Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 712615 12 pageshttpdxdoiorg1011552013712615

Research ArticleSwitched Two-Level119867

infinand Robust Fuzzy Learning

Control of an Overhead Crane

Kao-Ting Hung1 Zhi-Ren Tsai2 and Yau-Zen Chang1

1 Department of Mechanical Engineering Chang Gung University Taoyuan 33302 Taiwan2Department of Computer Science amp Information Engineering Asia University Taichung 41354 Taiwan

Correspondence should be addressed to Yau-Zen Chang zenmailcguedutw

Received 30 November 2012 Revised 9 February 2013 Accepted 25 February 2013

Academic Editor Peng Shi

Copyright copy 2013 Kao-Ting Hung et alThis 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

Overhead cranes are typical dynamic systems which can be modeled as a combination of a nominal linear part and a highlynonlinear part For such kind of systems we propose a control scheme that deals with each part separately yet ensures globalLyapunov stability The former part is readily controllable by the119867

infinPDC techniques and the latter part is compensated by fuzzy

mixture of affine constants leaving the remaining unmodeled dynamics or modeling error under robust learning control usingthe Nelder-Mead simplex algorithm Comparison with the adaptive fuzzy control method is given via simulation studies and thevalidity of the proposed control scheme is demonstrated by experiments on a prototype crane system

1 Introduction

Overhead cranes are used in workshops or harbors to trans-port massive goods within short distance The manipulationof overhead cranes is affected by the existence of unavoidabledisturbances such as friction winds unbalanced load andaccidental collision Besides change of payloads and stringlength can result in tremendous variations in system dynam-ics Due to these inherent problems most of the overheadcranes are still operated by skilled labors

An automatic crane system should be able to accuratelycarry payloads to the desired position as fast as possiblewithout swing Many works have been focused on automaticcontrol of the overhead crane in the literature For instancePark et al and Singhose et al [1 2] investigated the inputshaping control of the crane systems [3ndash5] used the variablestructure control with sliding modes to control the overheadcrane Moreno et al [6] used neural networks to tune theparameters of state feedback control law to improve theperformance of an overhead crane Lee and Cho [7] proposedan antiswing fuzzy controller to enhance a servo controllerthat was used for positioning Moreover Nalley and Trabia[8] adopted fuzzy control for both positioning control andswing damping Moreover a standard discrete-time fuzzymodel [9ndash13] and continuous-time fuzzy controller [14] have

been proposed in the literature While the controllers of[15ndash20] are based on the so-called Single Input Rule Modules(SIRMs) and [21] focused on the construction of a reduced-order model to approximate the original system

In the above researches [1 2] lack robustness consider-ation for external disturbances and plant uncertainty whilestability is not guaranteed in [6ndash8] Successful implemen-tation of these schemes might depend on unreliable andhard-to-obtain consequent parts (linguistic value) such asthe schemes of [3 14] and the dynamic importance degreedefined in [15] respectively

In this paper we model the nonlinear plant as a com-bination of a continuous-time linear nominal model andfuzzily blended supplemental affine terms These terms areadded mainly to account for dominant friction effects andresidual nonlinear dynamics The model not only simplifiessubsequent control design but also enhances system robust-ness because assumptions on the plant dynamics are sig-nificantly reduced The nominal model allows linear controltechniques specifically the119867

infinlinear control technique [22

23] to be applied to the nonlinear plantIn the closely related literature of [15ndash20 24ndash31] fuzzy

controllers are developed to simultaneously stabilize thesefuzzy linear models using the parallel distributed control(PDC) scheme that satisfies the linear matrix inequality

2 Mathematical Problems in Engineering

(LMI) relations However these control design strategies relyon accurate fuzzymodeling of the plant which usually resultsin a large number of fuzzy rules and hence complex andconservative designs

To further alleviate the requirement for accurate fuzzymodeling of the plant a two-level 119867

infinrobust nonlinear

control scheme is proposed The inner-level controller isresponsible for accurate servo control while the outer-levelcontroller compensates for unmodeled system dynamics andbounded disturbances Besides each part of the proposedcontrol laws can be independently designed satisfying itsown specification This incremental design procedure avoidssolving the problem at one time and allows each part to bedesignedwith different guidelines Also global stability of theclosed-loop system is ensured against bounded disturbanceswith guaranteed disturbance attenuation level

A particular switching controller is proposed in [32] fornonlinear systems with unknown parameters based on afuzzy logic approach The major difference between our pro-posed scheme and the controller of [32] is that the switchingof our scheme is between the inner-loop and the outer-loopcontrollers while the controller of [32] is switched constantlybetween many (which is 8 in the simulation example) linearcontrollers Furthermore the fuzzy terms in our controllerare dedicated for the compensation of highly nonlinear effectsthat deviate from the nominal linear dynamics Neverthelessin [32] a fuzzy plantmodel is required for the construction ofthe switching plant model which is then used for the model-based design of the switching controller The switchingTakagi-Sugeno fuzzy control proposed in [33] also requiresthe plant to be accurately represented by a fuzzy system

As the closed-loop stability is ensured by the outer-levelcontroller we are able to optimize the inner-level controllerby the Nelder-Mead simplex algorithm [34] based on actualclosed-loop control performance rather than deriving fromthe plant modelThe optimization algorithm converges fasterthan particle swarm optimization (PSO) [35] which is ade-quate for online applications This scheme which incorpo-rates online trials can be applied to many applications suchas self-guided robot and evolvable systems Furthermoreconsidering that the swing dynamics depend on both stringlength and load mass fuzzy rules are created to interpolatecontrol gains obtained from trial experiments [36ndash38]

In the following sections this paper is divided intofour parts Section 2 describes the plant model and theproblem Section 3 proposes the two-level control schemeand Section 4 evaluates the effectiveness of the proposedscheme using both simulation comparison with a recentlyproposed strategy in the literature and experimental studiesFinally Section 5 concludes the results

2 Problem Formulation

The plant under consideration is assumed to be a disturbednonlinear system which is affine in the input and containsuncertain dynamics

= 119891 (119909) + Δ119891 (119909 119905) + [119892 (119909) + Δ119892 (119905)] sdot 119906 + 119908 (1)

where Δ119891(119909 119905) and Δ119892(119905) are unknown system dynamicswhich are bounded in 119909 and 119905 119909 = [119909

1 1199092 119909

119899]119879

isin 119877119899times1 is

the state vector 119906 = [1199061 1199062 119906

119898]119879

isin 119877119898times1 is the nonlinear

input vector and 119908 isin 119877119899times1 denotes unknown and bounded

disturbance Furthermore nonlinear functions119891(119909) and119892(119909)are Lipschitz in 119909

Next we approximate the nonlinear system as a nominallinear system augmented with Takagi-Sugeno type fuzzyblending of affine terms Note that these affine terms whichare usually dominated by friction in many mechatronicsystems are added to the control-input term rather thanbeing added directly This form closely reflects the practicaleffects of friction on system dynamics Specifically the 119894thrule of the affine T-S fuzzy model is in the following form

Plant rule 119894IF 1199111(119905) is119872

1198941and sdot sdot sdot and 119911

119901(119905) is119872

119894119901

THEN = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 119888119894]

for 119894 = 1 2 119871(2)

In each rule 1199111(119905) 1199112(119905) and 119911

119901(119905) are the 119901 premise

variables which can be state variables or functions of statevariables 119872

119894119895is the fuzzy set corresponding to the 119895th

premise variable 119860 isin 119877119899times119899 is the system matrix and 119861 isin

119877119899times119898 denotes the control input matrix Moreover 119888

119894isin 119877119898times1

is the 119894th bias vector Δ119860(119905) isin 119877119899times119899 is the system uncertaintyand Δ119861(119905) isin 119877119899times119898 denotes the control input uncertainty

Defining 120583119894119895(sdot) as the membership function correspond-

ing to fuzzy set 119872119894119895 we have that 120583

119894119895(119911119895(119905)) is the grade

of membership of 119911119895(119905) in 119872

119894119895 Using the sum-product

composition the firing strength of the 119894th fuzzy rule isrepresented as 120603

119894= 120603119894(119911) equiv prod

119901

119895=1120583119894119895(119911119895(119905)) with 119911 equiv

[1199111(119905) 1199111(119905) 119911

119901(119905)]119879

By defining ℎ119894(119911) = 120603

119894sum119871

119895=1120603119895as the normalized firing

strength of the 119894th rule hencesum119871119894=1ℎ119894(119911) = 1 the overall fuzzy

system model is then inferred as the weighted average of theconsequent parts

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894] (3)

The proposed control scheme is of a two-level switchingstructure where the control input is composed of three parts119906119878 119906119867 and 119906

119891 defined as follows

119906 = (1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867+ 119906119891

= (1 minus 119868lowast

) sdot 119870119878sdot 119890 minus 119868

lowast

sdot 119870119867sdot 119909 minus

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894

(4)

where 119868lowast isin 0 1 is a switching function to be definedin Section 3 In (4) the first term 119906

119878= 119870119878sdot 119890 is a servo

controller located in the inner loop responsible for accuratetracking where 119890 = 119909

119903minus 119909 is the tracking error with 119909

119903

denoting the reference state trajectory The second term

Mathematical Problems in Engineering 3

119906119867= minus119870119867sdot 119909 is an119867

infinrobust controller in the outer loop to

ensure system stability And 119906119891= minussum

119871

119894=1ℎ119894(119911) sdot 119888

119894is a fuzzy-

combination term that compensates for nonlinear dynamicssuch as friction and other effects that deviate from nominallinear dynamics

Next let us define the modeling error 119890 mod as

119890 mod equiv 119891 (119909) + Δ119891 (119909 119905) + [119892 (119909) + Δ119892 (119905)] sdot 119906 + 119908 minus y (5)

where 119910 = 119860 sdot 119909 +Δ119860(119905) sdot 119909 + [119861 +Δ119861(119905)] sdot [119906 +sum119871119894=1ℎ119894(119911) sdot 119888119894]

Hence the closed-loop system formed by applying (4) to (1)can be expressed concisely as follows

= 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)]

sdot [(1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867+ 119906119891+

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894] + 119890 mod

= 119860 sdot 119909 + Δ119860(119905)sdot119909 + [119861 + Δ119861 (119905)]sdot[(1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867]

+ 119890 mod

(6)

3 The Proposed Two-Level Control Scheme

As shown in Figure 1 the overall control scheme is composedof an outer-level stabilizing controller and an inner-levelservo controller Each of the controllers is designed accordingto a switching condition defined by the deviation of trackingerrors from a prescribed reference vector 119909

119903(119905) That is

If 119890 = 1003817100381710038171003817119909119903minus 119909

1003817100381710038171003817le 120576119864

then 119868lowast = 0 otherwise 119868lowast = 1(7)

In the condition the threshold 120576119864is a user-defined positive

numberThe value of it for instance may be designed as 01timesmax119905(119909119903(119905))

The closed-loop system dynamics when 119890 gt 120576119864is

formed by assigning 119868lowast = 1 in (6) as follows

= [119860 + Δ119860 (119905)] sdot 119909 minus [119861 + Δ119861 (119905)] sdot 119870119867sdot 119909 + 119890 mod (8)

If uncertainties in the plant dynamic matrices Δ119860(119905) andΔ119861(119905) are bounded wemay introduce a time-varyingmatrix119865(119905) with 0 le 119865(119905) le 1 and constant matrices 119863 119864

1 and

1198642 such that

[Δ119860 (119905) Δ119861 (119905)119870119867] sdot 119909 = 119863 sdot 119865 (119905) sdot [119864

11198642119870119867] sdot 119909

+ [120575 (119905) 0]

(9)

with 120575(119905) being a bounded function in 119909

120575 (119905) le 119886 sdot 119909 where 119886 is a positive constant (10)

Using (9) the closed-loop system dynamics (8) can then bewritten as

= (119860 minus 119861 sdot 119870119867) sdot 119909 + 119863 sdot 119865 (119905) sdot (119864

1minus 1198642sdot 119870119867) sdot 119909 + 119890 mod

(11)

where 119890 mod = 119890 mod + 120575(119905)

Plant

Fuzzycompensator

++ +

minus

119909119903119906119878119870119878

119906119867

119870119867

119906119891

119909119890

119868lowast

Figure 1 The proposed two-level switching control scheme

When the system is under acceptable tracking that is119890 le 120576

119864 only the servo controller is in charge The closed-

loop system dynamics is then formed by assigning 119868lowast = 0 in(6) as follows

= [119860 + Δ119860 (119905)] sdot 119909 + [119861 + Δ119861 (119905)] sdot 119870119878sdot 119890 + 119890 mod (12)

