a novel fuzzy sliding-mode control for discrete-time uncertain

Research ArticleA Novel Fuzzy Sliding-Mode Control forDiscrete-Time Uncertain System

T H Yan1 B Wu1 B He2 W H Li3 and R B Wang1

1School of Mechanical amp Electrical Engineering China Jiliang University Hangzhou 310018 China2School of Information Science and Engineering College Ocean University of China Qingdao 266100 China3School of Mechanical Materials and Mechatronic Engineering University of Wollongong Wollongong NSW Australia

Correspondence should be addressed to T H Yan thyan163com

Received 19 April 2016 Revised 20 July 2016 Accepted 10 August 2016

Academic Editor Yan-Jun Liu

Copyright copy 2016 T H Yan et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers the sliding-mode control problem for discrete-time uncertain systems It begins by presenting a discretevariable speed reaching law and a discrete-time sliding-mode controller (DSMC) designed using the proposed reaching lawfollowed by an analysis of their stability and dynamic performance A sliding-mode controller with simple fuzzy logic is thenproposed to further strengthen the dynamic performance of the proposed sliding-mode controller Finally the presented DSMCand the DSMCwith fuzzy control for adjusting the parameters in this paper are compared with one of the previous proposed classicDSMC systems The results of this simulation show that the DSMC presented here can suppress chatter and ensure good dynamicperformances when fuzzy logic is used to tune the parameters

1 Introduction

A sliding-mode control (SMC) system has proved to bean effective control strategy for incompletely modeled oruncertain systems so much so that many relevant results canbe found in various publications for example [1ndash5] SMC isa popular robust strategy which can offer good robustnessagainst the parameter uncertainties and external disturbanceby limiting the effect in a bounded area The main advantageof SMC is that it can guarantee the stability of the system byswitching the control lawMovement in general SMC systemscan be divided into a reaching phase and a sliding mode [6]because the basic idea of SMC is to drive system states ontoa predefined switching surface and move on it to the pointof origin after the system state reaches the switching surfaceThe need to research SMC in discrete time (DSMC) is dueto the widespread use of computers for the control problemIn recent years DSMC has received a great deal of researchinterests A new method of DSMC is proposed in [7] Thealgorithm comprises not only functions in the sliding variablebut also functions in other variables It provided a generalmethod to design reaching law The ideal reaching law is

proposed in [8] which used LMI to list sufficient conditionsof switching surface existence in Delta operator system andthe result was compared with the index reaching law Fastconvergence and the attenuation of disturbance are studiedin [9ndash11] In [9] a new controller is present which is able tohandle the effect of interconnection for large-scale systemsand unmatched uncertainty It also shows a fast convergenceto the desired value In [10 11] quasi-sliding-mode controlwas proposed to the system which has eternal disturbanceFast convergence and small chatting are ensured by usingthese two algorithms The applications of DSMC in practicalsystems are also studied in [12 13] which study the SMC onservo system with external disturbance

However its major drawback in practical applications isthat the system trajectories of DSMC systems cannot reachtheir origin They only tend to reach a chattering stagesurrounding the origin [14ndash16] Numerous methods havebeen developed to eliminate this phenomenon in DSMC [17ndash19] for instance in [18] robustness is ensured and chatteringis eliminated by using an optimal integral sliding surface Anoutput-feedback quasi-sliding-mode control scheme is pro-posed in [19] which eliminates the chattering phenomenon

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1530760 9 pageshttpdxdoiorg10115520161530760

2 Mathematical Problems in Engineering

and also ensures the uncertainties and disturbances arerobust Bartoszewicz designed a quasi-sliding-mode con-troller for a discrete-time system in [20] that can reducechattering caused by high frequency Qu et al proposed aDSMCwith a disturbance compensator in [10] that obviouslyreduced chattering A lot of effort has beenmade to overcomethe chattering problem such as a state observer adaptivetechniques and an intelligent control method [21ndash25] Someother methods also can be applied into DSMC to overcomechattering like the neural network in [26] which designedthe adaptive adjusting parameters in scalar form to reducethe online computation burden and can adjust the slidingsurface dynamically and the genetic algorithm in [27] whichcan optimize the parameters in sliding-mode controller

Fuzzy control (FC) has been widely used in many prac-tical systems [28ndash33] It is a technique that requires expertknowledge to design a controller which is amodel-free designmethod that is insensitive to variations in parameters andexternal disturbances In [29] an adaptive fuzzy sliding-mode control is used for nonlinear systems to estimatethe unknown gain of the switching control A fuzzy basisfunction network is used to approximate the unknowndynamics of a robot with two arms [30] and a fuzzy adaptivestate observer is designed to estimate the unmeasured statein [32] Fuzzy control has been widely used to adaptiveparameters and optimal control [34ndash41] An adaptive fuzzydecentralized output-feedback controller was designed for aclass of switched nonlinear large-scale systems which containthe unknown nonlinearities and dead zone in [34] Chen etal [35] proposed an adaptive tracking control for a class ofnonlinear stochastic systems with unknown functions Thefuzzy-neural networks were used to approximate unknownfunctions In [36] Chen et al considered the problem ofobserver-based adaptive fuzzy control for a class of non-linear time-delay systems in nonstrict-feedback form Thefuzzy logic systems were used to approximate the unknownnonlinear functions In [37] a hybrid task-space trajectoryand force tracking based on fuzzy system and adaptivemechanism were proposed to compensate for the externalperturbation kinematics and dynamic uncertainties Liu etal [38] proposed an adaptive fuzzy optimal control for aclass of unknown nonlinear discrete-time systems whichcontained unknown functions and nonsymmetric dead zoneThe fuzzy logic systems were employed to approximatethe functions in the systems In [39] an adaptive fuzzyinverse compensation control method was proposed foruncertain nonlinear system and an adaptive fuzzy controllerwas developed to establish the close-loop system stabilityFor the Henon Mapping chaotic system Gao and Liu [40]proposed a direct heuristic dynamic programming to handlethe optimal tracking control and the fuzzy logic system isapplied to measure the long-term utility function In [41]an adaptive fuzzy controller was proposed for uncertainnonlinear systems with backlash The fuzzy logic systemswere used to approximate the unknown functions unknownbacklash and backlash inversionOne feature of fuzzy controlis that ldquoif-thenrdquo rules are based on expert knowledge [42 43]Fuzzy sliding-mode control (FSMC) combines two theoriesof FC and SMC which means the fuzzy control method

can optimize SMC dynamically in more detail [44ndash46] Forexample when a system trajectory is leaving the switchingsurface it can provide a large control force to drive thetrajectory back to the switching surface according to fuzzyrules

