Control of a Three-Dimensional String System

3rd IFAC International Conference on Intelligent Control and Automation Science. September 2-4, 2013. Chengdu, China

Control of a Three-Dimensional String System Wei He ∗ , Shuang Zhang ∗∗ ∗

Robotics Institute and School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China. (e-mail: [email protected]) ∗∗ Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576. (e-mail: [email protected])

Abstract: In this paper, the control design and stability analysis are presented for a three-dimensional string system with the payload dynamics. A set of partial-ordinary differential equations (PDEs-ODEs) are developed by using the Hamilton’s principle to describe the motion of the three-dimensional string system. The dynamic model considers the comprehensive effects of environmental loads, which are critical for the analysis of a string system. Based on the Lyapunov’s direct method and the properties of the string system dynamics, three boundary control inputs are applied at the boundary to suppress the vibrations of the system under the external disturbances. Uniformly boundedness of the three-dimensional dynamics with the proposed control is achieved. Exponential stability is proved via the Lyapunov’s direct method when there is no distributed disturbance. Simulation examples are provided by using the finite difference method, and some useful conclusions are drawn. Keywords: Partial differential equations (PDEs), boundary control, distributed parameter system, Lyapunov’s direct method. 1. INTRODUCTION Vibration suppression problems of flexible systems in industry such as oil drilling and gas exploration in offshore engineering have received increasing attention. Due to the effects of external loads induced by winds, currents, waves, etc., flexible systems may produce large vibrations in a three-dimensional space. Large vibrations would degrade the performance of the systems and produce premature fatigue problems. Therefore, accurate analysis and reliable improvement should be performed for flexible systems to predict extreme responses at different points under environmental loads. A flexible string system can be used to model a number of mechanical systems in industry, such as the crane cable for transferring the payload d’Andrea Novel and Coron (2000), thruster assisted mooring systems for positioning marine vessels, etc.. For dynamic analysis, the flexible string system is regarded as a distributed parameter system (DPS) which is mathematically represented by partial differential equations (PDEs) coupled with ordinary differential equations (ODEs). Many remarkable results Lin and Lewis (1994); Yang et al. (2005); Ge et al. (2010); Kim and Hong (2009); He et al. (2013c); Morgul (1994); Huang and Xu (2011); Choi et al. (2004); He et al. (2010); Do and Pan (2008); He and Ge (2012) have been presented for the DPSs. For purpose of suppressing vibrations of the flexible systems, in literatures, boundary control schemes which the 978-3-902823-45-8/2013 © IFAC

730

control inputs are implemented at the boundaries to control all the modes, haven been proposed on the basis of original PDE model Hong and Bentsman (1994); He et al. (2011a); Bentsman and Hong (1991); He et al. (2013b,a). A crane model is stabilized by using the angle feedback in Rahn et al. (1999). In Zhang et al. (2012), the vibrations of a nonuniform string is regulated by using boundary control with the consideration of the spatiotemporally varying tension and disturbance. The authors in Nguyen and Hong (2010) propose a boundary control law of an axially moving string based on the Lyapunov redesign method. Boundary control is designed for a cable with a gantry crane modeled by a string structure in Rahn et al. (1999), and the experiment is implemented to verify the control performance. For a nonlinear moving string in Li et al. (2008), exponentially stability is well achieved with a velocity feedback boundary control. In He and Ge (2012), robust adaptive boundary control is proposed for vibration suppression of a linear string system. In all the works mentioned above, the control design of flexible structures are restricted to one vertical plane, and only transverse deformation is taken into account. However, the string may move in a deflectable direction. The control performance will be affected if the coupling effects between motions in three directions are ignored. In addition, mathematical works in Tsay and Kingsbury (1988) shows that even slights pace curvature introduces significant changes in the beam natural frequencies and especially on mode shapes, i.e., the coupling of the out-of-