31 Design of the Outer-Level 119867infin

Stabilization ControllerThe119867

infinstabilization performance of 119906

119867is defined as follows

int

119905119891

0

[119909(119905)119879

sdot 119876 sdot 119909 (119905)] sdot 119889119905

119864 modle 1205882

(13)

where

119864 mod = int

119905119891

0

119890119879

mod sdot 119890 mod sdot 119889119905 (14)

119905119891is terminal time of control 119876 is a positive definite weight-

ing matrix and 120588 denotes prescribed attenuation level with1205882 being the attenuation disturbance level From the energy

viewpoint (13) confines the effect of 119890 mod on state 119909(119905) to beattenuated below a desired level If initial conditions are alsoconsidered the 119867

infinperformance in (13) can be modified as

follows

int

119905119891

0

(119909119879

sdot 119876 sdot 119909) 119889119905 le 119909119879

(0) sdot 119875 sdot 119909 (0) + 1205882

sdot 119864 mod (15)

where119876 and 119875 are symmetric and positive definite weightingmatrices The design of the stabilizing controller in the outerlevel corresponds to find a linear controller in the form of119906119867

= minus119870119867sdot 119909 such that the 119867

infinperformance (15) is

guaranteed to stabilize the closed-loop system (11)

Theorem 1 Assuming that the modeling error is boundedsuch that 119890 mod le 119890

119880 with 119890

119880being a positive constant

the 119867infin

control performance defined in (15) is guaranteedfor the closed-loop system (11) via the stabilizing control law119906119867

= minus119870119867sdot 119909 and the feed-forward fuzzy compensator

119906119891= minussum

119871

119894=1ℎ119894(119911) sdot 119888

119894 if there exist constant positive values

V 120588 positive-definite matrix 119875 and matrix 119870119867 such that the

following linear matrix inequality is satisfied

Φ equiv [120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (16)

4 Mathematical Problems in Engineering

where

119882 equiv 119875minus1

119884 equiv 119870119867sdot 119882 (17)

120601 equiv (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868 + V

2

sdot 119863 sdot 119863119879

(18)

The proof of Theorem 1 requires the following lemma

Lemma 2 (see [31 39]) Given constant matrices119863 and 119864 anda symmetric constant matrix 119878 of appropriate dimensions thefollowing inequality holds

119878 + 119863 sdot 119865 (119905) sdot 119864 + 119864119879

sdot 119865119879

(119905) sdot 119863119879

lt 0 (19)

if and only if for some V gt 0

119878 + [Vminus1 sdot 119864119879 V sdot 119863] sdot [119877 0

0 119868] sdot [

Vminus1 sdot 119864

V sdot 119863119879] lt 0 (20)

where 119865(119905) satisfies 119865(119905)119879 sdot 119865(119905) le 119877

Proof of Theorem 1 Considering a Lyapunov function candi-date composed of the Lyapunov function

119881 (119905) = 119909119879

(119905) sdot 119875 sdot 119909 (119905) (21)

its time derivative can be obtained as

(119909 (119905)) = 119879

sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot

= 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) sdot 119909

+ 119909119879

sdot [119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867)]119879

sdot 119875

+ 119875 sdot 119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867) sdot 119909

+ 119890119879

mod sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot 119890 mod

(22)

By Lemma 2 we have

le 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875] sdot 119909 + 120588

2

sdot 119890119879

mod sdot 119890 mod

= minus119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod

(23)

where

119876 equiv minus (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875]

(24)

According to (16) and (24) we have

119882119879

sdot 119876 sdot 119882 = minus (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868

+ [Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

V sdot 119863]

sdot [

Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)

V sdot 119863119879]

(25)

From (18) and (25) we have

120601 + Vminus2

sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

sdot (1198641sdot 119882 minus 119864

2sdot 119884) lt 0 (26)

Equation (26) can be represented in the standard LMI form

[120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (27)

If (16) holds then 119876 gt 0 Equation (23) can be rewritten as

le minus 119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod le minus120582min (119876) sdot 1199092

+ 1205882

sdot1003817100381710038171003817119890 mod

1003817100381710038171003817

2

le minus120582min (119876) sdot 1199092

+ 1205882

sdot 1198902

119880

(28)

where the property 119890 mod le 119890119880 is appliedWhenever 119909 gt (120588 sdot 119890

119880)radic120582min(119876) we have that lt 0

It is clear that if (16) is satisfied then the system (11) is UUBstable This completes the proof

32 Design of the Inner-Level Tracking Controller Once theouter-level stabilization controller 119906

119867= minus119870

119867sdot 119909 has been

designed we are able to put the system undergoing safe trialsTaking tracking performance together with control effort intoconsideration the overall performance index 119869 is defined asa weighted sum of the indices

119869 =

1

119905119891

sdot int

119905119891

0

119906 (119905) sdot 119889119905 +

1205961

119905119891

sdot int

119905119891

0

119890 (119905) sdot 119889119905 (29)

where 1205961is a weighting factor which is defined according

to practical trade-offs between desired tracking performanceand physical constrains

The inner-level controller 119906119878

= 119870119878sdot 119890 is designed

by searching for the gain matrix 119870119878such that the overall

performance index 119869 is minimized We propose to use theNelder-Mead simplexmethod [34] to guide theminimizationprocedure in this paper The method deals with nonlin-ear optimization problems without derivative informationwhich normally requires fewer steps to find a solution closeto global optimum when proper initial values are givenin comparison with the more powerful DIRECT (DIvidingRECTangle) algorithm or evolutionary computation tech-niques

The Nelder-Mead simplex method uses the concept ofa simplex which has 119873 + 1 vertices in 119873 dimensions foran optimization problem with119873 design parameters In each

Mathematical Problems in Engineering 5

step of the algorithm one of the four possible operations isconducted reflection expansion contraction and shrink Asthe method is sensitive to initial guess for an119873-dimensionalproblem we may start the algorithm with 119873 + 1 simplexeswith (119873 + 1)

2 randomly generated parameter sets for thevertices and after several steps collect the 119873 + 1 best solu-tions of the simplexes to form a simplex for final convergenceWith this strategy we have more initial guesses to avoidbeing trapped at local minimum Details are presented in thesubsequent case study

4 Case Study

In order to verify performance of the proposed controlscheme case studies of simulations and experiments areconducted In the simulations a comparison with the adap-tive fuzzy control method (AFCM) of [40] is made Inexperimental studies a two-dimensional prototype cranesystem is used

41 Simulation Study The crane system under control iscomposed of a motor-driven cart running along a horizontalrail a payload and a string carrying the payload which isattached to a joint on the cart We assume that the cart andthe load can move only in the vertical plane In the followingstudy the cart is of mass119872 = 678 kg the payload is of mass119898 = 15 kg and the string is of length 119897 = 05m Furthermore1199091is the cart position 120579 is the swing angle 119906 is the control

signal applied to the cart and 119909119903= [1 0 0 0]

119879 is the referenceinput The position of payload 119910 can be calculated from therelation119910 = 119909

1+119897sdotsin(120579) Besides we assume that the viscous

friction coefficient between the cart and the rail is1198631 and the

wind resistance coefficient between the air and the string is1198632Lagrange analysis of the simplified two-dimensional

crane system gives the dynamic equation

1= (119906 + 119898 sdot 119897 sdot

1205792

sdot sin (120579) + 119898 sdot 119892 sdot sin (120579) sdot cos (120579)

minus1198631sdot 1+ 1198632sdot120579 sdot cos (120579) ) (119872 + 119898 minus 119898 sdot cos2 (120579))

minus1

120579 = ((119898 sdot cos (120579) sdot 119906 + 1198982 sdot 119897 sdot

1205792

sdot sin (120579) sdot cos (120579)

+ (119872 + 119898) sdot 119898 sdot 119892 sdot sin (120579))

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

)

+ ( (minus1198631sdot 1sdot 119898 sdot cos (120579) + 119863

2sdot 119897

sdot [(119872 + 119898) minus 119898 sdot cos2 (120579)] sdot 120579)

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

) + 1199084

(30)

where119892 is the gravitational acceleration and1199084represents the

external disturbance

(1) Controller Design of the Proposed Control Strategy From(3) the overall fuzzymodel of the overhead crane system (30)is inferred to be

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

2

sum

119894=1

ℎ119894(119911) sdot 119888119894]

(31)

where 119909 = [1199091 1199092 1199093 1199094]119879

= [1199091 1 120579

120579]119879 is the state

vector And the matrices are

119860 =

[

[

[

[

[

[

[

0 1 0 0

0 minus

1198631

119872

119898 sdot 119892

119872

1198632

119872

0 0 0 1

0

1198631

(119897 sdot 119872)

minus (119872 + 119898) sdot 119892

119897 sdot 119872

(119872 + 119898) sdot 1198632

1198982sdot 119897 minus 119898 sdot 119897 sdot (119872 + 119898)

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

0

1

119872

0

minus

1

(119897 sdot 119872)

]

]

]

]

]

]

]

]

]

]

]

]

(32)

with 1198631= 588 119863

2= 001 119892 = 981 119911 = 119898 sdot 119897 sdot sin(119909

3) sdot 1199092

4

ℎ1= 05(1 + 119911) ℎ

2= 05(1 minus 119911) 119888

1= 1 and 119888

2= minus1 And

[ Δ119860(119905) Δ119861(119905) ] = 119863 sdot 119865(119905) sdot [ 11986411198642] where 119865(119905) = sin(119905)

119863 = [0 minus001 0 001]119879 1198641= [2 0 0 0] and 119864

2= 002

By selecting V = 3 and 120588 = 18 we are able to obtain

119875 =

[

[

[

[

903667 188347 130680 90783

188347 145588 03778 71938

130680 03778 514426 06680

90783 71938 06680 35618

]

]

]

]

(33)

and119870119867= [1250 3157 minus17665 1295] using the standard

LMI techniques The optimal servo control gains are foundto be 119870

119878= [114047 23047 6997 31522] by the simplex

method

(2) Controller Design of [40] For comparison purpose theadaptive fuzzy controller of [40] abbreviated as AFCMis implemented Design parameters of the AFCM includemembership functions of the antecedents in the fuzzy rulesvalues of the consequent forces and the fuzzy rule mapDetailed values obtained by the procedures described in [40]are shown in Figure 2

In the fuzzy rules each of the universe of discourseof the variables is divided into 6 linguistic values asNBNSZOPSPMPB which represent Negative Big

6 Mathematical Problems in Engineering

0 02 04 06 08 10

02

04

06

08

1M

embe

rshi

p gr

ades

Position error (m)

NSZOPS

PMPB

minus04 minus02

(a)

0 5 10 150

02

04

06

08

1

Mem

bers

hip

degr

ee

Swing angle (deg)

NSZOPS

PMPB

minus15 minus10 minus5

(b)

0 200 400 6000

02

04

06

08

1

12

Force (N)

Mem

bers

hip

degr

ee

NSZOPS

PMPB

minus600 minus400 minus200

(c)

Force Position error

PB PM PS ZO NS

PB PB PB PB NB NB

PS PB PS PS ZO PB

ZO PB PS PB ZO NB

NS PS PB NB NB NB

NB PS PB PB NS NB

Swin

g an

gle

(d)

Figure 2 Linguistic termsmembership functions and rule table of the fuzzy control rules for AFCM (a)Definition ofmembership functionsof position error (b) definition of membership functions of swing angle (c) consequent part membership function of control input 119906(119905) and(d) fuzzy rule map

Negative Small Zero Positive Small and Positive Big respec-tively

(3) Performance Comparison In order to compare relativeperformance of the two approaches a significant disturbanceof 119908 = [0 0 0 119908

4]119879 with

1199084=

120587

3

45 le 119905 le 65

0 otherwise(34)

is applied to the crane modelFrom the time history of the payload position of these

two approaches shown in Figure 3(a) it is clear that both cansuccessfully demonstrate stable tracking during 0 le 119905 lt 45However while the proposed approach remains stable andexhibits accurate tracking after 119905 ge 45 the controller ofAFCMcannot effectively compensate the applied disturbance1199084 shown in Figure 3(b) and eventually goes unstable Note

also that the control signal 119906(119905) generated by the proposedcontroller is much smoother and less violent than thatof AFCM further justifying it as a more efficient controlstrategy

42 Experimental Study Aprototype crane system shown inFigure 4(a) is built to test the proposed control strategy Asshown in the pictures of Figures 4(b) and 4(c) an encoderwith resolution of 2000 pulserev is installed in the hangingjoint to measure the swing angle 120579 To investigate robustnessof the control system the string length can vary between 05to 06m and the payload weight has three choices 05311041 and 1484 kg

The system is firstly identified using the parallel geneticalgorithms [41] as T-S type fuzzy combination of the follow-ing two rules

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

(LMI) relations However these control design strategies relyon accurate fuzzymodeling of the plant which usually resultsin a large number of fuzzy rules and hence complex andconservative designs

