Active Tendon Control of Stay Cable by a Giant Magnetostrictive Actuator Considering Time-Delay

Abstract

Active control of a stay cable through changing its tension is an effective way to lower the vibration amplitude. In this paper, active tendon control of a stay cable using a giant magnetostrictive actuator (GMA), which can generate axial deformation in a short period (normally less than 10 ms) under magnetic fields, is studied considering the time-delay. The bilinear controlled state equation of the small-sag cable is established by employing the Lagrange formulation. The linearization method for the bilinear controlled state equation is proposed and validated by numerical simulations. A GMA is designed and manufactured to actively control a stay cable model and the relationship between the output force and the input voltage is obtained by experiments. Based on the phase-shifting method, a multi-mode control algorithm with the time-delay compensation is proposed, and the time-delay compensation matrix based on the state feedback control algorithm is obtained. Based on a 1:20 scaled cable model, simulations and experiments are conducted to investigate the control performance of the proposed method under different external loads including the free vibration, harmonic excitation and random excitation. The results show that the tendon control of cable vibration by a GMA may be invalid and even amplify the vibration amplitude when the compensation of time-delay is not considered. Moreover, the excellent control performance can be achieved by using the proposed time-delay compensation method.

1. Introduction

Cables are very important structural members of cable-stayed bridges. However, large vibration of the cables is easily excited due to their light weight and low inherent damping, under external excitations such as wind, wind-rain combined and vehicles loads. Large vibrations of cables mainly include vortex-induced vibrations [1,2], wind-rain-induced vibrations [3,4,5,6], parameter vibrations [7,8], etc. Hence, the vibration control of cables is a key issue that needs to be solved urgently with more and more long-span cable-stayed bridges built.

Three methods for cable vibration control, including passive control, semi-active control, and active control, were proposed based on different vibration mechanisms. Among those passive methods, one of the effective methods in aerodynamic strategies is increasing the roughness of the cable, which can prevent the cable from forming the upper rain water rivulet, and then suppress the wind-rain induced vibration of the cable [9,10]. The other is the structural measures that can increase the natural frequency and enhance the inherent damping of the cable by providing a secondary cable between the cables [11]. In addition, several passive dampers were attached to cables near their anchorages in some cable-stayed bridges to suppress the cable vibrations, which can effectively improve the modal damping of the cables [12,13,14,15]. However, the passive damper can only be optimally adjusted to control a specific mode vibration of stay cables and may be ineffective in multi-mode vibration mitigation [14]. Therefore, developing semi-active control and active control methods of cables is more effective. Many new dampers for semi-active control of stay cables are developed and their properties are also studied, such as superelastic shape memory alloy (SMA) dampers [16], Magnetorheological (MR) dampers [17], and self-powered MR dampers [18].

With regard to active control for stay cables, the application of axial motion to control the lateral vibrations has been studied. Fujino et al. [19] proposed a modal control approach using the axial motion of the cable end induced by an actuator to suppress cable vibrations. Warnitchai et al. [20] conducted the research on the active control of the small sag cables under vertical sinusoidal oscillation by numerical simulation and experiments, discussing the influence of linear and nonlinear internal resonance on control. Then, Susumpow and Fujino [21] developed a globally asymptotic stable control strategy based on the bilinear control theory. Based on this strategy, an in-plane vibration control for stay cables applying piezoelectric actuators was investigated by Achkire and Preumont [22], and Gattulli et al. [23] verified the feasibility of multi-mode active control of cable vibrations using piezoelectric actuator via numerical simulations and experiments. Recently, an active control method of stay cable vibration using a Giant magnetostrictive actuator (GMA) is proposed by Huang et al. [24]. Although active control can effectively mitigate the vibration of the cable, it produces some problems in actual implementation. One of the most critical issues is the time-delay resulted from the control system. The time-delay mainly affects the control performance in two aspects. One is that the time-delay existed during the processing of signal transmissions, especially the time-delay caused by the wireless transmissions of sensor data and control commands when the wireless network is used for information transmission. The other one is the time-delay of signal processing and actuator response, especially response time from the actuator receiving the control signal to applying the control force on the cable. As explained, the performance of active vibration for cables cannot be guaranteed to be solid due to the time-delay of the control system. To eliminate the adverse effect of time-delay, many control algorithms considering the time-delay compensation have been proposed, including the Smith predictive control [25], the extended Kalman filter to obtain an optimal filtered estimation of structural responses [26], the Neural network predictor [27], the Neural Network Based Hysteresis Compensation [28], the polynomial modulating function compensation [29], the time-delay compensation controller based on regional pole-assignment method [30], the phase shift compensation method [31], etc.

