Stochastic optimal semi-active control of stay cables by using magneto-rheological damper

Abstract

Stochastic optimal semi-active control for stay cable multi-mode vibration attenuation by using magneto-rheological (MR) damper is developed. The Bingham model for an MR damper is used. The force produced by an MR damper is split into passive and active parts. The passive part is combined with structural damping forces into effective damping forces. The partially averaged Itô stochastic differential equations for controlled modal energies are derived by applying the stochastic averaging method for quasi-integrable Hamiltonian systems. Then the dynamical programming equation for controlled modal energies with an index involving control force is established by applying the stochastic dynamical programming principle, and a stochastic optimal semi-active control law is obtained by solving the dynamical programming equation. For controlled modal energies with an index not involving control force, bang-bang control law is obtained without solving a dynamical programming equation. A comparison between the two control laws shows that the stochastic optimal semi-active control strategy is superior to the bang-bang control strategy in the sense of higher control effectiveness and efficiency and less chattering.

Keywords

MR damper stay cable semi-active control stochastic optimal control

1. Introduction

Cables are critical structural components in cable-stayed bridges and suspension bridges. They are prone to vibration owing to their large flexibility, relative small mass and very low inherent damping. For example, under the combination of wind and rain, large-amplitude vibration of stay cables may be observed in a number of cable-stayed bridges (Mastsumoto et al., 1995, 2003; Wu et al., 2003). Such large-amplitude vibration may induce undue stresses and fatigue in the cables and their connections with the deck and the towers. To reduce cable vibration, transversely-attached passive viscous dampers have been widely used. To evaluate the effectiveness of the viscous dampers, some theoretical studies have been carried out. Pacheco et al. (1993) showed a universal curve for modal damping in stay cables with viscous damper. Complex modes and harmonic balance methods were used to study the nonlinear vibration of a cable-damper system, including the effects of cable sag and inclination and was verified in the laboratory with scaled cables and oil dampers (Yu and Xu, 1999; Xu and Yu, 1999). Abdel-Rohman and Spencer (2004) used a vertical viscous damper to control the galloping response of a suspended cable. Wang et al. (2005) designed a damper with optimal damping in assigned multiple modes based on optimal linear quadratic Gaussian (LQG) control strategy and gave the case studies of prototype cables on a real cable-stayed bridge to show the efficiency of the proposed method.

However, for long cables, passive dampers may provide insufficient supplemental damping to eliminate vibration. Recent studies on semi-active damping showed that smart dampers such as MR dampers can ideally exert dissipative forces and provide significant superior supplemental damping for a cable. Dyke et al. (1996) proposed a clipped optimal control strategy for MR dampers to reduce structural responses due to seismic loads based on acceleration feedback. Johnson et al. (2000, 2003) found that a semi-active damper could reduce the cable vibration much more than the optimal passive linear viscous damper for typical damper configurations. Neural network was also used to design the control strategies for MR dampers (Ni et al., 2002). Field vibration tests (Duan et al., 2002; Ko et al., 2002) of bridge stay cables incorporated with MR dampers were taken and experiments were also carried out by some researchers (Christenson et al., 2006; Duan et al., 2004).

Many engineering structures under random loading such as wind and earthquake ground motion can be modeled as quasi-Hamiltonian systems. A set of nonlinear stochastic optimal control strategies including the stochastic optimal semi-active control strategies have been developed by the second author of this paper and his co-workers (Cheng et al., 2006; Dong et al., 2004; Ying et al., 2003; Zhu and Ying, 1999; Zhu et al., 2001, 2004) for quasi-Hamiltonian systems. The optimal bounded control of quasi-Hamiltonian systems was also studied (Ying and Zhu, 2004; Ying et al., 2007).

In this paper the governing equation for a cable with an MR damper subject to distributed Gaussian white noise excitation is first derived. Then the equation is discretized as a quasi-Hamiltonian system by using the Galerkin method. The force produced by a Bingham model for an MR damper is split into passive and bounded semi-active parts and the partially averaged Itô stochastic differential equations for the modal energies of the system are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems. The dynamical programming equation is established based on the stochastic dynamical programming principle, from which the stochastic optimal semi-active control law is determined. Finally, the efficacy of the proposed semi-active control strategy is accessed by using the numerical results for an example and compared with that of bang-bang control strategy.