To further alleviate the requirement for accurate fuzzymodeling of the plant a two-level 119867

infinrobust nonlinear

control scheme is proposed The inner-level controller isresponsible for accurate servo control while the outer-levelcontroller compensates for unmodeled system dynamics andbounded disturbances Besides each part of the proposedcontrol laws can be independently designed satisfying itsown specification This incremental design procedure avoidssolving the problem at one time and allows each part to bedesignedwith different guidelines Also global stability of theclosed-loop system is ensured against bounded disturbanceswith guaranteed disturbance attenuation level

A particular switching controller is proposed in [32] fornonlinear systems with unknown parameters based on afuzzy logic approach The major difference between our pro-posed scheme and the controller of [32] is that the switchingof our scheme is between the inner-loop and the outer-loopcontrollers while the controller of [32] is switched constantlybetween many (which is 8 in the simulation example) linearcontrollers Furthermore the fuzzy terms in our controllerare dedicated for the compensation of highly nonlinear effectsthat deviate from the nominal linear dynamics Neverthelessin [32] a fuzzy plantmodel is required for the construction ofthe switching plant model which is then used for the model-based design of the switching controller The switchingTakagi-Sugeno fuzzy control proposed in [33] also requiresthe plant to be accurately represented by a fuzzy system

As the closed-loop stability is ensured by the outer-levelcontroller we are able to optimize the inner-level controllerby the Nelder-Mead simplex algorithm [34] based on actualclosed-loop control performance rather than deriving fromthe plant modelThe optimization algorithm converges fasterthan particle swarm optimization (PSO) [35] which is ade-quate for online applications This scheme which incorpo-rates online trials can be applied to many applications suchas self-guided robot and evolvable systems Furthermoreconsidering that the swing dynamics depend on both stringlength and load mass fuzzy rules are created to interpolatecontrol gains obtained from trial experiments [36ndash38]

In the following sections this paper is divided intofour parts Section 2 describes the plant model and theproblem Section 3 proposes the two-level control schemeand Section 4 evaluates the effectiveness of the proposedscheme using both simulation comparison with a recentlyproposed strategy in the literature and experimental studiesFinally Section 5 concludes the results

2 Problem Formulation

The plant under consideration is assumed to be a disturbednonlinear system which is affine in the input and containsuncertain dynamics

= 119891 (119909) + Δ119891 (119909 119905) + [119892 (119909) + Δ119892 (119905)] sdot 119906 + 119908 (1)

where Δ119891(119909 119905) and Δ119892(119905) are unknown system dynamicswhich are bounded in 119909 and 119905 119909 = [119909

1 1199092 119909

119899]119879

isin 119877119899times1 is

the state vector 119906 = [1199061 1199062 119906

119898]119879

isin 119877119898times1 is the nonlinear

input vector and 119908 isin 119877119899times1 denotes unknown and bounded

disturbance Furthermore nonlinear functions119891(119909) and119892(119909)are Lipschitz in 119909

Next we approximate the nonlinear system as a nominallinear system augmented with Takagi-Sugeno type fuzzyblending of affine terms Note that these affine terms whichare usually dominated by friction in many mechatronicsystems are added to the control-input term rather thanbeing added directly This form closely reflects the practicaleffects of friction on system dynamics Specifically the 119894thrule of the affine T-S fuzzy model is in the following form

Plant rule 119894IF 1199111(119905) is119872

1198941and sdot sdot sdot and 119911

119901(119905) is119872

119894119901

THEN = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 119888119894]

for 119894 = 1 2 119871(2)

In each rule 1199111(119905) 1199112(119905) and 119911

119901(119905) are the 119901 premise

variables which can be state variables or functions of statevariables 119872

119894119895is the fuzzy set corresponding to the 119895th

premise variable 119860 isin 119877119899times119899 is the system matrix and 119861 isin

119877119899times119898 denotes the control input matrix Moreover 119888

119894isin 119877119898times1

is the 119894th bias vector Δ119860(119905) isin 119877119899times119899 is the system uncertaintyand Δ119861(119905) isin 119877119899times119898 denotes the control input uncertainty

Defining 120583119894119895(sdot) as the membership function correspond-

ing to fuzzy set 119872119894119895 we have that 120583

119894119895(119911119895(119905)) is the grade

of membership of 119911119895(119905) in 119872

119894119895 Using the sum-product

composition the firing strength of the 119894th fuzzy rule isrepresented as 120603

119894= 120603119894(119911) equiv prod

119901

119895=1120583119894119895(119911119895(119905)) with 119911 equiv

[1199111(119905) 1199111(119905) 119911

119901(119905)]119879

By defining ℎ119894(119911) = 120603

119894sum119871

119895=1120603119895as the normalized firing

strength of the 119894th rule hencesum119871119894=1ℎ119894(119911) = 1 the overall fuzzy

system model is then inferred as the weighted average of theconsequent parts

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894] (3)

The proposed control scheme is of a two-level switchingstructure where the control input is composed of three parts119906119878 119906119867 and 119906

119891 defined as follows

119906 = (1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867+ 119906119891

= (1 minus 119868lowast

) sdot 119870119878sdot 119890 minus 119868

lowast

sdot 119870119867sdot 119909 minus

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894

(4)

where 119868lowast isin 0 1 is a switching function to be definedin Section 3 In (4) the first term 119906

119878= 119870119878sdot 119890 is a servo

controller located in the inner loop responsible for accuratetracking where 119890 = 119909

119903minus 119909 is the tracking error with 119909

119903

denoting the reference state trajectory The second term

Mathematical Problems in Engineering 3

119906119867= minus119870119867sdot 119909 is an119867

infinrobust controller in the outer loop to

ensure system stability And 119906119891= minussum

119871

119894=1ℎ119894(119911) sdot 119888

119894is a fuzzy-

combination term that compensates for nonlinear dynamicssuch as friction and other effects that deviate from nominallinear dynamics

Next let us define the modeling error 119890 mod as

119890 mod equiv 119891 (119909) + Δ119891 (119909 119905) + [119892 (119909) + Δ119892 (119905)] sdot 119906 + 119908 minus y (5)

where 119910 = 119860 sdot 119909 +Δ119860(119905) sdot 119909 + [119861 +Δ119861(119905)] sdot [119906 +sum119871119894=1ℎ119894(119911) sdot 119888119894]

Hence the closed-loop system formed by applying (4) to (1)can be expressed concisely as follows

= 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)]

sdot [(1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867+ 119906119891+

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894] + 119890 mod

= 119860 sdot 119909 + Δ119860(119905)sdot119909 + [119861 + Δ119861 (119905)]sdot[(1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867]

+ 119890 mod

(6)

3 The Proposed Two-Level Control Scheme

As shown in Figure 1 the overall control scheme is composedof an outer-level stabilizing controller and an inner-levelservo controller Each of the controllers is designed accordingto a switching condition defined by the deviation of trackingerrors from a prescribed reference vector 119909

119903(119905) That is

If 119890 = 1003817100381710038171003817119909119903minus 119909

1003817100381710038171003817le 120576119864

then 119868lowast = 0 otherwise 119868lowast = 1(7)

In the condition the threshold 120576119864is a user-defined positive

numberThe value of it for instance may be designed as 01timesmax119905(119909119903(119905))

The closed-loop system dynamics when 119890 gt 120576119864is

formed by assigning 119868lowast = 1 in (6) as follows

= [119860 + Δ119860 (119905)] sdot 119909 minus [119861 + Δ119861 (119905)] sdot 119870119867sdot 119909 + 119890 mod (8)

If uncertainties in the plant dynamic matrices Δ119860(119905) andΔ119861(119905) are bounded wemay introduce a time-varyingmatrix119865(119905) with 0 le 119865(119905) le 1 and constant matrices 119863 119864

1 and

1198642 such that

[Δ119860 (119905) Δ119861 (119905)119870119867] sdot 119909 = 119863 sdot 119865 (119905) sdot [119864

11198642119870119867] sdot 119909

+ [120575 (119905) 0]

(9)

with 120575(119905) being a bounded function in 119909

120575 (119905) le 119886 sdot 119909 where 119886 is a positive constant (10)

Using (9) the closed-loop system dynamics (8) can then bewritten as

= (119860 minus 119861 sdot 119870119867) sdot 119909 + 119863 sdot 119865 (119905) sdot (119864

1minus 1198642sdot 119870119867) sdot 119909 + 119890 mod

(11)

where 119890 mod = 119890 mod + 120575(119905)

Plant

Fuzzycompensator

++ +

minus

119909119903119906119878119870119878

119906119867

119870119867

119906119891

119909119890

119868lowast

Figure 1 The proposed two-level switching control scheme

When the system is under acceptable tracking that is119890 le 120576

119864 only the servo controller is in charge The closed-

loop system dynamics is then formed by assigning 119868lowast = 0 in(6) as follows

= [119860 + Δ119860 (119905)] sdot 119909 + [119861 + Δ119861 (119905)] sdot 119870119878sdot 119890 + 119890 mod (12)

31 Design of the Outer-Level 119867infin

Stabilization ControllerThe119867

infinstabilization performance of 119906

119867is defined as follows

int

119905119891

0

[119909(119905)119879

sdot 119876 sdot 119909 (119905)] sdot 119889119905

119864 modle 1205882

(13)

where

119864 mod = int

119905119891

0

119890119879

mod sdot 119890 mod sdot 119889119905 (14)

119905119891is terminal time of control 119876 is a positive definite weight-

ing matrix and 120588 denotes prescribed attenuation level with1205882 being the attenuation disturbance level From the energy

viewpoint (13) confines the effect of 119890 mod on state 119909(119905) to beattenuated below a desired level If initial conditions are alsoconsidered the 119867

infinperformance in (13) can be modified as

follows

int

119905119891

0

(119909119879

sdot 119876 sdot 119909) 119889119905 le 119909119879

(0) sdot 119875 sdot 119909 (0) + 1205882

sdot 119864 mod (15)

where119876 and 119875 are symmetric and positive definite weightingmatrices The design of the stabilizing controller in the outerlevel corresponds to find a linear controller in the form of119906119867

= minus119870119867sdot 119909 such that the 119867

infinperformance (15) is

guaranteed to stabilize the closed-loop system (11)

Theorem 1 Assuming that the modeling error is boundedsuch that 119890 mod le 119890

119880 with 119890

119880being a positive constant

the 119867infin

control performance defined in (15) is guaranteedfor the closed-loop system (11) via the stabilizing control law119906119867

= minus119870119867sdot 119909 and the feed-forward fuzzy compensator

119906119891= minussum

119871

119894=1ℎ119894(119911) sdot 119888

119894 if there exist constant positive values

V 120588 positive-definite matrix 119875 and matrix 119870119867 such that the

following linear matrix inequality is satisfied

Φ equiv [120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (16)

4 Mathematical Problems in Engineering

where

119882 equiv 119875minus1

119884 equiv 119870119867sdot 119882 (17)

120601 equiv (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868 + V

2

sdot 119863 sdot 119863119879

(18)

The proof of Theorem 1 requires the following lemma

Lemma 2 (see [31 39]) Given constant matrices119863 and 119864 anda symmetric constant matrix 119878 of appropriate dimensions thefollowing inequality holds

119878 + 119863 sdot 119865 (119905) sdot 119864 + 119864119879

sdot 119865119879

(119905) sdot 119863119879

lt 0 (19)

if and only if for some V gt 0

119878 + [Vminus1 sdot 119864119879 V sdot 119863] sdot [119877 0

0 119868] sdot [

Vminus1 sdot 119864

V sdot 119863119879] lt 0 (20)

where 119865(119905) satisfies 119865(119905)119879 sdot 119865(119905) le 119877

Proof of Theorem 1 Considering a Lyapunov function candi-date composed of the Lyapunov function

119881 (119905) = 119909119879

(119905) sdot 119875 sdot 119909 (119905) (21)

its time derivative can be obtained as

(119909 (119905)) = 119879

sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot

= 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) sdot 119909

+ 119909119879

sdot [119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867)]119879

sdot 119875

+ 119875 sdot 119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867) sdot 119909

+ 119890119879

mod sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot 119890 mod

(22)

By Lemma 2 we have

le 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875] sdot 119909 + 120588

2

sdot 119890119879

mod sdot 119890 mod

= minus119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod

(23)

where

119876 equiv minus (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875]

(24)

According to (16) and (24) we have

119882119879

sdot 119876 sdot 119882 = minus (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868

+ [Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

V sdot 119863]

sdot [

Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)

V sdot 119863119879]

(25)

From (18) and (25) we have

120601 + Vminus2

sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

sdot (1198641sdot 119882 minus 119864

2sdot 119884) lt 0 (26)

Equation (26) can be represented in the standard LMI form

[120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (27)

If (16) holds then 119876 gt 0 Equation (23) can be rewritten as

le minus 119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod le minus120582min (119876) sdot 1199092