In this paper, theoretical and experimental studies on active control of a cable model with time-delay effect are carried out. Firstly, the time-delay control equation of the small-sag cable was established and the related control algorithm was proposed. Secondly, a giant magnetostrictive actuator (GMA) for the active control of a stay cable was developed and the relationship between input voltage and the output force of GMA is determined. Thirdly, a multi-modes time-delay compensation method for the active cable control is proposed with help of the phase shift method. Finally, the control performance of the proposed system is validated by numerical simulations and experiments under different excitation conditions.

2. The State Equation of a Small Sag Cable with Axial Force

The mechanical model of a small sag cable with axial control force is shown in Figure 1. In Figure 1, the left end of the cable is fixed and the right end of the cable is assumed to be a rolling bearing. H^s is the static axial tension, U(t) is the control force provided by the GMA to the cable along the axial direction. In addition σ^(s) is the static axial stress. L is the length of the cable and ρ is the mass per volume of the cable.

The separation of variables method is employed to obtain the in-plane vertical displacement of the cable as the following form,

$$w(x,t)={\displaystyle \sum _n{\varphi }_i(x)q_i(t)}$$

(1)

where ϕ_i(x) is the i-th mode function and is assumed with a form of ϕ_i(x) = sin(iπx/L), q_i is the i-th modal displacement.

Employing the Lagrange formulation and ignoring the second and higher order nonlinear terms, the governing equations of the transverse motion of the small-sag cable can be obtained as follows [21],

$$\begin{array}{l}\left\{\begin{array}{l}{\ddot{q}}_1\\ {\ddot{q}}_2\\ ⋮\\ {\ddot{q}}_n\end{array}\right\}+\left[\begin{array}{llll}2{\zeta }_1{\omega }_1& 0 & \cdots & 0 \\ 0 & 2{\zeta }_2{\omega }_2& \cdots & 0 \\ ⋮& ⋮& \cdots & \cdots \\ 0& \cdots & \cdots & 2{\zeta }_{\mathrm{n}}{\omega }_n\end{array}\right]\left\{\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ ⋮\\ {\dot{q}}_n\end{array}\right\}+\left[\begin{array}{llll}{\omega }_1^2& 0 & \cdots & 0 \\ 0 & {\omega }_2^2& \cdots & 0\\ ⋮& ⋮& \cdots & \cdots \\ 0 & 0 & \cdots & {\omega }_n^2\end{array}\right]\left\{\begin{array}{l}{q}_1\\ q_2\\ ⋮\\ q_n\end{array}\right\}\\ +\left[\begin{array}{llll}{\alpha }_1& 0 & \cdots & 0 \\ 0 & {\alpha }_2& \cdots & 0 \\ ⋮& ⋮& \cdots & \cdots \\ 0 & 0 & \cdots & {\alpha }_n\end{array}\right]U(t)\left\{\begin{array}{l}{q}_1\\ q_2\\ ⋮\\ q_n\end{array}\right\}-\left[\begin{array}{l}{\beta }_1\\ {\beta }_2\\ ⋮\\ {\beta }_n\end{array}\right]\ddot{U}(t)=\left[\begin{array}{l}{F}_1(t)/m_1\\ F_2(t)/m_2\\ ⋮\\ F_n(t)/m_n\end{array}\right]\end{array}$$

(2)

where ω_n is the n-th undamped natural circular frequency of the cable, β_n and α_n are the elements of coefficient matrixes, respectively. m_i is the modal mass of the i-th mode. F_i is the i-th generalized modal force acting on the small-sag cable. The aforementioned parameters can be expressed as follows,

$${\omega }_n{}^2=\frac{{\sigma }^{(s)}}{\rho }\left[1+\frac{2{\lambda }^2{\left[1+{(-1)}^{n+1}\right]}^2}{n^4{\pi }^4}\right]{(\frac{n\pi }{l})}^2$$

(3)

$${\beta }_n=\frac{2\left[1+{(-1)}^{n+1}\right]L^2}{{(n\pi )}^3}\frac{mg\cos \theta }{{\left({\sigma }^{\left(s\right)}A\right)}^2}$$

(4)

$${\alpha }_n=\left[\frac{1}{\rho A}{(\frac{n\pi }{L})}^2\right]$$

(5)

$$m_i=\frac{1}{2}\rho AL$$

(6)

$$F_i={\displaystyle {\int }_0^{L}Y{\varphi }_i}dx$$

(7)

where Y is the external excitation in y direction. λ² is dimensionless parameter of geometric and elastic effects and expressed as follows:

$${\lambda }^2=\left[E/{\sigma }^{(s)}\right]{\left[\rho gL\cos \theta /{\sigma }^{(s)}\right]}^2$$

