university of south wal es...2.2 impulse response determination by the method of correlation 10 2.3...
TRANSCRIPT
University of South Wal_es
2053153
DIGITAL PROCESSING OF SYSTEM RESPONSES
David Rees, B.Sc., C.Eng., M.I.E.E.
A dissertation submitted to the Council for National Academic Awards for the degree of Doctor of Philosophy
The Polytechnic of Wales, Faculty of Engineering Department of Electrical Engineering, May, 1976.
DECLARATION
This dissertation has not been nor is being currently submitted for the awa^d of any other degree or similar qualification.
David Rees
ACKNOWLEDGEMENTS
The author wishes to thank Dr. D.W.F. James, Director,
Polytechnic of Wales for permitting the investigation to be
undertaken.
I am indebted to my supervisors, Dr. J.D. Lamb, formerly of
the University of Wales, Institute of Science and Technology, and
currently of the University of Rhodesia, and Mr. D.M. Dummer, Head
of Department of Electrical Engineering at the Polytechnic of
Wales, for their helpful guidance, encouragement and enthusiasm
during the course of this investigation.
I am grateful to a number of my colleagues, particularly Mr.
W.J. Lambert, who developed the hybrid computer facility at the
Department of Electrical Engineering, and Dr. T.L. Harcombe, for
interesting and stimulating discussion.
I am thankful to Mr. D. Tonge of the Department of Mathematics
at the Polytechnic for formalising the presentation of the analysis
for the polynomial-type nonlinearity, and to Mr. G. Draper, Computer
Programmer at the Polytechnic, for developing some of the computer
graph plotting programs.
My final thanks are due to my wife for typing the thesis and
for the patience she has shown during the many hours I have spent on
the investigation, which otherwise would have been spent with her.
CONTENTS
Page
Synopsis x
Nomenclature x ii
Chapter 1 Introduction
1.1 Measurement of System Characteristic 1
1.2 Frequency Response Analysers 2
1.3 Correlation and the use of Pseudo Random Binary
Sequences 2
1.4 Problems of Impulse Response Interpretation 4
1.5 Frequency Response Measurement using Pseudo
Random Binary Sequences 4
1.6 Problems of Nonlinearities 5
1.7 Objectives of the Investigation 6
Chapter 2 Theoretical Basis of System Response Determination
2.1 Frequency Response Determination using the Frequency
Response Analyser 8
2.2 Impulse Response Determination by the Method of
Correlation 10
2.3 Properties of Pseudo Random Binary Sequences 12
2.3.1 Autocorrelation Function 12
2.3.2 Prime Factors 14
2.3.3 Power Spectrum 14
2.4 Use of prbs to Determine the System Impulse
Response 18
iv
Page
2.5 Time Domain to Frequency Domain Transformation 19
2.6 Frequency Response Determination u^ing prbs and
the Discrete Fourier Transform 19
2.7 Errors of Discrete Fourier Transform Mechanisation
Procedure 23
2.7.1 Zero-Order Hold Function 23
2.7.2 Resolution, Frequency Range, Sampling Rate and
Sampling Time 25
2.8 Discrete Correlation 26
2.9 Time Response of a Linear Second Order System to
an Input Sequence x(k) 28
2.10 Summary 30
Chapter 3 Computer Program Details
3.1 Introduction 31
3.2 Description of Computer Facilities 32
3.3 Hybrid Computer Program 35
3.4 Fast Frequency Response Identification Program 39
3.4.1 Subroutine PRBS 40
3.4.2 Subroutine PRIMS 46
3.4.3 Subroutine FASTM 46
3.4.4 Subroutine SAHLD 47
3.4.5 Subroutine SSIML 47
3.4.6 Subroutine ANGLE 47
3.5 Supporting Programs 47
3.5.1 Additional Programs for Determining the
Discrete Fourier Transform 48
Page3.5.1.1 Subroutine TRIGF 48
3.5.1.2 Subroutine NLOGN 49
3.5.2 Program for the Determination of H(u>)Indirectly 49
3.5.2.1 Subroutine CROSS 49
3.5.2.2 Subroutine PINT 49
3.5.3 Miscellaneous Programs 50
3.6 Summary 51
Chapter 4 Experimentation and Error Analysis with Linear
System
4.1 Introduction 52
4.2 Experimental Arrangement 52
4.3 Error Criteria 55
4.4 Spectral Estimates under Noise-Free Measurement
Conditions 56
4.4.1 Spectrum of prbs 56
4.4.2 System Dynamics 56
4.4.3 Sequence Lengths 56
4.4.4 Sampling Frequency 61
4.4.5 Time Position of Sample 67
4.5 Spectral Estimates under Noisy Measurement
Conditions 67
4.5.1 Quantisation Noise 67
4.5.2 External Noise Sources 70
4.5.3 Multiple Period Averaging 78
4.5.4 Theoretical Estimation of Noise Error 78
4.6
4.6.1
4.6.2
4.6.3
4.6.4
4.7
Chapter 5
5.1
5.2
5.3
5.4
5.4.1
5.4.2
5.4.3
5.4.5
5.4.6
5.4.7
5.5
5.6
5.7
Chapter 6
Comparison of Procedures
Radix-2 fft
Model Estimates
Computation Time
Measurement Time
Conclusions
Derivation of New Test Signal
Introduction
Illustration of the Problem
Analysis of Polynomial Nonlinearities
Properties of Prime Sinusoid Signal
Harmonic Rejection
Spectral Distribution
Amplitude Probability Distribution
Autocorrelation Function
Noise Rejection Capability
Sampling Frequency
Selection of an Optimum 'set 1 of Prime Sinusoids
Selecting Pre-defined Amplitude Distribution
Summary
Application to System with Saturation Nonlinearity
Page 84
84
89
91
95
97
98
98
102
104
105
110
111
111
in117
120
123
124
6.1 Introduction 127
6.2 Problems with prbs 129
6.3 Frequency Response Analyser Results ^37
6.4 Use of Prime Sinusoid Signals 138
vn
Page
Chapter 7 Applications to other Nonlinearities
7.1 Introduction 146
7.2 Dead Space Nonlinearity 147
7.3 Backlash Nonlinearity 150
7.4 Direction Dependent Nonlinearity 150
7.5 D.C. Servo System 163
Chapter 8 Conclusions and Further Work 170
References 175
Appendices i
Appendix 1 Computer System Configuration
1.1 IBM 1130 i
1.2 Interdata 80 i
1.3 Solatron HS7/3A i
1.3.1 HS7 Integrator Mode Control ii
1.4 Functional Description of Hybrid CALL Functions iii
Appendix 2 Computer Program Details iv
2.1 Hybrid Program Listing v
2.2 FFRIP Program Listing vii
2.3 FFRIP Program Input and Output Data xviii
2.4 Listings of Supporting Subroutines xxxi
2.5 Subroutines CALL Procedure xxxv
Appendix 3 Response DATA for a Second Order Dynamic System
as Measured by FFRIP xlii
vm
Page Appendix 4
4.1 Derivation of the Output from a Polynomial-Type
Nonlinearity when Applying a Composite Sinusoid
Signal xlviii
4.1.1 An Example for Odd and Even Powers li
4.2 Autocorrelation of a Composite Prime Sinusoid
Signal liii
Appendix 5
5.1 Sinusoidal Describing Function liv
5.2 Gaussian Input Describing Function liv
Appendix 6 Paper presented at the IEE Conference on "The Use
of Digital Computers in Measurement", University
of York, September, 1973. Iviii
SYNOPSIS
This thesis describes an investigation into the development of
techniques for the measurement of system dynamic characteristics
based on digital processing methods. The techniques are developed
to meet the requirements of rapid measurement time, noise and harmonic
rejection capability and ease of interpretation of results.
A computational procedure using spectral methods and based on
the fast Fourier transform is proposed, which considers a pseudo
random binary sequence as a series of sine waves of 'discrete 1
frequencies of well defined amplitudes and phase relationships.
Three computational algorithms have been considered,
(a) the discrete Fourier transform,
(b) the radix-2 fast Fourier transform, and
(c) the mixed radix fast Fourier transform.
It has been shown that if the radix-2 fast Fourier transform is used
unacceptable results are obtained. However, because the sequence
length of pseudo-random binary sequences can be expressed as
multiples of prime numbers the mixed-radix fast Fourier transform is
suitable and has been mechanised successfully. Errors when using the
procedure are presented taking quantisation levels, sampling rates,
sequence lengths and smoothing techniques into consideration, both
with and without the presence of noise in the response.
A detailed comparison is made between the crosscorrelation
function and fast Fourier transform methods of mechanisation, by
comparing modelling estimates given by both procedures for different
system conditions.
The application of the technique to nonlinear systems has shown,
however, the procedure can produce results which are unacceptable.
The analysis of a polynomial-type nonlinearity, when subjected to a
composite sinusoid signal has led to the derivation of a new test
signal. This signal consists of an assemblage of discrete sinusoids
of frequencies which are odd and prime number multiples of some
fundamental which is itself excluded from the signal. The properties
of the new signal are derived. It is shown that an optimum 'set'
of prime sinusoids can be selected to minimise the harmonics
generated by a cubic nonlinearity, and that the amplitude probability
distribution of the new signal can be modelled to a pre-definedshape.
The new signal is successfully applied to a range of highly nonlinear
systems, both real and simulated, with and without memory and is
shown to be superior to a pseudo random binary sequence in terms of
accuracy and noise immunity.
Theoretical modelling predictions are presented, based on the
single input describing function for the frequency response analyser
results and upon the Gaussian input describing function for the
results derived from the new test signal. Both of these estimates
have been compared with the models obtained experimentally using
Powell's optimisation procedure.
XI
NOMENCLATURE
Ai Amplitude of i th frequency.
A Saturation amplitude.
Polynomial coefficients. + h
a.j Amplitude of i harmonic.
), B(jo)) Real and imaginery parts respectively of F(ju),
a,b Cartesian co-ordinate vector quantities
a Amplitude of prbs.
bmls Binary maximum length sequence.
"c Number of combinations of m out of a total p.
D Algebraic operator.
dft Discrete Fourier transform.
f Cyclical frequency.
F Objective function.
f 1 , f«, fo» f* Time dependent functions.
FRA Frequency response analyser.
fft Fast Fourier transform.
f(t) Non-periodic time function.
F(ju) Fourier transform of time function f(t).
h(r) System impulse response.
H(u) System frequency response.
i, j» k, m, p Harmonic number integer variables.
m-sequence Maximum length binary sequence.
K Gain ratio of test signal.
k Number of prime frequencies.
N Sample size, Number of primes, Noise.
XII
NAt Data record length time.
NT Integration time.
n Number of shift register stages.
prbs Pseudo random binary sequence.
p Probability distribution function.
r, q, p, m Register polynomial delay integer values.
u Step input magnitude.
R Magnitude ratio.
R. Theoretical magnitude ratio.
R^ Measured magnitude ratio.
s Time displacement.
t Time variable.
T Period.
T,, T2 System time constants.
X(t) System input signal.
X*(t) System input sampled data signal.
X*(u) Fourier transform of X*(t).
X-,, Xp, Xn n independent variables.
X(w) Fourier transform of X(t).
x(k) Input sequence.
Y(t) System output signal.
Y*(t) System output sampled data signal.
Y*(u>) Fourier transform of Y*(t).
y(k) Output sequence.
X+(^) Complex conjugate of X(~).
Z(kAt) Discrete correlation function.
Z (kAt) Discrete circular correlation function.
xm
z, y Number of permutations.
a Standard deviation.2a Variance.2 ar Variance of test signal.2ae Variance of system error.
T Time displacement.
At Clock period.
e, <{>, Phase angle.
4>(o> k ) Power contributed by the k harmonic.
4>(m) Eulers Phi function.
4>vv(T) Autocorrelation function.
<!>V/Y( T ) Crosscorrelation function.
e Error.
CD System natural frequency.
5 System damping factor.
® Modulo-2 addition.
Fourier transform pair.
xiv
CHAPTER 1
Introduction
1.1 Measurement of System Characteristic
An essential requirement in the design, development and operation
of control systems is the determination of the system characteristic
which can be expressed either in the time domain as an impulse response
function or in the frequency domain as a frequency response function.
The determination of the system characteristic is the first of a
four-stage process necessary in order to implement a control system
on an industrial plant. These stages are, identification of the
system characteristic in order to obtain a model of the process,
simulation of the model, design of the control system and final
tuning of the implemented control scheme.
Important requirements for the identification procedure are that
it should enable the system characteristic to be measured with speed
and accuracy and also that the measuring procedure should be either
negligibly affected by, or overcomes the effect of, any noise present
in the measurement channel. In addition to this, identification is
often undertaken on systems that are nonlinear except within carefully
controlled regions of operation, so that measurements may be
corrupted by harmonic generation. The main theme of this thesis is
the development of digital processing methods and their application
to the identification problem in order to meet the requirements of
speed and accuracy and also to provide noise and harmonic rejection
capability.
1.2 Frequency Response Analysers
These requirements for the measurement procedure have led, in
the past decade, to the development of frequency response analysers
based on pulse rate techniques which are mechanised to reduce the
effects of noise on the measurements taken and also to remove the
effect of harmonic? generated by the presence of nonlinearities in
the system . Test instruments of this type excite the system under
test by a sine wave of known frequency and amplitude and calculate
the frequency response data from measurement of amplitude and phase
of the output of the system. The frequency is then changed and the
procedure is repeated. For systems with long time constants this
procedure can be most time consuming, particularly if a complete
description in the presence of noise is required. In addition to
this, for reasons of plant safety, it is often unacceptable to
disturb many processes with a sinusoidal signal for long periods of
time.
1.3 Correlation and the use of Pseudo Randon: Binary Sequences
In view of the system testing requirements considered in
paragraph 1.1 recent efforts have been concentrated on obtaining the
frequency response or its time domain equivalent, the impulse function,2 with minimum test time using the process of correlation . The
original correlation method for estimating the impulse response of a3 process used random noise for system excitation , but this was not
widely accepted because of the difficulties of;
i. on-line application to processes,
ii. performing the necessary calculations, and
iii. obtaining reliable estimates in reasonable experimentation
time .
These difficulties have been removed by the use of pseudo random
binary sequences (prbs) as excitation and consequently widespread
attention has been devoted in the literature i'i the past fifteen years
or so, to the use of these signals in system identification inc.^
These sequences are of maximum length and therefore are referred to
in the literature as m sequences.
There are numerous advantages in using prbs rather than Gaussian
signals for measuring the response of a system, which may be
summarised as follows:
(a) prbs can be generated with ease.
(b) wide ranging spectral characteristics can be obtained by
a change in clock speed.
(c) the system response can be determined very quickly. On the
other hand, white noise test signals require long averaging
times to reduce statistical errors to acceptable levels.
(d) the signal is periodic so that system response can be
extracted with ease, even in a noisy environment.
(e) if there are no corrupting noise sources, then there is no
need to perform any type of averaging.
(f) if the signals measured are corrupted by noise, simple
periodic averaging can be used to recover the periodic
signals of interest. This cannot be accomplished with
conventional spectral analysis or correlation using random
signals.
(g) as the signal is deterministic, it can be re-created when
both the time at which the sequence started and the sequence
length are known.
1.4 Problems of Impulse Response Interpretation
Obtaining the impulse response function using the correlation
process still requires the system designer to interpret the function
and in this area some difficulty has been found. It has been shown
experimentally that attempts to obtain impulse response equivalents
of a system can produce results swamped by information which is20 unimportant in relation to the performance of that system . In
addition to this and perhaps for historical reasons associated with
design techniques and ease of manipulation, the designer invariably
"thinks" in the frequency domain so that interpretation of the
impulse response is consequently made more difficult for him.
1.5 Frequency Response Measurement using Ps.eudo Random Binary Sequence^
To transform from the time to the frequency domain the Fourier
integral of the impulse response is required. The accuracy of this
transform depends upon the "goodness" of the crosscorrelation
function as a measure of the impulse response and on the availability
of the whole response so that the effects of the rectangular
information window are minimised. A procedure to avoid such problems
has been described where prbs was regarded as a series of sine waves21 of discrete, well defined and readily varied frequencies . This led
to a rapid technique for frequency response determination, which
avoided calculating the crosscorrelation function and the subsequent
Fourier integral. By adopting this procedure the amount of test time
required is reduced considerably from that needed when single21 sinusoids are used. In this paper by Lamb, the frequency
characteristic was obtained by determining the Fourier integral using
numerical methods. Whilst this avoided the problems of aliasing
that occurs when continuous data is digitized, computationally it
was not the most efficient method because a heavy time penalty is
incurred. As the response to prbs is transformed directly the fast22 Fourier transform (fft) should be incorporated for most effective
mechanisation.
Binary maximum length sequences are of period 2n - 1 (n integer)
and thus cannot be used directly with the most efficient form of the
fft, the radix-2 transform, since this assumes data sequences having
a period 2n . In order to use the radix-2 fft, additional experimental
equipment must be introduced, as proposed in the method of Nichols and23 Dennis . Basically, this consists of dividing down the master-clock
frequency differently for an m sequence generator and the fft data
acquisition sampler such that the spectral lines of the m sequencer
and the fft coincide.
1.6 Problem of Nonlinearities
The procedure using prbs for system identification is effective
in linear and approximately linear systems. Problems, however, have24 25 been shown to occur in the presence of nonlinearities, ' and also
in the presence of deterministic noise, for example in the form of an
20 approximate sinusoid . Much effort has been devoted to selecting01-
test signals based on periodic pseudo random signals, but evidence
so far suggests that with nonlinear systems the selection requires a
good knowledge of the nonlinearity, or is restricted to particular
27 types of nonlinearity . For example, an improvement is achieved in
model estimates by selecting an antisymmetric prbs rather than normal
prbs to identify systems containing direction dependentpp nonlinearities .
1.7 Objectives of the Investigation
As can be deduced from the preceding sections the question of
noise rejection, duration of testing, ease of interpretation and effect
of nonlinearities are all very relevant in the quest for suitable test
signals and processing techniques. It is the intention of this
investigation to develop system testing methods that are compatible
with these considerations and which are an advancement on present
methods. In general, the results to be presented consist of a
combination of digital, hybrid and analog computer studies, including
some field measurements, in order to develop and validate techniques
and procedures. The significant contribution of this work is:-
(a) General computer procedure for system identification
based on the fast Fourier transform - mixed radix algorithm
using prbs and a composite sinusoid signal for system?Q "3(1excitation^' .
(b) Estimation of errors by experimental means when this
procedure is applied to noisy systems.
(c) A detailed comparison of the model estimates obtained from
time and frequency domain identification procedures.
(d) Derivation from theoretical considerations of a new test
signal consisting of an assemblage of sinusoids which has
advantages over prbs when testing nonlinear systems .
(e) The use of a digital computer algorithm to derive a
predefined amplitude probability distribution for the test
signal.
The following chapter contains the theoretical background
of the investigation and the mathematical basis of the computer
procedures adopted in later chapters.
CHAPTER 2
Theoretical Basis of System Response Determination
2.1 Frequency Response Determination using the Frequency Response
Analyser .
Modern frequency response analysers comprise of a programmable
generator that provides a stimulating signal, a correlator to analyse31 the response of the system and a display to present the results .
The instrument operates as shown in Fig. 2.1 and consists of a two-
phase oscillator, generating an in-phase and quadrature signal, the
in-ph.ase signal providing the system excitation. The system output
is correlated with the system input and the quadrature signal, so
that if the system input is A sin tut and the output is R sin(wt +e),
the integrator outputs are:-
- rNT Jc
rNT R a R sin(wt+6)sin tot dt = — cos 9 = —
NT ^0 22 2.1
and
1 f NT R b— ) R sin(ut+e)cos wt dt = - sin e = -NT J 0 22 2.2
where R = /a +b and 9 = tan" — and NT is the integration time ofa= *£2+b2 and 9 = tan" 1
N cycles of the waveform of period T. Equations 2.1 and 2.2 give in
cartesian co-ordinates the system frequency characteristics H(co) at
the frequency w rad/s.
The system frequency estimate is unaffected by distortion
within the system due to harmonics generated by the presence of
nonlinearities. This is mechanised in the instrument by correlating
sinw
t
OSCILLATOR
A sinwt
SYSTEM.
R si n
(h)t
+e)
AVERAGER
r ^
XAVERAGER
D I S P L A Y
COSut
Fig.
2.1
Freq
uenc
y Response A
naly
jer
Bloc
k Diag
ram
the system output with the outputs of the two-phase oscillator, over
an integer multiple of the system excitation period, thus averaging
any harmonic content to zero.
The correlation process rejects noise; the degree of rejection
depending on the averaging time. Elsden and Ley have derived
curves which are reproduced in Fig. 2.2 which show how the noise
rejection capability of a frequency response analyser (FRA) depends
upon integration time.
2.2 Impulse Response Determination by the Method of Correlation
There are numerous publications that deal with the basis of
32 this method, one example is Truxal
If a linear system having an impulse response h(t) is activated
by an input signal X(t) then the output is given by the convolution
integral as:-
Y(t) = f hCTjWt-nJd-n 2 3 A»
Function Y(t) is said to be the convolution of the functions X(t)
and h(t).
The crosscorrelation function as measured over a time T of two
time varying signals X(t) and Y(t) is given by:-
1 f TT) = - X(t-T
T J 0: )Y(t)dt 2.4
Rewriting this equation by substituting eqn. 2.3 in 2.4 gives:--
"-TJ;2.5
10
1-1
- - - ENVELOPE OF CURVE
N= NUMBER OF CYCLES
6 7
NORMALISED FREQUENCY
Fig. 2.2 Noise and Harmonic Rejection for the Frequency Response Analyser
The transfer function (H, (u)) of the correlator is plotted versus the normalised frequency. The graph shown is the response for the multiplication by sin cot. A similar graph is obtained for multiplication by cos tot. Further details may be found in the paper by Elsden and Ley.
11
and by interchanging the order of the integration becomes,
r ] rT<J> XY (T) = h( T i)dTl - { X(t- T )X(t- Tl )dt}
iia> T 0
f"= j h(Ti)«frxx (T-Ti)dT i 2.6
where <(>.,.. (T-TJ) is the autocorrelation function of the input signal,
of argument {T-TJ). Comparing eqn. 2.6 with the convolution integral,
eqn. 2.3, it may be seen that if a signal whose autocorrelation
function is <J>\/\/(T) is applied to a system with impulse response h(t),
the crosscorrelation function of the input and output signals is
equal to the time response, of the system when subjected to an input
signal <J>V\/(T). If <|>VV(T) = <S(T) (unit impulse), as is the case with
white noise, then ^(T) is the system impulse response h(t), provided
II(T) = 0 for i>T. In order to meet this condition I is chosen to be
greater than the system settling time, which is often taken to be five
times the largest time constant in the system.
The mechanisation of such a scheme to obtain the system weighting
function is shown in Fig. 2.3.
2.3 Properties of Pseudo Random Binary Sequences
2.3.1. Autocorrelation Function
The only requirement set down for the form of the input signal
in order to determine h(t), as can be concluded from eqn. 2.6 is that
<f> xx (T) should approximate to a dirac-delta function. Signals that
have this property are binary maximum length sequences (bmls) and are
generated by using a shift register with modulo-2 addition in the
feedback circuit. For an n-stqge shift register the sequence maximum
12
White Noise Input
.lit) SYSTEM h(t)
DELAY
Output y(t)
AVERAGER
Fig. 2.3 Experimental Procedure for Evaluating
the Impulse Response of a System.
13
length N is given by 2n -l and has an autocorrelation function of the
form shown in Fig. 2.4. The period of the sequence depends on both
sequence length ^nd clock frequency. Properties of these sequencesOO Ofl
have beer, extensively covered in the literature ' so that only
details relevant to this investigation will be covered here.
2.3.2. Prime Factors
The characteristic polynomial of the shift register, and the
initial contents of the register determine the m-sequence generated.
For each sequence length, a number of m-sequences can be generated
by suitable feedback connections, the number of which is given by:-
U(2n-l)}/n
where (f>(m) is the Euler's Phi function, defined as the number of
positive integers less than m and relatively prime to m, including 1.
Table 2.1 gives the factors of all values of m used in this
investigation and the resultant number of m-sequences available.
2.3.3. Power Spectrum
The power spectrum $xx (cu) of prbs is a line spectrum with
harmonic separation of j^-f rad/s and its amplitude envelope is given
by16
a 2 (N+l) N « £
IT k-1 2.7
r\ j r
where $yv(<\) is the power in volts contributed by the k harmonic,
a is the amplitude,
and k is the k harmonic line.
Figure 2.5 gives the form $w(<j3) and it can be seen that nodes occur
(continued on page 18)
14
-aI
T=(2n -l)At
Fig. 2.4 Autocorrelation Function for a Periodic Pseudo Random
Binary Sequence of Amplitude a, of Period T and Clock
Speed I/At.
15
n
3
4
5
6
7
8
9
10
n
N=2 n -l
7
15
31
63
127
255
511
1023
2047
Factors
7
3.5
31
3.3.7
127
3.5.17
7.73
3.11.31
23.89
Number of Codes
2
2
6
6
18
16
48
60
176
Table 2.1 Table of Prime Factors and Number of Codes for
Binary Maximum Length Sequences.
16
a 2 (N+l)N2
--.-.
x\
--, «
1
rt
r ( sin xf3dB .x *^^^\ *S"^\ -*^
Ii1 '". 1 ' i» 1f! !••.Lt\} i
£1T £TT DTT
3At At At
rad/s
Fig. 2.5 Power Spectrum of a Binary Maximum Length Sequence
17
rt
at ~ rad/s and that the effective frequency band covered, measured
up to the 3dB point is -n^-r to -^r rad/s. This means that prbs has
one of the properties that is desirable for a test signal in that the
signal spectral components are constant or "white" over the frequency
range of interest.
2.4 Use of prbs tc Determine the System Impulse Response.
When prbs is used as the system input in a correlation experiment
to determine h(i), the crosscorrelation function (^(T) is given by ;
for T5At o 2a fc r TAth(t)- h(s)
N ->nds
N JQ 2.8
and for Tin the rai:ge 0<-r<At
a 2 (N+l) At
2 2At
a 2 f T(T) -— h(5)ds
N JOh(-
M i n2.9
provided h(r) = 0 for t<T
where T = NAt the sequence period.
a2 f TThe term - t h(s)ds is constant, and will approximate to zero
N Jo
for large values of N. On this assumption, for the case of x5
Nh ( T ) =
a 2 (N+l)At 2.10
18
2.5 Time Domain to Frequency Domain Transformation.
The time and frequency domains are related by the Fourier
transform, which for a non-periodic time function f(t) is given by:-
+« F(jco) = f(t) exp(-joit) dt
» —oo 2.11
where, in general F(J<D) is complex and may be expressed in terms of
its real and imaginary part by:-
F(jo>) = A (u) + jB(w) 2 ^
so that
A(u>) =( f(t)cos(ujt)dtJ-» 2.13
andB(o>) = - f(t)s1n(ut)dt
J_co 2.14
and , 5_____5- j«o)| =/A(o))^ + B(u)^
e(u) = tan" B(co) 9 ,"7T7—Y ^ • 'A(u>)
Having measured h( T ) of a system, H(jw ) can be evaluated by solving
eqns. 2.13 to 2.15.
