In previous work (Kaszyński, 2003; Jaskula, & Kaszyński, 2004; Piskorowski, 2006), the
following equation was proposed for defining a new class of low-pass time-varying filters
x’’(t)+2ξ(t)ω(t)x ‘(t)+ ω2(t)x(t) = ω2(t)u(t)
(12)
where u(t) is the signal to be filtered, x(t) represents the output signal and ξ(t) and ω(t)
represent time-varying parameters. If the coefficients of (12) are compared to the coefficients
defined in (7) and (8) for (1), it can be concluded that the system defined in (1) will
approximate the behaviour of (12) when the maximum magnitude of the logarithmic
derivative of function ω(t) and the maximum magnitude of function ξ´(t)/(ξ2(t) − 1) are
small compared to 2ξ(t)ω(t) and ω(t), respectively. In other words, ξ(t) and ω(t) have to vary
slowly.
Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes
349
3. Stability properties of the proposed model
The stability properties of (1) when a1(t) and a0(t) are defined according to (7) and (8) are
determined solely by the modes used in the formulation of these coefficients. In the general
case (i.e. when time-varying parameters ξ(t) and ω(t) vary arbitrarily), a suitable measure of
the stability of the modes given in (4) and (5) can be obtained if their Lyapunov exponents
are computed. The Lyapunov exponents L1 and L2 associated to modes x1(t) and x2(t) in (4)
and (5) are given by
L = lim sup t− ln x (t) = lim Re t− ∫ ω (t) ⎡ −ξ (t) + (ξ (t) − )1/2
1
1
2
1 ⎤ dt (13)
1
1
t→∞
⎣
⎦
L = lim sup t− ln x (t) = lim Re t− ∫ ω (t) ⎡ −ξ (t) − ( ξ (t) − )1/2
1
1
2
1
⎤ dt (14)
2
2
t→∞
⎣
⎦
In order to assure the stability of (1), L1 < 0 and L2 < 0. Therefore, it suffices that the real part
of those time averages remains negative. This means that function ξ(t) may take arbitrary
negative values and the stability of the system will be still guaranteed, provided that the
average value of ω(t) [-ξ(t) ± (ξ2(t) − 1)1/2 ] is negative over the whole time interval in which
(1) is operating. This gives added flexibility when implementing a control for modulating
ξ(t) and ω(t) according to a predefined rule.
An important issue behind the proposed model in (1) is the possibility that it might be
bounded-input bounded-output (BIBO) stable when the right hand of (1) contains a non-
zero term. In order to guarantee this type of stability, it suffices that the homogeneous
response of (1) is exponentially asymptotically stable and that the coefficients a0(t) and a1(t)
are bounded for all t (Anderson & Moore, 1969). In this case, both conditions are satisfied
and therefore the proposed model has BIBO stability. The proposed modes in expressions
(4) and (5) have exponential asymptotic stability provided that ω(t) and ξ(t) remain positive
for all t. In order to guarantee the boundedness of the coefficients of (1), ω(t) hast to be
different from zero for all t, whereas ξ(t) has to be different from 1 for all t.
In the particular case of the system governed by equation (11), the considerations given in
(Anderson & Moore, 1969) to guarantee its BIBO stability when the right hand of (11) is non-
zero are also applicable. It should be noticed that if the dynamics of the system should
include a real pole in the left plane whose location is not known, function x(t) = e -∫σ(t)dt has
also exponential asymptotic stability provided that σ(t) > 0 for all t. Moreover, the average
value of σ(t) will define the Lyapunov exponent associated to that solution.
The results obtained so far should be compared against previous work. In (Kaszyński, 2003),
the following conditions for assuring the stability of (12) were given
ω(t) > 0
(15)
ξ(t) > 0
(16)
│ω’(t)│<│2ξ(t)ω2(t)│
(17)
Such conditions were obtained using the second Lyapunov method. Conditions (15) and
(16) are sufficient in (1) to ensure stability for all t. With these conditions, it is guaranteed
that the arguments of the integrals in (13) and (14) are always negative, and therefore the
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New Approaches in Automation and Robotics
Lyapunov exponents computed in this way will be also negative. Condition (17) imposes an
additional constraint on (12): in order to ensure stability, ω(t) may not be varied above the
limit imposed by the product 2ξ(t)ω2(t), thus limiting the tunability range of this parameter.