10.3182/20130902-3-CN-3020.00170

IFAC ICONS 2013 September 2-4, 2013. Chengdu, China

plane wave types, and extensional and flexural waves exhibits in the flexible beams Do and Pan (2009). To improve accuracy and reliability analysis, modeling and control of the string system in a three-dimensional space is necessary. In this paper, not only the transverse displacement of the string system is regarded, but also the axial deformation is under consideration, which leads to a more precise model for the string system. Due to the coupling between transverse and longitudinal displacements, the control design for the linear model of the string system He and Ge (2012); Queiroz et al. (2000); Nguyen and Hong (2010); Zhang et al. (2012); He et al. (2011b); Ge et al. (2011) cannot be straightforwardly used. To the best of our knowledge, the result is the first complete solution of boundary control to a nonlinear string system in three-dimensional space for vibrations reduction under the distributed disturbance. Considering the distributed disturbance, the string system is governed by nonhomogeneous PDEs, which would make the system model quite different with the previous work. In this paper, the control problem of a string system in three-dimensional space is addressed by using boundary control method. In He and Ge (2012), two boundary control laws are proposed to suppress both the transverse and the longitudinal vibrations for a two-dimensional riser system in marine engineering. However, in threedimensional space, strong system couplings along X, Y and Z axis lead to the high nonlinearities and result in a more complex mathematical model, shown in Section 2.1. Furthermore, the payload dynamics and the external loads along the string are also under consideration, making the control design more difficult. 2. PROBLEM FORMULATION A typical string system is shown in Fig. 1. x(s, t) and y(s, t) are the transverse displacements in the X and Y directions at the position s for time t, z(s, t) is the longitudinal displacement of the string at the position s for time t. dx (t), dy (t), dz (t) are the unknown time-varying boundary disturbances on the tip payload in X, Y, Z directions, fx (s, t), fy (s, t), fz (s, t) are the unknown spatiotemporally varying distributed disturbances along the string in X, Y, Z directions, and ux (s, t), uy (s, t), uz (s, t) are the boundary control force applied on the tip payload in X, Y, Z directions. The another boundary of the string is fixed at origin. Remark 1. For clarity, notions (·)0 = ∂(·)/∂x, (·)00 = ˙ = ∂(·)/∂t and (·) ¨ = ∂ 2 (·)/∂t2 are used ∂ 2 (·)/∂x2 , (·) throughout this paper.

Boundary Disturbances

Z

Longitudinal Distributed Disturbance f z( s, t )

Ek (t) =

ρ 2

Z

L

Transverse Distributed Disturbances f x( s, t )

f y( s, t )

y(s,t)

o Mn 2 2 2 [x(L, ˙ t)] + [y(L, ˙ t)] + [z(L, ˙ t)] + 2

(1) 731

dy( t )

Y

x(s,t)

L

X

Fig. 1.

A nonlinear three-dimensional string system.

where s and t represent the independent spatial and time variables respectively, L is the length of the string, M is the mass of the payload, and ρ denotes the mass per unit length of the string. The potential energy Ep (t) due to the tension T and the axial stiffness EA can be obtained from ¾2 Z ½ EA L 0 1 0 1 0 2 2 Ep (t) = z (s, t) + [x (s, t)] + [y (s, t)] ds 2 0 2 2 Z Ln o T 2 2 + [x0 (s, t)] + [y 0 (s, t)] ds (2) 2 0 The virtual work done by unknown distributed external loads fx (s, t), fy (s, t), fz (s, t) on the string and timevarying disturbance dx (t), dy (t), dz (t) on the tip payload is given by Z δWf (t) =

L

[fx (s, t)δx(s, t) + fy (s, t)δy(s, t) 0

+fz (s, t)δz(s, t)] ds + dx (t)δx(L, t) +dy (t)δy(L, t) + dz (t)δz(L, t) (3) To reduce the vibrations, boundary control force u(t) = [ux (t), uy (t), uz (t)] is introduced at the boundary of the string system. The virtual work done by the control can be expressed as δWm (t) = ux (t)δx(L, t) + uy (t)δy(L, t) + uz (t)δz(L, t)(4) Thus, the total virtual work done on the system is given by δW (t) = δWf (t) + δWm (t). By using the extended Rt Hamilton’s principle t12 [δEk (t) − δEp (t) + δW (t)]dt = 0, we obtain the governing equations of the system as ρ¨ x = T x00 + EA [x00 z 0 + x0 z 00 ] + +

+

0

ux( t ) dz( t )

¤ EA £ 00 02 x y + 2x0 y 0 y 00 2

3EA 02 00 x x + fx 2

ρ¨ y = T y 00 + EA [y 00 z 0 + y 0 z 00 ] +

n o 2 2 2 [x(s, ˙ t)] + [y(s, ˙ t)] + [z(s, ˙ t)] ds

dx( t )

uy( t ) z(s,t)

2.1 Dynamics of the three-dimensional string system In this section, equations of motion of the three-dimensional string is derived by using the Hamilton’s principle Goldstein (1951). The kinetic energy of the string system Ek (t) can be represented as

uz( t )

(5) ¤ EA £ 02 00 x y + 2x0 y 0 x00 2

3EA 02 00 y y + fy 2

ρ¨ z = EAz 00 + EAx0 x00 + EAy 0 y 00 + fz

(6) (7)

