Research Article Adaptive Second-Order Sliding Mode Control …downloads.hindawi.com/journals/mpe/2015/319495.pdf · 2019. 7. 31. · robustness against uncertainties [ ]. Sliding

Research ArticleAdaptive Second-Order Sliding Mode Control Design for a Classof Nonlinear Systems with Unknown Input

You Zheng1 Jianxing Liu2 Xinyi Liu3 Dandan Fang4 and L Wu1

1Ningbo University of Technology Ningbo 315000 China2School of Astronautics Harbin Institute of Technology Harbin 150000 China3Laboratoire OPERA Universite de Technologie de Belfort-Montbeliard 90010 Belfort France4Ningbo Academy of Smart City Development Ningbo 315000 China

Correspondence should be addressed to Jianxing Liu jxliuhiteducn

Received 7 March 2015 Accepted 20 April 2015

Academic Editor Xinggang Yan

Copyright copy 2015 You Zheng 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 adaptive second-order sliding mode controller is proposed for a class of nonlinear systems with unknown input The proposedcontroller continuously drives the sliding variable and its time derivative to zero in the presence of disturbances with unknownboundaries A Lyapunov approach is used to show finite time stability for the system in the presence of a class of uncertainty Anillustrative simulation example is presented to demonstrate the performance and robustness of the proposed controller

1 Introduction

Sliding mode control (SMC) has gained much attention dueto its attractive characteristics of finite time convergence androbustness against uncertainties [1ndash3] Sliding mode controlhas been thoroughly studied fromboth practical and theoret-ical point of view for example successfully applied for controlof time delay systems [4 5] control and observation of fuelcell power systems [6ndash11] control and observation of powerconverters [12ndash16] and adaptive control of flexible spacecraft[17 18] Robustness of a control system is essential basedon the reason that various uncertainties exist in practicalsystems These nonlinear systems are with structured andunstructured uncertainties and external disturbances suchas load variation However the price of using sliding modecontrol to achieve robustness to these disturbances is controlchattering problem [19ndash22] chattering is undesired becauseit may cause high frequency dynamics and even instabil-ity Therefore in order to achieve the optimal operationperformance of such uncertain systems suitable model andcontroller development efforts are needed [23ndash26]

There are several ways to avoid chattering problem whenusing sliding mode control The conventional SMC uses a

control law with large control gains yielding the undesiredchattering while the control system is in the sliding mode Toeliminate the chattering the discontinuous control functionis replaced by ldquosaturationrdquo or continuous ldquosigmoidrdquo functionsin [27 28] However such approach constrains the slidingsystemrsquos trajectories not to the sliding surface but to itsvicinity losing the robustness to the disturbances Higherorder sliding mode control techniques are used in [29ndash32]which allow driving the sliding variable and its consecutivederivatives to zero in the presence of the disturbancesHowever the main challenge of high order sliding modecontrollers is the use of high order time derivatives of slidingvariable It is interesting to note that the popular super-twisting algorithm [1] only requires the measurement of thesliding mode variable without its time derivative Recentlyadaptive sliding mode controller has been proposed to tunethe controller gains with respect to disturbances [33 34]Intelligent controllers such as fuzzy neural network [3536] and adaptive fuzzy sliding mode controller [37 38]were proposed to reduce the chattering However all theaforementioned literatures address the case based on theassumption that the boundary of the disturbances is knownThis boundary cannot be easily obtained in practical cases

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 319495 7 pageshttpdxdoiorg1011552015319495

2 Mathematical Problems in Engineering

The overestimating of the disturbance boundary yields tolarger than necessary control gains while designing thesuper-twisting control law [32]

In this paper an adaptive controller gain super-twistingalgorithm (ASTW) is proposed for nonlinear system withunknown input Based on the Lyapunov theory the proposedcontrol law continuously drives the sliding variable and itstime derivative to zero in the presence of bounded unknowninput but without knowing the boundary The stability andthe robustness of the control system are proven and thetracking performance is ensured

2 Problem Formulation

The super-twisting control law (STW) is effective to removethe chattering when the relative degree equals one It gener-ates the continuous control function that drives the slidingvariable and its time derivative to zero in finite time in thepresence of bounded unknown disturbance The main dis-advantage of STW algorithm is that it requires the boundedvalue of [39] (120590 is the sliding variable) Unfortunately theknowledge of is often unavailable The overestimating ofthe will yield to larger than necessary control gains whiledesigning the STW control law In this note an adaptive-gainapproach will be adopted when designing STW control lawwhich does not require the bounded value of The idea ofadaptive-gain approach is to increase the control gain 120572 120573dynamically until the STW controller converges Once the120590 gt 0 then gains will start reducing this gains reductionwill be reversed as soon as the sliding variable 120590 and its timederivative start deviating from the equilibrium point 120590 = = 0

Consider a class of 119899th-order uncertain nonlinear systemwhich is represented in a state-space form as

= 119891 (119909) + 119887 (119909) 119906 + 119889 (1)

where 119909 = [1199091 119909

2 119909

119899]

119879= [119909

1

1 119909

1

119899minus1]

119879isin R119899 is

the state vector 119906 isin R is the control function 119889 isin R119899 is theexternal disturbance and 119891(119909) isin R119899 and 119887(119909) isin R119899 are thesmooth vector fields

Let the desired state vector be 119909119889 the tracking error and

the sliding-surface function are defined as

119890 (119909 119905) = 119909 minus 119909

119889

120590 (119909 119905) = 119888

119879119890

(2)

where 119888 = [1198881 119888

2 119888

119899]

119879 The sliding mode will be obtainedin finite time if an appropriate control law is applied In thesliding mode the error dynamics will be

119888

119899119890

(119899minus1)

1+ 119888

119899minus1119890

(119899minus2)

1+ sdot sdot sdot + 119888

1119890

1= 0

(3)

The constants 1198881 119888

2 119888

119899are chosen to be positive such that

the eigenvalue polynomial 120593(120582) = 119888119899120582

119899minus1+ 119888

119899minus1120582

119899minus2+ sdot sdot sdot + 119888

1

is HurwitzThe choice of 119888 decides the convergence rate of thetracking error

Assumption 1 The relative degree of system (1) with thesliding variable 120590with respect to 119906 equals one in other wordsthe control function 119906 has to appear explicitly in the first totalderivative

Under Assumption 1 the dynamics of 120590 can be computedas follows

= 119892 (119909) 119906 + 120588 (119909 119905) (4)

where 120588(119909 119905) = 119888119879119891(119909) + 119888119879119889 minus 119888119879119889and 119892(119909) = 119888119879119887(119909) = 0

