Robust Adaptive Fuzzy Control Design for 3-D Tower Crane

Robust Adaptive Fuzzy Control Design for 3-D Tower Crane withTime Delayed States

Kuan-Wei Wu∗, Mansour Karkoub, and Tzu-Sung Wu

Abstract—Tower cranes are very complex mechanical systemsand have been the subject of research investigations for severaldecades. The cranes system has potentially six degrees of freedombut only three actuators. Also, the actuators are far from thepayload which makes the system non-colocated. The dynamicmodel describing the motion of the payload from point to point isaffected by uncertainties, time delayed and external disturbanceswhich may lead to inaccurate positioning, reduce safety andefficacy of the overall system. The proposed control law isbased on indirect adaptive fuzzy control. A fuzzy model is usedto approximate the dynamics of the nonlinear MIMO system;then, two on-line estimation schemes are developed to overcomethe nonlinearities and identify the gains of the delayed stateuncertainties, simultaneously, such that the payload swing can bereduced as small as possible when the load is moved from pointto point. The adaptive fuzzy scheme uses a Variable Structure(VS) scheme to resolve the system uncertainties, time delayeduncertainty and the external. Therefore, the effect can be reducedto any prescribed level to achieveH∞ tracking performance.The control laws are derived based on a Lyapunov criterionand the Riccati-inequality such that all states of the system areuniformly ultimately bounded (UUB). The effectiveness of theproposed control schemes for the three dimension (3-D) towercrane system is verified by simulations and experimental results.

Index Terms– Tower crane systems; adaptive fuzzy control;H∞ control; variable structure scheme.

I. I NTRODUCTION

Tower cranes are common industrial structures that are usedto lift and move heavy payload in the construction of high-risebuildings , factories, and harbors. Moving the load from pointto point is the most time consuming task and tend to disturbcrane operations, causing the behavior of the pendulum motionof the payload and the vibration of the crane which may lead toinaccurate positioning, reduce safety and efficacy of the overallsystem. Moreover, tower cranes are usually operated manuallyuntil recently.It is difficult to control by manually operatedwhen the cranes became larger and the motion expected to befaster. Such problems decrease the work efficiency and in somecases cause damage to the loads and cause safety accidents.Consequently, many researchers have proposed automatic con-trol scheme to minimize the swing and eliminate the negativelyeffects cause by load swing in dynamic processes (see e.g., [1]-[4] and the references therein). In [1]-[2], the authors proposedgain-scheduling feedback control methods applied to towercranes which enables the controller feedback gains can becalculated for varying load weight and cable length. Inputshaping control method is adopted in [3]-[4] to reduce the

Department of Mechanical Engineering, Texas A & M University at [email protected]

vibratory response of overhead cranes systems such that thepayload swing can be declined substantially. However, mostof these methods needs accurate system model and onerousmatrix computation which were greatly affected by systemlinearization and system parameters uncertainty.Furthermore,the backlash phenomenon also exists in tower cranes.

Backlash is a phenomenon which can severely limit thecontrol system performance. Typically backlash appears inmechanical couplings because of the gap between the motor-side and the load-side contact points. As backlash increases(gap increases) the time-delay due to backlash increases andachieving stability and tracking performance is more difficult[5]-[7]. The existence of time delays in the system degradesthe control performance and sometimes makes the closed-loopstabilization difficult if not impossible. This also complicatesthe dynamics of the system because it involves infinite-dimensional functional differential equations. Inherently, thedynamical model of tower cranes with time-delays is highlynonlinear and also classified as underactuated. Thus, it isnot justifiable to use a linear controller to deal with thecontrol problem of the tower crane with time-delays. Severalcontrol schemes have surfaced over the years to deal withtime delayed systems, e.g., sliding-mode control [8], adaptiveneural network control [9]-[10],and adaptive fuzzy control[11]-[12].

The basic sense of adaptive control is on-line to estimatethe uncertainties of system, nonparametric uncertainty andparametric uncertainty, based on the measured signals. How-ever, conventional adaptive controller can not deal with thesystems with unknown dynamic structure. A adaptive fuzzycontrol system provide the advantage that both numerical andqualitative information are used in the construction and thetraining stages in which the fuzzy logic system is constructedfrom a collection of fuzzy IF-THEN rules, and the trainingalgorithm adjusts the parameters of the fuzzy logic system tomatch the input/output data. As pointed out in [13], the benefitof using a fuzzy model-based design can be straightforwardused to obtain controller and observer using fuzzy inferences,a direct adaptive fuzzy controller with state observer [14],an adaptive fuzzy neural controller with state observer [15],and a mixedH2/H∞ fuzzy output feedback control designfor nonlinear dynamic systems using an LMI approach wasproposed in[16]. A Fuzzy adaptive control scheme integrat-ing variable structure (VS) has been developed as a robuststrategy to treat uncertain control systems which may eveninclude delayed states. This technique usually leads to goodresults in trajectory tracking problems with good data fitting

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)Volume 2, 2020

ISSN: 2769-2507 26

for unknown and nonlinear systems (see, for example, [17]-[19],[20]-[22]). This is achieved through repetitive adjustmentof the system parameters for better tracking performance inthe presence of incomplete knowledge of the plant or evendelayed states [17], [18], [19].