(8)

where E is the modulus of elasticity of the stay cable and θ is the inclination angle of the stay cable. Due to vibrations of the cable being multi-modal vibrations and low-order modes being the primary vibrations, only the first three modes are considered in this paper. The state variables of the in-plane vibration control of the cable are defined as,

$$z_i=q_i-{\beta }_{i}U\left(t\right), z_{i+3}={\dot{q}}_i-{\beta }_i\left[\dot{U}\left(t\right)-2{\mathsf{\xi }}_{yi}{\omega }_{yi}U\left(t\right)\right](i=1,2,3)$$

(9)

where

$${\dot{z}}_1=z_4-2{\mathsf{\xi }}_{y1}{\omega }_{y1}{\beta }_{1}U\left(t\right), {\dot{z}}_2=z_5-2{\mathsf{\xi }}_{y2}{\omega }_{y2}{\beta }_{2}U\left(t\right), {\dot{z}}_3=z_6-2{\mathsf{\xi }}_{y3}{\omega }_{y3}{\beta }_{3}U\left(t\right)$$

$${\dot{z}}_4=-2{\mathsf{\xi }}_{y1}{\omega }_{y1}{z}_4-{\omega }_{y1}{}^2{z}_1-{\alpha }_1{z}_{1}U\left(t\right)-\left(1-4{\mathsf{\xi }}_{y1}{}^2\right){\beta }_1{\omega }_{y1}{}^{2}U\left(t\right)-{\alpha }_1{\beta }_1{U}^2(t)+F_1\left(t\right)/m_1$$

$${\dot{z}}_5=-2{\mathsf{\xi }}_{y2}{\omega }_{y2}{z}_5-{\omega }_{y2}{}^2{z}_2-{\alpha }_2{z}_{2}U\left(t\right)-\left(1-4{\mathsf{\xi }}_{y2}{}^2\right){\beta }_2{\omega }_{y2}{}^{2}U\left(t\right)-{\alpha }_2{\beta }_2{U}^2(t)+F_2\left(t\right)/m_2$$

$${\dot{z}}_6=-2{\mathsf{\xi }}_{y3}{\omega }_{y3}{z}_6-{\omega }_{y3}{}^2{z}_3-{\alpha }_3{z}_{3}U\left(t\right)-\left(1-4{\mathsf{\xi }}_{y3}{}^2\right){\beta }_3{\omega }_{y3}{}^{2}U\left(t\right)-{\alpha }_3{\beta }_3{U}^2(t)+F_3\left(t\right)/m_3$$

The quadratic term α_iβ_iU²(t) is neglected in the controller design, and the state equation of the first three modes of the cable can be obtained as follow,

$$\dot{Z}=AZ+BU\left(t\right)+DF(t)$$

(10)

where Z is the state vector of the in-plane vibration control system of the small-sag cable, A and B are the parameter matrix and control matrix of the cable system, respectively. D is the action position matrix of external excitation; F is the generalized force matrix. The state vector is expressed as,

$$Z={\left[\begin{array}{cccccc}{z}_1& z_2& z_3& z_4& z_5& z_6\end{array}\right]}^T$$

The term of vector DF(t) in Equation (10) is assumed to be negligible when the optimal control of a civil structure is subjected to external loads. Thus, Equation (10) can be simplified as:

$$\dot{Z}=AZ+B_{2}U(t)$$

(11)

3. Simplification and Verification of Bilinear Controlling Equations

In Equation (11), the control matrix B = B₁Z + B₂ is a linear function of the state variable Z, and the multiplication of B and U(t) is a nonlinear term for the control system, which can be referred as a bilinear control system. Susumpow et al. [21] employed the direct Lyapunov method to derive the optimal control algorithm for bilinear systems. The optimal control force in the algorithm is expressed as [21]:

$$U(t)=-\frac{1}{R}{Z}^{\mathrm{T}}S(B_{1}Z+B_2)$$

(12)

where R is the weight coefficient greater than zero, S is a symmetric positive definite matrix and can be obtained from:

$$SA+A^{T}S=-Q$$

(13)

where Q is also a symmetric positive definite matrix.

In this paper, the research focuses on the influence of time delay on the controlled cable and the time delay compensation method. Due to the small-sag cable, bilinear control is considered as a weak nonlinear and simplified method for linearizing the bilinear system is adopted. A simplified linearization method is proposed for Equation (10), that is, B₁ = 0 is assumed, so that Equation (11) can be described as

$$\dot{Z}=AZ+B_{2}U(t)$$

(14)

Since the mode-based time delay compensation method is used in the follow-up study, the control force of Equation (14) is determined by the modal control of state equation.

