Ji Wang Axial Vibration Suppression College of Automotive

Ji WangState Key Laboratory of

Mechanical Transmission, and

College of Automotive, Engineering,

Chongqing University,

Chongqing 400044, China

e-mail: [email protected]

Shumon KogaDepartment of Mechanical and

Aerospace Engineering,

University of California, San Diego,

La Jolla, CA 92093-0411


Yangjun PiState Key Laboratory of

Mechanical Transmission, and

College of Automotive, Engineering,

Chongqing University,

Chongqing 400044, China


Miroslav KrsticFellow ASME

Department of Mechanical and

Aerospace Engineering,

University of California, San Diego,

La Jolla, CA 92093-0411


Axial Vibration Suppressionin a Partial Differential EquationModel of Ascending MiningCable ElevatorLifting up a cage with miners via a mining cable causes axial vibrations of the cable.These vibration dynamics can be described by a coupled wave partial differentialequation-ordinary differential equation (PDE-ODE) system with a Neumann intercon-nection on a time-varying spatial domain. Such a system is actuated not at the movingcage boundary, but at a separate fixed boundary where a hydraulic actuator acts on afloating sheave. In this paper, an observer-based output-feedback control law for the sup-pression of the axial vibration in the varying-length mining cable is designed by the back-stepping method. The control law is obtained through the estimated distributed vibrationdisplacements constructed via available boundary measurements. The exponential stabil-ity of the closed-loop system with the output-feedback control law is shown by Lyapunovanalysis. The performance of the proposed controller is investigated via numerical simu-lation, which illustrates the effective vibration suppression with the fast convergence ofthe observer error. [DOI: 10.1115/1.4040217]

Keywords: wave equation, PDE-ODE, moving boundaries, backstepping, vibrationcontrol

1 Introduction

1.1 Control of Mining Elevators. Compliant varying-lengthcable systems are widely applied in numerous industries, such aselevators and hoisters [1,2]. The cable’s property of “compliance”or its ability to “stretch” and contract causes mechanical vibra-tions especially in the ascending process when the vibratoryenergy is increasing [3,4], which leads to imprecise positioningand premature fatigue fracture. For the safe manipulation, vibra-tion suppression is important to avoid serious hazards. Hence, aneffective and feasible control design for vibration suppression ofthe varying-length compliant cable system lifting a cage shown inFig. 1 is required, where the vibration dynamics of the cable ele-vator is modeled as a distributed parameter system [2–5].

1.2 Control of Vibrating String/Cable Systems With Fixedand Moving Boundaries. The vibrating string is an infinite dimen-sional system described by a wave partial differential equation(PDE). Most of the existing studies about vibration control of com-pliant strings focus on the fixed length. An active boundary controlscheme was proposed in Refs. [6–8] to suppress the vibrations andregulate the transport velocity of the axially moving string system.In Refs. [9] and [10], the boundary control based on an integral bar-rier Lyapunov function was used to suppress the undesirable vibra-tions of the compliant string system. To guarantee stability underthe uncertainty of the model, a robust adaptive boundary controlwas developed in Ref. [11] for a class of compliant string systemsunder the unknown spatiotemporally varying distributed disturbanceand the time-dependent boundary disturbance. In Ref. [12], anadaptive boundary controller was designed to suppress vibrations

and control tension of a flexible marine riser. A cooperative controllaw based on the novel integral-barrier Lyapunov function was pro-posed for a nonuniform fixed-length gantry crane where the uncer-tain parameters were handled by two adaption laws in Ref. [13]. An“impedance matching” method [14], where the transfer functions ofdistributed parameter systems can be obtained through Laplacetransforms and the pole and zero locations are to be matched by thecontrol input design, was successfully applied in a fixed domainwave PDE, which is a linear time-invariant system describing avibrating string with a constant length L.

The time-varying length has a significant role on the vibrationdynamic characteristics of compliant string systems [3,15] andmakes the design of the controller more challenging. There arerelatively few studies dealing with vibration control problems ofvarying-length cables. The control problems for horizontally andvertically translating media with the varying length were investi-gated in Ref. [16]. A boundary control scheme was designed tosuppress the vibrations for a nonlinear varying length drilling risersystem in Ref. [17]. In Ref. [18], a boundary control law wasdeveloped to stabilize the transverse vibrations of a nonlinear ver-tically moving string system with the varying length. However, inthe literature, the actuators are required to follow the movingcage, which is difficult to achieve in the practical implementationdue to the inconvenient installation.

From a practical point of view, a control system where controlis applied through the fixed boundary opposite to the instability isneeded in the mining cable elevator. This is a more challengingtask than the classical collocated “boundary damper” feedbackcontrol [19]. In Ref. [19], a control problem for the stabilizationof an one-dimensional hyperbolic equation, which contains theinstability at its free end and the control input on the opposite end,was dealt with by using the backstepping method [20]. In Ref.[21], the first global result was proposed for hyperbolic equationswhere the actuator is not collocated with the source of the instabil-ity. In Refs. [22] and [23], adaptive control laws were developed

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedOctober 10, 2017; final manuscript received May 2, 2018; published online June 4,2018. Assoc. Editor: Davide Spinello.

Journal of Dynamic Systems, Measurement, and Control NOVEMBER 2018, Vol. 140 / 111003-1Copyright VC 2018 by ASME

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for one-dimensional hyperbolic equations, which had an actuatoron one boundary and the unknown anti-damping on the otherboundary. However, the spatial domain of their system is limitedto be constant in time.

1.3 Control of Partial Differential Equation-OrdinaryDifferential Equation Systems. The mathematical model ofvibration control for a varying-length string with a moving cagecan be formulated as stabilizing a coupled wave partial differentialequation-ordinary differential equation (PDE-ODE) system on atime-varying spatial domain with an uncontrolled Neumann typeinterface, as shown in this paper. For control design of PDE-ODEcascades, compensation of actuator dynamics governed by a heatPDE and a wave PDE was developed by Krstic [24] and [25],respectively. Designs of the boundary observer and the output feed-back controller for a class of hyperbolic PDE-ODE cascade systemswere developed in Refs. [26] and [27]. As a more challenging prob-lem, coupled PDE-ODE systems where the PDE state and the ODEstate act back simultaneously have been studied. In Ref. [28] and[29], the coupled heat PDE-ODE systems with Dirichlet typeuncontrolled interconnections were stabilized via the backsteppingtransformations. By the decomposition of the wave equation intotwo transport equations, the stabilization of a nonlinear ODE withactuator dynamics governed by a wave PDE through the Dirichletinterconnection on the moving boundary was developed in Refs.[30] and [31] based on the predictor-based feedback control. How-ever, the compensation of actuator dynamics of a wave PDEthrough the Neumann type interconnection is more challenging. InRef. [32], a PDE-ODE cascade system was extended from theDirichlet type interconnection to the Neumann type interconnec-tion. The control designs of coupled heat-ODE and wave-ODE sys-tems including the Neumann type interconnections were furtherdeveloped in Refs. [33] and [34], respectively, which focused onthe fixed domain PDE with the Dirichlet type actuation.

