# Detection of Short-Step Pulses  Using Practical Test-Functions and Resonance Aspects

Authors: Alexandru Toma; Cristian Morarescu
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

An important aspect in modeling dynamic phenomena consists in measuring
with higher accuracy some physical quantities corresponding to the
dynamic system. Yet for measurements performed on limited time interval
at high working frequency, certain intelligent methods should be added. The
high working frequency requires that the measurement and data processing
time interval should be greater than the time interval when the step input is
received, so as to allow an accurate measurement. This paper will show that
an intelligent processing method based on oscillating second-order systems
working on limited time interval can differentiate between large step inputs
(which are active on the whole limited time interval) and short step inputs
(which are active on a time interval shorter than the limited working period).
Some resonance aspects (appearing when the input frequency is close
to the working frequency of the oscillating second-order system) will be also
presented.


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## Body

## 1. Introduction

Filtering and sampling devices usually consist of asymptotically stable systems, sometimes an integration of the output over a
certain time interval being added. Yet such structures are very sensitive at
random variations of the integration period, being recommended for the signal
which is integrated to be approximately equal to zero at the end of the
integration period. For this reason, oscillating systems for filtering the
received signal should be used, so as the filtered signal and its slope to be
approximately zero at the end of a certain time interval (at the end of an
oscillation).  For avoiding instability of such oscillating
systems on extended time intervals, certain electronic devices (gates)
controlled by computer commands should be added, so as to restore the initial
null conditions for the oscillating system before a new working cycle to start
[1].The filtering performances of asymptotically stable
systems are determined by their transfer function. A filtering and sampling
device consisting of low-pass filters of first or second order having the
transfer function(1.1)H(s)=1T0s+1(for a first-order system)
and(1.2)H(s)=1T02s2+2bT0s+1(for a second-order system)
attenuates an alternating signal of angular
frequency ω≫ω0=1/T0 about ω/ω0 times (for a first-order system) or about (ω/ω0)2 times (for a second-order system). The
response time of such systems at a continuous useful signal is about 4−6T0 (5T0 for the first-order system and 4T0/b for the second-order system). If the signal
given by the first- or second-order system is integrated over such a period, a
supplementary attenuation for the alternating signal of about 4−6ω/ω0 can be obtained.But such structures are very sensitive at the random
variations of the integration period (for unity-step input, the signal which is
integrated is equal to unity at the sampling moment of time), and the use of
oscillators with a very high accuracy cannot solve the problem due to
switching phenomena appearing at the end of the integration period (when an
electric current charging a capacitor is interrupted).These random variations cannot be avoided if we use
asymptotically stable filters. By the other hand, an improvement in an
electrical scheme used for integrators in analog signal processing (see
[2, 3]) cannot lead to a significant
increasing in accuracy, as long as such electronic devices perform the same
task (the system has the same transfer function). There are also known
techniques for reducing the switching noise in digital systems, but such
procedures can be applied only after the analog signal is filtered and sampled,
so as to be prepared for further processing. So we must give attention to some
other kind of transfer functions and to analyze their properties in case of
filtering and sampling procedures, similar to wavelets analysis presented in
[4, 5].Mathematically, an ideal solution consists in using an
extended Dirac function for multiplying the received signal before the
integration (see [1]),
but is very hard to generate such extended Dirac functions (a kind of acausal
pulses) using nonlinear differential equations for (i) symmetrical pulses (see
[6]) or (ii)
asymmetrical pulses (see [7] for more details).A heuristic algorithm for generating practical test
functions using MATLAB procedures was presented in [6]. First it has been shown
that ideal test functions cannot be generated by differential equations, being
emphasized the fact that differential equations can only generate functions
similar to test functions (defined as practical test functions). Then a
step-by-step algorithm for designing the most simple differential equation able
to generate a practical test function was presented, based on the invariance
properties of the differential equation and on standard MATLAB procedures. The
result of the algorithm consists in a system working at the stability limit
from initial null conditions, on limited time intervals, the external signal
representing the free term in the differential equation corresponding to the
input of the oscillating system. Such a system could be built using standard
components and operational amplifiers.  However, the previously mentioned
study [6] did not
investigate the behavior of such an oscillating system for an input represented
by a short-step pulse. These aspects will be
studied in this paper. Finally, supplementary resonance aspects (appearing when
the input frequency is close to the working frequency of the oscillating
second-order system) will be also presented.