In order to reduce the chattering and accelerating con-vergent speed in discrete-time uncertain systems a newreaching law is proposed for discrete-time systems withexternal disturbances It begins by introducing a discrete-time system with external disturbances the proposed SMCand then a FMSC is proposed to improve the reaching phaseThis method guarantees the robust behavior of the systemwhereas FSMC improves system performance based on aperformance analysis Simulated examples are given to verifythe proposed method followed by the conclusions

2 System Description

The general description of a single-input uncertain system isas follows

= 119860119909 + 119861119906 + 119863119891 (1)

where 119909 = [119909

1119909

2sdot sdot sdot 119909

119899]

119879isin 119877

119899 is the state 119906 isin 119877

1 is thecontrol input and 119891 isin 119877

119899 denotes the external disturbance119860 119861 and 119863 are known real constant matrices of appropriatedimensions Without losing any generality it is assumed thatthe pair (119860 119861) is completely controllable and the disturbance119891 is bounded that is

1003816

1003816

1003816

1003816

119891 (119905)

1003816

1003816

1003816

1003816

le 119872 (2)

where119872 is a positive and known constant It should be notedthat the uncertainty means the system contains externaldisturbance in our system

In order to alleviate chattering the sample hold effectsshould be taken into account so by applying zero-order-hold(ZOH) sampling [47] the discrete-time description of system(1) can be written as

119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (3)

where 119909(119896) = 119909(119896119879) 119866 = 119890

119860119879119867 = int

119879

0119890

119860120591119889120591119861 119906(119896) = 119906(119896119879)

119889(119896) = int

119879

0119890

119860120591119863119891((119896 + 1)119879 minus 120591)119889120591 and 119879 is the sampling

periodThe goal here is to design a suitable FSMC controller for

system (3) The switching function 119904(119896) is defined as

119904 (119896) = 119862

119879119909 (119896) =

119899

sum

119894=1

119888

119894119909

119894(119896) (4)

where119862 = [119888

1119888


119899]

119879 is designed to drive the system tothe origin when travelling along the switching surface

119878 = 119909 (119896) | 119862

119879119909 (119896) = 0 (5)

The control object is to drive the system to the switchingsurface 119878 defined in (5) and travel to the origin point alongthe switching surface 119878

Mathematical Problems in Engineering 3

3 Fuzzy Sliding-Mode Control Design

Thevariable speed reaching law is a kind of common reachinglaw [48] that can effectively overcome the chattering problemits continuous form is

119904 = minus120576 1199091sgn (119904) (6)

and its discrete form is

119904 (119896 + 1) minus 119904 (119896) = minus120576119879 119909 (119896)

1sgn (119904 (119896)) (7)

where 120576 gt 0 119909(119896)1is the state norm and 119879 is the sampling

periodFrom this reaching law we get the following controller for

system (3)

119906 (119896) = (119862

119879119867)

minus1

(119904 (119896) minus 120576119879 119909 (119896)

1sgn (119904 (119896))

minus 119862

119879119866119909 (119896) minus 119862

119879119889 (119896))

(8)

The variable speed reaching law (7) can overcome thechattering problem but there are two shortages one is thatthe chattering will be serious at the beginning of the reachingphase and the other is that the reaching speed is too slownear the switching surfaceThese two shortages will seriouslyaffect the system performance

31 The New Reaching Law Design The discrete form of theimproved variable speed reaching law is

119904 (119896 + 1) minus 119904 (119896) = minus120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) (9)

where 120572 120573 gt 0From this reaching law we get the following controller for

system (3)

119906 (119896) = (119862

119879119867)

minus1

(119904 (119896)

minus 120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) minus 119862

119879119866119909 (119896)

minus 119862

119879119889 (119896))

(10)

To ensure the system can meet its performance require-ments we analyzed the system as follows

32 Stability Analysis To design a sliding-mode controllerfor discrete-time uncertain system the following two con-ditions should be met to guarantee the system trajectoryreaching the switching surface and converge to zero in alimited time [49]

(119904 (119896 + 1) minus 119904 (119896)) 119904 (119896) lt 0

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) gt 0

(11)

Substituting the reaching law (9) to condition (11) yields

(119904 (119896 + 1) minus 119904 (119896)) 119904 (119896)

= minus120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) 119904 (119896)

= minus120572119879 |119904 (119896)| log2(|119904 (119896)|

120573+ 1) lt 0

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896)

= (2119904 (119896) minus 120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))) 119904 (119896)

= 2 |119904 (119896)|

2minus 120572119879 |119904 (119896)| log

2(|119904 (119896)|

120573+ 1)

(12)

When 119904(119896) is very small by using the equivalent infinitesimalwe can get

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) asymp 2 |119904 (119896)|

2minus 120572120573119879 |119904 (119896)|

2

= (2 minus 120572120573119879) |119904 (119896)|

2

(13)

To guarantee (119904(119896 + 1) + 119904(119896))119904(119896) gt 0 120572120573119879must be less than2 When 119904(119896) is big we get the following formula

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) gt 2 |119904 (119896)|

2minus 120572119879 |119904 (119896)|

2

= (2 minus 120572119879) |119904 (119896)|

2

(14)

So if 120572119879 lt 2 then (119904(119896 + 1) + 119904(119896))119904(119896) gt 0 is guaranteed Sowhen 120572120573119879 lt 2 and 120572119879 lt 2 the system trajectory can reachthe switching surface and converge to zero in a limited time

Specifically when 119904(119896) = 0 we can get 119904(119896 + 1) = 0 from(9) So in the next time 119904(119896)will be zero all the time if there isno disturbanceTherefore the proposed reaching law (9) canmeet conditions (11) which means that the switching surfaceis existent and the system trajectory will reach the surface ina limited time

According to the abovementioned analysis the systemtrajectory will enter a region near the switching surface andwill not go away So we can conclude that system (3) withcontroller (10) is stable and it will reach the surface in alimited time

33 Dynamic Performance Analysis From reaching law (9)we can obtain the following two formulas

119904 (119896 + 1) = minus120572119879 log2(|119904 (119896)|

120573+ 1) when 119904 (119896) = 0

+

119904 (119896 + 1) = 120572119879 log2(|119904 (119896)|

120573+ 1) when 119904 (119896) = 0

minus

(15)

