On the sufficiently small sampling period for the convenient tuning of
fractional order hold circuits
R. Bárcena* and M. De la Sen**
*Departamento de Electrónica y Telecomunicaciones, Universidad del País Vasco,
E. U. Ingeniería Tca. Industrial. Plaza de La Casilla, 3. Aptdo. 48012 de Bilbao, Spain
E-mail: jtpbarur@lg.ehu.es
**Instituto de Investigación y Desarrollo de Procesos, Universidad del País Vasco,
Facultad de Ciencias, Leioa (Bizkaia). Aptdo. 644 de Bilbao, Spain
Abstract. Remarkable improvements in the performance of digitally handled systems may be
achieved by using properly adjusted Fractional Order Hold (FROH) circuits. Nevertheless,
the tuning methods formerly proposed in the bibliography are exclusively applicable for
certain range of sufficiently small sampling periods. In this paper, such a range is
analytically obtained and its significance for the applicability of such methods is discussed.
1. Introduction.
The location of the plant zeros critically influences the performance that can be achieved
when operating such a system –see, e. g., [1] and references therein-. In recent years,
important attention has been paid to the behaviour of the discrete zeros of the digitally
managed systems, depending on the used signal reconstruction method and sampling period
–see, [2], [3] and references therein-. In this context, it has been shown –see [4]- that it is
always possible to obtain discrete zeros with larger convergence abscissa than the zeros
obtained by using a Zero Order Hold (ZOH) or First Order Hold (FOH), by means of an
appropriate choice of negative values of the adjustable gain of a Fractional Order Hold
(FROH) circuit. Moreover, an analytical technique for obtaining the optimum FROH
parameter for each discretization has been proposed [4]. In particular, such a method leads to
a mathematical expression in closed form for the optimum FROH parameter, when applied
to the discretization process of zero-free second-order continuous-time plants. The
magnitude of the discrete zeros obtained in such a way is around 50 % smaller than the
corresponding ZOH (or FOH) ones. Thus, important improvements on the performance of
the handled systems have been obtained in many cases –see an application example in [5]-.
At first, a theoretical constraint exists on the maximum sampling period when such a
FROH circuit tuning method is applied. This maximum value depends exclusively on the
continuous-time parameters of the plant to be discretized. In this paper, such a limiting
sufficiently small sampling period is studied, as well as the drawback that such a restriction
may entail to the use of conveniently adjusted FROH devices. The previous results related to
the problem are reviewed in the preliminary section 2. The main result on the sufficiently
small sampling period for the convenient tuning of FROH devices when zero-free second-
1
order systems are manipulated, is presented in section 3. The drawback that such restriction
may entail in practice is discussed in section 4. Finally, an application example is presented
in section 5 and the conclusion is drawn in section 6.
2. Preliminaries and previous results.
In order to design a digital control scheme, the focus is on the discrete-time system
composed of a hold circuit, the nth order continuous-time plant and a sampler in series. When
a fractional-order hold (FROH) is used, the plant input is
 u( KT ) − u(( K − 1)T ) 
u(t ) = u( KT ) + β 
 (t − KT )
T

where KT ≤ t < (K + 1)T
(1)
where K is a non-negative integer, T the sampling period and β the device adjustable gain.
In such cases, the root-locus (RL) approach, with the parameter β used as the generalised
gain, have been applied to study the location of the discrete-time zeros on the complex plane.
General –for any continuous plant- properties of such RL have been detected. Based on such
properties, the following previous results about the stability properties of the FROH
discretization zeros have been presented and proved in [4].
Result 1: Assume that a given nth order continuous-time plant has a relative degree p, such
that 1≤p≤n. Then, it is always possible to obtain FROH discrete zeros with larger
convergence abscissa –i. e. more stable ones- than the ZOH (β=0) and FOH (β=1) ones, by
means of an appropriate choice of negative values of β when T→+0.
Result 2: Assume that a given nth order continuous-time plant has a relative degree p=2≤n
and is operated by a FROH. Then, the breakaway point located between both starting points
of the Complementary Root Locus (CRL) describing the FROH zeros evolution with β has
the best convergence position of such a locus for any value of β when T→+0.