2. Equations of controlled system

An inclined cable with an MR damper attached to the cable near the lower anchorage is shown in Figure 1. The cable has a sag 1 at the mid-span along x-axis, chord length 2 , inclination angle $θ$ , static tension $T_{0}$ in the longitudinal x-direction, cable mass 5 per unit length and the inherent damping coefficient 6 per unit length of cable. The damper is attached at the location of $x_{d}$ from the cable lower end and normal to x-axis. The semi-active control force $f_{d}$ is produced by an MR damper. As pointed out by Johnson et al. (2003), the phenomena that cause rain-wind induced vibration, including the aerodynamic forces, motion of water rivulets, and the nonlinear coupling with the cable motion are not all understood. In the absence of a suitable model, the distributed excitation $f (x, t)$ is modeled as a separable random field, i.e.,

(1)

where

α (x)

is a deterministic function of the position along the cable and

W (t)

is simply modeled as a zero-mean Gaussian white noise with intensity

2 D

. Assume that: (i) the static profile of the cable is a second-order parabola

y_{0} (x)

; (ii) the cable vibrates only in the xy-plane and its motion in the x-direction is negligibly small; (iii) the elastic cable is perfectly flexible so that it is capable of developing stresses only in the direction normal to the cross section and (iv) the sag-to-span ratio

d / L

is sufficiently small with respect to unity. By introducing the following normalized variables:

(2)

and omitting the overbars for convenience, the in-plane transverse cable motion

y (x, t)

, relative to the static profile, is governed by the following nondimensional governing equation (Irvine, 1981)

(3)

together with the boundary conditions

(4)

Figure 1.

Profile of inclined cable with sag and an magneto-rheological damper.

In Equation (3), $δ (\cdot)$ is the Dirac delta function and the sag-extensibility parameter $λ^{2}$ is defined as the ratio of the elastic-to-catenary stiffness as follows

(5)

where 18 is the gravity acceleration; 19 is the Young’s modulus; 20 is the cross-sectional area of the cable;

(6)

is the static length of the cable.

For flat-sag cables,

(7)

is the peak sag of the parabolic static profile

(8)

According to the Galerkin method, the dynamic response $y (x, t)$ may be approximated by using a finite series

(9)

where n is the number of degrees-of-freedom considered,

q_{i} (t)

are generalized displacement and the

φ_{i} (x)

are a set of continuous shape functions with piecewise continuous slope satisfying the following geometric boundary conditions

(10)

Johnson et al. (2000, 2003) showed that introducing a static deflection shape as an additional shape function will provide an excellent control-oriented model. The static deflection shape function is

(11)

and the remaining shape functions are

(12)

Substituting the shape functions into Equation (9) and then into Equation (3) leads to the following matrix equation:

(13)

where

M

C

, and

K

are generalized mass, damping and stiffness matrices, and

\bpsi_{e}

and

\bpsi_{d}

denote the generalized external force amplitude vector and the damper amplitude vector, respectively. Their expressions are

(14a)

(14b)

(14c)

(14d)

(14e)

Several mechanical models, such as the Bingham viscous-plastic model and the Bouc-Wen model, for MR dampers have been proposed (Spencer et al., 1997). Here the Bingham model is used for simplicity. According to the Bingham model, the force produced by an MR damper is

(15)

where

c_{d}

is the viscous damping coefficient and

{\tilde{f}}_{d} (\tilde{E})

is the frictional force related to fluid yield stress and controlled by the applied voltage

\tilde{E}

. The first term on the right hand side of Equation (15) denotes a passive force independent of the external voltage and the second term denotes an active force depending on the external voltage, i.e.

(16)

where

(17)

By combining $u_{p}$ with the cable structural damping force, Equation (14) can be rewritten as

(18)

where

C_{p} = C + c_{d} \bpsi_{d} \bpsi_{d}^{T \nolimits}

. Introduce a modal transform

(19)

where

Q = [Q_{1,} Q_{2},\dots, Q_{n}]^{T}

;

\buTheta

is an

n \times n

matrix that satisfies

\buTheta^{T} M \buTheta = I_{n \times n}

and

\buTheta^{T} K \buTheta = Diag \buTheta (ω_{i}^{2})

with

| ω_{i}^{2} M - K | = 0

. Then the equation for the 40 dominating modes can be written as

(20)

where

\buOmega^{2} = \buTheta^{T \nolimits} K \buTheta

;

\buGamma = \buTheta^{T \nolimits} C_{p} \buTheta

(i = 1, 2, \dots, n)

Equation (20) can be re-written as a controlled, stochastically excited and dissipated Hamiltonian system

(21)

with separable Hamiltonian

H = \sum_{i = 1}^{n} H_{i}

where

(22)

denotes the energy of the i-th mode.

3. Semi-active control strategy

3.1. Stochastic averaging

Equation (21) can be further converted into the following Itô stochastic differential equation:

(23)

where

B (t)

is a standard Wiener process. To simplify Equation (23) and reduce its dimension, the stochastic averaging method for quasi-integrable Hamiltonian systems (Zhu et al., 1997) is applied to Equation (23) except the control force term. The obtained partially averaged Itô stochastic differential equations for modal energies

H_{i}

are

(24)

where

(25)

3.2. Optimal control law

3.2.1. Stochastic optimal semi-active control law

Equation (24) implies that $H = [H_{1,} H_{2},\dots, H_{n}]^{T}$ is a vector of controlled diffusion processes and the stochastic dynamical programming principle can be applied. The objective here is to seek the optimal feedback control law $u_{a}^{*}$ to minimize a semi-infinite time-interval performance index

(26)

Based on the stochastic dynamical programming principle, the following dynamical programming equations is established

(27)

where

(28)

is the optimal average cost function and 49 is the value function.

