Automatica 93 (2018) 231–243

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Automatica

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Active vibration control of a flexible rod moving in water: Applicationto nuclear refueling machines✩

Umer Hameed Shah, Keum-Shik Hong *School of Mechanical Engineering, Pusan National University, 2 Busandaehak-ro, Guemjeong-gu, Busan 46241, Republic of Korea

a r t i c l e i n f o

Article history:Received 11 March 2017Received in revised form 5 November 2017Accepted 14 January 2018Available online 30 March 2018

Keywords:Nuclear reactorsVibration controlFluid–structure interactionDistributed-parameter systemsLyapunov stability

a b s t r a c t

This paper addresses a simultaneous control of the positions of the bridge and trolley and the vibrationsof the load of a nuclear refueling machine (RM) that transports nuclear fuel rods to given locations in thenuclear reactor. Hamilton’s principle is used to develop the equations of motion of the RM. The lateraland transverse vibrations of the fuel rods during their transportation in water are analyzed. In derivingthe control law, the nonlinear hydrodynamic forces acting on the rod are considered. Then, a boundarycontrol scheme is developed, which suppresses the lateral and transverse vibrations simultaneously inthe course of the transportation of the fuel rod to the desired locations. Furthermore, Lyapunov function-based stability analyses are performed to prove the uniform ultimate boundedness of the closed loopsystem aswell as the simultaneous control of the positions of the bridge and trolley under the influence ofnonlinear hydrodynamic forces. Finally, experimental and simulation results are provided to demonstratethe effectiveness of the proposed control scheme.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

The refueling machine (RM) in Fig. 1 is a type of overheadcrane that transports nuclear fuel rods to the desired locationsin the reactor of a nuclear power plant. This paper discusses asimultaneous control problem of the positions of the trolley andthe bridge of the RMand the residual vibrations of the fuel rod at itstarget position. Themain difference of the RM fromordinary cranesis that it should carry the fuel rods in water to minimize possibleradiation leaks from the fuel rods to the environment. During thetransference of a fuel rod, the rod is subjected to vortex-inducedforces as well as the drag forces caused by the movements of theRM. Consequently, these hydrodynamic forces affect the lateraland transverse deflections of the rod. If a rod undergoes a certainlevel of deflection, it can damage the fissile material inside therod. Certainly, the insertion of a damaged fuel rod into a reactorcore incurs a very serious safety hazard (Kim, 2010). Therefore, itis absolutely necessary to suppress unnecessary vibrations of therod to ensure the reactor’s safety. The main objective of this paper

✩ This work was supported by the National Research Foundation (NRF)of Korea under the Ministry of Science and ICT, Korea (grant no. NRF-2017R1A2A1A17069430 and NRF-2017R1A4A1015627). The material in this paperwas not presented at any conference. This paper was recommended for publicationin revised formbyAssociate Editor YoshihikoMiyasato under the direction of EditorThomas Parisini.

* Corresponding author.E-mail addresses: umershah@pusan.ac.kr (U.H. Shah), kshong@pusan.ac.kr

(K.-S. Hong).

is to develop a simultaneous control strategy for moving a fuel rodto its desired location inwater so that the suppression of the lateraland transverse vibrations is achieved at its target position.

The works on the control of nuclear fuel rods under the influ-ence of hydrodynamic forces are rare in the literature. Also, theexisting works discussed only the dynamics of the rod upon thecoolant flow (axial flow) after core-insertion (Chen, 1975; Pavlica& Marshall, 1966). The present paper scrutinizes the dynamicresponse of the fuel rod against the cross flow in the course ofits transportation. Recently, the authors have developed an un-derwater command shaping method for the suppression of thelateral vibrations of the rod generated by the movements of theRM (Shah, Hong, & Choi, 2017). The developed method there wassuccessful when the vortex-induced vibration of the rodwas small.However, it cannot be used when the speed of the RM becomeshigh. This is because the vortex-induced forces become high whenthe rod speed becomes high, and it causes the rod to vibratein the transverse direction, besides its lateral vibrations. In thispaper, a more realistic control model including the vortex-inducedhydrodynamic forces in the reactor as well as the inline forces is tobe developed, and a newcontrol law that suppresses the transversevibrations aswell as the lateral vibrations of the rod in the presenceof hydrodynamic forces will be developed.

The control problems of flexible systems in air have been ex-tensively studied over the past decades (Bialy, Chakraborty, Cekic,& Dixon, 2016; Canciello & Cavallo, 2017; Hong & Bentsman, 1994;Nguyen & Hong, 2012b). Particularly, the control problem of aflexible inverted pendulumon amoving cart in air is closely related

https://doi.org/10.1016/j.automatica.2018.03.0480005-1098/© 2018 Elsevier Ltd. All rights reserved.

232 U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243

Fig. 1. 3D schematic of the refueling machine (MFA denotes the master fuelassembly).

to the control problem of an overhead crane in air (Lin & Chao,2009; Park, Chung, Youm, & Lee, 2000). In comparison to thecontrol problem in air, the dynamics of an object in water aremorecomplicated. The effects of added mass, buoyancy force, viscousdamping force, nonlinear drag force, vortex shedding, etc. need tobe additionally considered. Several researchers have investigatedthe dynamics of flexible beams and cylinders in fluids, specificallyin water (Han & Xu, 1996; Troesch & Kim, 1991). In particular, thevortex shedding phenomenon has received a significant attention:Vortex shedding denotes the oscillating flow existing behind thebody in fluid, which is related to the transverse force (also calledthe lift force, but the word ‘‘lift’’ is reserved for the lifting motionof a crane) resulting in the transverse vibration (i.e., the vortex-induced vibration) of the rod (Williamson & Govardhan, 2004).Several experimental (Baarholm, Larsen, & Lie, 2006; Marcollo& Hinwood, 2006) and theoretical (Lie & Kaasen, 2006; Willden& Graham, 2001) results on the vortex-induced vibration havebeen reported with the aim of providing a realistic model for thetransverse force (Facchinetti, de Langre, & Biolley, 2004; Marzouk,Nayfeh, Akhtar, & Arafat, 2007).

Also, the dynamics of deep-sea oil exploration systems werediscussed in relation to the hydrodynamic forces acting along thelength of a beam (Furnes, 2000; He, Ge, How, Choo, & Hong, 2011;How, Ge, & Choo, 2009). In their formulation, one endwas assumedfixed to the seabed and the other, assuming a tip mass, was free(i.e., a vessel to which the riser was attached). In these studies,only the control of the beam itself was considered (not the vessel).However, the present paper considers a hybrid model of the RMsystem consisting of two lumped masses (i.e., the bridge and thetrolley) and a flexible master fuel assembly (one end of the flexiblebeam is affixed to the RM and the other is free), which vibrates inthe lateral and transverse directions when the RM moves. Severalcontrol techniques applicable to under-actuated mechanical sys-tems have been reported in the literature (Gao, Lam, &Wang, 2006;Gao, Sun & Shi, 2010; He & Ge, 2016; Hong, Sohn, & Hedrick, 2002;Li, Yan, & Shi, 2017; Liberzon, Nesic, & Teel, 2014; Ngo & Hong,2012a, b; Parisini & Zoppoli, 1994; Park, Chwa, & Hong, 2007; Pin &Parisini, 2011; Shah & Hong, 2014; Teel, Forni, & Zaccarian, 2013;Zhao, Liu, Guo, & Fu, 2017).

