Electrical Power and Energy Systems 32 (2010) 170–177

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

An intelligent maximum power extraction algorithm for hybridwind–diesel-storage system

Elkhatib Kamal *, Magdy Koutb, Abdul Azim Sobaih, Belal AbozalamIndustrial Electronics and Control Department, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 October 2008Received in revised form 26 June 2009Accepted 3 July 2009

Keywords:Integral controlTakagi–Sugeno (T–S) fuzzy modelWind energy conversion system

0142-0615/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijepes.2009.07.005

* Corresponding author. Fax: +20 483660716.E-mail address: elkateb.kamal@gmail.com (E. Kam

This paper focuses on the development of maximum wind power extraction algorithms for variable speedwind 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 withnonlinear systems; (ii) the powerful LMI approach to obtain control gains; (iii) the high performance ofintegral controller. The algorithm maximizes the power coefficient for a fixed pitch and suddenly loadchanges. Moreover, it reduces the voltage ripple and stabilizes the system over a wide range of windspeed variations. The control scheme is tested for different real profiles of wind speed pattern and pro-vides satisfactory results.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In remote areas and small islands, diesel generators are oftenthe main source of electric power. Diesel fuel has several draw-backs: it is expensive because the transportation to remote areasadds extra cost, and it causes air pollution by engine exhaust. Pro-viding a feasible, economical, and environmentally friendly solu-tion to diesel generators is important. A hybrid system of windpower and diesel generators can benefit islands or other isolatedcommunities and increase fuel savings. Wind is, however, a naturalenergy source that produces a fluctuating power output. Theexcessive fluctuation of power output adversely affects the qualityof power in the distribution system, particularly frequency andvoltage [1,2].

Variable speed operation and direct-drive generators have beenthe recent developments in wind turbine drive trains. Comparedwith constant speed operation, variable speed operation of windturbines provides 10–15% higher energy output, lower mechanicalstress and less power fluctuation. In order to fully realize the ben-efits of variable speed wind power generation systems (WPGS), it iscritical to develop advanced control methods to extract maximumpower output of wind turbines at variable wind speeds. A WPGSneeds a power electronic converter, often called an inverter, toconvert variable-frequency, variable-voltage power from a genera-tor into constant-frequency constant-voltage power, and to regu-late the output power of the WPGS. Traditionally a gearbox isused to couple a low speed wind turbine rotor with a high speed

ll rights reserved.

al).

generator in a WPGS. Recently much effort has been placed onthe use of a low speed direct-drive generator to eliminate the gear-box [3]. Optimum wind energy extraction is achieved by runningthe wind turbine generator (WTG) in variable speed, variable-fre-quency mode. The rotor speed is allowed to vary in sympathy withthe wind speed, by maintaining the tip speed ratio to a value thatmaximizes aerodynamic efficiency. In order to achieve this ratio,the permanent magnet synchronous generator load line shouldbe matched very closely to the maximum power line of the windturbine generator [4].

The problem of wind energy conversion system output powercontrol has been considered extensively [5–12]. Maximization ofthe wind energy conversion efficiency based on a brushless doublyfed reluctance generator is discussed in Ref. [5]. Ref. [6] maximizespower based on a standard V/Hz converter and controls the fre-quency 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 gridthough a rectifier-inverter branch. The firing angle of the inverteris used to control the slip power. Ref. [8] presents a hill-climbsearching (HCS) control for the maximum wind turbine powerat variable wind speeds. Ref. [9] present control of the powersmoothing system compensates for the effects of wind variationand load disturbances. Refs. [10–12] in investigate robustnessand power quality performance of a simple wind–diesel system.

The main contribution of this research is to maximize the en-ergy from the real profiles of wind speed using the proposed fuzzyintegral linear matrix equalities (FILME). Also, it provides a robustcontroller that stabilizes the HWDSS and overcomes the systemnonlinearity. In addition, it guarantees good robustness and

Fig. 2. Power–wind speed characteristics.

E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177 171

performance of the controller. Finally, the proposed algorithm uti-lizing FILME is simple and leads to robust control performance, itreduces the voltage ripple on the main bus voltage, which basedon the Takagi–Sugeno (T–S) fuzzy model and linear matrixinequalities [13–15].