Assumption 2 The first-order time derivative of the uncer-tain function 120588(119909 119905) isin R is bounded

120588 (119909 119905) le 120575 (5)

for some unknown constants 120575 gt 0

Assumption 3 The uncertain function 119892(119909 119905) isin R ispresented as

119892 (119909 119905) = 119892

0(119909 119905) + Δ119892 (119909 119905) (6)

where 1198920(119909 119905) gt 0 is the nominal parts of 119892(119909 119905) and Δ119892(119909 119905)

is the parameter variation such that1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

Δ119892 (119909 119905)

119892

0(119909 119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120578 lt 1 (7)

With Assumptions 2 and 3 (4) can be written as follows

= 120588 (119909 119905) + 119892

1(119909 119905) 120596 (8)

where 1 minus 120578 le 1198921(119909 119905) = 1 + Δ119892(119909 119905)119892

0(119909 119905) le 1 + 120578 and

120596 = 119892

0(119909 119905)119906 The solution of (8) is understood in the sense

of Filippov [40]

3 Adaptive-Gain STW Controller Design

The control objective is to drive the sliding variable 120590 andits derivative to zero in finite time without the controlgain overestimation based on Assumptions 2 and 3 satisfiedThe classical SMC can handle with the task to keep anoutput variable 120590 at zero when the relative degree of 120590 isone However the high frequency control switching leads tothe chattering effect which is exhibited by high frequencyvibration of the controlled plant and can be dangerous insome applications The second-order sliding mode (SOSM)controllers including the continuous STW control algorithmare able to remove the chattering effect while preserving themain sliding mode features and improving its accuracy inthe presence of unknown disturbance However it requiresthe knowledge of bounded disturbance Unfortunately theassumption of the disturbance is often unavailable in practicewhich leads to controller gain overestimation

In this note an adaptive-gain approach is used to solvethis problem with STW algorithm STW controller generatesthe continuous control function to remove the chatteringeffect while adaptive-gain approach allows controller gainnonoverestimation

Mathematical Problems in Engineering 3

The STW algorithm can be described by differentialinclusion as follows [1 32]

120596 = minus120582 |120590|

12 sign (120590) + V

V = minus120572 sign (120590) (9)

where the time varying controller gains

120582 = 120582 (120590 119905)

120572 = 120572 (120590 119905)

(10)

are to be defined laterSubstituting (9) into (8) it follows that

= minus120582119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(11)

where 120593 = 1198921(119909 119905)V + 120588(119909 119905) and 120601(119909 119905) = 119892

1(119909 119905)V + 120588(119909 119905)

The term 119892

1(119909 119905)V is assumed to be bounded with some

unknown constant 1205751 that is

1003816

1003816

1003816

1003816

120601 (119909 119905)

1003816

1003816

1003816

1003816

le 120575

1+ 120575 = 120575

2 (12)

for some unknown constant 1205752gt 0

In this paper the gains 120582(120590 119905) and 120572(120590 119905) are formulatedas

120582 (120590 119905) = 120582

0radic119897 (119905)

120572 (120590 119905) = 120572

0119897 (119905)

(13)

with 1205820 1205720arbitrary positive constants and a positive time

varying scalar 119897(119905) The dynamic law of the varying function119897(119905) is given by

119897 (119905)

119896 if |120590| = 0

0 otherwise(14)

where 119896 gt 0 is a positive constantNow the control objective is reduced to driving 120590 and its

derivative by (13) and (14) to zero in finite time with thecondition of bounded perturbations in (12) Thus the designof adaptive STW controller is formulated in the followingtheorem

Theorem 4 Consider system (11) suppose that (12) holdsThen for any initial conditions 119909(0) 120590(0) the adaptive-gainSTW control laws (13) (14) drive the sliding variable 120590 and itsderivative to zero in finite time

Proof A new state vector is introduced in order to presentsystem (11) in a more convenient form for Lyapunov analysis

120577 = [

120577

1

120577

2

] = [

119897

12(119905) |120590|

12 sign (120590)120593

] (15)

thus system (11) can be rewritten as

120577 =

119897

2

1003816

1003816

1003816

1003816

120577

1

1003816

1003816

1003816

1003816

[

minus120582

01

minus2120572

00

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

120577 + [

0

120601 (119909 119905)

] +

[

[

119897 (119905)

2119897 (119905)

120577

1

0

]

]

(16)

where 1205820= 120582

0119892

1(119909 119905) and 120572

0= 120572

0119892

1(119909 119905) Given that

119892

1(119909 119905) gt 0 it is easy to verify that 119860 is a Hurwitz matrixThen the following Lyapunov function candidate is

introduced for system (16)

119881 = 120577

119879119875120577

119875 =

1

2

[

[

4120572

0+ 120582

2

0minus120582

0

minus120582

02

]

]

(17)

Taking the derivative of (17)

119881 =

119897 (119905)

2

1003816

1003816

1003816

1003816

120577

1

1003816

1003816

1003816

1003816

120577

119879(119860

119879119875 + 119875119860) 120577 + 119902

1120601 (119909 119905) 120577

+

119897 (119905)

2119897 (119905)

120577

1119902

2120577

(18)

where 1199021= [minus120582

0 2] and 119902

2= [

4120572

0+ 120582

2

0 minus120582

0]

Since119860 is aHurwitzmatrix there exists a positive definitematrix 119876 such that 119860119879119875 + 119875119860 = minus119876 and 120582min(119875)120577

2le 119881 le

120582max(119875)1205772 Equation (18) can be rewritten as

119881 le minus

119897 (119905)

2

120582min (119876)

120582

12

max (119875)119881

12+

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875)119881

12+ Δ119881 (19)

where

Δ119881 =

119897 (119905)

2119897 (119905)

[(4120572

0+ 120582

2

0) 120577

2

1minus 2120582

0120577

1120577

2]

le

119897 (119905)

2119897 (119905)

[(4120572

0+ 120582

2

0+

120582

0

2

) 120577

2

1+

120582

0

2

120577

2

2]

=

119897 (119905)

2119897 (119905)

120577

119879Δ119876120577

Δ119876 =

[

[

[

[

4120572

0+ 120582

2

0+

120582

0

2

0

0

120582

0

2

]

]

]

]

(20)

With (20) (19) is present as

119881 le minus(

119897 (119905)

2

120582min (119876)

120582

12

max (119875)minus

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875))119881

12

+

119897 (119905)