The research objective of this paper is to control both the rigidbody motions of the RM and the lateral and transverse vibrationsof the flexible rod. One way of achieving the suppression of theflexible rod’s vibrations is to implement a distributed control,which requires the mounting of several actuators and sensorsalong the length of the rod. Since such distributed control is de-signed based on a finite number of system modes, it suffers from

the spillover problemcaused by the actuation of unmodeledmodesand, eventually therefrom, instability (Meirovitch & Baruh, 1983).Therefore, to avoid such situation, the present paper pursues amore easily implementable boundary control technique that re-quires actuation and sensing only at the boundaries (Choi, Hong,& Yang, 2004; Nguyen & Hong, 2010; Yang, Hong, & Matsuno,2004, 2005a, b). For hybrid rigid and flexible systems, severalboundary control schemes have been reported in the literature(Esterhuizen & Levine, 2016; Hasan, Aamo, & Krstic, 2016). Lot-fazar, Eghtesad, and Najafi (2008) reported a boundary controlscheme for simultaneous control of the lateral vibration and tra-jectory tracking of an Euler–Bernoulli beam.However, their controlmodel did not include the actuator’s dynamics. Nguyen and Hong(2012a) presented a boundary control scheme for suppressing thelongitudinal and transverse vibrations of an axially moving stringwhile achieving velocity tracking. He and Ge (2015) then reporteda boundary control scheme for suppressing the vibrations of twoflexible solar panels attached to a satellite (rigid body). Later,He and Zhang (2017) developed a boundary control scheme tosuppress the twist and bending phenomena in the flexible wingsystem of a robotic aircraft. Recently, a neural network basedadaptive control strategy was proposed to suppress the vibrationsof a flexible robotic manipulator in consideration of input dead-zone nonlinearity (He, Ouyang, & Hong, 2017). However, all thesestudies addressed control problems for systems operating in air.

As for the system operating in water, such control problemsof marine risers and mooring systems are more relevant to thepresent work. Several boundary control schemes for the suppres-sion of the lateral vibrations of a riser system, which utilize ac-tuation and sensing at the top boundary of the riser through themovement of the vessel, have been proposed (Do & Pan, 2008; Heet al., 2011; Nguyen, Do, & Pan, 2013). The riser control problemhowever is limited to the vibration suppression of the riser only(i.e., the position control of the vessel was not considered). How-ever, it is important to restrain the vessel’s movement to avoid anypossible damage to themooring lines. In thework ofHe, Zhang, andGe (2014), a thruster-assisted robust adaptive boundary controlscheme was proposed to limit the vessel’s movement. Consideringall these works, the following challenge for our control problem isstated: Suppression of the lateral and transverse vibrations of therod in the course of themovement of the RM in pursuit of its targetposition and in the presence of nonlinear hydrodynamic forces.

The contributions of this paper are the following: First, a hybridmodel of the RM (consisting of two lumped masses representingthe bridge and the trolley and one flexible system representingthe master fuel assembly affixed to the RM) is newly developed.Hamilton’s principle is utilized in obtaining the equations of mo-tion of the coupled system, where both movements of trolleyand bridge inflict the lateral and transverse vibrations of the rodunder the effect of inline and transverse forces. The hydrodynamicforces appear as nonlinear functions of the velocities of the RM andthe rod. Second, a boundary control scheme for the simultaneoussuppression of the lateral and transverse vibrations of the rod andthe position control of the RM is designed using the Lyapunovmethod. The overall control scheme consists of two control laws:One for the trolley and the other for the bridge. When the RM issent to a target positionwith the developed control laws, the lateraland transverse vibrations of the rod are shown satisfactorily sup-pressed in both simulation and experiment. Third, the proposedcontrol scheme guarantees the uniform ultimate boundedness ofthe closed-loop system from the sense that the position errors ofthe bridge and the trolley go to zero when the fluid-induced forcesin the nuclear reactor go to zero.

U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243 233

2. Problem formulation

In Fig. 1, the master fuel assembly (i.e., the fuel rod) is assumedto be an Euler–Bernoulli beamwith a circular cross-sectionmovingin water. Let l, d, I, E, and mr denote the length, the diameter,the area moment of inertia, the Young’s modulus, and the massof the rod, respectively; u(x, t) and w(x, t) imply the lateral andtransverse deflections of the rod; y(t) and z(t) indicate the dis-placements of the bridge and the trolley moving along the j andk axes; mb and mt be the masses of the bridge and trolley, andfy(t) and fz(t) denote the associated control inputs, respectively.The derivatives with respect to t and x are denoted by ˙and ′ forsimplicity.

2.1. Hydrodynamic forces

For a rod moving in water, the interaction between the rod andthe surrounding water generates the hydrodynamic forces in theinline and normal directions of the motion of the rod; namely,the inline and normal forces, fI (x, t) and fN (x, t), respectively. Forinstance, a rod moving along the j-axis generates the inline andnormal forces in the ij- and ik-plane, respectively. In this paper,the inline force is adopted from the Morison equation (Troesch &Kim, 1991) as follows.

fI (x, t) = (π/4)ρwCa(x)d2v(x, t)

+ (1/2)ρwCd(x)d v(x, t) |v(x, t)| , (1)

where ρw is the density of water, Ca is the added-mass coefficient,Cd is the drag coefficient, and v(x, t) is the velocity of the rod. Thenormal force is given as follows (How et al., 2009).

fN (x, t) = (1/2)ρwCN (t)d v2(x, t) cos(ωvt + φ), (2)

where CN (t) is the time-varying normal coefficient due to thevortex shedding, φ is the phase angle, and ωv is the frequency ofthe vortex shedding given by (Williamson & Govardhan, 2004)

ωv = 2πStv(x, t)/d, (3)

where St is the Strouhal number. The normal coefficient in (2) isgiven as follows.

CN (t) = a1 cos(ωvt) + a2 cos(ωvt + π/2), (4)

where a1 = 1.23 and a2 = 0.042 (Marzouk et al., 2007).This paper investigates the underwater responses of the rod

caused by planar movements of the crane. Therefore, the hydro-dynamic forces caused by a simultaneous movement of the bridgeand the trolley are to be included. First, assuming only a bridgemovement, the velocity of the rod is given by v(x, t) = y(t) +

u(x, t). Then, the inline (fI,y) and normal (fN,z) forces are obtainedas follows.

fI,y(x, t) = (π/4)ρwCa(x)d2(y(t) + u(x, t))

+ (1/2)ρwCd(x)d(y(t) + u(x, t)) |y(t) + u(x, t)| , (5)

fN,z(x, t) = (1/2)ρwCN (t)d(y(t) + u(x, t))2 cos(ωvt + φ). (6)

Second, assuming only a trolley movement, the velocity of the rodbecomes v(x, t) = z(t)+w(x, t). Then, the inline and normal forcesbecome the following.

fI,z(x, t) = (π/4)ρwCa(x)d2(z(t) + w(x, t))

+ (1/2)ρwCd(x)d(z(t) + w(x, t)) |z(t) + w(x, t)| , (7)

fN,y(x, t) = (1/2)ρwCN (t)d(z(t) + w(x, t))2 cos(ωvt + φ). (8)

Assumption 1. There exist positive constants f I , f N ∈ R+ such that∥fI (x, t)∥ ≤ f I and ∥fN (x, t)∥ ≤ f N , ∀(x, t) ∈ [0, l] × [0, ∞).

