Research ArticleExtended Nonsingular Terminal Sliding Surface forSecond-Order Nonlinear Systems
Ji Wung Jeong Ho Suk Yeon and Kang-Bak Park
Department of Control and Instrumentation Engineering Korea University 2511 Sejong-ro Sejong City 339-700 Republic of Korea
Correspondence should be addressed to Kang-Bak Park kbparkkoreaackr
Received 29 January 2014 Accepted 18 April 2014 Published 11 May 2014
Academic Editor Hui Zhang
Copyright copy 2014 Ji Wung Jeong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
An extended nonsingular terminal sliding surface is proposed for second-order nonlinear systems It is shown that the proposedsurface is a superset of a conventional nonsingular terminal sliding surface which guarantees that the system state gets to zeroin finite time The conventional nonsingular sliding surfaces have been designed using a power function whose exponent is arational number with positive odd numerator and denominatorThe proposed nonsingular terminal sliding surface overcomes therestriction on the exponent of a power function that is the exponent can be a positive real number Simulation results are providedto show the validity of the main result
1 Introduction
According to the progress of control schemes a variety ofnonlinear control systems have been proposed H-infinityoptimal control fuzzy control neural network control con-troller using genetic algorithms and so on [1ndash4] Slidingmode control (SMC) method which is also called a variablestructure system (VSS) is one of nonlinear robust controlschemes It has been widely used because of its invarianceproperties to parametric uncertainties and external distur-bances [5ndash8] In conventional sliding mode control systemssliding surfaces have been designed such that the overallsystem in the sliding mode is asymptotically stable Howeverthe asymptotic stability does not guarantee a finite timeconvergence
Although most of control systems proposed so far havebeen designed such that the closed-loop system is asymptoticfinite time stabilization is very important to many actualapplications such as motor power robot and aerospacesystems because in the actual applications themain objectiveof a control system is tomake systemrsquos state to a desired one ina finite time interval which is determined a prioriThusmanyrecent studies have focused on the finite time stabilization [9ndash13]
Finite time control systems based on slidingmode controlschemes have been called terminal sliding mode controlsystems since their sliding surfaces have been designed asterminal attractors [14] Recently terminal sliding modecontrol systems have been studied to achieve a finite timeconvergence in many applications [15ndash17] Conventional ter-minal slidingmode controllers used terminal sliding surfaceswhichwere designed using a power function of a system stateThey guaranteed that the system state reached zero in finitetimeOn the contrary however they suffered from singularityproblems and had restrictions on the range of the exponentof a power function The exponent should be a rationalnumber with positive odd numerator and denominator [18]In order to avoid the singularity problem in conventionalterminal sliding mode control systems nonsingular terminalsliding surfaces have been proposed very recently for robotmanipulators [19 20] However the same restriction on theexponent of a power function in the nonsingular slidingsurface still remained the exponent should be a rationalnumber with a positive odd numerator and denominator
Thus in this paper a novel nonsingular terminal slidingsurface is proposed for second-order nonlinear systems It isshown that the proposed scheme guarantees that the systemstate gets to zero in finite time and it does not suffer from
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 267510 6 pageshttpdxdoiorg1011552014267510
2 Abstract and Applied Analysis
singularity problems Furthermore the proposed nonsin-gular terminal sliding surface overcomes the restriction onthe exponent of a power function by being a superset of aconventional nonsingular terminal sliding surfaceThat is weextend the range of an exponent of a power function in thenonsingular terminal sliding surface from a rational numberwith an odd numerator and an odd denominator to a realnumber
Simulation results and experimental results are given toshow the validity of the main result
2 Main Results
Consider a second-order nonlinear system of the followingform
= 119891 (119909 119905) + 119892 (119909 119905) (119906 + 119889 (119909 119905)) (1)
where 119909 and are state variables 119891(119909 119905) is a nonlinearterm 119906 is a scalar input 119892(119909 119905) = 0 forall119909 isin 119877 forall119905 isin
119877
+ and 119889(119909 119905) represents the uncertainties and external
disturbances It is assumed that the following assumptionholds
Assumption 1 The uncertainty 119889(119909 119905) is bounded as fol-lows
|119889 (119909 119905)| le 119863 (119909 119905) forall119909 isin 119877 forall119905 isin 119877
+ (2)
where119863(119909 119905) is a known positive functionIn previous works on terminal sliding mode control
systems the conventional terminal sliding surfaces have beendesigned as
119904
1= + 119888
1119909
11990211199031
(3)
where 119888
1gt 0 0 lt 119902
1119903
1lt 1 and 119902
1and 119903
1are positive odd
integers [8ndash10] However though the conventional terminalsliding surface in (3) ensures finite time convergence it suffersfrom the singularity problem and has a restriction on theexponent of the power function [14] In the phase space with119909 and axes the set of singular points is a vertical axis exceptfor at the origin that is
119878 = (119909 ) | 119909 = 0 = 0 (4)
Recently a nonsingular terminal sliding surface wasproposed to overcome the singularity problem [12]
119904
2=
11990221199032+ 119888
2119909
(5)
where 119888
2gt 0 1 lt 119902
2119903
2lt 2 and 119902
2 119903
2are positive odd
integersHowever this surface still has the restrictions on the
exponent of the power function that is 1199022 119903
2should be
positive odd integers Thus we propose an extended nonsin-gular terminal sliding surface whose exponent can be any realnumber
119904 = ||
1119901 sgn () + 119888 sdot 119909(6)
where sgn(sdot) is the signum function 119888 is a positive constantand 12 le 119901 lt 1 is a real number
minus2minus15
minus1
minus05
0
05
1
15
2minus15 minus1 minus05 0 05 1 15
x1
x2
Figure 1 Example of a nonsingular terminal sliding surface
Figure 1 shows a typical nonsingular terminal slidingsurface Clearly for any 119888 gt 0 and 12 le 119901 lt 1 the phaseportrait is in the second and fourth quadrants in the phasespace with the axes of 119909
1= 119909 and 119909
2=
Remark 2 It is clear that the proposed nonsingular terminalsliding surface is stable because the sliding surface is in thesecond and fourth quadrants in the phase space with the axesof 119909 and where this property implies that 119909 sdot lt 0 for all119909 = 0
For the proposed nonsingular terminal sliding surface wederived the following theorem for the finite time convergence
Theorem3 Theproposed nonsingular terminal sliding surface(6) guarantees that the system state gets to zero in finite time inthe sliding mode 119904 = 0 and the relaxation time [18] is
119905
119903=
|119909(0)|
1minus119901
119888
119901(1 minus 119901)
(7)
Proof If the system is in the sliding mode and ge 0 theproposed sliding surface (6) can be rewritten as follows
(1119901)
+ 119888119909 = 0(8)
From the above equation if (0) ge 0 that is 119909(0) le 0the following equations can be easily derived