2119897 (119905)

120582max (Δ119876)

120582min (119875)119881

(21)

For simplicity we define

120574

1=

120582min (119876)

2120582

12

max (119875)

120574

2=

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875)

120574

3=

120582max (Δ119876)

2120582min (119875)

(22)


120574

1 1205742 1205743are all positive constants Thus (21) is simplified as

119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881(23)

where 119897 ge 0In order to show that the

119881 will be negative in finite timethe second time derivative of 119881 is calculated as

119881 le minus

119897 (119905) 119881

12minus (120574

1119897 (119905) minus 120574

2)

119881

2119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881

minus 120574

3

119897 (119905)

119897

2(119905)

119881

(24)

If

119881 gt 0 the positive term 120574

3(

119897(119905)119897(119905))

119881 is decreasingand the term (120574

1119897(119905) minus 120574

2)(

1198812119881

12) will be positive at

some time instant 119905 = 119905

1which dominates the positive

term 1205743(

119897(119905)119897(119905))

119881 since 119897(119905) is a monotonic nondecreasingfunction ( 119897(119905) ge 0) In this point we can conclude that afterthe time 119905 = 119905

1 119881 lt 0 That is to say119881 is not increasing faster

than a time linear function during time 119905 isin [0 1199051] which can

be formulated as

119881 le 119898119905 + 119899 (25)

Substituting (25) into (23) we have

119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

(26)

where 119897(0) gt 0 is the initial value of the scalar function (14)The positive term in (26) is bounded

120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

le 120574

3

119897 (119905)

max 119898 119899min 119896 119897 (0)

(27)

Now we can conclude that after some time 119905 gt 1199051the first

term in the right side of (26) will dominate the second termsuch that

119881 le minus120579 [120574

1119897 (119905) minus 120574

2] 119881

12

(28)

with 120579 isin (0 1) and 1205741119897(119905)minus120574

2gt 0Therefore 120590 and converge

to zero in finite time Theorem 4 is proven

Remark 5 In view of practical implementation the condition|120590| = 0 in (14) cannot be satisfied due to measurement noiseand numerical approximations The condition |120590| = 0 needsto be modified by dead-zone technique [41 42] such that thedynamic law (14) is practically implementable

119897 (119905) =

119896 if |120590| ge 120591

0 else(29)

where 120591 is a sufficiently small positive value

4 Simulation Results and Discussions

Consider the following nonlinear system [43]

1= 119909

2

2= 119909

3

3= minus119887

1(119905) 119909

2

1minus 119887

2(119905) 119909

2minus 119887

3(119905) 119909

3+ [3 + cos (119905)] 119906

+ 119889 (119905)

(30)

where 1198871(119905) = [1+03 sin(119905)] 119887

2(119905) = [15+02 cos(119905)] 119887

3(119905) =

[1 + 04 sin(119905)] and 119889(119905) = 005 sin(01119905) is considered as anexternal disturbance

The desired state trajectory is supposed to be 119909119879119889=

[sin(119905) cos(119905) minus sin(119905)] The sliding-surface function (2) ischosen and 119888119879 = [10 5 1] The initial state vector is 119909(119905

0) =

[0 0 minus035]

119879 The initial value of the sliding variable is takenas 120590(0) = minus535

According to (2) the sliding surface is given as120590 = 119888

1(119909

1minus 119909

1119889) + 119888

2(119909

2minus 119909

2119889) + 119888

3(119909

3minus 119909

3119889) (31)

The time derivative of 120590 is calculated as = 119892

1(119909 119905) 119906 + 120588 (119909 119905) (32)

where 1198921(119909 119905) = 119888

3(3 + cos 119905) and 120588(119909 119905) = minus119888119879119909

119889+ 119888

1119909

2+

119888

2119909

3+ 119888

3119889(119905) + 119888

3(119887

1(119905)119909

2

1minus 119887

2(119905)119909

2minus 119887

3(119905)119909

3)

The adaptive-gain STW control laws (9) (10) (11) and(13) are designed as

= minus120582 (120590 119905) 119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572 (120590 119905) 119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(33)

where the adaptive-gain 120582(120590 119905) and 120572(120590 119905) dynamics follow(29) and the values of the parameters of the adaptive-gain lawhave been taken as 120591 = 0005 120582

0= 25 120572

0= 8 and 119896 = 5

These parameters are tuned to get sufficiently accurate andfast convergence Finally the adaptive-gain dynamic law (29)becomes

119897 (119905) =

5 if |120590| ge 0005

0 else(34)

Figures 1ndash3 show the state performance of system (3) inthe presence of disturbance 119889(119905) The time response of thesliding surface function120590 is shown in Figure 4 It can be easilyfound that 120590 converges to zero in finite time From Figure 5we can see that the control input is smooth which is suitablefor real applications Figure 6 shows that the adaptive law of(34) is effective in the presence of external disturbance 119889(t)

Remark 6 In order to avoid high frequency control activitythe boundary layer technique is employed during the simula-tionThus the saturation function sat(120590Φ) is used to replacethe function sign(120590Φ) whereΦ is the boundary layer widththat is

sat( 120590Φ

) =

120590

Φ

if1003816

1003816

1003816

1003816

1003816

1003816

1003816

120590

Φ

1003816

1003816

1003816

1003816

1003816

1003816

1003816

ge 1

sign( 120590Φ

) else(35)


0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus06

minus04

minus02

0

02

Time (s)

16 18 208

112

15 2 25minus01

0

01

Stat

ex1

x1

e1

x1d

x1 minus x1d

Figure 1 The performance of state 1199091and its error 119890

1

0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

2

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus2

minus15minus1

minus050

051

Time (s)

15 2 25minus1

minus050

05

15 2 25minus05

0

05

Stat

ex2

x2

e2

x2d

x2 minus x2d


2

5 Conclusions

Adaptive second-order slidingmode control design is studiedfor a class of nonlinear systems with unknown inputs In realapplications the upper boundary of uncertainty is difficultto obtain which is required for calculating control gains ofsuper-twisting sliding mode control To solve this problem asimple control designmethod is proposed based onLyapunovfunction The idea is very simple the gains are increasedaccording to a dynamic law until the sliding mode is attained

0 1 2 3 4 5 6 7 8 9 10minus6minus4minus2

024

Time (s)


024

Time (s)

15 2 25minus4minus2

02

15 2 25minus4minus2

02

Stat

ex3

x3

e3

x3d

x3 minus x3d


3

0 1 2 3 4 5 6 7 8 9 10minus10

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Slid

ing

varia

ble

15 2 25minus2

minus15minus1

minus050

05

5 6 7 8 9minus2

024

120590

times10minus5

Figure 4 Sliding surface function of (31)