## 2. Modeling Transitions by Practical Test-Functions Integral Aspects

While this study is based on robust integral
procedures of practical test-functions for certain time intervals, certain
basic integral aspects of practical test functions should be mentioned. These
aspects are useful for modeling smooth transitions from a certain function of
time to another on a limited time interval [6].From basic mathematics, it is known that the productφ(t)g(t) between a function g(t) which belongs to C∞ class and a test-function φ(t) which differs to zero on (a,b) is also a test-function because(a)
it differs to zero only on the time interval(a,b) where φ(t) differs to zero (if φ(t) is null, then the product φ(t)g(t) is also null);(b)
the functionφ(t)g(t) belongs to the C∞ class of functions, while a derivative of a
certain order k can be written as(2.1)(φ(t)g(t))(k)=∑p=0kCkpφ(t)(p)g(t)(k−p)(a sum of terms represented by a
product of two continuous functions).Yet for practical cases (when phenomena must be
represented by differential equations), theφ(t) test functions must be replaced by a practical
test functions f(t)∈Cn on R (for a finite n-considered from now on as representing the order of the practical test function) having the
following properties:(a)
f is nonzero on (a), (b),(b)
f satisfies the boundary conditions f(k)(a)=f(k)(b)=0 for k=0,1,…,n,  and(c)
f restricted to (a,b) is the solution of an initial value problem
(i.e., an ordinary differential equation on (a,b) with initial conditions given at some point in
this interval).
The generation of such practical test functions is
based on the study of differential equations satisfied by these test functions,
with the initial moment of time chosen at a time moment close to the t=a moment of time (when the function begins to
present nonzero values).By using these properties of practical test-functions,
we obtain the following important result for a productf(t)g(t) between a function g(t) which belongs to C∞ class and a practical test-function of n order f(t) which differs to zero on (a,b):General property for product
The productg(t)f(t) between a function g(t)∈C∞ and a practical test-function f of order n is represented by a practical test function of
order n.
This is a consequence of the following two properties:
(a)
the productg(t)f(t) differs to zero only on the time interval (a,b) on which f(t) differs to zero;(b)
the derivative of orderk for the product g(t)f(t) is represented by the sum(2.2)(f(t)g(t))(k)=∑p=0kCkpf(t)(p)g(t)(k−p)
which is a sum of terms
representing products of two continuous functions for any k≤n,
(n being the order of the practical test-function f)—only for k>n discontinuous functions can appear in the
previous sum.
Now we will begin to study the integral properties of
practical test functions of certain order. For this, we note that the integralφ(t) of a test function ϕ(t) (which differs to zero on (a,b) interval) is a constant function on the time
intervals (−∞,a] and [b,+∞);
it presents a certain variation on the (a,b) time interval, from a constant null value to a
certain Δ quantity corresponding to the final constant
value. Moreover, all derivatives of order k≤n+1 for the integral function F(t) are equal to zero for t=a and t=b (this can be easily checked by taking into
account that all derivatives of order p for f(t) are equal to zero at these time moments, for p≤n,
and a derivative of order p for f(t) corresponds to a derivative of order p+1 for function F(t),
the integral function of f(t)). This suggests the possibility of using such