(1119901)
+ 119888119909 = 0
lArrrArr
119889119909
119889119905
= 119888
119901
(minus119909)
119901
lArrrArr
119889119909
(minus119909)
119901= 119888
119901
119889119905
lArrrArr minus
119889 (minus119909)
(minus119909)
119901= 119888
119901
119889119905
lArrrArr minusint
minus119909=0
minus119909=minus119909(0)
119889 (minus119909)
(minus119909)
119901= int
119905119903
0
119888
119901
119889119905
Abstract and Applied Analysis 3
lArrrArr
(minus119909)
(1minus119901)
1 minus 119901
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus119909=0
minus119909=minus119909(0)
= 119888
119901
119905
119903
lArrrArr 119905
119903=
(minus119909 (0))
(1minus119901)
119888
119901(1 minus 119901)
=
|119909 (0)|
(1minus119901)
119888
119901(1 minus 119901)
(9)where 119905
119903represents the relaxation time [18]
If (0) lt 0 that is 119909(0) gt 0 119905119903can be derived in a similar
way
minus (minus)
(1119901)
+ 119888119909 = 0
lArrrArr minus = 119888
119901
119909
119901
lArrrArr minus
119889119909
119889119905
= 119888
119901
119909
119901
lArrrArr minus
119889119909
119909
119901= 119888
119901
119889119905
lArrrArr minusint
0
119909(0)
119909
minus119901
119889119909 = int
119905119903
0
119888
119901
119889119905
lArrrArr minus
119909
(1minus119901)
1 minus 119901
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119909=0
119909=119909(0)
= 119888
119901
119905
119903
lArrrArr 119905
119903=
119909(0)
(1minus119901)
119888
119901(1 minus 119901)
=
|119909 (0)|
(1minus119901)
119888
119901(1 minus 119901)
(10)
This completes the proof
In the above theorem we proved that the system state getsto zero in finite time if it is in the sliding mode Thus in thefollowing theorem we propose a controller that guaranteesthat the sliding mode existence condition holds such that theoverall system will be in the sliding mode
Theorem 4 For system (1) with the proposed nonsingularterminal sliding surface (6) the following controller guaranteesthat the sliding mode existence condition holds
119906 =
1
119892 (119909 )
times (minus119891 (119909 ) minus 119888119901||
(2minus(1119901)) sgn () minus 119896
1119904 minus 119896
2sgn (119904))
(11)where 119896
1gt 0 119896
2gt |119892| (119863 + 120572) 120572 is a positive constant and
12 le 119901 lt 1
Proof Let the Lyapunov function candidate be
119881 (119904) =
1
2
119904
2
(12)
Applying (1) and (6) to 119889119881119889119905 =
119881 the following equationscan be obtained if gt 0 that is 119909 lt 0
119881 (119904) = 119904 119904 = 119904 (
1
119901
((1119901)minus1)
+ 119888)
= 119904 (
1
119901
((1119901)minus1)
(119891 + 119892 (119906 + 119889)) + 119888)
le 119904 (
1
119901
((1119901)minus1)
) (minus119896
1119904 minus 120572 sgn (119904))
le
1
119901
((1119901)minus1)
(minus119896
1119904
2
minus 120572 |119904|)
(13)
Similarly if lt 0 that is 119909 gt 0 the following equations canbe derived
119881 (119904) = 119904 119904 = 119904 (
1
119901
(minus)
((1119901)minus1)
(minus) (minus1) + 119888)
= 119904 (
1
119901
(minus)
((1119901)minus1)
(119891 + 119892 (119906 + 119889)) + 119888)
le 119904 (
1
119901
((1119901)minus1)
) (minus119896
1119904 minus 120572 sgn (119904))
le
1
119901
(minus)
((1119901)minus1)
(minus119896
1119904
2
minus 120572 |119904|)
(14)
Equations (13) and (14) can be represented as
119881 (119904) = minus
1
119901
||
((1119901)minus1)
(119896
1119904
2
+ 120572 |119904|) (15)
Here
119881(119904) lt 0 if || = 0 and 119904 = 0If || = 0 from (1) and (11) the following equation can be
obtained
=
119889
119889119905
= minus119896
1119904 minus 119896
2sgn (119904) + 119892 sdot 119889
(16)
which implies that
119904
119889
119889119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816 119909=0
le minus119896
1119904
2
minus 120572 |119904| (17)
Thus it is clear that if || = 0 and 119904 = 0 deviates from zerothat is the set of points || = 0 and 119904 = 0 is not an attractorIn phase space these points are all points on the horizontalaxis that save the origin
Therefore we can conclude that 119904 goes to zero
Remark 5 Since the proposed controller (11) has a term||
(2minus(1119901)) it is clear that in order to avoid the singularityproblem one should choose the control parameter 119901 suchthat 12 le 119901 lt 1 even though the finite time convergencecan be obtained for 0 lt 119901 lt 1 in (6)
3 Examples
To show the validity of the proposed method the simulationresults for the following system are given
= 20 || + 10 sin (5119905)radic|119909| + 10 (119906 + 119889)
(18)
where 119889(119909 119905) = 2radic119909 + sin(10119905) The control parameterswere chosen as follows 119888 = 5 119896
1= 2 119896
2= 10(2
radic|119909| + 1 +
120572) and 120572 = 1 When the initial condition is 119909(0) = 1 from
4 Abstract and Applied Analysis
0 01 02 03 04 05 06 07 08 09 1minus50
minus40
minus30
minus20
minus10
0
10
Time (s)
Slid
ing
varia
ble s(s)
Figure 2 Sliding variable (119904)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus2
0
2
4
6
8
10
12
14
x1
x2
Figure 3 Phase portrait (119909 versus )
(18) 119901 = 12 is obtained for the minimum relaxation timeClearly119901 = 12 is out of a conventional nonsingular terminalsliding surface In the following figures 119909
1 119909
2represent 119909
respectivelyFigures 2ndash5 show the simulation results Figure 2 repre-
sents the profile of the sliding variable 119904 It is shown thatthe system state stayed on the nonsingular terminal slidingsurface all the time after it hit the sliding surface for the firsttime
It is clear that the sliding mode existence condition 119904 sdot 119904 lt
0 was satisfied all the time and it was also known from thephase portrait given in Figure 3
Figure 4 shows that the output zeroed in finite timeThe control input signal can be seen in Figure 5To show the nonsingular property we simulated the
system with the initial condition near the vertical axis inthe phase space because from (4) a set of singular pointsfor the conventional terminal sliding surface is a vertical
0 01 02 03 04 05 06 07 08 09 1minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
Figure 4 Output (119909)
0 01 02 03 04 05 06 07 08 09 1minus80
minus60
minus40
minus20
0
20
40
Time (s)
Con
trol i
nput
(u)
Figure 5 Control input (119906) for signum function
axis in the phase space Figures 6 and 7 show the resultsFigure 6 represents that the system state zeroed even thoughit crossed over the vertical axis a set of singular pointsfor a conventional terminal sliding surface In the proof ofTheorem 4 we showed that the horizontal axis is not anattractor so that the system state on this axis moves out ofthe axis This is also verified from Figure 6 since the systemstate also crossed the horizontal axis The control input isshown in Figure 7 It can be seen that it did not suffer fromthe singularity problem
In addition we also applied the proposed method tothe actual DC motor system For the control unit in theexperimental system a TMS320F2812 DSP processor wasused The sampling time was set to 1msec The followingmodel was used for the DC motor system
= minus4065 + 4667119906 (19)
Figure 8 shows that the motor position converges zero infinite time
Abstract and Applied Analysis 5
minus008 minus007 minus006 minus005 minus004 minus003 minus002 minus001 0 001
minus12
minus10
minus8
minus6
minus4
minus2
0
2
x1
x2
Figure 6 Phase portrait (119909 versus )
0 01 02 03 04 05 06 07 08 09 1minus10
0
10
20
30
40
50
60
70
Time (s)
Con
trol i
nput
(u)
Figure 7 Control input (119906)
0 01 02 03 04 05 06 07 08 09 1
minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
(rad
ian)