We define the switching region as

119878

Δ= 119909 (119896) | minusΔ lt 119862

119879119909 (119896) lt Δ (16)

where Δ = 120572119879log2(|119904(119896)|

120573+ 1) The width of the switching

region is

2Δ = 2120572119879 log2(|119904 (119896)|

120573+ 1) (17)

Meanwhile the width in reaching law (7) is 2120576119879119909(119896)

1

Based on the functions 1198911(119909) = 119896 log

2(119909 + 1) and 119891

2(119909) = 119896119909


120572 = 5 120573 = 3120572 = 5 120573 = 1120572 = 3 120573 = 1

120572 = 1 120573 = 5120572 = 1 120573 = 3120572 = 1 120573 = 1

5 10 15 20 25 300t

minus05000510152025303540455055

s

Figure 1 The effect of 120572 and 120573 on the reaching rate

by comparing the width between reaching laws (9) and (7)we find that the width we proposed is smaller than (7) whichmeans the system trajectory will be in a smaller region whenit crosses the switching surface So the system with controller(10) has a smaller chatting than controller (8)

To analyze the parameters of the proposed reaching lawthe derived reaching rate is given here The derivative of acontinuous system is defined as follows [50]

119904 (119905) = limΔ119905rarr0

119904 (119905 + Δ119905) minus 119904 (119905)

Δ119905

(18)

Assuming

Δs (119896) = 119904 (119896 + 1) minus 119904 (119896) (19)

if the sampling period 119879 is small enough the followingequation can be obtained

119904 (119905) =

Δ119904

119879

=

119904 (119896 + 1) minus 119904 (119896)

119879

= minus120572 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))

(20)

For the first-order differential equation 119904 = 119891(119904) we givea numerical solution for (20) with different values of 120572 and 120573Here we assumed 119904(0) = 5 Figure 1 shows how 120572 and 120573 affectthe reaching rate

Figure 1 indicates that when 120572 and 120573 are increasing thereaching rate is faster but when 120573 is larger the reaching ratenear the sliding surface is smaller The results of simulatingsliding dynamics indicate that when 120572 and 120573 are too bigthe chattering problem will be terrible so based on theexperience of 120572 and 120573 the reaching law (9) we presented herehas further improvements in DSMC

34 Further Improvement Based on shortages of the reachinglaw (9) we used fuzzy control (FC) to adjust the parameters120572 and 120573 by referring to Figure 1

PS PBZONSNB

minus3 minus2 minus1 0 1 2 3 4minus4s

00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p

Figure 2 Membership function of input variables

PBPMPSPZ

08 10 12 14 16 1806120572 120573

00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p

Figure 3 Membership function of output variables

FC has replaced conventional technologies inmany fieldsOne major feature of FC is its ability to express ambiguousthinking so when the mathematical model of a system existsaccurately or exists but with uncertainties FC can deal withthe unknown process efficiently although it will sometimescomplicate the analysis due to the huge number of fuzzy rulesfor high-order systems To eliminate the chattering problemand accelerate the reaching rate in this study a fuzzy sliding-mode control method is proposedThemembership functionof our FSMCmethod is shown in Figures 2 and 3 it is a one-input-two-output FSMC and because there is only one inputthe number of fuzzy rules is greatly reduced Fuzzy controlrules are designed to map the input linguistic variables 119904 tothe output linguistic variables 120572 and 120573 as follows

120572 120573 = FSMC (119904) (21)

where FSMC(119904) means that 120572 and 120573 are the function ofswitching function 119904

Here the membership functions of 120572 and 120573 are normal-ized in the interval [06 18] such that 120572 120573 gt 0 and satisfycondition (11)


In our article the Mamdani fuzzy inference method [51]is used to determine the fuzzy rule such that

if 119904 is 119875

119894 then 120572 is 119872

119894and 120573 is 119873

119894 (22)

The FC we proposed has one input (119904) and two outputs(120572 120573) Linguistic variables which imply inputs are classified asNB NS ZO PS and PB and outputs are classified as PZ PSPM and PB Inputs are normalized in an interval of [minus4 4]and outputs are normalized in [06 18] as shown in Figures2 and 3 It is possible to assign a set of decision rules as shownin the following based on Figure 1 These rules contain therelationships between input andoutput that define the controlstrategy Each control input has five fuzzy sets so there areonly 5 fuzzy rules this overcomes the problem of complexfuzzy rules

The fuzzy rule set designed for FSMC is

if 119904 is NB then 120572 is PS and 120573 is PB

if 119904 is NS then 120572 is PS and 120573 is PM

if 119904 is ZO then 120572 is PZ and 120573 is PZ

if 119904 is PS then 120572 is PS and 120573 is PM

if 119904 is PB then 120572 is PS and 120573 is PB

(23)

Finally defuzzification should be designed to determinethe numerical value of 120572 and 120573 based on the fuzzy rule setHere the centre average method is used for defuzzification asfollows

120572 120573 =

int

119881V120583V (V) 119889V

int

119881120583V (V) 119889V

(24)

where119881 is the domain of linguistic variables 120572 and 120573 V is thepoint at which the membership function of 119872

119894(119873

119894) of 120572(120573)

achieves its maximum and 120583V(V) is the degree of membershipof 119904 to 119875

119894

From what has been discussed above a fuzzy sliding-mode controller can be obtained however the experience ofexperts should be referred to if the membership function andfuzzy rule set need to be adjusted for different problemsHereexperience was gained from Figure 1 which shows how 120572 and120573 affect system performance

4 Numerical Simulation

Consider a second-order systemwith time-varying uncertain[10] as follows

= [

0 1

5 minus2

] 119909 + [

0

1

] 119906 + [

0

1 + 22 cos (05120587119905)] (25)

When the sampling period 119879 = 01 the discrete-timedescription of system (25) can be obtained as

119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (26)

minus01314

s

s

01314

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35

Figure 4 Switching function with controller (28)

where

119866 = [

102351 009139

045696 084073

]

119867 = [

000470

009139

]

119889 (119896) = [

010080 000470

002351 009139

] [

0

1 + 22 cos (05120587119896119879)

]

(27)

Here the initial state was assumed to be 119909(0) = [21 1]

119879 and119862 = [1 1]

119879 It should be pointed out that a long samplingperiod was chosen to exhibit the system dynamics

In the following we used different controllers to comparethe controlling performance of system (26)

Case 1 Consider system (26) when controller (14) in [10] isimplemented with the expression that is