+ 1205882

sdot1003817100381710038171003817119890 mod

1003817100381710038171003817

2

le minus120582min (119876) sdot 1199092

+ 1205882

sdot 1198902

119880

(28)

where the property 119890 mod le 119890119880 is appliedWhenever 119909 gt (120588 sdot 119890

119880)radic120582min(119876) we have that lt 0

It is clear that if (16) is satisfied then the system (11) is UUBstable This completes the proof

32 Design of the Inner-Level Tracking Controller Once theouter-level stabilization controller 119906

119867= minus119870

119867sdot 119909 has been

designed we are able to put the system undergoing safe trialsTaking tracking performance together with control effort intoconsideration the overall performance index 119869 is defined asa weighted sum of the indices

119869 =

1

119905119891

sdot int

119905119891

0

119906 (119905) sdot 119889119905 +

1205961

119905119891

sdot int

119905119891

0

119890 (119905) sdot 119889119905 (29)

where 1205961is a weighting factor which is defined according

to practical trade-offs between desired tracking performanceand physical constrains

The inner-level controller 119906119878

= 119870119878sdot 119890 is designed

by searching for the gain matrix 119870119878such that the overall

performance index 119869 is minimized We propose to use theNelder-Mead simplexmethod [34] to guide theminimizationprocedure in this paper The method deals with nonlin-ear optimization problems without derivative informationwhich normally requires fewer steps to find a solution closeto global optimum when proper initial values are givenin comparison with the more powerful DIRECT (DIvidingRECTangle) algorithm or evolutionary computation tech-niques

The Nelder-Mead simplex method uses the concept ofa simplex which has 119873 + 1 vertices in 119873 dimensions foran optimization problem with119873 design parameters In each

Mathematical Problems in Engineering 5

step of the algorithm one of the four possible operations isconducted reflection expansion contraction and shrink Asthe method is sensitive to initial guess for an119873-dimensionalproblem we may start the algorithm with 119873 + 1 simplexeswith (119873 + 1)

2 randomly generated parameter sets for thevertices and after several steps collect the 119873 + 1 best solu-tions of the simplexes to form a simplex for final convergenceWith this strategy we have more initial guesses to avoidbeing trapped at local minimum Details are presented in thesubsequent case study

4 Case Study

In order to verify performance of the proposed controlscheme case studies of simulations and experiments areconducted In the simulations a comparison with the adap-tive fuzzy control method (AFCM) of [40] is made Inexperimental studies a two-dimensional prototype cranesystem is used

41 Simulation Study The crane system under control iscomposed of a motor-driven cart running along a horizontalrail a payload and a string carrying the payload which isattached to a joint on the cart We assume that the cart andthe load can move only in the vertical plane In the followingstudy the cart is of mass119872 = 678 kg the payload is of mass119898 = 15 kg and the string is of length 119897 = 05m Furthermore1199091is the cart position 120579 is the swing angle 119906 is the control

signal applied to the cart and 119909119903= [1 0 0 0]

119879 is the referenceinput The position of payload 119910 can be calculated from therelation119910 = 119909

1+119897sdotsin(120579) Besides we assume that the viscous

friction coefficient between the cart and the rail is1198631 and the

wind resistance coefficient between the air and the string is1198632Lagrange analysis of the simplified two-dimensional

crane system gives the dynamic equation

1= (119906 + 119898 sdot 119897 sdot

1205792

sdot sin (120579) + 119898 sdot 119892 sdot sin (120579) sdot cos (120579)

minus1198631sdot 1+ 1198632sdot120579 sdot cos (120579) ) (119872 + 119898 minus 119898 sdot cos2 (120579))

minus1

120579 = ((119898 sdot cos (120579) sdot 119906 + 1198982 sdot 119897 sdot

1205792

sdot sin (120579) sdot cos (120579)

+ (119872 + 119898) sdot 119898 sdot 119892 sdot sin (120579))

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

)

+ ( (minus1198631sdot 1sdot 119898 sdot cos (120579) + 119863

2sdot 119897

sdot [(119872 + 119898) minus 119898 sdot cos2 (120579)] sdot 120579)

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

) + 1199084

(30)

where119892 is the gravitational acceleration and1199084represents the

external disturbance

(1) Controller Design of the Proposed Control Strategy From(3) the overall fuzzymodel of the overhead crane system (30)is inferred to be

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

2

sum

119894=1

ℎ119894(119911) sdot 119888119894]

(31)

where 119909 = [1199091 1199092 1199093 1199094]119879

= [1199091 1 120579

120579]119879 is the state

vector And the matrices are

119860 =

[

[

[

[

[

[

[

0 1 0 0

0 minus

1198631

119872

119898 sdot 119892

119872

1198632

119872

0 0 0 1

0

1198631

(119897 sdot 119872)

minus (119872 + 119898) sdot 119892

119897 sdot 119872

(119872 + 119898) sdot 1198632

1198982sdot 119897 minus 119898 sdot 119897 sdot (119872 + 119898)

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

0

1

119872

0

minus

1

(119897 sdot 119872)

]

]

]

]

]

]

]

]

]

]

]

]

(32)

with 1198631= 588 119863

2= 001 119892 = 981 119911 = 119898 sdot 119897 sdot sin(119909

3) sdot 1199092

4

ℎ1= 05(1 + 119911) ℎ

2= 05(1 minus 119911) 119888

1= 1 and 119888

2= minus1 And

[ Δ119860(119905) Δ119861(119905) ] = 119863 sdot 119865(119905) sdot [ 11986411198642] where 119865(119905) = sin(119905)

119863 = [0 minus001 0 001]119879 1198641= [2 0 0 0] and 119864

2= 002

By selecting V = 3 and 120588 = 18 we are able to obtain

119875 =

[

[

[

[

903667 188347 130680 90783

188347 145588 03778 71938

130680 03778 514426 06680

90783 71938 06680 35618

]

]

]

]

(33)

and119870119867= [1250 3157 minus17665 1295] using the standard

LMI techniques The optimal servo control gains are foundto be 119870

119878= [114047 23047 6997 31522] by the simplex

method

(2) Controller Design of [40] For comparison purpose theadaptive fuzzy controller of [40] abbreviated as AFCMis implemented Design parameters of the AFCM includemembership functions of the antecedents in the fuzzy rulesvalues of the consequent forces and the fuzzy rule mapDetailed values obtained by the procedures described in [40]are shown in Figure 2

In the fuzzy rules each of the universe of discourseof the variables is divided into 6 linguistic values asNBNSZOPSPMPB which represent Negative Big

6 Mathematical Problems in Engineering

0 02 04 06 08 10

02

04

06

08

1M

embe

rshi

p gr

ades

Position error (m)

NSZOPS

PMPB

minus04 minus02

(a)

0 5 10 150

02

04

06

08

1

Mem

bers

hip

degr

ee

Swing angle (deg)

NSZOPS

PMPB

minus15 minus10 minus5

(b)

0 200 400 6000

02

04

06

08

1

12

Force (N)

Mem

bers

hip

degr

ee

NSZOPS

PMPB

minus600 minus400 minus200

(c)

Force Position error

PB PM PS ZO NS

PB PB PB PB NB NB

PS PB PS PS ZO PB

ZO PB PS PB ZO NB

NS PS PB NB NB NB

NB PS PB PB NS NB

Swin

g an

gle

(d)

Figure 2 Linguistic termsmembership functions and rule table of the fuzzy control rules for AFCM (a)Definition ofmembership functionsof position error (b) definition of membership functions of swing angle (c) consequent part membership function of control input 119906(119905) and(d) fuzzy rule map

Negative Small Zero Positive Small and Positive Big respec-tively

(3) Performance Comparison In order to compare relativeperformance of the two approaches a significant disturbanceof 119908 = [0 0 0 119908

4]119879 with

1199084=

120587

3

45 le 119905 le 65

0 otherwise(34)

is applied to the crane modelFrom the time history of the payload position of these

two approaches shown in Figure 3(a) it is clear that both cansuccessfully demonstrate stable tracking during 0 le 119905 lt 45However while the proposed approach remains stable andexhibits accurate tracking after 119905 ge 45 the controller ofAFCMcannot effectively compensate the applied disturbance1199084 shown in Figure 3(b) and eventually goes unstable Note

also that the control signal 119906(119905) generated by the proposedcontroller is much smoother and less violent than thatof AFCM further justifying it as a more efficient controlstrategy

42 Experimental Study Aprototype crane system shown inFigure 4(a) is built to test the proposed control strategy Asshown in the pictures of Figures 4(b) and 4(c) an encoderwith resolution of 2000 pulserev is installed in the hangingjoint to measure the swing angle 120579 To investigate robustnessof the control system the string length can vary between 05to 06m and the payload weight has three choices 05311041 and 1484 kg

The system is firstly identified using the parallel geneticalgorithms [41] as T-S type fuzzy combination of the follow-ing two rules

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119906119867= minus119870119867sdot 119909 is an119867

infinrobust controller in the outer loop to

ensure system stability And 119906119891= minussum

119871

119894=1ℎ119894(119911) sdot 119888

119894is a fuzzy-

combination term that compensates for nonlinear dynamicssuch as friction and other effects that deviate from nominallinear dynamics

Next let us define the modeling error 119890 mod as

119890 mod equiv 119891 (119909) + Δ119891 (119909 119905) + [119892 (119909) + Δ119892 (119905)] sdot 119906 + 119908 minus y (5)

where 119910 = 119860 sdot 119909 +Δ119860(119905) sdot 119909 + [119861 +Δ119861(119905)] sdot [119906 +sum119871119894=1ℎ119894(119911) sdot 119888119894]

Hence the closed-loop system formed by applying (4) to (1)can be expressed concisely as follows

= 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)]

sdot [(1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867+ 119906119891+

119871

sum

119894=1

ℎ119894(119911) sdot 119888119894] + 119890 mod

= 119860 sdot 119909 + Δ119860(119905)sdot119909 + [119861 + Δ119861 (119905)]sdot[(1 minus 119868lowast

) sdot 119906119878+ 119868lowast

sdot 119906119867]

+ 119890 mod

(6)

3 The Proposed Two-Level Control Scheme

As shown in Figure 1 the overall control scheme is composedof an outer-level stabilizing controller and an inner-levelservo controller Each of the controllers is designed accordingto a switching condition defined by the deviation of trackingerrors from a prescribed reference vector 119909

119903(119905) That is

If 119890 = 1003817100381710038171003817119909119903minus 119909

1003817100381710038171003817le 120576119864

then 119868lowast = 0 otherwise 119868lowast = 1(7)

In the condition the threshold 120576119864is a user-defined positive

numberThe value of it for instance may be designed as 01timesmax119905(119909119903(119905))

The closed-loop system dynamics when 119890 gt 120576119864is

formed by assigning 119868lowast = 1 in (6) as follows

= [119860 + Δ119860 (119905)] sdot 119909 minus [119861 + Δ119861 (119905)] sdot 119870119867sdot 119909 + 119890 mod (8)

If uncertainties in the plant dynamic matrices Δ119860(119905) andΔ119861(119905) are bounded wemay introduce a time-varyingmatrix119865(119905) with 0 le 119865(119905) le 1 and constant matrices 119863 119864

1 and

1198642 such that

[Δ119860 (119905) Δ119861 (119905)119870119867] sdot 119909 = 119863 sdot 119865 (119905) sdot [119864

11198642119870119867] sdot 119909

+ [120575 (119905) 0]

(9)

with 120575(119905) being a bounded function in 119909

120575 (119905) le 119886 sdot 119909 where 119886 is a positive constant (10)

Using (9) the closed-loop system dynamics (8) can then bewritten as

= (119860 minus 119861 sdot 119870119867) sdot 119909 + 119863 sdot 119865 (119905) sdot (119864

1minus 1198642sdot 119870119867) sdot 119909 + 119890 mod

(11)

where 119890 mod = 119890 mod + 120575(119905)

Plant

Fuzzycompensator

++ +

minus

119909119903119906119878119870119878

119906119867

119870119867

119906119891

119909119890

119868lowast

Figure 1 The proposed two-level switching control scheme

When the system is under acceptable tracking that is119890 le 120576

119864 only the servo controller is in charge The closed-

loop system dynamics is then formed by assigning 119868lowast = 0 in(6) as follows

= [119860 + Δ119860 (119905)] sdot 119909 + [119861 + Δ119861 (119905)] sdot 119870119878sdot 119890 + 119890 mod (12)

31 Design of the Outer-Level 119867infin

Stabilization ControllerThe119867

infinstabilization performance of 119906

119867is defined as follows

int

119905119891

0

[119909(119905)119879

sdot 119876 sdot 119909 (119905)] sdot 119889119905

119864 modle 1205882

(13)

where

119864 mod = int

119905119891

0

119890119879

mod sdot 119890 mod sdot 119889119905 (14)