0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

Time (s)

Con

trol i

nput

u

Figure 5 Control input 119906 versus time(s)


0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Vary

ing

func

tion

2 22 24 25104

106

114

l(t)

Figure 6 Time varying function 119897(119905)

and stops increasing thereafter The proposed approach hastwo advantages

(i) Only one parameter 119896 has to be tuned(ii) A priori knowledge of the uncertainty bound is not

required

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Zhejiang Provincial NaturalScience Foundation of China (no LQ14F010001) NingboCity Natural Science Foundation of China (no 2013A610114no 2014A610088 and no 2013A610150) Key Scientific andTechnological Projects of Ningbo (no 2014B92001) and theFundamental Research Funds for the Central Universities(Grant no HIT NSRIF201625)

References

[1] A Levant ldquoSliding order and sliding accuracy in sliding modecontrolrdquo International Journal of Control vol 58 no 6 pp 1247ndash1263 1993

[2] A Sabanovic ldquoVariable structure systems with sliding modesin motion controlmdasha surveyrdquo IEEE Transactions on IndustrialInformatics vol 7 no 2 pp 212ndash223 2011

[3] X-G Yan C Edwards and S K Spurgeon ldquoOutput feedbacksliding mode control for non-minimum phase systems withnon-linear disturbancesrdquo International Journal of Control vol77 no 15 pp 1353ndash1361 2004

[4] X-G Yan S K Spurgeon and C Edwards ldquoDecentralisedstabilisation for nonlinear time delay interconnected systemsusing static output feedbackrdquoAutomatica vol 49 no 2 pp 633ndash641 2013

[5] X-G Yan S K Spurgeon and C Edwards ldquoState and param-eter estimation for nonlinear delay systems using sliding modetechniquesrdquo IEEE Transactions on Automatic Control vol 58no 4 pp 1023ndash1029 2013

[6] R J Talj D Hissel R Ortega M Becherif and M HilairetldquoExperimental validation of a PEM fuel-cell reduced-ordermodel and a moto-compressor higher order sliding-modecontrolrdquo IEEE Transactions on Industrial Electronics vol 57 no6 pp 1906ndash1913 2010

[7] S Laghrouche J Liu F Ahmed M Harmouche and MWack ldquoAdaptive second-order sliding mode observer-basedfault reconstruction for pem fuel cell air-feed systemrdquo IEEETransactions on Control Systems Technology vol 23 no 3 pp1098ndash1109 2014

[8] W Garcia-Gabin F Dorado and C Bordons ldquoReal-timeimplementation of a sliding mode controller for air supply ona PEM fuel cellrdquo Journal of Process Control vol 20 no 3 pp325ndash336 2010

[9] J Liu S Laghrouche F S Ahmed and M Wack ldquoPEM fuelcell air-feed systemobserver design for automotive applicationsan adaptive numerical differentiation approachrdquo InternationalJournal ofHydrogenEnergy vol 39 no 30 pp 17210ndash17221 2014

[10] C Kunusch P F Puleston M A Mayosky and L FridmanldquoExperimental results applying second order sliding modecontrol to a PEM fuel cell based systemrdquo Control EngineeringPractice vol 21 no 5 pp 719ndash726 2013

[11] C Kunusch P F PulestonM AMayosky and J Riera ldquoSlidingmode strategy for PEM fuel cells stacks breathing controlusing a super-twisting algorithmrdquo IEEE Transactions on ControlSystems Technology vol 17 no 1 pp 167ndash174 2009

[12] J Liu S Laghrouche and M Wack ldquoObserver-based higherorder sliding mode control of power factor in three-phaseACDC converter for hybrid electric vehicle applicationsrdquoInternational Journal of Control vol 87 no 6 pp 1117ndash1130 2014

[13] Y B Shtessel A S Zinober and I A Shkolnikov ldquoSlidingmodecontrol of boost and buck-boost power converters using thedynamic sliding manifoldrdquo International Journal of Robust andNonlinear Control vol 13 no 14 pp 1285ndash1298 2003

[14] A Abrishamifar A A Ahmad and M Mohamadian ldquoFixedswitching frequency sliding mode control for single-phaseunipolar invertersrdquo IEEE Transactions on Power Electronics vol27 no 5 pp 2507ndash2514 2012

[15] J Liu S Laghrouche M Harmouche and MWack ldquoAdaptive-gain second-order sliding mode observer design for switchingpower convertersrdquoControl Engineering Practice vol 30 pp 124ndash131 2014

[16] E Vidal-Idiarte L Martinez-Salamero J Calvente and ARomero ldquoAn H

infincontrol strategy for switching converters

in sliding-mode current controlrdquo IEEE Transactions on PowerElectronics vol 21 no 2 pp 553ndash556 2006

[17] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010

[18] B Xiao Q Hu and Y Zhang ldquoAdaptive sliding mode faulttolerant attitude tracking control for flexible spacecraft underactuator saturationrdquo IEEE Transactions on Control SystemsTechnology vol 20 no 6 pp 1605ndash1612 2012

[19] V Parra-Vega and G Hirzinger ldquoChattering-free sliding modecontrol for a class of nonlinear mechanical systemsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 12pp 1161ndash1178 2001


[20] A Ferreira F J Bejarano and L M Fridman ldquoRobust controlwith exact uncertainties compensation with or without chatter-ingrdquo IEEE Transactions on Control Systems Technology vol 19no 5 pp 969ndash975 2011

[21] H Lee and V I Utkin ldquoChattering suppression methods insliding mode control systemsrdquo Annual Reviews in Control vol31 no 2 pp 179ndash188 2007

[22] I Boiko and L Fridman ldquoAnalysis of chattering in continuoussliding-mode controllersrdquo IEEE Transactions on AutomaticControl vol 50 no 9 pp 1442ndash1446 2005

[23] Y Zhao C Cachard and H Liebgott ldquoAutomatic needledetection and tracking in 3D ultrasound using an ROI-basedRANSAC and KalmanmethodrdquoUltrasonic Imaging vol 35 no4 pp 283ndash306 2013

[24] S Yin H Luo and S X Ding ldquoReal-time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[25] C Cachard Y Zhao and H Liebgott ldquoComparison of theexisting tool localisation methods on two-dimensional ultra-sound images and their tracking resultsrdquo IET Control Theory ampApplications vol 9 no 7 pp 1124ndash1134 2015