119906 (119896) = (119862

119879119867)

minus1

119862

119879119866 minus (1 minus 119902) 119904 (119896)

+ 120576119879 sgn (119904 (119896)) +

119896

sum

119894=1

[119904 (119894) minus (1 minus 119902119879) 119904 (119894 minus 1)

+ 120576119879 sgn (119904 (119894 minus 1))]

(28)

where 119902 = 5 and 120576 = 1 System performances are shownin Figures 4ndash6 and indicate that when the system trajectoryis close to the sliding surface chattering is small but thischattering cannot reach the origin it can only surround theorigin

Case 2 When controller (8) with parameters 120572 = 1 120573 = 1

is implemented for system (26) then system performances


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate

Figure 5 System states with controller (28)

u

1 2 3 4 50t (s)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

Figure 6 Controller (28)

are shown in Figures 7ndash9 respectively Unlike controller (28)the reaching rate accelerates at the beginning and chatteringdecreases but the reaching rate at the beginning has evidentlynot improved this can also be gleaned from Figure 1 where120572 = 120573 = 1

Case 3 When controller (8) is implemented for system (26)then by using FLC (21) to adjust parameters 120572 and 120573 theswitching function system states and controller are shownin Figures 10ndash12 Figure 10 shows that the dynamics of FSMChave improved the reaching rate in all regions and overcomethe chattering problem

From what has been discussed above the SMC andFSMC methods we proposed for discrete-time system withuncertainty are very capable and the proposed reaching lawperformed better than (28) By improving the reaching law

minus00232

s

00232

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25Sy

stem

stat

e


with fuzzy logic the system presents much better dynamicperformances at every stage

5 Conclusion

This paper has presented a new discrete reaching law anda discrete sliding-mode controller with fuzzy logical controlhas been designed by using the proposed reaching law andfuzzy set A comparison of three cases that is (1) the previousDSMC presented by [10] (2) the DSMC presented in thepaper and (3) aDSMCpresentedwith fuzzy control has beencarried out The experimental results show the obvious effec-tiveness of the proposed SMC especially with fuzzy controlIn the method presented the chattering phenomenon thatfrequently appears in a conventional SMCwas overcome andthe reaching rate has also been accelerated


u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35

Figure 10 Switching function with controller FSMC

t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate

Figure 11 System states with controller FSMC

1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u

Figure 12 Controller FSMC

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is partially supported by the Natural ScienceFoundation of China and Key Research and DevelopmentProgramofChina (51379198 2016YFC0301404 41176076 and31202036)

References

[1] F Valenciaga and R D Fernandez ldquoMultiple-input-multiple-output high-order sliding mode control for a permanentmagnet synchronous generator wind-based system with gridsupport capabilitiesrdquo IET Renewable Power Generation vol 9no 8 pp 925ndash934 2015

[2] W-J Chang P-H Chen and C-C Ku ldquoMixed sliding modefuzzy control for discrete-time non-linear stochastic systemssubject to variance and passivity constraintsrdquo IET ControlTheory amp Applications vol 9 no 16 pp 2369ndash2376 2015

[3] S Mobayen ldquoAn adaptive chattering-free PID sliding modecontrol based on dynamic sliding manifolds for a class ofuncertain nonlinear systemsrdquo Nonlinear Dynamics vol 82 no1-2 pp 53ndash60 2015

[4] T Yoshimura ldquoDesign of an adaptive fuzzy slidingmode controlfor uncertain discrete-time nonlinear systems based on noisymeasurementsrdquo International Journal of Systems Science vol 47no 3 pp 617ndash630 2016

[5] Y H Liu T G Jia Y G Niu and Y Y Zou ldquoDesign ofsliding mode control for a class of uncertain switched systemsrdquoInternational Journal of Systems Science vol 46 no 6 pp 993ndash1002 2015

[6] S Mobayen ldquoDesign of CNF-based nonlinear integral slidingsurface for matched uncertain linear systems with multiplestate-delaysrdquo Nonlinear Dynamics vol 77 no 3 pp 1047ndash10542014


[7] S Chakrabarty and B Bandyopadhyay ldquoA generalized reachinglaw for discrete time sliding mode controlrdquo Automatica vol 52pp 83ndash86 2015

[8] J Sun ldquoChattering-free sliding-mode variable structure controlof delta operator systemrdquo Journal of Computational and Theo-retical Nanoscience vol 11 no 10 pp 2150ndash2156 2014

[9] C H Chai and J H S Osman ldquoDiscrete-time integral slidingmode control for large-scale system with unmatched uncer-taintyrdquo Control Engineering and Applied Informatics vol 17 no3 pp 3ndash11 2015

[10] S C Qu X H Xia and J F Zhang ldquoDynamics of discrete-timesliding-mode-control uncertain systems with a disturbancecompensatorrdquo IEEE Transactions on Industrial Electronics vol61 no 7 pp 3502ndash3510 2014

[11] M Wu and J S Chen ldquoA discrete-time global quasi-slidingmode control scheme with bounded external disturbance rejec-tionrdquoAsian Journal of Control vol 16 no 6 pp 1839ndash1848 2014

[12] C Milosavljevic B Perunicic-Drazenovic and B VeselicldquoDiscrete-time velocity servo system design using sliding modecontrol approach with disturbance compensationrdquo IEEE Trans-actions on Industrial Informatics vol 9 no 2 pp 920ndash927 2013

[13] H-R Li Z-B Jiang and N Kang ldquoSliding mode disturbanceobserver-based fractional second-order nonsingular terminalsliding mode control for PMSM position regulation systemrdquoMathematical Problems in Engineering vol 2015 Article ID370904 14 pages 2015

[14] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[15] S J Wang L Hou L Dong and H J Xiao ldquoAdaptive fuzzysliding mode control of uncertain nonlinear SISO systemsrdquoProcedia Engineering vol 24 pp 33ndash37 2011

[16] N D That and Q P Ha ldquoDiscrete-time sliding mode controlwith state bounding for linear systems with time-varyingdelay and unmatched disturbancesrdquo IET Control Theory ampApplications vol 9 no 11 pp 1700ndash1708 2015

[17] S Govindaswamy T Floquet and S K Spurgeon ldquoDiscretetime output feedback sliding mode tracking control for systemswith uncertaintiesrdquo International Journal of Robust and Nonlin-ear Control vol 24 no 15 pp 2098ndash2118 2014