Let −α_j ± iβ_j and α_j ± iβj (i² = −1, j = 1,2,3) be the j-th complex conjugate eigenvalue and j-th eigenvector of eigenmatrix A in Equation (14), respectively. The following relationship can be obtained:

$${\mathsf{\psi }}^{-1}A\mathsf{\psi }=\Lambda =\left[\begin{array}{ccc}{\Lambda }_1& 0& 0\\ 0& {\Lambda }_2& 0\\ 0& 0& {\Lambda }_3\end{array}\right]$$

(15)

where

$$\mathsf{\psi }=\left[\begin{array}{cccccc}{a}_1& b_1& a_2& b_2& a_3& b_3\end{array}\right], {\Lambda }_j=\left[\begin{array}{cc}{\alpha }_j& {\beta }_j\\ -{\beta }_j& {\alpha }_j\end{array}\right](j=1,2,3)$$

The state vector of the system can be modally expressed as:

$${\rm Z}=\mathsf{\psi }\mathsf{\xi }\left(t\right)$$

(16)

where ξ(t) denotes the vector of modal coordinates, and it has following forms:

$$\mathsf{\xi }(t)={\left[\begin{array}{cccc}{\mathsf{\xi }}_1^{\mathrm{T}}(t)& {\mathsf{\xi }}_2^{\mathrm{T}}(t)& \cdots & {\mathsf{\xi }}_n^{\mathrm{T}}(t)\end{array}\right]}^T, {\mathsf{\xi }}_j^{\mathrm{T}}(t)=\left[\begin{array}{cc}{\mathsf{\xi }}_{j1}(t)& {\mathsf{\xi }}_{j2}(t)\end{array}\right], (j=1,2,3)$$

Substituting Equation (16) into Equation (14), the state equation expressed by the modal coordinates is:

$$\dot{\mathsf{\xi }}(t)=\Lambda \mathsf{\xi }(t)+U_m(t)$$

(17)

where

$$U_m(t)={\left[\begin{array}{ccc}{U}_{m1}^{\mathrm{T}}& U_{m2}^{\mathrm{T}}& U_{m3}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}={\mathsf{\psi }}^{-1}BU(t), U_{mj}^{\mathrm{T}}=\left[\begin{array}{cc}{u}_{mj1}& u_{mj2}\end{array}\right]$$

(18)

Equation (17) contains three pairs of independent equations, in which the j-th pair of equations describes the control of the j-th mode, namely:

$${\dot{\mathsf{\xi }}}_j(t)={\Lambda }_j{\mathsf{\xi }}_j(t)+U_{mj}(t)$$

(19)

According to the LQR control algorithm, the optimal control force in Equation (17) can be obtained as follows:

$$U_{mj}(t)=-R^{-1}P{\mathsf{\xi }}_j(t)=-G_j{\mathsf{\xi }}_j(t)$$

(20)

where, G_j is the feedback gain matrix, matrix P is the solution of the following Riccati equation [32]:

$$-P{\Lambda }_j-{\Lambda }_{{}^j}^{\mathrm{T}}P+P{R}^{-1}P-Q_1=0$$

(21)

where Q₁ is the semi-positive definite weight matrix and R is the weight coefficient.

So we have:

$$U_m(t)=\left[\begin{array}{c}{U}_{m1}\\ U_{m2}\\ U_{m3}\end{array}\right]=-\left[\begin{array}{c}{G}_1{\xi }_1(t)\\ G_2{\xi }_2(t)\\ G_3{\xi }_3(t)\end{array}\right]=-G\mathsf{\xi }(t)$$

(22)

According to Equations (18) and (20), the following equation can be obtained,

$$U(t)=-L^{+}G\mathsf{\xi }(t)=-L^{+}G{\mathsf{\psi }}^{-1}Z$$

(23)

where

$$L={\mathsf{\psi }}^{-1}B, L^{+}={(L^{\mathrm{T}}L)}^{-1}{L}^{\mathrm{T}}$$

where L⁺ is the pseudo inverse of L. To evaluate the efficiency of vibration suppression of the linear simplified model, the simulation analysis of vibration reduction under free vibration and harmonic excitation is carried out. In this paper, the A12 stay cable from the Dongting Lake Bridge [5] is taken as a prototype cable for simulations and experiments. A scaled-down cable model with a scale ratio of 1:20 is used in the simulations and experimental studies. The parameters of the cable model are listed in

Table 1. The main parameters of the scaled cable.

Diameter (mm)	Length (m)	Inclination (°)	Static Tension (N)	Mass Per Length (kg/m)	Damping Ratio of the First Three Modes (%)	Frequency of the First Three Modes (Hz)
1.58	6.1	0	219.1	1.335	0.6, 0.5, 0.5	1.02, 2.05, 3.08

.