Result 3: Assume that a given nth order continuous-time plant is zero-free (p=n≥1) and
operated by a FROH. Then, Result 1 is also true for finite sufficiently small sampling periods
T such that 0<T<TS2, where TS2 can be obtained from the continuous-time parameters of the
system. In addition, Result 2 is also true if 0<T<TS2 and p=n=2, that is, for the discretization
of zero-free second order continuous plants.
3. The main result on the sufficiently small T for zero-free second-order systems.
Theorem 1 below replaces and extends Result 3 above for the FROH discretization of
zero-free second-order systems.
2
Theorem 1: Maximum allowable sampling period. Assume that a given second-order
continuous-time plant is zero-free (p=n=2) and operated by a FROH. Then, Results 1 and 2
are also true for finite small sampling periods T such that 0<T<TS2, where
TS 2 =
π
I
(2)
being I the imaginary part of the poles of the continuous plant transfer function.
Proof: As presented in the previous section, the FROH tuning method described in [4] is
based on certain general properties concerning the generalised root locus (GRL) that
describes the evolution on the complex plane of the discrete zeros as a function of the
adjustable gain of the FROH circuit. In particular, such properties refer to the positions on
the real axis of the generalised starting and terminating points –see lemmas 1 and 3 in [4]-.
Now, these positions are obtained for the FROH discretization of a zero-free second-order
system described by
G( s) =
ω n2
s + 2δω n s + ω n2
(3)
2
where ωn is the undamped natural frequency and δ the damping ratio of the plant. Two cases
are considered separately.
(i) Complex poles: If –1<δ<1, the poles are complex conjugate and G(s) can be defined as
G( s) =
C1
(s + R)
2
(4)
+ I2
where C1 is the direct gain. I is the imaginary and (–R) the real part of the poles.
Discretizing the transfer function (4) by using a ZOH circuit –see [1]-, a discrete plant with
the following numerator is obtained
N T 0 ( z ) = z 2 ⋅ [ A + B ] + z ⋅  −2 Ae − R⋅T cos( I ⋅ T ) − B ( e − R⋅T cos( I ⋅ T ) + 1) + Ce − R ⋅T sin( I ⋅ T ) 
+  Ae −2⋅R⋅T + Be − R⋅T cos( I ⋅ T ) − Ce − R⋅T sin( I ⋅ T ) 
A=
where :
C1
C
C1 ⋅ R
; B = −A = − 2 1 2 ; C = −
2
R +I
R +I
I ⋅ ( R2 + I 2 )
(5)
(6)
2
On the other hand, if a FOH circuit is used, the numerator of the resulting discrete plant is
N T 1 ( z ) = ( A + D ) ⋅ ( z − 1) ⋅ ∆ + E ⋅ ∆ + ( B + F ) ⋅ ( z − 1) ⋅ ( z − e − R⋅T ⋅ cos( I ⋅ T ) )
2
+ ( C + G ) ⋅ ( z − 1) ⋅ ( e − R⋅T ⋅ sin( I ⋅ T ) )
2
∆ = z 2 − 2 ⋅ z − e − R⋅T ⋅ cos( I ⋅ T ) + e −2⋅R⋅T
where :
D=−
(7)
2 ⋅ C1 ⋅ R
T ⋅ ( R2 + I
)
2 2
;
E = A = −B =
2 ⋅ C1 ⋅ R
C1
; F = −D =
;
2
R + I2
T ⋅ ( R2 + I 2 )
2
G=

C1  R 2 − I 2
⋅
T ⋅ I  ( R 2 + I 2 )2

 (8)



It have been demonstrated –see [4]- that the GRL describing the evolution of the FROH
zeros on the complex plane can be expressed as
3
 N ( z ) − zN T 0 ( z ) 
1 + β  T1
=0
zN T 0 ( z )


(9)
where β is the device adjustable gain. Thus, the open-loop generalised transfer function is
 N ( z ) − z ⋅ N T 0 ( z )   ( z − 1) ⋅ ( z − zTP ) 
GOL ( z ) =  T 1
 =  z ⋅(z − z ) 
z ⋅ NT 0 ( z)
SP

 

(10)
given that there will always be one fixed starting point (β=0) located on the real axis at z=0
and one fixed terminating point (β=±∞) at z=1 [4]. From (5) and (7), one gets that
