Electrical Power and Energy Systems 32 (2010) 170–177
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
An intelligent maximum power extraction algorithm for hybrid
wind–diesel-storage system
Elkhatib Kamal *, Magdy Koutb, Abdul Azim Sobaih, Belal Abozalam
Industrial Electronics and Control Department, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt
a r t i c l e
i n f o
Article history:
Received 22 October 2008
Received in revised form 26 June 2009
Accepted 3 July 2009
Keywords:
Integral control
Takagi–Sugeno (T–S) fuzzy model
Wind energy conversion system
a b s t r a c t
This paper focuses on the development of maximum wind power extraction algorithms for variable speed
wind turbines in hybrid wind–diesel storage system (HWDSS). The propose algorithm utilizes Takagi–
Sugeno (T–S) fuzzy controller. This algorithm combines the merits of: (i) the capability for dealing with
nonlinear systems; (ii) the powerful LMI approach to obtain control gains; (iii) the high performance of
integral controller. The algorithm maximizes the power coefficient for a fixed pitch and suddenly load
changes. Moreover, it reduces the voltage ripple and stabilizes the system over a wide range of wind
speed variations. The control scheme is tested for different real profiles of wind speed pattern and provides satisfactory results.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
In remote areas and small islands, diesel generators are often
the main source of electric power. Diesel fuel has several drawbacks: it is expensive because the transportation to remote areas
adds extra cost, and it causes air pollution by engine exhaust. Providing a feasible, economical, and environmentally friendly solution to diesel generators is important. A hybrid system of wind
power and diesel generators can benefit islands or other isolated
communities and increase fuel savings. Wind is, however, a natural
energy source that produces a fluctuating power output. The
excessive fluctuation of power output adversely affects the quality
of power in the distribution system, particularly frequency and
voltage [1,2].
Variable speed operation and direct-drive generators have been
the recent developments in wind turbine drive trains. Compared
with constant speed operation, variable speed operation of wind
turbines provides 10–15% higher energy output, lower mechanical
stress and less power fluctuation. In order to fully realize the benefits of variable speed wind power generation systems (WPGS), it is
critical to develop advanced control methods to extract maximum
power output of wind turbines at variable wind speeds. A WPGS
needs a power electronic converter, often called an inverter, to
convert variable-frequency, variable-voltage power from a generator into constant-frequency constant-voltage power, and to regulate the output power of the WPGS. Traditionally a gearbox is
used to couple a low speed wind turbine rotor with a high speed
* Corresponding author. Fax: +20 483660716.
E-mail address: elkateb.kamal@gmail.com (E. Kamal).
0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijepes.2009.07.005
generator in a WPGS. Recently much effort has been placed on
the use of a low speed direct-drive generator to eliminate the gearbox [3]. Optimum wind energy extraction is achieved by running
the wind turbine generator (WTG) in variable speed, variable-frequency mode. The rotor speed is allowed to vary in sympathy with
the wind speed, by maintaining the tip speed ratio to a value that
maximizes aerodynamic efficiency. In order to achieve this ratio,
the permanent magnet synchronous generator load line should
be matched very closely to the maximum power line of the wind
turbine generator [4].
The problem of wind energy conversion system output power
control has been considered extensively [5–12]. Maximization of
the wind energy conversion efficiency based on a brushless doubly
fed reluctance generator is discussed in Ref. [5]. Ref. [6] maximizes
power based on a standard V/Hz converter and controls the frequency to achieve the desired power at a given turbine speed.
Ref. [7] maximizes power based on controlling the slip power,
which is extracted from the rotor circuits and fed to the grid
though a rectifier-inverter branch. The firing angle of the inverter
is used to control the slip power. Ref. [8] presents a hill-climb
searching (HCS) control for the maximum wind turbine power
at variable wind speeds. Ref. [9] present control of the power
smoothing system compensates for the effects of wind variation
and load disturbances. Refs. [10–12] in investigate robustness
and power quality performance of a simple wind–diesel system.
The main contribution of this research is to maximize the energy from the real profiles of wind speed using the proposed fuzzy
integral linear matrix equalities (FILME). Also, it provides a robust
controller that stabilizes the HWDSS and overcomes the system
nonlinearity. In addition, it guarantees good robustness and
E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177
performance of the controller. Finally, the proposed algorithm utilizing FILME is simple and leads to robust control performance, it
reduces the voltage ripple on the main bus voltage, which based
on the Takagi–Sugeno (T–S) fuzzy model and linear matrix
inequalities [13–15].