[18] N N Sun Y G Niu and B Chen ldquoOptimal integral slidingmode control for a class of uncertain discrete-time systemsrdquoOptimal Control Applications ampMethods vol 35 no 4 pp 468ndash478 2014

[19] M-C Pai ldquoDiscrete-time output feedback quasi-sliding modecontrol for robust tracking and model following of uncertainsystemsrdquo Journal of the Franklin Institute vol 351 no 5 pp2623ndash2639 2014

[20] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[21] M Mirzaei F S Nia and H Mohammadi ldquoApplying adaptivefuzzy sliding mode control to an underactuated systemrdquo inProceedings of the 2nd International Conference on ControlInstrumentation and Automation (ICCIA rsquo11) pp 654ndash659Shiraz Iran December 2011

[22] C-M Lin and C-F Hsu ldquoAdaptive fuzzy sliding-mode con-trol for induction servomotor systemsrdquo IEEE Transactions onEnergy Conversion vol 19 no 2 pp 362ndash368 2004

[23] Y M Fang J T Fei and K Q Ma ldquoModel reference adaptivesliding mode control using RBF neural network for active

power filterrdquo International Journal of Electrical Power amp EnergySystems vol 73 pp 249ndash258 2015

[24] C-F Hsu B-K Lee and C-W Chang ldquoDesign of an intelligentexponential-reaching sliding-mode control via recurrent fuzzyneural networkrdquo in Proceedings of the 11th IEEE InternationalConference on Control and Automation (ICCA rsquo14) pp 568ndash573Taichung Taiwan June 2014

[25] T-C Lin and M-C Chen ldquoAdaptive hybrid type-2 intelligentsliding mode control for uncertain nonlinear multivariabledynamical systemsrdquo Fuzzy Sets and Systems vol 171 no 1 pp44ndash71 2011

[26] G-XWen C L P Chen Y-J Liu and Z Liu ldquoNeural-network-based adaptive leader-following consensus control for second-order non-linear multi-agent systemsrdquo IET Control Theory ampApplications vol 9 no 13 pp 1927ndash1934 2015

[27] B Feng J Yang J G Ren andD S Zhang ldquoControl parametersoptimization for servo feed system using an improvedgeneticalgorithmrdquo in Proceedings of the 11th World Congress on Intel-ligent Control and Automation (WCICA rsquo14) pp 4865ndash4870Shenyang China June 2014

[28] P Balasubramaniam P Muthukumar and K Ratnavelu ldquoThe-oretical and practical applications of fuzzy fractional integralsliding mode control for fractional-order dynamical systemrdquoNonlinear Dynamics vol 80 no 1-2 pp 249ndash267 2015

[29] T Yoshimura ldquoAdaptive fuzzy sliding mode control for uncer-tain multi-input multi-output discrete-time systems using aset of noisy measurementsrdquo International Journal of SystemsScience vol 46 no 2 pp 255ndash270 2015

[30] W D Zheng G Wu G Xu and X C Chen ldquoAdaptivefuzzy sliding-mode control of uncertain nonlinear systemrdquo inProceedings of the International Conference on ManagementEducation Information and Control vol 125 pp 707ndash711 2015

[31] Y M Li and S C Tong ldquoHybrid adaptive fuzzy controlfor uncertain MIMO nonlinear systems with unknown dead-zonesrdquo Information Sciences vol 328 pp 97ndash114 2016

[32] W X Shi ldquoObserver-based indirect adaptive fuzzy control forSISO nonlinear systems with unknown gain signrdquo Neurocom-puting vol 171 pp 1598ndash1605 2016

[33] S Wen and Y Yan ldquoRobust adaptive fuzzy control for a classof uncertain MIMO nonlinear systems with input saturationrdquoMathematical Problems in Engineering vol 2015 Article ID561397 14 pages 2015

[34] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man amp Cybernetics Systems vol 46 no 1 pp 37ndash472015

[35] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014

[36] B Chen C Lin X Liu and K Liu ldquoObserver-based adaptivefuzzy control for a class of nonlinear delayed systemsrdquo IEEETransactions on Systems Man amp Cybernetics Systems vol 46no 1 pp 27ndash36 2015

[37] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrainedmultilegged robot system modeling and fuzzy control withuncertain kinematics and dynamics incorporating foot forceoptimizationrdquo IEEE Transactions on Systems Man amp Cybernet-ics Systems vol 46 no 1 pp 1ndash15 2016

[38] Y J Liu Y Gao S C Tong and YM Li ldquoFuzzy approximation-based adaptive backstepping optimal control for a class of


nonlinear discrete-time systems with dead-zonerdquo IEEE Trans-actions on Fuzzy Systems vol 24 no 1 pp 16ndash28 2016

[39] G Y Lai Z Liu Y Zhang C L P Chen S Xie and Y-J LiuldquoFuzzy adaptive inverse compensation method to tracking con-trol of uncertain nonlinear systems with generalized actuatordead zonerdquo IEEE Transactions on Fuzzy Systems 2016

[40] Y Gao and Y-J Liu ldquoAdaptive fuzzy optimal control usingdirect heuristic dynamic programming for chaotic discrete-time systemrdquo Journal of Vibration and Control vol 22 no 2pp 595ndash603 2016

[41] Y-J Liu and S Tong ldquoAdaptive fuzzy control for a class of non-linear discrete-time systems with backlashrdquo IEEE Transactionson Fuzzy Systems vol 22 no 5 pp 1359ndash1365 2014

[42] T-S Li S-C Tong and G Feng ldquoA novel robust adaptive-fuzzy-tracking control for a class of nonlinearmulti-inputmulti-output systemsrdquo IEEE Transactions on Fuzzy Systems vol18 no 1 pp 150ndash160 2009

[43] W-J Chang and F-L Hsu ldquoSliding mode fuzzy control forTakagi-Sugeno fuzzy systems with bilinear consequent partsubject to multiple constraintsrdquo Information Sciences vol 327pp 258ndash271 2016

[44] A F Amer E A Sallam and W M Elawady ldquoAdaptive fuzzyslidingmode control using supervisory fuzzy control for 3 DOFplanar robot manipulatorsrdquo Applied Soft Computing vol 11 no8 pp 4943ndash4953 2011

[45] Q Chen Y-R Nan H-H Zheng and X-M Ren ldquoFull-orderslidingmode control of uncertain chaos in a permanentmagnetsynchronous motor based on a fuzzy extended state observerrdquoChinese Physics B vol 24 no 11 Article ID 110504 pp 1ndash6 2015