This paper addresses the problem of designing robust track-ing controls for the tower crane systems with plant uncer-tainties, external disturbances and time delayed uncertaintyin which each uncertainty is assumed to be bounded by anunknown gain. VS andH∞ scheme are capable of tacklingnonlinear system with parameter uncertainties and externaldisturbances, and adaptive fuzzy control is independent ofsystem model, the tower crane system model is built to analyzesystem control characteristics. This technique will overcomemodeling inaccuracies, such as drag and friction losses, effectof time delayed uncertainty, as well as parameter uncertaintiessuch that the payload swing can be reduced as small aspossible when the load is moved from point to point. Theadaptive fuzzy scheme uses two on-line estimations, whichallows for the inclusion of identified gains of the delayed stateuncertainties and training of the weights of the fuzzy system,simultaneously. The adaptive fuzzy scheme fuses a VS schemeto resolve the system uncertainties, time delayed uncertaintyand the external disturbances the effect can be reduced to anyprescribed level to achieveH∞ tracking performance. Thecontrol laws are derived based on a Lyapunov criterion andthe Riccati-inequality such that all states of the system areUUB. Simulations and experimental results are presented toverify that the proposed method can reduce payload swing inthe process of drive a tower crane.

II. 3-D TOWER CRANE MODELING AND PROBLEM

FORMULATION

3-D tower crane model is a complex system, shown inFig. 1. The height and length of jib arm of tower crane areboth 1.2 m, the cable lengths is set at0.45 m, and tower andtrolley are driven by DC motor. The jib is capable of4.53 radrad rotation about the tower, the trolley moves radially alongthe jib via a lead screw, and a hoisting motor controls thesuspension cable length. The left and right sides of the tower,equipped with two limit switch, used to protect the jib’srotation about the tower. Tower and jib encoder connector formeasuring their the location, payload encoder for measure theangle of the payloadα andβ.

Fig. 1. 3-D tower crane.

Table I summarize the parameters of the tower crane. It

should be noted that most of these limits are enforced via soft-ware and are not the physical limitations of the system. Theselimitations are enforced to more closely match the operationalparameters of full-sized tower cranes. Fig. 2 shows a sketchof the three dimension tower crane payload configuration. TheX-axis is collinear in the direction of approach of the jib, theY -axis is perpendicular to the jib and parallel to the floor, andtheZ-axis is determined by the right-hand rule.

Mp

g

Z Y

-

X

xp

Mu

u

MF

O

Fig. 2. The general 3-D tower crane model.

TABLE IPARAMETERS OF THE3-D TOWER CRANE SYSTEM

Parameter NotationPayload length LMass oftrolley MMass of payload mTower Motor Mθ

Jib Motor MF

Cable Motor Mp

Slew Angle of Tower θγOut-of-Plane Angle αIn-Plane Angle βDistance Between Trolley and Tower xp

Payload Mass Relative toJ0 mrTrolley Mass Relative toJ0 MrMotor equivalent moment of inertia J0Acceleration Gain for Trolley Servo KmxAcceleration Gain for Tower Servo Kmr

The tower crane system is driven by three control inputstransmitted to the actuators. The three inputs are the voltageinput to motor,Mθ, causing rotation angleθγ , the input voltageto motor,MF , causing the trolley positionxp on the jib, andthe voltage input to motor,Mp, changing the cable lengthL. Five outputs are measured using encoder: tower angle, jibtranslation, cable length, and swaying angles. The five outputsarexp, θγ , L, α, andβ, wherexp is the distance between thetrolley and tower,θγ is the slew angle of the tower,L is thelength of the steel cable connecting the payload to trolley, andα andβ are deflection angles of the payload inY Z-plane andXZ-plane, respectively. To simplify the dynamic model of thetower crane motion, the following assumptions are made.

Assumption 1: The elasticity of the steel cable is neglectedand the cable is considered as a massless rigid rod, the hookand payload assembly are modeled as a point mass, and thepayload is much heavier than the cable.


ISSN: 2769-2507 27

According to the above descriptions and assumption, the idealtower crane model should be derived using Lagrange approachas follows [1]: The payload and trolley positions at sphericalcoordinates can be represented as:

Pp(t)= (xp(t)−L cos(α)(t) sin(β)(t))~i

=L sin(α)(t)~j−L cos(α)(t) cos(β)(t)~k (1)

Tp(t)=xp(t)~i (2)

respectively, where~i, ~j, and ~k are the unit vectors lyingalong the coordinate axesX , Y , andZ, respectively. Let thevelocity and acceleration of trolley beυx(t) = xp(t)~i andax(t) = xp(t)~i, respectively. Further, define the velocity andacceleration of the jib to beωr(t) = θr(t)~k andα(t) = θr(t)~k,respectively. All variables defined previously are functions oftime. To simplify the notation, the time argumentt will bedropped out from subsequent equations. Let the Lagrangianfunction be given by

L=Kh−Ph (3)

whereKh is the total kinetic energy given by

Kh=1

2m(Pp · Pp)+

1

2M(Tp · Tp)+

1

2J0θ

2r (4)

andPh the potential energy of the whole system given by

Ph=−mgL cosα cosβ (5)

where· is inner product, respectively. In order to calculate thekinetic energy of the payload, the velocity ofPp is given by:

Pp=∂Pp

∂t+ωr×Pp (6)

where × is outer product. Substituting the payload positionPp in (1) and the jib velocityωr into (6) leads to:

Pp=(xp − L cosα sin β − Lθr sinα+ Lα sinα sin β

−Lβ cosα cos β)~i+ (L sinα+ xpθr + Lα cosα

−Lθr sin β)~j+ (Lα sinα cosβ + Lβ cosα sin β

−L cosα cos β)~k (7)