This paper is organized as follows: Section 2 provides systemmodel. Section 3, presents the design of the proposed FILME con-troller. Section 4 shows the stability and robustness conditionsfor the proposed algorithm. Section 5 presents simulation of thewind turbine. Finally, concluding remarks are made in Section 6followed 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 drasti-cally affected by the turbine’s tip speed ratio (TSR), which is de-fined as the ratio of turbine rotor tip speed to the wind speed. Ata given wind speed, the maximum turbine energy conversion effi-ciency occurs at an optimal TSR. Therefore, as wind speed changes,the turbine’s rotor speed needs to change accordingly in order tomaintain the optimal TSR and thus to extract the maximum powerfrom the available wind resources [8]. The expression for aerody-namic power (Pa) captured by the wind turbine is given by thenonlinear expression [16].

Pa ¼ 0:5CpðkÞqpR2V31 ð1Þ

where q is the air density (kg/m3), R is the rotor radius (m), V1 is thewind speed (m/s), and Cp is the power coefficient defined by the fol-lowing relation [17].

Cp ¼ ð0:44� 0:0167bÞ sinpðk� 3Þ

15� 0:3b

� �� 0:00184ðk� 3Þb ð2Þ

where b is the blade pitch angle of the wind turbine, k is TSR and isgiven by [16]:

k ¼ xtRV1

ð3Þ

where xt is the rotational speed of the blades.Referring to (2), optimal TSR kopt can be obtained as follow:

kopt ¼15� 0:3b

p

� �cos�1 0:00184bð15� 0:3bÞ

pð0:44� 0:167bÞ

� �þ 3 ð4Þ

Thus the maximum power captured from the wind is given by:

PaðmaxÞ ¼ 0:5CpðmaxÞðkopt; bÞqpR2V3 ð5Þ

Fig. 1. Power coefficient Cp versus TSR k.

A typical Cp � k curve is shown in Fig. 1. It can be seen that thereis a maximum power coefficient CpðmaxÞ. Normally, a variable speedwind turbine follows the CpðmaxÞ to capture the maximum power upto the rated speed by varying the rotor speed to keep the system atkopt , then operates at the rated power with power control duringthe periods of high wind by the active control of the blade pitch an-gle or the passive regulation based on aerodynamic stall. A typicalpower–wind speed curve is shown in Fig. 2.

2.2. System description

The underlying hybrid wind–diesel system is illustrated inFig. 3. The hybrid generation system is composed of a wind turbinecoupled with a synchronous generator, a diesel-induction genera-tor, and an energy storage system. In the given system, the windturbine drives the synchronous generator that operates in parallelwith the storage battery system. When the wind-generator aloneprovides sufficient power for the load, the diesel engine is discon-nected from the induction generator. The PEI connecting the loadto the main bus is used to fit the frequency of the power supplyingthe load as well as the voltage.

The dynamics of the system can be characterized by the follow-ing equations [2]:

_x ¼ AxðtÞ þ BuðtÞ; y ¼ CxðtÞ; ð6Þ

Fig. 3. Structural diagram of hybrid wind–diesel storage system.

172 E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177

where xðtÞ ¼ ½Vb xs �T , uðtÞ ¼ ½ Efd Iref �T ,

A ¼1 10 1

� � Lf

s0do

xsLmd

Lf

s0do

LmdxsðLdi

sd�raisqxs

ÞPind�Pload

JsxsVb� Ds

Js

24

35

B ¼1 � Vc

Jsxs

0 � VcJsxs

" #; C ¼

1 00 1

� �;

where Vc is the AC side line-to-line voltage, Efd is the SG field volt-age, xs is the bus frequency (or angular speed of SG) Js, Ds are theinertia and frictional damping of SG, isd, isq are the direct and quad-rature current component of SG, Ld, Lf are the stator d-axis and rotorinductance of SG, Lmd is the d-axis field mutual inductance, s0do is thetransient open circuit time constant ra is the rotor resistance of SG,Pind is the power of the induction generator, Pload is the power of theload, Iref is the direct-current set point, and Vb is the bus voltage. Eq.(6) indicates that the model is the linear form for fixed matrices A, Band C. However, matrices A and B are not fixed, but change as func-tions of state variables, thus making the model nonlinear. Also, thismodel is only used as a tool for controller design purposes. The usedsystem 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 inputsare the excitation field voltage (Efd) of the SG and the direct-currentset point (Iref ) of the converter. The measurements are the voltage

Table 1System parameters.

