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Abstract The paper describes the application of thecorrelation technique in the linear system frequency responseanalysis. An example of the system with non-rational transferfunction is used to demonstrate the merits of the developedfrequency response analyzer implemented inMATLABSimulink environment.Keywords Frequency response analysis, zero-order hold,

PAD approximation,MATLABSimulink.

I. INTRODUCTIONREQUENCY response function measurements are aninteresting intermediate step in the identification

process with nonparametric results. However, there aremethods in the literature [1] that provide parametricmodels from available nonparametric data.

Frequency response analysis offers some useful insightsinto stability and other characteristics of the controlsystem. Frequency response allows us to understand thesystem behavior in the presence of more complex inputs.However, the frequency response methods, may be lessintuitive than other design methods. The main

disadvantage of frequency response analysis is that manyindustrial processes do not admit sinusoidal inputs duringnormal operation, especially because experiment repeatingfor many frequencies may lead to long research period.

The layout of the paper is as follows: in Section 2 wegive a brief review of the frequency response analysis based on the using the correlation method; Section 3addresses to MATLABSimulink implementation of theconsidered frequency response analyzer. Section 4contains an example of using the developed analyzer.

II. FREQUENCY RESPONSE ANALYSIS

A. The Frequency Response of Continuous-Time LinearTime-Invariant Systems

Consider a continuous-time, linear time-invariant (LTI)

Prof. Dr. Milica B. Naumovi is with the Faculty of ElectronicEngineering, University of Ni, Serbia (phone: +381 18 529 441; fax:+381 18 588 399; e-mail: milica.naumovic@elfak.ni.ac.rs).

Prof. Dr. ir. Robin De Keyser is with the Department of ElectricalEnergy, Systems, and Automation, Faculty of Engineering, GhentUniversity, Belgium (phone: +32-9-264 5585; fax: +32-9-264 5603; e-mail: rdk@autoctrl.ugent.be).

Faiber I. Robayo is with the Department of Electrical Energy,Systems, and Automation, Ghent University, Ghent 9000, Belgium(phone: +32-9-264 5608; fax: +32-9-264 5603; e-mail:

faiber@autoctrl.ugent.be)Drd. ir. Clara-Mihaela Ionescu is with the Department of Electrical

Energy, Systems, and Automation, Ghent University, Ghent 9000,Belgium (phone: +32-9-264 5608; fax: +32-9-264 5603; e-mail:clara@autoctrl.ugent.be).

system between the input ( )u t and the output ( )y t as

shown in Fig. 1. The function ( )G j is called the

frequency response of the system, where ( )U j , ( )Y j ,

and ( )G j are the Fourier transform of input ( )u t , output

( )y t , and impulse response ( )g t , respectively.

Fig. 1. Relationships between inputs and outputs in aLTI system.

The frequency response analysis is a simple method forobtaining detailed information about the considered linearsystem. Recall that in the case of a mono-frequentsinusoidal input signal

( ) sin( )u t a t = , (1)

the uncorrupted, steady-state response of a linear, stablesystem with transfer function ( )G s is

( ) sin( )y t b t = + , (2)

whose evaluation is shown in Fig. 2(a). By measuring thesignals with an oscilloscope, the amplitude and phase ofthe system frequency response can be promptly computedgiving

( )b

G ja

= ,o

( ) ( ) 360t

G j tT

= = =

, (3)

where oT and t are the signal period and the delay

time, respectively. This experiment should be repeated fora number of frequencies in a certain frequency band. Asshown in Figs. 2(d) and (e), the frequency response datacan be visualized in two different ways: via the NYQUISTdiagram or via theBODE plots.

B. Frequency Response Analysis by the Correlation

Method

The problems of non-linear distortion, as well asnoise corruption of the output signals, that usually occur inengineering systems (see Fig. 2(c)), can be overcame inthe measurement scheme given in Fig. 3, by applying timeaverages. This set-up visualizes a correlation method inorder to see how much energy is at the frequency inquestion [2]. Briefly, the correlation frequency analyzerfunctions as follows [3], [4].

Developing Frequency Response Analyzer inMATLABSimulink Environment

Milica B. Naumovi, Member, IEEE, Robin De Keyser, Faiber I. Robayo,and Clara-Mihaela Ionescu, Student Member, IEEE

F

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The measured system output is multiplied with a sineand cosine signal of the frequency of the system input .The products are then integrated for a specifiedmeasurement time mT . As the averaging time increases,

the contribution of all unwanted frequency components in( )y t goes to zero, and the integrator outputs

( s m( )y T , c m( )y T ) become constant values that depend

only on the gain and phase of the considered systemtransfer function at the test frequency.

For specific values of integration time, the sine channeloutput s m( )y T , given by

m

m m

s m0

0 0

( ) ( )sin( ) d

sin( ) sin( ) d ( )sin( )d ,

T

T T

y T y t t t

b t t t n t t t

= =

= + +

(4)

represents a measurement of the real part of the systemfrequency response at the test frequency . Namely, theprevious equation (4) can be rewritten in the form

m m

stnd rd

s m

m0 0

1 term 2 term 3 term

( )

cos cos(2 )d ( )sin( ) d .2 2

T T

y T

b bT t t n t t t

=

+ +

(5)

Note that in particular for om 2kT

T = , 1,2,3,k= , the

second term in Eq. (5) approaches zero. Also, it can beshown that the average of the integrated noise (the thirdterm in Eq. (5)) is zero, which implies that, selecting thesufficient number of test signal periods, the desiredaccuracy can be reached even in case of low signal-to-noise ratio [3]. Therefore, the sine channel output in Fig. 3becomes

ms m( ) cos2

bTy T . (6)

Fig. 3. Block-diagram of the correlation frequencyresponse analyzer.

Similarly, the cosine channel output c m( )y T can be

evaluated for om 2

kTT = , 1,2,3,k= as

(a) (b) (c)

(d)

Fig. 2. Responses of a LTI system to sine wave input;

(a) Evaluation of sinusoidal response;(b) Noise signal;(c) Output signal corrupted by noise;(d) Presentation of frequency response data - NYQUIST Plot;(e) Presentation of frequency response data - BODE Plot.

(e)

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polynomial. High-order PAD approximations producetransfer functions with clustered poles. Because such poleconfigurations tend to be very sensitive to perturbations,PAD approximations with order 10n > should be

avoided. The approximation 9,10 9,10( ) 1 ( )G s R Ts s =

for the transfer function h0 ( )G s is adopted in this paper.

In the case of a relative low sampling frequency of 10samples per cycle of the sinusoidal input ( )x t , Fig. 6

visualizes the output of the sampler/zero-order hold

h0 ( )x t , as well as the response of the function

9,10 ( )G s corrected by T according to sampling theory.

Fig. 6. Responses of ZOH to sine wave input.

B. Frequency responses Results and Discussion

To obtain the frequency responses (magnitude and phase) of the zero-order hold, consider the followingexpressions:

h02

( ) sincs s

G j

=

,2

sT

= ,

and (14)

h0 0 sin( ) 0arg ( ) , sin( ) 0s

ss

G j = + =