Form (2), the velocity of the trolley can be obtained as:

Tp = xp~i (8)

Let the generalized force and generalized displacements begiven by F = [MF 0 Mθ 0]⊤ and qt = [xp β θr α]⊤,respectively. Then, the Lagrange’s equation is given by

d

dt(

∂L

∂ ˙qtjr

)− ∂L

∂qtjr

= Fjr , jr = 1, 2, 3, 4 (9)

Using (4)-(9) and assuming small angles and non-varyinglength steel cable, the following equations of motion areobtained for the tower crane:

Mxp +mgβ = MF (10)

Lβ + gβ − xp + Lθrα = 0 (11)

(J0 +Mx2p)θr −mgxpα = Mθ (12)

Lα+ gα+ xpθr − Lθrβ = 0 (13)

Dividing equations (10) and (12) byM andJ0, respectively,leads to

xp +mtgβ = MF (14)

(1 +Mrx2p)θr −mrgxpα = Mθ (15)

wheremt =mM

, Mr = MJ0

, MF = MF

M, andMθ = Mθ

J0. Let

the motor torquesMF andMθ be given by:

MF = KmxVax (16)

Mθ = KmrVaθ (17)

respectively, whereKmx andKmr are acceleration gains oftrolley and arm, respectively;Vax and Vaθ are the motorinput voltages of trolley and arm, respectively. Equations(11) and (14) represent the translational motion equationsand (13) and (15) the rotational motions. It is worth notingthat tower crane’s motion is affected by wind gusts, friction,and other disturbances which are not included in equationsof motion (10)-(17). Therefore, letW1, W2, W3 andW4 bethe disturbances acting on the tower crane system and letx11 = xp, x12 = xp, x21 = β, x22 = β, x31 = θr, x32 = θr,x41 = α, x42 = α, andu = [u1 u2]

⊤ = [Vax Vaθ]⊤ as the

control inputs to the tower crane. Then, equations (11), (13),(14), and (15) described above can be rewritten as follows:

x12 +mtgx21 +W1 = Kmxu1 (18)

Lx22 + gx21 − x12 + Lx32x41 +W2 = 0 (19)

(1 +Mrx211)x32 −mrgx11x41 +W3 =Kmru2 (20)

Lx42 + gx41 + x11 x6 − Lx32x21 +W4 = 0 (21)

By using some algebraic manipulations, equations (18)-(21)can be rewritten as follows:

x12=Kmxu1 −mtgx21 −W1 (22)

x22=Kmx

Lu1 − mtg

Lx21 − g

Lx21 − Kmrx41u2

(1 +Mrx211)

− mrgx11x241

(1 +Mrx211)+

W3x41

1 +Mrx211

− W1 +W2

L(23)

x32=Kmru2 +mrgx11x41

(1 +Mrx211)

− W3

(1 +Mrx211)

(24)

x42=(Kmru2 +mrgx11x41)x21

(1 +Mrx211)

− (x11 + Lx21)W3

L(1 +Mrx211)

− gx41

L− (Kmru2 +mrgx11x41)x11

L(1 +Mrx211)

− W4

L(25)

Assumption 2: During system operation, the length of thesteel cable,L, is kept fixed, i.e., the motorMp is stopped.

Assumption 3: Actuator nonlinearities are ignored in thederivation of the control model.

Due to backlash and similar phenomena, time delays mustbe included in the dynamic model. Therefore, the equationsof motion of the tower crane system are rewritten as follows:

x11 = x12

x12 = f1(x) +∑2

r=1 g1r(x)ur + h1(t)x11(t− τ1) + d1x21 = x22

x22 = f2(x) +∑2

r=1 g2r(x)ur + h2(t)x21(t− τ2) + d2x31 = x32

x32 = f3(x) +∑2

r=1 g3r(x)ur + h3(t)x11(t− τ3) + d3x41 = x42

x42 = f4(x) +∑2

r=1 g4r(x)ur + h4(t)x11(t− τ4) + d4xq(t) = φq(t), t ∈ [−τ , 0]

(26)

wheref1(x) = −mtgx3, f2(x) = −mrgx1x2

7

1+Mrx21

−(1+mt)gx3

L, f3(x) =

mrgx1x7

1+Mrx21

, f4(x) = mrgx1x3x7

1+Mrx21

− gx7

L− mrgx

2

1x7

L(1+Mrx21), g11(x) = Kmx,

g12(x) = 0, g21(x) = Kmx

L, g22(x) = − Kmrx7

1+Mrx21

, g31(x) = 0,


ISSN: 2769-2507 28

g32(x) = Kmr

1+Mrx2

1

, g41(x) = 0, and g42(x) = Kmr(Lx3−x1)

L(1+Mrx2

1)

are unknown but bounded nonlinear continuous functions,x = [x⊤

1 x⊤2 x⊤

3 x⊤4 ]

⊤ = [x11 x11 x21 x21 x31 x31 x41 x41 ]⊤ =

[x11 x12 x21 x22 x31 x32 x41 x42 ]⊤ = [x1 x2 · · · x8]

⊤ is thesystem state vector,ur is therth control input,d1 = −W1, d2 =W3x7

1+Mrx21

− W1+W2

L, d3 = − W3

(1+Mrx21), d4 = − (x1+Lx3)W3

L(1+Mrx21)− W4

L

are theexternal disturbance,τq, q = 1, 2, 3, 4, is the known,bounded time delay in the state, i.e.,0 < τq < τ forany positive constantτ , φq(t), q = 1, 2, 3, 4, is an initialcontinuous function on[−τ , 0], andhq(t), q = 1, 2, 3, 4, is thenonlinear time-varying continuous function which representsthe gain of the delayed state uncertainty associated with thedelayed state of the system and is assumed to be bounded, i.e.,|hq(t)| < ϕq, ∀t, whereϕq, q = 1, 2, 3, 4, is an unknown butpositive constant. For convenience and no loss of generality,let