Rated power 1 [MW]Blade radius 37.38 [m]Air density 0.55 [kg/m3]Rated wind speed 12.35 [m/s]Cut-in speed 4 [m/s]Cut-out speed 25 [m/s]Blade pitch angle 0o

Rated line ac voltage 230 [V]AC rated current 138 [A]DC rated current 239 [A]Rated load power 40 [kW]The inertia of SG 1.11 [kg m2]Rated power of IG 55 [kW]The inertia of the IG 1.40 [kg m2]Torsional damping 0.557 [Nm/rad]Rotor resistance of SG 0.96 [X]Stator d-axis inductance of SG 2.03 [mH]Rotor inductance of SG 2.07 [mH]d-Axis field mutual inductance 1.704 [mH]The transient open circuit time constant 2.16 [ms]

Fig. 4. The wind–battery control system.

amplitude (Vb) and the frequency (f) of the AC bus. The wind speed(V1) and the load (V2) are considered to be disturbances. The windturbine generator and the battery-converter unit run in parallel,serving the load. From the control point of view, this is a coupled2 � 2 multi-input-multi-output nonlinear system.

3.2. Takagi–Sugeno’s fuzzy plant model

The Takagi–Sugeno fuzzy model represents a nonlinear systemby partitioning the system into sub-systems and then combiningthem with linguistic rules. In this paper, three linear sub-systemsare considered for the nonlinear state-space models (6). The con-tinuous fuzzy dynamic model, proposed by Takagi–Sugeno is de-scribed by fuzzy IF–THEN rules, which represent local linearinput–output relations of nonlinear systems [21]. The ith rule ofthis fuzzy model is given by, Plant Rule i:

IF q1ðxðtÞÞ is Ni1 AND . . . AND qwðxðtÞÞ is Ni

w

Then _xðtÞ ¼ AixðtÞ þ BiuðtÞ; y ¼ CixðtÞ ð7Þ

where NiX 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 andBi 2 Rnxm system matrices of appropriate dimensions, p is the num-ber of IF–THEN rules (p ¼ 3). q1ðxðtÞÞ; . . . ; qwðxðtÞÞ are the premisevariables. The plant dynamics is then described by,

_xðtÞ ¼Xp

i¼1

hiðxðtÞÞ AixðtÞ þ BiuðtÞ½ �; ð8Þ

where hiðxðtÞÞ ¼jiðxðtÞÞPp

i¼1jiðxðtÞÞ

;

jiðxðtÞÞ ¼ Pw

X¼1Ni

XðxðtÞÞ; hi > 0;Xp

i¼1

hiðxðtÞÞ ¼ 1

3.3. Fuzzy controller

Three controllers are designed for the three linear sub-systems,and then the total control output is obtained by defuzzification. Astate-feedback by linear matrix equalities (LME) is used to designcontroller for each sub-system. The control is performed so thatthe power coefficient is maximized, thus the maximum power cap-tured from the wind is obtained.

The jth rule of fuzzy controller is given by:Plant Rule jth:

IF f 1ðxðtÞÞ is Mj1 AND . . . AND f wðxðtÞÞ is Mj

w

Then uðtÞ ¼ GjxðtÞ þ r; ð9Þ

where Mj/ is a fuzzy set / ¼ 1;2; . . . ;w, j ¼ 1;2; . . . ; c, r is the refer-

ence input, f1ðxðtÞÞ; . . . ; fwðxðtÞÞ are the premise variables, c is thenumber of IF–THEN rules (c ¼ 5), and Gj are local feedback gains.The inferred output of the fuzzy controller is given by:

uðtÞ ¼Xc

j¼1

mjðxðtÞÞ GjxðtÞ þ r� �

; ð10Þ

where mjðxðtÞÞ ¼-jðxðtÞÞPcj¼1-jðxðtÞÞ

; -jðxðtÞÞ ¼Yw/

Mj/;

mj > 0;Xc

j¼1

mj ¼ 1

Fig. 5. Membership functions of states.