119905119891is terminal time of control 119876 is a positive definite weight-

ing matrix and 120588 denotes prescribed attenuation level with1205882 being the attenuation disturbance level From the energy

viewpoint (13) confines the effect of 119890 mod on state 119909(119905) to beattenuated below a desired level If initial conditions are alsoconsidered the 119867

infinperformance in (13) can be modified as

follows

int

119905119891

0

(119909119879

sdot 119876 sdot 119909) 119889119905 le 119909119879

(0) sdot 119875 sdot 119909 (0) + 1205882

sdot 119864 mod (15)

where119876 and 119875 are symmetric and positive definite weightingmatrices The design of the stabilizing controller in the outerlevel corresponds to find a linear controller in the form of119906119867

= minus119870119867sdot 119909 such that the 119867

infinperformance (15) is

guaranteed to stabilize the closed-loop system (11)

Theorem 1 Assuming that the modeling error is boundedsuch that 119890 mod le 119890

119880 with 119890

119880being a positive constant

the 119867infin

control performance defined in (15) is guaranteedfor the closed-loop system (11) via the stabilizing control law119906119867

= minus119870119867sdot 119909 and the feed-forward fuzzy compensator

119906119891= minussum

119871

119894=1ℎ119894(119911) sdot 119888

119894 if there exist constant positive values

V 120588 positive-definite matrix 119875 and matrix 119870119867 such that the

following linear matrix inequality is satisfied

Φ equiv [120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (16)

4 Mathematical Problems in Engineering

where

119882 equiv 119875minus1

119884 equiv 119870119867sdot 119882 (17)

120601 equiv (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868 + V

2

sdot 119863 sdot 119863119879

(18)

The proof of Theorem 1 requires the following lemma

Lemma 2 (see [31 39]) Given constant matrices119863 and 119864 anda symmetric constant matrix 119878 of appropriate dimensions thefollowing inequality holds

119878 + 119863 sdot 119865 (119905) sdot 119864 + 119864119879

sdot 119865119879

(119905) sdot 119863119879

lt 0 (19)

if and only if for some V gt 0

119878 + [Vminus1 sdot 119864119879 V sdot 119863] sdot [119877 0

0 119868] sdot [

Vminus1 sdot 119864

V sdot 119863119879] lt 0 (20)

where 119865(119905) satisfies 119865(119905)119879 sdot 119865(119905) le 119877

Proof of Theorem 1 Considering a Lyapunov function candi-date composed of the Lyapunov function

119881 (119905) = 119909119879

(119905) sdot 119875 sdot 119909 (119905) (21)

its time derivative can be obtained as

(119909 (119905)) = 119879

sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot

= 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) sdot 119909

+ 119909119879

sdot [119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867)]119879

sdot 119875

+ 119875 sdot 119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867) sdot 119909

+ 119890119879

mod sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot 119890 mod

(22)

By Lemma 2 we have

le 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875] sdot 119909 + 120588

2

sdot 119890119879

mod sdot 119890 mod

= minus119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod

(23)

where

119876 equiv minus (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875]

(24)

According to (16) and (24) we have

119882119879

sdot 119876 sdot 119882 = minus (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868

+ [Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

V sdot 119863]

sdot [

Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)

V sdot 119863119879]

(25)

From (18) and (25) we have

120601 + Vminus2

sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

sdot (1198641sdot 119882 minus 119864

2sdot 119884) lt 0 (26)

Equation (26) can be represented in the standard LMI form

[120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (27)

If (16) holds then 119876 gt 0 Equation (23) can be rewritten as

le minus 119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod le minus120582min (119876) sdot 1199092

+ 1205882

sdot1003817100381710038171003817119890 mod

1003817100381710038171003817

2

le minus120582min (119876) sdot 1199092

+ 1205882

sdot 1198902

119880

(28)

where the property 119890 mod le 119890119880 is appliedWhenever 119909 gt (120588 sdot 119890

119880)radic120582min(119876) we have that lt 0

It is clear that if (16) is satisfied then the system (11) is UUBstable This completes the proof

32 Design of the Inner-Level Tracking Controller Once theouter-level stabilization controller 119906

119867= minus119870

119867sdot 119909 has been

designed we are able to put the system undergoing safe trialsTaking tracking performance together with control effort intoconsideration the overall performance index 119869 is defined asa weighted sum of the indices

119869 =

1

119905119891

sdot int

119905119891

0

119906 (119905) sdot 119889119905 +

1205961

119905119891

sdot int

119905119891

0

119890 (119905) sdot 119889119905 (29)

where 1205961is a weighting factor which is defined according

to practical trade-offs between desired tracking performanceand physical constrains

The inner-level controller 119906119878

= 119870119878sdot 119890 is designed

by searching for the gain matrix 119870119878such that the overall

performance index 119869 is minimized We propose to use theNelder-Mead simplexmethod [34] to guide theminimizationprocedure in this paper The method deals with nonlin-ear optimization problems without derivative informationwhich normally requires fewer steps to find a solution closeto global optimum when proper initial values are givenin comparison with the more powerful DIRECT (DIvidingRECTangle) algorithm or evolutionary computation tech-niques

The Nelder-Mead simplex method uses the concept ofa simplex which has 119873 + 1 vertices in 119873 dimensions foran optimization problem with119873 design parameters In each

Mathematical Problems in Engineering 5

step of the algorithm one of the four possible operations isconducted reflection expansion contraction and shrink Asthe method is sensitive to initial guess for an119873-dimensionalproblem we may start the algorithm with 119873 + 1 simplexeswith (119873 + 1)

2 randomly generated parameter sets for thevertices and after several steps collect the 119873 + 1 best solu-tions of the simplexes to form a simplex for final convergenceWith this strategy we have more initial guesses to avoidbeing trapped at local minimum Details are presented in thesubsequent case study

4 Case Study

In order to verify performance of the proposed controlscheme case studies of simulations and experiments areconducted In the simulations a comparison with the adap-tive fuzzy control method (AFCM) of [40] is made Inexperimental studies a two-dimensional prototype cranesystem is used

41 Simulation Study The crane system under control iscomposed of a motor-driven cart running along a horizontalrail a payload and a string carrying the payload which isattached to a joint on the cart We assume that the cart andthe load can move only in the vertical plane In the followingstudy the cart is of mass119872 = 678 kg the payload is of mass119898 = 15 kg and the string is of length 119897 = 05m Furthermore1199091is the cart position 120579 is the swing angle 119906 is the control

signal applied to the cart and 119909119903= [1 0 0 0]

119879 is the referenceinput The position of payload 119910 can be calculated from therelation119910 = 119909

1+119897sdotsin(120579) Besides we assume that the viscous

friction coefficient between the cart and the rail is1198631 and the

wind resistance coefficient between the air and the string is1198632Lagrange analysis of the simplified two-dimensional

crane system gives the dynamic equation

1= (119906 + 119898 sdot 119897 sdot

1205792

sdot sin (120579) + 119898 sdot 119892 sdot sin (120579) sdot cos (120579)

minus1198631sdot 1+ 1198632sdot120579 sdot cos (120579) ) (119872 + 119898 minus 119898 sdot cos2 (120579))

minus1

120579 = ((119898 sdot cos (120579) sdot 119906 + 1198982 sdot 119897 sdot

1205792

sdot sin (120579) sdot cos (120579)

+ (119872 + 119898) sdot 119898 sdot 119892 sdot sin (120579))

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

)

+ ( (minus1198631sdot 1sdot 119898 sdot cos (120579) + 119863

2sdot 119897

sdot [(119872 + 119898) minus 119898 sdot cos2 (120579)] sdot 120579)

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

) + 1199084

(30)

where119892 is the gravitational acceleration and1199084represents the

external disturbance

(1) Controller Design of the Proposed Control Strategy From(3) the overall fuzzymodel of the overhead crane system (30)is inferred to be

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

2

sum

119894=1

ℎ119894(119911) sdot 119888119894]

(31)

where 119909 = [1199091 1199092 1199093 1199094]119879

= [1199091 1 120579

120579]119879 is the state

vector And the matrices are

119860 =

[

[

[

[

[

[

[

0 1 0 0

0 minus

1198631

119872

119898 sdot 119892

119872

1198632

119872

0 0 0 1

0

1198631

(119897 sdot 119872)

minus (119872 + 119898) sdot 119892

119897 sdot 119872

(119872 + 119898) sdot 1198632

1198982sdot 119897 minus 119898 sdot 119897 sdot (119872 + 119898)

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

0

1

119872

0

minus

1

(119897 sdot 119872)

]

]

]

]

]

]

]

]

]

]

]

]

(32)

with 1198631= 588 119863

2= 001 119892 = 981 119911 = 119898 sdot 119897 sdot sin(119909

3) sdot 1199092

4

ℎ1= 05(1 + 119911) ℎ

2= 05(1 minus 119911) 119888

1= 1 and 119888

2= minus1 And

[ Δ119860(119905) Δ119861(119905) ] = 119863 sdot 119865(119905) sdot [ 11986411198642] where 119865(119905) = sin(119905)

119863 = [0 minus001 0 001]119879 1198641= [2 0 0 0] and 119864

2= 002

By selecting V = 3 and 120588 = 18 we are able to obtain

119875 =

[

[

[

[

903667 188347 130680 90783

188347 145588 03778 71938

130680 03778 514426 06680

90783 71938 06680 35618

]

]

]

]

(33)

and119870119867= [1250 3157 minus17665 1295] using the standard

LMI techniques The optimal servo control gains are foundto be 119870

119878= [114047 23047 6997 31522] by the simplex

method

(2) Controller Design of [40] For comparison purpose theadaptive fuzzy controller of [40] abbreviated as AFCMis implemented Design parameters of the AFCM includemembership functions of the antecedents in the fuzzy rulesvalues of the consequent forces and the fuzzy rule mapDetailed values obtained by the procedures described in [40]are shown in Figure 2

In the fuzzy rules each of the universe of discourseof the variables is divided into 6 linguistic values asNBNSZOPSPMPB which represent Negative Big

6 Mathematical Problems in Engineering

0 02 04 06 08 10

02

04

06

08

1M

embe

rshi

p gr

ades

Position error (m)

NSZOPS

PMPB

minus04 minus02

(a)

0 5 10 150

02

04

06

08

1

Mem

bers

hip

degr

ee

Swing angle (deg)

NSZOPS

PMPB

minus15 minus10 minus5

(b)

0 200 400 6000

02

04

06

08

1

12

Force (N)

Mem

bers

hip

degr

ee

NSZOPS

PMPB

minus600 minus400 minus200

(c)

Force Position error

PB PM PS ZO NS

PB PB PB PB NB NB

PS PB PS PS ZO PB

ZO PB PS PB ZO NB

NS PS PB NB NB NB

NB PS PB PB NS NB

Swin

g an

gle

(d)

Figure 2 Linguistic termsmembership functions and rule table of the fuzzy control rules for AFCM (a)Definition ofmembership functionsof position error (b) definition of membership functions of swing angle (c) consequent part membership function of control input 119906(119905) and(d) fuzzy rule map

Negative Small Zero Positive Small and Positive Big respec-tively

(3) Performance Comparison In order to compare relativeperformance of the two approaches a significant disturbanceof 119908 = [0 0 0 119908

4]119879 with

1199084=

120587

3

45 le 119905 le 65

0 otherwise(34)

is applied to the crane modelFrom the time history of the payload position of these

two approaches shown in Figure 3(a) it is clear that both cansuccessfully demonstrate stable tracking during 0 le 119905 lt 45However while the proposed approach remains stable andexhibits accurate tracking after 119905 ge 45 the controller ofAFCMcannot effectively compensate the applied disturbance1199084 shown in Figure 3(b) and eventually goes unstable Note

also that the control signal 119906(119905) generated by the proposedcontroller is much smoother and less violent than thatof AFCM further justifying it as a more efficient controlstrategy

42 Experimental Study Aprototype crane system shown inFigure 4(a) is built to test the proposed control strategy Asshown in the pictures of Figures 4(b) and 4(c) an encoderwith resolution of 2000 pulserev is installed in the hangingjoint to measure the swing angle 120579 To investigate robustnessof the control system the string length can vary between 05to 06m and the payload weight has three choices 05311041 and 1484 kg

The system is firstly identified using the parallel geneticalgorithms [41] as T-S type fuzzy combination of the follow-ing two rules

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where

119882 equiv 119875minus1

119884 equiv 119870119867sdot 119882 (17)

120601 equiv (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868 + V

2

sdot 119863 sdot 119863119879

(18)

The proof of Theorem 1 requires the following lemma

Lemma 2 (see [31 39]) Given constant matrices119863 and 119864 anda symmetric constant matrix 119878 of appropriate dimensions thefollowing inequality holds