[26] S Yin X Zhu and O Kaynak ldquoImproved pls focused on key-performance-indicator-related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[27] H Elmali and N Olgac ldquoSliding mode control with pertur-bation estimation (SMCPE) a new approachrdquo InternationalJournal of Control vol 56 no 4 pp 923ndash941 1992

[28] J A Burton and A S I Zinober ldquoContinuous approximationof variable structure controlrdquo International Journal of SystemsScience vol 17 no 6 pp 875ndash885 1986

[29] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[30] J Liu Y Zhao B Geng and B Xiao ldquoAdaptive second ordersliding mode control of a fuel cell hybrid system for electricvehicle applicationsrdquoMathematical Problems in Engineering Inpress

[31] V I Utkin and A S Poznyak ldquoAdaptive sliding mode controlwith application to super-twist algorithm equivalent controlmethodrdquo Automatica vol 49 no 1 pp 39ndash47 2013

[32] Y Shtessel M Taleb and F Plestan ldquoA novel adaptive-gainsupertwisting sliding mode controller methodology and appli-cationrdquo Automatica vol 48 no 5 pp 759ndash769 2012

[33] J-J Slotine and S S Sastry ldquoTracking control of nonlinearsystems using sliding surfaces with application to robotmanip-ulatorsrdquo International Journal of Control vol 38 no 2 pp 465ndash492 1983

[34] W-D Chang R-C Hwang and J-G Hsieh ldquoApplication of anauto-tuning neuron to slidingmode controlrdquo IEEE Transactionson Systems Man and Cybernetics Part C Applications andReviews vol 32 no 4 pp 517ndash522 2002

[35] T-S Chiang C-S Chiu and P Liu ldquoAdaptive TS-FNN controlfor a class of uncertain multi-time-delay systems the expo-nentially stable sliding mode-based approachrdquo InternationalJournal of Adaptive Control and Signal Processing vol 23 no4 pp 378ndash399 2009

[36] F Da and W Song ldquoFuzzy neural networks for direct adaptivecontrolrdquo IEEE Transactions on Industrial Electronics vol 50 no3 pp 507ndash513 2003

[37] V Nekoukar and A Erfanian ldquoAdaptive fuzzy terminal slidingmode control for a class ofMIMOuncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 179 no 1 pp 34ndash49 2011

[38] HAlli andOYakut ldquoFuzzy sliding-mode control of structuresrdquoEngineering Structures vol 27 no 2 pp 277ndash284 2005

[39] J Davila L Fridman and A Levant ldquoSecond-order sliding-mode observer for mechanical systemsrdquo IEEE Transactions onAutomatic Control vol 50 no 11 pp 1785ndash1789 2005

[40] A F Filippov ldquoDifferential equations with discontinuous right-hand siderdquo Matematicheskii Sbornik vol 93 no 1 pp 99ndash1281960

[41] J-J E Slotine and W Li Applied Nonlinear Control vol 1Prentice Hall Englewood Cliffs NJ USA 1991

[42] A Pisano and E Usai ldquoGlobally convergent real-time differen-tiation via second order sliding modesrdquo International Journal ofSystems Science vol 38 no 10 pp 833ndash844 2007

[43] Y-J Huang T-C Kuo and S-H Chang ldquoAdaptive sliding-mode control for nonlinear systemswith uncertain parametersrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 38 no 2 pp 534ndash539 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The overestimating of the disturbance boundary yields tolarger than necessary control gains while designing thesuper-twisting control law [32]

In this paper an adaptive controller gain super-twistingalgorithm (ASTW) is proposed for nonlinear system withunknown input Based on the Lyapunov theory the proposedcontrol law continuously drives the sliding variable and itstime derivative to zero in the presence of bounded unknowninput but without knowing the boundary The stability andthe robustness of the control system are proven and thetracking performance is ensured

2 Problem Formulation

The super-twisting control law (STW) is effective to removethe chattering when the relative degree equals one It gener-ates the continuous control function that drives the slidingvariable and its time derivative to zero in finite time in thepresence of bounded unknown disturbance The main dis-advantage of STW algorithm is that it requires the boundedvalue of [39] (120590 is the sliding variable) Unfortunately theknowledge of is often unavailable The overestimating ofthe will yield to larger than necessary control gains whiledesigning the STW control law In this note an adaptive-gainapproach will be adopted when designing STW control lawwhich does not require the bounded value of The idea ofadaptive-gain approach is to increase the control gain 120572 120573dynamically until the STW controller converges Once the120590 gt 0 then gains will start reducing this gains reductionwill be reversed as soon as the sliding variable 120590 and its timederivative start deviating from the equilibrium point 120590 = = 0

Consider a class of 119899th-order uncertain nonlinear systemwhich is represented in a state-space form as

= 119891 (119909) + 119887 (119909) 119906 + 119889 (1)

where 119909 = [1199091 119909

2 119909

119899]

119879= [119909

1

1 119909

1

119899minus1]

119879isin R119899 is

the state vector 119906 isin R is the control function 119889 isin R119899 is theexternal disturbance and 119891(119909) isin R119899 and 119887(119909) isin R119899 are thesmooth vector fields

Let the desired state vector be 119909119889 the tracking error and

the sliding-surface function are defined as

119890 (119909 119905) = 119909 minus 119909

119889

120590 (119909 119905) = 119888

119879119890

(2)

where 119888 = [1198881 119888

2 119888

119899]

119879 The sliding mode will be obtainedin finite time if an appropriate control law is applied In thesliding mode the error dynamics will be

119888

119899119890

(119899minus1)

1+ 119888

119899minus1119890

(119899minus2)

1+ sdot sdot sdot + 119888

1119890

1= 0

(3)

The constants 1198881 119888

2 119888

119899are chosen to be positive such that

the eigenvalue polynomial 120593(120582) = 119888119899120582

119899minus1+ 119888

119899minus1120582

119899minus2+ sdot sdot sdot + 119888

1

is HurwitzThe choice of 119888 decides the convergence rate of thetracking error

Assumption 1 The relative degree of system (1) with thesliding variable 120590with respect to 119906 equals one in other wordsthe control function 119906 has to appear explicitly in the first totalderivative

Under Assumption 1 the dynamics of 120590 can be computedas follows

= 119892 (119909) 119906 + 120588 (119909 119905) (4)

where 120588(119909 119905) = 119888119879119891(119909) + 119888119879119889 minus 119888119879119889and 119892(119909) = 119888119879119887(119909) = 0