0 10 20 30 401.3

1.4

1.5

1.6

1.7

1.8

1.9

Rot

or s

peed

(m/s

)

Time (Sec)

wtwopt

Fig. 7. Rotor speed tracking.

0 10 20 30 400.2

0.3

0.4

0.5

0.6

0.7

Time(Sec)

Pow

er (M

w)

Fig. 8. Per unit wind turbine produced power.

E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177 173

3.4. General design approach (GDA)

It is applicable to those T–S fuzzy plant models with the numberof rules and the rule antecedents of the fuzzy controller are differ-ent from that of the T–S fuzzy plant model. In order to carry out theanalysis, the closed-loop fuzzy system should be obtained first.

Referring to (8) and (10), the fuzzy control system is given by:

_XðtÞ ¼Xp

i¼1

Xc

j¼1

hiðxðtÞÞmjðxðtÞÞ HijXðtÞ þ Bir� �

ð11Þ

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 forstates Vb and xs is depicted in Fig. 5. Each membership functionalso represents model uncertainty for each sub-system.

4. Stability and robustness for the proposed algorithm

A proof of the stability and robustness conditions for the plantdynamics described by (8) is shown in the appendix. The main re-sult is summarized in the following lemma.

Lemma. Under GDA, the fuzzy control system as given by (11) isstable if

l½THijT�1� � �m ð12Þ

where m nonzero positive constant, T is a transformation matrix. Theanalysis given in the appendix indicates that kxðtÞk will go to its

0 10 20 30 408

9

10

11

Win

d Sp

eed(

m/s

)

Time (Sec)

Fig. 6. The real profile of wind speed.

steady state faster if we use larger values of f. Calculation of Gj ofthe fuzzy controller that satisfies the stability and robustness con-ditions is formulated as an LME problem.

If TT ¼ T; P ¼ TT

Gj ¼ �RBTj P 8j; ð13Þ

10 20 30 40229.998

229.9985

229.999

229.9995

230

230.0005

230.001

Time (Sec)

Bus

volta

ge (v

)

Fig. 9. Bus voltage.

0 10 20 30 400

1

2

3

Roto

r spe

ed (m

/s)

Time (Sec)

wt

wopt

Fig. 12. Rotor speed tracking.

0.5

0.6

0.7

(Mw

)

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 trans-formation matrix (T) should be found in such a way that the uncer-tainty free system is stable [22]. Using (11)

PHij þ HTijP < 0

PAi þ ATi P � 2xPBRBT P ¼ �rI P > 0; 8i; j; ð14Þ

where r is robustness index.

5. Simulation and experimental results

The proposed controller for the HWDSS is tested for many casesof wind speed variations. Three wind speed signals are tested inthis section to prove the effectiveness of the proposed algorithm.

Case 1 Real profile of wind speed signalIn this case, the ratedpower of IG (Pind) is 55 kW and the real profile of windhave been used to test the control system is consideredas shown in Fig. 6. The rotor speed is shown in Fig. 7 (solidline) and the dash curve in the same figure represents theactual rotor speed. The proposed controller is provide bet-ter disturbance rejection than the control of the powersmoothing system compensates for the effects of windvariation and load disturbances that reported in [9]. Theproduced power curve as shown in Fig. 8. Fig. 9 shows

0 10 20 30 4015

20

25

30

35

40

45

50

Load

pow

er (k

w)

Time (Sec)

Fig. 10. The power of the load (Pload).

0 10 20 30 408

9

10

11

12

Win

d Sp

eed(

m/s

)

Time (Sec)

Fig. 11. The real profile of wind speed.

0 10 20 30 400.2

0.3

0.4

Time(Sec)

Pow

er

Fig. 13. Per unit wind turbine produced power.