IFAC ICONS 2013 September 2-4, 2013. Chengdu, China

∀(s, t) ∈ (0, L) × [0, ∞), and the boundary conditions of the system as x(0, t) = y(0, t) = z(0, t) = 0

(8)

EA 0 [x (L, t)]3 − dx (t) 2 EA 0 +EAx0 (L, t)z 0 (L, t) + x (L, t)[y 0 (L, t)]2 (9) 2

ux (t) = M x ¨(L, t) + T x0 (L, t) +

EA 0 uy (t) = M y¨(L, t) + T y (L, t) + [y (L, t)]3 − dy (t) 2 EA 0 [x (L, t)]2 y 0 (L, t) (10) +EAy 0 (L, t)z 0 (L, t) + 2

Remark 3. This is a reasonable assumption as the disturbances fx (s, t), fy (s, t), fz (s, t), dx (t), dy (t), and dz (t) have finite energy and hence are bounded, i.e. fx (s, t), fy (s, t), fz (s, t), dx (t), dy (t), dz (t) ∈ L∞ . The exact values for fx (s, t), fy (s, t), fz (s, t), dx (t), dy (t), and dz (t) is not required.

3. BOUNDARY CONTROL DESIGN

0

uz (t) = M z¨(L, t) + EAz 0 (L, t) +

EA 0 [x (L, t)]2 − dz (t) 2

EA 0 [y (L, t)]2 (11) 2 Remark 2. With consideration the nonlinear string system in three-dimensional space, the governing equations of the string system are described as three nonlinear nonhomogeneous PDEs (5) - (7). Due to the existence of the nonlinear nonhomogeneous PDEs in our system, the model in our paper differs from the system governed by the homogeneous PDEs or the linear PDEs in Krstic and Smyshlyaev (2008); Rahn et al. (1999). As a consequence, the control schemes in these papers are not suitable for our system.

In this section, under the external loads, the control laws ux (t), uy (t) and uz (t) at the boundary of the string are proposed to suppress the vibrations in all the three directions. The control laws are given as: ux (t) = −2k1 x(L, ˙ t) − sgn[x0 (L, t) + x(L, ˙ t)]d¯x

+

2.2 Preliminaries Lemma 1. Queiroz et al. (2000) Let φ(s, t) ∈ R be a function defined on s ∈ [0, L] and t ∈ [0, ∞) that satisfies the boundary condition φ(0, t) = 0, ∀t ∈ [0, ∞) then the following inequalities hold: Z

L

Z φ2 ds ≤ L2

0

L

(12)

φ02 ds,

∀x ∈ [0, L]

(13)

φ02 ds,

∀x ∈ [0, L]

(14)

0

Z φ2 ≤ L

L

0

Property 1. Queiroz et al. (2000): If the kinetic energy of the system (5) - (11), given by Eq. (1) is bounded ∀(s, t) ∈ [0, L]×[0, ∞), then x(s, ˙ t), x˙ 0 (s, t), x˙ 00 (s, t), y(s, ˙ t), 0 00 0 y˙ (s, t), y˙ (s, t), z(s, ˙ t), z˙ (s, t) and z˙ 00 (s, t) are bounded ∀(s, t) ∈ [0, L] × [0, ∞). Property 2. Queiroz et al. (2000): If the potential energy of the system (5) - (11), given by Eq. (2) is bounded ∀(s, t) ∈ [0, L] × [0, ∞), then x0 (s, t), x00 (s, t), y 0 (s, t), y 00 (s, t), z 0 (s, t) and z 00 (s, t) are bounded ∀(s, t) ∈ [0, L] × [0, ∞).

−M x˙ 0 (L, t) uy (t) = −2k2 y(L, ˙ t) − sgn[y 0 (L, t) + y(L, ˙ t)]d¯y −M y˙ 0 (L, t)

(16)

uz (t) = −2k3 z(L, ˙ t) − sgn[z 0 (L, t) + z(L, ˙ t)]d¯z −M z˙ 0 (L, t)

(17)

where k1 , k2 , k3 are the control gains. Remark 4. The control design is based on the PDE model (5), (6) and (7), and the spillover problems associated with traditional truncated model-based approaches Balas (1978); Vandegrift et al. (1994) caused by ignoring highfrequency modes in controller and observer design are avoided. Remark 5. In the proposed controller (15), (16) and (17), x(L, t), y(L, t) and z(L, t) can be measured through position sensors at the top boundary of the string. In practice, the effect of measurement noise from sensors is unavoidable, which will affect the the controller implementation, especially when the high order differentiating terms with respect to time exist. In our proposed controller, x(L, ˙ t), y(L, ˙ t) and z(L, ˙ t) with only one time differentiating with respect to time can be obtained through a backwards difference algorithm of the values of x(L, t), y(L, t) and z(L, t). It is noted that differentiating twice and three times positions x(L, t), y(L, ... with respect ... t) and z(L, t) to time ¨(L, t), x (L, t), y¨(L, t), y (L, t), z¨(L, t), ... to obtain x and z (L, t) respectively, are undesirable in practice due to noise amplification. For these cases, observers are needed to design to estimate the state values according to the boundary conditions.