During the simulations, the weighting matrix Q is the identity matrix selected as I_6×6 and weighting factor R is selected as 0.001. Simulations of free vibration without and with active tendon control, including linear LQR control and bilinear control, are conducted with the initial condition of Z₀ = [0.005, 0.005, 0.005, 0, 0, 0]^T, and the mid-span displacement response of the cable in different control methods is shown in Figure 2a. It can be seen from Figure 2a that the root mean square (RMS) of the mid-span displacement responses under no control, linear LQR control and bilinear control cases are 0.00307 m, 0.00132 m, and 0.00120 m, respectively. The results reflect that the vibrations of the cable are dramatically mitigated by the controller. The displacement responses are reduced about 53.1% and 61.2% with the LQR control and the bilinear control, respectively. Besides, the maximum active forces of the two control algorithms are 15.3 N and 18.5 N, respectively, as shown in Figure 2b.

A harmonic excitation force, whose amplitude is around three N and the frequency is set to the fundamental frequency of the cable, is applied at the node of 1/14 span length. The mid-span displacement of the cable of the first 40 s and the steady-state response from 80 s to 100 s are, respectively, shown in Figure 3. It shows the RMS of the mid-span displacement responses under no control, linear LQR control and bilinear control cases are 0.00615 m, 0.00173 m, and 0.00161 m, respectively. The reduction ratios of the two control strategies at the mid-span are 72.13% and 73.82%, respectively.

The results indicate that both the linear LQR control and bilinear control algorithms are valid and effective to free vibration and harmonic excitation. Although the former has a slightly worse control effect than the latter, this can still verify that the linear simplified model is feasible and applicable in engineering practices.

4. The Time-Delay Compensation Method of the Controlled Cable

When the control system has time delay, the Equation (14) can be written as:

$$\dot{Z}=AZ+B_{2}U(t\text{-}\tau )$$

(24)

where, τ is the time delay of the system.

For a multi-modal control system, the control force of the j-th mode without time-delay can be described as:

$$U_{mj}(t)=-G_{j1}{\xi }_{j1}(t)-G_{j2}{\xi }_{j2}(t)$$

(25)

where, ${\mathsf{\xi }}_{j1}(t)$ and ${\mathsf{\xi }}_{j2}(t)$ represent the modal displacement and velocity for the j-th mode, $G_{j1}$ and $G_{j2}$ are the gain coefficients of the modal displacement and velocity without time-delay, respectively.

When the time-delay of displacement and velocity in the system are uniformly assumed to be τ, the Equation (25) can be transformed into the following:

$$U_{mj}(t\text{-}\tau )=-G_{j1}{\xi }_{j1}(t-\tau )-G_{j2}{\mathsf{\xi }}_{j2}(t-\tau )$$

(26)

Then, modifying the gain coefficient with time-delay compensation, the Equation (26) can be rewritten as:

$${U^{\prime }}_{mj}(t)=-{G^{\prime }}_{j1}{\mathsf{\xi }}_{j1}(t-\tau )-{G^{\prime }}_{j2}{\xi }_{j2}(t-\tau )$$

(27)

where G^′_j1 and G^′_j2 are the modified gain coefficients, respectively.

In order for the time-delay system equivalent to be ideal, let Equation (25) be equal to Equation (27), namely:

$$U_{mj}(t)={U^{\prime }}_{mj}(t)=-G_{j1}{\xi }_{j1}(t)-G_{j2}{\xi }_{j2}(t)=-{G^{\prime }}_{j1}{\xi }_{j1}(t-\tau )-{G^{\prime }}_{j2}{\xi }_{j2}(t-\tau )$$

(28)

Equation (28) shows that the delay compensation can be realized by adjusting the gain coefficient. In this paper, the phase shifting method is used to determine the modified gain coefficients.

Since there is a time-delay in displacement and velocity during the control process, the phases of displacement and velocity are delayed compared to the state without time-delay, as illustrated in Figure 4.

When the phase difference caused by the time-delay is considered, the control force of the j-th modal displacement can be factorized as G^′_j1cos(ω_jτ) ξ_j1(t) and -G^′_j1sin(ω_jτ) ξ_j2(t)/ω_j in ξ_j1 and ξ_j2 coordinates, respectively. Similarly, the control force of the j-th modal velocity can be divided into G^′_j2sin(ω_jτ)ω_jξ_j1(t) and G^′_j2cos(ω_jτ) ξ_j2(t). Therefore, the Equation (27) can be written as [33]:

$$\begin{array}{l}{U^{\prime }}_{mj}(t)=-{G^{\prime }}_{j1}\cos ({\omega }_j\tau ){\xi }_{j1}(t)+\frac{{G^{\prime }}_{j1}\sin ({\omega }_j\tau )}{{\omega }_j}{\xi }_{j2}(t)-{G^{\prime }}_{j2}\sin ({\omega }_j\tau ){\omega }_j{\xi }_{j1}(t)-{G^{\prime }}_{j2}\cos ({\omega }_j\tau ){\xi }_{j2}(t)\\ =-\left[{G^{\prime }}_{j1}\cos ({\omega }_j\tau )+{G^{\prime }}_{j2}\sin ({\omega }_j\tau ){\omega }_j\right]{\xi }_{j1}(t)-\left[-\frac{{G^{\prime }}_{j1}\sin ({\omega }_j\tau )}{{\omega }_j}+{G^{\prime }}_{j2}\cos ({\omega }_j\tau )\right]{\xi }_{j2}(t)\end{array}$$

(29)

According to Equations (28) and (29), we can obtain:

$${G^{\prime }}_j=\left[\begin{array}{l}{G^{\prime }}_{j1}\\ {G^{\prime }}_{j2}\end{array}\right]={\left[\begin{array}{ll}\cos {\omega }_j\tau & {\omega }_j\sin {\omega }_j\tau \\ -(1/{\omega }_j)\sin {\omega }_j\tau & \cos {\omega }_j\tau \end{array}\right]}^{-1}\left[\begin{array}{l}{G}_{j1}\\ G_{j2}\end{array}\right]=D_j\left[\begin{array}{l}{G}_{j1}\\ G_{j2}\end{array}\right]=D_j{G}_j$$

(30)

Considering the first three modes, there is:

$$G^{\prime }=D_{e}G$$

(31)

where

$$D_e=\left[\begin{array}{ccc}{D}_1& 0& 0\\ 0& D_2& 0\\ 0& 0& D_3\end{array}\right]$$

When the time delay compensation is considered, Equation (23) can be rewritten as:

$$U(t)=-L^{+}{G}^{\prime }{\mathsf{\psi }}^{-1}Z$$

(32)

5. Giant Magnetostrictive Actuator and Its Dynamic Model

The active control force of the cable is provided by a (giant magnetostrictive actuator) GMA. The magnetostrictive rod employed in the GMA is made of Tb0.3Dy0.7Fe1.95 alloy, with a length of 120 mm and a diameter of 20 mm, as shown in Figure 5a. The specific structure of the designed GMA is shown in Figure 5b. The length and inner diameter of the excitation coil frame are 150 mm and 24 mm, respectively. Moreover, the number of excitation coil turns is selected as 1100 to cover the range of the control force. In addition, the manufactured GMA is illustrated in Figure 5c and the bias magnetic field is generated by a bias coil whose turns are 1300 and frame diameter is 52 mm. A six MPa preload is applied to the magnetostrictive rod by a preload spring.

Employing the cable active control model in Section 6, the relationships of the control force vs. the input voltage of GMA are obtained as follows [24]: ascending stage

$$U=-0.0114V_{\mathrm{in}}^3-0.0078V_{\mathrm{in}}^2+2.5955V_{\mathrm{in}}+0.3784$$

(33)

descending stage

$$U=0.0058V_{\mathrm{in}}^3-0.0993V_{\mathrm{in}}^2+1.2425V_{\mathrm{in}}+8.3992$$

(34)

where U is the control force produced by the GMA in the axial direction of the cable, V_in is the input voltage.

6. Simulation for the Active Control of Cable with the Time-Delay

6.1. Free Vibration

The cable model in simulation is the same as shown in Table 1. The initial conditions of free vibration is Z₀ = [0.005, 0.005, 0.005, 0, 0, 0]^T. Four conditions, including without control, control without time-delay, non-compensation control with time delay and time-delay compensation control, are simulated. The weighting matrix Q = I_6×6 and weighting factor R = 0.001. When the time-delay is set to 0.11 s, the displacement response of the cable at mid-span is illustrated in Figure 6. From Figure 6, the RMS of displacements at mid-span are 0.00307 m, 0.00132 m, 0.00276 m and 0.00143 m in the four cases, respectively. For the three control strategies, the mitigating ratios of the cables at mid-span are 57.13%, 9.87%, and 53.42%, respectively. The results show the control effect of time-delay compensation control is close to the control state when time-delay is not considered. Moreover, it is implied that non-compensation control with time delay cannot effectively suppress the vibration of the cable when the time-delay reaches 0.11 s.

The results of time-delay compensation control with different time-delay amounts under the free vibration are summarized in

Table 2. Response reduction at mid-span of three control strategies under free vibration.