10 20 30 40

229.4

229.6

229.8

230

230.2

230.4

Time (Sec)

Bus

volta

ge (v

)

Fig. 14. Bus voltage.

the voltage profile is nearly constant and the voltage rip-ple is reduced to 93% compared with the Fuzzy-LQR con-troller [1,20].

Case 2 Real profile of wind speed signal and suddenly load chan-gesIn this case, suddenly load changes are considered asshown in Fig. 10 since the parameter Pload take differentvalues. Fig. 11 shows the real profile of wind speed signal.

0 10 20 30 406.2

6.4

6.6

6.8

7

7.2

7.4

7.6

Win

d Sp

eed

(m/s

)

Time (Sec)

Fig. 16. Wind speed.

0 10 20 300.5

1

1.5

2

2.5

Time (Sec)

Roto

r spe

ed (m

/s)

wtwopt

Fig. 17. Rotor speed tracking.

0 10 20 30 400.1

0.12

0.14

0.16

0.18

0.2

Time(Sec)

Pow

er (M

w)

Fig. 18. Per unit wind turbine produced power.

0 10 20 30 40

30

35

40

45

50

Load

pow

er (k

w)

Time (Sec)

Fig. 15. The power of the load (Pload).

10 20 30229.7

229.8

229.9

230

230.1

230.2

230.3

230.4

Time (Sec)

Bus

vol

tage

(v)

Fig. 19. Bus voltage.

E. Kamal et al. / Electrical Power and Energy Systems 32 (2010) 170–177 175

The rotor speed is shown in Fig. 12 (solid line) and thedash curve in the same figure represents the actual rotorspeed. The proposed controller is provide better distur-bance rejection than the hill-climb searching method thatreported in [8]. The produced power curve as shown inFig. 13. Fig. 14 shows the voltage profile is nearly constantand the voltage ripple is reduced to 93% compared withthe Fuzzy-LQR controller [1,20].

Case 3 Random variation of wind speed signalIn this case, therated power of IG is 40 kW and load power take differentvalues considered as shown in Fig. 15, the wind speed sig-nal is considered as a sine wave as shown in Fig. 16. Therotor speed to capture the maximum power from thewind turbine is shown in Fig. 17 (solid line). It is clear thatthe dash curve in Fig. 17 which represents the actual rotorspeed coincides with the solid curve. As the wind speedranges between the cut-in and rated speed of the windturbine, the produced power curve take almost the windspeed curve as shown in Fig. 18. The power generated atwind speed of 7.6 m/s is 0.19 MW. Comparing this valuewith that obtained using Fuzzy-LQR controller [1] whichis 0.02 MW, it is clear that a 95% increase is obtained inthe maximum value. Fig. 19 shows the voltage profile isnearly 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 gi-ven in Refs. [8,9], Refs. [1,20], it could be seen that the proposedcontroller has the following advantages:

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 windspeeds.

(ii) The generated power is increased up to 95% compared with[1].

(iii) The algorithm is more robust in the presence of highnonlinearity.

(iv) Bus voltage is nearly constant and voltage ripple is reducedto 93% compared with [1,20].

6. Conclusion

This paper presents a hybrid power system consisted of a windturbine, a diesel generation unit and energy storage devices. Boththe wind power generator and the SG operate at variable speed soas to maximize the wind energy capture as a force source and min-imize the diesel fuel consumption for economic purpose. Both typesof generation units are connected to the ac load system through PEIto stabilize the system frequency. The control is performed so thatthe power coefficient is maximized. The operating principles havebeen discussed and the simulation model of the systems has beendeveloped. The proposed algorithm utilizing FILME is simple andleads to robust control performance. Simulation results have con-firmed that, maximum power conversion efficiency obtained in-creases to the order of 95% compared with previous methods andvoltage ripple reduced to 93%. Maximum power control of hy-brid-wind power generation with storage battery is achieved.