Figure 8 Output angle (119909)
0 01 02 03 04 05 06 07 08 09 1minus3
minus2
minus1
0
1
2
3
4
5
Time (s)
Con
trol i
nput
(u)
Figure 9 Control input (119906)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus05
0
05
1
15
2
25
3
35
4
x
x2
Figure 10 Phase portrait (119909 versus )
The control input signal is given in Figure 9 It representsthat the control signal is bounded all the time
Figure 10 shows a phase portrait It is clear that the slidingmode existence condition 119904 sdot 119904 lt 0 was satisfied all the time
4 Conclusions
In this paper the extended nonsingular terminal sliding sur-face for second-order nonlinear systems has been proposedIt has been shown that the proposed nonsingular terminalsliding surface guarantees finite time convergence and issingularity-free Furthermore the exponent of the powerfunction in the proposed sliding surface can be a real numberin contrast to conventional nonsingular terminal slidingsurfaces where the exponent should be a rational numberwith an odd numerator and odd denominator Simulationand experimental results have shown the validity of the mainresult
6 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by a Korea University Grant
References
[1] T Basar and P Bernhard Hinfin Optimal Control and RelatedMinimax Design Problems A Dynamic Game Approach 2008
[2] J H Lilly Fuzzy Control and Identification 2010[3] J Sarangapani Neural Network Control of Nonlinear Discrete-
Time Systems 2006[4] K F Man K S Tang andW A Halang Genetic Algorithms for
Control and Signal Processing 2011[5] J Y Hung W Gao and J C Hung ldquoVariable structure control
A surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993
[6] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999
[7] NHung J S Im S-K JeongH-K Kim and S B Kim ldquoDesignof a slidingmode controller for an automatic guided vehicle andits implementationrdquo International Journal of Control Automa-tion and Systems vol 8 no 1 pp 81ndash90 2010
[8] B-H Lee J-H Choo S Na P Marzocca and L LibresculdquoSliding mode robust control of supersonic three degrees-of-freedom airfoilsrdquo International Journal of Control Automationand Systems vol 8 no 2 pp 279ndash288 2010
[9] Y Wu B Wang and G D Zong ldquoFinite-time tracking con-troller design for nonholonomic systemswith extended chainedformrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 52 no 11 pp 798ndash802 2005
[10] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006
[11] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009
[12] C B Njima W B Mabrouk H Messaoud and G GarcialdquoFinite time stabilization of the four tanks system extensions tothe uncertain systemsrdquo Abstract and Applied Analysis vol 2014Article ID 964143 7 pages 2014
[13] Y Yang and G Chen ldquoFinite time stability of stochastic hybridsystemsrdquo Abstract and Applied Analysis vol 2014 Article ID867189 7 pages 2014
[14] S Janardhanan and B Bandyopadhyay ldquoOn discretization ofcontinuous-time terminal sliding moderdquo IEEE Transactions onAutomatic Control vol 51 no 9 pp 1532ndash1536 2006
[15] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009
[16] L Hui and J Li ldquoTerminal sliding mode control for spacecraftformation flyingrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 45 no 3 pp 835ndash846 2009
[17] J Li and L Yang ldquoFinite-time terminal sliding mode trackingcontrol for piezoelectric actuatorsrdquo Abstract and Applied Anal-ysis vol 2014 Article ID 760937 9 pages 2014
[18] K-B Park and J-J Lee ldquoComments on lsquoa robust MIMOterminal sliding mode control scheme for rigid robotic manip-ulatorsrsquordquo IEEE Transactions on Automatic Control vol 41 no 5pp 761ndash762 1996
[19] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control and its application for robot manipulatorsrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo01) pp 545ndash548 May 2001
[20] M Jin J Lee P H Chang and C Choi ldquoPractical nonsingularterminal sliding-mode control of robot manipulators for high-accuracy tracking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3593ndash3601 2009
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
singularity problems Furthermore the proposed nonsin-gular terminal sliding surface overcomes the restriction onthe exponent of a power function by being a superset of aconventional nonsingular terminal sliding surfaceThat is weextend the range of an exponent of a power function in thenonsingular terminal sliding surface from a rational numberwith an odd numerator and an odd denominator to a realnumber
Simulation results and experimental results are given toshow the validity of the main result
2 Main Results
Consider a second-order nonlinear system of the followingform
= 119891 (119909 119905) + 119892 (119909 119905) (119906 + 119889 (119909 119905)) (1)
where 119909 and are state variables 119891(119909 119905) is a nonlinearterm 119906 is a scalar input 119892(119909 119905) = 0 forall119909 isin 119877 forall119905 isin
119877
+ and 119889(119909 119905) represents the uncertainties and external
disturbances It is assumed that the following assumptionholds
Assumption 1 The uncertainty 119889(119909 119905) is bounded as fol-lows
|119889 (119909 119905)| le 119863 (119909 119905) forall119909 isin 119877 forall119905 isin 119877
+ (2)
where119863(119909 119905) is a known positive functionIn previous works on terminal sliding mode control
systems the conventional terminal sliding surfaces have beendesigned as
119904
1= + 119888
1119909
11990211199031
(3)
where 119888
1gt 0 0 lt 119902
1119903
1lt 1 and 119902
1and 119903
1are positive odd
integers [8ndash10] However though the conventional terminalsliding surface in (3) ensures finite time convergence it suffersfrom the singularity problem and has a restriction on theexponent of the power function [14] In the phase space with119909 and axes the set of singular points is a vertical axis exceptfor at the origin that is
119878 = (119909 ) | 119909 = 0 = 0 (4)
Recently a nonsingular terminal sliding surface wasproposed to overcome the singularity problem [12]
119904
2=
11990221199032+ 119888
2119909
(5)
where 119888
2gt 0 1 lt 119902
2119903
2lt 2 and 119902
2 119903
2are positive odd
integersHowever this surface still has the restrictions on the
exponent of the power function that is 1199022 119903
2should be
positive odd integers Thus we propose an extended nonsin-gular terminal sliding surface whose exponent can be any realnumber
119904 = ||
1119901 sgn () + 119888 sdot 119909(6)
where sgn(sdot) is the signum function 119888 is a positive constantand 12 le 119901 lt 1 is a real number
minus2minus15
minus1
minus05
0
05
1
15
2minus15 minus1 minus05 0 05 1 15
x1
x2
Figure 1 Example of a nonsingular terminal sliding surface
Figure 1 shows a typical nonsingular terminal slidingsurface Clearly for any 119888 gt 0 and 12 le 119901 lt 1 the phaseportrait is in the second and fourth quadrants in the phasespace with the axes of 119909
1= 119909 and 119909
2=
Remark 2 It is clear that the proposed nonsingular terminalsliding surface is stable because the sliding surface is in thesecond and fourth quadrants in the phase space with the axesof 119909 and where this property implies that 119909 sdot lt 0 for all119909 = 0
For the proposed nonsingular terminal sliding surface wederived the following theorem for the finite time convergence
Theorem3 Theproposed nonsingular terminal sliding surface(6) guarantees that the system state gets to zero in finite time inthe sliding mode 119904 = 0 and the relaxation time [18] is
119905
119903=
|119909(0)|
1minus119901
119888
119901(1 minus 119901)