ϑq = ϕ2q (27)

whereϑq is an unknown positive constant.Define

xτ = [x⊤1 (t− τ1) x

⊤2 (t− τ2) · · · x

⊤4 (t− τ4)]

⊤

Thus, equation (26) can be rewritten as

x = Aox+B(F(x) +G(x)u+ a⊤xτ + d)

= Aox+B(

F(x) +G(x)u+ (IH)⊤xτ + d)

(28)

where

F(x) =

f1(x)f2(x)f3(x)f4(x)

, G(x) =

g11(x) g12(x)g21(x) g22(x)g31(x) g32(x)g41(x) g42(x)

,

Ao = diagAo1,Ao2,Ao3,Ao4, B = diagB1,B2,B3,B4], u = [u1 u2]

⊤ is the control input,d = [d1 · · · d4]⊤,

a = diaga1(t) a2(t) · · · a4(t), H = diagh1(t), h2(t),h3(t), h4(t), and I = diagI1, I2, I3, I4, for which aq(t)

= [hq(t) 0]⊤, Iq = [1 0]⊤, Bq = [0 1]⊤, andAoq =

[

0 10 0

]

,

q = 1, . . . , 4.The proposed robust control law for payload positioning is based

on VS adaptive fuzzy control. The purpose of the paper is to synthe-size a adaptive fuzzy VS control scheme for payload positioning ofthe tower crane so that the tower crane system states can asymptoti-cally track the given desired trajectories. Tracking performance mustbe achieved in the presence of the unknown uncertain parameters, theunknown delayed state uncertainties, and the external disturbances.All signals in the overall system are UUB such that the payload swingcan be reduced as much as possible when the load is moved fromone point to another.

Let x=[x⊤d1

· · ·x⊤d4]⊤=[xd1 xd1 x12 xd2 x13 xd3 x14 xd4 ]

⊤=[xd1

xd2xd3xd4xd5xd6xd7xd8 ]⊤ be the tracking reference signal and

assume that there is a compact setΩd such thatxd ∈ Ωd, ∀t ≥ 0.Define the tracking error ase1 = x1 − xd1 , . . . , e4 = x4 − xd4

ande = x− xd = [e⊤1 · · · e⊤

4 ]⊤ = [e1 e2 · · · e8]⊤. Then, the error

dynamic equation can be obtained as

e = Aoe+B(

F(x) +G(x)u+ a⊤xτ + d − xd

)

(29)

where xd = [xd1 xd2 xd3 xd4 ]⊤. Choose a feedback gain matrix

K = diagK1,K2,K3,K4 with Kq = [Kq1 Kq2 ]⊤ such that the

characteristic polynomial ofAq = Aoq −BqK⊤q , q = 1, . . . , 4, to

be Hurwitz. DefineA = diagA1,A2,A3,A4 and rewrite (29)as

e = Ae+B(

Ke+ F(x) +G(x)u+ a⊤xτ + d− xd

)

(30)

Remark 1 : The input matrixG(x) as previously described, isassumed nonsingular and bounded, and satisfiesσ0I4×2 ≤ G(x) ≤σ1I4×2 for all x ∈ Ux, whereI4×2 is 4× 2 identity matrix, σ0,σ1 are positive constants, andUx ⊂ R8 is some compact set. Also,it is assumed that the external disturbance satisfiesd ∈ L2, i.e.,∫∞

0d⊤d dt ≤ ∞.

III. H∞ BASED ADAPTIVE FUZZY CONTROLLER DESIGN

A adaptive fuzzy system is defined as a Fuzzy Logic System(FLS) equipped with a learning algorithm, where the fuzzy systemis constructed from a set of fuzzy IF-THEN rules using fuzzy logicprinciples in the following form,

Rj : IF x1 is Cj1 andx2 is Cj

2 and· · · andx8 is Cj8 ,

THEN w is Ej , j = 1, 2, · · · , N (31)

wherexi, i = 1, 2, . . . , 8, and w are the input and output of thefuzzy logic system, respectively,N is the number of fuzzy rules,andCj

i , i = 1, 2, . . . , 8, andEj are fuzzy sets. Using the strategyof singleton fuzzifier, product inference engine, and center-averagedefuzzifier, the output of fuzzy system can be inferred as

w(x) =

∑N

j=1

∏8i=1 µC

ji

(xi)wj

∑N

j=1(∏8

i=1 µCji

(xi))(32)

whereµC

ji

(xi) is the membership function of thejth fuzzy set for

the input andwj = maxw∈R µEj (w) in which µEj (w) is themembership function of thejth fuzzy set for the output. Define thefuzzy basis functions as

ξj =

∏

i=1 µCji

(xi)∑N

j=1(∏8

i=1 µCji

(xi))(33)

Denotingξ(x) = [ξ1(x) ξ2(x) · · · ξN (x)]⊤, the learning algorithmadjusts the parameters of the fuzzy system based on the traininginformationθ = [w1 w2 · · ·wN ]⊤. Then (32) can be written as

w(x) = θ⊤ξ(x) (34)

Since fq and gqr, q = 1, . . . , 4, r = 1, 2, in (26) are unknown,they can be approximated by the fuzzy system (34) and expressed asfollows:

fq(x) = θ⊤fqξfq

(x)

gqr(x) = θ⊤gqrξgqr

(x)(35)

whereθfq andθgqr are the adjustable parameter vectors,ξfq(x) and

ξfqr(x) are the vector of fuzzy basis functions. Define

F(x) =[

f1(x) · · · f4(x)]⊤

= Ξf (x)Θf ,

G(x) =

g11(x) g12(x)...