2.6 Frequency Response Determination using prbs and the Discrete
Fourier Transform.2
Since prbs has a line spectrum with a ( s-1 " x ) amplitude
envelope the spectrum characteristics can be used to advantage to
determine the frequency characteristic directly without using the
Fourier integral. The basic identification arrangement is shown in
19
Fig. 2.6.
A signal X(t) is applied to a system with impulse response h(t)
to obtain an output signal Y(t). Both input and output are sampled
at the clock frequency to give two sequences X(kAt), Y(kAt), where
At is the sampling interval.
The sampled values ?re given by:-
CO
X*(t) = £ X(kAt)6(t-kAt)
Y*(t) = I Y(kAt)6(t-kAt)k=-«> 2.16
These are transformed using the discrete Fourier transform to obtain
the spectra of the input and output signals.
The spectrum computed by the discrete Fourier transform relates
to the Fourier transform of the system input and output, X(ai) and
Y(w)respectively by the relationships
1 ~ X*(o») = — z X(w -
i =' » At
1 oo £iriY*(u) = — Z Y(u> -
At i=-°° At 2.17
provided the following conditions apply :-
i. the time function X(t) and Y(t) are periodic.
ii. X(t) and Y(t) are band limited.
iii. the sampling rate must be at least twice the largest
frequency component of X(t) and Y(t).
iv. the rectangular information window for X(t) and Y(t) must
be non-zero over exactly one period of X(t) and Y(t).
20
Bseudo Random Input Signal
X(t)
X*(t) ^
SYSh(t),
\f x(«)
Pseudo Random Sampled Data
Signal
.6 Basic Identification
TEN _Y(t)H(«)
Y*(t) }
Y(«)
[ Y*(o>)
Output Sampled Data
Signal
Using the Discrete Fourier
System Output Signal
Transform.
If these conditions are satisfied then,
Atx(aj) =0 |<o|WAt
/ x _ At Y*(u>) |co|<ir/At
0 |u|WAt
2.18
so that the frequency response estimate
2.19
If the system input signal X(t) has period NAt, the discrete Fourier
transform of the input and output sequences are given by
X* =n=-» k=0
* Y(kAt)exp(-j2j!«l) n=- k=0
and the system frequency response estimate for discrete non-zero
frequencies is given by:-
n .,, 0<n<N/22.21
With prbs as a system input the discrete Fourier transform of theO/7
input and output sequences are determined. Barker and Davy have
derived theoretically the discrete Fourier transform for three types
of m-sequences (pseudo random binary, antisymmetric pseudo random
binary, and anti -symmetric pseudo random ternary sequences) so that
it is considered unnecessary to cover the theory here. However, one
22
point to observe in the case of anti-symmetric signals is that
eqn. 2.21 need only be determined for odd frequencies, since only
odd harmonic lines are present.
2.7 Errors of Discrete Fourier Transform Mechanisation Procedure
2.7.1. Zero-Order Hold Function
In the practical mechanisation of the identification scheme the
system input is obtained from a sample and hold device, as shown in
Fig. 2.7, so that the system input can be regarded as a prbs sampled
data signal convolved with a zero-order hold function.
Since the impulse response of a zero-order hold function is
h(t) = u(t) - u(t - At) 2.22
with Fourier transformation
H(a) ) = At~T 2.23
an exact expression may be obtained for X(u) and used in preference to
the approximation used in eqn. 2.18.
This is:-
X(u) = H(o>)
= At2.24
A similar argument is applied to the system output signal. Equation
2.16 expresses the system output as a periodic sequence of variable
height impulses. In o>"der to obtain an accurate expression for the
output spectra the system output signal is considered as that obtained
from a sampled data signal convolved with a zero-order hold function»
23
Pseudo Random ___JLHL Sampled Data X*(u>)
Signal
ZERO-ORDER HOLD h(t), H(o))
X(t) Pseudo Random ~5!a" Input Signal
Fig. 2.7 Experimental Arrangement for Identification Scheme.
24
so that
Y(o.) = H(u) Y*{u>)
- At~2~ 2.25
With prbs as the system input, if the discrete Fourier transform of an
unfiltered sequence is taken, the resultant spectrum will have a flat
envelope. In order to represent the input spectrum in terms of the
actual amplitude envelope of the input signal ,eqn. 2.24 is modified to
-,2
X(u) = At " ' ——' ^ wAt
- ~T J 2.26
so that
Vicf H
2.27
2.7.2 Resolution, Frequency Range, Sampling Rate, and Sampling Time
The discrete Fourier transform produces N outputs x*(k) which wek
have seen are the values of the Fourier transform at f = -r, where
is the record length. The N outputs span a frequency range from
N-l 1 zero to Hz and the frequency resolution is , so that to
achieve good resolution in the frequency domain, N should be chosen to
be large. The frequency components x*(k) for real data covering the
range -^ <ks=N-l are the complex conjugates of the positive frequencyN components x*(k) for the range Ckks-^-.
In order to avoid aliasing {spectral overlap or folding due to
sampling) the sampling frequency is chosen to be at least twice the
25
maximum frequency to be estimated, or alternatively the waveform
being analysed should be band limited at a frequency f, where
f = jj-r prior to sampling. As a general rule v^e frequency response
is virtually unaffected by aliasing up to half the sampling frequency,
if the sampling frequency is chosen so that the system frequency
response is attenuating at a rate of 40dB/decade cor frequencieso/-
greater than this .
To avoid a linear phase shift in the estimates the output
samples must be taken halfway between the epochs of the test signal.
Whilst this is advantageous in order to reduce the amount of
computation required it is not always possible to implement in a
field testing application. If this is the case, oqn. 2.25 can be
modified to compensate for a linear phase-shift by using the
shifting property of the Fourier transform, giving,
At Y(u) = At exp(-jmu)At) - exp(-j^i)Y*(u>)
T~ 2.28
where m is in the range 0<m<l. The time of sampling is given by
kmAt with k in the range 0<k<N-l,
so that,
W . . exp(osin(-rrn)W TO' 2.29
2.8 Discrete Correlation
The discrete correlation of two periodic time functions
X(kAt) and Y(kAt) with period N is defined as
26
N-l Z(kAt)= z Y(iAt)Xp+i)At)
i=o 2.30
The discrete cor-elation theorem transform pair is given by
N-l _ 4- n n
i=o NAt NAt 2.31
where the notation T^ indicates a Fourier transform pair, and
X (-r) is the complex conjugate of X(-). From this relationship,
eqn. 2.31, it can be seen that the process of correlation may be
effected in the frequency domain. To determine the crosscorrelation
function by Fourier transform methods the discrete Fourier transform
of x(k) and y(k) is determined and the product of Y(-R^r-) and the
conjugate of X(-A) is evaluated. The inverse discrete Fourier
transform of the product X+(ir) Y(-) yields the correlation|r
function. The inverse discrete Fourier transform is given by
N-l Z(kAt) =i z Z( *Uexp(j £A k=0, 1, ' N-l
N n=Q NAt N 2<32
In determining the correlation function by this method the usual
correlation function is not obtained but rather a "circular"
correlation function defined by the relationship
Z (kAt) = Z(kAt)+Z((N-l-k)At) c 2.33
Consequently, in order for Z (kAt) and Z(kAt) to be equal the sequence
period must be at least twice the system decay time.
27
2.9 Time Response of a Linear Second Order System to an Input
Sequence x(k)
A linear second order system with an input X(t) and an output
Y(t) is described by the differential equation
d2Y x ->; dY + 2 Y 2 Y——K + C.t,U n —— + tl>_ I = U>_ A
dt2 n dt n n 2.34
where £ is the damping factor and to the natural frequency in rad/s.
With £ in the range 0<?<1, and Y(0) = yQ and Y(0) = y,, the solution
of this equation for a step input of magnitude u is
Y(t) = (u-y^f^t) + yi f2 (t) +y0
where
f (t) = 1 - -r-L exp(-£unt)sin(co.,X^? t + tan~ 1v-^} 1 n 5
and1 X 2~
V"~* 2.35
and the derivative of the system output, Y(t), is
Y(t) = (u-yQ )f3 (t) + yi f<
where5(un / 7 -l
f3 (t) = T-^-exp(-5o)nt) sin(o)n/l-rt + tan '
/ 9 1 v\ F-a>nexp(-Ca) n t)cos(con /l-C t + tan" -^-)
andi 2
f4 (t) = exp(-Sa)nt)cos(un /l-? t) - 5wnexp(-Cwnt)sin(a)n /l-e t)
2.36
28
If the system input is a periodic sequence x(k) of length N and
period NAt which is held constant between the clock period (zero-A. L
order hold function) the output sequence of the k sample y(k)
measured at the halfway point between the epochs of the test signal
is given by
y(k) = (x(k) - Y((k-l)At))f.,(kAt - ^)
+ Y((k-l)At)f~(kAt - ^J) + Y((k-l)At)* ^ 2.37
where
y(k) = Y(kAt - 4|)
and
Y((k-l)At) = (x(k-l) - Y((k-2)At)f2 ((k-l)At)
+ Y((k-2)At)f2 ((k-l)At) + Y((k-2)At)
Y((k-l)At) = (x(k-l) - Y((k-2)At))f3 ((k-l)At)
+ Y((k-2)At)f4 ((k-l)At)4 2.38
With k=l and the system stationery
Y((k-l)At) = Y((k-2)At) = Y((k-l)At) = 02.39
For a recurring sequence the following recurrence relationships apply
for k=l,
Y((k-l)At) = Y(NAt)
Y((k-l)At) = Y(NAt)
and .Y((k-2)At) = Y((N-l)At)
2.40
29
2.10 Summary
In this chapter the theoretical basis for system response
determination has been presented with particular emphasis on the use
of the discrete Fourier transform (dft) to determine the frequency
response. The mechanisation errors resulting from the use of the
discrete Fourier transform have also been estimated. A mathematical
model of the response of a second order underdamped system to an
input sequence is presented.
The next chapter will give the computer programs for the
implementation of the equations presented in this chapter, programs
which will be used in the collection and analysis of experimental
data.
30
CHAPTER 3
Computer Program Details
3.1 Introduction
In this chapter the details of programs developed for use in
this investigation are presented. They can be divided into three
broad categories:-
i. A program to generate, excite and collect data from
simulated and physical systems (HYBRID PROGRAM).
ii. A program to implement the dft procedure on the data
collected, to determine the characteristic function of
the system (FFRIP - PROGRAM).
iii. Supporting programs that enable the assessment and
comparison of alternative procedures and computational
techniques for determining the system characteristic.
All the programs were written in Fortran IV computer language.
They were written to run on two computer systems, an IBM 1130 and an
Interdata 80 so that the FORTRAN instruction subset used was compatible
with both machines. The FORTRAN available on the IBM 1130 at the time
the programs were written did not have the facility of logical
expression, logical arithmetic, complex arithmetic and dynamic
dimensioning of arrays. These were available on the Interdata 80.
In the following sections the overall structure of the programs
will be described, with special emphasis on some of the features which
are of particular interest. Before considering any program details a
summary of computer hardware facilities will be given.
31
3.2 Description of Computer Facilities
The computer systems used were an IBM 1130 digital computer and
an Interdata 80 digital computer, and a Solart"on HS7 analogue
computer. The HS7 was interfaced to bcth the IBM 1130 and Interdata
80 computers. Details of the computer system configurations are given
in Appendix 1.1.
Simulation and data collection exercises were carried out on both
the IBM 1130/HS7 and Interdata/HS7 systems. The same hybrid interface
was used for both systems and this was arranged by means of a 100-way
logic switch. In order to appreciate the programming details for
driving the hybrid system a brief summary of the interface hardware
will be necessary. There are four separate data transfer modes:
these are:-
i. Analogue to digital conversion (ADC). Voltages at the
analogue patch panel are connected to a 16 channel
multiplexer, whose output provides the input to the ADC.
ii. Logic Sense Inputs (LSI). There are 16 logic line inputs
to the digital computer which are terminated on the logic
patch panel of the HS7.
iii. Digital to analogue conversion (DAC). There are 4 DAC
channels which are terminated on the anlogue patch panel
and they provide voltages which are proportional to a
digital number.
iv. Logic Sense Outputs (LSO). There are 16 logic line outputs
from the digital computer which are terminated on the logic
patch panel of the HS7.
32
A block diagram of the system is shown in Fig. 3.1. The
multiplexer channel is selected by means of an address matrix which
switches a solid' state switch connecting it to the ADC input. Thel ?
ADC givec a 12 bit together with a sign bit output. Since 2 = 4096
the smallest discrimination that can be used in measuring the input
voltage is - m.n. or approximately 23 mV (1 m.u. equals 100 volts).
No scaling is performed on the resultant binary number. To change to
floating point format the integer ADC number is divided by 4095
(the. floating point number is then in machine units). In this
investigation all calculations involving ADC values were made by the
use of floating point arithmetic.
The DAC channels are selected by an address register and a binary
decoder. The data transferred is 11 bits together with a sign bit
giving an integer equivalent of 2047 (least significant bit = 46 mV).
Logic signals from the digital computer can be provided under
program control and may be used to control directly the HS7 operation
controlling the logic levels of two logic lines. The mode control
logic is given in Appendix 1.
The LSI signals are used to signify to the digital program the
presence or absence of some condition on the analogue computer. For
example, some inputs can be set by means of logic switches on the
HS7 and used to signify that the setting-up procedure for the
simulation problem has been completed.
All the data transfer modes are undertaken by assembler written
programs and these are called by FORTRAN CALL statements with
appropriate arguments. The list of hybrid call statements available
with a functional description and argument list is given in Appendix 1.4.
33
CO
i
DI6ITAL
' *
ANALOGUE
TO
DIGITAL
CONVERSION
JMULTIPLEXER
1
COMPUTER IBM
1130 OR INTERDATA
80
f f
DIGITAL
TO
ANALOGUE
CONVERSION
if
DE-MULTIPLEXER
LOGIC
SENSE
INPUTS J
LOGIC
SENSE
OUTPUTS
ANALOGUE PATCH
LOGIC
PATCH
ANALOGUE COMPUTER SOLARTRON
HS7
Fig.
3.
1 Hybrid Computer S
yste
m.
3.3 Hybrid Computer Program
The function of the hybrid program is the generation and
application of test signals to dynamic systems, either simulated or
real, and the measurement of the system response data resulting from
the signal applied. The program as written allows a number of test
signals to be selected, each signal having different spectral
characteristics. The test signals provided were:-
i. single sinusoid
ii. pseudo random binary sequence
iii. inverse-repeat pseudo random binary sequence
iv. prime sinusoid.
The choice of test signal will be discussed in detail in later
chapters, where it will be shown that the choice depends upon the
characteristics of the system being investigated.
In addition to this, the program allows the user to specify the
sequence length, the clock frequency and in the case of prbs the
choice of characteristic polynomial. It has been shown that in the
measurement of 2nd-order Volterra kernels by crosscorrelation there
are wide variations in the performance of anti-symmetric signals
based on m-sequences, even between signals of the same levels and*?f\
characteristic polynomials of the same order. For this reason the
program is structured in a way which enables the user to specify the
polynomial coefficients of the sequence used. Also since one of the
aims of the investigation is to study the effects of quantisation and
external noise on the system estimates, the program allows the user to
specify the quantisation and averaging periods desired. The flow
diagram for the program is shown in Fig. 3.2 and the listing of the
(continued on page 39)
35
READ PROGRAM INPUT PARAMETERS
DELT=CLOCK PERIODVIN =TEST SIGNAL AMPLITUDEMl =TEST SIGNAL OPTIONN =REGISTER LENGTHIP =NUMBER OF PRIMESJl =QUANTISATION INDEXKll ^PERIODIC AVERAGING INDEX
NO
RESET HS7 AND SET INTO HOLD MODE SELECT MULTIPLEXER CHANNEL 1. COMPUTE SEQUENCE LENGTH ISEQL.
Ml=l
GENERATE PRBS
Ml =2
GENERATE INVERSE REPEAT
GENERATESINGLE SINUSOID
PRINT TEST SIGNAL
INVERT TEST SIGNAL DATA TO DAC FORMAT
SET HS7 INTO COMPUTE MODE
DO 40 L=1,ISBQL
OUTPUT Lth SEQUENCE DATA
AND DELAY "DELT"s
OUTPUT i Mth-a S£gUENCE DATA
1 F
™xsw*:&&faismm
DELAY
DIGITAL CONVERSION
DELAY"DELT"
50"
fSET HS7 INTO HOLD
MODE. QUANTISE DATA TO J1 BITS AND CON VERT ADC VALUES-MU.
DECREMENT Kll BY 1
DETERMINE AVERAGE
PRINT SAMPLED DATA
PUNCH DATA OUTPUT ON CARDS
Fig. 3.2 Hybrid Program Flow Diagram
38
program is given in Appendix 2.1 together with a description of the
parameters.
The prbs and prime sinusoid signals are generated by subroutine
PRBS and PRIMES respectively. Since these subroutines are also
included in FFRIP the program details will be reserved for a later
section and included with the other FFRIP subroutines.
3.4 Fast Frequency Response Identification Program
The mathematical techniques which form the basis of the Fast
Frequency Response Identification Program (FFRIP) have already been
presented in Chapter 2. FFRIP is based on the fast Fourier transform
using pseudo random binary signals and prime sinusoid signals to
excite systems. The program incorporates all the test signals available
in the hybrid program with an additions! option included, permitting
the user to input any other test signal. The program determines the
spectrum of the system input and output in polar co-ordinates and
estimates the frequency transfer function. In addition the
crosscorrelation function of the system input-output is calculated.
The program normalises the frequency response estimate to the first
line frequency of the test signal.
In order to assist in program validation and familiarisation a
system digital computer option is provided for a second order
underdamped system. This is used to obtain some of the results
presented in Chapter 4 (Section 4.4).
As was discussed in the previous chapter, the length of a prbs
sequence is odd and can be expressed as a product of prime numbers°8
(Table 2.1). This means that the mixed radix fast Fourier transform"1
39
could be used to compute the discrete Fourier transform. Whilst this?? ^Q algorithm is not as efficient computationally as the radix-2 fft '
it nevertheless provides a significant time advantage over the dft
(see chapter 4 Table 4.7). The radix-2 fft could of course only be
used with data lengths of power of 2.
The program flow diagram is shown in Fig. 3.3 with the program
listing and parameters documented in Appendix 2.2. A sample of the
program input and output data is also provided for the test signals
(a) prbs
(b) inverse - repeat, and
(c) prime sinusoids (Appendix 2.3).
The program incorporates six subroutines
(1) PRBS
(2) PRIMS
(3) FASTM
(4) SAHLD
(5) SSIML
(6) ANGLE
which are considered in the following sections. The calling sequence
for the respective subroutines with the argument lists given in
Appendix 2.5.
3.4.1. Subroutine PRBS
The subroutine PRBS computes a pseudo random binary maximum
length sequence of length 2n - 1 where n is in the range 3 to 10
inclusive.
An important feature of the subroutine is that it readily allows
(continued on page 44)
40
ENTER"~^\ FFRIP_)
READ INPUT PARAMETERS M =TEST SIGMTOPTION NN =OUTPUT SIGNAL OPTION ZETA -DAMPING FACTOR WN =NATURAL FREQUENCY TIM =TIME INTERVAL BETWEEN SAMPLES VIN ^AMPLITUDE OF TEST SIGNAL NPRIN =NO. OF ESTIMATES PRINTED NUMI ^SEQUENCE LENGTH NFACI =NUMBER OF FACTORS L___ =FArTOR<; DF NIIMT
M=l\
GENERATE PRBS. AND
SEQUENCE VECTOR
ARRAY
M=2i
GEN IN-RE & SEQ
VEC AR
f
IRATE 3 PRBS JENCE TOR RAY
' W&tXUSKSSSSS&it
GENEF PR-SK & SEQl VEC'
ARl
rSsnfflEsss*«razs
?ATE WSOID JENCE FOR <AY__«j
M=41 f
.
GENERATE SINUSOID &
SIGNAL j VECTOR
M=5 1
SPECIFIED SEQUENCE
VECTOR
Tf
PRINT TEST SIGNAL
DETERMINE SIGNAL MEAN
READ IN FROM CARDS SYSTEM OUTPUT DATA
GENERATE SYSTEMj OUTPUT DATA FOR j 2ND ORDER SYSTEM !
!<9«— -—»"•—— ———— - - - -- — -
PRINT SYSTEMOUTPUT DATA &
DETERMINE OUTPUTSIGNAL MEAN
CALL FASTMDETERMINE SPECTRAL
ESTIMATES FORSYSTEM INPUT
CALL SAHLD ADJUST SPECTRALESTIMATES FOR
£ERO-HOLD FUNCTION
CALL FASTM DETERMINE SPECTRAL
ESTIMATES FOR SYSTEM OUTPUT
CALL SAHLD ADJUST SPECTRAL ESTIMATES FOR
ZERO-HOLD FUNCTION
COMPUTE CROSS SPECTRAL ESTIMATE
CALL FASTM DETERMINE CIRCULAR CROSSCORRELATION
FUNCTION
CALCULATE FACTOR RELATING $ to h(r)
DO 70 K=1,NPRIN
DETERMINE MAGNITUDEOF SYSTEM
INPUT & OUTPUT
CALL ANGLE DETERMINE PHASE
OF SYSTEM INPUT & OUTPUT
DETERMINE TRANSFERFUNCTION
H(u) & h(x)
PRINT RESULTS
Fig. 3.3 FFRIP Program Flow Diagram
43
the user to specify the characteristic polynomial of the sequence.
The polynomial equation for a prbs generated with an n-stage register
is
(I 0 — © Dr © Dq © DP © Dm ) x = 0 3.1
where m, p, q, r represent those stages that are fed back through
modulo-two gates. (Fig. 3.4). The sign '©' denotes modulo-two
addition and the symbol D 1 is an algebraic operator, the effect of
which is to delay by i digits the variable it operates on.
The polynomial coefficients for the sequences generated by
subroutine PRBS are given in Table 3.1. Using this data, the
polynomial for a 9-stage register is given by
D9 © D4 © D® = I © D4 © D9 3.2
This means that the output of stage 4 and stage y are fed back to the
modulo-two gate. The coefficients for the registers are initialised
by means of DATA statements. To change the characteristic polynomial
of a sequence simply requires the changing of the DATA card for that
register generating the sequence (see Appendix 2.4). Davies gives
a complete list of irreducible polynomials up to 1Q degree.
Register Length
3456789
10
Polynomial
3, 1, 04, 1, 05, 2, 0.6, 1, 07, 1, 08, 7, 2, 1, 09, 4, 0
10, 3, 0
Table 3.1 Polynomial Coefficients used by PRBS
44
10 S
tage R
egister
tn
Feedback p
aths
activated b
y m-
sequ
ence
polynomial coefficients s
tore
d in
coe
ffic
ient
arr
ay.
Fig. 3.
4 10
Stage S
hift R
egister
Pseu
do R
ando
m Bi
nary S
equence
Gene
rato
r
3.4.2. Subroutine PRIMS
PRIMS generates.a signal which is given by
3 3°-°
Ngiving a real sequence X = E X(nAt)
n=l
The subroutine also provides an integer array output corresponding
to the index i including i = 0. The array is used both in the
subroutine and in the main-line FFRIP program either as a pointer to
the i frequency or to data corresponding to the i frequency.
3.4.3. Subroutine FASTM
FASTM computes the discrete Fourier transform of N data points
x , n = 0, 1, 2,""', N - 1, using the fast Fourier transform mixed
radix algorithm where N is assumed highly composite
(N-N, *N2 x N3 x..". y.
The transform of x is defined by
* N" 1 2 knx. = z xn exp(-j -n—) k = 0, 1, 2,""- N - 1 ~ . k n=Q n N d.4
The inverse transform is given by
M_l
3.5
and is evaluated by setting the subroutine argument SIGN to + 1.40 A number of fft programs are available in the literature ana
one of these, incorporating the mixed radix algorithm, was adopted
for use in this investigation with some minor modifications. There
46
are a number of publications which provide a detailed treatment of
fft algorithms and show why they are computationally so35 40 41 efficient ' ' . (Chapter 4 Table 4.7 gives a comparison of
computation times).
3.4.4. Subroutine SAHLD
The subroutine SAHLD adjusts the spectrum estimates obtained
from the dft to take account of the frequency characteristic of the
sample and hold function of the ADC and DAC. The program implements
eqn. 2.23 and determines
7* /2im x At sin (wl ft , . un w*,2^z > = — r — exP(-J ~N) Z *te)~ 3.6
where Z* () is the dft of N data points and Z*m () is the
modified transform, and the subroutine output.
3.4.5. Subroutine SSIML
SSIML generates a sequence y(k) which is the output of a second
order underdamped system to an input x(k) where the input sequence
is applied through a zero-order hold function. The equations
implemented are those derived in Chapter 2, eqns. 2.37 to 2.40.
3.4.6. Subroutine ANGLE
ANGLE calculates the phase in degrees of a two-dimensional
vector defined in cartesian co-ordinates.
3.5. Supporting Programs
In order to investigate alternative procedures for determining
the system characteristic, as well as providing a means of measuring
47
the "reliability" of the estimates a number of other programs were
written. In addition to this, alternative techniques were considered
within the same procedure. For example the dft was calculated using
either,
(a) dft algorithm
(b) radix-2 fft, or
(c) the mixed radix fft algorithm.
System identification was also performed using the method of correlation,
with the corresponding spectral estimates evaluated using the Fourier
integral. The listing for the subroutines are documented in Appendix
2.4 and the calling procedures in Appendix 2.5. Some of the programs
used will now be considered.
3.5.1. Additional Programs for Determining the Discrete Fourier
Transform
3.5.1.1 Subroutine TRIGF
Subroutine TRIGF computes one value of the Fourier transform by42 the trignometric formulae for the sum of sine and cosine terms. It
provides two outputs, S and C which are respectively the sine and
cosine transform at frequency <D rad/s. The outputs are defined by
n-1S = I x. sin uk
k=o K
n-1and C = £ x. cos wk
k=o k 3.7
The subroutine TRIGF was a replacement for the FASTM subroutine and
was used for some of the experimental work described in Chapter 4.