This is true if the velocity of the RM and the turbulence/dis-turbance in the nuclear reactor are finite. Consequently, the bound-edness of the velocity of the RM will be shown later.

2.2. Equations of motion

In deriving the equations of motion, the friction forces in thebridge and trolley system are assumed negligible. The kinetic en-ergy T of the bridge, trolley, and flexible rod system is given asfollows.

T (t) = (1/2)((mb + mt )y2(t) + mt z2(t)

+ m∫ l

0

((y(t) + u(x, t))2 + (z(t) + w(x, t))2

)dx, (9)

where m = (mr/l) + (π/4)ρwCa(x)d2 is the combined mass perunit length of the rod and the fluid itself displaced by the rod. Thepotential energy U of the rod is given as follows.

U(t) = (EI/2)∫ l

0(u′′2(x, t) + w′′2(x, t))dx. (10)

Assumption 2 (Queiroz, Dawson, Nagarkatti, & Zhang, 2000, p. 230).If the kinetic energy T of the system in (9) is bounded for ∀t ∈

[0, ∞), then u(x, t) and w(x, t) are bounded for ∀t ∈ [0, ∞).

Assumption3 (Queiroz et al., 2000, p. 229). If the potential energyUin (10) is bounded for ∀t ∈ [0, ∞), then u′′(x, t), u′′′(x, t), w′′(x, t),and w′′′(x, t) are bounded for ∀t ∈ [0, ∞).

Let c be the viscous damping coefficient of water. Then, thedamping forces in opposition to the motions of the rod alongthe lateral and the transverse directions are given by −cu(x, t)and −cw(x, t), respectively (He et al., 2011), which will be addedin the equations of motion. Now, using Hamilton’s principle, thefollowing equations of motion are obtained.

M1y(t) + m∫ l

0u(x, t)dx = fy(t), (11)

EIu′′′′(x, t) + cu(x, t) + mu(x, t) =

−my(t) − fD,y(x, t) − fN,y(x, t), (12)

M2z(t) + m∫ l

0w(x, t)dx = fz(t), (13)

EIw′′′′(x, t) + cw(x, t) + mw(x, t) =

−mz(t) − fD,z(x, t) − fN,z(x, t), (14)

where fD,y = ρwCd(x)d(y(t) + u(x, t)) |y(t) + u(x, t)| /2, fD,z =

ρwCd(x)d(z(t) + w(x, t)) |z(t) + w(x, t)| /2, M1 = mb + mt +ml,and M2 = mt + ml. Eqs. (11) and (12) represent the dynamicsof the bridge and the lateral dynamics of the rod, whereas (13)and (14) indicate the dynamics of the trolley and the transversedynamics of the rod, respectively. Substituting y(t) of (11) into (12),the following equation representing the coupled lateral dynamicsof the RM and the rod is obtained.

EIu′′′′(x, t) + cu(x, t) + mu(x, t) = −fD,y(x, t) − fN,y(x, t)

− (m/M1)(fy(t) − m

∫ l

0u(x, t)dx

). (15)

Similarly, substituting z(t) in (13) into (14) reveals the followingequation for the coupled transverse dynamics of the RM and therod.

EIw′′′′(x, t) + cw(x, t) + mw(x, t) = −fD,z(x, t) − fN,z(x, t)

− (m/M2)(fz(t) − m

∫ l

0w(x, t)dx

). (16)

234 U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243

The boundary conditions are obtained as follows.

u(0, t) = u′(0, t) = u′′(l, t) = u′′′(l, t) =

w(0, t) = w′(0, t) = w′′(l, t) = w′′′(l, t) = 0. (17)

3. Control formulation

The control objective is to transport the fuel rod to a targetposition within the nuclear reactor and to suppress its lateraland transverse vibrations under the effect of inline and normalhydrodynamic forces during the course of its transportation. Basedon the Lyapunov direct method, control inputs to the bridge fy(t)and to the trolley fz(t) are designed such that the uniform ultimateboundedness of the lateral and transverse rod vibrations and theconvergence of the crane’s position errors to zero are achieved.

As shown in (15) and (16), themovements of the bridge and thetrolley are coupled with the lateral and transverse dynamics of therod. To achieve the stability of the coupled system, not only thepositions/velocities of the bridge and trolley need to be bounded,but also the boundedness of the lateral and transverse vibrationsneeds to be assured. For a given target position (yd, zd) of the RM,the position errors, ey and ez , are defined as follows.

ey(t) = y(t) − yd, ez(t) = z(t) − zd, (18)

where the desired positions and the respective initial positionerrors at t = 0 of the bridge and trolley are assumed to be boundedas follows.

|yd| ≤ ξ1, |zd| ≤ ξ2,⏐⏐ey(0)⏐⏐ ≤ ξ3, and |ez(0)| ≤ ξ4, (19)

where ξi(i = 1, 2, 3, 4) are positive constants.

3.1. Stability analysis

In this paper, the feedback control law is designed in sucha way that the kinetic and potential energies are bounded andconverge near to zero through the Lyapunov redesign method. Tostabilize the given system in (15)–(17), the following control lawsare proposed.

fy(t) = −k1ey(t) − k2y(t) − k3u′′′(0, t), (20)

fz(t) = −k4ez(t) − k5z(t) − k6w′′′(0, t), (21)

where ki(i = 1, 2, . . . , 6) are positive control gains. For notationalconvenience, instead of u(x, t) and u′(x, t), u and u′ are used, andsimilar abbreviations are employed subsequently. In implement-ing (20)–(21), the measurement of u′′′(0, t) and w′′′(0, t) have tobe done (for instance, strain gauge sensors attached at the topend part of the rod can be used) besides the measurement of thedisplacements/velocities of the bridge and trolley for feedback.

The following lemma is utilized in analyzing the stability of theproposed feedback control system.