119878 + 119863 sdot 119865 (119905) sdot 119864 + 119864119879

sdot 119865119879

(119905) sdot 119863119879

lt 0 (19)

if and only if for some V gt 0

119878 + [Vminus1 sdot 119864119879 V sdot 119863] sdot [119877 0

0 119868] sdot [

Vminus1 sdot 119864

V sdot 119863119879] lt 0 (20)

where 119865(119905) satisfies 119865(119905)119879 sdot 119865(119905) le 119877

Proof of Theorem 1 Considering a Lyapunov function candi-date composed of the Lyapunov function

119881 (119905) = 119909119879

(119905) sdot 119875 sdot 119909 (119905) (21)

its time derivative can be obtained as

(119909 (119905)) = 119879

sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot

= 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) sdot 119909

+ 119909119879

sdot [119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867)]119879

sdot 119875

+ 119875 sdot 119863 sdot 119865 (119905) sdot (1198641minus 1198642sdot 119870119867) sdot 119909

+ 119890119879

mod sdot 119875 sdot 119909 + 119909119879

sdot 119875 sdot 119890 mod

(22)

By Lemma 2 we have

le 119909119879

sdot (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875] sdot 119909 + 120588

2

sdot 119890119879

mod sdot 119890 mod

= minus119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod

(23)

where

119876 equiv minus (119860 minus 119861 sdot 119870119867)119879

sdot 119875 + 119875 sdot (119860 minus 119861 sdot 119870119867) +

1

1205882sdot 119875119879

sdot 119875

+ [Vminus1 sdot (1198641minus 1198642sdot 119870119867)119879

V sdot 119875 sdot 119863]

sdot [

Vminus1 sdot (1198641minus 1198642sdot 119870119867)

V sdot 119863119879 sdot 119875]

(24)

According to (16) and (24) we have

119882119879

sdot 119876 sdot 119882 = minus (119860 sdot 119882 minus 119861 sdot 119884)119879

+ 119860 sdot 119882 minus 119861 sdot 119884 +

1

1205882sdot 119868

+ [Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

V sdot 119863]

sdot [

Vminus1 sdot (1198641sdot 119882 minus 119864

2sdot 119884)

V sdot 119863119879]

(25)

From (18) and (25) we have

120601 + Vminus2

sdot (1198641sdot 119882 minus 119864

2sdot 119884)119879

sdot (1198641sdot 119882 minus 119864

2sdot 119884) lt 0 (26)

Equation (26) can be represented in the standard LMI form

[120601 (119864

1sdot 119882 minus 119864

2sdot 119884)119879

(1198641sdot 119882 minus 119864

2sdot 119884) minusV2 sdot 119868

] lt 0 (27)

If (16) holds then 119876 gt 0 Equation (23) can be rewritten as

le minus 119909119879

sdot 119876 sdot 119909 + 1205882

sdot 119890119879

mod sdot 119890 mod le minus120582min (119876) sdot 1199092

+ 1205882

sdot1003817100381710038171003817119890 mod

1003817100381710038171003817

2

le minus120582min (119876) sdot 1199092

+ 1205882

sdot 1198902

119880

(28)

where the property 119890 mod le 119890119880 is appliedWhenever 119909 gt (120588 sdot 119890

119880)radic120582min(119876) we have that lt 0

It is clear that if (16) is satisfied then the system (11) is UUBstable This completes the proof

32 Design of the Inner-Level Tracking Controller Once theouter-level stabilization controller 119906

119867= minus119870

119867sdot 119909 has been

designed we are able to put the system undergoing safe trialsTaking tracking performance together with control effort intoconsideration the overall performance index 119869 is defined asa weighted sum of the indices

119869 =

1

119905119891

sdot int

119905119891

0

119906 (119905) sdot 119889119905 +

1205961

119905119891

sdot int

119905119891

0

119890 (119905) sdot 119889119905 (29)

where 1205961is a weighting factor which is defined according

to practical trade-offs between desired tracking performanceand physical constrains

The inner-level controller 119906119878

= 119870119878sdot 119890 is designed

by searching for the gain matrix 119870119878such that the overall

performance index 119869 is minimized We propose to use theNelder-Mead simplexmethod [34] to guide theminimizationprocedure in this paper The method deals with nonlin-ear optimization problems without derivative informationwhich normally requires fewer steps to find a solution closeto global optimum when proper initial values are givenin comparison with the more powerful DIRECT (DIvidingRECTangle) algorithm or evolutionary computation tech-niques

The Nelder-Mead simplex method uses the concept ofa simplex which has 119873 + 1 vertices in 119873 dimensions foran optimization problem with119873 design parameters In each

Mathematical Problems in Engineering 5

step of the algorithm one of the four possible operations isconducted reflection expansion contraction and shrink Asthe method is sensitive to initial guess for an119873-dimensionalproblem we may start the algorithm with 119873 + 1 simplexeswith (119873 + 1)

2 randomly generated parameter sets for thevertices and after several steps collect the 119873 + 1 best solu-tions of the simplexes to form a simplex for final convergenceWith this strategy we have more initial guesses to avoidbeing trapped at local minimum Details are presented in thesubsequent case study

4 Case Study

In order to verify performance of the proposed controlscheme case studies of simulations and experiments areconducted In the simulations a comparison with the adap-tive fuzzy control method (AFCM) of [40] is made Inexperimental studies a two-dimensional prototype cranesystem is used

41 Simulation Study The crane system under control iscomposed of a motor-driven cart running along a horizontalrail a payload and a string carrying the payload which isattached to a joint on the cart We assume that the cart andthe load can move only in the vertical plane In the followingstudy the cart is of mass119872 = 678 kg the payload is of mass119898 = 15 kg and the string is of length 119897 = 05m Furthermore1199091is the cart position 120579 is the swing angle 119906 is the control

signal applied to the cart and 119909119903= [1 0 0 0]

119879 is the referenceinput The position of payload 119910 can be calculated from therelation119910 = 119909

1+119897sdotsin(120579) Besides we assume that the viscous

friction coefficient between the cart and the rail is1198631 and the

wind resistance coefficient between the air and the string is1198632Lagrange analysis of the simplified two-dimensional

crane system gives the dynamic equation

1= (119906 + 119898 sdot 119897 sdot

1205792

sdot sin (120579) + 119898 sdot 119892 sdot sin (120579) sdot cos (120579)

minus1198631sdot 1+ 1198632sdot120579 sdot cos (120579) ) (119872 + 119898 minus 119898 sdot cos2 (120579))

minus1

120579 = ((119898 sdot cos (120579) sdot 119906 + 1198982 sdot 119897 sdot

1205792

sdot sin (120579) sdot cos (120579)

+ (119872 + 119898) sdot 119898 sdot 119892 sdot sin (120579))

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

)

+ ( (minus1198631sdot 1sdot 119898 sdot cos (120579) + 119863

2sdot 119897

sdot [(119872 + 119898) minus 119898 sdot cos2 (120579)] sdot 120579)

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

) + 1199084

(30)

where119892 is the gravitational acceleration and1199084represents the

external disturbance

(1) Controller Design of the Proposed Control Strategy From(3) the overall fuzzymodel of the overhead crane system (30)is inferred to be

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

2

sum

119894=1

ℎ119894(119911) sdot 119888119894]

(31)

where 119909 = [1199091 1199092 1199093 1199094]119879

= [1199091 1 120579

120579]119879 is the state

vector And the matrices are

119860 =

[

[

[

[

[

[

[

0 1 0 0

0 minus

1198631

119872

119898 sdot 119892

119872

1198632

119872

0 0 0 1

0

1198631

(119897 sdot 119872)

minus (119872 + 119898) sdot 119892

119897 sdot 119872

(119872 + 119898) sdot 1198632

1198982sdot 119897 minus 119898 sdot 119897 sdot (119872 + 119898)

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

0

1

119872

0

minus

1

(119897 sdot 119872)

]

]

]

]

]

]

]

]

]

]

]

]

(32)

with 1198631= 588 119863

2= 001 119892 = 981 119911 = 119898 sdot 119897 sdot sin(119909

3) sdot 1199092

4

ℎ1= 05(1 + 119911) ℎ

2= 05(1 minus 119911) 119888

1= 1 and 119888

2= minus1 And

[ Δ119860(119905) Δ119861(119905) ] = 119863 sdot 119865(119905) sdot [ 11986411198642] where 119865(119905) = sin(119905)

119863 = [0 minus001 0 001]119879 1198641= [2 0 0 0] and 119864

2= 002

By selecting V = 3 and 120588 = 18 we are able to obtain

119875 =

[

[

[

[

903667 188347 130680 90783

188347 145588 03778 71938

130680 03778 514426 06680

90783 71938 06680 35618

]

]

]

]

(33)

and119870119867= [1250 3157 minus17665 1295] using the standard

LMI techniques The optimal servo control gains are foundto be 119870

119878= [114047 23047 6997 31522] by the simplex

method

(2) Controller Design of [40] For comparison purpose theadaptive fuzzy controller of [40] abbreviated as AFCMis implemented Design parameters of the AFCM includemembership functions of the antecedents in the fuzzy rulesvalues of the consequent forces and the fuzzy rule mapDetailed values obtained by the procedures described in [40]are shown in Figure 2

In the fuzzy rules each of the universe of discourseof the variables is divided into 6 linguistic values asNBNSZOPSPMPB which represent Negative Big

6 Mathematical Problems in Engineering

0 02 04 06 08 10

02

04

06

08

1M

embe

rshi

p gr

ades

Position error (m)

NSZOPS

PMPB

minus04 minus02

(a)

0 5 10 150

02

04

06

08

1

Mem

bers

hip

degr

ee

Swing angle (deg)

NSZOPS

PMPB

minus15 minus10 minus5

(b)

0 200 400 6000

02

04

06

08

1

12

Force (N)

Mem

bers

hip

degr

ee

NSZOPS

PMPB

minus600 minus400 minus200

(c)

Force Position error

PB PM PS ZO NS

PB PB PB PB NB NB

PS PB PS PS ZO PB

ZO PB PS PB ZO NB

NS PS PB NB NB NB

NB PS PB PB NS NB

Swin

g an

gle

(d)

Figure 2 Linguistic termsmembership functions and rule table of the fuzzy control rules for AFCM (a)Definition ofmembership functionsof position error (b) definition of membership functions of swing angle (c) consequent part membership function of control input 119906(119905) and(d) fuzzy rule map

Negative Small Zero Positive Small and Positive Big respec-tively

(3) Performance Comparison In order to compare relativeperformance of the two approaches a significant disturbanceof 119908 = [0 0 0 119908

4]119879 with

1199084=

120587

3

45 le 119905 le 65

0 otherwise(34)

is applied to the crane modelFrom the time history of the payload position of these

two approaches shown in Figure 3(a) it is clear that both cansuccessfully demonstrate stable tracking during 0 le 119905 lt 45However while the proposed approach remains stable andexhibits accurate tracking after 119905 ge 45 the controller ofAFCMcannot effectively compensate the applied disturbance1199084 shown in Figure 3(b) and eventually goes unstable Note

also that the control signal 119906(119905) generated by the proposedcontroller is much smoother and less violent than thatof AFCM further justifying it as a more efficient controlstrategy

42 Experimental Study Aprototype crane system shown inFigure 4(a) is built to test the proposed control strategy Asshown in the pictures of Figures 4(b) and 4(c) an encoderwith resolution of 2000 pulserev is installed in the hangingjoint to measure the swing angle 120579 To investigate robustnessof the control system the string length can vary between 05to 06m and the payload weight has three choices 05311041 and 1484 kg

The system is firstly identified using the parallel geneticalgorithms [41] as T-S type fuzzy combination of the follow-ing two rules

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

step of the algorithm one of the four possible operations isconducted reflection expansion contraction and shrink Asthe method is sensitive to initial guess for an119873-dimensionalproblem we may start the algorithm with 119873 + 1 simplexeswith (119873 + 1)

2 randomly generated parameter sets for thevertices and after several steps collect the 119873 + 1 best solu-tions of the simplexes to form a simplex for final convergenceWith this strategy we have more initial guesses to avoidbeing trapped at local minimum Details are presented in thesubsequent case study

4 Case Study

In order to verify performance of the proposed controlscheme case studies of simulations and experiments areconducted In the simulations a comparison with the adap-tive fuzzy control method (AFCM) of [40] is made Inexperimental studies a two-dimensional prototype cranesystem is used

41 Simulation Study The crane system under control iscomposed of a motor-driven cart running along a horizontalrail a payload and a string carrying the payload which isattached to a joint on the cart We assume that the cart andthe load can move only in the vertical plane In the followingstudy the cart is of mass119872 = 678 kg the payload is of mass119898 = 15 kg and the string is of length 119897 = 05m Furthermore1199091is the cart position 120579 is the swing angle 119906 is the control

signal applied to the cart and 119909119903= [1 0 0 0]

119879 is the referenceinput The position of payload 119910 can be calculated from therelation119910 = 119909