Time-Delay Amounts (s)	Reduction without Time-Delay (%)	Reduction without Compensation of Time-Delay (%)	Reduction with Time-Delay Compensation (%)
0.00	57.13
0.05		56.06	55.37
0.07		55.76	54.41
0.09		42.12	51.90
0.11		9.87	53.34
0.13		−12.26	54.23

. It can be seen that the control effect of the cable vibration gradually decreases as the time-delay increases under non-compensation control with time delay. In particular, when the time-delay increases to a certain value, the response of the cable is greater than that without control state, meaning the control is woefully inefficient. On the contrary, when the phase-shift method for time-delay compensation is applied, satisfactory results demonstrate the cable vibration is significantly suppressed.

6.2. Harmonic Excitation

A harmonic excitation force is applied on the node of L/14 span, whose amplitude is 3N and frequency is set to the fundamental frequency of the cable. Other simulation conditions are same as the free vibration case. When the time-delay is set to 0.11 s, the displacement responses of the cable at mid-span are shown in Figure 7. The RMS of displacements in the four cases are 0.00615 m, 0.00173 m, 0.00258 m and 0.00179 m, respectively. For the three control strategies, the response reduction ratios of the cables at mid-span are 72.13%, 58.04%, and 70.89%, respectively.

With the change of the time-delay, the reductions at mid-span of three control strategies under harmonic excitation are summarized in

Table 3. Response reduction at mid-span of three control strategies under harmonic excitation.

Time-Delay Amounts (s)	Reduction without Time-Delay (%)	Reduction without Compensation of Time-Delay (%)	Reduction with Time-Delay Compensation (%)
0.00	72.13
0.09		64.62	71.08
0.11		58.04	70.89
0.13		31.41	69.15
0.15		divergence	67.62

. The results indicated that the control effect of the cable vibration gradually decreases as the time-delay increases for the condition without the time-delay compensation. Notably, when the time-delay is increased to 0.15 s, the response of the cable under no compensation is divergence. While the method of time-delay compensation is applied, the reduction in cable still reaches 67.62%, even though the time-delay increased to 0.17 s.

6.3. Random Excitation

The random excitation used in the simulation is Gaussian white noise excitation. The RMS of random excitation is 0.81 N, the peak value is 2.48 N, and the applied position is 1/14 span length of the cable. Other simulation conditions are the same as the free vibration case. When the time-delay is set to 0.11 s, the displacement responses at 1/2 span length are shown in Figure 8. The RMS of displacements in the four cases are 0.00516 m, 0.00189 m, 0.00594 m and 0.00205 m, respectively. The mitigating ratios of the three control strategies at the mid-span are 63.37%, −15.12%, and 59.12%, respectively.

Considering the change of the time-delay, the Mitigating ratios at mid-span of three control strategies under random excitation are summarized in

Table 4. Response reduction at mid-span of three control strategies under random excitation.

The Time-Delay (s)	Mitigating Ratio without Time-Delay (%)	Mitigating Ratio without Compensation with Time-Delay (%)	Mitigating Ratio with Time-Delay Compensation (%)
0.00	63.37
0.05		62.47	60.86
0.07		60.66	60.49
0.9		58.70	59.31
0.11		−15.12	59.12
0.13		divergence	57.67

. The results also indicated that the cable vibration is valid to be suppressed using the method of time-delay compensation.

7. Active Control Experiment of Cable with the Time-Delay

7.1. Experimental Set-Up

The cable model used in the experiment is the same as the simulation model, as shown in Figure 9a. The cable is made by steel wire with a diameter of 1.58 mm and a length of 6.1 m. To achieve the identical dynamic parameters of the practical cable, some additional mass blocks are uniformly arranged along the span of the cable model, and each mass block is 0.192 kg. The other detailed parameters are shown in Table 1. The active axial control force is generated by GMA (the required current and voltage of the actuator are 0.6 A and 15 V, respectively, and therefore the power consumption is 9 W) as shown in Figure 9b. In addition, there are two laser displacement sensors (model: optoNCDT14, which is manufactured by με Company with the sampling frequency 1000 Hz) set at 1/2 and 1/4 span length, respectively, to measure the displacement responses and a vibration exciter (model: KDJ-20, manufactured by Yangzhoukedong Technology Electronics Company in Yangzhou, China) connected at the point that is 1/14 span length away from the fixed support to provide harmonic and random excitation. The control system is shown in Figure 9c, where the active control can be come true by Simulink (developed by the MathWorks Company in Massachusetts, USA) and dSPACE (developed by the German dSPACE Company in Paderborn, German), and the amount of time-delay is adjusted by the Simulink module illustrated in Figure 9d. The module “Transport Delay” can adjust the amount of the time-delay, and the control force (the module “kongzhili”) is outputted based on the LQR algorithm considering the time-delay compensation. According to the relationship between force and voltage, the output voltage (the module “dianya”) is computed and applied to the cable by the actuator.