This paper is organized as follows: Section 2 provides system
model. Section 3, presents the design of the proposed FILME controller. Section 4 shows the stability and robustness conditions
for the proposed algorithm. Section 5 presents simulation of the
wind turbine. Finally, concluding remarks are made in Section 6
followed by the list of references.
2. System model
2.1. The wind turbine characteristics and modeling
The mechanical output power at a given wind speed is drastically affected by the turbine’s tip speed ratio (TSR), which is defined as the ratio of turbine rotor tip speed to the wind speed. At
a given wind speed, the maximum turbine energy conversion efficiency occurs at an optimal TSR. Therefore, as wind speed changes,
the turbine’s rotor speed needs to change accordingly in order to
maintain the optimal TSR and thus to extract the maximum power
from the available wind resources [8]. The expression for aerodynamic power (P a ) captured by the wind turbine is given by the
nonlinear expression [16].
Pa ¼ 0:5C p ðkÞqpR2 V 31
A typical C p k curve is shown in Fig. 1. It can be seen that there
is a maximum power coefficient C pðmaxÞ . Normally, a variable speed
wind turbine follows the C pðmaxÞ to capture the maximum power up
to the rated speed by varying the rotor speed to keep the system at
kopt , then operates at the rated power with power control during
the periods of high wind by the active control of the blade pitch angle or the passive regulation based on aerodynamic stall. A typical
power–wind speed curve is shown in Fig. 2.
2.2. System description
The underlying hybrid wind–diesel system is illustrated in
Fig. 3. The hybrid generation system is composed of a wind turbine
coupled with a synchronous generator, a diesel-induction generator, and an energy storage system. In the given system, the wind
turbine drives the synchronous generator that operates in parallel
with the storage battery system. When the wind-generator alone
provides sufficient power for the load, the diesel engine is disconnected from the induction generator. The PEI connecting the load
to the main bus is used to fit the frequency of the power supplying
the load as well as the voltage.
The dynamics of the system can be characterized by the following equations [2]:
x_ ¼ AxðtÞ þ BuðtÞ;
y ¼ CxðtÞ;
ð1Þ
3
where q is the air density (kg/m ), R is the rotor radius (m), V 1 is the
wind speed (m/s), and C p is the power coefficient defined by the following relation [17].
C p ¼ ð0:44 0:0167bÞ sin
pðk 3Þ
15 0:3b
0:00184ðk 3Þb
ð2Þ
where b is the blade pitch angle of the wind turbine, k is TSR and is
given by [16]:
k¼
xt R
ð3Þ
V1
where xt is the rotational speed of the blades.
Referring to (2), optimal TSR kopt can be obtained as follow:
kopt ¼
15 0:3b
p
cos1
0:00184bð15 0:3bÞ
þ3
pð0:44 0:167bÞ
ð4Þ
Fig. 2. Power–wind speed characteristics.
Thus the maximum power captured from the wind is given by:
PaðmaxÞ ¼ 0:5C pðmaxÞ ðkopt ; bÞqpR2 V 3
Fig. 1. Power coefficient C p versus TSR k.
171
ð5Þ
Fig. 3. Structural diagram of hybrid wind–diesel storage system.
ð6Þ
172
E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177
where xðtÞ ¼ ½ V b
xs T , uðtÞ ¼ ½ Efd Iref T ,
amplitude (V b ) and the frequency (f) of the AC bus. The wind speed
(V 1 ) and the load (V 2 ) are considered to be disturbances. The wind
turbine generator and the battery-converter unit run in parallel,
serving the load. From the control point of view, this is a coupled
2 2 multi-input-multi-output nonlinear system.
2 Lf
3
Lf
1 1 s0do xs Lmd s0do Lmdxs ðLd isdraixsq Þ
4
5
s
A¼
0 1 Pind Pload Ds
J s xs V b
Js
"
#
1 JsVxc s
1 0
;
C
¼
;
B¼
0 1
0 J Vxc s
3.2. Takagi–Sugeno’s fuzzy plant model
s
where V c is the AC side line-to-line voltage, Efd is the SG field voltage, xs is the bus frequency (or angular speed of SG) J s , Ds are the
inertia and frictional damping of SG, isd , isq are the direct and quadrature current component of SG, Ld , Lf are the stator d-axis and rotor
inductance of SG, Lmd is the d-axis field mutual inductance, s0do is the
transient open circuit time constant r a is the rotor resistance of SG,
Pind is the power of the induction generator, Pload is the power of the
load, Iref is the direct-current set point, and V b is the bus voltage. Eq.
