research article switched two-level and robust...
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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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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
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Decision SciencesAdvances in
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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
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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
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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
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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