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7
Research Article Extended Nonsingular Terminal Sliding Surface for Second-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; [email protected] Received 29 January 2014; Accepted 18 April 2014; Published 11 May 2014 Academic Editor: Hui Zhang Copyright © 2014 Ji Wung Jeong et al. is is an open access article distributed under the Creative Commons Attribution License, which 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 proposed surface is a superset of a conventional nonsingular terminal sliding surface which guarantees that the system state gets to zero in finite time. e conventional nonsingular sliding surfaces have been designed using a power function whose exponent is a rational number with positive odd numerator and denominator. e proposed nonsingular terminal sliding surface overcomes the restriction on the exponent of a power function; that is, the exponent can be a positive real number. Simulation results are provided to show the validity of the main result. 1. Introduction According to the progress of control schemes, a variety of nonlinear control systems have been proposed: H-infinity optimal control, fuzzy control, neural network control, con- troller using genetic algorithms, and so on [14]. Sliding mode control (SMC) method, which is also called a variable structure system (VSS), is one of nonlinear robust control schemes. It has been widely used because of its invariance properties to parametric uncertainties and external distur- bances [58]. In conventional sliding mode control systems, sliding surfaces have been designed such that the overall system in the sliding mode is asymptotically stable. However, the asymptotic stability does not guarantee a finite time convergence. Although most of control systems proposed so far have been designed such that the closed-loop system is asymptotic, finite time stabilization is very important to many actual applications, such as motor, power, robot, and aerospace systems because, in the actual applications, the main objective of a control system is to make system’s state to a desired one in a finite time interval which is determined a priori. us, many recent studies have focused on the finite time stabilization [913]. Finite time control systems based on sliding mode control schemes have been called terminal sliding mode control systems since their sliding surfaces have been designed as terminal attractors [14]. Recently, terminal sliding mode control systems have been studied to achieve a finite time convergence in many applications [1517]. Conventional ter- minal sliding mode controllers used terminal sliding surfaces which were designed using a power function of a system state. ey guaranteed that the system state reached zero in finite time. On the contrary, however, they suffered from singularity problems and had restrictions on the range of the exponent of a power function. e exponent should be a rational number with positive odd numerator and denominator [18]. In order to avoid the singularity problem in conventional terminal sliding mode control systems, nonsingular terminal sliding surfaces have been proposed very recently for robot manipulators [19, 20]. However, the same restriction on the exponent of a power function in the nonsingular sliding surface still remained: the exponent should be a rational number with a positive odd numerator and denominator. us, in this paper, a novel nonsingular terminal sliding surface is proposed for second-order nonlinear systems. It is shown that the proposed scheme guarantees that the system state gets to zero in finite time and it does not suffer from Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 267510, 6 pages http://dx.doi.org/10.1155/2014/267510

Transcript of Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa...

Page 1: Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa › 2014 › 267510.pdf · Research Article Extended Nonsingular Terminal Sliding

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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Page 2: Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa › 2014 › 267510.pdf · Research Article Extended Nonsingular Terminal Sliding

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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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa › 2014 › 267510.pdf · Research Article Extended Nonsingular Terminal Sliding

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

Page 4: Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa › 2014 › 267510.pdf · Research Article Extended Nonsingular Terminal Sliding

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

Page 5: Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa › 2014 › 267510.pdf · Research Article Extended Nonsingular Terminal Sliding

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

Page 6: Research Article Extended Nonsingular Terminal Sliding ...downloads.hindawi.com › journals › aaa › 2014 › 267510.pdf · Research Article Extended Nonsingular Terminal Sliding

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

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International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of