System (1) with coefficients a1(t) and a0(t) as defined in (7) and (8) does not show the
shortcoming described above.
To further demonstrate that the dynamics of (1) and (12) are different, the sum of the
Lyapunov exponents will be computed. According to (Nemytskii & Stepanov, 1989), the
sum of the Lyapunov exponents ν of a regular n-th order LTV system of the form (11) is
given by
1
−
t
ν = − lim sup t ∫ a
τ dτ
(18)
0
n 1
− (
)
t→∞
If the coefficient associated to the x’(t) term of (12) is compared to (7), it will be verified that
the only way to make the sum of Lyapunov exponents equal for both systems is forcing the
product 2ξ(t)ω(t) to be the same in both systems and cancelling the time variations of ξ(t)
and ω(t). This means that if the sum of Lyapunov exponents for (1) and (12) is meant to be
the same, both systems must be LTI and have the same set of eigenvalues.
4. Performance evaluation
In order to verify the stability properties of the proposed model, its response will be tested
for a predefined test signal (a step function). Therefore, x(t) will be estimated numerically
from the following equation
x ‘’(t)+ a1(t)x’(t) + a0(t)x(t) = a0(t)u(t)
(19)
where coefficients a1(t) and a0(t) are defined as in (7) and (8) and u(t) is the step function.
The obtained response will be compared to the response obtained from a reference LTI
system and from the LTV filter based on equation (12) subject to the same parameter
variation. It will be assumed that the LTI reference filter is a second-order lowpass
Butterworth filter with poles located on −5 ± 5j. Furthermore, it will be assumed that ω(t)
and ξ(t) are given by the following expressions
ω(t) = [50+50e−40t + 100e−20t]1/2
(20)
ξ(t) = (10 + 10 e−20t)/2ω(t)
(21)
Although it has been already demonstrated a long time ago that the eigenvalues obtained
from an arbitrary LTV system by means of the well-known characteristic equation for LTI
systems do not convey any valid information regarding the stability properties of the system
or its solution (Vinogradov, 1952), with the definition given for ω(t) and ξ(t) in (20) and (21)
the instantaneous location of the roots of the characteristic equation associated to (12) is
modulated exponentially in time from −10 ± 10j at t =0 to −5 ± 5j within a relatively short
time interval. With a time constant of 0.05 seconds for the modulation factor, the roots
associated to the characteristic equation of (12) should be very close to the poles of the
reference LTI system after 0.2 seconds.
The simulated responses obtained in Mathematica for the LTI reference system, the filter
described based on (12) and the proposed LTV model are depicted in Figure 3. From the
Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes
351
simulation results it can be seen that the responses of both LTV systems become identical to
the response of the LTI reference system as t increases. Although the response generated by
(12) rises faster compared to the response generated by (19), the proposed model produces a
smaller overshoot.
In Figure 4, the response to the step function of the systems defined by (19) and (12) is
depicted. In this particular case, it was assumed that ω(t) and ξ(t) were varied according to
the following relations
ω(t) = [50 + 90 sin(10πt) + 40.5 sin2(10πt)]1/2
(22)
ξ(t) = [10 + 9 sin(10πt) ]/ 2ω(t)
(23)
Output
1
LTI
0.8
0.6
LTV
0.4
LTV2
0.2
t
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 3. Responses of (12) and (19) to a step response with an exponential variation of ω(t)
and ξ(t). In this figure, the acronym LTV stands for the response of (12) whereas the
acronymLTV2 stands for the response of (19)
With the chosen variation parameters for ω(t) and ξ(t), the roots of the characteristic
equation associated to (12) will vary periodically according to the following relation
p1,2 = −5 ± 5j +(−4.5 ± 4.5j) cos(10πt)
(24)
where p1 and p2 stand for the roots of the characteristic equation of (12). With this variation
scheme, the condition given in (17) for (12) is not valid anymore. However, in the case of
(19), the stability measures given in (13) and (14) for equation (1) (L1 = L2 = −5), the
exponential asymptotic nature of its homogeneous response and the boundedness of its
coefficients guarantee that the output of (19) will be bounded (Anderson & Moore, 1969).