(6) indicates that the model is the linear form for fixed matrices A, B
and C. However, matrices A and B are not fixed, but change as functions of state variables, thus making the model nonlinear. Also, this
model is only used as a tool for controller design purposes. The used
system parameters are shown in Table 1 [18–20].
3. Fuzzy-ilme controller design
3.1. Control structure
Fig. 4 depicts the input and output relationship of the wind–
battery system from the control point of view. The control inputs
are the excitation field voltage (Efd ) of the SG and the direct-current
set point (Iref ) of the converter. The measurements are the voltage
The Takagi–Sugeno fuzzy model represents a nonlinear system
by partitioning the system into sub-systems and then combining
them with linguistic rules. In this paper, three linear sub-systems
are considered for the nonlinear state-space models (6). The continuous fuzzy dynamic model, proposed by Takagi–Sugeno is described by fuzzy IF–THEN rules, which represent local linear
th
input–output relations of nonlinear systems [21]. The i rule of
this fuzzy model is given by, Plant Rule i:
IF q1 ðxðtÞÞ is Ni1 AND . . . AND qw ðxðtÞÞ is Niw
_
Then xðtÞ
¼ Ai xðtÞ þ Bi uðtÞ; y ¼ C i xðtÞ
ð7Þ
where N iX is a fuzzy set X ¼ 1; 2; . . . ; w, i ¼ 1; 2; . . . ; p, xðtÞ 2 Rnx1 is
the state vector, uðtÞ 2 Rnx1 is the input vector, Ai 2 Rnxn and
Bi 2 Rnxm system matrices of appropriate dimensions, p is the number of IF–THEN rules (p ¼ 3). q1 ðxðtÞÞ; . . . ; qw ðxðtÞÞ are the premise
variables. The plant dynamics is then described by,
_
xðtÞ
¼
p
X
hi ðxðtÞÞ½Ai xðtÞ þ Bi uðtÞ;
ð8Þ
i¼1
where hi ðxðtÞÞ ¼
Table 1
System parameters.
ji ðxðtÞÞ
p
P
;
ji ðxðtÞÞ
i¼1
Rated power
Blade radius
Air density
Rated wind speed
Cut-in speed
Cut-out speed
Blade pitch angle
Rated line ac voltage
AC rated current
DC rated current
Rated load power
The inertia of SG
Rated power of IG
The inertia of the IG
Torsional damping
Rotor resistance of SG
Stator d-axis inductance of SG
Rotor inductance of SG
d-Axis field mutual inductance
The transient open circuit time constant
1 [MW]
37.38 [m]
0.55 [kg/m3]
12.35 [m/s]
4 [m/s]
25 [m/s]
0o
230 [V]
138 [A]
239 [A]
40 [kW]
1.11 [kg m2]
55 [kW]
1.40 [kg m2]
0.557 [Nm/rad]
0.96 [X]
2.03 [mH]
2.07 [mH]
1.704 [mH]
2.16 [ms]
w
ji ðxðtÞÞ ¼ P NiX ðxðtÞÞ; hi > 0;
X¼1
p
X
hi ðxðtÞÞ ¼ 1
i¼1
3.3. Fuzzy controller
Three controllers are designed for the three linear sub-systems,
and then the total control output is obtained by defuzzification. A
state-feedback by linear matrix equalities (LME) is used to design
controller for each sub-system. The control is performed so that
the power coefficient is maximized, thus the maximum power captured from the wind is obtained.
th
The j rule of fuzzy controller is given by:
th
Plant Rule j :
IF f 1 ðxðtÞÞ is Mj1 AND . . . AND f w ðxðtÞÞ is Mjw
Then uðtÞ ¼ Gj xðtÞ þ r;
ð9Þ
where Mj/ is a fuzzy set / ¼ 1; 2; . . . ; w, j ¼ 1; 2; . . . ; c, r is the reference input, f1 ðxðtÞÞ; . . . ; fw ðxðtÞÞ are the premise variables, c is the
number of IF–THEN rules (c ¼ 5), and Gj are local feedback gains.