This subroutine proved more efficient in terms of computation time
48
than the fft algorithms when only a small number of frequencies
were of interest (refer to Chapter 4 Table 4.7).
3.5.1.2. Subroutine NLOGN
Subroutine NLOGN computes the discrete Fourier transform by the4? n log(n) or radix-2 fft method. This method requires that the data
consists of 2n values where n is some positive integer. The
subroutine calculates either the forward transform (eqn. 3.4) or the
reverse transform (eqn. 3.5) depending on the value of SIGN.
3.5.2. Program for the Determination of H(CJ) Indirectly
Indirect determination of the frequency response of a system
necessitates two distinct operations. Firstly, the determination of
the crosscorrelation function of the system input and output signals.
and secondly the evaluation of the Fourier integral of the
crosscorrelation function, assuming of course that the crosscorrelation
function approximates to the impulse response function of the system.
In order to implement this procedure subroutines CROSS and FINT were
developed.
3.5.2.1. Subroutine CROSS
CROSS computes the crosscorrelation function of two real numbered
vectors, evaluating eqn. 2.30. The number of points to be correlated
must be specified by the user.
3.5.2.2. Subroutine FINT
computes the Fourier integral for an arbitrary time
function, implementing eqns. 2.13 to 2.15. Simpsons rule is used
for solving the integral, consequently the number of data point must.
be odd.
49
3.5.3. Miscellaneous Programs
A number of other programs were developed and used, and they
included:
i. A parameter estimation program. This program is based on
the optimisation algorithm of POWELL and is used to
determine the transfer function coefficients of a system
model from the system estimates. A best model is definedk 2
as that which minimises the cost function z c where em=l
is the error between the measured estimates and the
estimates calculated by the theoretical model. The
algorithm requires an initial estimation of the coefficients
and then iterates until no further reduction can be made in
the cost function. The algorithm was also used to select a
pre-defined amplitude distribution for a test signal (see
Chapter 5). The program allowed a maximum of twenty
parameters to be optimised.
ii. A program that combined the operations of the HYBRID program
(3.3) and FFRIP (3.4). This program, because of the amount
of core required used a number of disc files, which were
employed for both temporary and permanent storage of data.
It allowed sequences of up to 1023 to be analysed, and was
used extensively for the experimentation work undertaken
in this investigation.
ill. An error analysis program that evaluated the mean-square
and bias errors for the measured estimates.
iv. A statistical analysis program which measured the mean,
standard deviation and amplitude probability functions of a
50
time function.
v. A program to evaluate the harmonics generated when
subjecting a nonlinear element to a composite sinusoid test
signal.
vi. Graph plotting programs for the CALCOMP PLOTTER.
3.6 Summary
In this chapter details of programs for both frequency and time
domain computational algorithms have been considered, and a fast
frequency response identification program has been presented.
The next and subsequent chapters will use these programs for the
identification of dynamic systems under different measurement
conditions.
51
CHAPTER 4
Experimentation and Error Analysis with
Linear System
4.1 Introduction
In this chapter the accuracy of the discrete Fourier transform
procedure when applied to a linear system is examined for different
system conditions and test signal spectral characteristics. The effect
of sequence length and clock period is investigated, and the dependence
of the estimation procedure on system dynamics, quantisation, random
and deterministic noise and fft computational algorithm is examined.
A detailed comparison is presented between the crosscorrelation
function and the fft methods of mechanisation by comparing the
parameter modelling estimates given by both procedures for different
system conditions. The theoretical results are shown with all the
frequency response illustrations for comparison purposes.
4.2 Experimental Arrangement
The dynamic system under test is a second order transfer function
of the form
2__
S2 + 2Cun S + cojj 4.1
and is simulated on the analogue computer. Figure 4.1 gives the
experimental arrangement used, including the scaled machine diagram.
The arrangement allows the damping factor g to be varied and for
noise to be added to the system output. For the experimental studies
the natural frequency o> was fixed at 10 rad/s and $ was set to 0.3
52
SET
BY POTENTIOMETER
INPUT
FROM
S
DAC
INPUT
TO
ADC
NOISE
INPUT
Fig. 4.1
Experimental Arrangement
except where otherwise stated.
The psuedo random binary test signal is generated by software
using subroutine PRBS and applied through a digital to analog
converter (DAC) to the system input. The sample and hold function
is provided by the DAC. The output response is sampled at the half
way point between *he epochs of the test signal at the prbs clock
frequency —p. The sampled data signals are then processed using the
program FFRIP to obtain the spectral estimates. Since the prbs signal
is generated within the digital computer the need for sampling the
input signal is removed. The analogue computer simulation is replaced
by the digital subroutine SSIML for some of the experimental work
undertaken. This proved necessary due to problems of amplifier
saturation arising from inadequate scaling, due to the fact that it
is not always easy to predict maximum values for an input signal which
is essentially random. This problem could have been overcome by
re-scaling the analogue simulation, but the alternative of digital
simulation is used when necessary, as this removed completely any
possibility of errors arising from inadequate scaling. Digital
simulation is used when measuring the spectral estimates of high
resonant second order systems, and when lower clock frequencies are
used. Digital simulation is not adopted for all the experimental
work, since one of the aims of this investigation is to assess the
errors arising from using the procedure on-line so that, the effects
of quantisation noise, random noise and deterministic noise could be
investigated. Discussion of errors arising from saturation is
presented in Chapter 6.
The mixed-radix fft algorithm is used in all experimental
54
studies with the exception of those discussed in Section 4.6.
4.3 Error Criteria
In order to assess the accuracy of the spectral estimates
quantitatively, root mean square error values and bias error estimates
are obtained for both magnitude and phase information. The criteria
used are
em =
A/z
4.2
kEbm=
bp = —————— R ————— 4.3
where Rt and R are the theoretical and measured amplitude ratiosl« III
respectively and <f>. and * are the theoretical and measured phases\f ill
respectively. The error criteria are evaluated over a frequency
range of twice the bandwidth. This means that for the case of a
system with £ = 0.3, and test signal parameters of N = 1023 and
At = 20 ms the number of spectral lines covered is 89.
55
4.4 Spectral Estimates under Noise-Free Measurement Conditions
4.4.1 Spectrum of prbs
Table 4.1 gives the spectral characteristics of a prbs sequence
up to spectral line 30 generated by a 9-stage shift register starting
full (all stages set to 1). It also includes the measured and
theoretical response estimates for a second order system when the
clock period was 20 ms. A complete response description including
both measured and calculated system responses in both frequency and
time domains up to line 249 is given in Appendix 3.
These results illustrate the accuracy of the procedure and
verifies as correct the implementation of the procedure. This
conclusion is also confirmed by the values obtained for the error
estimates (see Table 4.2 and entry under 5 = 0.3). An important
asset of the technique is seen in its ability to give a complete
response description with minimum measurement time.
4.4.2 System Dynamics
The ability of the dft procedure to determine accurately spectral
estimates for wide ranging dynamic conditions is investigated. A
sample of the results obtained for a range of values of £ are shown
in Fig. 4.2. The results demonstrate the ability of the technique to
measure accurately highly resonant and damped response modes. Table
4.2 gives the measured error estimates for ? in the range 0.05 to
1.0 and indicates that the errors are negligible within this range
even although they increase with reduced damping.
4.4.3 Sequence Lengthy
Figure 4.3 illustrates the system frequency characteristic
(continued on page 61)
56
Line No
01234567
J910111-2131415161718192021222324252627282930
Spectrum of prbs
R
0.0041.0001.0001.0001.0001.0011.0010.999
* (cleg)
0.0-72.625.663.0
-137.8-0.7-2.158.4
0.999 -104.71.000 34.50.999 130.70.999 138.20.999 -132.50.998 175.10.998 26.40.998 64.90.997 -38.50.997 -30.10.9960.9970.9960.9950.9940.9940.9930.9930.9930.9910.9900.9900.990
-161.010.633.5
-64.6139.284.5
-33.1-160.1
12.8-116.6-37.7135.599.1
System Measured Frequency Response
**
1.0001.0031.0121.0281.0521.0821.1211.1701.2291.3001.3831.4761.5751.6661.7311.7441.6921.5771.4271.2651.1130.9790.8620.7630.6790.6080.5470.4960.4510.4120.378
4> vm(deg)
0.0-2.1-4.3-6.5-8.9
-11.5-14.4-17.6-21.3-25.6-30.7-36.8-44.2-53.0-63.3-74.9-86.9-98.4
-108.7-117.5-124.8-130.7-135.6-139.6-143.1-145.9-148.4-150.4-152.3-153.8-155.2
System Theoretical Frequency Response ,
Rt *t (deg)
1.000 0.01.003 -2.11.013 -4.31.028 -6.51.0511.0821.121
-8.9-11.5-14.4
1.170 -17.61.229 -21.31.300 -25.61.383 I -30.71.4761.5741.6661.7311.7451.6921.5781.4271.2661.1140.9780.8620.7630.6790.6080.5470.4950.4510.4120.378
-36.8-44.2-53.0-63.3-74.9-86.8-98.4
-108.7-117.5-124.8-130.7-135.6-139.7-143.1-145.9 •!-148,3-150.4-152.3-153.9-155.3
Table 4.1 Measured and Theoretical Frequency Responses of Second
Order System Including Test Signal Spectrum
System Parameters :- £ = 0.3 <on = 10 rad/s
Test Signal Parameters:- prbs s N = 511, At = 20 ms
57
= 0.05 = 0,2
5 = 1 0
Fig. 4.2 Spectral Estimates for Different Values of Damping
Factor
System Parameters :- O.OSsCsl.O, un = 10 rad/s
Test Signal Parameters:- prbs, M=511, At=20 ms
58
Damping Factor
0.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00
Root Mean Square Errors
rm
0.00420.00290.00250.00220.00200.00190.00180.00170.00160.00150.00150.00140.00140.00130.00130.00130.00130.00120.00110.0010
e p
0.1383
Bias Errors
bm
0.00283
0.0622 0.002320.0518 0.002110.04480.04000.03700.03500.03360.03260.03160.03110.03050.03020.02980.02870.02850.02820.02700.02730.0264
0.001950.001820.001720.001630.001550.001460.001380.001330.001290.001210.001170.001100.001060.001020.000960.000920.00087
bp-0.0412-0.0115 ;-0.0129-0.0123-0.0134-0.0131-0.0121-0.0119-0.0108-0.0103-0.0100-0.0100-0.0096-0.0091-0.0100-0.0099-0.0095-0.0083-0.0080-0.0083
Table 4.2 Error Estimates (eqn. 4.2 and 4.3) for Second Order
System for Different Damping Factors
System Parameters :-
Test Signal Parameters:-
0.05^1.0; con =10 rad/s
prbs; N=511, At=20 ms, k=52
59
0 1 2 3 4 f 5--- ' -
DEG
234 f,,_ 5
N = 511 N = 255
0 1 2 3 *f50 1 Z 3
N = 63 N = 31
Fig. 4.3 Spectral Estimates for Different Sequence Lengths
System Parameters :- 5 = 0.3, u> n = lOrad/s
Test Signal Parameters:- prbs, 31$N$5"I1, At = 50 ms
60
obtained from test signal sequence lengths of 511, 225, 63 and 31, with
the clock period in each case set to 50 ms. It can be seen that
inadequate frequ^-.cy resolution is provided with N equal to 63 and 31.
Attempts to improve the resolution by reducing the clock frequency
would have resulted in aliasing corrupting the spectral estimates at
the higher frequencies (see Fig. 4.4).
4.4.4 Sampling Frequency
For this investigation the period of the sequence NAt is varied
by changing the clock frequency with the sequence length kept constant.
The sampling period is varied between 5 and 120 ms. This is done for
sequence lengths of 255 and 511. Specimen results of the spectral
estimates obtained are shown in Fig. 4.4 for the clock periods of
20 ms and 120 ms. For the sake of clarity only every fifth estimate
has been plotted for the cases of At = 120 ms. The effects of
sampling frequency on the error estimates are shown in Fig. 4.5a to
Fig 4.5d. It can be seen that the effects of aliasing become
pronounced with At greater than 60 ms and gets rapidly worse for
higher periods. Also, as one would expect the errors at lower
sampling frequencies are greater for the shorter sequence. From these
results it can be concluded that provided the sampling frequency is
greater than seven times the bandwidth of the system being tested then
aliasing will be negligible or totally absent. This rule will in
general be true for higher order systems as well as systems possessing
second order dynamics. The one exception will be with systems
possessing higher resonances.
(continued on page 67)
61
120 ms 511
At = 20 ms N = 255
At N
120 ms 255
Fig. 4.4 Spectral Estimates for Different Sampling Periods
System Parameters:- 5= 0.3, u = 10 rad/s
Test Signal :- prbs
62
a CM
to •o"
COo
o« ^
o
o o
0-00 0-02 0.04
seconds
N=255
0.12
Fig. 4.5a. Magnitude RMS Error Estimates for a Second Order
System as a Function of Sampling Period.
System Parameters:- E; = 0.3, u, = 10. rad/s
Test Signal :- prbs ^-49
63
mCD O
toO
O•
O
CM O
OO
CM O
C3-
N=255
N=511
0-00 0-02 0-04 0.06 0-08 0-10 0-12
^seconds
Fig. 4.5b. Magnitude Bias Error Estimates for Second Order
System as a Function of Sampling Period.
System Parameters:- 5 = 0.3, wn = 10 rad/s
Test Signal :- prbs k=tf,o
64
°0-00
Fig. 4.5c. Phase RMS Error Estimate? for a Second Order System
as a Function of Sampling Period
System Parameters:- £ = 0.3, u> = 10 rad/s
Test Signal :- prbs |<=40
65
N=255
0.02 0-04 0.06 0-OB 0-10 0.12. seconds
Fig. 4.5d. Phase Bias Error Estimates for a Second Order Syster
as a Function of Sampling Period
System Parameters:- E, = 0,3, <D = 10 rad/s
Test Signal :- prbs k=40
66
4.4.5 Time Position of Sample
As was concluded in Section 2.7.2. the time when sampling
commences is important since the spectrum of any time sequence is
dependent on the time position of the Input record. A waveform of
constant shape will always have the same energy, but how this energy
is distributed between the sine and cosine terms depends on the phase
shift of the time position of the sampled data. This is illustrated
in Fig. 4.6 where it can be seen that a linear phase error as a
function of frequency is present. This occurred by sampling the
system output signal at the epochs of the test signal e.g. by shifting
the time position of the input record by —Ev As expected the
magnitude ratio is constant as the energy in any line is the same
irrespective of the time position of the sequence.
4.5 Spectral Estimates Under Noisy Measurement Conditions
4.5.1. Quantisation Noise
This investigation covers the effect of introducing quantisation
noise by changing the quantisation resolution of the measuring analogue
to digital converter (ADC). A 12 bit ADC is used and the quantisation
level is determined by performing shift-right and shift-left
operations on the sampled data. Figure 4.7 illustrates both amplitude
ratio and phase characteristics of the system as estimated using the
procedure with N = 1023 and At = 20 ms for quantisation of 12, 9, 6
and 3 bits respectively. The degradation of the excellent results
obtained with 12 bit quantisation for reduced values of quantisation is
evident.(continued on page 70)
67
"•DEG
Fig. 4.6 Spectral Estimates with System Output
Sequence Incorrectly Time Positioned
System Parameters
Test Signal Parameters:- prbs, N = 1023,
At = 20 ms
:- E.= 0.3, to = 10 rad/s
68
01 2 3 4 f 5 q I 2 3 4 f 5
QUANTISATION = 12 BITS QUANTISATION = 9 BITS
QUANTISATION = 6 BITS QUANTISATION = 3 BITS
Fig. 4.7 Spectral Estimates for Different ADC Quantisations
System Parameters :•
Test Signal Parameters:- prbs, N = 1023, At = 20 ms
= 0.3 S uj = 10 rad/s
69
4.5.2 External Noise Sources
The effects on the spectral estimates of three noise sources
added to the system output before sampling are studied. These are:-
(a) "white" noise having a rectangular low pass spectrum with
a bandwidth of 27kHz.
(b) "pink" noise having a power spectrum decreasing at 3dB/
octave from 3Hz to 20kHz.
(c) "sinusoidal" noise at a frequency of 4.5Hz.
In all cases a signal to noise ratio of 10 was established and
each noise source is examined with respect to quantisation. Figures
4.8 to 4.10 illustrate the spectral estimates for quantisation of 12,
9, 6 and 3 for each of the noise sources and Fig. 4.11 gives the
corresponding error analysis results.
It is evident that in the absence of noise, quantisation of 6
bits or more is satisfactory. The results as assessed by the bias
and root mean square error estimates indicate that the "white"
noise and the "pink" noise produce a scatter which is almost uniform
throughout the spectrum and that due to the single sinusoid the
scatter is principally evident around the sinusoid frequency and in
fact peaks rapidly at this frequency (Fig. 4.10.),This confirms results
obtained in a field test application of the technique where the system
signal being examined had present a predominantly sinusoidal noise39 superimposed on the response produced by the prbs. It is also
observed when coarse quantisation is used (3 to 5 bits) that the
resultant errors become reduced in the presence of "white" noise which
is acting as a dither signal at this time Fig. 4.11.(continued on page 73)
70
QUANTISATION = 12 BITS QUANTISATION = 9 BITS
4,50 1 2 3 4
QUANTISATION = 6 BITS QUANTISATION = 3 BITS
Fig. 4.8 Spectral Estimates for Different ADC Quantisations
Showing the Effect of "Hhite" Noise
System Parameters :- 5 = 0.3, wn =10 rad/s
Test Signal Parameters:- prbs, N = 1023, At = 20 ms,
S/N = 10
71
P DEGQUANTISATION = 12 BITS QUANTISATION = 9 BITS
345012345
QUANTISATION = 6 BITS QUANTISATION = 3 BITS
Fig. 4.9. Spectral Estimates for Different ADC Quantisations
Showing the Effect of "Pink 15 Noise
System Parameters :- £ = 0.3, to = rad/s
Test Signal Paramters:- prbs, N = 1023, At = 20 ms,
S/N = 10.
72
QUANTISATION = 12 BITS QUANTISATION = 9 BITS
QUANTISATION = 6 BITS QUANTISATION = 3 BITS
Fig. 4.10. Spectral Estimates for Different ADC Quantisations
Showing the Effect of Sinusoidal Noise
System Parameters :- c = 0.3, wn = 10 rad/s
Test Signal Parameters:- prbs, N = 1023, At = 20 ms
S/N = 10
73
4-
X
NO N
OISE
SINUSOIDAL NOISE
PINK N
OISE
WHITE
NOISE
^3-00
4 -CTJ
s-cr
oB-OT
9-00
QU
ANTI
SATI
ON B
ITS
Fig.
4.11 a.
Magnitude
RMS
Error
Esti
mate
s for
Various
Noise
Forms
as a
Fun
ctio
n of Q
uant
isat
ion
System P
aram
eter
s :-
5
= 0.3, un
=
10 r
ad/s
Test S
ignal
Para
mete
rs:-
prbs,
N=10
23,
At =
20
ms,
k =
89
en
A
NO N
OIS
EO
SINU
SOID
AL N
OISE
•f
PINK N
OISE
X
WHITE
NOISE
3-G
TJrcr-co
H-co
QUANTISATION B
ITS
12-GO
Fig.
4.lib.
Magn
itud
e Bi
as Er
ror
Estimates
for
Various
Noise
Forms
as a
Function
of Q
uantisation
System P
arameters
:- ?
= 0.3, «
„ =
10 r
ad/s
Test S
igna
l Parameters:- prbs,
N =
T023
, At
= 2
0 ms
, k
= 89
A oNO N
OISE
SINUSOIDAL N
OISE
PINK N
OISE
WHITE
NOISE
CTt
4 O
T5-0
0fr
-OO
1~£O
8^00
9-00
QU
ANTI
SAlM
^IT
SH
-00
12-C
t)
Fig. 4.lie
Phas
e RM
S Error
Esti
mate
s fo
r Various
Nois
e Fo
rms
as a
Function
of Q
uant
isat
ion
Syst
em P
aramet
ers
:- ?
= 0.3, u
> =
10 r
ad/s
Test S
igna
l Pa
rame
ters
;:-
prbs,
N =
T023,
At =
20
ms,
k =
89
A
NO N
OISE
O
SINUSOIDAL N
OISE
+
PINK N
OISE
X
WHITE
NOISE
9-00
10
.00
11-00
QUANTISATION BITS
Fig. 4.lid
Phase
Bias Error
Estimates
for
Various
Noise
Forms
as a
Function o
f Quantisation
System Parameters
:- £
= 0.3, j
= 10 r
ad/s
Test S
ignal
Parameters:- prbs,
N =
1023,
At =
20 m
s, k
= 89
4.5.3. Multiple Period Averaging
It is known that the effect of Gaussian noise on the measurement
process can be reduced by the use of multiple period averaging. The
benefits of this filtering process are investigated by averaging the*
sampled output data Z (t) over k complete periods. The sampled data
was averaged in preference to averaging the dft spectral estimates as
the latter would have required excessive computer time without
yielding any additional benefits. This is because for any N time
points exactly N values are obtained in the spectrum, and since no
new information about the signal is added by the dft, each spectral
line will have no more statistical certainty than the sample points in
the time domain from which the spectral line was computed.
The spectral estimates for four different averaging periods are
shown in Fig. 4.12 and the error criteria results are given in Fig.
4.13 for a white noise source with a signal/noise ratio of 5. The
results confirm that the effect of noise on the estimates can be
considerably reduced by multiple period averaging.
4.5.4 Theoretical Estimation of Noise Errors
It has been shown experimentally that the effect of discrete
operations such as quantisation and the corruption of the system
output signal by external noise sources result in errors in the system
estimate. Barker and Davy have shown, quoting the work of Bendat
and Piersol, that when the noise signal is Gaussian and contains no2 systematic component with frequency |N|J has a chi-squared
distribution with two degrees of freedom and in the absence of
aliasing , has an expected value of(continued on page 84)
78
AVERAGING INDEX = 1 AVERAGING INDEX = 4
AVERAGING INDEX = 8 AVERAGING INDEX = 10
Fig. 4.12 Spectral Estimates with White Noise Added to System
Output. Showing the Effect of Periodic Averaging
System Parameters :- € = 0.3, w = 10 rad/s
Test Signal Parameters:- prbs, N = 1023, At = 20 ms,
S/N = 5
79
o U7
03 O
to
en o
n uo
CM o CO °iT
oT2.00
3.0C
A.00
S.OO
6.00
7.00
8-00
9.00
10-00
11-0
0AV
ERAG
ING
INDEX
(k)
Fig. 4.13a
Magnitude
RMS
Erro
r Estimates
for
White
Nois
e as a
Function
of M
ultiple
Period A
veraging Index
System P
arameters
:- 5
= 0.3, w
=
10 r
ad/s
Test S
ignal
Parameters:- p
rbs, N=1023,
At =
20
ms,
k =
89,
S/N
= 5
00
9.00
10
-00
11.0
0 AVERAGING
INDEX
(k)
Fig.
4.13b
Magnitude
Bias
Error E
stimates f
or W
hite N
oise
as
a Function o
f Mu
ltip
le P
erio
d Av
erag
ing
Index
Syst
em P
aram
eter
s :- 5
= 0
.3,
o> =
10 r
ad/s
Test S
igna
l Pa
rame
ters
:- p
rbs, N
= 1023,
At =
20
ms,
k =
89,
S/N
= 5
CO ro
"VOO
2.00
3.00
4.00
5.00
5-00
7.00
8.00
9-00
10-00
11-00
AVERAGING
INDEX
(k)
Fig. 4.13c
Phase
RMS
Error
Estimates
for
White
Noise
as a
Function
of M
ultiple
Period A
veraging Index
System P
arameters
:- %
= 0.3, u
-
10 r
ad/s
Test S
ignal
Parameters:- p
rbs, N
= 1023,
At =
20
ms,
k =
89,
S/N
= 5
o
o C3. I
.00
O
(O o_
I
9-00
10-00
11-00
AVERAGING
INDEX
(k)
Fig. 4.13d
Phase
Bias
Err
or E
stim
ates
for
White N
oise a
s a
Function o
f Mu
ltip
le P
erio
d Averaging
Inde
x Sy
stem
Parameters
:- £
= 0.
3, o
> =
10 r
ad/s
Test S
igna
l Parameters:- p
rbs, N
= 1023,
At =
20
ms,
k =
89.
S/N
= 5
4.4where
Nk is the discrete Fourier transform of the sampled noise
sequence n
and
Pnn(w) is the bilateral power spectral density function of n(t).
The result enabled Barker and Davy to arrive at the following
conclusions:-
(a) The expected value of |N^| is inversely proportional to the
test signal input amplitude 'a' and At 2 but virtually
independent of N for cases of interest.
(b) If the error due to noise E. , is considered to be a random
complex variable, with constant magnitude |E. | and random
phased, then the greatest error in the magnitude estimate
occurs when the phase estimate is small, and the greatest
error in the phase estimate occurs when the error in the
magnitude estimate is small. This conclusion assumes that
the system estimates are greater than the noise-level
estimates.
The experimental results obtained when random noise was added to the
system output (Fig. 4.8, 4.9) indicates the trend that is stated in
(b), thus confirming the general conclusion.
4.6. Comparison of Procedures
4.6.1. Radix-2 fft
In an attempt to benefit from the computational speed of the
radix two algorithm the data record of length (2n - 1) is extended
84
to 2 by attaching one more sample to both the sequence and the
system output. The mixed-radix transform program is replaced by the
appropriate radix-2 program. Table 4.3 gives the errors resulting
from using this method for three different cases, namely attaching
either a 1, -1 or 0 to the pseudo random binary sequence. The
degradation of the frequency response function Is clearly shown in
Fig. 4.14 with the error criteria increased by a factor of 10 when
compared with the use of the mixed-radix algorithm. It can also be
seen that the degradation increases with increase in frequency. This
is as one would expect, as the degree of mismatch between the spectral
lines of the prbs clock frequency and the sampling spectral lines of
the radix-2 fft increases linearly with frequency, so that the
spectral estimates are increasingly affected by the distorted convolved
frequency function with increased frequency. This distortion, as
reflected by the spectral estimates, will become more pronounced with
shorter sequences so that to reduce distortion over the relevant
frequency range the sequence length should be increased.
The radix-2 algorithm when used on data that had been corrupted
with white noise gave spectral estimates as shown in Fig. 4.15. The
performances of the two algorithms are evaluated by comparing the
error criteria for the same data set and the results are as shown in
Table 4.4. These show increased degradation when the radix-2
algorithm is used. However, this is only marginal and suggests that
for noisy data the radix-2 fft could be used, without the need for
matching the periods of the prbs and the fft provided N is large.