2.3 Assumptions

3.1 Boundary control under external disturbances

Assumption 1. For the unknown distributed disturbances fx (s, t), fy (s, t), fz (s, t) and unknown boundary disturbances dx (t), dy (t), dz (t), we assume that there exist constants f x , f y , f z , dx , dy , dz ∈ R+ , such that fx (s, t) ≤ f x , fy (s, t) ≤ f y , fz (s, t) ≤ f z , ∀(s, t) ∈ [0, L] × [0, ∞) and dx (t) ≤ dx , dy (t) ≤ dy , dz (t) ≤ dz , ∀t ∈ [0, ∞)

Consider the Lyapunov function candidate as

732

(15)

V (t) = V1 (t) + V2 (t) + V3 (t) where

(18)

IFAC ICONS 2013 September 2-4, 2013. Chengdu, China

V1 (t) =

Z L ¢ α x˙ 2 + y˙ 2 + z˙ 2 ds + T (x02 + y 02 )ds 2 0 0 ¶2 Z Lµ α x02 y 02 0 + EA z + + ds (19) 2 2 2 0 α ρ 2

Z

L

¡

α α ˙ t)]2 + M [y 0 (L, t) + y(L, ˙ t)]2 V2 (t) = M [x0 (L, t) + x(L, 2 2 α ˙ t)]2 (20) + M [z 0 (L, t) + z(L, 2 Z L V3 (t) = βρ s (xx ˙ 0 + yy ˙ 0 + zz ˙ 0 ) ds (21)

Proof: We differentiate Eq. (18) with respect to time to obtain

V˙ (t) = V˙ 1 (t) + V˙ 2 (t) + V˙ 3 (t)

(31)

Using governing equation, the boundary conditions and the proposed control laws (15), (16), (17), and applying inequality |φ1 φ2 | ≤ 1δ φ21 + δφ22 , δ > 0, we have

0

where α and β are two positive constants. Lemma 2. The Lyapunov function candidate given by (18), is upper and lower bounded as 0 ≤ λ1 [θ(t) + V2 (t)] ≤ V (t) ≤ λ2 [θ(t) + V2 (t)] where λ1 and λ2 are positive constants, and Z

L

θ(t) =

2

2

2

0 2

0 2

(22)

0 2

[(x) ˙ + (y) ˙ + (z) ˙ + (x ) + (y ) + (z )

(23)

Proof: From the definition of V1 (t), by using inequalities |φ1 φ2 | ≤ 1δ φ21 + δφ22 , 2[z 0 (s, t)]2 ≤ [x0 (s, t)]2 and 2[z 0 (s, t)]2 ≤ [y 0 (s, t)]2 Narasimha (1968), let the positive 1 δ satisfying T − EA 2δ ≥ 0 and 4 − δ ≥ 0, we have 0 ≤ γ1 θ(t) ≤ V1 (t) ≤ γ2 θ(t) where γ1 and γ2 are defined as EA 1 1 α min[ρ, T − , EA, EA( − δ)] 2 2δ 2 4 α EA 1 γ2 = max[ρ, T + , EA, EA( + δ)] 2 2δ 4 γ1 =

(24)

(25) (26)

We can also obtain that Z L |V3 (t)| ≤ βρL [x˙ 2 + x02 + y˙ 2 + y 02 + z˙ 2 + z 02 ]ds 0

≤ β1 θ(t) (27) where β1 = βρL. Then, we have −β1 θ(t) ≤ V3 (t) ≤ β1 θ(t). Considering β is a small positive weighting conγ1 stant satisfying0 < β < ρL , we have 0 < β1 < γ1 . Let β2 = γ1 − β1 , β3 = γ2 − β1 , we further have 0 ≤ β2 V1 (t) ≤ V1 (t) + V3 (t) ≤ β3 V1 (t) (28) Given the Lyapunov function candidate in Eq. (18), we obtain 0 ≤ λ1 [θ(t) + V2 (t)] ≤ V (t) ≤ λ2 [θ(t) + V2 (t)] (29) where λ1 = min(β2 , 1) = β2 and λ2 = max(β3 , 1) = β3 are positive constants. Lemma 3. The time derivative of the Lyapunov function in (18) is upper bounded with V˙ (t) ≤ −λV (t) + ε where λ and ε are two positive constants.