Given that (12) is expected to show an unstable behaviour, backward differentiation
formulae were chosen to solve numerically equations (19) and (12). From the results of the
simulation it can be seen that both LTV systems reach in a finite time their steady-state
behaviour. However, the system described by (12) displays a somewhat larger overshoot
before it reaches steady-state behaviour.
It is noteworthy to consider the role of the scaling coefficient of the input function u(t) in
equations (12) and (19). According to condition (17), the output of equation (12) should be
unstable if the coefficients given in expressions (23) and (24) are considered. However, the
simulation results of Figure 4 show that the output of the filter described by equation (12) is
352
New Approaches in Automation and Robotics
stable. If the scaling coefficient of the input function u(t) present in equation (12) is removed
and the following equation is instead considered
x
‘’(t) + 2ξ(t)ω(t)x ‘(t) + ω2(t)x(t) = u(t)
(25)
it is possible to demonstrate that the stability condition given in (17) is indeed broken.
Output
1.75
1.5
LTI
1.25
1
LTV
0.75
0.5
LTV2
0.25
t
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 4. Responses of (12) and (19) to a step response with a periodic variation of ω(t) and
ξ(t).
Output
0.05
0.04
LTI
0.03
LTV
0.02
LTV2
0.01
t
0.5
1
1.5
2
2.5
3
Figure 5. Responses of (25) and (26) to a step response with a periodic variation of ω(t) and
ξ(t)
In Figure 5, the response of equation (25) with periodic coefficients ω(t) and ξ(t) as defined in
(23) and (24) to a unit step function is depicted. This response is compared to the response of
the LTV differential equation
x
‘’(t) + a1(t)x ‘(t) + a0(t)x(t) = u(t)
(26)
and to the response of the lowpass filter described by the following LTI differential equation
x
‘’(t) + 50x ‘(t) + 10x(t) = u(t)
(27)
In expression (26), coefficients a0(t) and a1(t) are defined as given in (8) and (7). From Figure
5 it may be noticed that the responses of equations (25) and (26) are oscillatory and do not
track the input function u(t). In this case, function u(t) is assumed to be a step function.
Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes
353
LTV
0.0155
0.015
0.0145
0.014
0.0135
t
5.5
6
6.5
7
Figure 6. Time series considered for the output of (25)
LTV2
0.048
0.046
0.044
0.042
t
5.5
6
6.5
7
Figure 7. Time series considered for the output of (26)
A detailed analysis of a portion of the time series obtained from (25) and (26) gives more
information on the results obtained through simulation with Mathematica. The time series
considered for analysis appear in Figures 6 and 7. In a first attempt, Lyapunov exponents
may be determined to establish the nature of the behaviour displayed by the systems
represented by equations (25) and (26). However, given the periodic character of their time
series, other methods should be used to assess the stability of (25) and (26) (Sprott, 2003).
In order to determine whether the stability condition given in (17) is broken or not for
equation (25), the power spectra of the time series considered should be obtained. According
to (Sigeti, 1995), the power spectrum of a given signal with positive Lyapunov exponents
has an exponential high-frequency falloff relationship. Such characteristic in the frequency
domain is due to the fact that the function which defines the signal under consideration has
singularities in the complex plane when the time variable t is seen as a complex variable and
not as a real one (Sigeti, 1995). When the Fourier transform is computed for such a signal,
the singularities must be avoided in the complex plane through an adequate integration
path and in this way exponential terms appear on its associated Fourier transform (Sigeti,
1995). In the presence of noise, the exponential frequency falloff relationship will be
noticeable up to a given frequency and afterwards it will decay as a power of f-n where f is
the frequency and n a natural number (Lipton & Dabke, 1996). These phenomena are also
observable in chaotic systems as well, independently of the appearance of attractors or not
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New Approaches in Automation and Robotics
in their dynamic behaviour (van Wyk &. Steeb, 1997). When there are no singularities in the
complex variable t present in a given signal and in the absence of noise, its power spectrum
will decay at high frequencies as a power of f-n as well (Sigeti & Horsthemke, 1987).