85
2nth data point added
1-1
0
Root Mean Square Errors
7m
0.0400
0.0292
C.0318
£P
3.60
5.36
4.90
Bias Errors
bm
-0.006
0.0121
0.008
bp0.752
0.191
0.411
Table 4.3 Errors Arising from Using Radix-2 fft on Extended prbs data
System Parameters:- £=0.3, un=10 rad/s
Test Signal Parameters:- prbs; N=1024, At=20ms, k=89
fftalgorithm
radix-2
mixed-radix
N
1024
1023
Root Mean Square Errors
S,
0.058
0.027
ep
6.5
5.06
Bias Errors
bm
0.056
0.049
BP-0.30
-0.40
Table 4.4 Comparison of Radix-2 and Mixed-Radix fft for Noisy Data
System Parameters:- £=0.3, un =10 rad/s.
Test Signal Parameters:- prbs, N=1024, At=20 ms, S/N = 10,
k=89
86
csr
2"th DATA = 1 2nth DATA = -1
2nth DATA = 0
Fig. 4.14 Spectral Estimates Using Radix-2 FFT Showing the
Effect of Adding 1, -1. 0 to the 2nth Data Point
System Parameters :- £ = 0.3, con = 10 rad/s
Test Signal Parameters:- prbs, N=1024, At = 20 ms
87
''DECMIXED RADIX N=1023
RADIX-2 N=1024
Fig. 4.15 Spectral Estimates Using Radix-2 and Mixed Radix
fft with White Noise Added
System Parameters :- c = 0.3, con = 10 rad/s
Test Signal Parameters:- prbs, At = 20 ms, S/N = 10
88
4.6.2. Model Estimates
A comparison is made of four procedures used for obtaining the
system characteristic by comparing the parameter estimates derived
from the experimental spectral estimates for each procedure. This is
done for a second order system under different measurement conditions.
The procedures used were, for the frequency domain,
(a) discrete Fourier transform
(b) Fourier integral of the crosscorrelation function
and for the time domain
(a) direct crosscorrelation of the input and output time
sequences
(b) indirect crosscorrelation through the discrete Fourier
transform.
Figure 4.16 illustrates the system characteristic obtained from
using these procedures.
The procedure used to determine the model estimates incorporates
Powell's optimisation procedure and the cost functions minimised were,
in the frequency domain,
Ff = I1 = 1
and in the time domain,
4.6
with k set to 89 data points, and h t (t) and hm (t) are the theoretical
and measured impulse responses respectively.
89
0 I 2 3 4 f 5 2 3 4 f. 5
DISCRETE FOURIER TRANSFORM FOURIER INTEGRAL OF $XY
°'6 t seconds l>2
DIRECT CROSSCORRELATION
0 0-6. . 1-2 t secondsCROSSCORREI1ATION VIA DISCRETE FOURIER
TRANSFORM
Fig. 4.16 System Characteristic for Second Order System Using Different Procedures
90
The estimates obtained for quantisation of 12 and 5 bits and for the
"no noise" and "white noise" cases are shown for the frequency and
time domain models in Table 4.5 and 4.6 respectively. The cost
function values Ft and Ff are also included as they give a measure
of the "goodness" of fit. It can be seen from Table 4.5 and 4.6
respectively that for the ideal measurement situation of 12 bits
quantisation and with no measurement noise the model estimates for the
different procedures compare with each other and show excellent
agreement with the simulated model. The same can be said for the case
of quantisation of 12 bits and a signal/noise = 10, however with
quantisation reduced to 5 bits the frequency-domain approach of the
dft gives more accurate estimates of the system model than those of
the time-domain method.
4.6.3. Computation Time
Table 4.7 gives the approximate times for computing the frequency
and impulse response estimates for different procedures and sequence
lengths with prbs as system input. The time advantage of using the
radix-2 fft is readily seen, with the time differential increasing with
sequence length. It can also be seen that the mixed-radix fft can be
used to advantage to reduce the computation time required for
computing the crosscorrelation function. It was shown in Section 2.7
that the crossccrrelation function evaluated by the dft gives a
circular correlation function which means that the input of the system
tested must decay to zero by the delay time (JJAt) otherwise $ will
be in error. In order to reflect this requirement the comparison in
Table 4.7 is made between ^ delay times for direct correlation with N
(continued on page 95)
91
System Conditions
No Noise Quantisation = 12
No Noise Quantisation = 5
White Noise S/N=10 Quantisation = 12
White Noise S/N=10 Quantisation = 5
£
0.29881
0.30045
0.29064
0.30692
"n
10.0086
10.0186
10.0121
9.9914
Ff
4.8591
1.9923 x TO3
2.268 x TO 3
3.481 x 10 3
(a) discrete Fourier transform
System Conditions
No Noise Quantisation = 12
No Noise Quantisation = 5
White Noise S/N=10 Quantisation = 12
White Noise S/N=10 Quantisation = 5
5
0.29916
0.2983
0.2968
0.2986
wn
10.0223
10.0368
10.026
10.007
Ff
0.2818
1.1963 x 102
1.574 x 102
3.401 x 102
(b) Fourier integral of 4.
Table 4.5 Model Estimates from Frequency Response Data for Different
Procedures
92
System Conditions
No Noise Quantisation = 12
No Noise Quantisation = 5
White Noise S/N=10 Quantisation = 12
White Noise S/N=10 Quantisation = 5
€
0.30024
0.32026
0.29796
0.31981
un
10.02136
10.02430
10.02332
10.0104
Ft
0.052335
1.2283
0.404112
1.3999
(a) direct crosscorrelation
System Conditions
No Noise Quantisation = 12
No Noise Quantisation = 5
White Noise S/N=10 Quantisation = 12
White Noise S/N=10 Quantisation = 5
e
0.29931
0.31926
0.29699
0.31873
wn
10.0195
10.0224
10.02128
10.00889
Ft
0.049469
1.15261
0.406477
1.39966
(b) indirect crosscorrelation through dft
Table 4.6 Model Estimates from Impulse Response Data for Different
Procedures.
93
Tech
niqu
e
dft
fft
fft
mixed
radix
No.
of
ops.
P N2
N Lo
g2N
m N
E r.
1=1
n
N=l 023/1 024
Approx.
P 106
104
4.5
x 10
4
Approx.
Rela
tive
Time 100 1 5
N=511/512
Approx.
P
2.6
x 10
5
4.5
x 10
3
4.0
x 10
4
Approx.
Relative
Time 58 1 9
N=255/256
Approx.
P
6.5
x 10
4
2 x
103
6.4
x 10
3
Approx.
Rela
tive
Time 32 1 3.2
N=63/64
Approx.
P
4 x
103
378
819
Appr
ox.
Rela
tive
Time 10 1 2.2
(a)
Operations required t
o determine
frequency
response
Technique
direct
correlation
fft
radi
x-2
fft
mixe
d radix
No.
of
ops.
P
(N/2
)2
3 N
Log2
N
m 3
N I
ri1=
1
N=l 0
23/1
024
Approx.
P
2.6
x 10
5
3 x
104
13.5 x
104
Appr
ox.
Rela
tive
Time 19 1 4.5
N=511/512
Appr
ox,
P
6.5
x 10
4
13.5 x
103
1.2
x 10
5
Approx.
Rela
tive
Time 5 1 9
N=255/256
Approx.
P
6.5
x 10
4
6 x
103
19.2 x
10
3
Approx.
Rela
tive
Time 10 1 3.2
N=63/64
Approx.
P
1.00 x
10
3
1.1
x 10
3
2.5
x 10
3
Approx.
Relative
Time 0.9
1 2.2
(b)
Oper
atio
ns required t
o determine
impulse
response
Table
4.7
Comparative
Time
s Required f
or C
ompu
ting
the
System
Characteristic f
rom
Syst
em R
esponses t
o Pseudo R
andom
Binary S
igna
ls.
delay values for the dft. When the same algorithm is used an
important point to observe is that these times are considerably
greater than those required to obtain the system frequency response
estimates by the dft. They also indicate that it might be
advantageous to apply the method in reverse by taking the inverse
discrete Fourier transform in cases where an estimate of the system
impulse response is required.
4.6.4 Measurement Time
The total measurement time with prbs as the test signal input is
equal to twice the sequence period. One complete sequence is used to
initialise the tested system with the measurements taken on the second
sequence. A comparison was made between this method of system
identification and that of the frequency response analyser. A modern
frequency response analyser, for example the Solatron 1172, provides
full programming capability with the user specifying the lower
frequency, the higher frequency and the incremental frequency. The
measurement sequence of the frequency response analyser is divided
between a delay time and measurement time. At the start of each
measurement sequence the measurement is delayed by a time t-|, to enable
the transients caused by the change in input to decay in order to
obtain a valid result. The time t-|, can be set to O.ls, Is, 10s, and
100s. The actual measurement time at frequencies below lOHz is
completed in one cycle of the output waveform if minimum integration
time is selected. Above lOHz, the measurement time varies between
O.ls and 0.2s depending on the frequency, so that the measurement
time Tm for fn <10Hz is
m 14.7
95
where n is the number of measurements taken and f. is the frequency at
the i measurement. If t^ =0.1 and the lower frequency f, = 0.054Hz
and the higher frequency fh = 5.0Hz with N measurements taken at
equi-spacod frequencies, the computed measurement times are as shown
in Table 4.8. These values compare with a measurement time of 40s for
a prbs of length 10?3 and clock period of 20 ms. The break even mark
is seen to be around 30 measurement points which would give, with
equi-spaced frequencies selected, inadequate resolution around
resonant frequencies. In order to improve the resolution more
measurements would need to be taken around the resonant frequency which
would lead to increased measurement time. A further consideration
which makes the time of Table 4.8 optimistic is that with the frequency
response analyser additional time is required for noting the results
and changing the frequency.
Compu
No. of measurements
points N
90 80 70 60 50 40 30 20 10
Time seconds
103 92 82 72 62 52 43 34 25
ted Frequency Response Analyser Measurements
Taken at Equi-Spaced Frequencies
System Parameters:- c = 0.3 o)n = 10 rad/s
96
4.7 Conclusions
It has been shown how the advantages of the fast Fourier
transform may be applied to the procedure for obtaining system
frequency responses using prbs. In this way gains are made in the
reduction of test time and in reduced computational requirements.
The errors introduced by the presence of noise of various forms
have been demonstrated and results obtained show how these errors
can be minimised with careful selection of quantisation, sampling
frequency and sequence length.
So far only linear systems have been considered. The following
chapter considers the effects of system nonlinearities on the spectral
estimates given by the dft procedure using prbs as system input.
Errors are evident and prompted by these, an analysis is given from
which a new form of test signal is evolved.
97
CHAPTER 5
Derivation of New Test Signal
5.1 Introduction
The experimental results presented so far have been restricted to
those obtained from linear systems. This chapter will illustrate the
difficulties encountered in obtaining meaningful system estimates when
using the fast frequency response procedure to test nonlinear systems.
A new test signal is proposed which will extend the usefulness of the
procedure to include a wide range of systems that exhibit pronounced
nonlinear characteristics.. Properties of the new signal are also
presented.
5.2 Illustration of the Problem
To identify the difficulties encountered in attempting to
determine frequency responses of an essentially nonlinear system
consider the results shown in Fig. 5.1. These are frequency response
measurements made on an electro-mechanism* forming part of the
shipborne elevation control system, firstly as assessed by a
conventional frequency response analyser and secondly, by direct
analysis (via the Fourier integral) of the response to a pseudo random
binary sequence. It can be seen that the frequency response analyser
gives a satisfactory measurement of the frequency characteristic of
the system. However, it is evident that the procedure using prbs is
unsatisfactory, the primary reason for this being the harmonics
generated by the nonlinearity and introduced into the frequency
*Elevation Servo Unit: MU35A No. 1258214Makers: Evershed & Vignoles Ltd.,
Servo Amplifier : MK146AAMakers: Hartley Electromotives Ltd.
98
O
CM
R o CM'
o
to o CO o
o
co o
prbs
m
A FR
A
<D<D
<D00
^
A
^
CD
CO
&
*
A
V
o o
A o
AO
A
O
Q
CDO
CD
O
^
w o
§3.0
0 I'.
OO
2
.00
3"
.00
4.00
5
.00
,
6-00
7"
.00
THz
Fig
. 5.
1 Fr
eque
ncy
Resp
onse
s of
Ele
vatio
n C
ontro
l Sy
stem
usi
ng F
requ
ency
Res
pons
e
Ana
lyse
r an
d Ps
eudo
Ran
dom
Bin
ary
Sequ
ence
s
response estimates.
Another illustration of the difficulty of applying the technique
to nonlinear systems is shown in Fig. 5.2. This time the system
consists of a D.C. Servo motor with a large amount of backlash (20°)
and Coulomb friction in the mechanical drive. The results obtained
were from closed loop measurements. It can be seen that the estimates
evaluated are unsatisfactory in that they provide no 'meaningful 1
measure of the system characteristic. Table 5.1 gives a measure of
the harmonic content determined from the single sinusoidal test
signal of the frequency response analyser and is sufficient to suggest
that the harmonics may in some way be contributing to the errors and
scatter obtained using the fast frequency response procedure.
Superimposed on Fig. 5.2 are the results obtained over a wide frequency
range using the frequency response analyser.
n
MnxlOO "?
1
100
2
0.48
3
11.6
4
0.49
5
4.34
6
1.46
7
2.4
8
1.46
9
2.8
Mn = amplitude ratio of the n harmonic
Table 5.1 Harmonic Content for DC Servo System with Backlash and
Coulomb Friction at Resonant Frequency (2.59 Hz) Measured
by Frequency Response Analyser
No attempt has been made at this stage to match statistically
the characteristics of the two form of stimuli as it is sufficient for
the purpose of this section merely to illustrate the problems
associated with rapid frequency response determination techniques. The
100
o.ooo (Pto
2.00 4.00 6.00fHz 8.00
o«<*•_
I
o o
o o
CSI.i
to
o o
•
•«*• OOJ
CS&jQP^O
A O O A
O
O
OCD 0
0
CD©x
CD CDo
O
©° CD
O
oFig. 5.2 Frequency Responses of DC Servomotor with Backlash and
Coulomb Friction using prbs and Frequency Respojisej\na}y^er 101
following sections will show that significant improvement can be
obtained by changing the form of the test signal, whilst still
retaining the considerable advantage in test time.
5.3 Analysis of Polynomial Nonlinearities
Nonlinearities can be divided into two groups: those with
'memory 1 (e.g., hysteresis) and those without such a property
(e.g. saturation). It is convenient from the analytical point of
view to consider in this section the latter type only. From the
analytical point of view, it is more convenient to consider the
latter type. Although the analysis is undertaken on this basis, it has
then been found that the signal so derived has been applied with equal
success to systems with memory-type nonlinearities.
Assume that the form of the nonlinearity is described by a
polynomial expression:-O O ri
y = aQ + a,x + a2x + a.o< + —— + a xp 5.1
where x is the input signal to the nonlinearity and y the resulting
output.
Suppose that the input signal (x) is made up of sine waves
having discrete frequencies (o>.) which are some integer multiple of
the fundamental frequency (u-j) and which are all in phase.
Then: -N
x = I sin(u>.t) 5.2 1
102
substituting 5.2 in 5.1 yields
N N N y = aQ + a.j E sinc^t + a 2 E E sinw,.t sinu.t + ——
N N N
D • i • i -i 1 i n1=1 0=1 P=i P 5>3
Using the trignometric identities for sine products and letting
cosmz( e i' ej' —-ep ) = cos h* + "j* +. — -y^ 5 4and
'"*- ' J r I J r (- i
where the argument on R.H.S. contains m negative terms and
z = 1,2 —— number of permutation.
It can be shown (Appendix 4) for the p term of the polynomial
nonlinearity that with p even -
a^ r N N N N r £ p^-lf * .1 E — *[(-l) Z
Pcm \ PC^ £ ^.^o /o rt _ ... n ^J /_! \P' *-
,p -
n7ii r"Z:
5.6
and with p odd -
a pX
r N N N N P
a r N N N N r= -P- f E E E E | (-12p"Hi=j j=i k=i p=i
P Cm
m=0
5.7
Examination of these relationships yields the following:-
(a) provided the test signal (x) consists of frequencies at
odd multiples only of the fundamental frequency, then there
103
can be no harmonic corruption at these frequencies
produced by even power nonlinear elements.
(b) odd power elements generate odd harnonics, hence errors
from this source remain. However, the influence of the odd
power elements can be reduced by restricting the allowable
frequency content of x to those frequencies which are prime
number multiples of the fundamental, which is itself removed.
Conclusion (a) confirms that an antisymmetric pseudo random sequence,
for example an inverse repeat sequence, should be better than the
normal prbs, and conclusion (b) suggests that by further restrictions
on the frequency content the effect of odd power nonlinear elements
can be reduced.
From the above conclusions the test signal
1 ky = -p- z s i nw. tk i4l 1
if 2 i=prime 5.8
is proposed as suitable for testing systems with pronounced
nonlinearities. For convenience this new signal will be called prime
sinusoid signal.
5.4 Properties of Prime Sinusoid Signal
The investigation so far presented suggests potential advantages
for the prime sinusoid signal in the testing of nonlinear systems.
It is also of importance that whilst it is proposed to discard
the prbs signal, the processing to be carried out is identical to that
used with prbs with its well known noise rejection capabilities. These
features will be examined in this section together with an investigation
104
of the statistical properties of the prime sinusoid signal.
5.4.1 Harmonic Rejection
The improvement over prbs in the harmonic rejection properties
of the new signal can be judged from the results of Table 5.2. The
table composes, as a line harmonic improvement ratio, the harmonic
contribution at particular line frequencies produced by a cubic
nonlinearity subjected to prbs, inverse repeat prbs, a 21 prime
sinusoid signal and a 10 prime sinusoid signal at frequencies
selected from the first 21 primes. In the case of normal and inverse
repeat prbs the signals were assumed to be band limited having a cut
off frequency at the 21st prime line. It can be seen from Table 5.3
and 5.4 that in order to minimise the number and power of the
harmonics generated, the test signal should contain few sinusoids.
Unfortunately, the number of sinusoids chosen is governed by the
number of points required for satisfactory description of the
frequency response characteristic.
In order to ascertain a measure of the extent of harmonic
distortion (in terms of the number of harmonics generated) as a
function of the number of sinusoids making up the test signal, a test
signal consisting of from 3 to 21 prime frequencies was applied to a
cubic nonlinearity. The results are shown in Fig. 5.3.
The data was fitted to three mathematical models, which were;
(a) an exponential function,
(b) a square function,
and (c) a cubic function of the number of prime frequencies k,
making up the test signal. (continued on page 110)
105
LINE NO. J
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
LINE HARMONIC RATIO
A. prbsJ
A. prime-21J
21
14
38
19
30
21
30
24
26
28
29
26
28
27
28
28
26
24
28
23
22
A- inv.rep.J
A- prime-k:1J
5.3
3.4
9.6
4.6
7.5
5.4
7.6
6.1
6.6
7.3-
7.4
6.8
7.0
7.0
7.2
7.2
6.7
6.3
7.2
6.0
6.0
A- prime-10J
A. prime-2TJ
0.59
-
0.37
-
0.43
-
0.26
0.19
0.13
-
-
-
-
0.19
0.18
-
0.16
-
0.20
-
——— -ju ————Input for 21 prime signal X(t) = y sin^t; Output = z A-sinu-t
.ill Y(t) J=l J
Table 5.2 Harmonics Generatedi=prime
by Cubic Nonlinearity Represented asan Harmonic Ratio of 21 Prime Signal for prbs-Normal and
Inv.Rep. and 10 Prime Signal 106
LINE No.j
13579
1113151719212325272931 33353739414345474951535557
AJ-36
66165
8195
249180146309240182342258210366345 251363369254417402252414389251396366248
LINE No.j
59616365676971737577798183858789 9193959699
101103105107109111113115
»j3783962313123842183063481742102821381351868966
12041156930
-18
21-12-54-24-51-81-42
LINE No.j
117119121123125127129131133135137138141143145147 149151153155-57
159-67
163165167169171173
ftj-57-102-75-67-i20-90-61-135-111-66-138
104-79-141-132-78 -129-138-84-111-129-70-99-114-57-72-93-54-63
LINE No.j
175177179181183185187189191193195197189201203205 207209211213215217219221223225229231233
I
-72-43-51-63-37-39-51-27-33-45-18-21-38-16-15-27 -12-9-21-13-3-12-10-3-6-6-3-3_i
J —————— , —— __ —————————————— - ^ Input X(t) = " s-Inu-t Output Y(t) = >E A^sin^t
i=prime j=l 141
Table 5.3 Harmonics Generated by Cubic Nonlinearity
107
LINE No.j
13579
1113151719212325272931333537394143454749515355
AJ-534
723996249257132610144191617960932
1986705
101423131602156222652094125032672700181532012304200637022532
LINE No.j
57596163656769717375777981838587899193959799
101103105107109111
AJ2008414333602601384634262633461443262772425141612940547238103251563.142753686506452323531615056913555616254033953
LINE No.j
113115117119121123125127129131133135137139141143145147149151153155157159161163165167
AJ
6750519040656495578745716516644745687578650743477389689748177857648547648187719754757590788453398217818150258844
Input X(t) =i=prime HI
sinu>.jt Output Y(t) = _J '
j T £ Table 5.4 Low Frequency Harmonics Generated by Cubic Nonlinearity
108
o o0_
a oo. t-
a. to
APPROXIMATES TO
y = 178 + 0.6774 k'
CJz
Oo
o Io
o o
"VTob .OO 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20-00 22.00
NUMBER OF PRIME FREQUENCIES IN TEST SIGNAL (k)
Fig. 5.3 Harmonics Generated by a Cubic Nonlinearity as a Function
of Number of Prime Frequencies in Test Signal
109
The best fit was given by a cubic model of the form:-
y = 178 + 0.67746k3 5.9
for k in the ranye 3gk$21.
The significance of this model is that it provides a measure of the
extent of spectral distortion expected, when testing nonlinear systems
using a prime sinusoid signal. This estimate provides only a guide
since few, if any, nonlinearities met in engineering systems fall
strictly into the cubic category. Also, systems invariably behave as
low-pass filters so that the higher frequency harmonics will be
attenuated.
A prime sinusoid signal of 21 frequencies was used for most of
the experimental work presented in subsequent chapters, although in
some areas of relevance the sequence of 169 prime lines was used.
5.4.2 Spectral Distribution
Already indicated in the previous section was the flexibility of
changing the signal spectral characteristics and the influence of test
signal badnwidth on harmonic corruption when testing nonlinear systems.
Which ever bandwidth is selected, the spectrum has a rectangular
envelope which, whilst not being essential if some alternative
envelope shape is desired, enables simpler computation than with prbs
with its ever changing envelope. Further, with regard to the phase of
the sinusoids making up the test signal, there is ready simplification
in comparison with prbs. In the prime sinusoid test signal the
sinusoids are given the same relative phasing. As with prbs, the
actual location of the spectral lines is most readily and easily
varied, by changing the clock'rate.
110
5.4.3 Amplitude Probability Distribution
Having selected the rectangular spectrum level, the zero phasing
and the number of frequencies making up the test signal, the signal
itself can be analysed with respect to its ampntude-time and
amplitude probability density function. For the signal of 21 lines
mentioned previously,the former is shown in Fig. 5.4a and the latter
is shown in Fig. 5.4b. With the number of signal prime frequencies
increased to 169 the amplitude probability distribution is as shown
in Fig. 5.5. This result suggests as the number of frequencies is
increased the distribution approaches that of a Gaussian distribution.
5.4.5 Autocorrelation Function
The test signal is no longer prbs and hence will not have the
autocorrelation function required for impulse response estimation by
crosscorrelation between system stimulus and response. Appendix 4
contains the general derivation of the expression for the
autocorrelation function of the prime sinusoid which is:-
N a, 26 YV (T) = Z -i- costo-t 5.10 XX i T 1 i 1
if2 i=prime
where a. and 03. are the amplitude and frequency of the component
associated with the prime number i. The form of equation 5.9 is shown
graphically in Fig. 5.4c.
5.4.6 Noise Rejection Capability
Table 5.5 illustrates the accuracy of the prime sinusoid technique
in the absence of noise. Reference 45 contains an application of
rapid frequency response measurement using prbs to a system with a
(continued on page 117)
111
ro
Fig. 5.
4 Ch
arac
teri
stic
s of
21
Prime
Sinusoid S
igna
l (a)
Time
History
€1.1
RELATIVE PROBABILITY 15-00 20.00 25.00
HI
AUTOCORRELATION FUNCTION-0
... ....... i.40-0
., .. j.200..00o. - i200.i40
-&•XX
H
TJ
tn
o-JSUnr* ft>-I
nt/5o-h
fD
CO _1,3cCA O _J.Q.
to
Oo o -s
orfr _i.
10
-0.12 -0.08 -0.04 °0 00 AMPLITUDE
0.04 0.08 0.12
Fig. 5.5 Amplitude Probability Distribution of 169 Prime Sinusoid
Signal
115
LineNumber
2
357
111317192329313741434753596167717379
TheoryM <fr
1.0001.0041.0191.0411.1111.1621.3071.3921.5971.7191.6251.1390.8660.7600.5990.4340.3330.3050.2440.2140.2000.167
-2.2-3.4-5.8-8.2
-14.0-17.4-25.9-31.5-46.6-80.9-93.6
-123.3-135.3-139.6-146.2-152.8-157.1-158.3-161.1-162.5-163.1-164.8
Prime Sinusoid M <j>
1.0021.0211.0391.1131.1621.2981.3831.5851.7271.6511.1710.8940.7840.6120.4460.3400.3120.2500.2170.2030.169
-3.6-5.8-8.2
-13.7-17.3-25.7-31.2-45.4-79.0-91.4
-122.0-134.0-138.8-145.4-152.3-156.9-157.9-160.8-162.2-162.5-164.4
prbs. M 4
1.0001.0051.0211.0481.1371.1771.3071.3911.5901.7291.6761.1600.9090.7900.6100.4440.3340.3060.2510.2200.2040.168
-2.0-3.7-5.2-7.9
-13.3-17.0-25.5-31.1-45.4-78.5-92.2
-122.7-135.6-138.7-147.0-153.6-157.3-159.9-161.2-162.8-162.0-164.3
Fundamental Frequency f^ = 0.05233 Hz
Table 5.5 Frequency Response for Linear System using Different
Test Signals
System Parameters :- € = 0.3, u>n = 10 rad/s
Test Signal Parameters:- N = 1023, At = 20 ms.
116
large amount of noise present, predominantly of a deterministic nature
in the form of a monotone 'dither 1 signal but with also a certain
amount of btt>ad band noise. For comparison purposes tests have been
carried out using simulated linear models possessing a lightly damped
resonance and subjected to a low frequency monotonic noise signal of
frequency approximately 2Hz. It is evident from the results in Fig.