[46] S Masumpoor H Yaghobi and M A Khanesar ldquoAdap-tive sliding-mode type-2 neuro-fuzzy control of an inductionmotorrdquo Expert Systems with Applications vol 42 no 19 pp6635ndash6647 2015

[47] B Wang X H Yu and G R Chen ldquoZOH discretization effecton single-input sliding mode control systems with matcheduncertaintiesrdquo Automatica vol 45 no 1 pp 118ndash125 2009

[48] L Z Song H Wen and Q H Yao ldquoVariable rate reachinglaw for discrete time variable structure control systemsrdquo AsianJournal of Control vol 3 pp 16ndash21 1999

[49] S Sarpturk Y Istefanopulos andO Kaynak ldquoOn the stability ofdiscrete-time sliding mode control systemsrdquo IEEE Transactionson Automatic Control vol 32 no 10 pp 930ndash932 1987

[50] Y Zhang Q Zhu G Ma and Y Guo ldquoAn improved reachinglaw of SMC for spacecraft tracking systemrdquo in Proceedings of theIEEE International Conference on Information and Automationpp 907ndash911 Lijiang China August 2015

[51] E H Mamdani and S Assilian ldquoAn experiment in linguisticsynthesis with a fuzzy logic controllerrdquo International Journal ofHuman-Computer Studies vol 51 no 1 pp 1ndash13 1975

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


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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


and also ensures the uncertainties and disturbances arerobust Bartoszewicz designed a quasi-sliding-mode con-troller for a discrete-time system in [20] that can reducechattering caused by high frequency Qu et al proposed aDSMCwith a disturbance compensator in [10] that obviouslyreduced chattering A lot of effort has beenmade to overcomethe chattering problem such as a state observer adaptivetechniques and an intelligent control method [21ndash25] Someother methods also can be applied into DSMC to overcomechattering like the neural network in [26] which designedthe adaptive adjusting parameters in scalar form to reducethe online computation burden and can adjust the slidingsurface dynamically and the genetic algorithm in [27] whichcan optimize the parameters in sliding-mode controller

Fuzzy control (FC) has been widely used in many prac-tical systems [28ndash33] It is a technique that requires expertknowledge to design a controller which is amodel-free designmethod that is insensitive to variations in parameters andexternal disturbances In [29] an adaptive fuzzy sliding-mode control is used for nonlinear systems to estimatethe unknown gain of the switching control A fuzzy basisfunction network is used to approximate the unknowndynamics of a robot with two arms [30] and a fuzzy adaptivestate observer is designed to estimate the unmeasured statein [32] Fuzzy control has been widely used to adaptiveparameters and optimal control [34ndash41] An adaptive fuzzydecentralized output-feedback controller was designed for aclass of switched nonlinear large-scale systems which containthe unknown nonlinearities and dead zone in [34] Chen etal [35] proposed an adaptive tracking control for a class ofnonlinear stochastic systems with unknown functions Thefuzzy-neural networks were used to approximate unknownfunctions In [36] Chen et al considered the problem ofobserver-based adaptive fuzzy control for a class of non-linear time-delay systems in nonstrict-feedback form Thefuzzy logic systems were used to approximate the unknownnonlinear functions In [37] a hybrid task-space trajectoryand force tracking based on fuzzy system and adaptivemechanism were proposed to compensate for the externalperturbation kinematics and dynamic uncertainties Liu etal [38] proposed an adaptive fuzzy optimal control for aclass of unknown nonlinear discrete-time systems whichcontained unknown functions and nonsymmetric dead zoneThe fuzzy logic systems were employed to approximatethe functions in the systems In [39] an adaptive fuzzyinverse compensation control method was proposed foruncertain nonlinear system and an adaptive fuzzy controllerwas developed to establish the close-loop system stabilityFor the Henon Mapping chaotic system Gao and Liu [40]proposed a direct heuristic dynamic programming to handlethe optimal tracking control and the fuzzy logic system isapplied to measure the long-term utility function In [41]an adaptive fuzzy controller was proposed for uncertainnonlinear systems with backlash The fuzzy logic systemswere used to approximate the unknown functions unknownbacklash and backlash inversionOne feature of fuzzy controlis that ldquoif-thenrdquo rules are based on expert knowledge [42 43]Fuzzy sliding-mode control (FSMC) combines two theoriesof FC and SMC which means the fuzzy control method

can optimize SMC dynamically in more detail [44ndash46] Forexample when a system trajectory is leaving the switchingsurface it can provide a large control force to drive thetrajectory back to the switching surface according to fuzzyrules

In order to reduce the chattering and accelerating con-vergent speed in discrete-time uncertain systems a newreaching law is proposed for discrete-time systems withexternal disturbances It begins by introducing a discrete-time system with external disturbances the proposed SMCand then a FMSC is proposed to improve the reaching phaseThis method guarantees the robust behavior of the systemwhereas FSMC improves system performance based on aperformance analysis Simulated examples are given to verifythe proposed method followed by the conclusions

2 System Description

The general description of a single-input uncertain system isas follows

= 119860119909 + 119861119906 + 119863119891 (1)

where 119909 = [119909

1119909


119899]

119879isin 119877

119899 is the state 119906 isin 119877

1 is thecontrol input and 119891 isin 119877

119899 denotes the external disturbance119860 119861 and 119863 are known real constant matrices of appropriatedimensions Without losing any generality it is assumed thatthe pair (119860 119861) is completely controllable and the disturbance119891 is bounded that is

1003816

1003816

1003816

1003816

119891 (119905)

1003816

1003816

1003816

1003816

le 119872 (2)

where119872 is a positive and known constant It should be notedthat the uncertainty means the system contains externaldisturbance in our system

In order to alleviate chattering the sample hold effectsshould be taken into account so by applying zero-order-hold(ZOH) sampling [47] the discrete-time description of system(1) can be written as

119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (3)

where 119909(119896) = 119909(119896119879) 119866 = 119890

119860119879119867 = int

119879

0119890

119860120591119889120591119861 119906(119896) = 119906(119896119879)

119889(119896) = int

119879

0119890

119860120591119863119891((119896 + 1)119879 minus 120591)119889120591 and 119879 is the sampling

periodThe goal here is to design a suitable FSMC controller for

system (3) The switching function 119904(119896) is defined as

119904 (119896) = 119862

119879119909 (119896) =

119899

sum

119894=1

119888

119894119909

119894(119896) (4)

where119862 = [119888

1119888


119899]

119879 is designed to drive the system tothe origin when travelling along the switching surface