−( −(

0

+(x0 )4 + (y 0 )4 + (x0 y 0 )2 ]ds

βρ V˙ (t) ≤ −( − αδ1 ) 2

(30)

733

βρ − αδ3 ) 2

Z

L

x˙ 2 ds − (

0

Z

L

βρ − αδ2 ) 2

Z

L

y˙ 2 ds

0

z˙ 2 ds

0

βL βEA − βLδ4 − ) 2 2δ9

Z

L

x02 ds

0

Z L βEA βL − βLδ5 − −( ) y 02 ds 2 2δ10 0 Z L βEA − βLδ6 ) −( z 02 ds 2 0 Z L 3βEAL − δ9 βEA) x04 ds −( 8 0 Z L 3βEAL −( − δ10 βEA) y 04 ds 8 0 Z 3βEA L 0 0 2 − (x y ) ds − αk1 [x0 (L, t) + x(L, ˙ t)]2 4 0 −αk2 [y 0 (L, t) + y(L, ˙ t)]2 − αk3 [z 0 (L, t) + z(L, ˙ t)]2 Z L Z L α βL βL α +( + ) ) f 2 ds + ( + f 2 ds δ1 δ4 0 x δ2 δ5 0 y Z L βL α βρL )[x(L, ˙ t)]2 ) +( + f 2 ds − (αk1 − δ3 δ6 0 z 2 βρL βρL −(αk2 − )[y(L, ˙ t)]2 − (αk3 − )[z(L, ˙ t)]2 2 2 | 3α − βL|EA βLT 0 −(αT − αk1 − 2 )[x (L, t)]2 − 2δ7 2 | 3α − βL|EA βLT 0 −(αT − αk2 − 2 − )[y (L, t)]2 2δ8 2 βLEA 0 )[z (L, t)]2 −(αEA − αk3 − 2 3α αEA 3βLEA − − δ7 | − βL|EA)[x0 (L, t)]4 −( 2 8 2 αEA 3βLEA 3α −( − − δ8 | − βL|EA)[y 0 (L, t)]4 2 8 2 3βEAL 0 )[x (L, t)]2 [y 0 (L, t)]2 −(αEA − 4 ≤ −λ3 [θ(t) + V2 (t)] + ε (32)

where δ1 - δ10 are positive constants, and

IFAC ICONS 2013 September 2-4, 2013. Chengdu, China

½

βρ βρ βρ − αδ1 , − αδ2 , − αδ3 , 2 2 2 βEA βL βEA βL − βLδ4 − , − βLδ5 − , 2 2δ9 2 2δ10 βEA 3βEA 3βEAL − βLδ6 , , − δ9 βEA, 2 4 8 ¾ 3βEAL 2k1 2k2 2k3 − δ10 EA, , , > 0 (33) 8 M M M αL βL2 2 αL βL2 2 ε≤( + )f x + ( + )f y δ1 δ4 δ2 δ5 αL βL2 2 + )f z ∈ L∞ (34) +( δ3 δ6 The design constants k1 , k2 , k3 , α, β, δ7 , δ8 are selected to satisfy the following conditions: λ3 = min

βρL ≥0 2 βρL αk2 − ≥0 2 βρL αk3 − ≥0 2 | 3α − βL|EA βLT αT − αk1 − 2 ≥0 − 2δ7 2 | 3α − βL|EA βLT αT − αk2 − 2 ≥0 − 2δ8 2 βLEA ≥0 αEA − αk3 − 2 αEA 3βLEA 3α − − δ7 | − βL|EA ≥ 0 2 8 2 3α αEA 3βLEA − − δ8 | − βL|EA ≥ 0 2 8 2 3βEAL αEA − ≥0 4 αk1 −

(35) (36) (37) (38) (39) (40) (41) (42) (43)

From Ineqs. (29), we have V˙ (t) ≤ −λV (t) + ε where λ = λ3 /λ2 > 0 and ε > 0.