N T 1 ( z ) − z ⋅ N T 0 ( z ) = ( z − 1) ⋅ {z 2 ⋅ [ B + F − B + D ]
+ z ⋅  − ( 2 ⋅ D + B + F − B ) ⋅ e − R⋅T ⋅ cos( I ⋅ T ) + (C + G − C ) ⋅ e − R⋅T ⋅ sin( I ⋅ T ) − ( B + F ) 
(11)
+  ( B + F ) ⋅ e − R⋅T ⋅ cos( I ⋅ T ) − (C + G ) ⋅ e − R⋅T ⋅ sin( I ⋅ T ) + D ⋅ e − R⋅T }
Hence, we have from expressions (10), (5), (6), (8) and (11) that the non-fixed starting (zSP)
and terminating (zTP) points are located on the negative real axis respectively in:
zSP (C1 , R, I , T ) = zSP =
A ⋅ e −2⋅R⋅T + B ⋅ e − R⋅T ⋅ cos( I ⋅ T ) − C ⋅ e − R⋅T ⋅ sin( I ⋅ T )
−2 ⋅ A ⋅ e − R⋅T ⋅ cos( I ⋅ T ) − B ⋅ ( e − R⋅T ⋅ cos( I ⋅ T ) + 1) + C ⋅ e − R⋅T ⋅ sin( I ⋅ T )
zTP (C1 , R, I , T ) = zTP =
( B + F ) ⋅ e − R⋅T ⋅ cos( I ⋅ T ) − (C + G ) ⋅ e − R⋅T ⋅ sin( I ⋅ T ) + D ⋅ e −2⋅R⋅T
D ⋅ e − R⋅T ⋅ cos( I ⋅ T ) − G ⋅ e − R⋅T ⋅ sin( I ⋅ T ) + ( B + F )
(12)
(13)
Next, the conditions that should be fulfilled to allow the application of the analytical method
for the FROH circuits right tuning are established. In particular, the non-fixed starting and
terminating points of the GRL under study must be located on the negative real axis in such a
way that zTP-zSP<0, zSP<0 and zTP<0, if the sampling period is sufficiently small –see lemma 6
in [4]-. Since, by using (12) and (13), lim zSP = −1, lim zTP = −2 and lim ( zTP − zSP ) = −1 -all being
T→0
T →0
T→0
negative values-, the following limiting conditions are imposed to obtain the limiting
sampling period TS2 that defines the application range (0, TS2] for the FROH tuning method:
zTP − zSP = 0
⇒
zTP = zSP
(14)
zSP = 0
(15)
zTP = 0
(16)
Due to the characteristics of the expressions (12) and (13), the resulting equations (14)-(16)
belong to the transcendental type. In practice, a large fraction of equations involving
transcendental functions can not be solved exactly in symbolic form. However, in this case,
by taking T =
k ⋅π
I
(∀ integer k) in the expression (14), an exact solution is obtained that
implies zSP = zTP = ( −1)k e
−
k ⋅π ⋅ R
I
. Since the interest focuses on the limiting value of (0, TS2], the
sampling period as small as possible must be considered, so we make k=1 and thus
T=TS 2 =
π
. On the other hand, expressions (15) and (16) have no root inside (0, π ] . In this
I
I
4
way, expression (2) gives the largest sampling period for which the FROH tuning method
presented in [4] is applicable.
(ii) Real poles: Proceeding in the same fashion as in case (i) with |δ|≥1, it clearly follows
that no restriction about the maximum value of the sampling period for the FROH analytical
tuning arises when discretizing zero-free second-order plants with real poles, since TS2→∞
in such a case. The same conclusion is inferred by observing expression (2) as I→0.
4. Restrictions on the practical applicability of the FROH tuning method.
It will be next demonstrated that it always exists a more restrictive condition than that in
Theorem 1, imposed by the Shannon’s sampling theorem –see, e. g., [1], [6], [7]- on the
sampling period of a digital system.
Theorem 2: Consider a zero-free second-order continuous-time plant digitally controlled.
The system output is sampled and the input generated by a FROH hold device. Then, the
maximum allowable sampling period TM, imposed theoretically to prevent aliasing in the
plant output sampling process, is always smaller than the maximum sampling period TS2 in
Theorem 1.
