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Problem statement: Most of the common gantry crane results in a sway motion when transporting the load as fast as possible. In addition, precise cart position control of gantry crane must required a zero or near zero residual sway. Approach: In this

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American J. of Engineering and Applied Sciences 2 (1): 241-251, 2009ISSN 1941-7020© 2009 Science Publications
Corresponding Author:
Mohd Ashraf Ahmad, Faculty of Electrical and Electronics Engineering, University Malaysia Pahang,Lebuhraya Tun Razak, 26300, Kuantan, Pahang, Malaysia
241
Hybrid Fuzzy Logic Control with Input Shaping for Input Tracking andSway Suppression of a Gantry Crane System
1
M.A. Ahmad
and
2
Z. Mohamed
1
Faculty of Electrical and Electronics Engineering,University Malaysia Pahang, Lebuhraya Tun Razak, 26300, Kuantan, Pahang, Malaysia
2
Faculty of Electrical Engineering, University Technology Malaysia,81310, UTM Skudai, Johor Bahru, Malaysia
Abstract:
Problem statement:
Most of the common gantry crane results in a sway motion whentransporting the load as fast as possible. In addition, precise cart position control of gantry crane mustrequired a zero or near zero residual sway.
Approach:
In this study, the development of hybrid controlschemes for input tracking and anti-sway control of a gantry crane system was investigated. To studythe effectiveness of the controllers, a Proportional-Derivative (PD)-type fuzzy logic control wasdeveloped for cart position control of a gantry crane. It was then extended to incorporate input shapercontrol schemes for anti-sway control of the system.
The positive and new modified SpecifiedNegative Amplitude (SNA) input shapers were designed based on the properties of the system forcontrol of system sway. The new SNA was proposed to improve the robustness capability whileincreasing the speed of the system response.
Results:
Simulation results of the response of the gantrycrane with the controllers were presented in time and frequency domains. The performances of the of the hybrid control schemes were examined in terms of input tracking capability, level of swayreduction and robustness to parameters uncertainty.
Conclusion:
A significant reduction in the systemsways had been achieved with the hybrid controllers regardless of the polarities of the shapers.
Key words:
Gantry crane, anti-sway control, input shaping, PD-type fuzzy logic controller
INTRODUCTION
The main purpose of controlling a gantry crane istransporting the load as fast as possible without causingany excessive sway at the final position. However, mostof the common gantry crane results in a sway motionwhen payload is suddenly stopped after a fast motion.The sway motion can be reduced but will be timeconsuming. Moreover, the gantry crane needs a skilfuloperator to control manually based on his or herexperiences to stop the sway immediately at the rightposition. The failure of controlling crane also mightcause accident and may harm people and thesurrounding.The requirement of precise cart position control of gantry crane implies that residual sway of the systemshould be zero or near zero. Over the years,investigations have been carried out to devise efficientapproaches to reduce the sway of gantry crane. Theconsidered sway control schemes can be divided intotwo main categories: Feed-forward control andfeedback control techniques. Feed-forward techniquesfor sway suppression involve developing the controlinput through consideration of the physical and swayingproperties of the system, so that system sways atdominant response modes are reduced. This methoddoes not require additional sensors or actuators anddoes not account for changes in the system once theinput is developed. On the other hand, feedback-controltechniques use measurement and estimations of thesystem states to reduce sways. Feedback controllers canbe designed to be robust to parameter uncertainty. Forgantry crane, feed-forward and feedback controltechniques are used for sway suppression and cartposition control respectively. An acceptable systemperformance without sway that accounts for systemchanges can be achieved by developing a hybridcontroller consisting of both control techniques. Thus,with a properly designed feed-forward controller, thecomplexity of the required feedback controller can bereduced.Various attempts in controlling gantry cranessystem based on feed-forward control schemes wereproposed. For example, open loop time optimal
Am. J. Engg. & Applied Sci., 2 (1): 241-251, 2009
242strategies were applied to the crane by manyresearchers such as discussed in
[1]
. They came out withpoor results because feed-forward strategy is sensitiveto the system parameters (e.g., rope length) and couldnot compensate for wind disturbances. Another feed-forward control strategy is input shaping
[2-4]