(44)

With the above lemmas, we are ready to present the following stability theorem of the closed-loop string system. Theorem 1. For the system dynamics described by (5) (7) and boundary conditions (8) - (11), under Assumption 1, and the control laws (15), (16), (17), given that the initial conditions are bounded, and that the required state information x(L, t), y(L, t), z(L, t), x(L, ˙ t), y(L, ˙ t) and z(L, ˙ t) are available, the closed loop system is uniformly bounded as |x(s, t)| ≤ D, |y(s, t)| ≤ D and |z(s, t)| ≤ D, ∀s ∈ [0, L], where r ³ L ε´ D= , ∀s ∈ [0, L] (45) V (0)e−λt + λ1 λ λ and ε are two positive constants. Proof: Multiplying Eq. (30) by eλt yields ∂ (V eλt ) ≤ εeλt ∂t

(46) 734

Integration of the above inequalities, we obtain ³ ε ε ´ −λt ε V (t) ≤ V (0) − e + ≤ V (0)e−λt + ∈ L∞(47) λ λ λ which implies V (t) is bounded. Utilizing Ineq. (14) and Eq. (22), we have 1 2 x (s, t) ≤ L 1 2 y (s, t) ≤ L

Z

L

[x0 (s, t)]2 ds ≤ θ(t) ≤

1 V (t) ∈ L∞(48) λ1

[y 0 (s, t)]2 ds ≤ θ(t) ≤

1 V (t) ∈ L∞(49) λ1

0

Z

L

0

Z L 1 1 2 z (s, t) ≤ [z 0 (s, t)]2 ds ≤ θ(t) ≤ V (t) ∈ L∞(50) L λ 1 0 From the above three inequalities and inequality (47), we can state the V1 (t) is bounded ∀t ∈ [0, ∞). Since V1 (t) is bounded, x(s, ˙ t), x0 (s, t), x00 (s, t), y(s, ˙ t), y 0 (s, t), y 00 (s, t), 0 z(s, ˙ t) and z (s, t) are bounded ∀(s, t) ∈ [0, L] × [0, ∞). From Eq. (1), the kinetic energy of the system is bounded and using Property 1, x˙ 0 (s, t) and y˙ 0 (s, t) are bounded ∀(s, t) ∈ [0, L] × [0, ∞). From the boundedness of the potential energy Eq. (2), we can use Property 2 to conclude that z 00 (s, t) is bounded ∀(s, t) ∈ [0, L] × [0, ∞). Finally, using Assumption 1, Eqs. (5), (6), (7) through (8), (9), (10), (11) and the above statements, we can conclude that x ¨(s, t), y¨(s, t) and z¨(s, t) are also bounded ∀(s, t) ∈ [0, L]× [0, ∞). From Lemma 3 and the above proof, it is shown the deflection x(s, t), y(s, t) and z(s, t) are uniformly bounded ∀(s, t) ∈ [0, L] × [0, ∞). Remark 6. From above stability analysis, x(s, ˙ t), y(s, ˙ t), z(s, ˙ t), x(s, t), y(s, t) and z(s, t) are all bounded ∀(s, t) ∈ [0, L] × [0, ∞), and we can conclude the control inputs of ux (t), uy (t) and uz (t) are bounded ∀t ∈ [0, ∞). 3.2 Boundary control without the distributed disturbances In this section, when the distributed disturbances fx (s, t) = fy (s, t) = fz (s, t) = 0, we analyze the exponential stability of the system by using the same Lyapunov function candidate Eq. (18) and control laws (15), (16) and (17) of Section 3.1. Theorem 2. For the system dynamics described by (5) (7) and boundary conditions (8) - (11), using the proposed control law (15), (16) and (17), then the exponential stability under the condition fx (s, t) = fy (s, t) = fz (s, t) = 0 can be achieved as |x(s, t)| ≤ D0 , |y(s, t)| ≤ D0 and |z(s, t)| ≤ D0 , ∀s ∈ [0, L], where r D0 =

L V (0)e−λt λ1

(51)

Furthermore, we have lim |x(s, t)| = lim |y(s, t)| = lim |z(s, t)| = 0 (52)

t→∞

t→∞

t→∞

Proof: If fx (s, t) = fy (s, t) = fz (s, t) = 0, we have ε = 0. From Ineq. (30), we obtain the time derivation of the Lyapunov function candidate (18) as V˙ (t) ≤ −λV (t) (53) λt where λ = λ3 /λ2 . Multiplying Eq. (53) by e yields

IFAC ICONS 2013 September 2-4, 2013. Chengdu, China

∂ (V eλt ) ≤ 0 ∂t

(54)

Integration of the above inequality, we obtain V (t) ≤ V (0)e−λt ∈ L∞

(55)

which implies V (t) is bounded. Using Ineq. (48) - (50), we obtain |x(s, t)| ≤ D0 ,q |y(s, t)| ≤ D0 and |z(s, t)| ≤ D0 , ∀s ∈ [0, L], where D0 =

L λ1 V

(0)e−λt , and

lim |x(s, t)| = lim |y(s, t)| = lim |z(s, t)| = 0 (56)

t→∞

t→∞

t→∞

From the above proof, we have the conclusion that the displacements x(s, t), y(s, t) and z(s, t) exponentially converge to zero at the rate of convergence λ as t → ∞ by using the proposed control laws, when external loads fx (s, t) = fy (s, t) = fz (s, t) = 0.