Lemma 1 (Hardy, Littlewood, & Polya, 1959). Let ϕ(x, t) ∈ R bea function defined on x ∈ [0, l] and t ∈ [0, ∞) that satisfies theboundary condition ϕ(0, t) = 0 for ∀t ∈ [0, ∞), then the followinginequalities hold.∫ l

0ϕ2(x, t)dx ≤ l2

∫ l

0ϕ′2(x, t)dx, (22)

ϕ2(x, t) ≤ l∫ l

0ϕ′2(x, t)dx, ∀x ∈ [0, l]. (23)

If ϕ(x, t) satisfies the boundary conditions ϕ(0, t) = 0 and ϕ′(0, t) =

0 for ∀t ∈ [0, ∞), then the following inequalities also hold.∫ l

0ϕ′2(x, t)dx ≤ l2

∫ l

0ϕ′′2(x, t)dx, (24)

ϕ2(x, t) ≤ l3∫ l

0ϕ′′2(x, t)dx, ∀x ∈ [0, l]. (25)

Based on the total mechanical energy of the system, the follow-ing Lyapunov function candidate is introduced.

V (t) = V0(t) + V1(t) + V2(t), (26)

where

V0(t) = T + U = (1/2)(EI

∫ l

0(u′′2

+ w′′2)dx

+ (mb + mt )y2(t) + m∫ l

0(y(t) + u)2dx

+mt z2(t) + m∫ l

0(z(t) + w)2dx

), (27)

V1(t) = α1

∫ l

0u2dx + α2

∫ l

0w2dx + α3

∫ l

0u′′2dx

+ α4

∫ l

0w′′2dx + (α5/2)e2y(t) + (α6/2)e2z (t), (28)

V2(t) = mβ

∫ l

0(uu + ww)dx + β1

∫ l

0u(u − y(t))dx

+ β2

∫ l

0u(u + y(t))dx + β3

∫ l

0w(w − z(t))dx

+ β4

∫ l

0w(w + z(t))dx + β5

∫ l

0y(t)(y(t) + u)dx

+ β6

∫ l

0z(t)(z(t) + w)dx. (29)

Lemma 2. The Lyapunov function candidate in (26) is upper andlower bounded as follows.

0 ≤ γ1W1(t) ≤ V (t) ≤ γ2W2(t), (30)

where

W1(t) = y2(t) + z2(t) +

∫ l

0(u2

+ w2+ u′′2

+ w′′2)dx, (31)

W2(t) = y2(t) + z2(t) +

∫ l

0(u2

+ w2+ u′′2

+ w′′2)dx

+ e2y(t) + e2z (t), (32)

and γ1 and γ2 are positive constants.

Proof. See Appendix A.

Lemma 3. The time derivative of the Lyapunov function candidate in(26) can be upper bounded as follows.

V (t) ≤ −γV (t) + ε, (33)

where γ and ε are positive constants.

Proof. See Appendix B.

U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243 235

Lemma 4 (Queiroz et al., 2000). Let V (t) ∈ R be a non-negativefunction on t ∈ [0, ∞) that satisfies the differential inequality V (t) ≤

−γV (t) + ε where γ and ε are positive constants, then for ∀t ∈

[0, ∞)

V (t) ≤ V (0)e−γ t+ (ε/γ )(1 − e−γ t ). (34)

Theorem 1. Consider the plant given in (15)–(17) under Assump-tions 1–3. With the control laws (20) and (21), (ey(t), ez(t), u(x, t),w(x, t)) = (0, 0, 0, 0) of the closed loop system is uniformly ulti-mately bounded from the sense that the position errors of the bridgeand trolley and the lateral and transverse deflections of the rod are allbounded and furthermore they converge to zero when the hydrody-namic forces go to zero.

Proof. Lemma 2 shows that (26) is positive definite and de-crescent. Lemma 3 indicates that it is really a Lyapunov functionunder control laws (20)–(21) that assures the uniform ultimateboundedness of the closed loop system. What left is to show theexistence of a positive ε such that ε is bounded and actually goesto zero as the parameters in the control loop go to zero.

The following inequalities for lateral and transverse deflectionsare obtained from Lemma 1 and (26)–(28).(

EI + 2α3

2l3

)u2(x, t) ≤

(EI2

+ α3

)∫ l

0u′′2(x, t)dx ≤ V (t), (35)(

EI + 2α4

2l3

)w2(x, t) ≤

(EI2

+ α4

)∫ l

0w′′2(x, t)dx ≤ V (t), (36)

for ∀x ∈ [0, l], where the first inequalities in (35)–(36) are evidentfrom (25) and the second inequalities are from the first term of(27) and the third/fourth terms of (28), respectively. Eqs. (35)–(36)and Lemma 4 reveal that, for ∀x ∈ [0, l], the lateral and transversevibrations of the rod are uniformly (in time) bounded as follows.

|u(x, t)| ≤

√(2l3/(EI + 2α3))(V (0)e−γ t + (ε/γ )), (37)

|w(x, t)| ≤

√(2l3/(EI + 2α4))(V (0)e−γ t + (ε/γ )), (38)

for ∀x ∈ [0, l], where ε is defined in (B.13). The boundedness of ε isshown in (B.13). The convergence to zero when the hydrodynamicforces go to zero is to be shown. Eqs. (26) and (28) reveal that

(α5/2)e2y(t) ≤ V (t), (39)

(α6/2)e2z (t) ≤ V (t). (40)

Considering (39)–(40) and Lemma 4, the position errors of the RMare uniformly bounded as follows.⏐⏐ey(t)⏐⏐ ≤

√(2/α5)(V (0)e−γ t + (ε/γ )), (41)

|ez(t)| ≤

√(2/α6)(V (0)e−γ t + (ε/γ )). (42)

The inequalities (41)–(42) show that ey(t) and ez(t) are bounded,which implies that y(t) and z(t) are bounded. Also, the bounded-ness of V (t) implies that y(t) and z(t) are bounded (see (27) orLemma 2), which implies that u(x, t) and w(x, t) are bounded byAssumption 2, and again u′′(x, t), w′′(x, t), u′′′(x, t), and w′′′(x, t)are all bounded by Assumption 3. Also, for a cantilever beamunder distributed loads, u′′′(x, t) = (1/EI)

∫ l0 fh,y(x, t)dx and

w′′′(x, t) = (1/EI)∫ l0 fh,z(x, t)dx (Meirovitch, 1967, p.136). Since the

hydrodynamic forces are assumed bounded, u′′′(x, t) and w′′′(x, t)are bounded. Therefore, from (20)–(21), the control inputs arebounded.

Table 1Experimental/simulation parameters.