. Inputshaping is implemented in real time by convolving thecommand signal with an impulse sequence. The processhas the effect of placing zeros at the locations of theflexible poles of the srcinal system. An IIR filteringtechnique related to input shaping has been proposedfor controlling suspended payloads
[5]
. Input shaping hasbeen shown to be effective for controlling oscillation of gantry cranes when the load does not undergohoisting
[6,7]
. Experimental results also indicate thatshaped commands can be of benefit when the load ishoisted during the motion
[8]
.Investigations have shown that with the inputshaping technique, a system response with delay isobtained. To reduce the delay and thus increase thespeed of the response, negative amplitude input shapershave been introduced and investigated in vibrationcontrol. By allowing the shaper to contain negativeimpulses, the shaper duration can be shortened, whilesatisfying the same robustness constraint. A significantnumber of negative shapers for vibration control havealso been proposed. These include negative Unity-Magnitude (UM) shaper, Specified-Negative-Amplitude(SNA) shaper, negative Zero-Vibration (ZV) shaper,negative Zero-Vibration-Derivative (ZVD) shaper andnegative Zero-Vibration-Derivative-Derivative (ZVDD)shaper
[9-11]
. Comparisons of positive and negative inputshapers for vibration control of a single-link flexiblemanipulator have also been reported
[11]
.On the other hand, feedback control which is wellknown to be less sensitive to disturbances andparameter variations
[12]
is also adopted for controllingthe gantry crane system. Recent research on gantrycrane control system was presented by
[13]
. The authorhad proposed proportional-derivative PD controllers forboth position and anti-sway controls. Furthermore, afuzzy-based intelligent gantry crane system has beenproposed
[14]
. The proposed fuzzy logic controllersconsist of position as well as anti-sway controllers.However, most of the feedback control systemproposed needs sensors for measuring the cart positionas well as the load sway angle. In addition, designingthe sway angle measurement of the real gantry cranesystem, in particular, is not an easy task since there is ahoisting mechanism.This study presents investigations into thedevelopment of hybrid control schemes for inputtracking and anti-sway control of a gantry crane system.A nonlinear overhead gantry crane system is consideredand the dynamic model of the system is derived usingthe Euler-Lagrange formulation. Hybrid controlschemes based on feed-forward with collocatedfeedback controllers are investigated. In this study,feed-forward control based on input shaping withpositive Zero-Sway-Derivative-Derivative (ZSDD) inputshapers and new modified SNA Zero-Sway-Derivative-Derivative (ZSDD) input shapers are considered. A newmodified shaper from the previous SNA input shapers
[11]
is proposed where more negative impulses are added toimprove the robustness of the controller while increasingthe speed of the system response. To demonstrate theeffectiveness of the proposed control schemes, a PD-typeFuzzy Logic controller is developed for control of cartmotion of the gantry crane. This is then extended toincorporate the proposed input shapers for control of sway of hoisting rope. Simulation exercises areperformed within the gantry crane simulationenvironment. Performances of the developed controllersare examined in terms of input tracking capability, levelof sway reduction and robustness to errors in swayfrequency. In this case, the robustness of the hybridcontrol schemes is assessed with up to 30% errortolerance in sway frequencies. Simulation results in timeand frequency domains of the response of the gantrycrane to the unshaped input and shaped inputs withpositive and modified SNA input shapers are presented.Moreover, a comparative assessment of the effectivenessof the hybrid controllers with positive and negative inputshapers in suppressing sway and maintaining the inputtracking capability of the gantry crane is discussed.
The gantry crane system:
The two-dimensional gantrycrane system with its payload considered in this study isshown in Fig. 1, where x is the horizontal position of the cart, l is the length of the rope,
θ
is the sway angleof the rope, M and m is the mass of the cart and payloadrespectively. In this simulation, the cart and payload canbe considered as point masses and are assumed to movein two-dimensional, x-y plane. The tension force thatmay cause the hoisting rope elongate is also ignored. Inthis study the length of the cart, l = 1.00 m, M = 2.49 kg,m = 1.00 kg and g = 9.81 m s
−
2
is considered.
Modeling of the gantry crane:
In this research, themathematical modeling of the gantry crane system isconsidered as a basis of a simulation environment fordevelopment and assessment of the input shapingcontrol techniques. The Euler-Lagrange formulation isconsidered in characterizing the dynamic behavior of the crane system incorporating payload.