(7)
Proof If the system is in the sliding mode and ge 0 theproposed sliding surface (6) can be rewritten as follows
(1119901)
+ 119888119909 = 0(8)
From the above equation if (0) ge 0 that is 119909(0) le 0the following equations can be easily derived
(1119901)
+ 119888119909 = 0
lArrrArr
119889119909
119889119905
= 119888
119901
(minus119909)
119901
lArrrArr
119889119909
(minus119909)
119901= 119888
119901
119889119905
lArrrArr minus
119889 (minus119909)
(minus119909)
119901= 119888
119901
119889119905
lArrrArr minusint
minus119909=0
minus119909=minus119909(0)
119889 (minus119909)
(minus119909)
119901= int
119905119903
0
119888
119901
119889119905
Abstract and Applied Analysis 3
lArrrArr
(minus119909)
(1minus119901)
1 minus 119901
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus119909=0
minus119909=minus119909(0)
= 119888
119901
119905
119903
lArrrArr 119905
119903=
(minus119909 (0))
(1minus119901)
119888
119901(1 minus 119901)
=
|119909 (0)|
(1minus119901)
119888
119901(1 minus 119901)
(9)where 119905
119903represents the relaxation time [18]
If (0) lt 0 that is 119909(0) gt 0 119905119903can be derived in a similar
way
minus (minus)
(1119901)
+ 119888119909 = 0
lArrrArr minus = 119888
119901
119909
119901
lArrrArr minus
119889119909
119889119905
= 119888
119901
119909
119901
lArrrArr minus
119889119909
119909
119901= 119888
119901
119889119905
lArrrArr minusint
0
119909(0)
119909
minus119901
119889119909 = int
119905119903
0
119888
119901
119889119905
lArrrArr minus
119909
(1minus119901)
1 minus 119901
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119909=0
119909=119909(0)
= 119888
119901
119905
119903
lArrrArr 119905
119903=
119909(0)
(1minus119901)
119888
119901(1 minus 119901)
=
|119909 (0)|
(1minus119901)
119888
119901(1 minus 119901)
(10)
This completes the proof
In the above theorem we proved that the system state getsto zero in finite time if it is in the sliding mode Thus in thefollowing theorem we propose a controller that guaranteesthat the sliding mode existence condition holds such that theoverall system will be in the sliding mode
Theorem 4 For system (1) with the proposed nonsingularterminal sliding surface (6) the following controller guaranteesthat the sliding mode existence condition holds
119906 =
1
119892 (119909 )
times (minus119891 (119909 ) minus 119888119901||
(2minus(1119901)) sgn () minus 119896
1119904 minus 119896
2sgn (119904))
(11)where 119896
1gt 0 119896
2gt |119892| (119863 + 120572) 120572 is a positive constant and
12 le 119901 lt 1
Proof Let the Lyapunov function candidate be
119881 (119904) =
1
2
119904
2
(12)
Applying (1) and (6) to 119889119881119889119905 =
119881 the following equationscan be obtained if gt 0 that is 119909 lt 0
119881 (119904) = 119904 119904 = 119904 (
1
119901
((1119901)minus1)
+ 119888)
= 119904 (
1
119901
((1119901)minus1)
(119891 + 119892 (119906 + 119889)) + 119888)
le 119904 (
1
119901
((1119901)minus1)
) (minus119896
1119904 minus 120572 sgn (119904))
le
1
119901
((1119901)minus1)
(minus119896
1119904
2
minus 120572 |119904|)
(13)
Similarly if lt 0 that is 119909 gt 0 the following equations canbe derived
119881 (119904) = 119904 119904 = 119904 (
1
119901
(minus)
((1119901)minus1)
(minus) (minus1) + 119888)
= 119904 (
1
119901
(minus)
((1119901)minus1)
(119891 + 119892 (119906 + 119889)) + 119888)
le 119904 (
1
119901
((1119901)minus1)
) (minus119896
1119904 minus 120572 sgn (119904))
le
1
119901
(minus)
((1119901)minus1)
(minus119896
1119904
2
minus 120572 |119904|)
(14)
Equations (13) and (14) can be represented as
119881 (119904) = minus
1
119901
||
((1119901)minus1)
(119896
1119904
2
+ 120572 |119904|) (15)
Here
119881(119904) lt 0 if || = 0 and 119904 = 0If || = 0 from (1) and (11) the following equation can be
obtained
=
119889
119889119905
= minus119896
1119904 minus 119896
2sgn (119904) + 119892 sdot 119889
(16)
which implies that
119904
119889
119889119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816 119909=0
le minus119896
1119904
2
minus 120572 |119904| (17)
Thus it is clear that if || = 0 and 119904 = 0 deviates from zerothat is the set of points || = 0 and 119904 = 0 is not an attractorIn phase space these points are all points on the horizontalaxis that save the origin
Therefore we can conclude that 119904 goes to zero
Remark 5 Since the proposed controller (11) has a term||
(2minus(1119901)) it is clear that in order to avoid the singularityproblem one should choose the control parameter 119901 suchthat 12 le 119901 lt 1 even though the finite time convergencecan be obtained for 0 lt 119901 lt 1 in (6)
3 Examples
To show the validity of the proposed method the simulationresults for the following system are given
= 20 || + 10 sin (5119905)radic|119909| + 10 (119906 + 119889)
(18)
where 119889(119909 119905) = 2radic119909 + sin(10119905) The control parameterswere chosen as follows 119888 = 5 119896
1= 2 119896
2= 10(2
radic|119909| + 1 +
120572) and 120572 = 1 When the initial condition is 119909(0) = 1 from
4 Abstract and Applied Analysis
0 01 02 03 04 05 06 07 08 09 1minus50
minus40
minus30
minus20
minus10
0
10
Time (s)
Slid
ing
varia
ble s(s)
Figure 2 Sliding variable (119904)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus2
0
2
4
6
8
10
12
14
x1
x2
Figure 3 Phase portrait (119909 versus )
(18) 119901 = 12 is obtained for the minimum relaxation timeClearly119901 = 12 is out of a conventional nonsingular terminalsliding surface In the following figures 119909
1 119909
2represent 119909
respectivelyFigures 2ndash5 show the simulation results Figure 2 repre-
sents the profile of the sliding variable 119904 It is shown thatthe system state stayed on the nonsingular terminal slidingsurface all the time after it hit the sliding surface for the firsttime
It is clear that the sliding mode existence condition 119904 sdot 119904 lt
0 was satisfied all the time and it was also known from thephase portrait given in Figure 3
Figure 4 shows that the output zeroed in finite timeThe control input signal can be seen in Figure 5To show the nonsingular property we simulated the
system with the initial condition near the vertical axis inthe phase space because from (4) a set of singular pointsfor the conventional terminal sliding surface is a vertical
0 01 02 03 04 05 06 07 08 09 1minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
Figure 4 Output (119909)
0 01 02 03 04 05 06 07 08 09 1minus80
minus60
minus40
minus20
0
20
40
Time (s)
Con
trol i
nput
(u)
Figure 5 Control input (119906) for signum function
axis in the phase space Figures 6 and 7 show the resultsFigure 6 represents that the system state zeroed even thoughit crossed over the vertical axis a set of singular pointsfor a conventional terminal sliding surface In the proof ofTheorem 4 we showed that the horizontal axis is not anattractor so that the system state on this axis moves out ofthe axis This is also verified from Figure 6 since the systemstate also crossed the horizontal axis The control input isshown in Figure 7 It can be seen that it did not suffer fromthe singularity problem
In addition we also applied the proposed method tothe actual DC motor system For the control unit in theexperimental system a TMS320F2812 DSP processor wasused The sampling time was set to 1msec The followingmodel was used for the DC motor system
= minus4065 + 4667119906 (19)
Figure 8 shows that the motor position converges zero infinite time
Abstract and Applied Analysis 5
minus008 minus007 minus006 minus005 minus004 minus003 minus002 minus001 0 001
minus12
minus10
minus8
minus6
minus4
minus2
0
2
x1
x2
Figure 6 Phase portrait (119909 versus )