5.6 that the prime sinusoid test signal is superior to prbs in its
ability to remove the influence of the monotonic noise and is still
suitable for detection of the true response mode of the system. The
irregular line spacing could conceivably cause areas to be inadequately
described. However, no problems have been met so far in this area.
Noise can also be of broad based form and hence tests have been
carried out on a linear model with 'white 1 noise of differing levels
added to the system output giving a range of signal/noise ratios. The
relative values of the RMS errors and bias between measured and true
amplitude ratio and phase are given in Table 5.6 for normal prbs,
inverse repeat prbs and prime sinusoid.
The superiority of the prime sinusoid signal in this area is
evident from these results.
5.4.7 Sampling Frequency
The criterion for avoiding aliasing in the case of the prime
sinusoid signal is much simpler than that when prbs is used, due to
the rectangular envelope of the power spectrum. Aliasing will not
occur if the highest prime frequency of the signal is chosen to be
less than l/(2At) Hz where At is the clock period.
(continued on page 120)
117
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USING
PRIME
SINUSOID
Fig. .5.
6 Fr
eque
ncy
Response o
f 4t
h Order
Syst
em I
n Presence of Sinusoidal'Noise (2Hz)
S/N RATIO
158
65
15
865
15865
Magnitude
RMS Error BIAS Error
Phase
RMS Error BIAS Error
prbs Test Signal
0.02450.03330.05520.0432
-0.0116-0.0228-0.02640.0264
2.766
4.29710.841
14.411
-0.333-9.904-4.414
-4.733
Inverse-repeat prbs Test Signal
0.0236 0.07580.03010.132
0.0166 0.06280.008-0.108
1.610
3.15423.264
4.446
0.2299 0.1225
-0.095
0.599
Prime Sinusoid Test Signal
0.00430.00690.01410.0102
-0.0004-0.0015-0.0102-0.0085
0.55971.514
1.4281.899
-0.219-0.361
-0.009-0.457
Table 5.6 Comparison of Error Measurements (eqn. 4.2 and 4.3) for
Different Signal/Noise Ratios using prbs-normal, prbs-
ihverse-repeat and Prime Sinusoid Test Signals.
119
5.5 Selection of an Optimum 'Set' of Prime Sinusoid^
In the previous sections it has been proposed that a prime
sinusoid signal, when used to obtain the spec-oral estimates of a
nonlinear system, will provide complete immunity from harmonics
generated by even nonlinearities and will reduce the number of
harmonics generated by odd power nonlinearities. In this section
consideration will be given to the possibility of selecting an optimum
set of prime sinusoids tailored to a cubic nonlinearity such that a
minimum number of harmonics distort the spectral estimates.
For a given frequency range the number of frequencies evaluated
is dependent on the number- that will give sufficient spectral data to
adequately describe the system being tested. For example, let us
assume, that the system under test has a frequency response range of
73 spectral lines and that 10 frequency points (all of course being
prime frequencies) will give sufficient information to describe the
system. The function minimised is defined as the sum of harmonics
generated at the particular test frequencies selected. For such a
case the value of the minimisation function is shown in Table 5.7 for
different sets of frequencies. It can be seen that an optimum set
does exist (signal 5 consisting of prime frequency lines 3, 5, 13, 17,
19, 23, 37, 47, 67, 73} giving a 40% reduction in the harmonics
generated compared with the 'worst case' 10 frequency prime sinusoid
signal (Table 5.7 test signal 1) and a reduction of 91% compared with
a 21 prime frequency signal. The way the harmonics are distributed
across the frequencies for the optimum signal is shown in Table 5.8.
Also included for comparison purposes in Table 5.8 are percentage
harmonic reduction figures when using an optimum 10 prime signal(continued on page 123)
120
ro
TEST
SIGNAL
NUMBER 1 2 3 4 5 6 7 8
10 P
rime S
inusoid
Sign
alSpectrum L
ine
Number K
5 3 3 3 3 3 3 3
11 5 7 5 5 7 7 5
17 7 13 13 11 19 13 13
23 11 19 17 17 29 19 17
31 13 29 19 23 37 23 19
41 53 37 23 31 43 29 23
47 59 43 29 41 53 43 37
59 61 53 37 47 61 53 47
67 67 61 53 59 71 61 67
73 71 71 67 67 73 71 73
COST
FUNCTI
ON
894
750
714
692
690
666
594
546
MX(t) =
E sinu.t
,th
Y(t)
=
z A.sinco.t
j=k
J n
C.F.
=
z A.
j=k
JM
= 10
U" s
pectral
line
Table
5.7
HarmonicsGenerated b
y a
Cubi
c No
nl i n
ear! ty
'wit
h a
10 P
rime
Sinusoid
Signal f
or D
ifferent
Combinations o
f Line F
requ
enci
es
LINE NUMBER
HARMONICS HITH 21 PRIME
HARMONICS WITH 10 PRIME
% LINE HARMONIC REDUCTION
3
66
48
27
5
165
39
76
13
180
51
71
17
303
54
82
19
240
60
75
23
342
7?.
79
37
369
57
84
47
414
54
87
67
384
66
83
73
348
45
87
No. of harmonics at spectral _ No. of harmonic at spectralline M for 21 prime______ line M for 10 prime______
- NQ _ Qf harmonics at line M for 21 prime
Table 5.8 Line Harmonics for Cubic Nonlinearity using Optimum 10
Prime Sinusoid Signal
122
instead of a 21 prime sinusoid signal.
5.6 Selecting Predefined Amplitude Distribution
Occasionally the need arises for test signals with a certain
amplitude distribution and frequency spectrum. One application is in
fatigue testing where it is known that fatigue life is a complex
function of these two quantities. Also, when tasting nonlinear
systems to obtain linear estimates for modelling, the linear estimates
obtained are very much dependent on the amplitude distribution of the
test signal (see Chapter 6) so that it is desirable when modelling
such systems to have a test signal whose amplitude distribution
corresponds to the distribution of the normal operating signal. It is
difficult to achieve this in practice and in many situations impossible
since so little is known about the 'statistics 1 of the normal
operating record. However, there are situations where the desired
spectrum and amplitude distribution is known. A technique is proposed
for such cases.
Firstly, the desired line spectra for the test signal is
mechanised by selecting those prime frequencies covering the frequency
range of interest. The desired power spectrum determines the amplitude
of the respective sinusoids making up the signal. The phases of the
respective sinusoids are then chosen, using a minimisation algorithm,
so that the amplitude distribution corresponds to the desired shape.
The mechanisation procedure is shown in Fig. 5.7. The best model fitk=m „
is defined as that which minimises Z e,: where £ k is the error ofk=l
4-Uthe 'k' amplitude window of the amplitude distribution. The hill-
climbing algorithm used incorporated Powell's method. The application
of this method is still in progress but an example of the results will
123
be included here. The results obtained when shaping the amplitude
distribution of a 10 prime sinusoid signal, each sinusoid being of
equal amplitude, to be a rectangular amplitude distribution is shown
on Fig. 5.8 and indicates the improvement obtained by an optimum
selection of sinusoid phases. A signal of greater spectral content
will allow for an enhanced matching of the desired and actual
distributions but does necessitate a more time-consuming runj of the
estimation algorithm.
5.7 Summary
A new test signal has been derived consisting of an assemblage
of discrete sinusoids of frequencies which are odd and prime number
multiples of some fundamental which is itself excluded from the
signal. Properties of the new signal have been presented and the
selection of an optimum set of frequencies have been considered. An
algorithm for selecting a pre-defined amplitude distribution for the
new signal has also been presented.
The next chapter will consider the application of the new test
signal to a nonlinear system.
124
r-o
en
en O Q.
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Q.
Ct-
C a. ro t/1
ct- o- c:
PROBABILITY
AMPLIT
UDE
DISTRI
BUTI
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READ INPUT DATA NUMBER OF PRIMES = n SEQUENCE LENGTH = N SINUSOID AMPLITUDES = A,—A AMPLITUDE RANGE = V '
fZERO ALL PHASES <j= 0, j = 1 --n
DETERMINE AMPLITUDE DISTRIBUTION p(a)FOF J AMPLITUDE WINDOWSIN'THE RANGE + V
DETERMINE OST FUNCTION
SELECT NEW SET OF X. VALUES USING POWELLJ
NYURTHEREDUCTION
IN C
OPTIMUM SET OF PHASES
BEST MATCH
Fig. 5.8 Amplitude Distribution Shaping Mechanisation Procedure
126
CHAPTER 6
Application to System with Saturation
Nonlinearity
6.1 Introduction
What has been achieved so far is the development of a test
signal from theoretical considerations and the justification of its
application to linear systems with corrupting noise present. Already
certain advantages over both the use of normal and inverse-repeat prbs
have been made evident. It is now necessary to justify the analysis
of Chapter 5 by applying the technique to systems which are known to
be nonlinear. At this point it should be stressed that in this
investigation the purpose of testing nonlinear systems is. in osder to
obtain a linear representation for the system. Consequently the
linear model obtained will be dependent on the particular input signal
used. No attempt has been made to model the nonlinearities. This
approach is justified in that the work has been carried out with
industrial applications in mind for which complex models are too
intractible for purposes of control system design.
In this section, a simulated feedback system with a saturation
nonlinearity in the feedforward path will be subjected to various
signals. In its linear range, the system has second-order dynamics.
(Fig. 6.la). Saturation is chosen as the first nonlinearity for
consideration since it is invariably present in engineering control
systems (amplifier saturation etc.). It is also an odd function
nonlinearity and has a dominant cubic term. (This can be seen from
Fig. 6.1b which gives a polynomial approximation to a saturation
127
r(t) +A
c(t)
System Parameters:- 5=0.3, con =10 rad/s,.A = + 0.2
Fl'9- 6.la Block Diagram of System Investigated
y = a9th Order Polynomial Fit
+a.,x +a,x +a 5x +agx +a ?x +agx +3gX
3.20
a = 1.057xlO" 6 a, = 1.123 a, = -2.36xlO" 6 a, = -2.009X10" 10 I c. J
a4 = 6.570xlO" 7 a g = -3.56xlO" 3 a 6 = -3.678xlO"8
a ? = 4.95xlO" 3 a 8 = -9.98xlO" 10 a g = -3.151xlO~ 4
Fig.6.1b Polynomial Approximation for Saturation Nonlinearity
128
nonlinearity. The polynomial coefficients are also presented). A
stringent test on the usefulness of the respective test signals for
measuring the spectral estimates of nonlinear systems is thus provided.
The signals used will be normal and inverse-repeat prbs, monotones from
the frequency response analyser, and the prime sinusoid signal, and in
each case a range cf signal levels will be studied. Theoretical
modelling predictions will be presented based upon the single input
describing function technique for the frequency response analyser
47 48 results ' and upon the Gaussian input describing function for the
prime sinusoid results ' both of which are to be compared with the
models obtained experimentally using Powell's Optimisation Procedure.
A summary of the p?rameter estimation procedure using the single
input and Gaussian input describing function is given in Appendix 5.
6.2 Problems with prbs
The frequency response results presented in Fig. 6.2 show that
rapid degradation occurs in the performance of normal prbs as the
signal level is increased. Similarly, the degradation is evident in
Fig. 6.3 which shows the results of using the inverse-repeat sequence.
At low signal amplitude the results depicted in Fig. 6.2a are
reasonably acceptable with regard to scatter in both amplitude ratio
and phase, but deterioration in the scatter becomes very rapid with
increasing signal amplitude. If the response information from which
Fig. 6.2a was derived is analysed by first forming the crosscorrelation
function and then obtaining the Fourier integral transform of this,
large divergencies in the two sets of supposed frequency responses
are evident, confirming the presence of nonlinear behaviour. This(continued on page 132)
129
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9
ee
« 0e
<fe e
0 «> oe
0 « «GO O
Oj^ Tp OO ^*
0 ° tf> o^Asfi,^
(d) prbs 0.4muFig. 6.2 Amplitude Ratio of System with Saturation Nonlinearity
using prbs of Various Signal Levels.Test Signal Parameters:- N=1023, At=20 ms.
130
'V.oo o'.to I'.EO
PDEG
2.40 a.£o.p t'.oo'Hz
(a) inv-rep prbs 0.05mu
e c
o •«0 B
0 00 I'.CO Z'.40 l.ta f 4.03T Hz
(b) inv-rep prbs 0.15mu
.00 0.00 1.60 2.40 3.ZO f 4.00T Hz
e •• o •«.
>DEG'
z'.40 s'.zo -f 4'.oo
(d) inv-rep prbs 0.4mulj (c) inv-rep prbs O.Smu • Fig. 6.3 Amplitude Ratio and Phase of System with Saturation
Nonlinearity using Inverse-Repeat prbs of Various Levels
Test Signal Parameters:- N=1022, At=20 ms
131
latter frequency response is shown in Fig. 6.4. The same trend is
obtained with an inverse-repeat sequence as can be seen from Fig. 6.5
and 6.6 which give the impulse responses and the frequency responses
respectively. A number of inverse-repeat sequences were used all
having different characteristic polynomials, including the optimal?fi
coefficient set as deduced by Barker et al,., but no
significant difference was found between the spectral estimates
computed. To provide a basis for comparison Fig. 6.6a gives the
frequency response data (derived via the Fourier integral) for the
system when operating in its linear region. Although the prbs used is
such as to provide a good 'impulse' response estimate for an
equivalent linear system there are errors, particularly around the low
frequency region, which are not accounted for or reproduced by either
the use of the frequency response analyser or the prime sinusoid test
signal (see subsequent sections).3fi The suggestion by Barker and Davy that the use of antisymmetric
pseudorandom signals in two separate experiments with differing
amplitudes is valid in the case where there is no feedback around a
predominantly third order nonlinearity? is correct, but no simple
relationship is evident in the feedback situation. This is
demonstrated by the experimental results of Fig. 6.7 which was
obtained by applying the relationship
2 „ (27tk) 2 „ (M) u(2irfc) 3 1 H2(NAt) a 2 H1(NAtlH _ ————— —^
a l " a 2 6.1
for the inverse-repeat frequency response measurement of the two
amplitudes, a ] =0.15 and a2 =-0.2 respectively.(continued on page 137)
132
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Fig. 6.4 Frequency Response of Second Order System with
Saturation Nonlinearity Obtained from Fourier
Integral of the Crosscorrelation Function.
133
h(t)'
aoa tot seconds
(b) inv-rep prbs 0.15
TJOO *w
h(t)
o.oo a.ta a. BO i.ro
t seconds (a) inv-rep prbs 0.05 mu
h(t) 1
A
\0.00 O.JO O.M I-M t.M '.*» «••»
t seconds
h(t)«
r
o-*o a.ia i.to i.oo i.«ii t.ao
t seconds(c) inv-rep prbs 0.3 mu (d) inv-rep prbs 0.4 mu
Fig. 6.5 Impulse Response of Second Order System with Saturation
Nonlinearity Obtained by Crosscorrelation
Test Signal Parameters:- N=1022, At=20 ms.
134
DEG
Hz
(a) inv-rep prbs 0.05mu (b) inv-rep prbs 0.15mu
v (c) inv-rep prbs 0.3mu Y (d) inv-rep prbs 0.4mu
Fig. 6.6 Frequency Response of Second Order System with Saturation Nonlinearity Obtained from Fourier Integral of the Crosscorrelation Function
Test Signal Parameters:- N=1022, At=20 ms135
o
o-*•*
CM
oID
O COf
o
o o
oo
oQ
O <D
<D
O
<DG<D
2*.40 fHz 3^20"°XJ.OO 0.80 1.60
Fig. 6.7 Frequency Response Estimate for Saturation
Nonlinean'ty using Barker's Error Correcting
Procedure.
136
6.3 Frequency Response Analyser Results
The results of Fig. 6.8a were obtained using the monotone of the
frequency response analyser with progressively increasing signal
amplitudes. The trend is for a decreased bandwidth and peak
amplitude ratio to be observed as amplitude increases, in accordance
with the severity of excursions into the saturation region. Table
6.1 contains typical assessments of the harmonic content generated by
a fundamental of 0.780 Hz and input amplitude of 0.5.
Modelling based upon the results of Fig. 6.8a using Powell's
procedure and the theoretical procedure of the single input
describing function is summarised in tabular form in Table 6.2. It
should be noted that there are constraints on the theoretical
estimates obtained. Since the measurements were made on a closed loop
system the actual input amplitude to the saturation nonlinearity is a
function of frequency and is not constant, so that the assumption of
a constant amplitude equal to the sinusoidal test signal amplitude is
not valid. However, it does give estimates that indicate the correct
trend for the system parameters as a function of test signal amplitude.
Notice that the minimisation algorithm has in one case included
magnitude information only and in the other both magnitude and phase.
The parameter F is the objective function to be minimised. What is
clearly apparent are the different values of damping ratio predicted
by the tv/o minimisation functions used, thus indicating the extent of
nonlinear behaviour. However, these results give a good linear system
model. Although there are numerical differences between theory and
model parameters they do exhibit similar trends of decreasing
natural frequency co and increasing damping ratio 5.
137
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MOO
1
100
2
0.47
3
3.63
4
0.316
5
0.316
6
0.316
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0.316
8
0.316
"
9
0.316
Mn = amplitude ratio of the n harmonic
Table 6.1 Harmonic Content Measured by Frequency Response Analyser
for Second Order System with Saturation Nonlinearity
6.4 Use of Prime Sinusoid Signals
As with the frequency response analyser experiments, the 21 line
prime sinusoid signal has been assembled and used with a range of
amplitudes on the system with and without the saturation nonlinearity.
The trend is exactly of the same form as that obtained using the
monotone of the frequency response analyser (Fig. 6.8b). Modelling
using Powell's optimisation procedure for parameter estimation has
been carried out in this instance but theoretical modelling parameters
have been derived using the Gaussian input describing function
technique (Table 6.2). It is evident from these results that a
superior agreement between theory and model has been achieved using
the prime sinusoid approach, compared with the single sinusoid method.
The statistical characteristics of the prime sinusoid signal
applied to the system input, and the signal at the nonlinearity input
are summarised in Table 6.3. Also shown are the measurement of the
system output amplitude at line frequencies up to the second harmonic.
The output at these frequencies are generated by the combination of
the input prime frequencies. As expected these components exist but
only that of the fundamental is significant and this increases with(continued on page 141)
138
CO
o if) ""^
R o
o — ."
o 1/3 0 '
C! o
o0
°
A
oS
x ".
a°
**
* °
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• *
o*
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".
O
AO
"*"
*
S
X
^A
Q
<?
y•i
* rf
i
^"
O
0.0
0
0-8
0
1.60
2
.40
f
3.2
0
°00
"*•
0
I*5* 0
0**,
O
lin
ear
o
o o L/5
.1 0 O O :G o
o o
B 0
X
V_.
= 0.
25m
u °
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J
0 A
o A
V.
=
O.SO
mu
"•1
<f>
V.
= 0.
40m
u o
. v
T a
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x o
4.
V i
= O.
BOmu
g
+..*
°
DEG o
X
o Oe.
*
G)
LO
*
0
~"
if.
*
0
0
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°
A
X
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2 x
0
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1 |
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o'.eo
t'.
jo
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cofL
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10
Hz
l!°o
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K =
0.2
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* t
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A
K =
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'x t
o
4-
K =
0.50
^ ^
A
**
x X
K
= 0.
60*
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80x
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K =
1 . 0
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•*• X*
0
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A S
A
O
^ *
5 *
$ 9$
^
j 9
° 5
1 (a
) fre
quen
cy r
espo
nse
anal
yser
(b
) pr
ime
sinu
soid
Fig
. 6.
8 Fr
eque
ncy
Res
pons
es o
f Se
cond
Ord
er S
yste
m w
ith S
atur
atio
n N
onlin
ear
ity f
or
Var
ious
Am
plitu
des
of
Test
Sig
nal .
Frequency Response Analyser Results
r=KX(t)K
0.2
0.25
0.3
0.4
0.5
Describing Function
wn
10.000
9.465
8.78
7.863
6.646
5
0.309
0.316
0.341
0.394
0.451
Powel 1 Modelling Procedure
Objective Function F-jwn
10.00
7.96
7.39
6.61
6.01
£
0.297
0.403
0.414
0.451
0.493
h0.0002
0.0118
0.0234
0.0307
0.077
Objective Function F?
%
10.02
7.95
7.92
7.00
6.49
£
0.295
0.260
0.256
0.308
0.299
F2
3.15
239.2
550.7
890.2
1500.5
X(t) = sintut
Prime Sinusoid Results
r=KX(t)
K
0.25
0.4
0.5
0.6
0.8
1.0
Gaussian Input Describing Function
"n
10.00
9.95
9.8
9.6
8.9
7.12
5
0.3
0.301
0.306
0.312
0.335
0.420
Powell Modelling Procedure
Objective Function F-|
w n
9.524
9.418
9.200
8.964
8.463
7.913
£
0.313
0.369
0.403
0.429
0.464
0.480
F l
0.003
0.044
0.062
0.069
0.025
0.111
Objective Function Fpw n
9.45
8.60
8.00
7.56
6.90
6.43
?
0.311
0.379
0.431
0.489
0.522
0.540
h 2
53.6
801.1
1370.3
1611.4
1843.5
1855.6
23 23 X(t) = ( I sino).t)/21 ; F ] = E (Rt (o) i )-Rm («) 1 ))
if2 i=prime=m
= F23E (
i=m
Table 6.2 Modelling Parameters for Second Order System with Saturation
Nonlinearity.
140
increasing amplitude of test signal. The magnitude of the
fundamental gives a measure of the extent of nonlinear operation of
the system.
Test Signal
r = Kx(t)K
1.0
0.8
0.6
0.5
0.4
0.25
ar
0.1543
0.1234
0.0926
0.0771
0.0617
0.0385
ffe
0.1875
0.1503
0.1127
0.0953
0.0755
0.0458
As J^l
0.7526
0.9408
1.254
1.483
1.873
3.021
Harmonic Content of System Response (Mn/M3 x 100)
Mean
0.9
4.1
5.1
0.0
0.5
2.3
Fundamental
54.1
51.9
44.7
37.3
25.9
6.3
Second Harmonic
0.1
0.2
0.2
0.3
0.1
0.2
23x(t) = ( E sinco,t)/21
ill 1if 2 i=prime
Table 6.3 Statistical Characteristics of Test Signal
It was demonstrated in Chapter 5 that the prime sinusoid signal,
whilst being capable of providing analytic rejection of the influence
of even power nonlinearities, would be influenced by the presence of
odd power terms. What has been shown so far is that for a saturation
nonlinear system, the prime sinusoid signal is suitable as a test
signal for obtaining the systems spectral estimates. Since saturation
is predominantly describable by odd power terms, the results obtained
suggest that the prime sinusoid signal is particularly suitable for
testing systems possessing odd 'power nonlinearities. Nevertheless,
141
degradation of performance is to be anticipated for test signals
containing a large number of frequencies. For this reason the study
has been repeated using the 169 spectral line sequence. The results
for varying levels of amplitude are shown in Fig. 6.9. Whilst those
of Fig. 6.9a are reasonable, degradation is rapid with increase of
input level. This degradation is indicated by ti:o measurements made
of the fundamental and second harmonic present in the response for
each case (see Table 6.4). Notwithstanding this problem, most
significant progress has been made in the rapid determination of
meaningful frequency responses for a system with saturation
nonlinearity. To emphasis this, Fig. 6.10 has been prepared which
summarises:-
(a) the unsatisfactory nature of the result when using inverse
repeat prbs;
(b) the Fourier integral of the input-output crosscorrelation
function;
(c) the result of using the 21 prime sinusoid signal, and
(d) the result of using the frequency response analyser.
The comparison is made on an equal RMS power basis.
The next chapter will consider the application of the prime
sinusoid and prbs signals to nonlinear systems other than those with
saturation, including those which possess memory.
142
o
CO o CD
O
1.00
0
.80
1-
60(a
) K
. 1
2. 4
0 f
3.20
Hz
O o'
0
O0
0
oO
0
0
°C.G
O(c
) K
= 4
l.6Q
z.
40 f
3.
20Hz
o
o(D
OO
o
e e
"0-0
0 0.
80
1.60
(b)
K =
22
'.4C
fu
3.2
3Hz
0 03
0
0
o
o
I.CO
0.
80
1.60
2.
40 f
3.
20(d
) K
= 6
HzFr i
g. 6.
9 Fr
eque
ncy
Response o
f Se
cond
Order S
yste
m with S
atur
atio
n Nonlinearity u
sing
169
Prime S
inus
oid
Sign
al of
Several Amplitudes.
c«o iso t.4o f 3'.to H2
P DEG -1o
(a) inv-rep prbs
K DEG
(b) Fourier integral of.crosscorrelation function
o9
Ro
w
oVt>"
0
S;V
0M
r»Cl
oP>
t> O
O?o
o
*DEG ?'0 «7
C3 Cu
V
F
o «>R~s
«> 0 O
« «•" 0
O
"- • -
.oa o'.co t'.eo t.io-f a'.eo 'bHz 3
W
0a
?•>• r.
C3
• 0G.
0 S
• fi *DEG 7
e« o• 0 0» «. «
(c) 21 prime sinusoid §
0 *
» e•
«•
.00 o'.aa I'.so z'.o~r~~:ujflr Hz
o o
o
e
*
• a
0
(d) frequency response analyse
iq. 6.10 Frequency Response of Second Order System with Saturation
Nonlinearity using Different Procedures.
144
r = K X (t)
K
1.0
2.0
4.0
6.0
^ x 100%
Harmonic Content
Fundamental
15.4
38.4
58.3
58.1
Second
15.2
34.1
42.1
37.7
171X(t) = ( z sino3,t)/169 if! ]
i-f2 i=prime
Table 6.4 Harmonic Content for Second Order System
with Saturation Nonlinearity Measured with
169 Primes
145
CHAPTER 7
Application to Other Nonlinearities
7.1 Introduction
In addition to saturation there are several other types of
nonlinearities often encountered in systems who^e dynamics are to be
measured. It is proposed to consider a selection of these
nonlinearities as part of a linear system in order to evaluate the
ability of the fast frequency response procedure to measure the
spectral characteristics of a wide range of nonlinear systems using
different test signals. It is again important to recognise that the
objective of the measurement procedure is to obtain linear estimates
and consequently a linear model for the systems under test. The
systems considered include both simulated and real systems.
7.2 Dead-Zone Nonlinearity
As a further example of a nonlinearity without memory a dead-
zone element is examined in a similar manner to that previously
adopted except that the modelling procedures have not been pursued.