Proof: It is well known –see e.g. [1], [6], [7]- that, in choosing the sampling period for a
control system, the sampling frequency should be at least greater than twice the highestfrequency component of significant amplitude of the signal being sampled. On the other
hand, the cutoff frequency (ω0) of a plant is usually defined as the frequency at which the
log-magnitude of the response is 3 dB smaller than the log-magnitude obtained at ω=0. Thus,
if a generic second-order plant given by the transfer function (3) is considered, the
3
bandwidth obtained is [0, ω0] –see e.g. [7]-, where ωo = ωn 1 − 2δ 2 + 1010 − 4δ 2 + 4δ 4 . So, in
the less restrictive case, the Shannon’s sampling theorem imposes a maximum value to the
sampling period given by
TM =
π
=
ωo
π
(17)
3
10
ω n 1 − 2δ + 10 − 4δ + 4δ
2
2
4
On the one hand, if the poles of the transfer function (3) are p1,2 = −δω n ± ωn δ 2 − 1 , it
follows that -1<δ<1 to warrant such poles to be complex, as it was imposed in equation (4).
Hence, the poles become p1,2 = − R ± Ij = −δω n ± ω n 1 − δ 2 j . Thus, it results from eqn. (2) that
TS2 =
π
π
=
I ωn 1 − δ 2
5
(18)
On the other hand, suppose that if −1 < δ < 1 , it holds that
F1 (δ ) =
π
1−δ
2
π
>
3
10
= F2 (δ )
(19)
1 − 2δ 2 + 10 − 4δ 2 + 4δ 4
Readily, if expression (19) holds, the next inequality holds
3
3δ 4 − 4δ 2 + 1010 > 0
(20)
3
3
10
Define F (δ ) = δ 2 ( 3δ 2 − 4 ) + 1010 . Thus, F (δ ) = 0 if and only if δ 1,22 = 4 ± 16 − 12 ⋅ 10 which
6
always leads to complex solutions, so that F (δ ) > 0 or F (δ ) < 0 for all real δ ∈ [ −1,1] . It is
obvious that sign ( F (δ ) ) = sign ( F ( 0 ) ) = 1 ⇒ F (δ ) > 0 and expression (20) is true for all real
δ ∈ [ −1,1] , what implies that expression (19) hold. Alternatively, expression (19) can be
trivially probed by construction –see figure 1- for any δ ∈ [ −1,1] .
-Figure 1Thus, from expressions (17), (18) and (19), we have that TM<TS2 for any ωn≠0 and
−1 < δ < 1 . With |δ|≥1 -real poles-, TM<TS2 for any ωn, given that TS2→∞ as I→0.
From Theorem 2, the following corollary is readily obtained.
Corollary 1: In the FROH discretization of second-order zero-free systems, it is clearly
demonstrated that TS2 never supposes a factual drawback in practice for the analytical FROH
tuning procedure presented in Result 2 –see [4]-.
Remark 1: As a matter of fact, commonly a sampling rate that is much higher than the
theoretical minimum governed by the sampling theorem must be selected. Then, notice that,
if Theorem 2 holds when twice the break frequency is considered for obtaining TM -the less
restrictive case-, TS2>TM will clearly be true under more pragmatic assumptions, like real
control schemes -where sampling rates between six and ten times the system bandwidth is
contemplated- or digital signal processing systems –see, e. g., [1] and [7]-.
5. Remotely radio controlled vehicle. An illustrative example.
The use of remotely controlled vehicles for reconnaissance for U.N. peacekeeping
missions is nowadays a possibility under study. The desired speed is transmitted by radio to
the vehicle. The zero-free second-order transfer function
Gveh ( s ) =
4
4
=
s 2 + 2 s + 4 ( s + 1)2 + (1.73)2
6
(21)
describes conveniently –see [8] and [9] for details- the behaviour of the vehicle in order to
design a speed control system. By using expressions (3), (4) and (21), the continuous
parameters are ωn=2 rad/s., δ=0.5, C1=4, R=1 and I=1.73.