Frequency (Hz)
Magnitude
0
2.86448 × 10−2
5
1.32849 × 10−3
10
8.75641 × 10−5
15
2.82938 × 10−6
20
4.9275 × 10−8
25
1.02782 × 10−9
Table 1. Discrete power spectrum for the output of equation (25)
Frequency (Hz)
Magnitude
0
8.78076 × 10−2
5
4.50563 × 10−3
10
8.13031 × 10−4
15
5.94339 × 10−5
20
7.58989 × 10−6
25
2.38456 × 10−6
30
8.0822 × 10−7
35
3.04263 × 10−7
40
1.23233 × 10−7
45
5.27326 × 10−8
50
2.35502 × 10−8
55
1.08831 × 10−8
60
5.1718 × 10−9
Table 2. Discrete power spectrum for the output of equation (26)
Given that the numerical solutions obtained for equations (25) and (26) are periodic, their
power spectra turn out to be discrete. In Tables 1 and 2 the magnitude of the harmonic
components of the responses computed via Mathematica has been tabulated. The data given
in Table 1 was used to obtain the best fit in Mathematica using routine NonlinearFit[ ] for
expressions
y = A1e B1f
(28)
and
y = A2fB2
(29)
where A1, A2, B1 and B2 are fitting parameters. If the data given in Table 1 is considered from
f = 15 Hz for the fitting process, the constants A1 and B1 which fit best expression (28) are
equal to 0.535465 and -0.810056 respectively. With the same data, the constants obtained for
the best fit of expression (29) are A2 = 175010 and B2 = −9.17964. In Figure 8 expressions (28)
and (29) are plotted together with the original data and it can be seen that the exponential
curve matches better the obtained data from equation (25) at high frequencies. The same
procedure was carried out with the data presented in Table 2. The coefficients obtained for
expression (28) were A1 =0.0244961 and B1 = −0.401458 whereas for expression (29) the
coefficients were A2 = 10292.2 and B2 = −7.00509. From Figure 9 it can be seen that
Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes
355
Output LTV
0.001
DFT
0.00001
A
B
1 f
1
1.
10 7
A2 fB2
1.
10 9
f
0
5
10
15
20
25
Figure 8. Power spectrum obtained for the time series of (25)
Output LTV2
1
DFT
0.01
0.0001
A
B
1 f
1
1.
10 6
A2 fB2
1.
10 8
f
0
10
20
30
40
50
60
Figure 9. Power spectrum obtained for the time series of (26)
expression (29) gives the best fit for the data obtained from equation (26) at frequencies
greater than 15 Hz. Given that condition (17) is broken, it can be thus safely concluded that
the response obtained from equation (26) is unstable when ω(t) and ξ(t) are defined as given
in expressions (23) and (24). In the case of equation (26), under the same conditions, it turns
out that its response is bounded. The response of equation (26) is bounded for a bounded
input because the conditions given in (Anderson & Moore, 1969) for BIBO stability are
enforced.
5. Conclusions
In this chapter, a strategy for the formulation of a LTV scalar dynamical system with
predefined dynamic behaviour was presented. Moreover, a model of a second-order LTV
system whose dynamic response is fully adaptable was presented. It was demonstrated that
the proposed model has a exponentially asymptotically stable behaviour provided that a set
of stability constraints are observed. Moreover, it was demonstrated that the obtained
system is BIBO stable as well. Finally, it was shown via simulations that the response of the
proposed model reaches with a smaller overshoot its steady-state response compared to the
response of a LTV lowpass filter proposed previously.
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