Parameter Unit Value

Rod length (l) m 1.0Rod diameter (d) m 0.008Rod mass (mr) kg 0.038Trolley mass (mt) kg 3.2Bridge mass (mb) kg 6.8Young’s modulus (E) GPa 3.89Density of water (ρw) kg/m3 1000Density of fuel rod (ρr ) kg/m3 1204.5Viscous damping coefficient (c) N-s/m 0.2Added-mass coefficient (Ca; see Han & Xu, 1996) – 0.93Drag coefficient (Cd; see Shah & Hong, 2014) – 1.28Strouhal number (St; see How et al., 2009) – 2.0

Finally, from (37)–(38) and (41)–(42), we obtain

limt→∞

⏐⏐ey(t)⏐⏐ ≤

√(2/k1)(ε/γ ), (43)

limt→∞

|ez(t)| ≤

√(2/k4)(ε/γ ), (44)

limt→∞

|u(x, t)| ≤

√(2l3/(EI + 2α3))(ε/γ ), (45)

limt→∞

|w(x, t)| ≤

√(2l3/(EI + 2α4))(ε/γ ). (46)

From (43)–(46), it can be seen that ey(t), ez(t), and u(x, t) andw(x, t), for ∀x ∈ [0, l], will converge to a small neighborhoodof zero if ε is small and the values of k1, k4, α3, α4, and γ arelarge. It is noted that α3, α4, and γ are the design parameters,but ε is dependent on α7, α8, β, β5, β6, δi (i = 1–4), and thehydrodynamic force, see (B.13) in Appendix B. The hydrodynamicforces, in turn, are dependent on y(t) and z(t), see (5)–(8). First,loosely analyzing, if the bridge and trolley velocities get to zero(i.e., the hydrodynamic forces are zero), then ε = 0. Therefore,there exists a time T0 such that, for ∀t ≥ T0, all the errors belong toa small neighborhood of zero.Mathematically, as seen in (B.13), ε isbounded by the values of α7, α8, β, β5, β6, and δi (i = 1–4), whereδi are pre-determined values but α7, α8, β, β5, and β6 belong tosome ranges. If using the physical parameter values in Table 1,(B.18)–(B.21) give the ranges of β, β5, and β6 as follows; 0 < β <

14.1, 0 < β5 < 0.00038, and 0 < β6 < 0.588. And α7 and α8 are inour disposal allowing any positive real number. An extreme casewill be to set them to α7 = α8 = β = β5 = β6 = 0, in which caseε becomes 0. However, this is not desirable, because the controlgains k3 and k6 in (B.11) are dependent on β5 and β6. Henceforth,the selection of small values of β5 and β6 are desirable to keep thecapability of vibration suppression in associationwith u′′′(0, t) andw′′′(0, t) terms in (20)–(21). In this work, the values of β5 and β6are set to β5 = 0.00017 and β6 = 0.259, which is a half point inindividual ranges. Therefore, in view of (B.11), k3 = 0.0023 andk6 = 3.5. □

4. Experiment and simulation

This section presents the experimental and simulation resultsof the developed control laws in Section 3. Fig. 2 shows theexperimental setup consisting of an Inteco 3D Crane combinedwith awater tank, inwhich a flexible rod is transported inwater bythe crane. To simulate the lateral and transverse deflections of therod upon the movements of the bridge and trolley, (15) and (16)were solved numerically by implementing a finite difference algo-rithm. Simulations were made using MATLAB and the parametersgiven in Table 1.

236 U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243

Fig. 2. Experimental setup.

Fig. 3 compares the simulated (black dashed lines) and experi-mental (blue solid lines) responses when a rod is transported to atarget position by following a trapezoidal command (see Fig. 3a).The target distance was 0.4 m and the maximum velocity of thebridgewas 0.1m/s. The bridge is shown to reach the target positionin 4.3 s, see Fig. 3b. The lateral and transverse deflections of the endpoint of the rod upon the trapezoidal velocity command are shownin Fig. 3c and d, respectively. It can be seen that the rod exhibitssignificant oscillations in response to the open loop trapezoidalcommand.

In Shah et al. (2017), a shaped command profile in water(called the underwater command shaping) was developed (theblue solid line in Fig. 4a),which can transport a flexible rod inwaterwith minimal lateral vibrations u(x, t). However, the developedunderwater (UW) shaped command may be effective only if thehydrodynamic forces are small (i.e., the bridge/trolley velocitiesare low and the lateral and transverse vibrations of the rod arenegligible). Fig. 4b compares the simulation results of the bridgedynamics (15) under the assumption of fN (x, t) = 0: The blackdashed line in Fig. 4b is the lateral deflection of the tip point ofthe rod upon the trapezoidal velocity profile (the dashed black linein Fig. 4a). The blue solid line in Fig. 4b is the lateral deflectionof the rod upon the UW-shaped velocity profile in Fig. 4a. Thisconfirms that the UW-shaped velocity profile provides less vibra-tions than the conventional trapezoidal velocity profile. Now, tosee the importance of the existence of the hydrodynamics forcesin (15), the same simulation has been done by assuming v(x, t) =

y(t) (i.e, the bridge velocity is small and therefore the lateral roddeflection is negligible, i.e., u = 0): The green dash–dot line inFig. 4b depicts this response. This reveals that a new control lawdifferent from Shah et al. (2017) needs to be developed, whichjustifies the contribution of this paper.

The boundary feedback control strategy in this paper is todrive the RM according to the UW-shaped velocity command(a reference signal) and simultaneously suppress the lateral andtransverse vibrations of the rod. The last terms in (20)–(21) are thefeedback terms for compensating the shear forces occurring at thetop end of the rod. These terms will actively suppress the lateral

and transverse vibrations, but in doing so the shaped commandwill be altered. Such change in the shaped command consequentlywill prevent the RM from reaching the original target positions.The first terms in (20)–(21) are the feedback signals for the RM’sposition errors, which intend to make the RM’s position errors tozero.

Fig. 5 demonstrates the results of the proposed control schemein both experiment (blue solid line) and simulation (red dash–dot line). Fig. 5a depicts the control input to the bridge, fy(t), andFig. 5b compares the lateral deflection upon the open loop trape-zoidal velocity profile in Fig. 3a and that of the proposed controllaws (20)–(21) upon the UW-shaped velocity profile in Fig. 4a. Aquick suppression of the residual vibration is shown. Fig. 5c showsthe control input to the trolley, fz(t), and Fig. 5d compares thetransverse deflections of the end point between the trapezoidalvelocity profile and the proposed control scheme. The red dash–dot lines in Fig. 5b and d represent the simulated endpoint lateraland transverse vibrations, respectively, under the proposed controlscheme. In experimenting/simulating control laws (20)–(21), thefollowing control gains were used.

k1 = 0.002, k2 = 0.0035, k3 = 0.0023, k4 = 0.0008,

k5 = 0.0026, and k6 = 3.5. (47)

In this process, some design parameters have been chosen asβ1 = 0.704, β2 = 0.705, β3 = 0.005, β4 = 1.405, β5 = 0.00017and β6 = 0.259, and the satisfaction of the following ranges of thedesigned parameters was also checked.

0 < α1 < 0.0138, 0 < α2 < 0.0138, 0 < α3 < 0.0824,

0 < α4 < 0.0824, α5 > 0.0022, α6 > 7.58,

0 < β < 14.1, 0 < δ1 < 0.2,

0 < δ2 < 0.2, 0 < δ3 < 0.458, 0 < δ4 < 1.158, (48)

which are from (B.11) and (B.14)–(B.27) in Appendix B.Fig. 6 illustrates the 3D plots of the lateral response of the entire

rod upon the trapezoidal velocity command (without control) andthat of the proposed boundary control scheme. Similarly, Fig. 7shows the 3D plots of the transverse response of the entire rodupon the two cases. It is seen that the lateral and transversevibrations were quickly suppressed at the target point by theproposed scheme. Fig. 8 illustrates the coupling effect between thedynamics of the bridge and trolley, in which Fig. 8b shows that thetrolley does not move in the case of the trapezoidal velocity profilewhereas it moves with the proposed boundary control law even ifonly the bridge is moved. This is because the proposed control lawtries to suppress the transverse vibration of the rod by moving thetrolley even in the presence of no trolley command.