Many reasons recommend to make use of a digital controller. Therefore, the transfer
function Gveh(s) is discretized by using an appropriate sampling period T. The frequency
response of Gveh(s) is presented in figure 2. By observing figure 2 or by using expression (17)
, we have that ωo =2.54 rad/s. and TM=1.24 s. From equation (2) in Theorem 1, TS2=1.81 s.
and TS2>TM do not suppose a factual drawback –see corollary 1- for the analytical FROH
tuning procedure given by Result 2. Anyway, due to realistic assumptions, a significantly
smaller sampling period T=0.1 s. is utilized for the control system design.
-Figure 2Then, if a FROH device described by (1) is used to generate de continuous control
signal, the evolution of the discrete zeros on the complex plane with β can be described by
the following GRL, obtained from expressions (9), (5), (6), (8), (11) and shown in figure 3.
 ( z-1)( z + 1.8525 ) 
 0.339z 2 +0.289z-0.628 
=0
1+ β 
 = 1 + β  z z+0.935
2
z
+0.935z
(
) 



(22)
-Figure 3From Result 2, the optimum stability point (Breakaway Point in Figure 3) zOPT= -0.4727
and the optimum β, βOPT= -0.3173 have been obtained. Notice that the FROH device
adjusted to βOPT provides a zero magnitude that is 50.53% smaller than the magnitude
corresponding to the ZOH zero zZOH=zSP= -0.9354. Note that, if the poles and zeros are
cancelled by the digital controller, then the closed-loop response is much more oscillatory in
the ZOH case, since the controller possesses a poorly damped pole which corresponds to the
zero zZOH. Once the β-gain of the FROH is tuned to the value of βopt, the most stable
discretization zeros arise for the given sampling period. Thus, the performance of the
controlled system can be significantly improved –see also application example in [5]-.
In particular, suppose that the classical model reference control scheme of pole-zero
placement proposed in [1][10] is used. The desired closed loop performance is described by
the characteristic polynomial AM ( z ) = z 2 + 0.2 z + 0.05 of the reference model, whose zeros are
still unspecified. Now, assume that all the discrete plant poles and zeros are cancelled by the
controller in order to freely choose the reference model. The closed-loop step responses,
when ZOH and FROH with βOPT devices are used, are presented in figures 4 and 5,
respectively, and the improvement obtained in the system performance is clearly shown.
Anyway, it clearly turns out that the zero cancellation with the ZOH device is highly
7
unsuitable, but the optimal FROH is not a good choice either in this case, due to oscillatory
effects arisen between sampling instants –see [1][7] for details on such effects-.
-Figures 4 and 5An alternative procedure for improving the performance of the closed-loop has been
proposed in [5], when the cancellation of the discrete zeros are discouraged. Such procedure
consists in synthesising an appropriate pole-zero configuration for the reference model by
selecting an appropriate gain β. First, the FROH are placed properly -considering the
position of the roots of the desired characteristic polynomial on the complex plane- by using
the CRL of the FROH zeros with β -see figure 3-. Then, those zeros are transmitted to the
reference model. In this case, it was found that the best closed loop response corresponds
again to βOPT=- -0.3173. The step responses obtained by the transmission to the reference
model of the ZOH and the optimum FROH zeros are drawn in figures 6 and 7, respectively.
-Figures 6 and 7Comparing both figures, the improvement on the controlled system performance when
using a properly adjusted FROH is clearly observed. In addition, the control signals referred
to such responses are displayed in figures 8 and 9. The ZOH control signal is more exigent
for the electronic circuitry than the FROH one, since the power dissipation is higher when
the ZOH device is utilized.
A similar example, concerning a digitally controlled hard disk and explained in detail, has
been presented in [5] and supplementary application cases have been described in [4] [11].
-Figures 8 and 9-
6. Conclusion.
The real operability of a novel procedure that allows digitally handled systems to notably
enhance their response performances has been discussed. As a result, it has been shown that
there exits no constraint in practice for the application of such a method, when the
continuous-time system behaviour can be described as a zero-free second-order plant. Such a
case is very common in the automatic control and signal processing environments.
Acknowledgments: The authors would like to thank the anonymous referees for their useful
comments concerning improvements of the original version of the manuscript. This work has
been partially supported by the University of the Basque Country through Project I06.I06EB8235/2000 and MCYT (Project DPI2000-0244).