0 01 02 03 04 05 06 07 08 09 1minus10
0
10
20
30
40
50
60
70
Time (s)
Con
trol i
nput
(u)
Figure 7 Control input (119906)
0 01 02 03 04 05 06 07 08 09 1
minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
(rad
ian)
Figure 8 Output angle (119909)
0 01 02 03 04 05 06 07 08 09 1minus3
minus2
minus1
0
1
2
3
4
5
Time (s)
Con
trol i
nput
(u)
Figure 9 Control input (119906)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus05
0
05
1
15
2
25
3
35
4
x
x2
Figure 10 Phase portrait (119909 versus )
The control input signal is given in Figure 9 It representsthat the control signal is bounded all the time
Figure 10 shows a phase portrait It is clear that the slidingmode existence condition 119904 sdot 119904 lt 0 was satisfied all the time
4 Conclusions
In this paper the extended nonsingular terminal sliding sur-face for second-order nonlinear systems has been proposedIt has been shown that the proposed nonsingular terminalsliding surface guarantees finite time convergence and issingularity-free Furthermore the exponent of the powerfunction in the proposed sliding surface can be a real numberin contrast to conventional nonsingular terminal slidingsurfaces where the exponent should be a rational numberwith an odd numerator and odd denominator Simulationand experimental results have shown the validity of the mainresult
6 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by a Korea University Grant
References
[1] T Basar and P Bernhard Hinfin Optimal Control and RelatedMinimax Design Problems A Dynamic Game Approach 2008
[2] J H Lilly Fuzzy Control and Identification 2010[3] J Sarangapani Neural Network Control of Nonlinear Discrete-
Time Systems 2006[4] K F Man K S Tang andW A Halang Genetic Algorithms for
Control and Signal Processing 2011[5] J Y Hung W Gao and J C Hung ldquoVariable structure control
A surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993
[6] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999
[7] NHung J S Im S-K JeongH-K Kim and S B Kim ldquoDesignof a slidingmode controller for an automatic guided vehicle andits implementationrdquo International Journal of Control Automa-tion and Systems vol 8 no 1 pp 81ndash90 2010
[8] B-H Lee J-H Choo S Na P Marzocca and L LibresculdquoSliding mode robust control of supersonic three degrees-of-freedom airfoilsrdquo International Journal of Control Automationand Systems vol 8 no 2 pp 279ndash288 2010
[9] Y Wu B Wang and G D Zong ldquoFinite-time tracking con-troller design for nonholonomic systemswith extended chainedformrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 52 no 11 pp 798ndash802 2005
[10] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006
[11] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009
[12] C B Njima W B Mabrouk H Messaoud and G GarcialdquoFinite time stabilization of the four tanks system extensions tothe uncertain systemsrdquo Abstract and Applied Analysis vol 2014Article ID 964143 7 pages 2014
[13] Y Yang and G Chen ldquoFinite time stability of stochastic hybridsystemsrdquo Abstract and Applied Analysis vol 2014 Article ID867189 7 pages 2014
[14] S Janardhanan and B Bandyopadhyay ldquoOn discretization ofcontinuous-time terminal sliding moderdquo IEEE Transactions onAutomatic Control vol 51 no 9 pp 1532ndash1536 2006
[15] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009
[16] L Hui and J Li ldquoTerminal sliding mode control for spacecraftformation flyingrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 45 no 3 pp 835ndash846 2009
[17] J Li and L Yang ldquoFinite-time terminal sliding mode trackingcontrol for piezoelectric actuatorsrdquo Abstract and Applied Anal-ysis vol 2014 Article ID 760937 9 pages 2014
[18] K-B Park and J-J Lee ldquoComments on lsquoa robust MIMOterminal sliding mode control scheme for rigid robotic manip-ulatorsrsquordquo IEEE Transactions on Automatic Control vol 41 no 5pp 761ndash762 1996
[19] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control and its application for robot manipulatorsrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo01) pp 545ndash548 May 2001
[20] M Jin J Lee P H Chang and C Choi ldquoPractical nonsingularterminal sliding-mode control of robot manipulators for high-accuracy tracking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3593ndash3601 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
lArrrArr
(minus119909)
(1minus119901)
1 minus 119901
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus119909=0
minus119909=minus119909(0)
= 119888
119901
119905
119903
lArrrArr 119905
119903=
(minus119909 (0))
(1minus119901)
119888
119901(1 minus 119901)
=
|119909 (0)|
(1minus119901)
119888
119901(1 minus 119901)
(9)where 119905
119903represents the relaxation time [18]
If (0) lt 0 that is 119909(0) gt 0 119905119903can be derived in a similar
way
minus (minus)
(1119901)
+ 119888119909 = 0
lArrrArr minus = 119888
119901
119909
119901
lArrrArr minus
119889119909
119889119905
= 119888
119901
119909
119901
lArrrArr minus
119889119909
119909
119901= 119888
119901
119889119905
lArrrArr minusint
0
119909(0)
119909
minus119901
119889119909 = int
119905119903
0
119888
119901
119889119905
lArrrArr minus
119909
(1minus119901)
1 minus 119901
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119909=0
119909=119909(0)
= 119888
119901
119905
119903
lArrrArr 119905
119903=
119909(0)
(1minus119901)
119888
119901(1 minus 119901)
=
|119909 (0)|
(1minus119901)
119888
119901(1 minus 119901)
(10)
This completes the proof
In the above theorem we proved that the system state getsto zero in finite time if it is in the sliding mode Thus in thefollowing theorem we propose a controller that guaranteesthat the sliding mode existence condition holds such that theoverall system will be in the sliding mode
Theorem 4 For system (1) with the proposed nonsingularterminal sliding surface (6) the following controller guaranteesthat the sliding mode existence condition holds
119906 =
1
119892 (119909 )
times (minus119891 (119909 ) minus 119888119901||
(2minus(1119901)) sgn () minus 119896
1119904 minus 119896
2sgn (119904))
(11)where 119896
1gt 0 119896
2gt |119892| (119863 + 120572) 120572 is a positive constant and
12 le 119901 lt 1
Proof Let the Lyapunov function candidate be
119881 (119904) =
1
2
119904
2
(12)
Applying (1) and (6) to 119889119881119889119905 =
119881 the following equationscan be obtained if gt 0 that is 119909 lt 0
119881 (119904) = 119904 119904 = 119904 (
1
119901
((1119901)minus1)
+ 119888)
= 119904 (
1
119901
((1119901)minus1)
(119891 + 119892 (119906 + 119889)) + 119888)
le 119904 (
1
119901
((1119901)minus1)
) (minus119896
1119904 minus 120572 sgn (119904))
le
1
119901
((1119901)minus1)
(minus119896
1119904
2
minus 120572 |119904|)
(13)
Similarly if lt 0 that is 119909 gt 0 the following equations canbe derived
119881 (119904) = 119904 119904 = 119904 (
1
119901
(minus)
((1119901)minus1)
(minus) (minus1) + 119888)
= 119904 (
1
119901
(minus)
((1119901)minus1)
(119891 + 119892 (119906 + 119889)) + 119888)
le 119904 (
1
119901
((1119901)minus1)
) (minus119896
1119904 minus 120572 sgn (119904))
le
1
119901
(minus)
((1119901)minus1)
(minus119896
1119904
2
minus 120572 |119904|)
(14)
Equations (13) and (14) can be represented as
119881 (119904) = minus
1
119901
||
((1119901)minus1)
(119896
1119904
2
+ 120572 |119904|) (15)
Here
119881(119904) lt 0 if || = 0 and 119904 = 0If || = 0 from (1) and (11) the following equation can be
obtained
=
119889
119889119905
= minus119896
1119904 minus 119896
2sgn (119904) + 119892 sdot 119889
(16)
which implies that
119904
119889