1.4 Results of the Paper

(1) Axial vibration dynamics of a mining cable elevator ismodeled as a PDE system by Hamilton’s principle inSec. 2.

(2) A state-feedback controller with explicit gain kernels isdesigned to stabilize the coupled wave PDE-ODE systemon a time-varying spatial domain with a Neumann typeinterconnection, which is shown in Sec. 3.

(3) A finite-dimensional observer with explicit gain kernels isalso designed to estimate the full distributed states of thevarying-length string only using measurable boundarystates in an anti-collocated setup, which is shown inSec. 4.

(4) The exponential stability of the observer-based output-feedback control system is proved via Lyapunov analysisin Sec. 5. The result is verified via numerical simulationsin Sec. 6 before the conclusions and future work inSec. 7.

1.5 Contributions of the Paper

(1) We extend the result in Sec. 6 of Ref. [32], which stabilizeda cascaded wave PDE-ODE system on a fixed domain via afull-state feedback controller to the stabilization of acoupled wave PDE-ODE system on a time-varying spatialdomain via an observer-based output-feedback controller.

(2) Compared with previous contributions for the wave PDE-ODE system with a Dirichlet type interconnection on a freeboundary studied in Refs. [30] and [31] which exploited thestabilization via predictor-based design, we deal with aproblem of a wave PDE coupled with an ODE in oneboundary through a Neumann type interaction at theinterface.

(3) This is the first contribution for the observer-based output-feedback stabilization of the coupled wave PDE-ODE sys-tem on a time-varying domain with the Neumann typeinterconnection.

(4) This is the first control design for the axial vibration sup-pression of the varying-length string with a payload, wherethe actuator acts at the boundary separated from thepayload.

1.6 Notation. Throughout this paper, the partial derivativesand total derivatives are denoted as

Fig. 1 The mining cable elevator: (a) original model and (b) simplified model

111003-2 / Vol. 140, NOVEMBER 2018 Transactions of the ASME


fx x; tð Þ ¼@f

@xx; tð Þ; ft x; tð Þ ¼

@f

@tx; tð Þ

_f l tð Þ; tð Þ ¼ _l tð Þfx l tð Þ; tð Þ þ ft l tð Þ; tð Þ

b0 xð Þ ¼ db xð Þdx

; _X tð Þ ¼ dX tð Þdt

2 Problem Formulation

A schematic of a mining cable elevator is depicted in Fig. 1.Because the catenary cable in Fig. 1(a) is much shorter than thevertical cable (comparing 70 m with 2000 m), we suppose that thevibrations on the catenary part are negligible, which gives the sim-plified model of a varying-length cable with a cage shown inFig. 1(b). Due to the help of the lateral guides, the transversevibrations in the vertical cable can be neglected since they aremuch smaller than the axial vibrations.

Two external forces are actuated. One is the motion controlforce Ua(t) driven by a motor and the other is the vibration controlforce Uv(t) manipulated by a hydraulic actuator at the floatingsheave. The axial transport motion z*(t) is rigid-body motionneglecting the compliant property of the cable in the fixed coordi-nate system O0, and _z�ðtÞ; €z�ðtÞ are the velocity and accelerationaccordingly. The dynamics of axial elastic deformations (vibra-tion displacements) u(x, t), with ut(x, t) being the vibration veloc-ity accordingly, are referred to the moving coordinate system Oassociated with the motion z*(t). Here, we assume that the motionstate z*(t) is controlled perfectly by the motion control force Ua(t)and acts as the known target hosting trajectory. Then the axialvibration dynamics u(x, t) on a prescribed time-varying domainl(t)¼L – z*(t) is derived by Hamilton’s principle [35] in thefollowing.

2.1 Modeling of Physical System. The kinetic energy Ek andthe potential energy Ep of the system Fig. 1(b) are represented as

Ek ¼1

2qðl tð Þ

0

ut x; tð Þ þ _z� tð Þ� �2

dxþ 1

2M ut 0; tð Þ þ _z� tð Þ� �2

þ 1

2

JD

R2D

_u l tð Þ; tð Þ þ _z� tð Þð Þ2 (1)

Ep ¼1

2EA

ðl tð Þ

0

u2x x; tð Þdxþ

ðl tð Þ

0

T xð Þux x; tð Þdx

þ qg

ðl tð Þ

0

u x; tð Þ þ z� tð Þ� �

dx

þMg u 0; tð Þ þ z� tð Þ� �

(2)

where EA¼E�Aa and T(x)¼ (Mþ qx)g is the static tension inthe cable.

The virtual work done by external forces is written as

dW ¼ UvðtÞdðuðlðtÞ; tÞ þ z�ðtÞÞ þ c1 _uð0; tÞduð0; tÞþ c2ð _uðlðtÞ; tÞ þ _z�ðtÞÞdðuðlðtÞ; tÞ þ z�ðtÞÞ (3)

Substituting Eqs. (1)–(3) into extended Hamilton’s principle

ðt2

t1

ðdEk � dEp þ dWÞdt ¼ 0 (4)

and apply the variational operation. Note that because the lengthof the cable l(t) changes with time, the domain of integration forthe spatial variable is time-dependent. The standard procedure forintegration by parts with respect to the temporal variable does notapply and some modifications are required. The use of Leibnitz’srule gives

ðl tð Þ

0

q ut x; tð Þ þ _z� tð Þ� �

dutdx

¼ 1

dt

ðl tð Þ

0

q ut x; tð Þ þ _z� tð Þ� �

dudx

�ðl tð Þ

0

q utt x; tð Þ þ €z� tð Þ� �

dudx

� _l tð Þq ut l tð Þ; tð Þ þ _z� tð Þð Þdu l tð Þ; tð Þ (5)

Integrating Eq. (5) from t1 to t2 yields

ðt2

t1

ðlðtÞ

0

qðutðx; tÞ þ _z�ðtÞÞdutdxdt

¼ �ðt2

t1

ðlðtÞ

0

qðuttðx; tÞ þ €z�ðtÞÞdudxdt

�ðt2

t1

_lðtÞqðutðlðtÞ; tÞ þ _z�ðtÞÞduðlðtÞ; tÞdt (6)

Applying Eq. (6) and following the standard procedure for inte-gration by parts with respect to the spatial variable, one obtainsfrom Eq. (4):

�qðuttðx; tÞ þ €z�ðtÞÞ þ EAuxxðx; tÞ � qgþ TxðxÞ ¼ 0 (7)

�Mðuttð0; tÞ þ €z� ðtÞÞ �Mgþ Tð0ÞþEAuxð0; tÞ þ c1utð0; tÞ ¼ 0

(8)

Uv tð Þ � EAux l tð Þ; tð Þ � T l tð Þð Þ � _l tð Þq ut l tð Þ; tð Þ þ _z� tð Þð Þ

� JD

R2D

€u l tð Þ; tð Þ þ €z� tð Þð Þ þ c2 _u l tð Þ; tð Þ þ _z� tð Þð Þ ¼ 0 (9)

Considering T(x)¼ (Mþ qx)g, we have the followingrelationships:

Tð0Þ ¼ Mg; TxðxÞ ¼ qg (10)

Inserting Eqs. (10), Eqs. (7)–(9) can be written as

�qðuttðx; tÞ þ €z�ðtÞÞ þ EAuxxðx; tÞ ¼ 0 (11)

�Mðuttð0; tÞ þ €z�ðtÞÞ þ EAuxð0; tÞ þ c1utð0; tÞ ¼ 0 (12)

Uv tð Þ � EAux l tð Þ; tð Þ � M þ ql tð Þð Þg� _l tð Þq ut l tð Þ; tð Þ þ _z� tð Þð Þ

� JD

R2D

€u l tð Þ; tð Þ þ €z� tð Þð Þ þ c2 _u l tð Þ; tð Þ þ _z� tð Þð Þ ¼ 0 (13)

2.2 Simplified Model for Controller Design. In the control-ler design, we assume the acceleration of the target reference €z�ðtÞis zero, which is reasonable because the velocity of the referencemotion z*(t) can be set to be uniform except for starting and stop-ping moments in the practical operation of the elevator.