integral functions for modeling smooth transitions from a certain state to
another in different kind of applications, when almost all derivatives of a
certain function are equal to zero at the initial moment of time.
For modeling such a transition, we analyze the general
case when a functionf and a finite number of its derivatives f(1),f(2),…f(n) present variations from null values to values Δ,Δ1,Δ2,…Δn on the time interval [−1,1]. We begin by looking for a function fn which should be added to the null initial
function so as to obtain a variation Δn for the derivative of n order.
By multiplying the bump-like function(2.3)φ(τ)={Cexp(1τ2−1),ifτ∈(−1,1),0,otherwise(a test-function on [−1,1]) with the variation Δn of the derivative of n order and by integrating this product n+1 times we obtain
(i)
after the first integration: a constant value equal
toΔn at the time moment t=1 (while the integral of the bump-like test
function on [−1,1] is equal to 1), and a null variation on (1,+∞);(ii)
after the second integration (when we integrate the
function obtained at previous step): a term equal toΔn(t−1) and a term equal to a constant value c11 (a constant of integration) on the time
interval (1,+∞);(iii)
after
then+1 integration: a term equal to Δn(t−1)n/n! and a sum of terms having the form c1i(t−1)i/i! for i∈N,⁢i<n (cni being constants of integration) on the time
interval (1,+∞).
All previous constants of integration are determined
by integrating the test function on[−1,1].
The procedure continues by looking for the other functions fn−1,fn−2… which must be added to the initial null
function. However, we must take care to the fact that the function fn previously obtained has nonzero variations dn−1,dn−2,…d1 for its derivatives of order n−1,n−2,…1 on the working interval and so we must
subtract these values from the set Δn−1,Δn−2,…Δ1 before passing to the next step.
Then we multiply the bump-like function with the
corrected value(2.4)Δn−1′=Δn−1−dn−1and by integrating this product n times we obtain in a similar manner a function
with a term equal to Δn−1′(t−1)n−1/(n−1)! and a sum of terms having the form c2i(t−1)i/i! for i∈N,i<n−1 (cni being constants of integration) on the time
interval (1,+∞).
It can be noticed that the result obtained after n integration possess the n−1 order derivative equal to Δn−1′,
a smooth transition for this derivative from the initial null value being
performed. So the second function which must be added to the initial null
function is the integral of n−1 order for the bump-like function multiplied by
this variation Δn−1−dn−1 (the function being noted as fn−1). This function fn−1 has a null value for the derivative of n order
for t>1,
so the result obtained at first step is not affected. We must take care again to the fact that
the function fn−1 previously obtained has nonzero variations dn−11,dn−21,…d11 for its derivatives of order n−1,n−2,…1 and so we must once again subtract these values
from the previously corrected set Δn−1−dn−1,Δn−2−dn−2,…Δ1−d1 before passing to the next step. Finally we
obtain all functions fn+1,fn,…f1 which represent the terms of function f modeling the smooth transition from an initial
null function to a function having a certain set of variations for a finite
number of its derivatives on a small time interval. The procedure can be also
applied for functions possessing a finite number of derivatives within a
certain time interval by time reversal (t being replaced with −t). More details regarding possible
applications of such a procedure can be found in [6].