119889119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816 119909=0
le minus119896
1119904
2
minus 120572 |119904| (17)
Thus it is clear that if || = 0 and 119904 = 0 deviates from zerothat is the set of points || = 0 and 119904 = 0 is not an attractorIn phase space these points are all points on the horizontalaxis that save the origin
Therefore we can conclude that 119904 goes to zero
Remark 5 Since the proposed controller (11) has a term||
(2minus(1119901)) it is clear that in order to avoid the singularityproblem one should choose the control parameter 119901 suchthat 12 le 119901 lt 1 even though the finite time convergencecan be obtained for 0 lt 119901 lt 1 in (6)
3 Examples
To show the validity of the proposed method the simulationresults for the following system are given
= 20 || + 10 sin (5119905)radic|119909| + 10 (119906 + 119889)
(18)
where 119889(119909 119905) = 2radic119909 + sin(10119905) The control parameterswere chosen as follows 119888 = 5 119896
1= 2 119896
2= 10(2
radic|119909| + 1 +
120572) and 120572 = 1 When the initial condition is 119909(0) = 1 from
4 Abstract and Applied Analysis
0 01 02 03 04 05 06 07 08 09 1minus50
minus40
minus30
minus20
minus10
0
10
Time (s)
Slid
ing
varia
ble s(s)
Figure 2 Sliding variable (119904)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus2
0
2
4
6
8
10
12
14
x1
x2
Figure 3 Phase portrait (119909 versus )
(18) 119901 = 12 is obtained for the minimum relaxation timeClearly119901 = 12 is out of a conventional nonsingular terminalsliding surface In the following figures 119909
1 119909
2represent 119909
respectivelyFigures 2ndash5 show the simulation results Figure 2 repre-
sents the profile of the sliding variable 119904 It is shown thatthe system state stayed on the nonsingular terminal slidingsurface all the time after it hit the sliding surface for the firsttime
It is clear that the sliding mode existence condition 119904 sdot 119904 lt
0 was satisfied all the time and it was also known from thephase portrait given in Figure 3
Figure 4 shows that the output zeroed in finite timeThe control input signal can be seen in Figure 5To show the nonsingular property we simulated the
system with the initial condition near the vertical axis inthe phase space because from (4) a set of singular pointsfor the conventional terminal sliding surface is a vertical
0 01 02 03 04 05 06 07 08 09 1minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
Figure 4 Output (119909)
0 01 02 03 04 05 06 07 08 09 1minus80
minus60
minus40
minus20
0
20
40
Time (s)
Con
trol i
nput
(u)
Figure 5 Control input (119906) for signum function
axis in the phase space Figures 6 and 7 show the resultsFigure 6 represents that the system state zeroed even thoughit crossed over the vertical axis a set of singular pointsfor a conventional terminal sliding surface In the proof ofTheorem 4 we showed that the horizontal axis is not anattractor so that the system state on this axis moves out ofthe axis This is also verified from Figure 6 since the systemstate also crossed the horizontal axis The control input isshown in Figure 7 It can be seen that it did not suffer fromthe singularity problem
In addition we also applied the proposed method tothe actual DC motor system For the control unit in theexperimental system a TMS320F2812 DSP processor wasused The sampling time was set to 1msec The followingmodel was used for the DC motor system
= minus4065 + 4667119906 (19)
Figure 8 shows that the motor position converges zero infinite time
Abstract and Applied Analysis 5
minus008 minus007 minus006 minus005 minus004 minus003 minus002 minus001 0 001
minus12
minus10
minus8
minus6
minus4
minus2
0
2
x1
x2
Figure 6 Phase portrait (119909 versus )
0 01 02 03 04 05 06 07 08 09 1minus10
0
10
20
30
40
50
60
70
Time (s)
Con
trol i
nput
(u)
Figure 7 Control input (119906)
0 01 02 03 04 05 06 07 08 09 1
minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
(rad
ian)
Figure 8 Output angle (119909)
0 01 02 03 04 05 06 07 08 09 1minus3
minus2
minus1
0
1
2
3
4
5
Time (s)
Con
trol i
nput
(u)
Figure 9 Control input (119906)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus05
0
05
1
15
2
25
3
35
4
x
x2
Figure 10 Phase portrait (119909 versus )
The control input signal is given in Figure 9 It representsthat the control signal is bounded all the time
Figure 10 shows a phase portrait It is clear that the slidingmode existence condition 119904 sdot 119904 lt 0 was satisfied all the time
4 Conclusions
In this paper the extended nonsingular terminal sliding sur-face for second-order nonlinear systems has been proposedIt has been shown that the proposed nonsingular terminalsliding surface guarantees finite time convergence and issingularity-free Furthermore the exponent of the powerfunction in the proposed sliding surface can be a real numberin contrast to conventional nonsingular terminal slidingsurfaces where the exponent should be a rational numberwith an odd numerator and odd denominator Simulationand experimental results have shown the validity of the mainresult
6 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by a Korea University Grant
References
[1] T Basar and P Bernhard Hinfin Optimal Control and RelatedMinimax Design Problems A Dynamic Game Approach 2008
[2] J H Lilly Fuzzy Control and Identification 2010[3] J Sarangapani Neural Network Control of Nonlinear Discrete-
Time Systems 2006[4] K F Man K S Tang andW A Halang Genetic Algorithms for
Control and Signal Processing 2011[5] J Y Hung W Gao and J C Hung ldquoVariable structure control
A surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993
[6] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999
[7] NHung J S Im S-K JeongH-K Kim and S B Kim ldquoDesignof a slidingmode controller for an automatic guided vehicle andits implementationrdquo International Journal of Control Automa-tion and Systems vol 8 no 1 pp 81ndash90 2010
[8] B-H Lee J-H Choo S Na P Marzocca and L LibresculdquoSliding mode robust control of supersonic three degrees-of-freedom airfoilsrdquo International Journal of Control Automationand Systems vol 8 no 2 pp 279ndash288 2010
[9] Y Wu B Wang and G D Zong ldquoFinite-time tracking con-troller design for nonholonomic systemswith extended chainedformrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 52 no 11 pp 798ndash802 2005
[10] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006
[11] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009
[12] C B Njima W B Mabrouk H Messaoud and G GarcialdquoFinite time stabilization of the four tanks system extensions tothe uncertain systemsrdquo Abstract and Applied Analysis vol 2014Article ID 964143 7 pages 2014
[13] Y Yang and G Chen ldquoFinite time stability of stochastic hybridsystemsrdquo Abstract and Applied Analysis vol 2014 Article ID867189 7 pages 2014
[14] S Janardhanan and B Bandyopadhyay ldquoOn discretization ofcontinuous-time terminal sliding moderdquo IEEE Transactions onAutomatic Control vol 51 no 9 pp 1532ndash1536 2006
[15] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009
[16] L Hui and J Li ldquoTerminal sliding mode control for spacecraftformation flyingrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 45 no 3 pp 835ndash846 2009
[17] J Li and L Yang ldquoFinite-time terminal sliding mode trackingcontrol for piezoelectric actuatorsrdquo Abstract and Applied Anal-ysis vol 2014 Article ID 760937 9 pages 2014
[18] K-B Park and J-J Lee ldquoComments on lsquoa robust MIMOterminal sliding mode control scheme for rigid robotic manip-ulatorsrsquordquo IEEE Transactions on Automatic Control vol 41 no 5pp 761ndash762 1996