Am. J. Engg. & Applied Sci., 2 (1): 241-251, 2009
243Fig. 1: Description of the gantry crane system.Considering the motion of the gantry crane systemon a two-dimensional plane, the kinetic energy of thesystem can thus be formulated as:
22
11TMxm(xi2l22222xlsin2xlcos)
= + + + θ +θ+ θ θ
& &
(1)The potential energy of the beam can beformulated as:
Umglcos
= − θ
(2)To obtain a closed-form dynamic model of thegantry crane, the energy expressions in (1) and (2) areused to formulate the Lagrangian
LTU
= −
. Let thegeneralized forces corresponding to the generalizeddisplacements
q{x,}
= θ
be
x
F{F,0}
=
. UsingLagrangian’s equation:
j jj
dLLFj1,2dtqq
∂ ∂− = =
∂ ∂
&
(3)the equation of motion is obtained as:
2x
F(Mm)xml(cossin)2mlcosmlsin
= + + θ θ−θ θθ θ+ θ
&& &&&& &&&
(4)
2x
F(Mm)xml(cossin)2mlcosmlsinl2lxcosgsin0
= + + θ θ−θ θ + θ θ+ θ θ+ θ+ θ+ θ =
&&& & &&&&& &&& &&&
(5)In order to eliminate the nonlinearity equation inthe system, a linear model of gantry crane system isobtained. The linear model of the uncontrolled systemcan be represented in a state-space form as shown inEq. 6 by assuming the change of rope and sway angleare very small:
xAxBuyCx
= +=
&
(6)with the vector
T
xxx
= θ θ
&&
and the matrices A andB are given by:
[ ] [ ]
0010000010mg1A,B,000MM(Mm)g1000MlMlC1000 ,D0
= =
+
− −
= =
(7)
MATERIALS AND METHODSPD-type fuzzy logic control scheme:
Fuzzy controlcan be viewed as a way of converting expert knowledgeinto an automatic control strategy without a detailedknowledge of the plant. The input is first fuzzified andthen processed by the fuzzy inference engine usingheuristic decision rules. FLC uses rules in the form of ‘‘IF [condition] THEN [action]’’ to linguisticallydescribe the input/output relationship. The membershipfunctions convert linguistic terms into precise numericvalues. The output of the fuzzy controller is obtained bya defuzzification process that converts the fuzzyquantities representing the control signal into a signalthat can be used as the control input to the plant.A PD-type fuzzy logic controller utilizing hubangle and hub velocity feedback is developed to controlthe rigid body motion of the system
[15]
. The hybridfuzzy control system proposed in this study is shown inFig. 2, where R
f
is the reference horizontal position, xand
x
&
represent horizontal position and velocity of thecart, respectively,
θ
and
θ
&
represent swing angle andswing velocity, respectively, whereas k
1
, k
2
and k
3
arescaling factors for two inputs and one output of thefuzzy logic controller used with the normalizeduniverse of discourse for the fuzzy membershipfunctions.
Am. J. Engg. & Applied Sci., 2 (1): 241-251, 2009
244Fig. 2: PD-type Fuzzy Logic control structure
In this study, the triangular membership functionsare chosen for inputs and output. Normalized universesof discourse are used for both hub angle and velocityand output torque. Scaling factors k
1
and k
2
are chosenin such a way as to convert the two inputs within theuniverse of discourse and activate the rule baseeffectively, whereas
k
3
is selected such that it activatesthe system to generate the desired output. Initially allthese scaling factors are chosen based on trial and error.To construct a rule base, the cart position error, cartposition error derivative and force input are partitionedinto five primary fuzzy sets as
[15]
:
Cart position error E = {NM NS ZE PS PM}Cart position error derivative V = {NM NS ZE PS PM}Force U = {NM NS ZE PS PM}where E, V and U are the universes of discourse cartposition, cart velocity and force input, respectively. Thenth rule of the rule base for the FLC, with cart positionerror and derivative of cart position error as inputs, isgiven by:R
n
: IF(e is E
i
) AND (
e
&
is V
j
) THEN (u is U
k
)where, R
n
, n = 1, 2,…N
max
is the nth fuzzy rule, E
i
, V
j
and U
k
, for i, j, k = 1,2,…,5 are the primary fuzzy sets.A PD-type fuzzy logic controller was designedwith 11 rules as a closed loop component of the controlstrategy for maintaining the cart position of gantrycrane system while suppressing the swaying effect. Therule base was extracted based on underdamped systemresponse and is shown in Table 1. The three scalingfactors, k
1
, k
2
and k
3
were chosen heuristically toachieve a satisfactory set of time domain parameters.These values were recorded as, k
1
= 0.05, k
2
= 0.001and k
3
= -350.
INPUT shaping control schemes:
Input shapingtechnique is a feed-forward control technique thatinvolves convolving a desired command with asequence of impulses known as input shaper.