119878 = 119909 (119896) | 119862

119879119909 (119896) = 0 (5)

The control object is to drive the system to the switchingsurface 119878 defined in (5) and travel to the origin point alongthe switching surface 119878




119904 = minus120576 1199091sgn (119904) (6)


119904 (119896 + 1) minus 119904 (119896) = minus120576119879 119909 (119896)

1sgn (119904 (119896)) (7)



system (3)

119906 (119896) = (119862

119879119867)

minus1

(119904 (119896) minus 120576119879 119909 (119896)

1sgn (119904 (119896))

minus 119862

119879119866119909 (119896) minus 119862

119879119889 (119896))

(8)



119904 (119896 + 1) minus 119904 (119896) = minus120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) (9)


system (3)

119906 (119896) = (119862

119879119867)

minus1

(119904 (119896)

minus 120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) minus 119862

119879119866119909 (119896)

minus 119862

119879119889 (119896))

(10)



(119904 (119896 + 1) minus 119904 (119896)) 119904 (119896) lt 0

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) gt 0

(11)


(119904 (119896 + 1) minus 119904 (119896)) 119904 (119896)

= minus120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) 119904 (119896)

= minus120572119879 |119904 (119896)| log2(|119904 (119896)|

120573+ 1) lt 0

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896)

= (2119904 (119896) minus 120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))) 119904 (119896)

= 2 |119904 (119896)|

2minus 120572119879 |119904 (119896)| log

2(|119904 (119896)|

120573+ 1)

(12)


(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) asymp 2 |119904 (119896)|

2minus 120572120573119879 |119904 (119896)|

2

= (2 minus 120572120573119879) |119904 (119896)|

2

(13)


(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) gt 2 |119904 (119896)|

2minus 120572119879 |119904 (119896)|

2

= (2 minus 120572119879) |119904 (119896)|

2

(14)





119904 (119896 + 1) = minus120572119879 log2(|119904 (119896)|

120573+ 1) when 119904 (119896) = 0

+

119904 (119896 + 1) = 120572119879 log2(|119904 (119896)|

120573+ 1) when 119904 (119896) = 0

minus

(15)


119878

Δ= 119909 (119896) | minusΔ lt 119862

119879119909 (119896) lt Δ (16)

where Δ = 120572119879log2(|119904(119896)|


region is

2Δ = 2120572119879 log2(|119904 (119896)|

120573+ 1) (17)


1


2(119909 + 1) and 119891

2(119909) = 119896119909


120572 = 5 120573 = 3120572 = 5 120573 = 1120572 = 3 120573 = 1

120572 = 1 120573 = 5120572 = 1 120573 = 3120572 = 1 120573 = 1

5 10 15 20 25 300t

minus05000510152025303540455055

s




119904 (119905) = limΔ119905rarr0

119904 (119905 + Δ119905) minus 119904 (119905)

Δ119905

(18)

Assuming

Δs (119896) = 119904 (119896 + 1) minus 119904 (119896) (19)


119904 (119905) =

Δ119904

119879

=

119904 (119896 + 1) minus 119904 (119896)

119879

= minus120572 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))

(20)




PS PBZONSNB


00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p


PBPMPSPZ

08 10 12 14 16 1806120572 120573

00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p



120572 120573 = FSMC (119904) (21)





if 119904 is 119875

119894 then 120572 is 119872

119894and 120573 is 119873

119894 (22)








(23)


120572 120573 =

int

119881V120583V (V) 119889V

int

119881120583V (V) 119889V

(24)


119894(119873

119894) of 120572(120573)


119894




= [

0 1

5 minus2

] 119909 + [

0

1

] 119906 + [

0

1 + 22 cos (05120587119905)] (25)


119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (26)

minus01314

s

s

01314

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


where

119866 = [

102351 009139

045696 084073

]

119867 = [

000470

009139

]

119889 (119896) = [

010080 000470

002351 009139

] [

0

1 + 22 cos (05120587119896119879)

]

(27)


119879 and119862 = [1 1]




119906 (119896) = (119862

119879119867)

minus1

119862

119879119866 minus (1 minus 119902) 119904 (119896)

+ 120576119879 sgn (119904 (119896)) +

119896

sum

119894=1


+ 120576119879 sgn (119904 (119894 minus 1))]

(28)





x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


u

1 2 3 4 50t (s)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r





minus00232

s

00232

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25Sy

stem

stat

e



5 Conclusion



u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u


Competing Interests


Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119904 = minus120576 1199091sgn (119904) (6)


119904 (119896 + 1) minus 119904 (119896) = minus120576119879 119909 (119896)

1sgn (119904 (119896)) (7)



system (3)

119906 (119896) = (119862

119879119867)

minus1

(119904 (119896) minus 120576119879 119909 (119896)

1sgn (119904 (119896))

minus 119862

119879119866119909 (119896) minus 119862

119879119889 (119896))

(8)



119904 (119896 + 1) minus 119904 (119896) = minus120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) (9)


system (3)

119906 (119896) = (119862

119879119867)

minus1

(119904 (119896)

minus 120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) minus 119862

119879119866119909 (119896)

minus 119862

119879119889 (119896))

(10)



(119904 (119896 + 1) minus 119904 (119896)) 119904 (119896) lt 0

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) gt 0

(11)


(119904 (119896 + 1) minus 119904 (119896)) 119904 (119896)

= minus120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896)) 119904 (119896)

= minus120572119879 |119904 (119896)| log2(|119904 (119896)|

120573+ 1) lt 0

(119904 (119896 + 1) + 119904 (119896)) 119904 (119896)

= (2119904 (119896) minus 120572119879 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))) 119904 (119896)

= 2 |119904 (119896)|

2minus 120572119879 |119904 (119896)| log

2(|119904 (119896)|

120573+ 1)

(12)


(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) asymp 2 |119904 (119896)|

2minus 120572120573119879 |119904 (119896)|

2

= (2 minus 120572120573119879) |119904 (119896)|

2

(13)


(119904 (119896 + 1) + 119904 (119896)) 119904 (119896) gt 2 |119904 (119896)|

2minus 120572119879 |119904 (119896)|

2

= (2 minus 120572119879) |119904 (119896)|

2

(14)





119904 (119896 + 1) = minus120572119879 log2(|119904 (119896)|

120573+ 1) when 119904 (119896) = 0

+

119904 (119896 + 1) = 120572119879 log2(|119904 (119896)|

120573+ 1) when 119904 (119896) = 0

minus

(15)