[19] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control and its application for robot manipulatorsrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo01) pp 545ndash548 May 2001
[20] M Jin J Lee P H Chang and C Choi ldquoPractical nonsingularterminal sliding-mode control of robot manipulators for high-accuracy tracking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3593ndash3601 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
0 01 02 03 04 05 06 07 08 09 1minus50
minus40
minus30
minus20
minus10
0
10
Time (s)
Slid
ing
varia
ble s(s)
Figure 2 Sliding variable (119904)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus2
0
2
4
6
8
10
12
14
x1
x2
Figure 3 Phase portrait (119909 versus )
(18) 119901 = 12 is obtained for the minimum relaxation timeClearly119901 = 12 is out of a conventional nonsingular terminalsliding surface In the following figures 119909
1 119909
2represent 119909
respectivelyFigures 2ndash5 show the simulation results Figure 2 repre-
sents the profile of the sliding variable 119904 It is shown thatthe system state stayed on the nonsingular terminal slidingsurface all the time after it hit the sliding surface for the firsttime
It is clear that the sliding mode existence condition 119904 sdot 119904 lt
0 was satisfied all the time and it was also known from thephase portrait given in Figure 3
Figure 4 shows that the output zeroed in finite timeThe control input signal can be seen in Figure 5To show the nonsingular property we simulated the
system with the initial condition near the vertical axis inthe phase space because from (4) a set of singular pointsfor the conventional terminal sliding surface is a vertical
0 01 02 03 04 05 06 07 08 09 1minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
Figure 4 Output (119909)
0 01 02 03 04 05 06 07 08 09 1minus80
minus60
minus40
minus20
0
20
40
Time (s)
Con
trol i
nput
(u)
Figure 5 Control input (119906) for signum function
axis in the phase space Figures 6 and 7 show the resultsFigure 6 represents that the system state zeroed even thoughit crossed over the vertical axis a set of singular pointsfor a conventional terminal sliding surface In the proof ofTheorem 4 we showed that the horizontal axis is not anattractor so that the system state on this axis moves out ofthe axis This is also verified from Figure 6 since the systemstate also crossed the horizontal axis The control input isshown in Figure 7 It can be seen that it did not suffer fromthe singularity problem
In addition we also applied the proposed method tothe actual DC motor system For the control unit in theexperimental system a TMS320F2812 DSP processor wasused The sampling time was set to 1msec The followingmodel was used for the DC motor system
= minus4065 + 4667119906 (19)
Figure 8 shows that the motor position converges zero infinite time
Abstract and Applied Analysis 5
minus008 minus007 minus006 minus005 minus004 minus003 minus002 minus001 0 001
minus12
minus10
minus8
minus6
minus4
minus2
0
2
x1
x2
Figure 6 Phase portrait (119909 versus )
0 01 02 03 04 05 06 07 08 09 1minus10
0
10
20
30
40
50
60
70
Time (s)
Con
trol i
nput
(u)
Figure 7 Control input (119906)
0 01 02 03 04 05 06 07 08 09 1
minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
(rad
ian)
Figure 8 Output angle (119909)
0 01 02 03 04 05 06 07 08 09 1minus3
minus2
minus1
0
1
2
3
4
5
Time (s)
Con
trol i
nput
(u)
Figure 9 Control input (119906)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus05
0
05
1
15
2
25
3
35
4
x
x2
Figure 10 Phase portrait (119909 versus )
The control input signal is given in Figure 9 It representsthat the control signal is bounded all the time
Figure 10 shows a phase portrait It is clear that the slidingmode existence condition 119904 sdot 119904 lt 0 was satisfied all the time
4 Conclusions
In this paper the extended nonsingular terminal sliding sur-face for second-order nonlinear systems has been proposedIt has been shown that the proposed nonsingular terminalsliding surface guarantees finite time convergence and issingularity-free Furthermore the exponent of the powerfunction in the proposed sliding surface can be a real numberin contrast to conventional nonsingular terminal slidingsurfaces where the exponent should be a rational numberwith an odd numerator and odd denominator Simulationand experimental results have shown the validity of the mainresult
6 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by a Korea University Grant
References
[1] T Basar and P Bernhard Hinfin Optimal Control and RelatedMinimax Design Problems A Dynamic Game Approach 2008
[2] J H Lilly Fuzzy Control and Identification 2010[3] J Sarangapani Neural Network Control of Nonlinear Discrete-
Time Systems 2006[4] K F Man K S Tang andW A Halang Genetic Algorithms for
Control and Signal Processing 2011[5] J Y Hung W Gao and J C Hung ldquoVariable structure control
A surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993
[6] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999
[7] NHung J S Im S-K JeongH-K Kim and S B Kim ldquoDesignof a slidingmode controller for an automatic guided vehicle andits implementationrdquo International Journal of Control Automa-tion and Systems vol 8 no 1 pp 81ndash90 2010
[8] B-H Lee J-H Choo S Na P Marzocca and L LibresculdquoSliding mode robust control of supersonic three degrees-of-freedom airfoilsrdquo International Journal of Control Automationand Systems vol 8 no 2 pp 279ndash288 2010
[9] Y Wu B Wang and G D Zong ldquoFinite-time tracking con-troller design for nonholonomic systemswith extended chainedformrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 52 no 11 pp 798ndash802 2005
[10] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006
[11] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009
[12] C B Njima W B Mabrouk H Messaoud and G GarcialdquoFinite time stabilization of the four tanks system extensions tothe uncertain systemsrdquo Abstract and Applied Analysis vol 2014Article ID 964143 7 pages 2014
[13] Y Yang and G Chen ldquoFinite time stability of stochastic hybridsystemsrdquo Abstract and Applied Analysis vol 2014 Article ID867189 7 pages 2014
[14] S Janardhanan and B Bandyopadhyay ldquoOn discretization ofcontinuous-time terminal sliding moderdquo IEEE Transactions onAutomatic Control vol 51 no 9 pp 1532ndash1536 2006
[15] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009
[16] L Hui and J Li ldquoTerminal sliding mode control for spacecraftformation flyingrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 45 no 3 pp 835ndash846 2009
[17] J Li and L Yang ldquoFinite-time terminal sliding mode trackingcontrol for piezoelectric actuatorsrdquo Abstract and Applied Anal-ysis vol 2014 Article ID 760937 9 pages 2014
[18] K-B Park and J-J Lee ldquoComments on lsquoa robust MIMOterminal sliding mode control scheme for rigid robotic manip-ulatorsrsquordquo IEEE Transactions on Automatic Control vol 41 no 5pp 761ndash762 1996
[19] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control and its application for robot manipulatorsrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo01) pp 545ndash548 May 2001
[20] M Jin J Lee P H Chang and C Choi ldquoPractical nonsingularterminal sliding-mode control of robot manipulators for high-accuracy tracking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3593ndash3601 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
minus008 minus007 minus006 minus005 minus004 minus003 minus002 minus001 0 001
minus12
minus10
minus8
minus6
minus4
minus2
0
2
x1
x2
Figure 6 Phase portrait (119909 versus )
0 01 02 03 04 05 06 07 08 09 1minus10
0
10
20
30
40
50
60
70
Time (s)