5. CONCLUSION In this paper, the dynamic equations of motion for a threedimensional string system have been derived, and then used for the design of the boundary control. The models are reliable since the comprehensive effects of the external loads have been taken into account, which guarantee an accurate analysis of the string system. With the proposed control scheme, all the signals of the closed-loop system have been proved to be uniformly bounded by using the Lyapunov’s direct method. Numerical simulations have been provided to illustrate the effectiveness of the proposed control. ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China under Grant 61203057, the Fundamental Research Funds for the China Central Universities of UESTC under Grant ZYGX2012J087 and the National Basic Research Program of China (973 Program) under Grant 2011CB707005.

4. NUMERICAL SIMULATIONS In this section, by using the finite difference method, simulations for a three-dimensional string system with a payload under the external loads are carried out to verify the effectiveness of the proposed control laws (15), (16) and (17). The string system, initially at rest, is excited by the boundary disturbance and the distributed transverse load. The external disturbances are s, given as fx (s, t) = fy (s, t) = 3+sin(πst)+sin(2πst)+sin(3πst) 50 3+sin(πst)+sin(2πst)+sin(3πst) fz (s, t) = s, and dx (t) = 55 1+0.2 sin(0.2πst)+0.3 sin(0.3πst)+0.5 sin(0.5πst) dy (t) = dz (t) = . 5 The corresponding initial conditions of the system are described as x(s, 0) = y(s, 0) = z(s, 0) = 0.2s, x(s, ˙ 0) = y(s, ˙ 0) = z(s, ˙ 0) = 0. Parameters of the nonlinear string system are listed as below: Table 1: Parameters of the string Parameter Description Value L Length of string 1m M Mass of the tip payload 5kg ρ Mass per unit length 0.1kg/m T Tension 15N EA Axial stiffness 0.3N The simulation results are shown in Figs. 2 - 5. For the case that the control inputs ux (t) = uy (t) = uz (t) = 0, the results are plotted in Figs. 2(a), 3(a) and 4(a), respectively. It can be seen that the external loads can cause large vibrations in X, Y and Z directions. When the proposed control (15), (16) and (17) are applied, displacements of the string in three directions are shown in Figs. 2(b), 3(b) and 4(b). We choose control gains as k1 = k2 = 3, k3 = 0.2. It can be observed that when the system is under the proposed control, displacements in the X, Y and Z directions are significantly suppressed. We can conclude that the proposed control can stabilize the string at the small neighborhood of its equilibrium position. The control inputs are given in Fig. 5 735

REFERENCES Balas, M.J. (1978). Feedback control of flexible systems. IEEE Transactions on Automatic Control, 23, 673–679. Bentsman, J. and Hong, K.S. (1991). Vibrational stabilization of nonlinear parabolic systems with Neumann boundary conditions. IEEE Transactions on Automatic Control, 36(4), 501–507. Choi, J.Y., Hong, K.S., and Yang, K.J. (2004). Exponential stabilization of an axially moving tensioned strip by passive damping and boundary control. Journal of Vibration and Control, 10(5), 661–682. d’Andrea Novel, B. and Coron, J.M. (2000). Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach. Automatica, 36, 587–593. Do, K. and Pan, J. (2008). Boundary control of transverse motion of marine risers with actuator dynamics. Journal of Sound and Vibration, 318, 768–791. Do, K. and Pan, J. (2009). Boundary control of threedimensional inextensible marine risers. Journal of Sound and Vibration, 327(3-5), 299–321. Ge, S.S., He, W., How, B.V.E., and Choo, Y.S. (2010). Boundary Control of a Coupled Nonlinear Flexible Marine Riser. IEEE Transactions on Control Systems Technology, 18(5), 1080–1091. Ge, S.S., Zhang, S., and He, W. (2011). Vibration Control of an Euler–Bernoulli Beam Under Unknown Spatiotemporally Varying Disturbance. International Journal of Control, 84(5), 947–960. Goldstein, H. (1951). Classical Mechanics. AddisonWesley, Massachusetts, USA. He, W. and Ge, S.S. (2012). Robust Adaptive Boundary Control of a Vibrating String Under Unknown TimeVarying Disturbance. IEEE Transactions on Control Systems Technology, 20(1), 48–58. He, W., Ge, S.S., Hang, C.C., and Hong, K.S. (2010). Boundary control of a vibrating string under unknown time-varying disturbance. In Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, 2584–2589.