1+119897sdotsin(120579) Besides we assume that the viscous

friction coefficient between the cart and the rail is1198631 and the

wind resistance coefficient between the air and the string is1198632Lagrange analysis of the simplified two-dimensional

crane system gives the dynamic equation

1= (119906 + 119898 sdot 119897 sdot

1205792

sdot sin (120579) + 119898 sdot 119892 sdot sin (120579) sdot cos (120579)

minus1198631sdot 1+ 1198632sdot120579 sdot cos (120579) ) (119872 + 119898 minus 119898 sdot cos2 (120579))

minus1

120579 = ((119898 sdot cos (120579) sdot 119906 + 1198982 sdot 119897 sdot

1205792

sdot sin (120579) sdot cos (120579)

+ (119872 + 119898) sdot 119898 sdot 119892 sdot sin (120579))

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

)

+ ( (minus1198631sdot 1sdot 119898 sdot cos (120579) + 119863

2sdot 119897

sdot [(119872 + 119898) minus 119898 sdot cos2 (120579)] sdot 120579)

times(1198982

sdot 119897 sdot cos2 (120579) minus 119898 sdot 119897 sdot (119872 + 119898))

minus1

) + 1199084

(30)

where119892 is the gravitational acceleration and1199084represents the

external disturbance

(1) Controller Design of the Proposed Control Strategy From(3) the overall fuzzymodel of the overhead crane system (30)is inferred to be

= 119860119909 + Δ119860 (119905) 119909 + [119861 + Δ119861 (119905)] sdot [119906 +

2

sum

119894=1

ℎ119894(119911) sdot 119888119894]

(31)

where 119909 = [1199091 1199092 1199093 1199094]119879

= [1199091 1 120579

120579]119879 is the state

vector And the matrices are

119860 =

[

[

[

[

[

[

[

0 1 0 0

0 minus

1198631

119872

119898 sdot 119892

119872

1198632

119872

0 0 0 1

0

1198631

(119897 sdot 119872)

minus (119872 + 119898) sdot 119892

119897 sdot 119872

(119872 + 119898) sdot 1198632

1198982sdot 119897 minus 119898 sdot 119897 sdot (119872 + 119898)

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

0

1

119872

0

minus

1

(119897 sdot 119872)

]

]

]

]

]

]

]

]

]

]

]

]

(32)

with 1198631= 588 119863

2= 001 119892 = 981 119911 = 119898 sdot 119897 sdot sin(119909

3) sdot 1199092

4

ℎ1= 05(1 + 119911) ℎ

2= 05(1 minus 119911) 119888

1= 1 and 119888

2= minus1 And

[ Δ119860(119905) Δ119861(119905) ] = 119863 sdot 119865(119905) sdot [ 11986411198642] where 119865(119905) = sin(119905)

119863 = [0 minus001 0 001]119879 1198641= [2 0 0 0] and 119864

2= 002

By selecting V = 3 and 120588 = 18 we are able to obtain

119875 =

[

[

[

[

903667 188347 130680 90783

188347 145588 03778 71938

130680 03778 514426 06680

90783 71938 06680 35618

]

]

]

]

(33)

and119870119867= [1250 3157 minus17665 1295] using the standard

LMI techniques The optimal servo control gains are foundto be 119870

119878= [114047 23047 6997 31522] by the simplex

method

(2) Controller Design of [40] For comparison purpose theadaptive fuzzy controller of [40] abbreviated as AFCMis implemented Design parameters of the AFCM includemembership functions of the antecedents in the fuzzy rulesvalues of the consequent forces and the fuzzy rule mapDetailed values obtained by the procedures described in [40]are shown in Figure 2

In the fuzzy rules each of the universe of discourseof the variables is divided into 6 linguistic values asNBNSZOPSPMPB which represent Negative Big

6 Mathematical Problems in Engineering

0 02 04 06 08 10

02

04

06

08

1M

embe

rshi

p gr

ades

Position error (m)

NSZOPS

PMPB

minus04 minus02

(a)

0 5 10 150

02

04

06

08

1

Mem

bers

hip

degr

ee

Swing angle (deg)

NSZOPS

PMPB

minus15 minus10 minus5

(b)

0 200 400 6000

02

04

06

08

1

12

Force (N)

Mem

bers

hip

degr

ee

NSZOPS

PMPB

minus600 minus400 minus200

(c)

Force Position error

PB PM PS ZO NS

PB PB PB PB NB NB

PS PB PS PS ZO PB

ZO PB PS PB ZO NB

NS PS PB NB NB NB

NB PS PB PB NS NB

Swin

g an

gle

(d)

Figure 2 Linguistic termsmembership functions and rule table of the fuzzy control rules for AFCM (a)Definition ofmembership functionsof position error (b) definition of membership functions of swing angle (c) consequent part membership function of control input 119906(119905) and(d) fuzzy rule map

Negative Small Zero Positive Small and Positive Big respec-tively

(3) Performance Comparison In order to compare relativeperformance of the two approaches a significant disturbanceof 119908 = [0 0 0 119908

4]119879 with

1199084=

120587

3

45 le 119905 le 65

0 otherwise(34)

is applied to the crane modelFrom the time history of the payload position of these

two approaches shown in Figure 3(a) it is clear that both cansuccessfully demonstrate stable tracking during 0 le 119905 lt 45However while the proposed approach remains stable andexhibits accurate tracking after 119905 ge 45 the controller ofAFCMcannot effectively compensate the applied disturbance1199084 shown in Figure 3(b) and eventually goes unstable Note

also that the control signal 119906(119905) generated by the proposedcontroller is much smoother and less violent than thatof AFCM further justifying it as a more efficient controlstrategy

42 Experimental Study Aprototype crane system shown inFigure 4(a) is built to test the proposed control strategy Asshown in the pictures of Figures 4(b) and 4(c) an encoderwith resolution of 2000 pulserev is installed in the hangingjoint to measure the swing angle 120579 To investigate robustnessof the control system the string length can vary between 05to 06m and the payload weight has three choices 05311041 and 1484 kg

The system is firstly identified using the parallel geneticalgorithms [41] as T-S type fuzzy combination of the follow-ing two rules

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 02 04 06 08 10

02

04

06

08

1M

embe

rshi

p gr

ades

Position error (m)

NSZOPS

PMPB

minus04 minus02

(a)

0 5 10 150

02

04

06

08

1

Mem

bers

hip

degr

ee

Swing angle (deg)

NSZOPS

PMPB

minus15 minus10 minus5

(b)

0 200 400 6000

02

04

06

08

1

12

Force (N)

Mem

bers

hip

degr

ee

NSZOPS

PMPB

minus600 minus400 minus200

(c)

Force Position error

PB PM PS ZO NS

PB PB PB PB NB NB

PS PB PS PS ZO PB

ZO PB PS PB ZO NB

NS PS PB NB NB NB

NB PS PB PB NS NB

Swin

g an

gle

(d)

Figure 2 Linguistic termsmembership functions and rule table of the fuzzy control rules for AFCM (a)Definition ofmembership functionsof position error (b) definition of membership functions of swing angle (c) consequent part membership function of control input 119906(119905) and(d) fuzzy rule map

Negative Small Zero Positive Small and Positive Big respec-tively

(3) Performance Comparison In order to compare relativeperformance of the two approaches a significant disturbanceof 119908 = [0 0 0 119908

4]119879 with

1199084=

120587

3

45 le 119905 le 65

0 otherwise(34)

is applied to the crane modelFrom the time history of the payload position of these

two approaches shown in Figure 3(a) it is clear that both cansuccessfully demonstrate stable tracking during 0 le 119905 lt 45However while the proposed approach remains stable andexhibits accurate tracking after 119905 ge 45 the controller ofAFCMcannot effectively compensate the applied disturbance1199084 shown in Figure 3(b) and eventually goes unstable Note

also that the control signal 119906(119905) generated by the proposedcontroller is much smoother and less violent than thatof AFCM further justifying it as a more efficient controlstrategy

42 Experimental Study Aprototype crane system shown inFigure 4(a) is built to test the proposed control strategy Asshown in the pictures of Figures 4(b) and 4(c) an encoderwith resolution of 2000 pulserev is installed in the hangingjoint to measure the swing angle 120579 To investigate robustnessof the control system the string length can vary between 05to 06m and the payload weight has three choices 05311041 and 1484 kg

The system is firstly identified using the parallel geneticalgorithms [41] as T-S type fuzzy combination of the follow-ing two rules

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Reference inputAFCMProposed control scheme

119910(119905)

(m)

(a)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0

500

1000

1500

002040608

11214

AFCMProposed control scheme

minus500

119906(119905)

1199084(119905)

(b)

Figure 3 Comparative simulation for the proposed control strategy and the AFCM (a) Position of payload 119910(119905) (b) control input 119906(119905) andthe external disturbance 119908

4(119905)

(a)

(b) (c)

Figure 4 Pictures of the experimental crane system (a) A whole view of the systemThe image was generated by overlapping five snapshotstaken during operation (b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang thepayload (c) Close view of an encoder also shown in (b) which is attached to the shaft for swing angle measurement

(i) Plant rule 1

If 1199092is11987211

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198881] (35)

(ii) Plant rule 2

If 1199092is11987221

then = 119860 sdot 119909 + Δ119860 (119905) sdot 119909 + [119861 + Δ119861 (119905)] sdot [119906 + 1198882] (36)

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 002 004 006 008 010

01

02

03

04

05

06

07

08

09

1

minus01 minus008 minus006 minus004 minus002

1199094

1198721111987221

Mem

bers

hip

grad

es

Figure 5 The antecedent membership functions11987211and119872

21 of

the fuzzy control law 119906119891

0 002 004 006 008 01

0

2

4

6

8

minus01 minus008 minus006 minus004 minus002

1199094

minus6

minus4

minus2

minus119906119891

Figure 6 The magnitude of minus119906119891as a function of 119909

4(cart velocity)

In the identification a set of commands are designed toperform various maneuvers satisfying persistent excitationrequirements for system identification The identified twoantecedentmembership functions of these two rules119872

11and

11987221 are shown in Figure 5 with

119860 =

[

[

[

[

0 1 0 0

minus239363 0 0 0

0 0 0 1

21681 0 0 0

]

]

]

]

119861 =

[

[

[

[

0

minus0295

0

01475

]

]

]

]

(37)

Furthermore

119863 =

[

[

[

[

0

minus01

0

001

]

]

]

]

1198641= [2 0 0 0]

1198642= 002 with 119865 (119905) isin [minus1 1]

(38)

These are used to define Δ119860(119905) and Δ119861(119905) according to (9)Interesting enough if we draw the magnitude ofsum119871

119894=1ℎ119894(119911) sdot 119888119894

versus 1199094 the velocity of the cart we are able to obtain the

relationship of Figure 6 which shows the behavior similar toa combination of Coulomb friction with Stribeck effects [42]

Next by selecting V = 1 and120588 = 054 we are able to obtain

119875 =

[

[

[

[

17070 03014 minus01362 minus02631

03014 00869 minus00188 minus00444

minus01362 minus00188 00298 00269

minus02631 minus00444 00269 00658

]

]

]

]

(39)

1198881= minus7772 119888

2= 40561 and119870

119867= [2623 802 2344 1492]

by the standard LMI techniquesFigure 7 shows the performance of the outer-level stabi-

lizing controller 119906119867 In this figure three cases were recorded

where impacts were applied to the payload at 172 145and 038 sec respectively The string length and payloadweight [length weight] of these cases were [05 0531][055 1041] and [06 1484] respectively According to theseexperimental results the stabilizing controller applied atthe outer level exhibits 119867

infinrobustness against significant

disturbances in spite of variations in the plant dynamicsNext considering that string length and payload weight

dominate system dynamics we implemented the servo con-trol law 119906

119878= 119870119878sdot 119890 as a fuzzy controller composed of four

fuzzy rules

Servo control rule 119894119895

If string length is 119860119894and payload weight is 119861

119895

then 119906119878= 119870119878119894119895sdot 119890 (40)

That is both string length and payload weight are fuzzifiedwith two membership functions 119860

1 1198602 1198611 and 119861

2 respec-

tively The corresponding membership grades of these fourfuzzy sets are shown in Figure 8

Furthermore by assigning 1205961= 10 in the definition of

the overall performance index 119869 defined in (29) the Nelder-Mead simplex method was applied to search for the bestgains 119870

119878119894119895in the four rules The learning history of gains is

depicted in Figure 9 Note that only the gains correspondingto [length weight] = [05 0531] underwent 116 steps all theother gains were initiated with the gains of fuzzy rule 1 henceless than 30 steps were required According to considerations

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9

Cart position

Time (s)

Disturbed at 119905 = 038 sDisturbed at 119905 = 145 s

Disturbed at 119905 = 172 s

minus05

005

1

(m)