5. Results and conclusions

This paper presented the position and vibration control prob-lem of a nuclear RM, which transports nuclear fuel rods to theirtarget positions in the nuclear reactor. First, a hybrid RM modelconsisting of two lumped masses (i.e., the bridge and the trolley)and a flexible fuel rod was developed in consideration of the non-linear hydrodynamic forces acting on the rod. Hamilton’s principlewas utilized in obtaining the equations of motion of the consid-ered system. Both the experimental and simulation results wereprovided to verify the developed RM model. Fig. 3 demonstratedthat the rod exhibits significant lateral and transverse vibrationswhen an open loop velocity command is used to transport the RMto the target position (0.4m along the j -axis; see the rod responsesin Fig. 3c and d). For validation purposes, the underwater shapedcommand developed in Shah et al. (2017) was simulated to drive

U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243 237

Fig. 3. Comparison of experimental and simulated responses of the rod to a trapezoidal velocity command: (a) Bridge velocity (trapezoidal command), (b) bridgedisplacement, (c) the lateral deflection of the end point, and (d) the transverse deflection of the end point.

the RM to the same target position, which successfully suppressedthe lateral vibrations of the rodunder the condition that the plant islinear (see the green dash–dot line in Fig. 4b). Fig. 4b also showed,however, that the underwater shaped command could not effec-tively suppress the rod vibrations in the presence of transversevibrations (i.e., the nonlinear dynamics with hydrodynamic forces,see the blue solid line in Fig. 4b). Therefore, the development of anewboundary feedback control scheme in this paper has been fullyjustified. Accordingly, Eqs. (20)–(21) consisting of two control lawswere developed, which suppress both the lateral and transverserod vibrations simultaneously, as the RM is transported to a targetposition (see the red dash–dot lines for simulation and the bluesolid lines for experiment in Fig. 5b and d).

In conclusion, the coincidence of experimental and simula-tion results demonstrates that the developed hybrid rigid–flexiblemodel for the nuclear RM is valid. Further, the developed boundarycontrol law can suppress the lateral and transverse vibrations of

the rod simultaneously while controlling the positions of the RM.A Lyapunov-function based stability analysis of the closed-loopsystem was performed to show that the rod vibrations as wellas the bridge and trolley position errors are uniformly ultimatelybounded in the presence of hydrodynamic forces. To verify theeffectiveness of the proposed boundary control scheme, experi-mental and simulation results were compared with the existingresult of Shah et al. (2017). In the future, the authors intend todevelop an adaptive boundary control scheme to handle systemuncertainties including the machine’s degradation and unknowndisturbances in the nuclear reactor.

Acknowledgments

The authors would like to thank the Associate Editor and theanonymous reviewers for their constructive comments in the revi-sion process.

238 U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243

Fig. 4. Simulated responses of the bridge dynamics (15) assuming fN = 0: (a) Bridge velocity, (b) lateral deflection (the black dashed line is upon the open loop trapezoidalcommand, the blue solid line is upon the UW-shaped command assuming v = y + u, and the green dash–dot line is upon the UW-shaped command assuming v = y).

Appendix A. Proof of Lemma 2

According to Young’s inequality (Hardy et al., 1959), V0(t) canbe upper and lower bounded as follows.

0 ≤ (1/2)((mb + mt )y(t)2 + mt z(t)2 + EI∫ l

0(u′′2

+ w′′2)dx)

≤ V0(t) ≤ (ml + ((mb + mt )/2))y2(t) + (ml + (mt/2))z2(t)

+ m∫ l

0(u2

+ w2)dx + (EI/2)∫ l

0(u′′2

+ w′′2)dx. (A.1)

By using Young’s inequality, all of the terms in V2(t) can beupper and lower bounded as follows.

−βm2

(∫ l

0(u2

+ w2)dx + l4∫ l

0(u′′2

+ w′′2)dx)

≤ βm∫ l

0(uu + ww)dx

≤βm2

(∫ l

0(u2

+ w2)dx + l4∫ l

0(u′′2

+ w′′2)dx)

, (A.2)

− (β1/2)(ly2(t) + 3l4

∫ l

0u′′2dx

)≤ β1

∫ l

0u(u − y(t))dx

≤ (β1/2)(ly2(t) + 3l4

∫ l

0u′′2dx

), (A.3)

− (β2/2)(ly2(t) + 3l4

∫ l

0u′′2dx

)≤ β2

∫ l

0u(u + y(t))dx

≤ (β2/2)(ly2(t) + 3l4

∫ l

0u′′2dx

), (A.4)

− (β3/2)(lz2(t) + 3l4

∫ l

0w′′2dx

)≤ β3

∫ l

0w(w − z(t))dx

≤ (β3/2)(lz2(t) + 3l4

∫ l

0w′′2dx

), (A.5)

− (β4/2)(lz2(t) + 3l4

∫ l

0w′′2dx

)≤ β4

∫ l

0w(w + z(t))dx

≤ (β4/2)(lz2(t) + 3l4

∫ l

0w′′2dx

), (A.6)

− (β5/2)(3ly2(t) +

∫ l

0u2dx

)≤ β5

∫ l

0y(t)(y(t) + u)dx

≤ (β5/2)(3ly2(t) +

∫ l

0u2dx

), (A.7)

− (β6/2)(3lz2(t) +

∫ l

0w2dx

)≤ β6

∫ l

0z(t)(z(t) + w)dx

≤ (β6/2)(3lz2(t) +

∫ l

0w2dx

). (A.8)

In view of (A.2)–(A.8), V2(t) can be upper bounded as follows.

V2(t) ≤ (1/2)(l(β1 + β2 + 3β5)y2(t) + l(β3 + β4 + 3β6)z2(t)

+ (βm + β5)∫ l

0u2dx + (βm + 3l4(β1 + β2))

∫ l

0u2dx

+ (βm + β6)∫ l

0w2dx + (βm + 3l4(β3 + β4))

∫ l

0w2dx

).

(A.9)

Similarly, V2(t) can be lower bounded as follows.

V2(t) ≥ −(1/2)(l(β1 + β2 + 3β5)y2(t) + l(β3 + β4 + 3β6)z2(t)

+ (βm + β5)∫ l

0u2dx + (βm + 3l4(β1 + β2))

∫ l

0u2dx

+ (βm + β6)∫ l

0w2dx + (βm + 3l4(β3 + β4))

∫ l

0w2dx

).

(A.10)

U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243 239

Fig. 5. Control performance of the developed control laws (20)–(21): (a) Control input to the bridge, (b) lateral deflection of the end point, (c) control input to the trolley,and (d) transverse deflection of the endpoint.

In view of (26), (A.1), and (A.9), V (t) can be upper bounded asfollows.