Table 1: Linguistic rules of Fuzzy Logic Controller
No. Rules1. If (e is NM) and(e
&
is ZE) then (u is PM)2. If (e is NS) and (e
&
is ZE) then (u is PS)3. If (e is NS) and (e
&
is PS) then (u is ZE)4. If (e
is ZE) and (e
&
is NM) then (u is PM)5. If (e
is ZE) and (e
&
is NS) then (u is PS)6. If (e is ZE) and (e
&
is ZE) then (u is ZE)7. If (e
is ZE) and (e
&
is PS) then (u is NS)8. If (e
is ZE) and (e
&
is PM) then (u is NM)9. If (e
is PS) and (e
&
is NS) then (u is ZE)10. If (e
is PS) and (e
&
is ZE) then (u is NS)11. If (e
is PM) and (e
&
is ZE) then (u is NM)
Fig. 3: Illustration of input shaping technique
The shaped command that results from the convolutionis then used to drive the system. Design objectives areto determine the amplitude and time locations of theimpulses, so that the shaped command reduces thedetrimental effects of system flexibility. Theseparameters are obtained from the natural frequenciesand damping ratios of the system. Thus, sway reductionof a gantry crane system can be achieved with the inputshaping technique. Figure 3 shows the input shapingprocess. Several techniques have been investigated toobtain an efficient input shaper for a particular system.A brief description and derivation of the controltechnique is presented in this study.Generally, a vibratory system of any order can bemodeled as a superposition of second order systemseach with a transfer function:
222
G(s)s2s
ω=+ ζω +ω
(8)Where:
ω
= The natural frequency of the vibratory system
ζ
= The damping ratio of the systemThus, the response of the system in time domaincan be obtained as:
( )
0
(tt)202
Ay(t)expsin1(tt)1
−ζω −
ω= ω −ζ −−ζ
(9)
Am. J. Engg. & Applied Sci., 2 (1): 241-251, 2009
245where, A and t
0
are the amplitude and the time locationof the impulse respectively. The response to a sequenceof impulses can be obtained by superposition of theimpulse responses. Thus, for N impulses, with
( )
2d
1
ω = ω −ζ
, the impulse response can be expressedas:
( )
d
y(t)Msint
= ω +β
(10)Where:M =
22NNiiiii1i1
BcosBsin
= =
φ + φ
∑ ∑
B
i
=
0
(tt)i2
Aexp1
−ζω −
ω−ζ
idi
t
φ = ω
A
i
and t
i
= The amplitudes and time locations of theimpulsesThe residual single mode sway amplitude of theimpulse response is obtained at the time of the lastimpulse, t
N
as:
2212
VVV
= +
(11)Where:
nNi
N(tt)in1di2i1
AVexpcos(t)1
−ζω −=
ω= ω−ζ
∑
nNi
N(tt)in2di2i1
AVexpsin(t)1
−ζω −=
ω= ω−ζ
∑
To achieve zero sway after the last impulse, it isrequired that both V
1
and V
2
in Eq. 11 areindependently zero. This is known as the zero residualsway constraints. In order to ensure that the shapedcommand input produces the same rigid body motion asthe unshaped reference command, it is required that thesum of amplitudes of the impulses is unity. This yieldsthe unity amplitude summation constraint as:
Nii1
A1
=
=
∑
(12)In order to avoid response delay, time optimalityconstraint is utilized. The first impulse is selected attime t
1
= 0 and the last impulse must be at theminimum, i.e., min (t
N
). The robustness of the inputshaper to errors in natural frequencies of the system canbe increased by taking the derivatives of V
1
and V
2
tozero. Setting the derivatives to zero is equivalent toproducing small changes in sway corresponding to thefrequency changes. The level of robustness can furtherbe increased by increasing the order of derivatives of V
1
and V
2
and set them to zero. Thus, the robustnessconstraints can be obtained as:
ii12iinn
dVdV0,0dd
= =ω ω
(13)Both the positive and modified SNA input shapersare designed by considering the constraints Eq.s. Thedesign of the positive and modified SNA input shapersis further discussed in this investigation.
Positive input shaper:
The positive input shapers havebeen used in most input shaping schemes. Therequirement of positive amplitude for the impulses is toavoid the problem of large amplitude impulses. In thiscase, each individual impulse must be less than one tosatisfy the unity magnitude constraint. In order toincrease the robustness of the input shaper to errors innatural frequencies, the positive ZSDD input shaper isdesigned by setting the second derivatives of V
1
and V
2
in Eq. 11 to zero. Simplifying
22in
dVd
ω
yields:
nNinNi
2N(tt)21iidi2i1n2N(tt)22iidi2i1n
dVAtesin(t);ddVAtecos(t)d
−ζω −=−ζω −=
= ωω= ωω
∑∑
(14)The positive ZSDD input shaper, i.e., four-impulsesequence is obtained by setting Eq. 11 and 14 to zeroand solving with the other constraint Eq.s. Hence, afour-impulse sequence can be obtained with theparameters as:
1234ddd12232323342323
23t0,t,t,t13KA,A13K3KK13K3KK3KKA,A13K3KK13K3KK
π π π= = = =ω ω ω= =+ + + + + += =+ + + + + +
(15)Where:
1
Ke
−ζπ−ζ
=
2dn
1
ω = ω −ζ

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