## 3. The Oscillating Second-Order System for the Case of Short-Step Inputs

After presenting basic aspects regarding integral
properties of practical test-functions, we will analyze the behavior of a
system able to generate a practical test-function for signal processing when
its command is represented by a short-step pulse.
Unlike aspects presented in previous paragraph, the step change appears for the
command functionu(t),
and the dynamical behavior on a limited time interval should be performed. We
are searching for a robust integral procedure (with null values of the function
which is integrated at the beginning and at the end
of the interval of integration) so as the sampled values to be further
processed for determining the amplitude and the time length of the
short-step pulse.For a robust filtering and sampling procedure based on
an integration on a limited time interval, a search for a system having the
following property was performed in a rigorous manner in [8]: starting to work from
initial null conditions, for a unity step input it must generate an output and
a derivative of this output equal to zero at a certain moment of time (the
condition for the derivative of the output to be equal to zero has been added
so as the slope and the first derivative of the slope for the signal which is
integrated to be equal to zero at the sampling moment of time, when the
integration is interrupted). It was finally shown that the simplest structure
possessing such properties is represented by an oscillating second-order system
having the transfer function(3.1)Hosc=1T02s2+1receiving a step input and
working on the time interval [0,2πT0].
For initial conditions equal to zero, the response of the oscillating system at
a step input with amplitude A will have the form(3.2)y(t)=A(1−cos(tT0)).By integrating this result on
the time interval [0,2πT0],
we obtain the result 2πAT0,
and we can also notice that the quantity which is integrated and its slope are
equal to zero at the end of the integration period. Thus the influence of the
random variations of the integration period (generated by the switching
phenomena) is practically rejected.This oscillating system attenuates about(ω/ω0)2 times such an input, and the influence of the
integrator consists in a supplementary attenuation of about(3.3)[(1/(2π))(ω/ω0)]times. The oscillations having
the form(3.4)yosc=asin(ω0t)+bcos(ω0t)generated by the input
alternating component have a lower amplitude and give a null result after an
integration over the time interval [0,2πT0].These results have shown that such a structure
provides practically the same performances as a structure consisting of an
asymptotically stable second-order system and an integrator (response time of
about6T0,
an attenuation of about (1/6)(ω/ω0)3 times for an alternating component having
frequency ω), moreover being less sensitive at the random
variations of the integration period. For restoring the initial null conditions
after the sampling procedure (at the end of the working period), some
electronic devices must be added. Yet the previous analysis is valid for
extended step inputs, which are active on the whole working interval (the
integration period).We will continue the analysis of this structure by
considering that the input is represented by a unity short-step pulse (instead
of a unity step-pulse) which differs to zero on the time interval[0,τ].
This means that the input u can be represented under the
following form:(3.5)u(t)=1,fort∈[0,τ],u(t)=0,fort>τ,or, using the Heaviside function(3.6)u(t)=h(τ−t)fort∈[0,∞),where h(τ) corresponds to the function 1/s if we apply the Laplace transformation.The transfer function of the second-order oscillating
system is(3.7)H(s)=1T02s2+1.On the time interval [0,τ], the output of the second-order oscillating
system is represented (using the Laplace transformation) as(3.8)y(s)=H(s)u(s)=1T02s2+11swhich corresponds to the
output(3.9)y(t)=(1−cos(tT0))which can be written
as(3.10)y(t)=1−cos(ω0t),where ω0=2π/T0.
When the action of the external unity pulse ceases (for t=τ) the output y(t) is(3.11)y(τ)=1−cos(ω0τ)and the derivative of y(t)
is(3.12)y′(τ)=ω0sin(ω0τ).These values,y(τ) and y′(τ), represent the initial values for the free
oscillations of the second-order oscillating system generated for t>τ (when the input command u(t)=0). These free oscillations have the angular
velocity ω0,
and thus the output y(t) for t>τ will have the form(3.13)y(t)=Csin(ω0t+ϕ),fort>τ.The quantities C and ϕ (amplitude and initial phase of free
oscillations) should be determined using the initial conditions for t=τ:(3.14)y(τ)=1−cos(ω0τ),y′(τ)=ω0sin(ω0τ).This implies
(3.15)1−cos(ω0τ)=Csin(ω0τ+ϕ),ω0sin(ω0τ)=Cω0cos(ω0τ+ϕ).By dividing the second equality withω0,
squaring both equalities and summing left-hand sides and right-hand sides of
both squared equalities we obtain(3.16)[1−cos(ω0τ)]2+sin⁡2(ω0τ)=C2and (while sin⁡2(ω0τ)+cos⁡2(ω0τ)=1)(3.17)2−2cos(ω0τ)=C2.While
(3.18)cos(ω0τ)=1−2sin2(ω0τ2)by substituting cos(ω0τ) with the right-hand side of the above
equality, the result is (3.19)4sin⁡2(ω0τ2)=C2.It results(3.20)C=2sin(ω0τ2).(C is a positive quantity, because ω0τ∈[0,2π]→ω0τ/2∈[0,π]→sin(ω0τ/2)≥0). The phase ϕ can be obtained using the two equations
determined by the initial conditions:(3.21)1−cos(ω0τ)=Csin(ω0τ+ϕ),sin(ω0τ)=Ccos(ω0τ+ϕ)(the second equation resulting
by dividing previous equation of y′(τ) to ω0).By dividing left-hand side of first equality to
left-hand side of second equality, and right-hand side of first equality to
right-hand side of second equality, it results(3.22)1−cos(ω0τ)sin(ω0τ)=tan(ω0τ+ϕ).Using equalities(3.23)cos(ω0τ)=1−2sin⁡2(ω0τ2),sin(ω0τ)=2sin(ω0τ2)cos(ω0τ2),it results by substituting cos(ω0τ),cos(ω0τ) with the above expressions(3.24)tan(ω0τ2)=tan(ω0τ+ϕ).So ϕ is obtained as(3.25)ϕ=−(ω0τ2)and the output y(t) corresponding to the free oscillations of the
system for t>τ (when the action of the external short-step
command u has ceased) can be written as(3.26)y(t)=2sin(ω0τ2)sin(ω0t−ω0τ2).