Remark 1. Although we derive the control design under theassumption €z�ðtÞ ¼ 0 in Eqs. (11)–(13), we conduct the simulationbased on both the simplified model and the accurate model with-out the assumption €z�ðtÞ ¼ 0. The first one is to verify our theoret-ical result shown later about exponential stability of the output-feedback closed-loop system. The second one is to show that ourcontrol design is effective on vibration suppression of the miningcable elevator considered in this paper.

Then, the vibration dynamics Eqs. (11)–(13) can be written as

�quttðx; tÞ þ EAuxxðx; tÞ ¼ 0 (14)

�Muttð0; tÞ þ EAuxð0; tÞ þ c1utð0; tÞ ¼ 0 (15)

Journal of Dynamic Systems, Measurement, and Control NOVEMBER 2018, Vol. 140 / 111003-3


Uv tð Þ � EAux l tð Þ; tð Þ � M þ ql tð Þð Þg� _l tð Þq ut l tð Þ; tð Þ þ _z� tð Þð Þ

� JD

R2D

€u l tð Þ; tð Þ þ c2 _u l tð Þ; tð Þ þ _z� tð Þð Þ ¼ 0 (16)

Define the vibration control force as

UvðtÞ ¼ Uv1ðtÞ þ Uv2ðtÞ (17)

choosing Uv2(t) as

Uv2 tð Þ ¼ M þ ql tð Þð Þgþ _l tð Þq ut l tð Þ; tð Þ þ _z� tð Þð Þ

þ JD

R2D

€u l tð Þ; tð Þ � c2 _u l tð Þ; tð Þ þ _z� tð Þð Þ (18)

Therefore, Eqs. (14)–(16) can be obtained as

quttðx; tÞ ¼ EAuxxðx; tÞ; 8ðx; tÞ 2 ½0; lðtÞ� � ½0;1Þ (19)

�Muttð0; tÞ þ EAuxð0; tÞ þ c1utð0; tÞ ¼ 0 (20)

Uv1ðtÞ ¼ EAuxðlðtÞ; tÞ (21)

2.3 Description in Coupled Partial Differential Equation-Ordinary Differential Equation System. The axial vibrationdynamic system Eqs. (19)–(21) is a wave PDE with boundary con-ditions (21) and (20) described as a second-order ODE in time,which makes the problem difficult. To reduce the order of theboundary conditions, we introduce new variables x1(t) and x2(t)defined by

x1ðtÞ ¼ uð0; tÞ (22)

x2ðtÞ ¼ utð0; tÞ (23)

as the vibration displacement and the vibration velocity of thepayload. Then, the following relation is obtained:

_x1ðtÞ ¼ x2ðtÞ (24)

_x2 tð Þ ¼ �EA

Mux 0; tð Þ � c1

Mut 0; tð Þ (25)

Let XðtÞ 2 R2�1 be a state variable defined by

XðtÞ ¼ ½x1ðtÞ; x2ðtÞ�T (26)

Through the definition Eq. (26), we rewrite Eqs. (19)–(21) as thefollowing coupled PDE-ODE system:

_XðtÞ ¼ AXðtÞ þ Buxð0; tÞ (27)

uð0; tÞ ¼ CXðtÞ (28)

utt x; tð Þ ¼EA

quxx x; tð Þ (29)

EAuxðlðtÞ; tÞ ¼ Uv1ðtÞ (30)

where

A ¼0 1

0�c1

M

24

35; B ¼ EA

M

0

�1

" #; C ¼ 1; 0½ � (31)

The Neumann interconnection in ODE Eq. (27) physicallyamounts to the force acting on the cage.

Remark 2. The cage-guide boundary is damped when the damp-ing coefficients c1> 0 in Eq. (8). We process the control designbased on a more general model where c1 is arbitrary. It means theuncontrolled boundary in the wave equation can be damped(c1> 0), undamped (c1¼ 0), or even anti-damped (c1< 0).

The more general wave PDE-ODE model is considered in thecontrol design as

_XðtÞ ¼ AXðtÞ þ Buxð0; tÞ (32)

uttðx; tÞ ¼ quxxðx; tÞ (33)

uð0; tÞ ¼ CXðtÞ (34)

uxðlðtÞ; tÞ ¼ UðtÞ (35)

8ðx; tÞ 2 ½0; lðtÞ� � ½0;1Þ, where q is an arbitrary positive con-stant. A 2 R2�2; B 2 R2�1; C 2 R1�2 satisfy that the pair [A, B]is controllable and pair [A, C] is observable and CB¼ 0. XðtÞ 2R2 is the ODE state and u(x, t) � R is the state of the wave PDE.UðtÞ ¼ 1=EAUv1ðtÞ is the control input to be designed.

Remark 3. Our control design is also applicable to other physi-cal problems described by the wave PDE-stable/unstable/antista-ble ODE coupled model, such as controlling the torsionalvibration dynamics of drill strings with stick-slip instabilities aris-ing in deep oil drilling [36].

In the control design of this paper, the time-varying spatial domainl(t) is assumed to have following properties, which are reasonable forthe string’s length of the ascending mining cable elevator:

ASSUMPTION 1. There exists a lower bound l> 0, s.t.lðtÞ � l; 8t � 0.

ASSUMPTION 2. The domain length l(t) of the wave PDE isdecreasing, i.e., _lðtÞ � 0.