The excitation in the experiment included three cases: free vibration, harmonic excitation and random excitation. For the free vibration experiment, the initial displacement of 0.011 m was applied at the middle of the span. In the harmonic excitation experiment, a harmonic excitation force with the frequency of first modal of the cable was provided by a vibration exciter connected at the point of around 1/14 span length. For the random excitation experiment, a Gaussian white noise with RMS value 0.81N and peak value 2.48N was applied at 1/14 span length of the cable.

For the active control of the cable with mode-based LQR algorithm, it is necessary to measure the cable response to determine all modal responses and then to determine the real-time state variables. However, the modal response cannot be directly measured in the practical control process, and it is very difficult to obtain all the modal responses of the cable accurately with limited sensors. As the displacement response of the cable is mainly the first mode component, the influence of the higher-order mode is ignored in this paper, and only the first-order modal response is considered. When n = 1, Equation (1) is simplified as follows:

$$w(x,t)={\varphi }_1(x)q_1(t)$$

(35)

when x = L/2, we have,

$$w(L/2,t)=q_1(t)$$

(36)

From the Equations (29) and (30), the displacement at the mid-span of the cable is equivalent to the first-order modal displacement. Therefore, the control force can be obtained by the displacement response at the mid-span in the experiment.

7.2. Experimental Results

The displacement history of the cable at mid-span under free vibration, harmonic excitation and random excitation, are illustrated in Figure 10, Figure 11 and Figure 12, respectively. Moreover, for the different time-delay amounts, the experimental RMS reductions at mid-span under free vibration, harmonic excitation and random excitation are, respectively, summarized in

Table 5. Experimental response reductions at mid-span under free vibration.

The Time-Delay (s)	Mitigating Ratio without Time-Delay (%)	Mitigating Ratio without Compensation with Time-Delay (%)	Mitigating Ratio with Time-Delay Compensation (%)
0.00	55.63
0.05		54.23	53.14
0.07		52.62	52.55
0.09		42.15	50.69
0.11		18.77	50.51
0.13		divergence	49.67

,

Table 6. Experimental response reductions at mid-span under harmonic excitation.

The Time-Delay (s)	Mitigating Ratio without Time-Delay (%)	Mitigating Ratio without Compensation with Time-Delay (%)	Mitigating Ratio with Time-Delay Compensation (%)
0.00	67.80
0.05		65.24	63.47
0.07		60.89	61.55
0.09		58.65	62.97
0.11		47.83	62.68
0.13		−18.26	60.72

and

Table 7. Experimental response reductions at mid-span under random excitation.

The Time-Delay (s)	Mitigating Ratio without Time-Delay (%)	Mitigating Ratio without Compensation with Time-Delay (%)	Mitigating Ratio with Time-Delay Compensation (%)
0.00	48.69
0.05		48.08	47.81
0.07		46.72	46.24
0.09		42.93	45.11
0.11		−35.18 (divergence)	45.93
0.13		divergence	44.57

. It can be seen that the cable vibration is significantly suppressed when the time-delay compensation is applied, which indicates the proposed compensation method is effective. Moreover, experimental results agree well with the simulation results. However, the experimental results are slightly lower than results from simulations. This may be due to the fact that the experiments only consider controlling the first mode.

8. Conclusions

In this paper, the simulation and experimental research on the active control of cables with time-delay is carried out. The main work includes the motion equation for the active control of cable, active control algorithm, GMA actuator and its mechanical model, time delay compensation algorithm, and mitigating performance evaluation of cable. Some conclusions are obtained as follows.

When the axial control force is used to control the cable lateral vibration, the controlled state equation is a bilinear equation due to the influence of the sag. In order to apply the time delay compensation method of the linear control system, the nonlinear term in the controlled state equation is ignored and the controlled equation is linearized. The simulation results show that, for the cable with small sag, the mitigating effect obtained by using the linearized control equation is only slightly lower than that by the bilinear controlled equation, which indicates that the linearization method is feasible.

A GMA was fabricated with Tb0.3Dy0.7Fe1.95 alloy magnetostrictive materials, which provides axial control force for cables. The relationship between input voltage and output force of GMA is obtained by experiment and applied to the control experiment. Based on the phase-shifting method, a multi-mode time-delay compensation algorithm is proposed, and the time-delay compensation matrix based on state feedback control is obtained.

Considering four different cases, simulations and experiments are conducted to investigate the mitigating performance under free vibration, harmonic excitation and random excitation. The results show that, with the increase in time delay, the mitigating ratios without time-delay compensation decrease significantly, while the excellent mitigating effect can be achieved by using the proposed time-delay compensation method.