## 4. Algorithm for Detecting Short-Step Pulses

While the signal processing system is linear, in case
of a short-step pulse of amplitudeA defined on time interval [0,τ] the output y(t) of the system will be multiplied by A as related to the output obtained in case of a
unity short-step input (presented in previous paragraph). Thus, the output y(t) will be(4.1)A(1−cos(ω0t)),fort∈[0.τ],y(t)=2Asin(ω0τ2)sin(ω0t−ω0τ2),fort>τ.This output,y(t), is equal to zero at two time moments t1 and t2 after time moment τ.
At time moment t1,(4.2)(ω0t1−ω0τ2)=π;and at time moment t2,(4.3)(ω0t2−ω0τ2)=2π.These imply
that(4.4)t1=τ2+πω0,t2=τ2+2(πω0).We must check whether botht1,t2 are greater than τ.
First we check the inequality t1=τ/2+π/ω0>τ.
This is equivalent to τ/2<π/ω0, τ<2π/ω0=T0,
where T0 represents the period of the second-order
oscillating system. It is obvious that τ<T0,
while we have considered that the short-step pulse
has nonzero values for t∈(0,T0).
Thus t1>τ,
and while t2>t1 it results that t2>t1>τ.The signal processing system will perform the
integration of outputy(t) on two different time intervals. The first
value I1 is obtained by an integration of y(t) on the time interval [0,t2].
It results(4.5)I1=∫0t2y(t)dt=∫0τA(1−cos(ω0t))dt+∫τt22Asin(ω0τ2)sin(ω0t−ω0τ2)dt,I1=A(τ−1ω0sin(ω0τ)+2A1ω0sin(ω0τ2)(−cos(2π)+cos(ω0τ2)),(while ω0t2=2π). Then we obtain(4.6)I1=Aτ−A1ω0sin(ω0τ)+A2ω0sin(ω0τ2)(cos(ω0τ2)−1)which can be finally written
as(4.7)I1=Aτ−A2ω0sin(ω0τ2).The second integralI2 is performed by integrating y(t) on the time interval [t1,t2]. On this time interval, the output y(t) is represented by a free
oscillation(4.8)y(t)=2Asin(ω0τ2)sin(ω0t−ω0τ2),so I2 is determined by(4.9)I2=∫t1t22Asin(ω0τ2)sin(ω0t−ω0τ2)dt;and taking into account
that(4.10)ω0t−ω0τ2=π,fort=t1,ω0t−ω0τ2=2π,fort=t2,it results I2 as(4.11)I2=−4Aω0sin(ω0τ2).The valuesI1,I2 allow us to
determine A,τ by robust integrations (the values of the
function y(t) which is integrated are zero at the beginning
and the end of the time interval used for integration). A quantity I0 can be determined as(4.12)I0=I1−I22=Aτ;and a quantity R can be determined as(4.13)R=−I22I0=(4A/ω0)sin(ω0τ/2)2Aτwhich can be written
as(4.14)R=sin(ω0τ/2)ω0τ/2=sinc(ω0τ2).This means that after performing the robust
integration ofy(t) in order to obtain the sampled values I1,I2 we can compute I0=I1−I2/2 and then I0=−I2/(2I0).
While(4.15)I0=sinc(ω0τ2),we can determine the time
interval τ of the received short-step pulse using I0 and a mathematical memory (the quantity ω0 being known).Next quantityA (the amplitude of this received short-step
pulse) can be determined using τ (determined at previous computation) and I0=Aτ.Taking into account the previous considerations, the
algorithm for detecting short-step pulses consists in
the steps shown in Algorithm1.Algorithm 1
Att=0 Start Integration for I1if output = 0thenStart Integration forI2if output = 0Stop Integration forI1 and Integration for I2DetermineA and τ using sampled values for I1 and I2