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Yang, K.J., Hong, K.S., and Matsuno, F. (2005). Boundary Control of a Translating Tensioned Beam with Varying Speed. IEEE/ASME Transactions on Mechatronics, 10(5), 594–597. Zhang, S., He, W., and Ge, S.S. (2012). Modeling and Control of a Nonuniform Vibrating String Under Spatiotemporally Varying Tension and Disturbance. IEEE/ASME Transactions on Mechatronics, 17(6), 1196–1203. (b)

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Displacements of the string at the X direction: (a) without control; (b)with the proposed control. (b)

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Displacements of the string at the Z direction: (a) without control; (b)with the proposed control.

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He, W., Ge, S.S., How, B.V.E., and Choo, Y.S. (2013a). Dynamics and Control of Mechanical Systems in Offshore Engineering. Springer, London, UK. He, W., Ge, S.S., How, B.V.E., Choo, Y.S., and Hong, K.S. (2011a). Robust Adaptive Boundary Control of a Flexible Marine Riser with Vessel Dynamics. Auotmatica, 47(4), 722–732. He, W., Ge, S.S., and Zhang, S. (2011b). Adaptive Boundary Control of a Flexible Marine Installation System. Automatica, 47(12), 2728–2734. He, W., Zhang, S., and Ge, S.S. (2013b). Boundary control of a flexible riser with application to marine installation. IEEE Transactions on Industrial Electronics, 60(12), 5802–5810. He, W., Zhang, S., and Ge, S.S. (2013c). Boundary outputfeedback stabilization of a Timoshenko beam using disturbance observer. IEEE Transactions on Industrial Electronics, 60(11), 5186–5194. Hong, K.S. and Bentsman, J. (1994). Direct adaptive control of parabolic systems: algorithm synthesis and convergence and stability analysis. IEEE Transactions on Automatic Control, 39(10), 2018–2033. Huang, D. and Xu, J.X. (2011). Steady-state iterative learning control for a class of nonlinear pde processes. Journal of Process Control, 21(8), 1155–1163. Kim, C.S. and Hong, K.S. (2009). Boundary Control of Container Cranes From the Perspective of Controlling an Axially Moving String System. International Journal of Control, Automation and Systems, 7(3), 437–445. Krstic, M. and Smyshlyaev, A. (2008). Boundary Control of PDEs: A Course on Backstepping Designs. Society for Industrial and Applied Mathematics, Philadelphia, USA. Li, T., Hou, Z., and Li, J. (2008). Stabilization analysis of a generalized nonlinear axially moving string by boundary velocity feedback. Automatica, 44(2), 498–503. Lin, J. and Lewis, F.L. (1994). A symbolic formulation of dynamic equations for a manipulator with rigid and flexible links. The International Journal of Robotics Research, 13(5), 454. Morgul, O. (1994). A Dynamic Control Law for the Wave Equation. Automatica, 30(11), 1785–1792. Narasimha, R. (1968). Non-linear vibration of an elastic string. Journal of Sound and Vibration, 8(1), 134–146. Nguyen, Q.C. and Hong, K.S. (2010). Asymptotic Stabilization of a Nonlinear Axially Moving String by Adaptive Boundary Control. Journal of Sound and Vibration, 329(22), 4588–4603. Queiroz, M.S., Dawson, D.M., Nagarkatti, S.P., and Zhang, F. (2000). Lyapunov Based Control of Mechanical Systems. Birkhauser, Boston, USA. Rahn, C., Zhang, F., Joshi, S., and Dawson, D. (1999). Asymptotically stabilizing angle feedback for a flexible cable gantry crane. Journal of Dynamic Systems, Measurement, and Control, 121, 563–565. Tsay, H.S. and Kingsbury, H. (1988). Vibrations of rods with general space curvature. Journal of sound and vibration, 124(3), 539–554. Vandegrift, M.W., Lewis, F.L., and Zhu, S.Q. (1994). Flexible-link robot arm control by a feedback linearization/singular perturbation approach. Journal of Robotic Systems, 11(7), 591–603.

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