Raton, Fla. ; London : Chapman & Hall/CRC , 2002
Khalil, Hassan K., Nonlinear systems, 3 ed (international ed.), Upper Saddle River, rd
N.J. London: Prentice Hall, c2002.
L. M. Li, S. A. Billings, Generalized frequency response functions and output
response synthesis for MIMO non-linear systems. International Journal of Control ,
79:1, 53 – 62
Mohammed Dahleh, Munther A. Dahleh, George Verghese, Lectures on Dynamic
Systems and Control. Department of Electrical Engineering and Computer Science,
Massachuasetts Institute of Technology
Penny, J. E. T. (John E. T.), Numerical methods using MATLAB, New York ; London:
Ellis Horwood , 1995.
Z. Q Lang, S. A. Billings, Output frequency characteristics of nonlinear systems.
Research report no.546, Sheffield: University of Sheffield , 1994
Z. Q Lang, S. A. Billings, Output frequencies of nonlinear systems. Research report
no.610, Sheffield: University of Sheffield , 1995
Z. Q Lang, S. A. Billings, Energy transfer Properties of nonlinear systems in the
frequency domain, International Journal of Control 78 (2005) 354-342
Z. Q Lang, S. A. Billings, R. Yue, J. Li, Output frequency response function of
nonlinear Volterra systems
Z.K. Peng, Z.Q. Lang and S.A. Billings, Crack detection using nonlinear output
frequency response functions. University of Sheffield
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