3 State-Feedback Control Design

In this section, we design the state-feedback controller, whichstabilizes the systems (32)–(35) with the full-state measurementsu(x, t) for 8x � [0, l(t)] and X(t). We seek an invertible transfor-mation that converts the (X, u)-system into the following stabletarget system (X, w), described as:

_XðtÞ ¼ ðAþ BKÞXðtÞ þ Bwxð0; tÞ (36)

wttðx; tÞ ¼ qwxxðx; tÞ (37)

wð0; tÞ ¼ 0 (38)

wxðlðtÞ; tÞ ¼ �dwtðlðtÞ; tÞ (39)

where d> 0 is a positive arbitrary damping gain. K is chosen tomake AþBK Hurwitz. Based on Refs. [25] and [37], the back-stepping transformation is formulated as

wðx; tÞ ¼ uðx; tÞ �ðx

0

cðx; yÞuðy; tÞdy

�ðx

0

hðx; yÞutðy; tÞdy� bðxÞXðtÞ (40)

where the kernel functions c(x, y) � R, h(x, y) � R, and bðxÞ 2R1�2 are to be determined. Taking second derivatives of Eq. (40)with respect to x and t, respectively, along the solution of Eqs.(32)–(35), we have



wtt x; tð Þ � qwxx x; tð Þ

¼ 2qd

dxc x; xð Þ

� �u x; tð Þ

þ q

ðx

0

hxx x; yð Þ � hyy x; yð Þ� �

ut y; tð Þdy

þ q

ðx

0

cxx x; yð Þ � cyy x; yð Þð Þu y; tð Þdy

þ 2qd

dxh x; xð Þ

� �ut x; tð Þ

� b xð ÞAB� qc x; 0ð Þ þ qhy x; 0ð ÞCB� �

ux 0; tð Þþ qh x; 0ð Þ � b xð ÞB� �

uxt 0; tð Þ

þ qb00 xð Þ � b xð ÞA2 � qcy x; 0ð ÞC�qhy x; 0ð ÞCA� �

X tð Þ ¼ 0

(41)

For Eq. (41) to hold, the following conditions must be satisfied:

d

dxc x; xð Þ ¼ 0 (42)

cxxðx; yÞ ¼ cyyðx; yÞ; (43)

d

dxh x; xð Þ ¼ 0; (44)

hxxðx; yÞ ¼ hyyðx; yÞ (45)

qhðx; 0Þ ¼ bðxÞB (46)

bðxÞAB ¼ qcðx; 0Þ � qhyðx; 0ÞCB (47)

qb00ðxÞ ¼ bðxÞA2 þ qcyðx; 0ÞCþ qhyðx; 0ÞCA (48)

Substituting the transformation Eq. (40) into Eqs. (36) and (38),and comparing them with Eqs. (32) and (34), we can chose b(x) tosatisfy

b0ð0Þ ¼ K � cð0; 0ÞC� hð0; 0ÞCA (49)

bð0Þ ¼ C (50)

By conditions Eqs. (42)–(45), c(x, y) and h(x, y) can be written as

cðx; yÞ ¼ mðx� yÞ (51)

hðx; yÞ ¼ nðx� yÞ (52)

Let D 2 R4�4 and K 2 R1�2 be defined as

D ¼0

1

qA2

I � 1

qBCAþ ABCð Þ

26664

37775 (53)

K ¼ 1

qCABC (54)

Solving Eqs. (46)–(50) with the help of Eqs. (51) and (52), theexplicit solutions of b(x), c(x, y), and h(x, y) are obtained as

bðxÞ ¼ ½C; K � K �eDx I0

� (55)

c x; yð Þ ¼1

qb x� yð ÞAB (56)

h x; yð Þ ¼1

qb x� yð ÞB (57)

where I 2 R2�2 is an identity matrix. For the mining elevatormodeled in Sec. 2, the solutions of gain kernels (55)–(57) are writ-ten as

b xð Þ ¼ �M

qk1 � k1 þ

qM

� �e

qMx; k2 � k2e

qMx

� (58)

c x; yð Þ ¼ k1 � k1 þqM

� �e

qM x�yð Þ (59)

h x; yð Þ ¼ k2 � k2eqM x�yð Þ (60)

where k1> 0, k2> 0 are controller gains such that K¼ [k1, k2]makes (AþBK) Hurwitz. For the boundary Eq. (39) to hold, thestate-feedback control law is given by

U tð Þ ¼ 1

N1

N2ut l tð Þ; tð Þ þ N3u l tð Þ; tð Þ

þ N4ux 0; tð Þ þ N5u 0; tð Þ þ N6X tð Þ

þðl tð Þ

0

N7u x; tð Þdxþðl tð Þ

0

N8ut x; tð Þdx

!(61)

where

N1 ¼ 1� dKB (62)

N2 ¼ �d (63)

N3ðlðtÞÞ ¼ cðlðtÞ; lðtÞÞ � qhxyðlðtÞ; lðtÞÞ (64)

N4ðlðtÞÞ ¼ dqhxðlðtÞ; 0Þ � dbðlðtÞÞB (65)

N5ðlðtÞÞ ¼ qdhxyðlðtÞ; 0Þ (66)

N6ðlðtÞÞ ¼ bxðlðtÞÞ þ dbðlðtÞÞA (67)

N7ðlðtÞ; xÞ ¼ cxðlðtÞ; xÞ þ qhxyyðlðtÞ; xÞ (68)

N8ðlðtÞ; xÞ ¼ hxðlðtÞ; xÞ þ dcðlðtÞ; xÞ (69)

In the same manner to obtain the direct transformation, we alsoobtain the inverse transformation

uðx; tÞ ¼ wðx; tÞ �ðx

0

uðx; yÞwðy; tÞdy

�ðx

0

kðx; yÞwtðy; tÞdy� aðxÞXðtÞ (70)

with

aðxÞ ¼ ½�C �K �eZx I0

� (71)

u x; yð Þ ¼1

qa x� yð Þ Aþ BKð ÞB (72)

k x; yð Þ ¼1

qa x� yð ÞB (73)



where

Z ¼ 0 qðAþ BKÞ2I 0

" #(74)

The detailed procedure to derive the inverse transformation (70) isshown in the Appendix.

4 Observer and Output-Feedback Control Design

In Sec. 3, a state-feedback controller is designed to stabilize theoriginal system exponentially. However, the designed state-feedback control law requires an infinite number of sensors toobtain the distributed states in a whole domain, which is not feasi-ble in practice. In this section, we propose an observer-based out-put feedback control law, which requires only a few boundaryvalues as available measurements. An exponentially convergentobserver is designed to reconstruct the distributed states using afinite number of available boundary measurements in Sec. 4.1 andthe output feedback control law based on the observer is proposedin Sec. 4.2. Suppose the available measurement of the system isX(t), which is not collocated with the actuator. In the mining cableelevator, the acceleration sensor placed at the cage with the inte-gration algorithm can be used to obtain X(t)¼ [u(0, t), ut(0, t)].Here, the initial condition of the vibration displacement at thecage can be obtained by the static equilibrium equation and theinitial velocity is zero.