## 5. The Problem of Initial Conditions and Resonance Aspects

In previous paragraphs has been analyzed the case when
a certain short-step pulse is received at the beginning of a working
interval—this pulse presenting nonzero values on a limited time interval(0,τ).
Yet in applications, the signal processing system can receive a sequence of
step pulses with different time lengths. Due to
this reason, the whole procedure must be adapted for the case when a certain
step input with amplitude D is received on a certain time interval (0,σ) (the first part of the working interval) and a
different step input with amplitude B appears on the time interval [σ,T0] (the second part of the time interval).Under these circumstances the outputy(t) could be represented by(5.1)y(t)=D(1−cos(ω0t)),fort<σ,y(t)=B(1−cos(ω0(t−σ)))+yosc,fort≥σ,where
(5.2)yosc=2Dsin(ω0σ2)sin(ω0t−ω0σ2)represents the free oscillations
of the system generated by the short-step pulse with amplitude D (which has ceased its action at time moment σ). It can be noticed that
it is very difficult to analyze these
outputs by performing certain integral operations
in order to determine the parameters D,B,σ.
However, the whole procedure can be simplified in a significant manner if we
observe that the amplitude D of the first step input can be predicted by
the signal processing system (by considering that the pulse with amplitude D has been received by the system on a previous
working interval and its action continues at the beginning of the analyzed
working interval). This suggests the possibility of adjusting the input command
for the second-order oscillating system by subtracting quantity D from u(t) and thus the output of the oscillating system
will become(5.3)y(t)=0,fort<σ,y(t)=(B−D)(1−cos(ω0(t−σ))),fort≥σ.The quantity yosc vanishes while the oscillating second-order
system has initial null conditions and a null command for t<σ.At first sight it looks like the system receives a
nonzero command at the end of the working time interval and the results
presented in previous paragraph cannot be applied. Yet we can observe that an
integration performed on the first time interval[0,σ] generates a null result. Due to this reason,
we have to adjust the integration procedure by(i)
starting the
integration forI1 and I2 at the beginning of a new working cycle, with
the input command considered as w(t)=u(t)−A (the previous value for the amplitude of the
received pulse is subtracted from received pulse u(t));(ii)
if a step variation of amplitude for the input
commandw(t) is detected then
(a)
stop the
action of adjusted input command w(t)=u(t)−A upon the oscillating system at time moment T0 (the output y(t) of the system will be represented by free
oscillations),(b)
continue the
integration for I1 and I2 after the time moment T0 (which would have been the end of the working
cycle in case that the step variation of input has been not detected);(iii)
stop the integration procedure forI1 when y(t) first time equals zero and the integration
procedure for I2 when y(t) second time equals zero.
Note that the time moments when the integration
procedures cease are not affected by noise, while the outputy(t) is represented just by free oscillations of
the second-order system (after the time moment T0 the action of u(t) upon the system ceases);(iv)
Determineσ using(5.4)I0=−I2/(2I0),I0=sinc(ω0τ2),T0−σ=τ,I0=Aτ,B−D=A,B=D+A.Note: by translating the time origin from the
beginning of a working interval to the time moment when the step change(B−D) appears, we can consider that a short-step
pulse with amplitude (B−D) acts upon the second-order system at time
moment zero, from null initial conditions, on the time interval (0,T0−σ)—thus the quantities τ and A corresponding, respectively, to the time
length of the received short-step pulse and to its amplitude in previous
paragraph should be replaced by T0−σ and (B−D).All presented aspects are valid if the system receives
a step pulse (which can be represented by an extended step pulse as presented
in [1]) or by a
short-step pulse (as presented in this study, in previous paragraphs). Filtering properties of the second-order
oscillating system were studied in [1]. However, we must study also resonance aspects. The
second-order system being an oscillating system, resonance aspects (appearing
when the input u(t) is represented by an alternating function Asin(ωt+φ) with ω≈ω0) are very important. Instead of damped proper
oscillations of angular frequency ω0 with zero limit (as for an asymptotic stable
second-order system), some proper oscillations of angular frequency ω0 and with constant amplitude can be noticed as
term in y(t).
These are added to the oscillations with angular frequency ω generated by the command function u(t),
both having a higher amplitude inversely proportional to ω2−ω02.
If ω=ω0 (the limit case), for a command
function(5.5)u(t)=Asin(ω0t+φ0),the output y(t) is represented by(5.6)y(t)=E1tsin(ω0t+φ1)+E2sin(ω0t+φ2).This function is hard to be
processed by a signal processing system, working on a limited time interval.
Moreover, in applications it is possible for the input u(t) to be represented by a sum of alternating
functions with different angular frequencies ωi≈ω0.
However, a signal processing procedure can be established for the case when u(t) is represented by a sum of alternating
functions with angular frequencies ωi≈ω0 by taking into account the fact that for an
input u(t) represented by an alternating function Asin(ωt+φ) the amplitude E of oscillations with angular frequency ω generated by this command function u(t) is determined by(5.7)E(ω)=Aω02ω2−ω02which is a very sharp function.
For this reason we can consider that in the case when the input u(t) is represented by a sum of oscillations with