...g41(x) g42(x)

= Ξg(x)Θg

whereΘf = [θ⊤f1θ⊤f2θ⊤f3θ⊤f4]⊤, Ξf (x) = diagξ⊤f1 , ξ

⊤f2, ξ⊤

f3,

ξ⊤f4, Ξg(x) = diagξ⊤g1 , ξ

⊤g2, ξ⊤

g3, ξ⊤

g4, and

Θg =

θ⊤g11 (x) θ⊤

g12(x)...

...θ⊤g41

(x) θ⊤g42

(x)

⊤

(36)

Furthermore, according to the universal approximation theorem, thereexists optimal approximation parametersΘ∗

f andΘ∗g which lead to


ISSN: 2769-2507 29

minimum approximation errors forF(x) andG(x), respectively, asfollows:

∆F(x) = F(x)− F(x|Θ∗f ) (37)

∆G(x) = G(x)− G(x|Θ∗g) (38)

where∆F(x) and ∆G(x) are the minimum approximation errorsassumed to be bounded, and

Θ∗f = arg min

Θf∈Ωf

supx∈Ux

|F(x|Θf )− F(x)|

Θ∗g = arg min

Θg∈Ωg

supx∈Ux

|G(x|Θg)−G(x)|

Define the estimated parameter errors as

Θf = Θ∗f −Θf (39)

Θg = Θ∗g −Θg (40)

The following VS scheme adaptive fuzzy-based control law can beobtained

u = G∗(x|Θg)

(

− F(x|Θf ) + xd −K⊤e+uh +us +ud

)

(41)

whereG∗ denotes the pseudo-inverse ofG, i.e., GG∗G = G, uh,us, andud are compensators described later for disturbance, uncer-tainties, and delay gain uncertainties, respectively. Then, substituting(41) into (28), the error dynamic equation can be obtained as

e=Ae+B(−Ξ⊤f Θf−Ξ

⊤g Θgu+∆F+∆Gu+uh

+us+ud+a⊤xτ+d)=Ae+B(−Ξ⊤

f Θf−Ξ⊤g Θgu

+∆F+∆GG∗(x|Θg)

(

− F(x|Θf ) + xd −K⊤e

+uh + us + ud

)

+ uh + us + ud + a⊤xτ + d) (42)

Throughout this paper, we need the following assumptions:

Assumption 4: There exist positive constantsκf > 0, κg >0, and a positive function 0 ≤ αg(x) < 1 suchthat |∆Fq| ≤ κf , |λq(∆G(x)G∗(x|Θg))| ≤ κg , andmaxq(

∑2r=1 |∆G(x)G∗(x|Θg)qr|) ≤ αg(x), where∆Fq is the

qth element of∆F, λq(∆G(x)G∗(x|Θg)) is theqth eigenvalue of∆G(x)G∗(x|Θg), and∆G(x)G∗(x|Θg)qr is theqrth element of∆G(x)G∗(x|Θg), q = 1, . . . , 4, r = 1, 2.

Theorem 1 : H∞ Performance: AnH∞ performance is consideredas follows [23]:∫ t

0

e⊤Qe dt ≤ e

⊤(0)Pe(0)+1

γfΘ

⊤f (0)Θf (0)+

1

γgΘ

⊤g (0)Θg(0)

+ν

2

4∑

q=1

∫ 0

−τq

e⊤q (ς)eq(ς) dς+ρ2

∫ t

0

(d⊤d+x

⊤dτ IϑI

⊤xdτ ) dt (43)

whereQ > 0, P = P⊤ > 0, andρ2 is the prescribed attenuationvalue which denote the worst case effect of the external disturbancesd and delay uncertainty boundϑ on tracking errore. The physicalmeaning of performance in (43) is that the effects ofd and ϑ one must be attenuated below a desired levelρ from the viewpoint ofenergy, no matter whatd andϑ are, i.e., theL2-gain fromd andϑto e must be equal to or less than a prescribed valueρ2. In general,ρ is chosen as a positive small value less than one for attenuation ofd andϑ.

Theorem 2: Consider the tower crane system (26) and the modelingsystem (29). Let the VS adaptive fuzzy-based control law be givenas in (41) with

uh = − 1

2(1− κg)R

−1B

⊤Pe (44)

us = −Me(x)

1− κg

sgn(B⊤Pe) (45)

ud = − 1

2ν(1− κg)ϑB

⊤Pe (46)

where Me , |(∆F + ∆GG∗(x|Θg)(−F + xd − K⊤e))q| is theabsolute value of theqth element of∆F + ∆GG∗(x|Θg)(−F +xd−K⊤e), q = 1, . . . , 4, R denotes the robustH∞ control gain, andP = P⊤ > 0 is a symmetric positive definite matrix and satisfyingthe Riccati-like equation

PA+A⊤P+Q+PB( 2ρ2I−R−1)B⊤P+νI ≤ 0 (47)

where0 < ρ < 1 denotes a prescribed attenuation level,ν is a positiveconstant and determined by the designer, andQ = Q⊤ > 0 is aweighting matrix. Moreover, the gain of delayed state uncertaintiesis updated by the following rule:

˙ϑ

⊤ =1

2νΓB

⊤Pee

⊤PB (48)

and the adaptive parameter adjustment laws are chosen as˙Θ

⊤f = γfΞ

⊤f B

⊤Pe (49)

˙Θ

⊤g = γgΞ

⊤g B

⊤Peu

⊤ (50)

whereΓ is positive constant matrix and determined by the designer,and γf > 0, γg > 0 are adaptation gains. Then, for anyt ≥ t0,e, ϑ(t), Θf and Θg are UUB and theH∞ tracking performancewithin the prescribed attenuation levelρ is achieved in the presenceof the unknown uncertain parameters, the unknown delayed stateuncertainties, and the external disturbances such that the payloadswing can be reduced as much as possible when the load is movedfrom point to point.

Proof: Consider the Lyapunov function candidate

V =1

2e⊤Pe+

1

2γfΘ

⊤f Θf +

1

2γgtr(Θ

⊤g Θg)

+1

2tr(ϑ

⊤Γ

−1ϑ) +

ν

2

4∑

q=1

∫ t

t−τq

e⊤q (ς)eq(ς) dς. (51)

The time derivative of V is

V =1

2e⊤Pe+

1

2e⊤Pe+

1

2γf( ˙ΘfΘ

⊤f +Θ

⊤f˙Θf ) +

1

2γgtr( ˙Θ⊤

g Θg

+Θ⊤g˙Θg) +

1

2tr( ˙ϑ⊤

Γ−1

ϑ+ ϑ⊤Γ

−1 ˙ϑ) +

ν

2(e⊤

e− e⊤τ eτ ) (52)

where eτ = [e⊤1 (t− τ1) · · · e⊤

4 (t− τ4)]⊤. Substituting (42) into (52)

leads to:

V =1

2e⊤(A⊤

P+PA)e+e⊤PBΞf (x)

⊤Θf

+e⊤PBΞg(x)

⊤Θgu+

ν

2(e⊤

e− e⊤τ eτ )+tr(ϑΓ−1 ˙

ϑ⊤)

+e⊤PB(∆F+∆GG

∗(x|Θg)(−F−xd−K⊤e))

+e⊤PB(I+∆GG

∗(x|Θg))(uh+us+ud)+e⊤PBa

⊤xτ

+e⊤PBd+

1

γf

˙Θ

⊤f Θf+

1

γgtr( ˙Θ⊤

g Θg) (53)

Substituting adaptive laws (49) and (50) into (53) leads to:

V ≤ 1

2e⊤(A⊤

P+PA)e+e⊤PB(IH)⊤xτ

+e⊤PB(∆F+∆GG

∗(x|Θg)(−F− xd −K⊤e))

+e⊤PB(I+∆GG

∗(x|Θg))(uh+us+ud)

+e⊤PBd +

ν

2(e⊤

e− e⊤τ eτ )+tr(ϑΓ−1 ˙

ϑ⊤) (54)


ISSN: 2769-2507 30

By completing the squares and using the control inputuh in (44) and|λq(∆G(x)G∗(x|Θg))| ≤ κg, we obtain the time derivative of theLyapunov function as follows:

V ≤ 1

2e⊤(A⊤

P+PA+ νI)e+1

2ρ2e⊤PBB

⊤Pe− ν

2(e⊤

τ eτ )

+e⊤PB(∆F+∆GG

∗(x|Θg)(−F− xd −K⊤e))

+1

2e⊤PB(

1

ρ2I−R

−1)B⊤Pe+ e

⊤PB(IH)⊤xτ

+e⊤PB(I+∆GG

∗(x|Θg))(us + ud) + tr(ϑΓ−1 ˙ϑ

⊤)(55)

Sincex = e + xd and xτ = eτ + xdτ , wherexdτ = [x⊤d1(t −

τ1) · · ·x⊤d4(t− τ4)]

⊤, we have

V =1

2e⊤(A⊤

P+PA+ νI)e+1

2e⊤PB(

1

ρ2I−R

−1)B⊤Pe

+e⊤PB(∆F+∆GG

∗(x|Θg)(−F− xd −K⊤e))

+e⊤PB(I+∆GG

∗(x|Θg))(us + ud) + ρ2d⊤d

2− ν

2(e⊤

τ eτ )

+e⊤PB(IH)⊤xdτ + e

⊤PB(IH)⊤eτ+tr(ϑΓ−1 ˙

ϑ⊤) (56)

Using the technique of squares completion and using the fact that˙ϑ =

˙ϑ, HI⊤IH = ϑ, and ‖a‖ <

√ϑ where ϑ is finite, and

substituting (46) and (48) into (56) leads to

V ≤ 1

2e⊤(A⊤

P+PA)e+1

2e⊤PB(

2

ρ2I−R

−1)B⊤Pe

+e⊤PB(∆F+∆GG

∗(x|Θg)(−F− xd −K⊤e))

+e⊤PB(I+∆GG

∗(x|Θg))us + ρ2d⊤d

2+

ρ2

2x

⊤dτ IH

×HI⊤xdτ − 1

2νe⊤PBϑB

⊤Pe+

1

2νe⊤PBϑB

⊤Pe

+1

2νtr(e⊤

PBϑB⊤Pe) +

1

2ρe⊤PBB

⊤Pe+

1

2νe⊤

e (57)