8
References
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1999, 44 (6) pp. 1229-1234.
[3] DE LA SEN, M., BARCENA, R. and GARRIDO, A. J.: ‘On the intrinsic limiting zeros
as the sampling period tends to zero’, IEEE Trans. Circuits Syst. Part I., 2001, 48 (7)
pp. 898-900.
[4] BARCENA, R. DE LA SEN, M. and SAGASTABEITIA, I.: ‘Improving the stability
properties of the zeros of sampled systems with fractional order hold’, IEE Proc.
Control Theory Appl., 2000, 147 (4) pp. 456- 464.
[5] BARCENA, R., DE LA SEN, M., SAGASTABEITIA, I. and COLLANTES, J.M.
‘Discrete control for a computer hard disk by using a fractional order hold device’, IEE
Proc. Control Theory Appl., 2001, 148 (2) pp. 117- 124.
[6] PHILLIPS, C. L. Digital control systems. Analysis and design. (Prentice-Hall,
Englewood Cliffs, New Jersey. 1990. 2nd edition).
[7] KUO, B. C. Digital control systems. (Oxford University Press. New York. 1992. 2nd
edition).
[8] YAN, J and SALCUDEAN, S. E.: ‘Teleoperation controller design’, IEEE Transactions
on Control Systems Technology, May 1996, pp. 244-247.
[9] DORF, R. C. and BISHOP, R. H.: Modern control systems (Prentice Hall, 2000, 9th edn.)
[10] ASTRÖM, K. J., WITTENMARK, B.: 'Self-tuning controllers based on pole-zero
placement', IEE Proc. D, Control Theory and Applications, 1980, 127 (3) pp. 120-130
[11] BÁRCENA, R., DE LA SEN, M. and SAGASTABEITIA,I.: ‘Fractional Order Hold
Tuning using Neural Networks’, Proceedings of the 2001 American Control
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9
Figure captions:
Figure 1. Functions F1(δ) and F2(δ) involved in inequality (19) when –1<δ<1.
Figure 2. Bode diagram for the remotely controlled reconnaissance vehicle.
Figure 3. Generalised CRL describing the FROH zeros evolution on the complex plane with
β for Gveh(s) and T=0.1s. (detail)
Figure 4. Unit-step response of the compensated system by using a ZOH device, when the
plant zero is cancelled by the digital controller.
Figure 5. Unit-step response of the closed-loop system by using a FROH device with
β=βopt=-0.3173, when the plant zero is cancelled by the digital controller.
Figure 6. Unit-step response of the compensated system by using a ZOH device, when the
discrete plant zero is transmitted to the reference model.
Figure 7. Unit-step response of the compensated system by using a FROH device with
β=βopt=-0.3173, when the discrete zero is transmitted to the reference model.
Figure 8. Control signal referred to the figure 6.
Figure 9. Control signal referred to the figure 7.
10
Figure 1
6
5
F1 ( δ)
4
F2 ( δ)
3
2
1
-1
-0.5
Damping ratio
11
0.5
1
Figure 2
−10
Magnitude (dB)
−15
−20
−25
−30
−35
−40
0
Phase (deg)
−45
−90
−135
−180
−1
10
0
10
Frequency (rad/s.)
12
1
10
Figure 3
1.5
1
Unity disk
Imag Axis
0.5
0
Ztp
Zsp
Zzoh
z=1
z=0
Zopt
−0.5
−1
−1.5
−2
−1.5
−1
−0.5
0
Real Axis
13
0.5
1
1.5
Figure 4
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Time (s.)
0.6
14
0.7
0.8
0.9
1
Figure 5
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Time (s.)
0.6
15
0.7
0.8
0.9
1
Figure 6
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Time (s.)
0.6
16
0.7
0.8
0.9
1
Figure 7
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Time (s.)
0.6
17
0.7
0.8
0.9
1
Figure 8
40
30
20
10
0
−10
−20
−30
−40
0
0.1
0.2
0.3
0.4
0.5
Time (s.)
0.6
18
0.7
0.8
0.9
1
Figure 9
40
30
20
10
0
−10
−20
−30
−40
0
0.1
0.2
0.3
0.4
0.5
Time (s.)
0.6
19
0.7
0.8
0.9
1