Assumption 2 The first-order time derivative of the uncer-tain function 120588(119909 119905) isin R is bounded

120588 (119909 119905) le 120575 (5)

for some unknown constants 120575 gt 0

Assumption 3 The uncertain function 119892(119909 119905) isin R ispresented as

119892 (119909 119905) = 119892

0(119909 119905) + Δ119892 (119909 119905) (6)

where 1198920(119909 119905) gt 0 is the nominal parts of 119892(119909 119905) and Δ119892(119909 119905)

is the parameter variation such that1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

Δ119892 (119909 119905)

119892

0(119909 119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

le 120578 lt 1 (7)

With Assumptions 2 and 3 (4) can be written as follows

= 120588 (119909 119905) + 119892

1(119909 119905) 120596 (8)

where 1 minus 120578 le 1198921(119909 119905) = 1 + Δ119892(119909 119905)119892

0(119909 119905) le 1 + 120578 and

120596 = 119892

0(119909 119905)119906 The solution of (8) is understood in the sense

of Filippov [40]

3 Adaptive-Gain STW Controller Design

The control objective is to drive the sliding variable 120590 andits derivative to zero in finite time without the controlgain overestimation based on Assumptions 2 and 3 satisfiedThe classical SMC can handle with the task to keep anoutput variable 120590 at zero when the relative degree of 120590 isone However the high frequency control switching leads tothe chattering effect which is exhibited by high frequencyvibration of the controlled plant and can be dangerous insome applications The second-order sliding mode (SOSM)controllers including the continuous STW control algorithmare able to remove the chattering effect while preserving themain sliding mode features and improving its accuracy inthe presence of unknown disturbance However it requiresthe knowledge of bounded disturbance Unfortunately theassumption of the disturbance is often unavailable in practicewhich leads to controller gain overestimation

In this note an adaptive-gain approach is used to solvethis problem with STW algorithm STW controller generatesthe continuous control function to remove the chatteringeffect while adaptive-gain approach allows controller gainnonoverestimation



120596 = minus120582 |120590|

12 sign (120590) + V

V = minus120572 sign (120590) (9)


120582 = 120582 (120590 119905)

120572 = 120572 (120590 119905)

(10)


= minus120582119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(11)

where 120593 = 1198921(119909 119905)V + 120588(119909 119905) and 120601(119909 119905) = 119892

1(119909 119905)V + 120588(119909 119905)

The term 119892



1003816

1003816

1003816

1003816

120601 (119909 119905)

1003816

1003816

1003816

1003816

le 120575

1+ 120575 = 120575

2 (12)



120582 (120590 119905) = 120582

0radic119897 (119905)

120572 (120590 119905) = 120572

0119897 (119905)

(13)



119897 (119905)

119896 if |120590| = 0

0 otherwise(14)





120577 = [

120577

1

120577

2

] = [

119897

12(119905) |120590|

12 sign (120590)120593

] (15)


120577 =

119897

2

1003816

1003816

1003816

1003816

120577

1

1003816

1003816

1003816

1003816

[

minus120582

01

minus2120572

00

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

120577 + [

0

120601 (119909 119905)

] +

[

[

119897 (119905)

2119897 (119905)

120577

1

0

]

]

(16)

where 1205820= 120582

0119892

1(119909 119905) and 120572

0= 120572

0119892

1(119909 119905) Given that

119892



119881 = 120577

119879119875120577

119875 =

1

2

[

[

4120572

0+ 120582

2

0minus120582

0

minus120582

02

]

]

(17)


119881 =

119897 (119905)

2

1003816

1003816

1003816

1003816

120577

1

1003816

1003816

1003816

1003816

120577

119879(119860

119879119875 + 119875119860) 120577 + 119902

1120601 (119909 119905) 120577

+

119897 (119905)

2119897 (119905)

120577

1119902

2120577

(18)

where 1199021= [minus120582

0 2] and 119902

2= [

4120572

0+ 120582

2

0 minus120582

0]


2le 119881 le


119881 le minus

119897 (119905)

2

120582min (119876)

120582

12

max (119875)119881

12+

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875)119881

12+ Δ119881 (19)

where

Δ119881 =

119897 (119905)

2119897 (119905)

[(4120572

0+ 120582

2

0) 120577

2

1minus 2120582

0120577

1120577

2]

le

119897 (119905)

2119897 (119905)

[(4120572

0+ 120582

2

0+

120582

0

2

) 120577

2

1+

120582

0

2

120577

2

2]

=

119897 (119905)

2119897 (119905)

120577

119879Δ119876120577

Δ119876 =

[

[

[

[

4120572

0+ 120582

2

0+

120582

0

2

0

0

120582

0

2

]

]

]

]

(20)


119881 le minus(

119897 (119905)

2

120582min (119876)

120582

12

max (119875)minus

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875))119881

12

+

119897 (119905)

2119897 (119905)

120582max (Δ119876)

120582min (119875)119881

(21)


120574

1=

120582min (119876)

2120582

12

max (119875)

120574

2=

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875)

120574

3=

120582max (Δ119876)

2120582min (119875)

(22)


120574


119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881(23)



119881 le minus

119897 (119905) 119881

12minus (120574

1119897 (119905) minus 120574

2)

119881

2119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881

minus 120574

3

119897 (119905)

119897

2(119905)

119881

(24)

If


3(

119897(119905)119897(119905))


1119897(119905) minus 120574

2)(

1198812119881




term 1205743(

119897(119905)119897(119905))




be formulated as

119881 le 119898119905 + 119899 (25)


119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

(26)


120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

le 120574

3

119897 (119905)

max 119898 119899min 119896 119897 (0)

(27)



119881 le minus120579 [120574

1119897 (119905) minus 120574

2] 119881

12

(28)





119897 (119905) =

119896 if |120590| ge 120591

0 else(29)




1= 119909

2

2= 119909

3

3= minus119887

1(119905) 119909

2

1minus 119887

2(119905) 119909

2minus 119887

3(119905) 119909

3+ [3 + cos (119905)] 119906

+ 119889 (119905)

(30)

where 1198871(119905) = [1+03 sin(119905)] 119887

2(119905) = [15+02 cos(119905)] 119887

3(119905) =




0) =

[0 0 minus035]



1(119909

1minus 119909

1119889) + 119888

2(119909

2minus 119909

2119889) + 119888

3(119909

3minus 119909

3119889) (31)