Knowing that ϑ = ϑ− ϑ and substituting (45) into (57) leads to

V ≤ 1

2e⊤(A⊤

P+PA+ νI)e+1

2e⊤PB(

2

ρ2I−R

−1)B⊤Pe

+ρ2d⊤d

2+

ρ2

2x

⊤dτ IϑI

⊤xdτ (58)

Hence, we have from (59)

V ≤ −1

2e⊤Qe+

1

2ρ2(d⊤

d+ x⊤dτ IϑI

⊤xdτ ) (59)

Therefore, whenever

‖e‖ ≥ρ√

d⊤d+ x⊤dτIϑI⊤xdτ

√

λmin(Q)(60)

we haveV ≤ 0, whereλmin(Q) denotes the minimum eigenvalue ofQ. In light of the Lyapunov stability theory of the delayed functionaldifferential equation and sinceρ is the design constant serving as anattenuation level, it can be concluded that for anyt ≥ t0, e, ϑ(t),Θf andΘg are UUB and theH∞ tracking performance is within theprescribed attenuation levelρ. This performance is achieved in thepresence of the unknown uncertain parameters, the unknown delayedstate uncertainties, and the external disturbances such that the payloadswing can be reduced as much as possible when the payload ismoved from point to point. Therefore, the overall system satisfies

the following relationship:∫ t

0

e⊤Qe dt ≤ e

⊤(0)Pe(0) +1

γfΘ

⊤f (0)Θf (0)

+1

γgΘ

⊤g (0)Θg(0) +

ν

2

4∑

q=1

∫ 0

−τq

e⊤q (ς)eq(ς) dς + ρ2

∫ t

0

(d⊤d

+x⊤dτ IϑI

⊤xdτ ) dt. (61)

This completes the proof.

The parameter update laws with projection algorithms [24],[25]are used to amend the self-adaptation law. In order to ensure theparametersΘf (t) andΘg(t) to be bounded for allt ≥ 0, combinetwo adaptation rules to keep estimated parameters inside a feasibleset. Reconsiderθgqr = [w1

gqr · · ·wNgqr ]

⊤ ∈ RN and ξgqr=

[ξ1gqr · · · ξNgqr ]⊤ ∈ RN for q = 1, . . . , 4, r = 1, 2 whereN is thenumber of fuzzy rules defined previously. Suppose the constraint setsΩ0f , Ωf , Ω0g , andΩg are specified asΩ0f , Θf |Θ⊤

f Θf ≤ αf,Ωf , Θf |Θ⊤

f Θf ≤ αf + σf, Ω0g , Θg|bjqr ≤ θjgqr ≤ cjqr,andΩg , Θg|bjqr − σg ≤ θjgqr ≤ cjqr + σg, respectively, whereαf , σf , σg, bjqr, cjqr, 1 ≤ j ≤ N , are all positive constants andcan be arbitrarily specified by the designer.Ωf , which is a closedball andΩg, which is a convex hypercube are constrained regionsfor approximation parametersΘf and Θg , respectively.Θf andΘg shall be updated with a smooth projection scheme delineatedas follows: small

Θf =

γf (Ξ⊤f B

⊤Pe− (‖Θf ‖2−αf )e⊤PBΞfΘf

σf‖Θf ‖2Θf )

If ‖Θf‖2 = αf ande⊤PBΞfΘf > 0,

γfΞ⊤f B

⊤Pe,otherwise.

(62)and

Θg =

γg(1 + (cjqr − θjgqr )/σg)ξj⊤

gqrB⊤Peu⊤

If (θjgqr > cjqr andξj⊤

gqrB⊤Pe > 0),

γg(1 + (θjgqr − bjqr)/σg)ξj⊤

gqrB⊤Peu⊤

If (θjgqr < bjqr andξj⊤

gqrB⊤Pe < 0),

γgΞ⊤g B

⊤Peu,

otherwise.(63)

for q = 1, . . . , 4, r = 1, 2, and1 ≤ j ≤ N .Remark 2 [26]: From Riccati-like equation (47) in Theorem 2, theattenuation levelρ2 to satisfies the following inequalities:

2

ρ2I−R

−1 ≤ 0 or ρ2I ≥ 2R (64)

such that the estimation and tracking errors of the closed-loopsystem with delayed states and outputs, modeling uncertainties, anddisturbances can be effectively minimize to achieveH∞ trackingperformance.

Based on the above analysis, the block diagram of the overall VSadaptive fuzzy control system is shown in Fig. 3.

IV. N UMERICAL SIMULATION AND EXPERIMENTAL

RESULTS

In this section, the tower crane system in Fig. 1 is used assimulation and experimental example. The experimental results arecarried out via the Matlab programming in a personal computer andmanufactured by the Quanser Inc. The parameters values of 3-D


ISSN: 2769-2507 31

+ ( )

,

,

= + [ ( ) + ( )

Tower Crane System

+ ( ) + ]

=

=

Fuzzy Logic System

= ( +( )

+ + + )

Control Law

, ,

=

=( )

( )

=( )

Adaptive Laws

=

=

=

Fig. 3. Tower crane control flow chart.