The inferred output of the fuzzy controller is given by:
uðtÞ ¼
c
X
mj ðxðtÞÞ Gj xðtÞ þ r ;
j¼1
w
Y
-j ðxðtÞÞ
Mj/ ;
; -j ðxðtÞÞ ¼
j¼1 -j ðxðtÞÞ
/
where mj ðxðtÞÞ ¼ Pc
mj > 0;
Fig. 4. The wind–battery control system.
c
X
j¼1
mj ¼ 1
ð10Þ
173
E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177
steady state faster if we use larger values of f. Calculation of Gj of
the fuzzy controller that satisfies the stability and robustness conditions is formulated as an LME problem.
If T T ¼ T; P ¼ TT
Gj ¼ RBTj P
8j;
ð13Þ
Rotor speed (m/s)
1.9
Fig. 5. Membership functions of states.
3.4. General design approach (GDA)
It is applicable to those T–S fuzzy plant models with the number
of rules and the rule antecedents of the fuzzy controller are different from that of the T–S fuzzy plant model. In order to carry out the
analysis, the closed-loop fuzzy system should be obtained first.
Referring to (8) and (10), the fuzzy control system is given by:
_
XðtÞ
¼
p X
c
X
i¼1
1.8
1.7
1.6
1.5
1.4
1.3
wt
w opt
0
10
20
30
40
30
40
Time (Sec)
Fig. 7. Rotor speed tracking.
hi ðxðtÞÞmj ðxðtÞÞ Hij XðtÞ þ Bi r
ð11Þ
j¼1
where Hij ¼ Ai þ BGj .
For each sub-space, different model (i ¼ 1; 2; 3), j ¼ 1; 2; 3; 4; 5
and (p ¼ 3) is applied. The degree of membership function for
states V b and xs is depicted in Fig. 5. Each membership function
also represents model uncertainty for each sub-system.
0.7
Power (Mw)
0.6
4. Stability and robustness for the proposed algorithm
A proof of the stability and robustness conditions for the plant
dynamics described by (8) is shown in the appendix. The main result is summarized in the following lemma.
0.5
0.4
0.3
Lemma. Under GDA, the fuzzy control system as given by (11) is
stable if
0.2
0
10
20
Time(Sec)
l½THij T 1 m
ð12Þ
Fig. 8. Per unit wind turbine produced power.
where m nonzero positive constant, T is a transformation matrix. The
analysis given in the appendix indicates that kxðtÞk will go to its
230.001
Bus voltage (v)
Wind Speed(m/s)
230.0005
11
10
9
8
0
230
229.9995
229.999
229.9985
10
20
30
Time (Sec)
Fig. 6. The real profile of wind speed.
40
229.998
10
20
Time (Sec)
Fig. 9. Bus voltage.
30
40
174
E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177
P > 0; R 2 jmxm are symmetric positive definite matrix. The transformation matrix (T) should be found in such a way that the uncertainty free system is stable [22]. Using (11)
Rotor speed (m/s)
wt
PHij þ HTij P < 0
PAi þ ATi P 2xPBRBT P ¼ rI
where
P > 0;
8i; j;
ð14Þ
r is robustness index.
5. Simulation and experimental results
The proposed controller for the HWDSS is tested for many cases
of wind speed variations. Three wind speed signals are tested in
this section to prove the effectiveness of the proposed algorithm.
Load power (kw)
50
w opt
2
1
0
0
10
20
30
40
Time (Sec)
Fig. 12. Rotor speed tracking.
0.7
0.6
Power (Mw)
Case 1 Real profile of wind speed signalIn this case, the rated
power of IG (P ind ) is 55 kW and the real profile of wind
have been used to test the control system is considered
as shown in Fig. 6. The rotor speed is shown in Fig. 7 (solid
line) and the dash curve in the same figure represents the
actual rotor speed. The proposed controller is provide better disturbance rejection than the control of the power
smoothing system compensates for the effects of wind
variation and load disturbances that reported in [9]. The
produced power curve as shown in Fig. 8. Fig. 9 shows
3
0.5
0.4
45
0.3
40
0.2
35
0
10
20
30
40
Time(Sec)
30
Fig. 13. Per unit wind turbine produced power.