V (t) ≤ (1/2)((2ml + mb + mt + l(β1 + β2 + 3β5))y2(t)

+ (2ml + mt + l(β3 + β4 + 3β6))z2(t)

+ (2α3 + EI + βm + 3l4(β1 + β2))∫ l

0u′′2dx

+ (2α4 + EI + βm + 3l4(β3 + β4))∫ l

0w′′2dx

+ (2(α1 + m) + βm + β5)∫ l

0u2dx + α5e2y(t)

+ (2(α2 + m) + βm + β6)∫ l

0w2dx + α6e2z (t)

). (A.11)

Similarly, V (t) can be lower bounded as follows.

V (t) ≥ (1/2)((mb + mt − l(β1 + β2 + 3β5))y2(t)

+ (mt − l(β3 + β4 + 3β6))z2(t)

+ (2α3 + EI − βm − 3l4(β1 + β2))∫ l

0u′′2dx

+ (2α4 + EI − βm − 3l4(β3 + β4))∫ l

0w′′2dx

+ (2α1 − βm − β5)∫ l

0u2dx

+ (2α2 − βm − β6)∫ l

0w2dx

). (A.12)

In view of (A.11) and (A.12), we obtain the following inequality.

0 ≤ γ1W1(t) ≤ V (t) ≤ γ2W2(t),

where

W1(t) = y2(t) + z2(t) +

∫ l

0(u2

+ w2+ u′′2

+ w′′2)dx,

W2(t) = y2(t) + z2(t) +

∫ l

0(u2

+ w2+ u′′2

+ w′′2)dx

+ e2y(t) + e2z (t),

240 U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243

Fig. 6. 3D plot of the lateral response: (a) Response to the trapezoidal velocitycommand (i.e., without control), (b) response of the proposed control scheme.

γ1 = min

⎛⎜⎜⎜⎜⎝(mb + mt − l(β1 + β2 + 3β5))/2,(mt − l(β3 + β4 + 3β6))/2,(2α1 − βm − β5)/2, (2α2 − βm − β6)/2,(2α3 + EI − βm − 3l4(β1 + β2))/2,(2α4 + EI − βm − 3l4(β3 + β4))/2

⎞⎟⎟⎟⎟⎠ ,

γ2 = max

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(2ml + mb + mt + l(β1 + β2 + 3β5))/2,(2ml + mt + l(β3 + β4 + 3β6))/2,(2(α1 + m) + βm + β5)/2,(2(α2 + m) + βm + β6)/2,(2α3 + EI + βm + 3l4(β1 + β2))/2,(2α4 + EI + βm + 3l4(β3 + β4))/2,(α5/2), (α6/2)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

provided that the following inequalities hold.

β1 + β2 + 3β5 ≤ (mb + mt )/l, β3 + β4 + 3β6 ≤ mt/l,

βm + β5 ≤ 2α1, βm + 3l4(β1 + β2) ≤ 2α3 + EI,

βm + β6 ≤ 2α2, βm + 3l4(β3 + β4) ≤ 2α4 + EI. (A.13)

The lemma is proved. □

Fig. 7. 3D plot of the transverse response: (a) Response to the trapezoidal velocitycommand (i.e., without control), (b) response of the proposed control scheme.

Appendix B. Proof of Lemma 3

Differentiating (26) with respect to time yields

V (t) = V0(t) + V1(t) + V2(t), (B.1)

where

V0(t) + V1(t) = (m + 2α1)∫ l

0uudx + (m + 2α2)

∫ l

0wwdx

+ y(t)(M1y(t) + m

∫ l

0udx

)+ (EI + 2α3)

∫ l

0u′′u′′dx

+ z(t)(M2z(t) + m

∫ l

0wdx

)+ (EI + 2α4)

∫ l

0w′′w′′dx

+ my(t)∫ l

0udx + mz(t)

∫ l

0wdx + α5eyey + α6ez ez, (B.2)

where ey = ey(t) and ez = ez(t), for notational convenience.Substituting y(t) from (11) and z(t) from (13) into (B.2) gives

V0(t) + V1(t) = y(t)fy(t) + z(t)fz(t) + α5eyey + α6ez ez

+ my(t)∫ l

0udx + mz(t)

∫ l

0wdx + B1 + B2, (B.3)

U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243 241

Fig. 8. Displacements of the bridge and trolley only when the bridge is moved: The coupling effect appearing in the trolley dynamics due to the hydrodynamic forces hasbeen shown in the proposed scheme.

where

B1 = (m + 2α1)∫ l

0uudx + (m + 2α2)

∫ l

0wwdx, (B.4)

B2 = (EI + 2α3)∫ l

0u′′u′′dx + (EI + 2α4)

∫ l

0w′′w′′dx. (B.5)

Substituting u from (12) and w from (14) into (B.4) yields

B1 = −α7(EI∫ l

0uu′′′′dx + c

∫ l

0u2dx + my(t)

∫ l

0udx +

∫ l

0ufh,ydx)

− α8

(EI

∫ l

0ww′′′′dx + c

∫ l

0w2dx

+mz(t)∫ l

0wdx +

∫ l

0wfh,zdx

), (B.6)

where α7 = (m+2α1)/m, α8 = (m+2α2)/m, fh,y = fD,y+ fN,y, andfh,z = fD,z + fN,z . Integrating (B.5) by parts and applying boundaryconditions (17), the following equation is obtained.

B2 = (EI + 2α3)∫ l

0uu′′′′dx + (EI + 2α4)

∫ l

0ww′′′′dx. (B.7)

Substituting (B.6) and (B.7) into (B.3), defining α3 = α1EI/m andα4 = α2EI/m, and using Young’s inequality yields the followinginequality.

V0(t) + V1(t) ≤ y(t)fy(t) + z(t)fz(t) − 2α1y(t)∫ l

0udx

− 2α2z(t)∫ l

0wdx − α7(c − δ1)

∫ l

0u2dx

− α8(c − δ2)∫ l

0w2dx + α5eyey + α6ez ez

+ (α7/δ1)∫ l

0f 2h,ydx + (α8/δ2)

∫ l

0f 2h,zdx. (B.8)

The third term of (B.1) is rewritten as follows.

V2(t) = βm∫ l

0(u2

+ uu + w2+ ww2)dx + 2(β1 + β2)

∫ l

0uudx

− (β1 − β2)y(t)∫ l

0udx − (β1 − β2)y(t)

∫ l

0udx

+ 2(β3 + β4)∫ l

0wwdx − (β3 − β4)z(t)

∫ l

0wdx

− (β3 − β4)z(t)∫ l

0wdx + 2β5ly(t)y(t)

+ 2β6lz(t)z(t) + β5(y(t)∫ l

0udx + y(t)

∫ l

0udx)

+ β6(z(t)∫ l

0wdx + z(t)

∫ l

0wdx). (B.9)

First substituting (B.8) and (B.9) into (B.1), then fy(t) from (20), fz(t)from (21), u from (12), and w from (14) into the resulting equation,and using Young’s inequality yields the following inequality.