1(119909 119905) 119906 + 120588 (119909 119905) (32)

where 1198921(119909 119905) = 119888

3(3 + cos 119905) and 120588(119909 119905) = minus119888119879119909

119889+ 119888

1119909

2+

119888

2119909

3+ 119888

3119889(119905) + 119888

3(119887

1(119905)119909

2

1minus 119887

2(119905)119909

2minus 119887

3(119905)119909

3)


= minus120582 (120590 119905) 119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572 (120590 119905) 119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(33)


0= 25 120572

0= 8 and 119896 = 5


119897 (119905) =

5 if |120590| ge 0005

0 else(34)



sat( 120590Φ

) =

120590

Φ

if1003816

1003816

1003816

1003816

1003816

1003816

1003816

120590

Φ

1003816

1003816

1003816

1003816

1003816

1003816

1003816

ge 1

sign( 120590Φ

) else(35)


0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus06

minus04

minus02

0

02

Time (s)

16 18 208

112

15 2 25minus01

0

01

Stat

ex1

x1

e1

x1d

x1 minus x1d


1

0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

2

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus2

minus15minus1

minus050

051

Time (s)

15 2 25minus1

minus050

05

15 2 25minus05

0

05

Stat

ex2

x2

e2

x2d

x2 minus x2d


2

5 Conclusions



024

Time (s)


024

Time (s)

15 2 25minus4minus2

02

15 2 25minus4minus2

02

Stat

ex3

x3

e3

x3d

x3 minus x3d


3

0 1 2 3 4 5 6 7 8 9 10minus10

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Slid

ing

varia

ble

15 2 25minus2

minus15minus1

minus050

05

5 6 7 8 9minus2

024

120590

times10minus5


0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

Time (s)

Con

trol i

nput

u



0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Vary

ing

func

tion

2 22 24 25104

106

114

l(t)




required



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120596 = minus120582 |120590|

12 sign (120590) + V

V = minus120572 sign (120590) (9)


120582 = 120582 (120590 119905)

120572 = 120572 (120590 119905)

(10)


= minus120582119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(11)

where 120593 = 1198921(119909 119905)V + 120588(119909 119905) and 120601(119909 119905) = 119892

1(119909 119905)V + 120588(119909 119905)

The term 119892



1003816

1003816

1003816

1003816

120601 (119909 119905)

1003816

1003816

1003816

1003816

le 120575

1+ 120575 = 120575

2 (12)



120582 (120590 119905) = 120582

0radic119897 (119905)

120572 (120590 119905) = 120572

0119897 (119905)

(13)



119897 (119905)

119896 if |120590| = 0

0 otherwise(14)





120577 = [

120577

1

120577

2

] = [

119897

12(119905) |120590|

12 sign (120590)120593

] (15)


120577 =

119897

2

1003816

1003816

1003816

1003816

120577

1

1003816

1003816

1003816

1003816

[

minus120582

01

minus2120572

00

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119860

120577 + [

0

120601 (119909 119905)

] +

[

[

119897 (119905)

2119897 (119905)

120577

1

0

]

]

(16)

where 1205820= 120582

0119892

1(119909 119905) and 120572

0= 120572

0119892

1(119909 119905) Given that

119892



119881 = 120577

119879119875120577

119875 =

1

2

[

[

4120572

0+ 120582

2

0minus120582

0

minus120582

02

]

]

(17)


119881 =

119897 (119905)

2

1003816

1003816

1003816

1003816

120577

1

1003816

1003816

1003816

1003816

120577

119879(119860

119879119875 + 119875119860) 120577 + 119902

1120601 (119909 119905) 120577

+

119897 (119905)

2119897 (119905)

120577

1119902

2120577

(18)

where 1199021= [minus120582

0 2] and 119902

2= [

4120572

0+ 120582

2

0 minus120582

0]


2le 119881 le


119881 le minus

119897 (119905)

2

120582min (119876)

120582

12

max (119875)119881

12+

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875)119881

12+ Δ119881 (19)

where

Δ119881 =

119897 (119905)

2119897 (119905)

[(4120572

0+ 120582

2

0) 120577

2

1minus 2120582

0120577

1120577

2]

le

119897 (119905)

2119897 (119905)

[(4120572

0+ 120582

2

0+

120582

0

2

) 120577

2

1+

120582

0

2

120577

2

2]

=

119897 (119905)

2119897 (119905)

120577

119879Δ119876120577

Δ119876 =

[

[

[

[

4120572

0+ 120582

2

0+

120582

0

2

0

0

120582

0

2

]

]

]

]

(20)


119881 le minus(

119897 (119905)

2

120582min (119876)

120582

12

max (119875)minus

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875))119881

12

+

119897 (119905)

2119897 (119905)

120582max (Δ119876)

120582min (119875)119881

(21)


120574

1=

120582min (119876)

2120582

12

max (119875)

120574

2=

120590

2

1003817

1003817

1003817

1003817

119902

1

1003817

1003817

1003817

1003817

120582

12

min (119875)

120574

3=

120582max (Δ119876)

2120582min (119875)

(22)


120574


119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881(23)



119881 le minus

119897 (119905) 119881

12minus (120574

1119897 (119905) minus 120574

2)

119881

2119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881

minus 120574

3

119897 (119905)

119897

2(119905)

119881

(24)

If


3(

119897(119905)119897(119905))


1119897(119905) minus 120574

2)(

1198812119881




term 1205743(

119897(119905)119897(119905))




be formulated as

119881 le 119898119905 + 119899 (25)


119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

(26)


120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

le 120574

3

119897 (119905)

max 119898 119899min 119896 119897 (0)

(27)



119881 le minus120579 [120574

1119897 (119905) minus 120574

2] 119881

12

(28)





119897 (119905) =

119896 if |120590| ge 120591

0 else(29)




1= 119909

2

2= 119909

3

3= minus119887

1(119905) 119909

2

1minus 119887

2(119905) 119909

2minus 119887

3(119905) 119909

3+ [3 + cos (119905)] 119906

+ 119889 (119905)

(30)

where 1198871(119905) = [1+03 sin(119905)] 119887

2(119905) = [15+02 cos(119905)] 119887

3(119905) =




0) =

[0 0 minus035]



1(119909

1minus 119909

1119889) + 119888

2(119909

2minus 119909

2119889) + 119888

3(119909

3minus 119909

3119889) (31)


1(119909 119905) 119906 + 120588 (119909 119905) (32)

where 1198921(119909 119905) = 119888

3(3 + cos 119905) and 120588(119909 119905) = minus119888119879119909

119889+ 119888

1119909

2+

119888

2119909

3+ 119888

3119889(119905) + 119888

3(119887

1(119905)119909

2

1minus 119887

2(119905)119909

2minus 119887

3(119905)119909

3)