4.1 Observer Design. The observer structure consists of acopy of the plant (32)–(35) plus the boundary state error injection,described as

_XðtÞ ¼ AXðtÞ þ Buxð0; tÞ þ LCðXðtÞ � XðtÞÞ (75)

uttðx; tÞ ¼ quxxðx; tÞ � D1ðXðtÞ � XðtÞÞ (76)

uð0; tÞ ¼ CXðtÞ � D2ðXðtÞ � XðtÞÞ (77)

uxðlðtÞ; tÞ ¼ UðtÞ (78)

The observer gains D1, D2, and L ¼ ½l1; l2�T are to be determined.Define the observer errors as

~uðx; tÞ ¼ uðx; tÞ � uðx; tÞ (79)

~XðtÞ ¼ XðtÞ � XðtÞ (80)

Then, subtracting Eqs. (75)–(78) from Eqs. (32)–(35) provides theobserver error system written as

_~XðtÞ ¼ ðA� LCÞ ~XðtÞ þ B~uxð0; tÞ (81)

~uttðx; tÞ ¼ q~uxxðx; tÞ þ D1~XðtÞ (82)

~uð0; tÞ ¼ D2~XðtÞ (83)

~uxðlðtÞ; tÞ ¼ 0 (84)

To convert the systems (81)–(84) into the following exponentiallystable target system described as:

_~XðtÞ ¼ ðA� LCÞ ~XðtÞ þ B ~wxð0; tÞ (85)

~wttðx; tÞ ¼ q ~wxxðx; tÞ (86)

~wð0; tÞ ¼ 0 (87)

~wxðlðtÞ; tÞ ¼ �d ~wtðlðtÞ; tÞ (88)

where L is chosen to make A� LC Hurwitz and d is an arbitrarypositive design parameter, the following direct and inverse trans-formations are formulated:

~uðx; tÞ ¼ ~wðx; tÞ �ðx

0

d0ðx; yÞ~wðy; tÞdy

�ðx

0

d1ðx; yÞ ~wtðy; tÞdy� CðxÞ ~XðtÞ (89)

~wðx; tÞ ¼ ~uðx; tÞ �ðx

0

d2ðx; yÞ~uðy; tÞdy

�ðx

0

d3ðx; yÞ~utðy; tÞdy� wðxÞ ~XðtÞ (90)

By matching Eqs. (81)–(84) and Eqs. (85)–(88), the followingconditions are obtained:

CðxÞAB ¼ qd0ðx; 0Þ (91)

qd1ðx; 0Þ ¼ CðxÞB (92)

qC00ðxÞ ¼ CðxÞðA� LCÞ2 þ D1 (93)

D2 ¼ �Cð0Þ (94)

C0ð0Þ ¼ 0 (95)

d1ðlðtÞ; lðtÞÞ ¼ �d (96)

d0ðlðtÞ; lðtÞÞ ¼ 0 (97)

d0xðlðtÞ; yÞ ¼ 0 (98)

d1xðlðtÞ; yÞ ¼ 0 (99)

The solutions of the gain kernels in Eq. (89) are obtained as

CðxÞ ¼ �½0; qd �½AB;B��1(100)

d0ðx; yÞ ¼ 0 (101)

d1ðx; yÞ ¼ �d (102)

The observer gains are obtained as

D1 ¼ ½0; qd �½AB;B��1ðA� LCÞ2 (103)

D2 ¼ ½0; qd �½AB;B��1(104)

Here, the matrix [AB, B] is invertible since the pair [A, B] iscontrollable.

For the mining elevator modeled in Sec. 2, the solutions of thegain kernels (100)–(102) are written as

C ¼ c1d

q;Md

q

" #(105)

d0 ¼ 0 (106)

d1 ¼ �d (107)

and then the observer gains Eqs. (103) and (104) are obtained as



D1 ¼ �d

qc1 l

2

1 � l2

� �þM l1l2 þ

c1l2M

� ��c1

c1

Mþ l1

� �"

þMc2

1

M2� l2

� �#(108)

D2 ¼ � c1d

q;�Md

q

" #(109)

A� LC can be Hurwitz by choosing positive parameters l1 > 0and l2 > 0.

4.2 Output-Feedback Control Design. To design theoutput-feedback controller, we consider the target ðX; wÞ-subsys-tem, which is constructed by the direct and inverse transforma-tions with the same gain kernels as the state feedback Eqs. (40)and (70). Hence, we introduce the following transformations fromðX; uÞ to ðX; wÞ described as:

wðx; tÞ ¼ uðx; tÞ �ðx

0

cðx; yÞuðy; tÞdy

�ðx

0

hðx; yÞutðy; tÞdy� bðxÞXðtÞ (110)


0


�ðx

0

kðx; yÞwtðy; tÞdy� aðxÞXðtÞ (111)

Taking time and spatial derivatives of Eq. (110) with the help ofgain kernels (55)–(57) and ðX; uÞ-system (75)–(78), we derive thefollowing coupled PDE-ODE ðX; wÞ-system:

_XðtÞ ¼ ðAþ BKÞXðtÞ þ Bwxð0; tÞþ ðLCþ Bcð0; 0ÞðC� D2ÞÞ ~XðtÞ (112)

wttðx; tÞ ¼ qwxxðx; tÞ � f1ðxÞ ~XðtÞ � f2ðxÞ ~wxð0; tÞ (113)

wð0; tÞ ¼ ðC� D2Þ ~XðtÞ (114)

wxðlðtÞ; tÞ ¼ �dwtðlðtÞ; tÞ (115)

where

f1ðxÞ ¼ bðxÞALCþ bðxÞLCðA� LCÞ

�ðx

0

hðx; yÞD1ðA� LCÞdy

�ðx

0

cðx; yÞD1dy (116)

f2ðxÞ ¼ �ðx

0

hðx; yÞD1Bdyþ bðxÞLCB (117)

By Eq. (115), the output-feedback control law is designed as

U tð Þ ¼ 1

N1

N2ut l tð Þ; tð Þ þ N3u l tð Þ; tð Þ

þN4 l tð Þð Þux 0; tð Þ þ N5 l tð Þð Þu 0; tð Þ

þN6 l tð Þð ÞX tð Þ þðl tð Þ

0

N7 l tð Þ; xð Þu x; tð Þdx

þðl tð Þ

0

N8 l tð Þ; xð Þut x; tð Þdx

!(118)

Remark 4. If time delay is considered, two ways can be used toaccommodate the delay: one is by incorporating damping into themodel and performing control design for that model, as was donein Refs. [38] and [39]. Another one is compensating a known timedelay at the input to a wave equation, as described in Ref. [40].

5 Stability Analysis

In this section, we establish the stability proof of the target sys-tem via Lyapunov analysis of PDEs. The equivalent stabilityproperty between the target system and the original system isensured due to the invertibility of the backstepping transforma-tion. The main theorem of this paper is stated in the following.

THEOREM 1. For any initial estimates ðuðx; 0Þ; Xð0ÞÞ compatiblewith the control law (118) and initial values (u(x, 0), ut(x, 0)),which belong to H1(0, L)� L2(0, L), the closed-loop system con-sisting of the plant (32)–(35) and the observer design (75)–(78)with the output-feedback control law (118) is exponentially stablein the sense of the norm

�ðlðtÞ

0

u2t ðx; tÞdxþ

ðlðtÞ

0

u2xðx; tÞdxþ

ðlðtÞ

0

u2t ðx; tÞdx

þðlðtÞ

0

u2xðx; tÞdxþ jXðtÞj2 þ jXðtÞj2

�1=2

(119)

Proof. First, we show the stability of ð ~X; ~wÞ-subsystem. Define

X1ðtÞ ¼ k~utðtÞk2 þ k~uxðtÞk2 þ j ~XðtÞj2 (120)