tower crane system are described as follows:Kmx = 0.9 m/s2V ,Kmr = 3.33 rad/s2V , M = 0.465 kg, m = 0.125 kg,mt = 0.268, Mr = 0.53, mr = 0.142, g = 9.81 m/s2, J0 =0.877 kg.m2, κf = κg = 0.1, ρ = 0.9, ν = 0.1, R = 0.1ρ2I4×4,Γ = 0.01I4×4 , γf = 10, γg = 0.01. The initial conditionsare chosen asx1(0) = x2(0) = x3(0) = x4(0) = x5(0) =x6(0) = x7(0) = x8(0) = 0, Θf (0) = 0.05I5000×1 , Θg(0) =[17I1×1250 0I1×1250 ; 3I1×1250 5I1×1250 ; 0I1×1250 19I1×1250 ;0I1×1250 0.1I1×1250 ], and ϑ(0) = 0.05I4×4. The feedback gainmatrix K = diagK1,K2,K3,K4 for K1 = [20.51 45.71],K2 = [30.01 1.16], K3 = [1.02 60.42], andK4 = [45.01 50.16]is chosen such that all the roots of the characteristic polynomialsof Aq = A0q − BqKq, q = 1, . . . , 4 have negative real parts.The external disturbances are given byd1 = 0.1 sin(t) exp(−0.2t),d2 = 0.1 cos(t) exp(−0.2t), d3 = 0.1 sin(t) exp(−0.2t), andd4 = 0.1 cos(t) exp(−0.2t) [24]. Furthermore, the tower cranemodel has a time-delay issue mentioned before which is in practice,a magnitude phenomenon. Accordingly, we prescribe the four delaysfor the crane system asτ1 = 0.2s, τ2 = 0.1s, τ3 = 0.15s, τ4 = 0.1s.The four corresponding time-varying gains which accompany thetime-delays areh1 = h3 = 0.01 sin(t) andh2 = h4 = 0.01 cos(t).The idea here is to give a command to the jib moving from theorigin O to 0.06 m (x1d = 0.06 m) and the tower rotating0.593 rad (x5d = 0.593 rad) from Y Z to XZ direction andhave the jib and tower reach the desired location with a zero finalspeed(x2d = x6d = 0), respectively. More importantly, we wantthe payload to reach the desired position with zero oscillation anglesand zero slew rates(x3d = x4d = x7d = x8d = 0). The weightingmatrix is chosenQ = 0.01I8×8 such that the positive definitesymmetric matricesP for ρ = 0.9 can be obtained as

P = diag

[

6.6069 1.51641.5164 2.3130

]

,

[

2.7306 0.63430.6343 2.4339

]

,

[

3.3820 0.82370.8237 2.3321

]

,

[

2.7684 0.65000.6500 2.4378

]

The results of simulation and experimental are depicted in Figs. 4-9. Fig. 4 shows the trajectories of statex1 and the reference signalxr1 = 0.06 m from the originO. Fig. 5 shows the trajectories of thejib position, x5, after motion of0.593 rad from the originO, i.e.,34 degrees. Figs. 6 and 7 show that the trajectories of payload swingin the anglex3, and out of plane anglex7. The input trajectories ofthe motor,u1 andu2, are shown in Figs. 8 and 9. The trajectories ofx1 approach to the reference signalxr1 = 0.06m in about20s forρ = 0.9 as shown Fig. 4. It is seen thatx5 converges to0.593 rad atsteady state in about8s for ρ = 0.9 as shown Fig. 5. By observing

Figs. 6 and 7, its are shown that the payload swing angle,x3 andx7,were close to experimental payload oscillations and both convergeto the neighborhood of zero. Since all states of system converge toreference signals at steady state, thus, the motor inputs are smallerthan transient state as shown Figs. 8 and 9. The results of simulationand experimental presented here reveal that the proposed schemeindeed improves the system performances including convergence ofthe estimations and tracking errors such that the payload swing issignificantly reduced as the payload is moved from point to point.

V. CONCLUSION

This study has successfully developed a robust adaptive fuzzycontroller applied to 3-D tower crane, include a crane positionadjustment, tracking and payload swing, experimental and simulationresults. The proposed control scheme is based on Lyapunov stabilityto suppress the influence of the external disturbances and eliminatethe fuzzy approximation errors and guarantees the swaying motionof the tower crane system to converge to the neighborhood of zero.It can be concluded from the work presented here that the proposedH∞ based adaptive fuzzy control scheme is effective in improving thesystem safety, smoothness of motion, precision of payload positioningand elimination of swaying angle. The results of simulation andexperimental show that the 3-D tower crane system which controlsthe position of the target for the rapid positioning and the positioningduring movement due to inertia caused by the payload swing issuppressed.

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ISSN: 2769-2507 32

0 5 10 15 20 25−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time (sec)

Pos

ition

(m)

ExperimentSimulation

Fig. 4. x1 (trolley position) tracking0.06 m .

0 5 10 15 20 25−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (sec)

An

gu

lar

Po

sitio

n (

rad

)

ExperimentSimuation

Fig. 5. x5 (jib position) tracking 0.593 rad.

0 5 10 15 20 25−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time (sec)

Ang

ular

Pos

ition

(rad

)

ExperimentSimuation

Fig. 6. x3-payload swing in-plane angle.

0 5 10 15 20 25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (sec)

An

gu

lar

Po

sitio

n (

rad

)

ExperimentSimuation

Fig. 7. x7-payload swing out-of-plane angle.

0 5 10 15 20 25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (sec)

Vo

ltag

e

ExperimentSimulation

Fig. 8. u1-trolley motor.

0 5 10 15 20 25−1

−0.5

0

0.5

1

1.5

Time (sec)

Vo

ltag

e

ExperimentSimuation

Fig. 9. u2-tower motor.

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ISSN: 2769-2507 33

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Robust Adaptive Fuzzy Control Design for 3-D Tower Crane