Con
trol i
nput
(u)
Figure 7 Control input (119906)
0 01 02 03 04 05 06 07 08 09 1
minus16
minus14
minus12
minus1
minus08
minus06
minus04
minus02
0
02
Time (s)
x1
(rad
ian)
Figure 8 Output angle (119909)
0 01 02 03 04 05 06 07 08 09 1minus3
minus2
minus1
0
1
2
3
4
5
Time (s)
Con
trol i
nput
(u)
Figure 9 Control input (119906)
minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0 02
minus05
0
05
1
15
2
25
3
35
4
x
x2
Figure 10 Phase portrait (119909 versus )
The control input signal is given in Figure 9 It representsthat the control signal is bounded all the time
Figure 10 shows a phase portrait It is clear that the slidingmode existence condition 119904 sdot 119904 lt 0 was satisfied all the time
4 Conclusions
In this paper the extended nonsingular terminal sliding sur-face for second-order nonlinear systems has been proposedIt has been shown that the proposed nonsingular terminalsliding surface guarantees finite time convergence and issingularity-free Furthermore the exponent of the powerfunction in the proposed sliding surface can be a real numberin contrast to conventional nonsingular terminal slidingsurfaces where the exponent should be a rational numberwith an odd numerator and odd denominator Simulationand experimental results have shown the validity of the mainresult
6 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by a Korea University Grant
References
[1] T Basar and P Bernhard Hinfin Optimal Control and RelatedMinimax Design Problems A Dynamic Game Approach 2008
[2] J H Lilly Fuzzy Control and Identification 2010[3] J Sarangapani Neural Network Control of Nonlinear Discrete-
Time Systems 2006[4] K F Man K S Tang andW A Halang Genetic Algorithms for
Control and Signal Processing 2011[5] J Y Hung W Gao and J C Hung ldquoVariable structure control
A surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993
[6] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999
[7] NHung J S Im S-K JeongH-K Kim and S B Kim ldquoDesignof a slidingmode controller for an automatic guided vehicle andits implementationrdquo International Journal of Control Automa-tion and Systems vol 8 no 1 pp 81ndash90 2010
[8] B-H Lee J-H Choo S Na P Marzocca and L LibresculdquoSliding mode robust control of supersonic three degrees-of-freedom airfoilsrdquo International Journal of Control Automationand Systems vol 8 no 2 pp 279ndash288 2010
[9] Y Wu B Wang and G D Zong ldquoFinite-time tracking con-troller design for nonholonomic systemswith extended chainedformrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 52 no 11 pp 798ndash802 2005
[10] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006
[11] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009
[12] C B Njima W B Mabrouk H Messaoud and G GarcialdquoFinite time stabilization of the four tanks system extensions tothe uncertain systemsrdquo Abstract and Applied Analysis vol 2014Article ID 964143 7 pages 2014
[13] Y Yang and G Chen ldquoFinite time stability of stochastic hybridsystemsrdquo Abstract and Applied Analysis vol 2014 Article ID867189 7 pages 2014
[14] S Janardhanan and B Bandyopadhyay ldquoOn discretization ofcontinuous-time terminal sliding moderdquo IEEE Transactions onAutomatic Control vol 51 no 9 pp 1532ndash1536 2006
[15] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009
[16] L Hui and J Li ldquoTerminal sliding mode control for spacecraftformation flyingrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 45 no 3 pp 835ndash846 2009
[17] J Li and L Yang ldquoFinite-time terminal sliding mode trackingcontrol for piezoelectric actuatorsrdquo Abstract and Applied Anal-ysis vol 2014 Article ID 760937 9 pages 2014
[18] K-B Park and J-J Lee ldquoComments on lsquoa robust MIMOterminal sliding mode control scheme for rigid robotic manip-ulatorsrsquordquo IEEE Transactions on Automatic Control vol 41 no 5pp 761ndash762 1996
[19] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control and its application for robot manipulatorsrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo01) pp 545ndash548 May 2001
[20] M Jin J Lee P H Chang and C Choi ldquoPractical nonsingularterminal sliding-mode control of robot manipulators for high-accuracy tracking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3593ndash3601 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was supported by a Korea University Grant
References
[1] T Basar and P Bernhard Hinfin Optimal Control and RelatedMinimax Design Problems A Dynamic Game Approach 2008
[2] J H Lilly Fuzzy Control and Identification 2010[3] J Sarangapani Neural Network Control of Nonlinear Discrete-
Time Systems 2006[4] K F Man K S Tang andW A Halang Genetic Algorithms for
Control and Signal Processing 2011[5] J Y Hung W Gao and J C Hung ldquoVariable structure control
A surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993
[6] K D Young V I Utkin and U Ozguner ldquoA control engineerrsquosguide to sliding mode controlrdquo IEEE Transactions on ControlSystems Technology vol 7 no 3 pp 328ndash342 1999
[7] NHung J S Im S-K JeongH-K Kim and S B Kim ldquoDesignof a slidingmode controller for an automatic guided vehicle andits implementationrdquo International Journal of Control Automa-tion and Systems vol 8 no 1 pp 81ndash90 2010
[8] B-H Lee J-H Choo S Na P Marzocca and L LibresculdquoSliding mode robust control of supersonic three degrees-of-freedom airfoilsrdquo International Journal of Control Automationand Systems vol 8 no 2 pp 279ndash288 2010
[9] Y Wu B Wang and G D Zong ldquoFinite-time tracking con-troller design for nonholonomic systemswith extended chainedformrdquo IEEE Transactions on Circuits and Systems II ExpressBriefs vol 52 no 11 pp 798ndash802 2005
[10] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006
[11] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009
[12] C B Njima W B Mabrouk H Messaoud and G GarcialdquoFinite time stabilization of the four tanks system extensions tothe uncertain systemsrdquo Abstract and Applied Analysis vol 2014Article ID 964143 7 pages 2014
[13] Y Yang and G Chen ldquoFinite time stability of stochastic hybridsystemsrdquo Abstract and Applied Analysis vol 2014 Article ID867189 7 pages 2014
[14] S Janardhanan and B Bandyopadhyay ldquoOn discretization ofcontinuous-time terminal sliding moderdquo IEEE Transactions onAutomatic Control vol 51 no 9 pp 1532ndash1536 2006
[15] L Wang T Chai and L Zhai ldquoNeural-network-based terminalsliding-mode control of robotic manipulators including actua-tor dynamicsrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3296ndash3304 2009
[16] L Hui and J Li ldquoTerminal sliding mode control for spacecraftformation flyingrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 45 no 3 pp 835ndash846 2009
[17] J Li and L Yang ldquoFinite-time terminal sliding mode trackingcontrol for piezoelectric actuatorsrdquo Abstract and Applied Anal-ysis vol 2014 Article ID 760937 9 pages 2014
[18] K-B Park and J-J Lee ldquoComments on lsquoa robust MIMOterminal sliding mode control scheme for rigid robotic manip-ulatorsrsquordquo IEEE Transactions on Automatic Control vol 41 no 5pp 761ndash762 1996
[19] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control and its application for robot manipulatorsrdquo inProceedings of the IEEE International Symposium onCircuits andSystems (ISCAS rsquo01) pp 545ndash548 May 2001
[20] M Jin J Lee P H Chang and C Choi ldquoPractical nonsingularterminal sliding-mode control of robot manipulators for high-accuracy tracking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3593ndash3601 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of