119878

Δ= 119909 (119896) | minusΔ lt 119862

119879119909 (119896) lt Δ (16)

where Δ = 120572119879log2(|119904(119896)|


region is

2Δ = 2120572119879 log2(|119904 (119896)|

120573+ 1) (17)


1


2(119909 + 1) and 119891

2(119909) = 119896119909


120572 = 5 120573 = 3120572 = 5 120573 = 1120572 = 3 120573 = 1

120572 = 1 120573 = 5120572 = 1 120573 = 3120572 = 1 120573 = 1

5 10 15 20 25 300t

minus05000510152025303540455055

s




119904 (119905) = limΔ119905rarr0

119904 (119905 + Δ119905) minus 119904 (119905)

Δ119905

(18)

Assuming

Δs (119896) = 119904 (119896 + 1) minus 119904 (119896) (19)


119904 (119905) =

Δ119904

119879

=

119904 (119896 + 1) minus 119904 (119896)

119879

= minus120572 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))

(20)




PS PBZONSNB


00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p


PBPMPSPZ

08 10 12 14 16 1806120572 120573

00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p



120572 120573 = FSMC (119904) (21)





if 119904 is 119875

119894 then 120572 is 119872

119894and 120573 is 119873

119894 (22)








(23)


120572 120573 =

int

119881V120583V (V) 119889V

int

119881120583V (V) 119889V

(24)


119894(119873

119894) of 120572(120573)


119894




= [

0 1

5 minus2

] 119909 + [

0

1

] 119906 + [

0

1 + 22 cos (05120587119905)] (25)


119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (26)

minus01314

s

s

01314

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


where

119866 = [

102351 009139

045696 084073

]

119867 = [

000470

009139

]

119889 (119896) = [

010080 000470

002351 009139

] [

0

1 + 22 cos (05120587119896119879)

]

(27)


119879 and119862 = [1 1]




119906 (119896) = (119862

119879119867)

minus1

119862

119879119866 minus (1 minus 119902) 119904 (119896)

+ 120576119879 sgn (119904 (119896)) +

119896

sum

119894=1


+ 120576119879 sgn (119904 (119894 minus 1))]

(28)





x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


u

1 2 3 4 50t (s)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r





minus00232

s

00232

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25Sy

stem

stat

e



5 Conclusion



u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u


Competing Interests


Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120572 = 5 120573 = 3120572 = 5 120573 = 1120572 = 3 120573 = 1

120572 = 1 120573 = 5120572 = 1 120573 = 3120572 = 1 120573 = 1

5 10 15 20 25 300t

minus05000510152025303540455055

s




119904 (119905) = limΔ119905rarr0

119904 (119905 + Δ119905) minus 119904 (119905)

Δ119905

(18)

Assuming

Δs (119896) = 119904 (119896 + 1) minus 119904 (119896) (19)


119904 (119905) =

Δ119904

119879

=

119904 (119896 + 1) minus 119904 (119896)

119879

= minus120572 log2(|119904 (119896)|

120573+ 1) sgn (119904 (119896))

(20)




PS PBZONSNB


00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p


PBPMPSPZ

08 10 12 14 16 1806120572 120573

00

02

04

06

08

10

The d

egre

e of m

embe

rshi

p



120572 120573 = FSMC (119904) (21)





if 119904 is 119875

119894 then 120572 is 119872

119894and 120573 is 119873

119894 (22)








(23)


120572 120573 =

int

119881V120583V (V) 119889V

int

119881120583V (V) 119889V

(24)


119894(119873

119894) of 120572(120573)


119894




= [

0 1

5 minus2

] 119909 + [

0

1

] 119906 + [

0

1 + 22 cos (05120587119905)] (25)


119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (26)

minus01314

s

s

01314

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


where

119866 = [

102351 009139

045696 084073

]

119867 = [

000470

009139

]

119889 (119896) = [

010080 000470

002351 009139

] [

0

1 + 22 cos (05120587119896119879)

]

(27)


119879 and119862 = [1 1]




119906 (119896) = (119862

119879119867)

minus1

119862

119879119866 minus (1 minus 119902) 119904 (119896)

+ 120576119879 sgn (119904 (119896)) +

119896

sum

119894=1


+ 120576119879 sgn (119904 (119894 minus 1))]

(28)





x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


u

1 2 3 4 50t (s)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r





minus00232

s

00232

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25Sy

stem

stat

e



5 Conclusion



u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u


Competing Interests


Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











if 119904 is 119875

119894 then 120572 is 119872

119894and 120573 is 119873

119894 (22)








(23)


120572 120573 =

int

119881V120583V (V) 119889V

int

119881120583V (V) 119889V

(24)


119894(119873

119894) of 120572(120573)


119894




= [

0 1

5 minus2

] 119909 + [

0

1

] 119906 + [

0

1 + 22 cos (05120587119905)] (25)


119909 (119896 + 1) = 119866119909 (119896) + 119867119906 (119896) + 119889 (119896) (26)

minus01314

s

s

01314

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


where

119866 = [

102351 009139

045696 084073

]

119867 = [

000470

009139

]

119889 (119896) = [

010080 000470

002351 009139

] [

0

1 + 22 cos (05120587119896119879)

]

(27)


119879 and119862 = [1 1]




119906 (119896) = (119862

119879119867)

minus1

119862

119879119866 minus (1 minus 119902) 119904 (119896)

+ 120576119879 sgn (119904 (119896)) +

119896

sum

119894=1


+ 120576119879 sgn (119904 (119894 minus 1))]

(28)





x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


u

1 2 3 4 50t (s)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r





minus00232

s

00232

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25Sy

stem

stat

e



5 Conclusion



u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u


Competing Interests


Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


u

1 2 3 4 50t (s)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r





minus00232

s

00232

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


x1

x2

1 2 3 4 50t (s)

minus20

minus15

minus10

minus05

00

05

10

15

20

25Sy

stem

stat

e



5 Conclusion



u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u


Competing Interests


Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u

1 2 3 4 50t (s)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r


minus00179

s

00179

s

1 2 3 4 50t (s)

minus05

00

05

10

15

20

25

30

35


t (s)0 1 2 3 4 5

x1

x2

minus25

minus20

minus15

minus10

minus05

00

05

10

15

20

25

Syste

m st

ate


1 2 3 4 50t (s)

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trolle

r

u


Competing Interests


Acknowledgments


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









a novel fuzzy sliding-mode control for discrete-time uncertain

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