The experimentation was conducted using an analogue computer with the
arrangement as shown in Figure 7.1. The nonlinearity is represented
by diodes suitably connected to the summing resistance of an
operational amplifier. The linear part of the forward path transfer
is identical to that considered when examining saturation.
Figure 7.2 shows the results obtained using a 21 line prime
sinusoid for varying amount of dead-zone. A decrease in bandwidth
and increase in damping is apparent as the amount of deadzone is
(continued on page 150)
146
+lmu
TEST
SIGN
ALFROM
DAC
CHANNEL
QC1
= QD
1 =
0.04
48 f
or a
= +
0.05
mu.
Dynamics o
f li
near
part;
5 =
0.3, u
=
10 r
ad/s
0 AD
C CHANNEL
Fig.
7.
1 Ex
perime
ntal
Arrangement
with D
ead-
Zone
Nonlinearity
o
oCM
oCO
o
o o
&
•V
X
oA
o A
+ oA
x !+xX
O A = 0.00
A A = 0.01
•f A = 0.03
X A = 0.05
oA4-
X+ X
0-00oQ
0.80 1 .60 2.40 3.20 'HZ 4.00 4.80
O O
oWJ-
o oo o
24- X
O
o o
P DEGo oo oCM.
4-
XAO
Fig. 7.2 Frequency Responses of Second Order System with
Dead-Zone Nonlinearity using Prime Sinusoid Test
Signal.
System Parameters :- Dynamics of linear part, C - 0.3,w = 10 rad/s.
Test Signal Parameters:- 21 Prime Sinusoid, N = 1023,At = 20 ms.
148
<?*>
V*» o• Jfco*
3.00 0.80
P DEG'
1.60 2.40 3.£0 - 4.00fHz
(a) prbs
2-
'DEG '
De e
O.tO 1.60 2.40 1.20
Hz
(b) Fourier Integral ofCrosscorrelation Function
P DEG
Om — *"
R0 o
o v>o'
o o
1O-«a
o0o'
C9c»e>
o (a17 IP
•*~'
O C7
O
?'
r<"
Roirt
a "
ca
!•
o•" "..• «»
o
.00 fl'.OO t'.SO 2.40 3.80 4.00 «)
T HZ gi
o
o••••.;s
* !f*• 0
D
* •• ." • % ••".•• A T*DEGo oow
«• ••
*
*
• •
e
a0
.00 O.BO 1 -*Q 1,*.* "3-tfl
fHz
'•-•.•«
e
• «
"••• . .
(c) 21 Prime Sinusoid (d) Frequency Response Analyser
Fig. 7.3 Frequency Responses of Second Order System with Dead-Zone Nonlinearity using Different Test Signals.
149
increased. This corresponds to the trend observed with saturation.
Also shown in Fig. 7.3 are the results obtained using prbs and
frequency response analyser. The Fourier integral of the
crosscorrelation function obtained when prbs was used is also
presented. Again, the unreliability of the estimates computed when
using prbs as the test signal is evident and the suitability of the
prime sinusoid signal is demonstrated.
7.3 Backlash Nonlinearity
As an example of a nonlinearity with memory a nonlinear system
is considered, with backlash placed after the feedforward loop
dynamics but before the feedback. The simulation arrangement is
shown in Fig. 7.4. The prime sinusoid results for a range of signal
levels are shown in Fig. 7.5a and for comparison purposes the results
of the frequency response analyser monotone amplitudes are shown in
Fig. 7.5b. Finally Fig. 7.6 illustrates the results for prbs, the
transformed crosscorrelation function, the prime sinusoid signal and
the frequency response analyser. From these results it can be
observed that whilst the scatter of the estimates using prbs is not
as large as that present when testing systems with saturation, scatter
is nevertheless present and is avoided when the prime sinusoid signal
is used. The validity of the prime sinusoid estimates are confirmed
by those obtained using the frequency response analyser.
7.4 Direction-dependent Nonlinearity
One of the most persistant nonlinearities, even when the
pertirbation signal amplitude is small, is that in which the dynamics
of the process are different according to whether the variable under
(continued on page 154) 150
TEST
SIGNA
FROM
DAC
CHANNEL
TO A
DCCHANNEL
QC1
= QD1
= 0.1
for
a -
+ 0.1
muDynamics o
f linear p
art: e= 0
.3,
u =
10 r
ad/s
Fig.
7.4
Experimental Arra
ngem
ent
with B
acklash
Nonlinearity.
BACKLASH S
IMULATION
o
r-o
Ro —• o (3 O~
O o"
o o
o T
8$
* X
yS
<?
K =
0.25
s
ce
xX
"x-
A
K =
0.50
-"
nS*
XX
+
K =
0.75
R
a X
K
= 1.
00
g o"
^ x
& °J
e °
+ *
£Q
„o D
t a
*AS
°^
O
Vi
=
0.21
82 v
±fi
A
V1
= 0.
1091
v
s^d
4. V.
=
0.05
36 v
6 1
QO
O
A
Si
*
°o'.C
O
o'.S
O
l'.S
O
2'.4
C
s'.2
0.f
4'.0
0 °o
'.00
o'.S
O
l'-6
0
2'.4
0.
3'.2
0 f
4'.0
0g
fHz
g Hz
o"
o o 0 LO.
1 O o o o o 0 a LO
i
mm
o'
A^
X
o5?
'":jj
o U'j .
&
O C.,' 4
w
""-^
s
o
o ^D
FPX
°o
*?
M x
^0
^
Xy
V"
©ft*
X
X
X
x
ea
*«B
9S
jj
*
(3 +
*
2°r
v 2
0*
O
(a)
21
Prim
e S
inus
oid
° £~
£
(b)
Freq
uenc
y Re
spon
se A
naly
ser
f *
° o
N =
1023
, A
t =
20 m
s.
+ +
Fig
. 7.
5 Fr
eque
ncy
Res
pons
es o
f Se
cond
Ord
er S
yste
m w
ith
Bac
klas
h N
onlin
earity
for
Rang
e of
Test
Sig
nal
Leve
lsSy
stem
Parameters:-
Dyna
mics
of
line
ar p
art
5= 0
.3,
<*>n =
. 10
rad
/s
§1.00
P DEG
vj (a) prbs
P DEG
3. 00 0.UQ [ .CO i.fO _rHz
(b) Fourier Integral ofCrosscorrelation Function
0.«Q I ,60
'DEG
1 (c) 21 Prime Sinusoid
P DEG
,CO t . tO 1 .TO r- 4.0fHz
(d) Frequency Response Analyser
Fig. 7.6 Frequency Responses of Second Order System with Backlash
Nonli.nearity using Different Procedures
153
investigation is increasing or decreasing. This situation occurs
widely in industry and one well known example is a temperature loop
operating well above ambient temperature, where, owing to heat
losses, the controlled temperature takes longer to reach the desired
temperature with an increase of setpoint compared with a decrease of
setpoint value. Godfrey and Briggs have investigated the effects of
such a nonlinearity on the correlation estimates and difference-28 equation models obtained using different m-sequences. In order to
evaluate the effects of such a nonlinearity on estimates obtained
using the alternative frequency domain approach a digital simulation
of direction dependent dynamic systems is implemented.
Two specific dynamic systems are considered, one possessing
first order dynamics described by the equations
Tl 7K + Y = X with Y P°s1tive
To S + Y = X with Y negative1 dt 7.1
and the second possessing second order dynamics described by the
equations
= X Wlth Y P°s1t1ve
= X w1th Y ne9ativen dt 7.2
The estimates obtained for the first order process using prbs,
inverse-repeat prbs and the 21 prime sinusoid signal are shown in? Fig. 7.7 for a system with j- equal to 3. Figure 7.8 gives the
frequency response estimates using the 21 prime sinusoid signal for
(continued on page 157)
154
oID
O OM
0 CO
oXT
o
o o
a o
cb.ooo o_
2.00
o a
oo
o o
*DEGo o
O prhs - normalA prbs - inverse repeat+ prime sinusoid
6-00 8-00'Hz
Fig. 7.7 Frequency Responses of First Order Process with Direction Dependent Dynamics using Different Test Signals
System Parameters :- T^ = O.ls, T2 = 0.3s Test Signal Parameters:- N = 511, At = 20 ms
155
oc J
R
oCOo"
oo~
oo
01,
A Tnj
•»• Tl
* T 1
So 0 T]* 0
ot o
A ^ 4 °UT^
* *0 o «** °° 0
A * * 0A 0 5^, . OQ o
^ " ** * AA * ' ** * **A A AA A AA
= O.ls= 0.5s= O.ls= O.ls= O.ls
*A
T 2 = O.ls
T9 - 0.5st
T2 = 0.3sT 2 = 0.4s
T2 = 0.5s
0-00 2. 00 4.DO e.OO £ 8-00' Hz
oo
a oo I
o o
*DEG'o ooCM
Fig. 7.8 Frequency Responses of First Order Process
for a Range of Direction Dependent Dynamics
using a Prime Sinusoid Signal
Test Signal Parameters:- 21 prime sinusoid,N = 511, At = 20 ms,
156
T2a range of ~ values.'l
In the case of a system possessing second-order dynamics two
possibilities are considered,
(a) a system with different natural frequencies for the
positive going and negative going directions
and
(b) a system with different damping factors for the positive
going and negative going directions.
The frequency responses obtained using prbs,inverse-repeat prbs and the
21 prime sinusoid signal are shown in Fig. 7.9 and Fig. 7.10 for
systems in category (a) and (b) respectively. The responses computedWn2
using the 21 prime sinusoid signal for different values of -^py are
shown in Fig. 7.11. As in the case of the previous nonlinearities
considered, the superior quality of the estimates measured by the
prime sinusoid signal is evident. Whilst there is a reduction of
scatter with inverse-repeat prbs compared with prbs (corroborating
the results obtained by Godfrey and Briggs) the estimates are still
unacceptable as a1 basis for obtaining a linear model for the
nonlinear system. This conclusion is confirmed by the modelling
estimates presented in Tables 7.1 and 7.2 respectively which have
been obtained using the hill-climbing algorithm of Powell. (The
accuracy of the model fit to the measured data is reflected by the
numerical values of the minimisation functions F-| and F 2 - For a
direct comparison of the respective F-j and F2 values obtained using
different test signals, the values for prbs and those for inverse-
repeat should be divided by 4 and 2 respectively.) A fact that clearly
emerges from these prime sinusoid estimates is that whether the cost(continued on page 163)
157
O prbs - normalA prbs - inverse repea+ prime sinusoid
, = 10 rad/s u o = 8 rad/s
Fig. 7.9 Frequency Responses of Second Order Process with Direction
Dependent Dynamics using Different Test Signals
(different to )
Test Signal Parameters:- N = 511 9 At = 20 ms.
158
O prbs - normalA prbs - inverse repeat4. prime sinusoid
nl = 10 rad/S
Fig. 7.10 Frequency Responses of Second Order Process
with Direction Dependent Dynamics using
Pi fferent Procedures.
(different ?}
Test Signal Parameters:- N = 511, At = 20 ms.
159
0r-j
oCO
o o"
oA
'goA
I
CD 'A (
+ '
X c
<!> (
un2 - 9 rad/s "]Dn2 = 8 rad/s*Vi2 = 7 rad/ so - = 6 rad/s"nZ = 5 rad/ s
wnl - 10 rad/s> 5 1 = 0.5
C2 = 0.5
o o
.00o o"
o o a
2.00 4.00 6-00 f 8.00T Hz
a a
a o
P DEGo oo oCM-
XooA+0 XA
5?
?S ® vo s I I il
Fig. 7.11 Frequency Responses of Second Order Process
for a Range of Direction Dependent Dynamics
using a Prime Sinusoid Signal.
Test Signal Parameters:- 21 prime sinusoid,N = 511, At = 20 rns,
160
TEST
SIGN
AL
PRIMES
PRIMES
PRIMES
PRIMES
PRIMES
PRBS
INV. REP
-fve g
oing
time
con
stan
t
Tl 0.1
0.5
0.1
0.1
0.1
0.1
0.1
+ve
going
time c
onstant
T2 0.1
0.5
0.3
0.4
0.5
0.3
0.3
Objective
Function i
ncludes
magn
itud
e &
phas
e "information
Te
0.0994
0.4986
0.1617
0.1778
0.1894
0.1176
0.1162
Fl 1.278
0.1588
265.8
•521.6
775.5
7290 931
Objective
Function i
ncludes
only m
agni
tude
information
Te
0.1024
0.5025
0.1465
0.1551
0.1607
0.1210
0.1341
F2
0.671
x 10"3
0.059
x 10"3
5.094
x 10"3
6.78
x 10"3
7.97
x 10"3
7.81
1.25
cr>
Table
7.1
Modelling
Results
for
Firs
t Order
System w
ith
Direction
Dependent
Dynamic
Responses
(Objective f
unctions
F-, and
Fo d
efined in Table
6.2)
TEST
SIGNAI
PRBS
INV
REP
PRIMES
PRBS
INV
REP
PRIMES
PRBS
INV
REP
PRIMES
PRIMES
PRIMES
PRIMES
+ve
going
parameters
«i 0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
unl 10 10 10 10 10 10 10 10 10 10 10 10
-ve
going
parameters
^ 0.7
0.7
0.7
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Wn2
10 10 10 8 8 8 6 6 6 9 7 5
Objective
Function i
ncludes
magnitude
& phase
information
Ee,
0.657
0.672
0.492
0.487
Un2
10.05
9.965
10.05
8.39
0.497
'8.215
1 .
0.491
1.838
24.192
0.446
0.499
0.473
0.4095
8.994
2.926
0.2212
8.118
9.487
8.534
7.763
F l 395
105
15.7
4263 599
4.71
1 54000*
92600*
83 1.171
2.042
269
Objective
Function includes
only m
agnitude i
nformation
?e
0.6415
0.6748
0.498
0.5924
0.5748
0.4965
0.758
0.787
0.482
0.499
0.491
0.412
Wn2
9.864
10.060
10.025
8.264
8.159
8.950
6.695
6.574
8.125
9.44
8.511
7.721
F2
0.0206
0.00824
0.000731
0.4212
0.0727
0.00156
1.974
0.3179
0.007655
0.000358
0.00395
0.00451
en
r\>
* hill climbing a
lgorithm f
ailed
to f
ind
a true m
inimum
Table
7.2
Modelling
Results
for
Second O
rder S
ystem
with D
irection D
ependent
Dynamic
Responses
(Objective f
unctions F
, and
F~ d
efined i
n Table
672)
function minimised includes both phase and magnitude information or
magnitude information alone, there is good agreement (within about
between the estimates computed using the respective minimisation
functions.
7.5 D.C. Servo-System
In order to demonstrate initially the problems associated with
the application of prbs to the rapid determination of nonlinear system
frequency responses Chapter 5 presented results which had been
obtained experimentally from a DC servomotor* in closed loop but with
a significant amount of backlash (20°) and Coulomb friction in the
final drive. (Fig. 7.12). Figure 5.2 illustrated the problem. It is
worthwhile returning to this example to examine what improvement, if
any, can be achieved in this instance by using the prime sinusoid
as a test signal. The application of the prime sinusoid signal in
this situation yields the results illustrated in Fig. 7.13, from
which it is evident that the wide scatter produced when prbs is used
has been removed and some meaningful measurement obtained.
Finally, the position control system of Fig. 7.14 with a phase
advance and first order lag network introduced into the forward path
was considered. The system was tested using (a) prbs (b) the
prime sinusoid signal and (c) frequency response analyser monotone
and the frequency responses obtained are shown in Fig. 7.15. The
system is evidently nonlinear as can be seen from the scatter when
prbs is used, although prbs does provide some meaningful measurement.
However, the estimates obtained using the 21 prime sinusoid signal are
*ES130 Feedback DC Servo System(continued on page 169)
163
ERROR
DETECTOR
AMPL
IFIE
Re. in
CTl
SERVO
AMPLIFIER
ARMATURE
CONTROLLED
MOTOR
AND
LOAD CHARACTERISTICS
BACKLASH A
ND
COULOMB
FRICTION
Fig.
7.12
D.C.
Mo
tor
with B
acklash
and
Coulomb
Friction i
n Final
Drive
o o•CM"
o in
o o
oLO
•o"
o o
•
o,
coCD
CD
CD
CD<D
'CD 0
CD
O CD CD
CD <r>
i————————;T _ j—————-i————————r'0-00 2.00 4.00 6-00 fy 8.00
o Hz o
O o
o oo co
i
o o
*
o to
o o
CslI
CD ® O CD
Fig. 7.13 Frequency Response of D.C. Servomotor with
Backlash and Coulomb Frictjon obtained using
Prime Sinusoid Test Signal
Test Signal Parameters:- 21 prime sinusoid,N - 1023, At = 20 ms.
165
VELOCITY
CONSTANT
PHASE-ADVANCE
PHASE
LAG
D.C. SERVO
MOTOR
CTl
cn
Kv1 +
0.1S
1 +
0.02S
11
+ 0.02S
10.16S
+ 1
]_ s
(kv
= 12
0)
Fig.
7.
14
Posi
tion
Control Sy
stem
with
Phas
e Advance
Compensation.
a prbs.. :-:_-:: '—+- frequency response analyse
o •
o *
.zo «oo
0.60 1.60 2.40 3.ZO 4.00 *.60 5.60 6.40 7.20 «.00M_
s.eo
(a) prbs
Fig. 7.15a. Frequency Responses of Position Control System using
Different Test Signals
Test Signal Parameters:- N = 1023, At = 20 ms.
167
& prime sinusoid —»- frequency response analyser
O.BO
O.BO
1 .SO
1-60
2.40
2.40
J.ZO 4.00
4.00
4.80,- 5.60fHz4,60 5.60
6.40
6.40
7.20 0.00
6.00
e a
(b) prime sinusoid Fig. 7.15b Frequency Responses of P o si ti on_Con t ro 1 Sy s tejri_u s i n g
Different Test Signals
Test Signal Parameters:- 21 prime sinusoid, N = 1023,At = 20 ms.
168
free from any pronounced scatter, thus confirming its superiority
over prbs for measuring the spectral characteristics of physical
systems.
169
CHAPTER 8
Conclusions and Further Work
This investigation has been concerned with the development of
techniques for the measurement of systems dynamic characteristics.
These techniques satisfy the requirements of rapid measurement time,
noise and harmonic rejection capability, and ease of interpretation
of results. It has been shown that these requirements, except for
the important one of harmonic rejection, have been satisfied by using
prbs in conjunction with the fast Fourier transform mixed radix
algorithm to obtain the frequency characteristic of a system. The
errors introduced by the presence of various forms of noise have
been demonstrated and the results obtained show how the errors may
be minimised with careful selection of .quantisation, sampling
frequency and sequence length.
The application of the fast frequency response procedure to
nonlinear systems using prbs as the test signal has shown that the
procedure can produce results which are unacceptable. This is due
to harmonic generation corrupting the frequency response data which
indicates the inability of the technique to reject or minimise
harmonics. The Fourier integral of the crosscorrelation function has
been shown to provide equally unacceptable spectral estimates.
However, this investigation has shown that the procedure can be
extended to the testing of nonlinear systems by using test
frequencies which are prime multiples of a fundamental frequency
which is itself excluded from the test signal. To minimise the
effect of the harmonics and sub-harmonics it has been shown how prbs
170
(either normal or inverse repeat) can be replaced by a test signal
which has the following advantages:-
(a) only those frequencies of interest need be considered;
(b) all amplitudes and phases may be identical, providing
reduced computational complexity.
(c) since tl,? number of samples may be made equal to 2n
(n integer) the radix-2 fft may be used if required.
(d) in a similar manner to prbs the location of each line in
the test signal spectrum may be selected by changing the
sampling frequency (or clock rate);
(e) improved noise-rejection capability;
(f) the ability to eliminate sinusoidal noise by selecting a
suitable clock frequency;
(g) the prime sinusoid signal is suitable for both linear and
nonlinear systems. Furthermore the results presented
suggest that meaningful and useful estimates can be
obtained for a wide range of nonlinearities without prior
knowledge of the specific nonlinearities that are present;
(h) as in prbs, the signal is periodic so that the system
response can be readily extracted, even in a noisy
environment. If the signals measured are particularly
noisy periodic averaging may be used.
It is evident from the experimental results presented that the choice
of the test signal depends on the characteristics of the system under
investigation. If the system is linear within well defined limits,
then normal prbs will give satisfactory spectral estimates, but with
pronounced nonlinear behaviour such estimates are heavily corrupted
171
by harmonic generation. With a nonlinearity that is predominantly
even-ordered then inverse-repeat prbs is suitable because of its
anti-symmetric properties. For systems with pronounced even and odd
power nonlinearity, prbs-normal and inverse repeat give unsatisfactory
results. However, the prime sinusoid signal has been shown to give
considerably improved estimates.
The mechanisation of the procedure has relied heavily on the
use of a digital computer, in particular the implementation of the
fft algorithm,and generation of the test signals. As far as the
generation of prbs is concerned, commercially available generators
are available, so that field testing is possible. Unfortunately,
however, generation of the prime sinusoid signal is not readily
achieved. It is feasible to pre-record signals with a limited range
of spectral characteristics using an instrument recorder for specific
field testing applications. This approach however, provides a poor
alternative to a ^commercially available instrument in that it lacks
versatility and flexibility. A further consideration for successful
implementation in field applications is the need to record the
response data with accurate timing. If this is not provided the
computed spectral estimates will be inaccurate. There is therefore
a requirement for the development of an instrument which mechanises
the procedure from test signal generation to fft computation and
presentation of the spectral estimates. A possible development is
the use of a microprocessor with a ROM capability where the
procedure implementation is represented by a firmware algorithm. It
is intended to embark upon a feasibility study to assess the
possibilities of such an implementation.
172
It is apparent from the results presented that when testing
nonlinear systems, the linear model is dependent on the amplitude
probability distribution of the test signal. The investigation has
presented the possibility of shaping the amplitude distribution of
the prime -sinusoid signal to a pre defined shape; however to fully
assess the potential of the prime-sinusoid in this area further work
will be necessary.51 It has been shown that higher harmonics generated by nonlinear
systems contain useful information which may be used as features in a
pattern recognition scheme to assess system quality and isolate the
locations of nonlinear faults. It is considered that the application
of the fast frequency response procedure would be of value in such a
role.
A further possible area of study is the extension of the fast
frequency response procedure to the testing and identification of
multi-variable dynamic systems. The application of correlation
techniques to the identification of such systems using several52 53
simultaneous pseudo-random signals is well established. ' The
author is unaware of any published material concerned with the direct
measurement of the frequency characteristic of respective channels of a
imuMi^variable system using simultaneous injection of test signals.
Finally, one of the objectives of this work has been the
determination of linear models of nonlinear systems by measurement and
the need now exists to evaluate the utility of the linear models
obtained. In particular, it is felt that the fast frequency response
procedure using the prime sinusoid as test signal, should be used as a
tool in the design process of nonlinear systems in order to assess the
173
procedure over a wide range of conditions which were not possible
within this investigation.
174
REFERENCES
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2. HUGHES, M.T.G. and NOTON, A.R.M.: 'The measurement of control
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6. BRIGGS, P.A.N. , HAMMOND, P.H., HUGHES, M.T.G. and PLUMB, G.O.:
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1964-65.
7. GORRON, E.R., CUMMINS, J.D. and HOPKINSON, A.: 'Identification of
some cross-flow heat exchanger dynamic responses by
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signals'. Atomic Energy Establishment, Winfrith, Dorset.
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175
8. CUMMINS, J.D. 'A note on errors and signal to noise ratio of
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9. GODFREY, K.R. and MURGATROYD, W.: 'Input transducer errors in
binary crosscorrelation experiments'. Proc. IEE, Vol. 112
No. 3. 1965.
10. DAVIES, W.D.T.: 'Identification of a system in the presence of
low-frequency drift 1 . Electronics Letters, Vol. 2, No. 9,
Sept. 1966.
11. DOUCE, J.L., NG, K.C. and WALKER, A.E.G.: 'System identification
in the presence of a ramp disturbance 1 .' Electronics Letters
(UK) Vol. 2. p 243, 1966.
12. DAVIES, W.D.T. and DOUCE, J.L.: 'On-line system identification
in the presence of drift 1 . IFAC Symposium on Problems of
Identification in Automatic Control Systems, Prague, June
1967.
13. DAVIES, W.D.T and SINCLAIR, P.A. 'Application of pseudo random
sequences to process plants - a case history 1 . IEE
Colloquim on Pseudo Random Signals Applied to Control
Systems'. London 1967.
14. WILLIAMS, B.J. and CLARKE, D.W. 'Plant modelling from prbs
experiments'. Control Vol. 12 pp 856-860, pp 947-951, 1968.
15. GODFREY, K.R.: 'Theory of the correlation method of dynamic
analysis and its application to industrial processes and
nuclear plant 1 . Trans. Inst. Meas. and Control. Vol. 2,
T65-72 May 1969.
176
16. DAVIES, W.D.T.: 'System identification for self-adaptive
control 1 . Wiley Interscience. 1970.
17. GODFREY, K.R. and SHACKCLOTH, B.: 'Dynamic modelling of a steam
reformer and the implementation of feed forward/feedback
control 1 . Measurement & Control, 3, pp T65-T72, 1970.
18. BANAZIEWICZ, H., WILLIAMSON, S.E. and LOVERING, W.F.: 'Use of
synchronised dither in pseudo random sequence testing'.
Electronics Letters, Vol. 9, No. 15, 26th July 1973.
19. EVANS, R.P. and WALKER, P.A.W.: 'Assessment of drift rejection
schemes applied to on-line crosscorrelatlon experiments'
Int. J. Control, Vol. 18, No. 1, 33-56, 1973.
20. LAMB, J.D.: 'The detection and filtering of system response
modes of low damping using prbs and crosscorrelation.'
Proc. U.K.A.C. Conf. Automatic Test Systems, pp 521-527,
1970.
21. LAMB, J.D.; 'System frequency response using p-n binary waveforms'
IEEE Trans. Automatic Control, AC-15, pp 478-480. 1970.
22. COOLEY, J.S. and TUKEY, J.W.:'An algorithm for the machine
calculation of complex Fourier Series'. Math. Computation,
19, pp 297-301, 1965.
23. NICHOLS, S.T. and DENNIS, L.P.: 'Estimating frequency response
function using periodic signals and the fft'. Electronic
Letters, 7, pp 662-663, 1971.
24. SIMPSON, R.J. and POWER, H.M.: 'Correlation techniques for the
identification of nonlinear systems'. Trans. Inst. Meas. and
Control, 5, pp Til2-117, 1972.
177
25. BARKER, T.D. and OBIDEGWU, S.N.: 'Effects of nonlinearities on
the measurement of weighting functions by crosscorrelation
using pseudo random signals'. Proc. IEE, Vol. 120, No. 10
Oct. 1973.