25
20
15
230.4
0
10
20
30
230.2
40
Bus voltage (v)
Time (Sec)
Fig. 10. The power of the load (P load ).
Wind Speed(m/s)
12
230
229.8
229.6
229.4
11
10
20
30
40
Time (Sec)
10
Fig. 14. Bus voltage.
9
8
0
10
20
30
Time (Sec)
Fig. 11. The real profile of wind speed.
40
the voltage profile is nearly constant and the voltage ripple is reduced to 93% compared with the Fuzzy-LQR controller [1,20].
Case 2 Real profile of wind speed signal and suddenly load changesIn this case, suddenly load changes are considered as
shown in Fig. 10 since the parameter P load take different
values. Fig. 11 shows the real profile of wind speed signal.
175
E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177
Load power (kw)
50
45
40
35
30
0
10
20
30
40
Time (Sec)
Fig. 15. The power of the load (P load ).
Case 3 Random variation of wind speed signalIn this case, the
rated power of IG is 40 kW and load power take different
values considered as shown in Fig. 15, the wind speed signal is considered as a sine wave as shown in Fig. 16. The
rotor speed to capture the maximum power from the
wind turbine is shown in Fig. 17 (solid line). It is clear that
the dash curve in Fig. 17 which represents the actual rotor
speed coincides with the solid curve. As the wind speed
ranges between the cut-in and rated speed of the wind
turbine, the produced power curve take almost the wind
speed curve as shown in Fig. 18. The power generated at
wind speed of 7.6 m/s is 0.19 MW. Comparing this value
with that obtained using Fuzzy-LQR controller [1] which
is 0.02 MW, it is clear that a 95% increase is obtained in
the maximum value. Fig. 19 shows the voltage profile is
nearly constant and the voltage ripple is reduced to 93%
compared with the adaptive fuzzy logic control [1,20].
Comparing the results of the proposed algorithm, with that given in Refs. [8,9], Refs. [1,20], it could be seen that the proposed
controller has the following advantages:
7.4
7.2
0.2
7
6.8
0.18
6.6
6.4
6.2
0
10
20
30
40
Time (Sec)
Power (Mw)
Wind Speed (m/s)
7.6
0.16
0.14
0.12
Fig. 16. Wind speed.
0.1
wt
w opt
10
20
30
40
Time(Sec)
Fig. 18. Per unit wind turbine produced power.
2
1.5
230.4
230.3
1
0.5
0
10
20
30
Time (Sec)
Fig. 17. Rotor speed tracking.
The rotor speed is shown in Fig. 12 (solid line) and the
dash curve in the same figure represents the actual rotor
speed. The proposed controller is provide better disturbance rejection than the hill-climb searching method that
reported in [8]. The produced power curve as shown in
Fig. 13. Fig. 14 shows the voltage profile is nearly constant
and the voltage ripple is reduced to 93% compared with
the Fuzzy-LQR controller [1,20].
Bus voltage (v)
Rotor speed (m/s)
2.5
0
230.2
230.1
230
229.9
229.8
229.7
10
20
Time (Sec)
Fig. 19. Bus voltage.
30
176
E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177
(i) It can control the plant well over a wide range of the wind
speeds.
(ii) The generated power is increased up to 95% compared with
[1].
(iii) The algorithm is more robust in the presence of high
nonlinearity.
(iv) Bus voltage is nearly constant and voltage ripple is reduced
to 93% compared with [1,20].
p
p
c
X
dkTxðtÞk X X
hi ðxðtÞÞmj ðxðtÞÞl½THij T 1 kTxðtÞk þ k
6
dt
i¼1 j¼1
i¼1
c
X
hi ðxðtÞÞmj ðxðtÞÞ½TBrk
where
l½THij T 1 ¼ limþ
Dt!0
Appendix. Proof of the stability and robustness conditions
Consider the Taylor series [21].