Appendix. Proof of the stability and robustness conditions

Consider the Taylor series [21].

xðt þ DtÞ ¼ xðtÞ þ _xðtÞDt þUðDtÞ ð15Þ

where UðDtÞ ¼ xðt þ DtÞ � xðtÞ � _xðtÞDt is the error term and Dt > 0

limDt!0þ

UðDtÞDt

¼ 0 ð16Þ

From (11) and (15) and multiplying a transformation matrixT 2 Rnxn of rank n to both sides and taking norm on both sides ofthe above equation, we have

kTðxðt þ DtÞÞk 6 kXp

i¼1

Xc

j¼1

hiðxðtÞÞmjðxðtÞÞðI

þ THijT�1DtÞkkTxðtÞk þ k

Xp

i¼1

�Xc

j¼1

hiðxðtÞÞmjðxðtÞÞ½TBr�Dtk þ kTUðDtÞk ð17Þ

where k � k denotes the L2 norm for vectors and L2 induced norm formatrices, from (17)

limDt!0þ

kTðxðt þ DtÞÞk � kTxðtÞkDt

� limDt!0þ

fXp

i¼1

Xc

j¼1

hiðxðtÞÞmjðxðtÞÞ

� ðkI þ THijT�1Dtk � 1Þ

� kTxðtÞkg=Dt þ imDt!0þfkXp

i¼1

�Xc

j¼1

hiðxðtÞÞmjðxðtÞÞ

� ½TBr�Dtk þ kTUðDtÞkg=Dt ð18Þ

From (16) and (18)

dkTxðtÞkdt

6

Xp

i¼1

Xc

j¼1

hiðxðtÞÞmjðxðtÞÞl½THijT�1�kTxðtÞk þ k

Xp

i¼1

�Xc

j¼1

hiðxðtÞÞmjðxðtÞÞ½TBr�k ð19Þ

where l½THijT�1� ¼ lim

Dt!0þ

kI þ THijT�1Dtk � 1

Dt

¼ gmaxTHijT

�1 þ ðTHijT�1Þ�

2

!ð20Þ

where gmaxð�Þ is the largest eigenvalue, * is the conjugate transpose,from (19)

dkTxðtÞkdt

6

Xp

i¼1

Xc

j¼1

hiðxðtÞÞmjðxðtÞÞðl½THijT�1�ÞkTxðtÞk þ k

Xp

i¼1

�Xc

j¼1

hiðxðtÞÞmjðxðtÞÞ½TBr�k ð21Þ

Let l½THiiT�1� � �m 8i ð22Þ

From (21) and (22)

ddtkTxðtÞkemðt�t0Þ�

�Xp

i¼1

Xc

j¼1

hiðxðtÞÞmjðxðtÞÞk½TBr�kemðt�t0Þ ð23Þ

where t0 < t is an arbitrary initial time, based on (23) there are twocases to investigate the system behavior.

(1) – r ¼ 0, (2) – r–0If the condition (22) is satisfied the closed-loop system (11) is

stable, and kxðtÞk ! 0 as t !1Proof for: (1) r ¼ 0

ddtkTxðtÞkemðt�t0Þ�

� 0

kTxðtÞk 6 kTxðt0Þke�mðt�t0Þ ð24Þ

Since n is positive value, kxðtÞk ! 0 as t !1 (2) r – 0, from (23)

ðkTxðtÞkemðt�t0ÞÞ 6 kTxðt0Þk þ kT BK

rkZ t

t0

emðs�t0Þds

where kT BK

rk 6 maxik½TBrkmax 6 kTBrk; then

kTxðtÞk 6 kTxðt0Þke�mðt�t0Þ þ kT BK

rknð1� e�mðt�t0ÞÞ ð25Þ

Since the right-hand side of (25) is finite if r is bounded, the sys-tem states (11) are also bounded.

The above analysis gives an upper bound of kTxðtÞk under differ-ent the two considered cases. The result is given by Eqs. (24) and(25). Similarly, a lower bound of kxðtÞk can be obtained by follow-ing the same analysis procedure with

xðt � DtÞ ¼ xðtÞ � _xðtÞDt þuðDtÞ ð26Þ

where uðDtÞ ¼ xðt � DtÞ � xðtÞ þ _xðtÞDt is the error term and Dt > 0,# is governed by

Let l½�THiiT�1� 6 # 8i ð27Þ

Since # is positive value

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