26. BARKER, T.D., OBIDESGIWI, S.N. and PRADESTHAYON, T.:'Performance
of anti-symmetric pseudo random signals in the measurement
of 2nd order volterra kernals by crosscorrelation. 1 Proc.
IEE, Vol. 119, No. 3. pp 353-362, March 1972.
27. GARDINAR, A.B.: 'Identification of processes containing single
values nonlinearities 1 . Int.J. Control, 18, pp 1029-1033,
1973.
28. GODFREY, K.R. and BRIGGS, P.A.N.: 'Identification of processes
with direction dependent dynamic responses'. Proc. IEE.
Vol. 119, No. 12, Dec. 1972. .
29. LAMB, J.D., and REES, D.: 'Digital processing of system responses
to pseudo random sequences to obtain frequency response
characteristics using the fast Fourier transform'. Proc.
IEE. Conf. 'The Use of Digital Computers in Measurement 1 .
Conf. Publication No. 103, 1973.
30. LAMB, J.D., and REES, D.: 'Rapid frequency response determination
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31. SOLATRON Publication: 'Operating manual for the 1172 frequency
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32. TRUXAL, J.G.:Automatic control systems synthesis. McGraw Hill,
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33. GOLOMB, S.W.: 'Shift register sequences', Holden-Day, 1967.
34. HOFFMAN de VUSME, G.: 'Binary sequences', English Universities
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178
35. BRIGHAM, E.I.: 'The fast Fourier transform 1 , Prentice Hall, 1974.
36. BARKER, T.D. and DAVY, B.A.: 'System identification using
pseudo random signals and the discrete Fourier transform 1 .
Proc. IEE. Vol. 122, No~. 3, pp. 305-311, 1975.
37. BENDAT, J.S. and PIERSOL, A.G.: 'Random data, analysis and
measurement procedures', Mi ley 1971.
38. SINGLETON, R.C.; 'An algorithm for computing the mixed radix
fast Fourier transform 1 , IEEE Trans. Audio and Electro-
acoustics, Vol. AU-17, No.: 2, 1969.
39. COOLEY, J.W., LEWIS, P.A.W. and WELCH, P.O.: 'The fast Fourier
transform and its applications', IEEE Trans. Education,
Vol. E-l?, No.l, pp. 23-24, March 1969.
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processing 1 . IEEE Press, 1972.
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McGraw Hill, 1969.
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computer programs', Hoi den-Day, 1967.
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44. POWELL, M.J.D.: 'An efficient method of finding the minimum of a
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Computer Journal, 7, pp 155-162.: 1964.
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the deterministic characteristics of pseudo random binary
sequences'. Proc. U.K.A.C. Conference on 'Automatic Test
Systems'. Univ. Birmingham, 1970.
179
46. DOUCE, J.L. and DAVALL, P.W.: 'Generation of signals with
specified statistics 1 . Proc. IEE. Vol. 120, No. 10,
Oct. 1973.
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EUP, 1960.
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systems'. Wiley, 1961.
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IEE. Monograph 4.9M, 1960.
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Winfrith, Dorset. Report No. 507, (SC. C8 W6833) Feb. 1965.
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N.P.L. Autonomies Division Report 14/66, 1966.
180
APPENDICES
APPENDIX 1
Computer System Configurations
1.1 IBM 1130
32K bytes of core store
3 disc drives 0.5M Bytes/disc
Card Reader/Punch 380/80 cards/minute
Paper Tape Reader/Punch
1403 Line Printer 600 lines/minute
Console typewriter
Calcomp 736 plotter
1.2 Interdata 80
48K bytes of MOS Memory
2 disc drives 1.25M Bytes/disc
DATA 100 line printer 400 lines/minute
Paper tape reader/punch
Card reader 600 cards/minute
Teletype
Beehive Visual Display Unit
1.3 Solatron HS7/3A
72 Amplifiers, 16 dual Integrators/Summers, 8 Summers
7 quarter square multipliers with a complement of
Diode function generators
Bridge limiters
Integrated circuit comparators
Solid state D/A switches
Buffered read relay switches
Logic modules (gates, bistables, monostables, registers)
1.3.1 HS7 Integrator Mode Contro1
Mode
RESET
COMPUTE
HOLD
Logic Line 1
1
1
0
Logic Line 2
1
0
0
n
1.4 Functional Description of Hybrid CALL Functions
Symbolic Name
CLEAR
LEAVE
FLSI
;FLSO
CLSO
FADSC
FADC
FDAC
DEL
FIX
Calling Procedure
CALL CLEAR
CALL LEAVE
CALL FLSI(J S K)
CALL FLSO(J,K,..L)
CALL CLSO
CALL FADSC(J)
CALL FADC(K)
CALL FDAC(J 5 K)
CALL DEL(J)
K = FIX(FK)
Operation
RESETS INTERFACE LOGIC
RESETS INTERRUPT FLAG
READS LOGIC SENSE INPUT LINES
SETS LOGIC OUTPUT LINES
RESETS LOGIC OUTPUT LINES
SELECTS MULTIPLEXER CHANNEL J
CONVERTS ANALOGUE VOLTAGE TO DIGITAL NUMBER
CONVERTS DIGITAL NUMBER TO ANALOGUE VOLTAGE
GIVES TIME DELAY
CONVERTS FLOATING POINT NUMBER TO INTEGER NUMBER K
Arguments
NONE
NONE
J LINE No.(O^J<15) K SET TO 1 if 0=0 K SET TO 2 if J=T 1
J No. OF LINES SET UJ$16
K...L LINES SET TO 1
NONE
0 CHANNEL No. 0£j£l5
K DIGITAL No. -4095£K£4095
J DAC CHANNEL No. O^J$3
K INTEGER No. CONVERTED -2047sK$+2047
TIME DELAY J MILLISECONDS
K=INTEGER(FK)*24 -1.0sFK$+1.0
TM
APPENDIX 2
Computer Program Details
Contents
2.1 Hybrid Program Listing
2.2 FFRIP Program Listing
2.3 FFRIP Program Input and Output Data
2.4 listings of Supporting Subroutines
2.5 Subroutines CALL Procedure
Note:- Punch tape copies of the programs will be made available by the author on request.
2.1 Hybrid Program Listing
cC HYBRID PPOflPAM FOR TESTING OF DYNAMIC SYSTEMSCC AUTHOR DcC DATA CARDS
c CA*D i cOMiAiMS THE FOLLOWING CONSTANTScc (I) LL NO. OF TEST HUNScc CARD 2 cOMTfliNjs THE FOLLOWING CONSTANTSc«c (D DFLT — CLOCK PERIOD — TTMF INTERVAL BETWEEN SAMPLESC (2) VTK' — AMPLITIDE OF TEST SIGNALC (3) M — TEST SIGN'AL OPTIONC M = l Pf-^S NOPMALc M=2 Psis INVERSE REPEATC M=3 PHTMF SINUSOIDC K=4 SINGLE SINUSOIDCC (4) ?• — PP-^S RFRISTFP LENGTH (N<10)r (5) T D -Mn.OF PRIMP FPFUUFNCTFSc (6) M] — PFRIOOIC AVFKAGING INOFXC (7) Jl — QUANTISATION INDEX Cc
DIMENSION A(2» bll) ,I^HTM12S1) >ISEQ( 5il)tNUM( 511) CALL CLEA*
2 FORMAT (15) DO 25 K^=l tLLCULL LF*VF
41 CALL FLSI (1 ,KD60 TO (&1 « 19) t*
AT (2FlO.<nbl5) I F ( •••! ) 1 Q . 1 o , 1 5
15 WHITF; (3.1 JDELT, VIN,M,J'J,TP,KII,JICALL CALLC»LL CL^O CALL FiDSc(l)
GO 10 ( ?(), ??,?.** 26) ,M 20 CALL ^P^S (M» A« VIM)
bO TO 34
NL = 2CALL v-HPS (N t AiVIN)MM=OUO 3t j=l.i«N
36 A(2,J)=i (1 , J)DO 3U K=1.2DO 3U J=l .^NI/^I=^N» (K-l ) + JIF(M)'-?. CS3,?S2
53 A(l ,v-..i)=-A (2.J)
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2.2 FFRIP Program Listing
CARD 1 CONTAINS THE FOLLOWING CONSTANTS
CC PROGRAM NAME FFRIP CC FOURIER ANALYSIS USING MIXFD RADIX FAST FOURIER TRANSFORMC FOR THE DETERMINATION OF FREQUENCY fiND IMPULSE RESPONSE FUNCTIONSC AUTHOR D PF.ES, GLAMORGAN POL YTECHNlC»DfPT. OF. ELECT. £NGCC DATA CARDS
C CARD 1 CONTAINS THE FOLLOWING CONSTANTS
C (1) M TEST SIGNAL OPTIONC M=l PPBS—NORMALC M=? PPdS — INVERSE REPEATC M='H PRIME SINUSOIDC f-i=4 SINGLE SINUSOIDC M=5 -"NY OTHER TEST SIGMAL SPECIFIED BY USER AND READ IN ONC DATA CARDSC (?) MM OUTPUT SIGNAL OPTIONC K'M=OOP1 OUTPUT DATA GENERATED BY ?ND ORDER SIMULATOR SUBR.C NN=? OUTPUT DATA READ IN UN CA^OSc (3) ZETA—DAMPING FACTOR--KEOUJHEO ONLY IF OPTJOM MM=Oc (4) WM—NATURAL FREOUE^CY-.PEQUIRFD ONLY IF OPTJOM MM=O
CARD 2 CONTAINS THE FOLLOWING CONSTANTS «•»CC (1) TI«—TIMF INTERVAL BFTwFFN SAMPLESC (?) VTi\i--iMPLITUD£ OF TFST SIGNALC (3) MPPI N--MUi/«FP OF FSTIMATFS TO B= PRJNTFD ( NPRT KIC (4) Ml.lMI—SEQUENCE LF K.'GTH/NO.OF StfPL^S (NUMl<512)c (t,) M__PK«S REGISTER LFUGTH (2»o-h.'-i=f\;u'«-i) c\!<io)c (^) N^ACI — THE NU^ER OF FACTORS OF NU^I <NFAci<6)C CARD 3 THE FACTORS OF NUMI,DFCPEAS ING n»DEP IS OPTIMUMC WITH OPTION <* SFLFTTFU L(l) SpT TO CO-PFSPoMU TO THEC LINE NO OF THF FkFQUENCY
T (i-fTTH MU«I=51 1 ,CA«D 3 l-'TLL CONTAIN 73,7) C (-vITH NUMI=:>10,CAKD 3 n'TLL CONTAIN 17,5,3.2) C
ntCARD 4 TO CARD NUMI/H*3 TO CONTAIN NUMI VALUES OF SYSTEM OUTPUT,
C CORRESPONDING TO NUMr VtLUFS OF TEST SIGNALC (fi DATA POINTS PER CARD)CC IF USER SPECIFIES A SIGNAL UNDER OPTION 4 THEN TEST SIGNAL DATACC CARDS MUST BE INSERTED AFTER CARD 3C
REAL Mor>X»MODY,MODTDIMEWS TOM A(2,511) .T(2..511)»TX(2,250) ,TY(2,250),TXT(2,^11)DIMti'MSlON L(10) ,IPRIM(251)Rt ALMr'.M M»NN»2ETA« WNREAL) (>, 1 ) TIM, VIfM,NPRIN»NUMI ,N,NFACIHLAU (?_< 5) (L ( I ) , 1 = 1 »NFACD
(3.2)W,NN,ZF.TA,WN
VII
WRITE <3,1>T.TM.VIM,NPRIN,NUMT,N.NFACI*'«IIt (3.5) (L(I) . I=1,NFACI)WRITE(3,21>FN=FLOAT(NUMI)TIM=TIMN'PR = f\iHR IN + 1GO TO <?0.?2.24,26,28),M
C OPTION 1 GENERATE PRBS SIGNAL * 20 CALL PRRS <N»A,VIN>
NL = 2DO 32 J=1,NPR
32 IPHIM(J)=J-1GO TO 34
c OPTION 2 GFNFRATE INVERSE REPEAT SIGNAL22 MN=2*«N-1
NL=2CALL PPRS (N.A.VJN)MM = UDO 3b J=1,MN
36 A(2,JJ =e (1,J>DO 30 K=l,2DO 30 J=1,MNMM = ('.N« (K-l ) + JIF <*,>•=?. e;3, 52
53 * (1 ,i..-) =-t (2, J)M = lGO TO 30
52 A (!,MMJ=A (2,J)M = 0
30 CONTINUE
IPHIM(1)=0
GENERATE ODD NUMBER SEQUENCE VFCTOR
Kl = 200 4b J=1.INP»H
60 TO 34
r* OPTIO.--J .? RFNFRAT'E PRJMF SINUSOID SIGNAL AND PRIHC-NUMBFR
24 CALL PRTWS (NPRIN, NUMI ,A, IPRIM, VjN) NL = 4 60 )0 34
C* OPTION * nr.NF.RATE SINGLE SINUSOID SEQUENCE
26 FIP=L(1)DO 7t> J=1»NUMI T1=PIJ*FLOAT(J)/FN FIP=J
75 A(l« J) =SP>J( NL=L (1) +1 IPHIM(1)=0
GO TO 34
C* OPTION 5 PFftD USER SPECIFTFD TFST SIGNAL
rf. 7) ML = 2
34 WRITE(3,A) (A(1»I) «I = liNUMI)SUM = II.ODO 10 I=1,NUMI
SUM=SUV+A 10 A(2tl)=o.
C* fEAD/tSENFRATE SYSTEM OUTPUT DiTA
IF (NfM-1 )54,54»5656 .
fiQ TO 4<354 CALL SSTMU(TXT«WN,ZETA,NUMI,TJM) 49 CONTINUE
»-<i rt (3«3i)h'KlTt (3.4) (TXT(ltJ) ,J=1.NUMI)TX(1»1)=SUM/FNTX(2»1)=0.
DFTER^JMF OFT OF TEST SIGNAL AND CORSECT FOP THE Z£PO OonER HOLD f UNCTION OF SAMPLING PKQCESS
CALL -AST'-i(4.T,L,NUMI.NFACT,-l-,0) CALL SfiHLD(T,TX,NPWlN,JPKIM,FN»TTM) DO 153 J=] .NPK1N
MOL= JPHTM (J+l ) PU=3.141S93°f: LOAT (MQL) J» = HI J/FN
TX(1,"OL)=S*TX(1»KOL)/WWW 153 TX(2,K.nL)=S*TX(a,KOL)/^i''W
00 40 J=1.NUMI A(1.J)=TXT(1 ,J) TXT(1.J)=T(1»J)
40 TXT (?t.J)=T(2»J) C*
SUM=0 .000 SO J=1,NUMI Sijv, = SiiM + A (1 , J)
50 A (^« J) =n.nTY<1,1)=SUM/FN TY(^,1)=0.
!.)F.Tt'^--r lM |r OFT OF SYSTE." nUTPUT DATA AND CORRECT FOR THE ZEPn pRDER HOLD HONcTlON OF SAMPLING PROCESS CALL FASTM(A,T,L»NUMI,NFACI,-1 .0) CALL SAriLn(T,TY,NPhIN',IPRIM,FN,TTM) UFTEn ; iT K| F CROSS COPRELATION FUNCTION USING FFT OU 60 J=1.NUMIA(l,J)=(TXT(l»J)«T(l,J)+TXT(2,J) e T<2,J) )/(FN*FN)
60 A(i^.j) = (TXT(2»J)*T(l,J)-TXT(l . j)«T(2»J) )/{FN»FN) FAST"(A,T,L»NUMI,NFACI.-1.0)
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SUBROUTINE P«BS(N,XSFQ,VIN) C*C* PSFUDO-PffNHOM BINARY SEQUENCE .GENERATOR" C» N — MO. OF PFGISTFP STAGES N<11 C* XSFQ-'-OrJ FXIT PRBS SEQUENCE ARDAY r* VIN--AMPLTTUDE OF bEUl'ENCE C*
TO CM/.MRF POLYNOMIAL ro^FFiciP^TS OF A REGISTER CHANGEDATA CAPO CORRtSHONDIlvG TO IT.FX.:TF RrGisTp.* OF LENGTH ^ is TO HAVT POLY\'OMIOL/0,0.1,1,0,1,0,0,1,0/ .THEN. REPLACE DATA CfiRD RELATING TO REG. 9 BY THF Ff)LLOi>; lUG STATEMENT
HE(f'..R).IE<9,8),IF(10.a)/0,0.1,l,0.1,0.0.1 f O-/
DIMENSION X5EO(2,102^) »TREG(10),IE(10,8)c*»r**«* POLYNOMT^L COEFFICIANTS FOR :- C •»'--c* REGISTER OF LENGTH 3
DATA JF(l.l),IE(2,l).IE(3,l),!F(4,l),IEf5, 1),IE(6,1),IF(7,))» 1IF. (M.!). IF (9.1), IE< 10. 11/1. n, 1,0,0, n, 0.0.0, O/
C* KF.GISTFR OK LENGTH *DATA TF(1.2>.IE(2.8),IE(3.?),IE(*.2),IF(5,2),IE{6,?).IF(7.2). llF(ti,if).IE{9,2),IE(10,2)/l,0,0»l.»0,0,0,O t O»0/
C* HEGISTFR OF LENGTH 5DATA IE(1.3),IE(2,3),IE(3»3)«IF{*.3),IE(5,3).IE(6,3)»lE(7.3)t HE(«.3).IEi9,3),IE(lO«3)/0«l,0»0,l,0,0,0»0.0/
C-> kFGISTr-R 'OF LENGTH 6DAlA ip (1 . A), IE (2, A), IE (3,4) , IF (*. 4), IE (5, 4), IF (6, 4), IE (7,4), lIE{b»4).IF<9,4),IEllO.<O/1.0,0.0«0.1.0.0»0»0/
C* ^hGISTpk OF LENbTH 7DATA TF(1.5)»IEK2,5).JE<3,5).I?:c4.5),IF(5,5>.IE<6.5).IF<7,5>. H F ( to » S ) . I F ( 9 , 5 ) «IE(10,b)/l,0,0,0»0,0,l,0»OfO/
C* Kfr,isTFK Of" LENGTH &DATA TF(1.6>,IE.{2,M,Tt'<3.f),TEf<t,fr)iIF(&,6).!E(6.6)«IF(7,fi), llt(b.h) , If. (9.6) ,lL(10.b)/l,l,OtO,0,0,l,l,0,0/
C« KLGISTER OF LENGTH VDATA JE(1.7).IE(2,7),IE(3.7),IE(4,7J.IE(6,7),IE(6,7).IE{7,7)»
1 I F, < a , 7 ) . I F. ( 9 , 7 ) .IE(10.7)/0.0,0»1«0»0»0,0»1.0/ C* Kf'GlSTER OF LENbTH 10
DATA TF(l»H>»It(2-.«>,IE(3,«)»IF(<nfl)tIF(5,ft),IF. (6.8),Iir (7,8), HE (8, 8). IF (9, 8). IE ( 10. 8)/0«0, 1.0, 0,0, 0,0,0, I/
DO 10 J=1.N10 IHt<i(J)=l
MM=N-2DO 20 J=1.ISEOL JK = 0DO 30 M=1.NJR = lH>- G ('•') "IE (M,KM) +JH IF (Jr(-if)3n,ll,30
11 JR=030 CONTINUE
IF ( J^)21,31,21 21 XSEQd ,J)=VIN
GO TO ??31 XSL>J(] ,J)=-VIN ?? NG=N-1
DO 10 L=1,NGLi=N+l-L
40 IREG(L) )=IP£G(L1-1) 20 IKti3(l)=JR
RETURN END
SUBROUTINE PR I MS < IP ,N , TEST , IPRIM ,r«c* GFNEHATF.S PRI^E NUMBER VFCTOS AND PRIMP- SINUSOID. TEST SIGNALrfr IP — NO, OF PRIME FREQUENCIES INCLUDED IN TEST SIGNAL IP<169 f-* KJ — T.EST SIGNAL SEQUENCE LFNGTH N<512c c TEST — T(T. C.T SIGNAL APKAY OF DATS POINTS
— PPTMF NUMBEH VFCTOR — AKHLIT'JPE OF SINUSOIDS
C*DIMENSION IPRIM (3sn> , TEST (2,511)
C EVALUATF PRIMESC AND e-MFR/,TESC PklrfE SINUSOIDALC TEST SIGNALC
PIJ=2. 0*3. 1415936
(?)=3 IPHIM(3>=5
, (6) =13
1 = 11
DO 35 J=3?»N Ir ( (J/?l )-3l-U)7
7 1; IF ( ( J/?o) *>?9-J) 76.35»3576 IF{ (j/?7)»23-J)77»35»3577 IF( (J/1Q)*19-J>78, 35.3578 IK r (J/l 7)»]7-J) 79.35.3579 1K( <j/i-3)*n-J)eO,35»3580 JF ( {J/i i )*i i-J)b] ,35,3581 IF < (J/7)*7-J) ti2«35,35 PH IF ( ( jx?) C -J) b3»35»35 P3 IF ( ( J/3) *?-J) «4,35»35 P4 IF ( ( J/?) »?-J) b5-35t35 85 1=1+1
IPHIM(I)=J 35 CO\7^:tlF
DU lu 1=1 «N
DO 20 L=1.IP
20 XC=XC+VT10 TEST (1 , T)=XC/FLUAT(IP)
END
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SUBKOUTTNF SSIML(A , «N, ZET A,NJUMI , Tl M) C« C* Slf-lULATo^ OF 2ND OKDER SYSTEM
C* T^A\'SFt^ FUNCTION Wf-J«»?/ ( S»e?+?c-ZF.TA«WN»S*WN»«-?)C* A — Tf-ST SIGNAL ARWflY OF D*Tfi POINTST* *rt — NATi.lPAL FKEUUtNCYr* ZETO-UAMPJNG FACTOR ZFT/KI.OC« NUwJ — MJ-HFR OF DATA POINTS MIJMT<512r* TI'-'.F-Tlvp IMTFUVAL BFTWFEN SAMPLFS
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1 CONTINUFDO 2 1,' J=1,NUMI
0 A.(1.J)=A(2,J)RETUkwEND
XVI
SUbKUUTTME SAHLU (T.TY.KPRTM. TP«TM.FN,TTM)
P* COUWECTS FOR ZERO ORUER MOLD FUNCTION Of SAMPLING PRQrF.SS
c«- T _ 2-iJ o^p/sY OF DATA POINTS- — INPUT ARRAY(-«. TY__2-D ^POftY COKktCTED DATA POINTS — OUTPUT ftPRAYr* NPrflN — ••>;<",. OF DATA PQlMTii CORRECTEDr *. IPPJM — PPJMF NUMBER VECTORCt FM —— TEST sTPMAL SEUUFN'CE LENGTH
T(2tbll),TY(2.250),lPRIN(l)
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27 PT=160.+P 50 RETURN
END
xvii
2.3 FFRIP Program Input and Output Data
2.3.1. Nomenclature
M Spectral line
f Frequence Hz
MOD(TX) Magnitude of prbs, normalised about first spectral line
PH(TX) Phase of prbs degrees
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PH(TY) Phase of system output degrees
MOD(TF) Measured magnitude ratio of system transfer function
PH(TF) Measured phase of system transfer function
T Time seconds
H(T) Measured system impulse response
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2.4 Listing of Supporting Subroutines
SUbKUUTlNE TRIfaF <LX , X , W ,S , C) N X(2»1024)
s=o.o c=o.oDO 10 1=1, LX C = C + CUSr\'W*X ( S=S+Si rMW*X (
10RETUkM
XXXI
SUtt^uuTJMF NLOG.N (rg»X »SIGN)NU^'ri^- OF SAMPLES = 2* fl'NSI ON IS LfiBFLLf.U -1 FuK TIMr FUh'CTTO>' TO
+1 F"K FRFOUFKrY FUNCTION TO TIME COMHUTFS DISCRETE FOU--1FP ToANSFO»f. OF y
00 1 I=1»N M (I >=?«*< N-I) [JO 4 L=l »N
DO * I»LOK=1
] ) =COS (V) (2) =SIM( V)
DO 2 I=1.L3HAF J=IS7AT+IJri = J + L
X ( ) « J-) =X ( ] , J-) -'i(l X(?.,;~.)=X (?.J)-a<2 X ( 1 » J ) = * ( 1 » J ) * (J ( 1 )
? CONTIrUF 00 3 J=?»M TI = IIF(^-••--(T) )
3 i^ = ^-t-(I) A K = < + '• ( T I )
00 7 J=1»LX IF(K-J) 5,9,9
9 MOLO( 1 ) =X (1 « J) HULU (>•) =X (?, J) X(l ,vM=X (1 ,K*1) x (2. J) =X (?,K*1 ) M 1 tr *1 ) =»-JLD ( 1 ) K (2,r<*l )=nOLD(2)
5 00 6 ]=1»N II = ! IF(«-'-(T)) 7.6,6
6 K=K-H(I)7 K=K+v(TT)
10 DO « 1=1X(1«I)=V
P X{?, I }=X
END
XXXI1
cp.ossiiSF.QL.SRE.sP. i SFO.MCOR.ACORL)CC EVALUATF. GROSS CoWRFLaTlQN nF SYSTEM- REPONSE WITH PRBS INPUTC C
REAL 1SEODIMENSION ISEU(1024),SHESP(1024).XCORL(MCOR)FSEQL= FLOAT(ISEQD
MH=iSrOL-Ml+l Cc DELAY GFNFRATORc SCOHL=P.O
DO 70 M3=l,ISE:QL
71.71.7?7? y,2=l71 CONTINUE
C C; FVflLUTP GD05S CORPrL«TIONC
70 ^CO»L= t;rO^L + e:Pfc'SP('>'3)fSO XCO^i (Ml ) =SCO*L/FSEQL
WtTEND
xxxm
C C C
C C C
SUBROUTINE PINT (F,M,T!NIT.TMAX.TTNCR,WLOW1.W^AX1,WTNC»].FJW,PHI) THE EVALUATION OF THF FOURJFR InTcGPAL FOR AW />RBITR/>WY FUNCTION
DIMENSION F (400) »FJW{400) »PHJ (4 on)L=MN=MU = 2. 0<>3. 141593
LTMI1= (I..MAX-WLOW) /WjNCR+1
DO 10 T=1.LJMI1W = W + Ml I NJ C R A = 0.08 = 0.0
DO 20 J=1.N.2
ATTON BY SIMPSO'-'S
C=F{J)»C05(W*T)
= F ( J)r J+l ) *S TIUCR) )
A=TINCP*A/3.H=-T • ;
IF (A) ?1 «??»?3
GO TO ?0 25 PH! (I)=+90.