(a)

0 1 2 3 4 5 6 7 8 9

Swing angle

Time (s)

minus40

minus20

020

(deg

)

(b)

0 1 2 3 4 5 6 7 8 9

Case 1Case 2Case 3

Time (s)

Control input

minus40

minus20

020

Mag

nitu

de

(c)

Figure 7 Performance of 119906119867in the face of impact during manipulationThe values of string length and payload weight [length weight] are

Case 1 [05 0531] Case 2 [055 1041] and Case 3 [06 1484] respectively

04 045 05 055 06 0650

02040608

1String length

Mem

bers

hip

grad

es

Fuzzy set 1198601Fuzzy set 1198602

(m)

(a)

04 06 08 1 12 14 16

Payload weight

002040608

1

Mem

bers

hip

grad

es

(kg)

Fuzzy set 1198611Fuzzy set 1198612

(b)

Figure 8 The antecedent membership functions of the fuzzy sets 1198601 1198602 1198611 and 119861

2

and procedures detailed in Remark 3 the gains are found tobe of the following values

11987011987811

= [56 44 34 23] for [lengthweight] = [05 0531]

11987011987821

= [62 40 53 39] for [lengthweight] = [06 0531]

11987011987812

= [53 48 42 24] for [lengthweight] = [05 1484]

11987011987822

= [56 41 46 38] for [lengthweight] = [06 1484] (41)

Remark 3 Four fuzzy rules defined in (40) each of themcontains a set of optimized control gains are designed tocompensate for the uncertainties in the weight of payload andstring length As shown in the learning history of Figure 10the gains of rule 1 took 50 iterations and those of the rest ofthe rules took only 12 iterationsThat is only the first gain setrequires complete search This is because the performance ofthe Nelder-Mead simplex algorithm is sensitive to initial trialvalues and the optimal control gains are close to each otherIf the search for the other sets begin with the optimized firstset less iteration is required Also an iteration of Figure 10

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 20 40 60 80 100 120

0

20

40

60

80

100

Count of steps

11987011987811

(a)

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

Count of steps

11987011987821

(b)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

70

80

11987011987812

(c)

0 20 40 60 80 100 120Count of steps

0

10

20

30

40

50

60

11987011987822

(d)

Figure 9 The learning history of gains guided by the simplex method showing convergence of the gain parameters In each of the 3dimensional cases the method starts with 4 simplexes and a new simplex is formed for the following steps by collecting the 4 best solutionsso far at the 30th step

5 10 15 20 25 30 35 40 45 500002

0004

0006

0008

001

0012

0014

0016

0018

Iteration

Learning curve using the simplex method

Rule 1Rule 2

Rule 3Rule 4

The o

vera

ll pe

rform

ance

inde

x119869

Figure 10 The simplex convergence history of the overall perfor-mance index 119869 in the four rules

corresponds to 1 to 3 steps in Figure 9 since only improvedstep is regarded as an effective iteration

In the experiments of automatic repetitive trials to findthe optimal gains reference state trajectories were designed

such that the payload moves smoothly forward withoutswinging back In fulfilling the requirement the referencetrajectories should be a function of the nature frequencythat in turn depends on both the string length and the loadweight Specifically for the payload position 119910 = 119909

1+ 119897 sdot sin 120579

to move in this way the trajectory of 120579 should contain integermultiple of a full nature-frequency cycle

Finally experiments were conducted to justify the controlperformance Three experiments were designed Case 1[length weight distance] = [06 1484 10] Case 2 [lengthweight distance] = [06 1041 08] and Case 3 [lengthweight distance] = [05 0531 06] The gains of Case 2are interpolated from four fuzzy rules to be 119870

119878= [587891

405352 492539 384648] The performance of the cranecontrol system is demonstrated in Figure 11 According to theexperimental results the proposed control strategy can guidethe payload smoothly forward without swinging back in areasonable period of time

5 Conclusions

By the antiswing control approach a two-level controlscheme is proposed for crane systems The plant is modeledas a combination of a nominal linear system and a T-Sfuzzy blending of affine terms This type of dynamic model

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 1 2 3 4 5 60

05

1Payload position

(m)

Time (s)

(a)

0 1 2 3 4 5 60

051

(m)

Cart position

Time (s)

(b)

0 1 2 3 4 5 6Time (s)

Swing angle

05

(deg

)minus5

(c)

Figure 11 Experimental results of the crane control system Case 1 [length weight distance] = [06 1484 10] Case 2 [length weightdistance] = [06 1041 08] Case 3 [length weight distance] = [05 0531 06]

significantly simplifies the subsequent analysis and controldesigns because assumptions on the plant dynamics can besignificantly reduced The proposed control scheme can alsobe applied to other nonlinear plants such as ships mobilerobots and aircrafts but is not applicable for systems withconsiderable time delay which is the issue to be addressed inour future investigation

In the scheme the outer-level control law serves asan 119867infin

robust controller which is responsible for closed-loop stability in the face of disturbances and plant dynamicvariations Optimal gains of the inner-loop servo controllaw are obtained using the Nelder-Mead simplex algorithmin a learning control manner Close observation of theobtained fuzzy model reveals that the fuzzy compensatormainly counteracts the effects of friction The dynamics ofCoulomb friction viscous friction and Stribeck effects aredistinguishable as functions of relative velocity

A simulation study shows superior performance of theproposed control strategy in compensating significant distur-bances Experimental results of a prototype two-dimensionalcrane control system also demonstrate smooth manipulationof the payload with119867

infinrobust stability The control strategy

can be extended to full dimensional crane systems and iswithin our plans of future research

Acknowledgments

The authors would like to express their sincere appreciationto the editor and all the reviewers for their helpful andconstructive comments Besides the authors are enormouslygrateful for the supports from theNational ScienceCouncil ofTaiwan under Grants nos NSC 101-2221-E-182-006 and 100-2221-E-182-008

References

[1] B J Park K S Hong and C D Huh ldquoTime-efficient inputshaping control of container crane systemsrdquo in Proceedings of

IEEE International Conference on Control Applications pp 80ndash85 2000

[2] W Singhose L Porter M Kenison and E Kriikku ldquoEffects ofhoisting on the input shaping control of gantry cranesrdquo ControlEngineering Practice vol 8 no 10 pp 1159ndash1165 2000

[3] D Liu J Yi D Zhao and W Wang ldquoAdaptive sliding modefuzzy control for a two-dimensional overhead cranerdquo Mecha-tronics vol 15 no 5 pp 505ndash522 2005

[4] M J Er M Zribi and K L Lee ldquoVariable structure controlof an overhead cranerdquo in Proceedings of IEEE InternationalConference on Control Applocations pp 398ndash402 September1998

[5] A M H Basher ldquoSwing-free transport using variable structuremodel reference controlrdquo in Proceedings of IEEE SoutheastConpp 85ndash92 April 2001

[6] L Moreno L Acosta J A Mendez S Torres A Hamilton andG N Marichal ldquoA self-tuning neuromorphic controller appli-cation to the crane problemrdquo Control Engineering Practice vol6 no 12 pp 1475ndash1483 1998

[7] H H Lee and S K Cho ldquoA new fuzzy-logic anti-swingcontrol for industrial three-dimensional overhead cranesrdquo inProceedings of IEEE International Conference on Robotics andAutomation pp 2956ndash2961 May 2001

[8] M J Nalley andM B Trabia ldquoControl of overhead cranes usinga fuzzy logic controllerrdquo Journal of Intelligent and Fuzzy Systemsvol 8 no 1 pp 1ndash18 2000

[9] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the TS fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[10] X Su P Shi L Wu and Y D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEE Transactions on Fuzzy Systems vol 20 pp 1114ndash1129 2012

[11] C H Sun S W Lin and Y T Wang ldquoRelaxed stabilizationconditions for switching T-S fuzzy systems with practicalconstraintsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 4133ndash4145 2012

[12] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[13] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[14] A Benhidjeb andG L Gissinger ldquoFuzzy control of an overheadcrane performance comparison with classic controlrdquo ControlEngineering Practice vol 3 no 12 pp 1687ndash1696 1995

[15] J Yi N Yubazaki and K Hirota ldquoAnti-swing and positioningcontrol of overhead traveling cranerdquo Information Sciences vol155 no 1-2 pp 19ndash42 2003

[16] X H Chang and G H Yang ldquoRelaxed results on stabi-lization and state feedback 119867

infincontrol conditions for T-S

fuzzy systemsrdquo International Journal of Innovative ComputingInformation and Control vol 7 no 4 pp 1753ndash1764 2011

[17] A Schwung T Gusner and J Adamy ldquoStability analysis ofrecurrent fuzzy systems a hybrid system and sos approachrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 423ndash4312011

[18] H K Lam ldquoPolynomial fuzzy-model-based control systemsStability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[19] J C Lo Y T Lin and W C Liao ldquoGeneralized stabilizingcontrollers for fuzzy systems via circle criterion-LMI andSOSrdquo in Proceedings of IEEE International Conference on FuzzySystems pp 1294ndash1298 2011

[20] K Tanaka H Ohtake T Seo and H O Wang ldquoAn SOS-basedobserver design for polynomial fuzzy systemsrdquo in AmericanControl Conference pp 4953ndash4958 2011

[21] X Su L Wu P Shi and Y D Song ldquo119867infinmodel reduction of T-

S fuzzy stochastic systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B vol 42 pp 1574ndash1585 2012

[22] F Li andX Zhang ldquoDelay-range-dependent robust119867infinfiltering

for singular LPV systems with time variant delayrdquo InternationalJournal of Innovative Computing Information and Control vol9 pp 339ndash353 2013

[23] R Yang H Gao and P Shi ldquoDelay-dependent robust119867infincon-

trol for uncertain stochastic time-delay systemsrdquo InternationalJournal of Robust and Nonlinear Control vol 20 no 16 pp1852ndash1865 2010

[24] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley NewYork NY USA 2001

[25] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[26] K Tanaka T Ikeda and H O Wang ldquoRobust stabilizationof a class of uncertain nonlinear systems via fuzzy controlquadratic stabilizability 119867

infincontrol theory and linear matrix

inequalitiesrdquo IEEE Transactions on Fuzzy Systems vol 4 no 1pp 1ndash13 1996

[27] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[28] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[29] B S Chen C S Tseng and H J Uang ldquoRobustness designof nonlinear dynamic systems via fuzzy linear controlrdquo IEEETransactions on Fuzzy Systems vol 7 no 5 pp 571ndash585 1999

[30] C S Tseng B S Chen and H J Uang ldquoFuzzy tracking controldesign for nonlinear dynamic systems via T-S fuzzy modelrdquoIEEE Transactions on Fuzzy Systems vol 9 no 3 pp 381ndash3922001

[31] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[32] H K Lam F H F Leung and Y S Lee ldquoDesign of a switchingcontroller for nonlinear systems with unknown parametersbased on a fuzzy logic approachrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 2 pp 1068ndash1074 2004

[33] C H Sun Y T Wang and C C Chang ldquoSwitching T-Sfuzzy model-based guaranteed cost control for two-wheeledmobile robotsrdquo International Journal of Innovative ComputingInformation and Control vol 8 pp 3015ndash3028 2012

[34] J A Nelder and R Mead ldquoA simplex method for functionminimizationrdquo Computer Journal vol 7 pp 308ndash313 1965

[35] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[36] Y C Ho W C Hsu and C C Chang ldquoMulti-category andmulti-standard project selection with fuzzy value-based timelimitrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 pp 971ndash989 2013

[37] C M Lin C F Hsu and R G Yeh ldquoAdaptive fuzzy sliding-mode control system design for brushless DC motorsrdquo Inter-national Journal of Innovative Computing Information andControl vol 9 pp 1259ndash1270 2013

[38] HM Lee C F Fuh and J S Su ldquoFuzzy parallel system reliabil-ity analysis based on level (120582 120588) interval-valued fuzzy numbersrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 8 pp 5703ndash5713 2012

[39] L K Wong F H F Leung and P K S Tarn ldquoLyapunov-function-based design of fuzzy logic controllers and its applica-tion on combining controllersrdquo IEEE Transactions on IndustrialElectronics vol 45 no 3 pp 502ndash509 1998

[40] C Y Chang ldquoAdaptive fuzzy controller of the overhead craneswith nonlinear disturbancerdquo IEEE Transactions on IndustrialInformatics vol 3 no 2 pp 164ndash172 2007

[41] Y Z Chang J Chang and C K Huang ldquoParallel geneticalgorithms for a neurocontrol problemrdquo in International JointConference on Neural Networks (IJCNN rsquo99) pp 4151ndash4155 July1999

[42] P A Bliman and M Sorine ldquoFriction modelling by hysteresisoperators application to Dahl sticktion and Stribeck effectsrdquo inProceedings of the Conference on Models of Hysteresis 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of