N1ðtÞ ¼ k~wtðtÞk2 þ k~wxðtÞk2 þ j ~XðtÞj2 (121)

where k~uðtÞk2is a compact notation for

Ð lðtÞ0

~uðx; tÞ2dx. In addi-tion, we employ a Lyapunov function

V1 ¼ ~XTðtÞP1

~XðtÞ þ /1E1ðtÞ (122)

where the matrix P1 ¼ PT1 > 0 is the solution to the Lyapunov

equation

P1ðA� LCÞ þ ðA� LCÞTP1 ¼ �Q1 (123)

for some Q1 ¼ QT1 > 0. The positive parameter /1 is to be chosen

later. E1(t) is defined as

E1 tð Þ ¼ 1

2k~wt tð Þk2 þ q

2k~wx tð Þk2

þ d1

ðl tð Þ

0

1þ xð Þ~wx x; tð Þ ~wt x; tð Þdx (124)

where the parameter d1 should satisfy

0 < d1 <1

1þ Lmin 1; qf g (125)

Then, we get

h11N1ðtÞ � V1ðtÞ � h12N1ðtÞ (126)

where

h11 ¼ min kmin P1ð Þ;/1

21� d1 1þ Lð Þð Þ;

/1

2q� d1 1þ Lð Þð Þ

�> 0 (127)



h12 ¼ max kmax P1ð Þ;/1

21þ d1 1þ Lð Þð Þ;

/1

2qþ d1 1þ Lð Þð Þ

�> 0 (128)

Time derivative of V1 along Eqs. (85)–(88) is obtained as

_V1 ¼ � dq/1 ~w2t l tð Þ; tð Þ � 1

2j _l tð Þj/1 ~w2

t l tð Þ; tð Þ

� q

2j _l tð Þj/1 ~w2

x l tð Þ; tð Þ � ~XT

tð ÞQ1~X tð Þ

þ 2BP1 ~wx 0; tð Þ ~X tð Þ þ d2

21þ l tð Þð Þ/1 ~w2

t l tð Þ; tð Þ

þ qd2 d1

21þ l tð Þð Þ/1 ~w2

t l tð Þ; tð Þ

� qd1

2/1 ~w2

x 0; tð Þ � d1

2/1k~wtk2 � d1

2q/1k~wxk2

þ j _l tð Þjdd1 1þ l tð Þð Þ/1 ~w2t l tð Þ; tð Þ (129)

Remark 5. Assumption 2 yields _lðtÞ ¼ �j _lðtÞj.Applying Young’s inequality to Eq. (129), the following

inequality is obtained:

_V1 � �1

2kmin Q1ð Þj ~X tð Þj2 � d1

2/1k~wtk2 � d1

2q/1k~wxk2

� dq� d1 1þ Lð Þ2

1þ qd2

� �� /1 ~w2

t l tð Þ; tð Þ

� j _l tð Þj 1

2þ d

2q

2� dd1 1þ Lð Þ

!/1 ~wt l tð Þ; tð Þ2

� qd1

2/1 �

2jP1Bj2

kmin Q1ð Þ

!~wx 0; tð Þ2 (130)

Therefore, combining with Eq. (125), the parameter d1 and /1 arechosen to satisfy the following:

0 < d1 <1

1þ Lmin 1; q;

2dq

1þ qd2;1þ qd

2

2d

( )(131)

/1 >4jP1Bj2

qd1kmin Q1ð Þ þ - (132)

with a positive parameter -. Then, we arrive at

_V1 � �r1V1 � -~wxð0; tÞ2 � �r1V1 (133)

where

r1 ¼1

h12

mind1

2/1;

d1

2q/1;

1

2kmin Q1ð Þ

�(134)

Next, we show the stability analysis of the ðX; wÞ-subsystem.Define

X2ðtÞ ¼ kutðtÞk2 þ kuxðtÞk2 þ jXðtÞj2 (135)

N2ðtÞ ¼ kwtðtÞk2 þ kwxðtÞk2 þ jXðtÞj2 (136)

Let V2 be a Lyapunov function written as

V2 ¼ XTðtÞP2XðtÞ þ /2E2ðtÞ (137)

where the matrix P2 ¼ PT2 > 0 is the solution to the following

Lyapunov equation:

P2ðAþ BKÞ þ ðAþ BKÞTP2 ¼ �Q2 (138)

for some Q2 ¼ QT2 > 0. The positive parameter /2 is to be chosen

later. Define E2(t) as

E2 tð Þ ¼ 1

2kwt tð Þk2 þ q

2kwx tð Þk2

þ d2

ðl tð Þ

0

1þ xð Þwx x; tð Þwt x; tð Þdx (139)

the parameter d2 must be chosen to satisfy

0 < d2 <1

1þ Lmin 1; qf g (140)

Similar with Eqs. (126)–(128), we get

h21N2ðtÞ � V2ðtÞ � h22N2ðtÞ (141)

where

h21¼min kmin P2ð Þ;/2

21�d2 1þLð Þð Þ; /2

2q�d2 1þLð Þð Þ

�> 0

(142)

h22¼max kmax P2ð Þ;/2

21þd2 1þLð Þð Þ; /2

2qþd2 1þLð Þð Þ

�> 0

(143)

Taking the time derivative of V2 along Eqs. (112)–(115), we get

_V2 ¼ /2q

ðl tð Þ

0

wt x; tð Þwxx x; tð Þdx

þ /2q

ðl tð Þ

0

wx x; tð Þwxt x; tð Þdx

� /2

ðl tð Þ

0

wt x; tð Þ f1 xð Þ ~X tð Þ þ f2 xð Þ ~wx 0; tð Þ�

dx

� /2j _l tð Þj 12

w2t l tð Þ; tð Þ � q/2j _l tð Þj 1

2w2

x l tð Þ; tð Þ

þ 1

2/2qd2 1þ l tð Þð Þw2

x l tð Þ; tð Þ � 1

2/2qd2w2

x 0; tð Þ

� 1

2/2qd2kwxk2 þ 1

2/2d2 1þ l tð Þð Þw2

t l tð Þ; tð Þ

� 1

2/2d2w2

t 0; tð Þ � 1

2/2d2kwtk2

þ /2_l tð Þd2 1þ l tð Þð Þwt l tð Þ; tð Þwx l tð Þ; tð Þ

þ _XT

tð ÞP2X tð Þ þ XT

tð ÞP2_X tð Þ (144)

Applying Young’s inequality to Eq. (144) as in Eq. (130), the fol-lowing inequality is obtained:



_V2 � �1

2/2d2 � B1 þ B2ð ÞL

� �kwtk2

� 1

2q/2d2kwxk2 � 1

2kmin Q2ð ÞjX tð Þj2

� 1

2q/2d2 �

1

2q2 � 4jP2Bj2

kmin Q2ð Þ

!w2

x 0; tð Þ

� /2 qd � d2

21þ Lð Þ 1þ qd2

� �� wt l tð Þ; tð Þ2

� /2j _l tð Þj 1

2þ qd2

2� dd2 1þ Lð Þ

� �wt l tð Þ; tð Þ2

þ 1

4/2

2 þ1

2/2

2 C� D2ð Þ2 A� LCð Þ2�

þ 4jP2 LCþ Bc 0; 0ð Þ C� D2ð Þ� �

j2

kmin Q2ð Þ

!j ~X tð Þj2

þ 1

4/2

2 ~w2x 0; tð Þ (145)

where Bi for i¼ 1, 2 are defined as

Bi ¼ maxx2½0;L�

fjfiðxÞj2g

Therefore, by choosing the parameter d2 and /2 as

0 < d2 <1

1þ Lmin 1; q;