GO TO 2B 23 Phim = (
GO Tu ?P. ?1 1FIB) 26 PHI (I
•30 TO ?°?7 PH! (I)=+1HO.+(ATAN(B/A) )*lPO./3. 141 5927 28 CONTINUE 10 CONTIMUF
) »] &U ,/3 . 1 41 5927
)*l»0./3. 1*15927
ENU
XXXTV
2.5 Subroutines CALL Procedure
2.5.1. Subroutine PRBS
i. Purpose
Computes a maximum length pseudo random binary sequence of
length 2 n -l where n is in the range 3 to 10 inclusive.
ii. Calling Sequence
CALL PRBS(N,XSEQ,VIN)
iii. Description of Parameters
N Sequence register length S^NglO.
XSEQ Two-dimensional array. The first column of the array
contains the prbs sequence on exit. The second column
is left unchanged.
VIN Amplitude of the prbs sequence.
iv. Additional Program Information
The sequence polynomial coefficients are stored in array
IE (I,J) where I corresponds to 10 (maximum number of stages
of prbs register) and J = N-3. To change the coefficients
of a sequence simply requires the changing of the DATA card
relating to that sequence. For example, for a register of
length 9 (J = 6) with feedback from registers 2, 3, 5 and 9
the DATA card (IE (1,6)) must be changed from
70,0,0,1,0,0,0,0,1,07 to 70,0,1,1,0,1,0,0,1,07.
2.5.2. Subroutine PRIMS
i. Purpose
Generated a prime sinusoid signal which is given by VIN • ,irv X(nAt) = 1™ _£ . sin(-1R—)
if2 i=prime
xxxv
Ngiving a real sequence x = z X(nAt)
n=l
i i. Calling Sequence
CALL PRIMS(IP,N,TEST,IPRIM,VIN)
iii. Description of Parameters
IP Number of sinusoids in the composite sinusoid test
signal.
N Sequence length.
TEST Two-dimensional array, the first column of which
contains on exit the prime sinusoid sequence.
IPRIM Array containing prime number integers.
VIN Test signal amplitude constant
2.5.3. Subroutine FASTM
i. Purpose
Computes the discrete Fourier transform of N data points
using the fast Fourier transform mixed radix algorithm,
ii. Calling Sequence
CALL FASTM(A,T,L, NUMI, NFACI, SIGN)
i i i. Description of Parameters
A Two-dimensional array containing the data points to be
transformed. If the data is real the second column of
the array (imaginery part) should be set to zero before
entry. Data in A array is destroyed on exit.
T Two-dimensional array containing the discrete Fourier
transform on exit. Real part in the first column,
imaginery part in the second column.
xxxvi
L Integer array containing the factors of NUMI
(decreasing order is optimum). It is required that
eich factor is less or equal to 100.
NUMI Number of data points to be transformed.
NFACI Number of factors of NUMI. NFACU15.
SIGN Set to -1 or +1 and determines whether the forward or
inverse transform is evaluated. With SIGN = -1 eqn.
3.4 is computed and with SIGN = +1 eqn. 3.5 is
evaluated.
2.5.4. Subroutine SAHLD
i. Purpose
Adjusts the spectrum estimates obtained from the discrete
Fourier transform to take account of the frequency
characteristic of the sample and hold function of the ADC
and DAC.
i i• Calling Sequence
CALL SAHLD(T, TX, NPRIN, IPRIN, FN, TIM)
i i i. Description of Parameters
T Two-dimensional array containing the data points to be
adjusted.
TX Two-dimensional array containing the corrected data
points
NPRIN Number of data points to be corrected
IPRIN Integer array containing prime numbers integers
FN Number of data points
TIM Sampling interval of the sample and hold device.
xxxvn
2.5.5. Subroutine SSIML
i. Purpose
Generates a sequence which is the output of a second-order
underdamped system to an inpi't sequence x(k), where the
input sequence is applied through a zero-order hold function.
ii. Calling Sequence
CALL SSIML(A, WN, ZETA, NUMI, TIM)
iii. Description of Parameters
A Two-dimensional array. On entry the first column of
the array contains the input sequence. On exit the
first column contains the output sequence.
WN Natural frequency of oscillation for the system rad/s.
ZETA Damping factor of the second orde"- system 0<5<l.f>
NUMI Length of data sequence.
TIM Time interval between samples.
2.5.6. Subroutine ANGLE
i. Purpose
Calculates the phase in degrees of a two-dimensional vector
defined in cartesian co-ordinates.
ii. Calling Sequence
CALL ANGLE(R, I, PT)
iii. Description of Parameters
R Real part of a two-dimensional vector.
I Imaginery part of a two-dimensional vector.
PT Subroutine output in degrees.
xxxvm
2.5.7. Subroutine TRIGF
i. Purpose
Computes one value of the discrete Fourier transform
providing two outputs, S and C which are the sine and
cosine transform respectively,
i i. Calling Sequence
CALL TRIGF(LX, X, W, S, C)
i i i. Description of Parameters
LX Number of data points
X Two-dimensional array of data points. If the data is
real the second column of the array should be set to
zero.
W Angular frequency in rad/unit time
S Sine transform subroutine output at frequency W.
C Cosine transform subroutine output at frequency W.
iv. Additional Program Information
The following program patch enables the subroutine to be
incorporated in FFRIP (replacing FASTM):-
DO 10 J = 1, NPRIN
W = PIJ * FLOAT (IPRIM(J))/FN
CALL TRIGF (NUMI,X,W,S,C)
A(1,J) - C
10 A (2,0) = S
2.5.8. Subroutine NLOGN
i. Purpose
Computes the discrete Fourier transform by the radix-2 fast
Fourier transform method.
XXXTX
i i. Calling Sequence
CALL NLOGN (N,X, SIGN)
i i i. Description of Parameters
N Positive integer
X Two-dimensional array of data points. The first
column contains the real part and the second column
the imaginery part of the data. If the data is real
the second column of the array should be set to zero.
SIGN Set to -1 or +1 and determines whether the forward or
inverse transform is evaluated. With SIGN = -1 eqn.
3.4 is computed, and with SIGN = +1 eqn. 3.5 is
evaluated.
2.5.9. Subroutine CROSS
i• Purpose
Computes the crosscorrelation function of two real numbered
vectors.
i i. Calling Sequence
CALL CROSS (ISEQL, SRESP, ISEQ, MCOR, XCORL)
iii. Description of Parameters
ISEQL Length of the sequences to be correlated
SRESP 1 Sequences to be correlated. One dimensional
ISEQ J arrays
MCOR Number of points to be correlated
XCORL Subroutine output containing the crosscorrelation
function.
xl
2.5.10. Subroutine FINT
i. Purpose^
Computes the Fourier integral for ar. arbitrary time
function,
i i. Calling Sequence
CALL FINT (F,M,TINIT, TMAX, TINCR, WLO*.1 !, WMAX1, WINCRl,
FJW, PHI)
iii. Description of Parameters
F Time function to be analysed m, ,m2 .. m
M Number of time function data points (M must be odd)
TINIT Lower time limit of F
TMAX Upper time limit of F
TINCR Increment of time between time function data
WLOWI Starting frequency Hz
WMAX Maximum frequency Hz
WINCRl Frequency increment Hz
FJW Subroutine output array containing the magnitude
PHI Subroutine output array containing the phase.
iv. Additional Program Information
For the data analysed in this investigation the subroutine
argument values were selected as follows:-
TINIT = 0
TINCR = At (clock period)
TMAX = MAt where MAt was chosen to be approximately equal
to five times the systems largest time constant
WLOWI = fl]j-N is the sequence length
WINCRl = WLOWI
and WMAX1 = -^
xli
APPENDIX 3
Response DATA for a Second Order Dynamic
System as Measured by FFRIP
System response data up to spectral line 249 is presented for:-
System Parameter :- N=511 , At=20 ms
Test Signal Parameter:- prbs, N=511 , At=20 ms
Nomenclature
M Spectral line
f Frequence Hz
MOD(TX) Magnitude of prbs, normalised about first spectral line
PH(TX) Phase of prbs degrees
MOD(TY) Magnitude of system output, normalised about first
spectral line
PH(TY) Phase of system output degrees
MOD(TF) Measured magnitude ratio of system transfer function
PH(TF) Measured phase of system transfer function
T Time seconds
H(T) Measured system impulse response
THOD(TF) Theoretical magnitude of system transfer function
TPH(TF) Theoretical phase of system transfer function
TH(T) Theoretical system impulse response
xlii
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APPENDIX 4
4.1 Derivation of the Output from a Polynomial-Type Nonlinearity
when Applying a Composite Sinusoid Signal
A nonlinearity described by a polynomial expression is given by
y = a + a-,x + a 9 x + a~x + —— + a x p A4.1w I C- O U*
LetN
x = _E sin( Wi t) A4.2
so that,
N N N y = a + a, E sinu-t + a 2 E E sin^t sinwjt + ——
N N N—— + a E E —— E sinto-t sinw-t —— sinw t
P 1=1 j=l p=l 1 J P A4.3
The output y consists of sine products of the formj_ U
sine-! sine 2 —— sine n for the n polynomial term.
-With n even^n/2t n n r sine, sine 2 —— sine p = -^ff- cos(e 1 + Q 2 — - + e n )
2 Ln terms each
- 005(8, +6 9 + — + e n i-e )+ — ) containing 1 * n-i n Qne _ ve e
+ cos(Q 1 +Q ? — +e n _ 2" e n-r9 n )+ "" } C 2 terms eachcontaining two -ve e
(-I) n/2 {cos(e 1
1 ncn terms containing n/2 -ve Q S 1 A4.4
xlviii
Note:- nc = ,, n! v , A4 5 r V: (n-r): m-J
Total number of cosine terms = 1 + n + n c + -- nc + ~ n cr n T 2 n1
n n
A4.6
Coefficient of cos(e-j — e n ) where m is the number of negative e 1 is
given by
H f2""1 A4.7
VJith n odd n-1
sine 2 — sinen = -^^ |"sin(9 1 +e 2 — +en )
^
(6i «r__ t9n_ 2 -on.ren ,-, =n-1
+(-1) 2 sin(e 1 +Q 2 - — + e n+1 — -e n )
,
Total number of sine terms = 1 + n + nc 2 — +
A4.8
= 2n-1 A4.9
Coefficient of sin (9j — ep ) where m is the number of negative e^
given by
n-1
2n-l A4.10
xlix
Let cosmi (er e 2 ,— e n ) = cos(e 1 +e 2— -on )
where the argument on the R.H.S. contains m negative terms
i = 1,2, — number of permutation
Similarly,
A4.12
With n even n ,n
sine, sine,, 12 •"v£f-i
1 n c I n/2
A4.13
With n odd
"- /2
sine, sine 2—s1nen =
-(£ -1)I I
:. m=0 Z=l
2 '
2=1
H)m ^ sinm1 0 1 ,-en )
A4.14
So that for the p ul term of the polynomial nonlinearity:-
with p even -
a - N N N N r -&E z — z (-!/
.1=1 j=l k=l p=l L
A4.15
and with p odd -
a XP =a r N N N Nr IN IN IN IN r
•_i ^ * s z (-1)'Li=j j=l k=l p=U
\mP" 2p
4.1.1 Example for Odd and Even Powers
With n odd and equal to 3
from equation A4.14
2 = 1
A4.16
1 1 "(-1 ) m sin . mi
wnere
S1' nmi e l ' 6 2' e 3^ ~ S1' n (t0 iie 2- G 3^ wni' cn contains m -ve terms
and i = number of permutation
Therefore, 3°1 f= - -^ z si
If 1= - ^ z si
3_z i
= - 4- sin(e, +e
- sin(-e-| +62+65)1
1 S e 2 ,e 3 )
li
With n even and equal to 4
from equation A4.134C
\2, 1 _. msin6
i x < i\ w ,tm=0 i=l
! 18 Li'l C°Soi ]
+ (-1) 2
4
2' 3 s 4 i=1
14? ?
z cos^^e,,i=l
/)s li l 1' 2 s 3'
60,83,6
3
4 i=l
1 [cos 01 (9 1 ,e 2 ,e 3 ,e4 )-cos 11 (e 1 ,9 2 ,8 3 ,e 4 )-cos 1 2(6^62,
-cos 13 (e 1 ,e 2 ,9 3 ,9 4 )-cos 14 (6 1 ,e 2 ,6 3 ,9
,62,63,64)]
= i rcos(6i+e2+e 3+e 4 )-cos(9i+e 2+e 3 -e 4 )-cos(9-|+e 2 -e 3 +e 4 )
+9 3+9
lii
4.2 Autocorrelation of a Composite Prime Sinusoid Signal
A prime sinusoid signal is given by
nX(t) = Z a.sinw.t A4.17
i=m '
where m = prime and m-=f 1> ro ^ 2
The autocorrelation of a function X(t) is
.TX(t)X(t~r)dt
"' A4.18
so that
T n n•L f (z a,sinoj.t)( z a.sin(u>. (t-r))dt
•i-T i=m i=mA4.19
Integrating ov_j r the period
(o, /- 2ir/(jJi n n<f>v,v,(i:) - ^r~ \ ( £ a.sinu.t)( z a-sinw. (t-r))dt
xx 2lT J o i=m 1 1 1=m 1 1A4.20
This integral separates into two terms-,
-i ? i = ' z a . sinoj.t sinw. (t-t)dt11 1 1 1011
W-, n n + ' £ x I a.a .sinw-t^ i=m q=m J 0 J
A4.21n af
The first integral reduces to z -*•i=m
and the second equals zero. Hence
„ a?
liii
APPENDIX 5
5 . 1 Sinusoidal Describing Function
The sinusoidal describing function is a method of representing a
nonlinearity by a linear gain when the input to the nonlinearity is
sinusoidal. It is defined as the ratio of the fundamental component
of the output of ths nonlinearity to the phasor representing the
sinusoidal input, which is
A5.1
where
N(ju>) = describing function
Y(jw) = fundamental component of output determined by
Fourier analysis
X(ow) = sinusoidal input signal
For a saturation nonlinearity with unity gain in the linear region
A5.2
where
A is the amplitude of the input signal
a is the saturation limit.
5.2 Gaussian Input Describing Function
The Gaussian input describing function is a method of representing
a nonlinearity by a linear gain when the input to the nonlinearity is49
random and Gaussian. The method is due to Booton and is well covered
in the literature41 ' 48 ' 50 so that only a summary will be given here.
Let the input to the nonlinear element f(X) be represented by X(t)
liv
and the output by Y(t) where both X(t) and Y(t) are random variables.
The error in approximating the true output Y(t) by an equivalent gain
K times the input X(t) will be
e(t) = Y - KX
or e(t) = f(X) - KX A5.3
It is now necessary to choose the value of K that will minimise the
statistical mean-square value of the error e(t)
e(t) = T (f(X)-KX) dt A5.41 Jo
For analytical convenience eqn. A5.4 is rewritten using the alternative
probabilistic description. So that if the input signal X(t) has a
probability distribution P-j(X) then the probability that the squared
error has an amplitude in the range
(f(X)-KX) 2 to (f(X+5X)-K(X+c5X)) 2 due to X in the range X to
X+6X is also P
Hence
=J C
(f(X)-KX) 2 P 1 (X}dX
The value of K required is that which minimises the error power, and
this is given by the value which makes
This value of K is called the equivalent gain K . Hence if A5.5 is
expanded and differentiated with respect to K, the equivalent gain is
K" 0 J-co
A5.6
where o2 is the mean power or variance of the input signal. On the
assumption of a gaussian input probability density distribution,
Iv
that is
T-2 ) A5.7 2o
then, on substitution and integration
Keq = ^727[" ^T1 e*P(- ~2> d * A5.8
For the particular nonlinearity of saturation, considered in this
paper, with saturation limits in the range of + A , then the
derivative *^J is unity in the range -Ag to Ag . So that eqn.A5.8
becomes
By making a change of variable, u = X/a/2 eqn. A5.9 reduces to the
well known error function,
exp(-u")du = erf (-I-) JO ^° A5.10
This integral is not directly integrable but its numerical solution
is widely tabulated for the argument A//2a.
The estimates obtained using the Gaussian input describing
function (Chapter 6) were arrived at by first measuring the variance
of the signal at the input to the nonlinearity. This was done by
sampling the error signal using the ADC and evaluating the variance of
the sampled values. Having obtained the variance and knowing the
saturation limits the equivalent gain for the saturation nonlinearity
was evaluated. The parameters for the second order linear model were
then determined using the relationships:
Ivi
to^o -n^ eq
and51 w_ i 1 nl
a, „ n2 A5.ll
whereto , and 5, are the natural frequency and damping factor
respectively of the linear part of the forward path transfer function
and co 2 and £o are the natural frequency and damping factor
respectively of the linear model for the nonlinear system.
Ivii
APPENDIX 6
Paper presented at the IEE Conference on "The Use of
Digital Computers in Measurement"", University of York, September,
1973. IEE Conference Publication Number 103.
Iviii
DIGITAL PROCESSING OF SYSTEM RESPONSES TO PSEUDO-RANDOM BINARY SEQUENCES
TO OBTAIN FREQUENCY RESPONSE CHARACTERISTICS USING THE FAST FOURIER TRANSFORM
J D Lamb and D Rees
1 Introduction
The study of system dynamics using frequency response characteristics as the signature is well established. Recently it has been shown how digital ly generated signals known as Pseudo-random Binary Sequences (PRBS) can be regarded as a series of sine waves of discrete, well defined and readily varied frequencies and hence the analysis of the response of a system to such a sequence can reveal the complete frequency response 1 . In doing so the amount of test time required is reduced considerably from that needed if single sinusoids are used as in frequency response analysers. Also the necessity for obtaining the complete impulse response by input-output cross-correlation prior to Fourier transformation is avoided. The response to PRBS is transformed directly and hence for most effective mechanisation the Fast Fourier Transform (FFT) should be incorporated. This paper will show how the procedure based on 2N data points produces errors if used directly, or requires additional equipment and computation2 . However, because the lengths of PRBS are in general (2^-1) arid can be expressed as multiples of prime numbers the mixed radix FFT algorithm can be applied 3 . The computational speed benefit over the continuous Fourier Transform will be illustrated by example as will the minor decrease in speed from that of the radix-2 transform. The investigation described here justifies the procedure, followed by an analysis of errors when used on-line. Quantisa tion levels, sampling rates, smoothing techniques and changes in system dynamics have been considered with respect to the presence of both random and deterministic noise in the system response. Sample results are con tained in this paper.
2 The Mixed Radix Fast Fourier Transform
As has been mentioned the period of PRBS obtained from shift registers has value (2^-l)At where N is the length of the shift register and At the period of the clock generating the sequence 4 . These values are odd and may be expressed as a product of prime numbers. Table I 5 lists a few such products for values of N available in commercial PRBS generators at the present time and including those suitable for dynamic testing of systems.
Nichols and Dennis 2 use PRBS to obtain frequency response via the radix 2 algorithm but to do so introduce additional equipment and an additional sequence of PRBS, the time advantage of the power of 2 algorithm then being reduced as illustrated in Table 2. This table compares numerical estimates of the number of operations required to perform in the usual manner the Discrete Fourier Transform, the radix 2 algorithm, and the mixed radix algorithm for a sequence of (2 10 -1). The algebraic relationships have been taken from Reference 6. Also given are actual timings obtained on an IBM 1130. It is important to notice that there is considerable benefit in adopting the mixed radix algorithm in this case. Later it will be shown
J D Lamb is with UWIST, CARDIFF, U.KD Rees is with Glamorgan Polytechnic, Treforest, Glamorgan, U.K
how attempts to use the radix 2 algorithm directly by adding a further data point to the PRBS produce errors in the results.
3 Implementation
It is intended to study in detail the effects of quantisation, sampling, averaging, and noise of both deterministic and random nature on the results obtained by using the mixed radix algorithm. Having done so under control led conditions the field application can be approached with optimised test parameters. Here, therefore, a hybrid computer system, including a Solartron HS7 analog computer and an IBM 1130 digital computer, has been used to achieve this controlled mechanisation of the procedure. Figure 1 gives the arrangement and Figure 2 the flow diagram for the computer prog ram. It can be seen from Figure 1 that the unwanted noise is added to the system output before being sampled and also that the PRBS is generated within the IBM 1130.
k Results
To assess previously mentioned factors of interest quantitatively, root mean square error values and estimates of bias are obtained between theor etical and experimental results for a linear second order system. These error criteria have been calculated up to a frequency of twice the system bandwidth (w ^2-4 Hz). The information presented here has been chosen from the wide range obtained to illustrate the sort of results to be expec ted for given situations. First the case with changing quantisation levels.
Figures 3/4 illustrate both amplitude ratio and phase characteristics of the system as estimated using the procedure with no noise present for the extremes of quantisation, 12 and 3 bits respectively. In each case the theoretical results are shown for comparison purposes. The degradation of the excellent results obtained with 12 bit quantisation for reduced values of quantisation is evident.
The effects of the addition of the noise signal of various forms are shown in Figure 5. Here both bias and root mean square error values have been produced for the whole quantisation range considered. As can be seen errors in magnitude only have been shown here. Three forms of noise have been studied (a) 'white' noise having a rectangular low pass spectrum with bandwidth 27 KHz (b) 'pink 1 noise which has a power spectrum decreasing at 3 db /octave from 3 Hz to 20 KHz and (c) sinusoidal noise at a frequency of 4-5 Hz. In all cases a signal to noise ratio of 10 has been established and the results in Figure 5 for a single PRBS period following the initial ising sequence- (see Figure 2). It is evident that in the absence of noise, quantisation of 6 bits or more is satisfactory. The results as assessed by the bias and rms estimates indicate that the sine wave noise and the 'pink 1 noise produce most scatter but whilst the 'pink' noise produces a scatter almost uniform throughout the spectrum that due to the single sinusoid is principally evident around that frequency and in fact peaks rapidly at the frequency of the sinusoid (Figure 6) . This confirms results obtained in a field test application of the technique where the system signal being examined had a predominantly sinusoidal noise present superimposed on the response produced by the PRBS?. It is also observed w'nen coarse quantisation is used (3-»-5 bits) that the resulting errors become reduced in the presence of 'white 1 noise which is acting as a dither signal at this time (Figure 5). It is not possible to include graphical information regarding sampling and averaging because of space
limitations. It is found however that an optimum sampling rate exists
and as expected increasing the computation and measurement time improves the performance of the procedure in the presence of noise. Also it is found that the time of sampling is important in that it can if incorrectly chosen introduce an effective time delay or phase error increasing linearly with frequency.
.5 Radix Two FFT
In an attempt to benefit from the computational speed >£ the radix two algorithm the data record of length (2N-1) was extended to 2N by atcaching one more sample to both the sequence and the system output. The mixed radix transform program was replaced by the appropriate radix 2 program. The tests carried out resulted in the timing shewn in Ta^le 2 confirming the time advantage. Unfortunately the frequency response characteristics showed bias and rms errors of approximately 10 times those of the noise free mixed radix procedure.(see Figure 7).
6 Conclusions
It has been shown how the benefits of the Fast Fourier Transform may be made available to the procedure for obtaining system frequency responses using PRBS. In this way gains are made not only in the reduction of test time but also in reduced computational requirements. The errors intro duced by the presence of noise of various forms have been demonstrated, and results obtained show how these errors can be minimised4 It is considered that the hybrid computer system has been developed to such an extent that unknown system identification may be approached with confidence.
7 Acknowledgements
The authors would like to thank Mr J James, formerly of UWIST and now with the Admiralty Research Laboratory, Mr W Lambert and Mr D Doyle of Glamorgan Polytechnic for their assistance in this project.
8 References
1. J D Lamb 'System Frequency Response using p-n Binary Waveforms' IEEE Trans. 1970 AC-15 pp 475-480
2. S T Nichols and L P Dennis 'Estimating frequency response function using periodic signals and the FFT' Electronics Letters, Vol 7, No: 22, pp 662-663, 1971
3. R C Singleton 'An algorithm for computing the mixed radix fast f-ourier transform' IEEE Trans. Audio and Electroacoustics, 1969 Vol AU-17, No: 2
*• W P T Davies 'System identification for self-adaptive control' Wiley, 1970
5. V W Peterson 'Error correcting codes' MIT Technical Press, 1961
6. J W Cooley, PAW Lewis, P D Welch 'The Fast Fourier Transform and its Applications' IEEE Trans. Education, Vol E-12, No: 1, March 1969, pp 23-24
7. J D Lanb 'Direct Frequency Response Determination Exploiting the Deterministic Characteristics of Pseudo-random Binary Sequences' Proc. UKAC Conference on 'Automatic Test Systems' Univ.Birmingham 1970
Register length Sequence length
N
34
567
8910
1112
Table 1 Factorisation of
2N-1 (M)
715
3163
127
255511
1023
2047
4095
N 2 -1 into Primes
Prime Products
M
73-5
313-3-7
1272Jt*,lJ
7-73
3-11.31
23-89
^\DFT
FFT radix 2
FFT mixed radix
FFT Nichols B Dennis
No: of operations
M2
M 1og 2?1 n
M Z r-. 1=1 1
2M log2M
Approx. No: for M=1023/1024
10 6104
4.5-10 1*
2.104
1
Time (relative)
100
1
5
2
Table 2 Comparison of Computation required of DFT/FFT Technique^
iVSTEIVtUNDERTEST
DIC.1TAL.__"COMPUTER ANALOG COMPUTER
DIGITALCOMPUTER'
FIG 1 Hybrid System Arrangement
o (O
It^I
,x
*•
n c •5! 2
i
X ».
• 7°X
L^
^*
'^
i?3"
-5 o
5 0
S •"
'.Z .o
• eraI CO!«= et
__!!!» i
CD ! 4.00
I ,.- ffi- L'.JSINUSD'!OflL^!JI5E;.i.5C/S.O :N3i :SE I >• II !
00 6-00 i 7.00 > S-CO : S.CD 1C.00 11.00, 12i | IOURNTISRTION E>ITS| | | i ; ' •-<--••--••—-—.—•—'—r-y—j-~f-—r:jr
FIG 5 ERROR ESTIMATES FOR VARIOUS NOISE FORMS AS A FUNCTION OF QUANTISATION. MIXED RADIX FFT.
1-0 -5
2.0 FREQ. (Hz)4.0 (b) 'PINK' NOISE
1-0
0 2-0 FREQ.(Hz) 4-0 (a) Sinusoidal noise 4-5 Hz
FIG 6 SYSTEM AMPLITUDE RATIO FOR 'PINK' AND SINUSOIDAL NOISE. 12 BIT QUANTISATION. MIXED RADIX FFT.
FIG 7 SYSTEM FREQUENCY RESPONSEUSING RADIX-2 FFT. NO NOISE AND 12 BIT QUANTISATION