V (t) ≤ −(k2 + ((α5 − k1)/2) − (β5/2)(l + (1/m)))y2(t)

− (k5 + ((α6 − k4)/2) − (β6/2)(l + (1/m)))z2(t)

− (α7(c − δ1) + (β5/2) − α1 − βm)∫ l

ou2dx

242 U.H. Shah, K.-S. Hong / Automatica 93 (2018) 231–243

− (α8(c − δ2) + (β6/2) − α2 − βm)∫ l

0w2dx

− (β(EI − δ3l4) − (l4/2)(βm + β1 − β2))∫ l

0u′′2dx

− (β(EI − δ4l4) − (l4/2)(βm + β3 − β4))∫ l

0w′′2dx

− ((α5 − k1)/2)e2y − ((α6 − k4)/2)e2z

+ ((α7/δ1) + (β/δ3) + (β5/2m))∫ l

0f 2h,ydx

+ ((α8/δ2) + (β/δ4) + (β6/2m))∫ l

0f 2h,zdx, (B.10)

provided that the following conditions hold.

k1 ≤ α5, k2 ≥ (k1 − α5 + β5(l + (1/m)))/2, k3 = (EI/m)β5,

k4 ≤ α6, k5 ≥ (k4 − α6 + β6(l + (1/m)))/2, k6 = (EI/m)β6,

β2 − β1 = (c/m)β5, (β5/2)((c/m) + 1 − l) = α1 + (βm/2),

β4 − β3 = (c/m)β6, (β6/2)((c/m) + 1 − l) = α2 + (βm/2),

and β1 + β2 = β3 + β4 = βc/2. (B.11)

Now, the inequality (B.11) can be written as follows.

V (t) ≤ −γW2(t) + ε, (B.12)

where

ε = ((α7/δ1) + (β/δ3) + (β5/2m))∫ l

0f 2h,ydx

+ ((α8/δ2) + (β/δ4) + (β6/2m))∫ l

0f 2h,zdx

≤ ((α7/δ1) + (β/δ3) + (β5/2m))lf2h,y

+ ((α8/δ2) + (β/δ4) + (β6/2m))lf2h,z < ∞, (B.13)

γ = min (σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ8) ,

σ1 = k2 + ((α5 − k1)/2) − (β5/2)(l + (1/m)) > 0,

σ2 = k5 + ((α6 − k4)/2) − (β6/2)(l + (1/m)) > 0,

σ3 = α7(c − δ1) + (β5/2) − α1 − βm > 0,

σ4 = β(EI − δ3l4) − (l4/2)(βm + β1 − β2) > 0,

σ5 = α8(c − δ2) + (β6/2) − α2 − βm > 0,

σ6 = β(EI − δ4l4) − (l4/2)(βm + β3 − β4) > 0,

σ7 = (α5 − k1)/2 > 0, σ8 = (α6 − k4)/2 > 0.

Considering the constants σi = (i = 1–8) above, the conditionsin (B.11), and the inequalities in (A.13), the conditions for theexistence of the parameters (i.e., α3, α4, β, β1, β2, β3, β4, β5, β6,

δ1, δ2, δ3, δ4) are obtained as follows.

β1 = (c/2)((β/2) − (β5/m)) > 0, (B.14)

β2 = (c/2)((β/2) + (β5/m)) > 0, (B.15)

β3 = (c/2)((β/2) − (β6/m)) > 0, (B.16)

β4 = (c/2)((β/2) + (β6/m)) > 0, (B.17)

0 < β5 ≤ m(mb + mt )/(l(c + 3m)), (B.18)

0 < β6 ≤ m(mt )/(l(c + 3m)), (B.19)

0 < 2β5/m ≤ β ≤ (2/c)(((mb + mt )/l) − 3β5), (B.20)

0 < 2β6/m ≤ β ≤ (2/c)((mt/l) − 3β6), (B.21)

0 < α3 ≤((c/m) − 1 − l)(EI)2

4m + 6cl4 − 2EI((c/m) − 1 − l), (B.22)

0 < α4 ≤((c/m) − 1 − l)(EI)2

4m + 6cl4 − 2EI((c/m) − 1 − l), (B.23)

0 < δ1 ≤ c − (((mα3/EI) + mβ − (β5/2))/α5), (B.24)

0 < δ2 ≤ c − (((mα4/EI) + mβ − (β6/2))/α6), (B.25)

0 < δ3 ≤ (β5c/2βm) + (EI/l4) − (m/2), (B.26)

0 < δ4 ≤ (β6c/2βm) + (EI/l4) − (m/2). (B.27)

From inequalities (30) and (B.12), we obtain

V (t) ≤ −γ V (t) + ε,

where γ = γ /γ2 > 0 and 0 < ε < ∞. The lemma is proved. □

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Umer Hameed Shah received the B.E. and M.S. degreesin mechanical engineering from the National Universityof Sciences and Technology, Pakistan, in 2005 and 2012,respectively, and the Ph.D. degree in mechanical engi-neering from the School of Mechanical Engineering, Pu-san National University, Busan, Korea, in 2018. Dr. Shahhas received the Outstanding Paper Award from the In-ternational Control Conference of the Chinese AutomaticControl Society, Taiwan, in 2015. His research interestsinclude dynamics and vibration control of underactuatedmechanical systems and control of underwater systems.

Keum-Shik Hong received the B.S. degree in mechanicaldesign and production engineering from Seoul NationalUniversity, Seoul, Korea, in 1979, the M.S. degree in me-chanical engineering fromColumbiaUniversity, NewYork,NY, in 1987, and both the M.S. degree in applied mathe-matics and the Ph.D. in mechanical engineering from theUniversity of Illinois at Urbana–Champaign, Champaign,IL, in 1991.

Dr. Hong joined the School of Mechanical Engineer-ing at Pusan National University (PNU) in 1993, wherehe became a Professor in 2004. His Integrated Dynamics

and Control Laboratory was designated as a National Research Laboratory bythe Ministry of Education, Science and Technology (MEST) of Korea in 2003. In2009, under the auspices of the World Class University Program of the MEST ofKorea, he established the Department of Cogno-Mechatronics Engineering, PNU.Dr. Hong serves as the Editor-in-Chief (EiC) of the International Journal of Control,Automation, and Systems and served as EiC of the Journal ofMechanical Science andTechnology from 2008 to 2011, as an Associate Editor for Automatica from 2000to 2006. He also served as the President of the Institute of Control, Robotics andSystems (ICROS) in 2015 and is President-Elect of the Asian Control Association.He has received many awards including the Presidential Award of Korea in 2007,the ICROS Achievement Award in 2009, the IJCAS Contribution Award in 2010, andthe IEEE Academic Award of ICROS in 2016. He is an ICROS Fellow and a memberof the IEEE, ASME, ICROS, KSME, KSPE, KIEE, KINPR and the National Academyof Engineering of Korea. His current research interests include nonlinear systemstheory, brain–computer interface, adaptive control, distributed parameter systems,autonomous systems, and innovative control applications in brain engineering.