2dq

1þ qd2;1þ qd2

2d

( )(146)

/2 ¼2

d2

max 2 B1 þ B2ð ÞL; q2þ 4jP2Bj2

qkmin Q2ð Þ

( )(147)

we arrive at

_V2 � �r2V2 þ n1j ~XðtÞj2 þ n2 ~w2xð0; tÞ (148)

where r2¼ l2/h22> 0, and

l2 ¼ min1

4/2d2;

1

2q/2d2;

1

2kmin Q2ð Þ

�(149)

n1 ¼1

4/2

2 þ1

2/2

2 C� D2ð Þ2 A� LCð Þ2

þ 4jP2 LCþ Bc 0; 0ð Þ C� D2ð Þ� �

j2

kmin Q2ð Þ (150)

n2 ¼1

4/2

2 (151)

Let V be the Lyapunov function of the overall ð ~X; ~w; X; wÞ-systemdefined as

V ¼ RV1 þ V2 (152)

Taking time derivative of Eq. (152) and using Eqs. (126), (133),and (148), we get

_V �� Rr1

2V1 � r2V2 �

Rr1h12

2� n1

� �j ~X tð Þj2

� R-� n2ð Þ~wx 0; tð Þ2 (153)

Therefore, choosing R sufficiently large, finally we arrive at

_V � �rV (154)

for some positive r. The differential inequality Eq. (154) deducesthat there exists a positive parameter g1> 0 such that

k~wtk2 þ k~wxk2 þ j ~XðtÞj2 þ kwtk2 þ kwxk2 þ jXðtÞj2

� g1ðk~wtð0Þk2 þ k~wxð0Þk2 þ j ~Xð0Þj2

þkwtð0Þk2 þ kwxð0Þk2 þ jXð0Þj2Þe�rt (155)

Therefore, the overall target system ð~w; ~X; w; XÞ is exponentiallystable. Due to the invertibility of the transformations Eqs. (90)and (110) as explicitly written in Eqs. (89) and (111), applyingPoincare’s, Young’s, and Cauchy–Schwartz inequalities to Eq.(155) in a similar manner as Theorem 16.1 in Ref. [37] yields

k~utk2 þ k~uxk2 þ j ~XðtÞj2 þ kutk2 þ kuxk2 þ jXðtÞj2

� g2ðk~utð0Þk2 þ k~uxð0Þk2 þ j ~Xð0Þj2

þkutð0Þk2 þ kuxð0Þk2 þ jXð0Þj2Þe�rt (156)

for some positive g2. Therefore, the exponential stability of theoverall original system ð~u; ~X; u; XÞ in the sense of Eqs. (120) and(135) is proved, which concludes Theorem 1 with the help of Eqs.(79) and (80).

6 Numerical Simulation

The simulation is performed based on the simplified model andthe accurate model. In detail, in the first case, the simplified model(27)–(30) under the designed state-feedback control law (61) andthe output-feedback control law (118) is conducted to verify thetheoretical result in Theorem 1. In the second case, the accuratemodel (11)–(13) with Eq. (18) under the designed output-feedback control law (118) is used to test the controller perform-ance on vibration suppression. Note that the control input Uv1

applied in both cases is Uv1¼EAU(t) where U(t) is the designedcontrol law and the constant EA¼E�Aa.

The physical parameters of the mining cable elevator used inthe simulation are shown in Table 1. To highlight the controllerperformance on vibration suppression, we make the dampingcoefficient c1 in the elevator be zero. The designed reference ofthe hoisting velocity _z�ðtÞ ¼ _lðtÞ is plotted in Fig. 2. The initialprofile of the vibration displacement is obtained by the force bal-ance equation at the static state, which is written as u(x,0)¼ –(qxgþMg)/EA. The initial velocity is defined as ut(x, 0)¼ 0because the initial velocity of the each point in the cable is zero.The initial conditions u(x, 0) and ut(x, 0) used in the simulationsatisfy the conditions in Theorem 1. The closed-loop responseswith the proposed control law (118) and the proportional–derivative (PD) control law, which is classically utilized in indus-tries, are examined to compare their performance to suppress theaxial vibrations of the mining cable. The PD control law is

UpdðtÞ ¼ kpuðlðtÞ; tÞ þ kd _uðlðtÞ; tÞ (157)

where kp and kd are gain parameters. The values of kp and kd aretuned to attain the efficient control performance. We have tested

Table 1 Physical parameters of the mining cable elevator

Parameters (units) values

Initial length L (m) 2000Final length (m) 200Cable effective steel area Aa (m2) 0.47� 10�3

Cable effective Young’s modulus E (N/m2) 1.03� 1010

Cable linear density q (kg/m) 8. 1Total hoisted mass M (kg) 15,000Gravitational acceleration g (m/s2) 9.8Maximum hoisting velocities Vmax (m/s) 15Total hoisting time tf (s) 150



different values of kp and kd, and the best regulating performanceis achieved with kp¼ 2000, kd¼ 7000 considering the overshootand adjusting time. The gains of the proposed controller d, d andK¼ [k1, k2] are chosen as d ¼ d ¼ 1 and [k1, k2]¼ [0.0035, 0.03]in the simulation. The numerical simulation is performed by thefinite difference method for the discretization in time and spaceafter converting the time-varying domain PDE to the PDE on afixed domain [0, 1] but with time-varying coefficients by introduc-ing g ¼ x=lðtÞ [4]. The time-step and space step are chosen as0.001 and 0.01, respectively.

6.1 The Vibration Suppression by the Proportional–Derivative Control and the Proposed Control Law. Figure 3shows the open-loop responses of the plant (27)–(30). It illustratesthat the large vibration is caused at both the cage and the midpointof the string during the total hoisting time. To suppress the vibra-tion, the closed-loop responses with the PD control law (157) andthe proposed control law are investigated and shown in Fig. 4. Itshows the vibration is suppressed and converges to zero on boththe proposed control law and the PD control. Moreover, it can beobserved that the responses with the proposed control law havefaster convergence and less overshoot than the responses with the

Fig. 3 The open-loop responses of the plant Eqs. (27)–(30).The large vibration is caused both at the moving cage and atthe midpoint of the cable: (a) the axial vibration at the movingcage and (b) the axial vibration at the midpoint.

Fig. 4 The closed-loop responses of the plant Eqs. (27)–(30)with the PD controller (157) (dashed line) and the proposedstate-feedback controller (61). While both controllers achievethe convergence to zero, the proposed controller achievesfaster convergence with less overshoot: (a) the axial vibrationat the moving cage and (b) the axial vibration at the midpoint ofthe cable.

Fig. 2 The hoisting velocity _z �(t)

Fig. 5 The responses of the closed-loop system Eqs. (27)–(30)and the observer design Eqs. (75)–(78) with the output-feedback control law (118). The observer achieves convergenceto the actual distributed state, and the associated output-feedback controller retains similar performance to the state-feedback: (a) the observer error of the axial vibration at the mid-point and (b) the axial vibration at the midpoint.