= minus120582 (120590 119905) 119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572 (120590 119905) 119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(33)


0= 25 120572

0= 8 and 119896 = 5


119897 (119905) =

5 if |120590| ge 0005

0 else(34)



sat( 120590Φ

) =

120590

Φ

if1003816

1003816

1003816

1003816

1003816

1003816

1003816

120590

Φ

1003816

1003816

1003816

1003816

1003816

1003816

1003816

ge 1

sign( 120590Φ

) else(35)


0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus06

minus04

minus02

0

02

Time (s)

16 18 208

112

15 2 25minus01

0

01

Stat

ex1

x1

e1

x1d

x1 minus x1d


1

0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

2

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus2

minus15minus1

minus050

051

Time (s)

15 2 25minus1

minus050

05

15 2 25minus05

0

05

Stat

ex2

x2

e2

x2d

x2 minus x2d


2

5 Conclusions



024

Time (s)


024

Time (s)

15 2 25minus4minus2

02

15 2 25minus4minus2

02

Stat

ex3

x3

e3

x3d

x3 minus x3d


3

0 1 2 3 4 5 6 7 8 9 10minus10

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Slid

ing

varia

ble

15 2 25minus2

minus15minus1

minus050

05

5 6 7 8 9minus2

024

120590

times10minus5


0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

Time (s)

Con

trol i

nput

u



0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Vary

ing

func

tion

2 22 24 25104

106

114

l(t)




required



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120574


119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881(23)



119881 le minus

119897 (119905) 119881

12minus (120574

1119897 (119905) minus 120574

2)

119881

2119881

12+ 120574

3

119897 (119905)

119897 (119905)

119881

minus 120574

3

119897 (119905)

119897

2(119905)

119881

(24)

If


3(

119897(119905)119897(119905))


1119897(119905) minus 120574

2)(

1198812119881




term 1205743(

119897(119905)119897(119905))




be formulated as

119881 le 119898119905 + 119899 (25)


119881 le minus (120574

1119897 (119905) minus 120574

2) 119881

12+ 120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

(26)


120574

3

119897 (119905)

119898119905 + 119899

119896119905 + 119897 (0)

le 120574

3

119897 (119905)

max 119898 119899min 119896 119897 (0)

(27)



119881 le minus120579 [120574

1119897 (119905) minus 120574

2] 119881

12

(28)





119897 (119905) =

119896 if |120590| ge 120591

0 else(29)




1= 119909

2

2= 119909

3

3= minus119887

1(119905) 119909

2

1minus 119887

2(119905) 119909

2minus 119887

3(119905) 119909

3+ [3 + cos (119905)] 119906

+ 119889 (119905)

(30)

where 1198871(119905) = [1+03 sin(119905)] 119887

2(119905) = [15+02 cos(119905)] 119887

3(119905) =




0) =

[0 0 minus035]



1(119909

1minus 119909

1119889) + 119888

2(119909

2minus 119909

2119889) + 119888

3(119909

3minus 119909

3119889) (31)


1(119909 119905) 119906 + 120588 (119909 119905) (32)

where 1198921(119909 119905) = 119888

3(3 + cos 119905) and 120588(119909 119905) = minus119888119879119909

119889+ 119888

1119909

2+

119888

2119909

3+ 119888

3119889(119905) + 119888

3(119887

1(119905)119909

2

1minus 119887

2(119905)119909

2minus 119887

3(119905)119909

3)


= minus120582 (120590 119905) 119892

1(119909 119905) |120590|

12 sign (120590) + 120593

= minus120572 (120590 119905) 119892

1(119909 119905) sign (120590) + 120601 (119909 119905)

(33)


0= 25 120572

0= 8 and 119896 = 5


119897 (119905) =

5 if |120590| ge 0005

0 else(34)



sat( 120590Φ

) =

120590

Φ

if1003816

1003816

1003816

1003816

1003816

1003816

1003816

120590

Φ

1003816

1003816

1003816

1003816

1003816

1003816

1003816

ge 1

sign( 120590Φ

) else(35)


0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus06

minus04

minus02

0

02

Time (s)

16 18 208

112

15 2 25minus01

0

01

Stat

ex1

x1

e1

x1d

x1 minus x1d


1

0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

2

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus2

minus15minus1

minus050

051

Time (s)

15 2 25minus1

minus050

05

15 2 25minus05

0

05

Stat

ex2

x2

e2

x2d

x2 minus x2d


2

5 Conclusions



024

Time (s)


024

Time (s)

15 2 25minus4minus2

02

15 2 25minus4minus2

02

Stat

ex3

x3

e3

x3d

x3 minus x3d


3

0 1 2 3 4 5 6 7 8 9 10minus10

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Slid

ing

varia

ble

15 2 25minus2

minus15minus1

minus050

05

5 6 7 8 9minus2

024

120590

times10minus5


0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

Time (s)

Con

trol i

nput

u



0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Vary

ing

func

tion

2 22 24 25104

106

114

l(t)




required



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus06

minus04

minus02

0

02

Time (s)

16 18 208

112

15 2 25minus01

0

01

Stat

ex1

x1

e1

x1d

x1 minus x1d


1

0 1 2 3 4 5 6 7 8 9 10minus15

minus1minus05

005

115

2

Time (s)

0 1 2 3 4 5 6 7 8 9 10minus2

minus15minus1

minus050

051

Time (s)

15 2 25minus1

minus050

05

15 2 25minus05

0

05

Stat

ex2

x2

e2

x2d

x2 minus x2d


2

5 Conclusions



024

Time (s)


024

Time (s)

15 2 25minus4minus2

02

15 2 25minus4minus2

02

Stat

ex3

x3

e3

x3d

x3 minus x3d


3

0 1 2 3 4 5 6 7 8 9 10minus10

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Slid

ing

varia

ble

15 2 25minus2

minus15minus1

minus050

05

5 6 7 8 9minus2

024

120590

times10minus5


0 1 2 3 4 5 6 7 8 9 10minus15

minus10

minus5

0

5

10

Time (s)

Con

trol i

nput

u



0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Vary

ing

func

tion

2 22 24 25104

106

114

l(t)




required



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Vary

ing

func

tion

2 22 24 25104

106

114

l(t)




required



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Adaptive Second-Order Sliding Mode Control …downloads.hindawi.com/journals/mpe/2015/319495.pdf · 2019. 7. 31. · robustness against uncertainties [ ]. Sliding