_ Dt þ UðDtÞ
xðt þ DtÞ ¼ xðtÞ þ xðtÞ
ð15Þ
_ Dt is the error term and Dt > 0
where UðDtÞ ¼ xðt þ DtÞ xðtÞ xðtÞ
lim
Dt!0þ
UðDtÞ
¼0
Dt
ð16Þ
From (11) and (15) and multiplying a transformation matrix
T 2 Rnxn of rank n to both sides and taking norm on both sides of
the above equation, we have
kTðxðt þ DtÞÞk 6 k
p X
c
X
i¼1
p
p
c
X
dkTxðtÞk X X
hi ðxðtÞÞmj ðxðtÞÞðl½THij T 1 ÞkTxðtÞk þ k
6
dt
i¼1 j¼1
i¼1
c
X
hi ðxðtÞÞmj ðxðtÞÞ½TBrk
ð21Þ
j¼1
Let
l½THii T 1 m 8i
ð22Þ
From (21) and (22)
p X
c
X
d
hi ðxðtÞÞmj ðxðtÞÞk½TBrkemðtt0 Þ
kTxðtÞkemðtt0 Þ dt
i¼1 j¼1
ð23Þ
where t0 < t is an arbitrary initial time, based on (23) there are two
cases to investigate the system behavior.
(1) – r ¼ 0, (2) – r–0
If the condition (22) is satisfied the closed-loop system (11) is
stable, and kxðtÞk ! 0 as t ! 1
Proof for: (1) r ¼ 0
d
kTxðtÞkemðtt0 Þ 0
dt
kTxðtÞk 6 kTxðt 0 Þkemðtt0 Þ
ð24Þ
Since n is positive value, kxðtÞk ! 0 as t ! 1 (2) r – 0, from (23)
K
ðkTxðtÞkemðtt0 Þ Þ 6 kTxðt 0 Þk þ kT B rk
Z
t
emðst0 Þ ds
t0
i
K
p
X
kTxðtÞk 6 kTxðt0 Þkemðtt0 Þ þ
hi ðxðtÞÞmj ðxðtÞÞ½TBrDtk þ kT UðDtÞk
ð17Þ
j¼1
where k k denotes the L2 norm for vectors and L2 induced norm for
matrices, from (17)
Dt!0
ð20Þ
where gmax ðÞ is the largest eigenvalue, * is the conjugate transpose,
from (19)
j¼1
i¼1
!
K
þ THij T DtÞkkTxðtÞk þ k
c
X
THij T 1 þ ðTHij T 1 Þ
2
where kT B rk 6 max k½TBrkmax 6 kTBrk; then
hi ðxðtÞÞmj ðxðtÞÞðI
1
limþ
kI þ THij T 1 Dtk 1
Dt
¼ gmax
6. Conclusion
This paper presents a hybrid power system consisted of a wind
turbine, a diesel generation unit and energy storage devices. Both
the wind power generator and the SG operate at variable speed so
as to maximize the wind energy capture as a force source and minimize the diesel fuel consumption for economic purpose. Both types
of generation units are connected to the ac load system through PEI
to stabilize the system frequency. The control is performed so that
the power coefficient is maximized. The operating principles have
been discussed and the simulation model of the systems has been
developed. The proposed algorithm utilizing FILME is simple and
leads to robust control performance. Simulation results have confirmed that, maximum power conversion efficiency obtained increases to the order of 95% compared with previous methods and
voltage ripple reduced to 93%. Maximum power control of hybrid-wind power generation with storage battery is achieved.
ð19Þ
j¼1
p X
c
X
kTðxðt þ DtÞÞk kTxðtÞk
hi ðxðtÞÞmj ðxðtÞÞ
limþ f
Dt
Dt!0
i¼1 j¼1
ð25Þ
Since the right-hand side of (25) is finite if r is bounded, the system states (11) are also bounded.
The above analysis gives an upper bound of kTxðtÞk under different the two considered cases. The result is given by Eqs. (24) and
(25). Similarly, a lower bound of kxðtÞk can be obtained by following the same analysis procedure with
_ Dt þ uðDtÞ
xðt DtÞ ¼ xðtÞ xðtÞ
ð26Þ
_ Dt is the error term and Dt > 0,
where uðDtÞ ¼ xðt DtÞ xðtÞ þ xðtÞ
# is governed by
ðkI þ THij T 1 Dtk 1Þ
kTxðtÞkg=Dt þ imDt!0þ fk
kT B rk
ð1 emðtt0 Þ Þ
n
p
X
Let
l½THii T 1 6 # 8i
ð27Þ
i¼1
c
X
Since # is positive value
hi ðxðtÞÞmj ðxðtÞÞ
References
j¼1
½TBrDtk þ kT UðDtÞkg=Dt
From (16) and (18)
ð18Þ
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