PD control law. Thus, the proposed control law shows better per-formance than the classical PD control.

6.2 The Responses With the Observer-Based Output-Feedback Control Law. With the available boundary measure-ments of the displacement and the velocity of the axial vibrationat the cage u(0, t) and ut(0, t), the estimated variables of the dis-tributed states required in the control law are obtained by the pro-posed observer (75)–(78). The closed-loop responses with anobserver-based output-feedback controller are simulated with theinitial observer error ~uðx; 0Þ ¼ 0:002ðmÞ uniformly. Then the ini-tial conditions of the observer used in the simulation are uðx; 0Þ ¼uðx; 0Þ þ 0:002 and Xð0Þ ¼ Xð0Þ þ ½0:002; 0�T, which satisfy theconditions in Theorem 1. The dynamics of the observer error andthe vibration displacement at the midpoint of the string are shownin Figs. 5(a) and 5(b), respectively. Because the locations of theactuator and the sensor are at the opposite boundaries, the stabili-zation and the estimation of the vibration at the midpoint x¼ l(t)/2is most challenging due to its accessibility. Figure 5(a) shows theobserver error converges to zero quickly, which implies that theestimation of the vibration displacements reconstructs their actualdistributed states. Figure 5(b) shows that the convergence to zeroof the vibration state at the midpoint is achieved with the output-

feedback control law as well, although the initial observer erroraffects the controller performance in the initial stage comparedwith the state-feedback response in Fig. 4.

6.3 The Responses of the Accurate Model. We process thecontroller design and stability analysis based on the simplifiedmodel. In this section, we test the performance of our controllerbased on the accurate model (11)–(13) with Eq. (18), whichincludes the boundary and distributed force disturbances from themotion acceleration €z�ðtÞ shown in Fig. 6. Applying the proposedoutput-feedback controller used in Sec. 6.2 and the PD controllerwith the coefficients kp¼ 3500, kd¼ 9000 which are adjusted toobtain efficient performance, the results under the two controllersare compared in Fig. 7. We can see that our controller also has thebetter performance on vibration suppression than the PD control-ler even though considering the boundary and distributed forcedisturbances from the motion acceleration €z�ðtÞ, which wouldintroduce saltation at 30 s and 120 s.

7 Conclusion and Future Work

In this paper, we propose an observer-based output-feedbackcontrol design for the axial vibration suppression of a varying-length mining cable with a cage. The dynamics is represented as acoupled wave PDE-ODE system with a Neumann type intercon-nection on a time-varying spatial domain. The proposed controldesign is practical for the installation of the actuator and sensor inthe mining cable elevator, in which the actuator is located not atthe moving cage but at the fixed boundary where a hydraulic actu-ator acts on a floating sheave. Exponential stability of the closed-loop system with the proposed observer-based output-feedbackcontrol law has been proved by Lyapunov analysis. The simula-tion results verify the theoretical analysis and illustrate that theproposed control law can effectively suppress the axial vibrationsof the mining cable elevator. In future work, uncertainties andunknown parameters in the dynamic model of the mining cableelevator will be dealt with in the control design.

Funding Data

� National Basic Research Program of China (973 Program)(Grant No. 2014CB049404).

� Fundamental Research Funds for the Central Universities(Grant No. 106112016CDJXY330002).

� China Scholarship Council (CSC) and Chongqing UniversityPostgraduates’ Innovation Project (CYD15023).

Nomenclature

Aa ¼ cross-sectional area of the cablec1 ¼ cage-guide damping coefficientc2 ¼ cable-head sheave damping coefficientE ¼ Young’s modulus of the cableg ¼ gravitational acceleration

JD ¼ moment of inertia of the druml(t) ¼ time-varying length of the cable

L ¼ initial length of the cableM ¼ mass of the load

RD ¼ radius of the drumu(x, t) ¼ axial vibration displacement

q ¼ linear density of the cable

Appendix

The inverse transformation is defined as


0


�ðx

0

kðx; yÞwtðy; tÞdy� aðxÞXðtÞ (A1)

Fig. 6 The hoisting acceleration

Fig. 7 The closed-loop responses of the accurate plant Eqs.(11)–(13) with Eq. (18) under the PD controller (157) (dashedline) and the proposed output-feedback controller (118): (a) theaxial vibration at the moving cage and (b) the axial vibration atthe midpoint of the cable



where kernel functions u(x, y) � R, k(x, y) � R and aðxÞ 2 R1�2

are to be determined. Taking second derivatives of Eq. (A1) withrespect to x and t respectively, and substituting them into Eq. (33),we get

uttðx; tÞ � quxxðx; tÞ¼ 2qðuyðx; xÞ þ uxðx; xÞÞwðx; tÞ

þ q

ðx

0

ðkxxðx; yÞ � kyyðx; yÞÞwtðy; tÞdy

þ q

ðx

0

ðuxxðx; yÞ � uyyðx; yÞÞwðy; tÞdy

þ 2qðkxðx; xÞ þ kyðx; xÞÞwtðx; tÞ� ðaðxÞ ~AB� quðx; 0ÞÞwxð0; tÞþ ðqkðx; 0Þ � aðxÞBÞwxtð0; tÞ

þ ðqa00ðxÞ � aðxÞ ~A2ÞXðtÞ ¼ 0 (A2)

where ~A ¼ Aþ BK. Recall the ODE (32) in the system (32)–(35)

_XðtÞ ¼ AXðtÞ þ Buxð0; tÞ¼ AXðtÞ þ Bwxð0; tÞ � Ba0ð0ÞXðtÞ¼ ðAþ BKÞXðtÞ þ Bwxð0; tÞ (A3)

Considering the boundary condition (34), we get

uð0; tÞ ¼ wð0; tÞ � að0ÞXðtÞ ¼ CXðtÞ (A4)

According to Eqs. (A2)–(A4), we get the following conditions onthe kernel functions in the inverse transformation:

uyðx; xÞ þ uxðx; xÞ ¼ 0 (A5)

kyðx; xÞ þ kxðx; xÞ ¼ 0 (A6)

uyyðx; yÞ � uxxðx; yÞ ¼ 0 (A7)

kyyðx; yÞ � kxxðx; yÞ ¼ 0 (A8)

a00ðxÞ � qaðxÞ ~A2 ¼ 0 (A9)

qkðx; 0Þ � aðxÞB ¼ 0 (A10)

quðx; 0Þ � aðxÞ ~AB ¼ 0 (A11)

�a0ð0Þ ¼ K (A12)

�að0Þ ¼ C (A13)

According to Eqs. (A9), (A12), and (A13), the solution of a(x) canbe obtained as

aðxÞ ¼ ½�C �K �eZx I0

� (A14)

where

Z ¼ 0 q ~A2

I 0

" #

Considering Eqs. (A10), (A11), and (A14), we can get

u x; yð Þ ¼1

q�C; �K�

eZ x�yð Þ I0

� ~AB (A15)

k x; yð Þ ¼1

q�C; �K�

eZ x�yð Þ I0

� B (A16)

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Ji Wang Axial Vibration Suppression College of Automotive