different angular frequencies ωi≈ω0,
the output y(t) of the second-order system will be represented
by an oscillation with the angular frequency ωj closest to ω0 (generated by the received oscillation with
angular frequency ωj) and a proper oscillation of the second-order
system (with angular frequency ω0). Thus y(t) could be represented by a sum of two
oscillations.The result of an integration of this outputy(t) on the working interval [0,2πT0] would depend just on the oscillation with
angular frequency ωj,
while the oscillation with angular frequency ω0 (with time period T0) gives a null result by an integration on
this period. The result of this integration could be used for determining
certain parameters for the received oscillation with angular frequency ωj.
However, such an integration is not a robust integration, while the signal
which is integrated is not equal to zero at the end of this working interval. A
possible solution of this problem would consist in disconnecting the input
signal after a certain time interval, so as to analyze (using robust
integrations) the free oscillations of the second-order system after this
moment (as presented in previous paragraph). A faster procedure could consist
in a previous adjustment of initial conditions, so as the free oscillations not
to appear.Theoretically, this can be done by using a set of
identical oscillating second-order systems (receiving the same input commandu(t)) with initial conditions adjusted to
different values. The system generating a single oscillation with angular
frequency ωj would be selected by checking the following
condition:(5.8)d2ydt2+ωj2y=0.However, the adjustment of two
initial conditions at different values for a second-order system requires a
large number of identical oscillators. Due to this reason, this method is
inconvenient. A better choice would be represented by a delay systems of first
order, with transfer function(5.9)H(s)=sssinφ+ωjcosφwould transform a received
oscillation sin(ωjt+φ) according to(5.10)V(s)=H(s)U(s)=sssinφ+ωjcosφssinφ+ωjcosφs2+ωj2=ss2+ωj2which corresponds to an output v(t)=cos(ω0t).
If this function v(t) represents the command for the second-order
oscillating system, the output y(t) (for initial null conditions) will
be(5.11)y(t)=(1/2)ωjtsin(ωjt)when ωj is very close to ω0 and can be approximated by this quantity. The
function y(t) presented in previous equation is suitable for
a sequence of robust integration procedures on half-period time intervals:(5.12)[0,π/ωj],[π/ωj,2π/ωj],etc.(it presents null values at the
beginning and at the end of these intervals integration). The results of these
procedures are proportional to the amplitude A of received oscillation with angular frequency ωj (as can be easily noticed).Unlike the possible solution based on adjustment of
two initial conditions, this procedure requires a set of signal processing
systems composed of different time-delaying systems adjusted according to a
single parameter (the phaseϕ,
while ωj is supposed to be known) and identical second-order
oscillating systems starting to work from initial null conditions. The number
of required systems is less than in previous case, while a single parameter has
to be adjusted at different values (the quantity φ in the time-delay systems). The corresponding
output is selected by checking whether the results of these procedures
(considered as positive quantities) vary according to a linear mathematical law
(as required by the integration of y(t)=(1/2)ωjtsin(ωjt) on a sequence of half-period time intervals).One major disadvantage of this method has to be
mentioned: the functiony(t)=(1/2)ωjtsin(ωjt) equals zero at the beginning of signal
processing time interval (t=0) and presents a small slope. So an extended
time interval is necessary for obtaining significant results using robust
integration procedures. If we need a fast signal processing, we can simply use
a set of identical oscillating second-order systems, with different initial
conditions. The greatest amplitude for the output oscillation (considered as a
sum of an oscillation with angular frequency ωj and an oscillation with angular frequency ω0, ωj≈ω0) corresponds to the case when the two
oscillations are in-phase. By detecting the output presenting in-phase
oscillations, we can establish the amplitude and phase for the received signal
using the initial conditions for the second-order system generating this
output.


## 6. Conclusions

An important aspect in modeling dynamic phenomena
consists in measuring with higher accuracy some physical quantities
corresponding to the dynamic system. Yet for measurements performed on limited
time interval at high working frequency, certain intelligent methods should be
added. The high working frequency requires that the measurement and data
processing time interval should be greater than the time interval when the step
input is received, so as to allow an accurate measurement. This paper has shown
that an intelligent processing method based on oscillating second-order systems
working on limited time interval can differentiate between large-step inputs
(which are active on the whole limited time interval) and short-step inputs
(which are active on a time interval shorter than the limited working period).
Some resonance aspects (appearing when the input frequency is close to the
working frequency of the oscillating second-order system) were also presented.
Possible applications could be represented by
processing the electric signal generated by transducers [9] and by advanced modeling of
traffic network [10].



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*Source: 543457-2008-09-01.xml*