The expression of $u_{a}^{*}$ can be obtained from minimizing the right hand side of Equation (27) with respect to $u_{a}$ . Suppose that the cost function 52 is of the form

(29)

where 53 is a positive real number. For symmetric bounded control forces,

(30)

u_{a}^{*}

is of the following form

(31)

in which

(32)

Note that the MR damper can only produce a dissipated force which satisfies the condition

(33)

It is seen from Equation (32) that if

(34)

the damper may exert the optimal control forces all the time. That means Equation (32) can be rewritten as

(35)

which dissipates the energy of the vibration and stabilizes the controlled system.

Substituting Equation (35) into Equation (27), and completing the averaging with respect to $Q$ and $P$ yields the following final dynamical programming equation:

(36)

where

Δ_{ii} = {[\buTheta^{T} \bpsi_{d} \bpsi_{d}^{T} \buTheta]}_{ii}

. The value function 58 can be obtained by solving the final dynamical programming Equation (36). Then the optimal control law can be obtained by substituting the resultant

\partial V / \partial H

into Equation (35). Actually, it is difficult to directly solve the partial differential equation so that an alternative approach is adopted to convert the equation into algebraic equations. Let

g (H)

be of the following polynomial form in

H_{i}

, i. e.,

(37)

Then the corresponding value function solution 62 of Equation (36) may be assumed to be of the polynomial form in $H_{i}$

(38)

Submitting Equations (37) and (38) into Equation (36) and vanishing the coefficients of the resultant polynomials in $H_{i}$ yield a series of algebraic equations. In Equation (38) there are only two parameters, and therefore only two algebraic equations have to be solved. Substituting $\partial V / \partial H_{i}$ into Equation (35) yields

(39)

which can be implemented by MR damper.

3.2.2. Bang-bang control law

If the performance index (26) does not depend explicitly on $u_{a}$ , i.e.,

(40)

The right-hand side of Equation (27) will be minimum when $| u_{a} | = u_{b}$ and $(\partial V / \partial H)^{T} (\partial H / \partial P) \buTheta^{T \nolimits} \bpsi_{d} u_{a}$ is negative. Thus, the optimal control law is

(41)

where ‘sgn’ denotes sign function. Take

L (H)

so that

\partial V / \partial H_{i} = \partial V / \partial H \gt 0

and Equation (41) can be rewritten as

(42)

which implies that optimal control

u_{a}^{*}

is a bang-bang control. It has a constant magnitude, in the opposite direction of

\overset{\cdot}{y} (x_{d},t)

and changes its direction at

\overset{\cdot}{y} (x_{d},t) = 0

. The bang-bang control satisfies

u_{a}^{*} \cdot \overset{\cdot}{y} (x_{d},t) \leq 0

, which implies that the controlled system is stable.

4. Performance criteria

A numerical method such as the Runge-Kutta method is used to predict the response of the controlled system described by Equation (18). To characterize the performance of the semi-active control strategy, two criteria are introduced. One is control effectiveness, which is defined as the relative reduction in the root mean square (RMS) of cable deflection, i.e.,

(43)

where the root mean square cable deflection is defined as (Johnson et al., 2000, 2003)

(44)

and subscripts 75 and 76 denote uncontrolled and controlled systems, respectively.

The other is controller efficiency, which is defined as the ratio of relative reduction in the RMS of cable deflection to the ratio of the RMS of the optimal control force to the RMS of excitation intensity, i.e.,

(45)

Obviously, higher values of 77 and $μ$ indicate a more effective and more efficient control strategy.

5. Case study

Numerical calculation is conducted on a stay cable with the following parameter values (Ni et al., 2002): cable length $L = 11.90 m$ ; mass $m = 0.391 kg / m$ per unit length; effective axial stiffness $EA = 6.0 \times 10^{5} N$ ; initial tension $T_{0} = 1.5 kN$ ; inclination angle $θ = 18 . 12^{\circ}$ . The damper is attached at $x_{d} = 0.05 L$ and the inherent damping coefficient $c = 0.001 N \cdot s / m$ . The excitation is uniformly distributed, i.e. the deterministic function in Equation (2) is $α (x) = 1$ . In the Bingham model of the MR damper, the damping coefficient is $c_{d} = 0.8 N \cdot s / m$ and the allowable maximum value of the MR damper force equals $30 N$ . One static deflection and the first eight vibration modes are considered so the number of degrees-of-freedom $n = 9$ .

Figures 2 to 7 show the numerical values of 112 and $μ$ , respectively in bang-bang control and stochastic optimal semi-active control for different nondimensional excitation intensity values. It can be seen from Figure 2 that there exists a most effective bang-bang control force for certain excitation intensity. Figures 2 to 7 indicate that both the control effectiveness and efficiency using bang-bang control and the stochastic optimal semi-active control are higher where there is less intensity of excitation. Table 1 gives a comparison between the bang-bang control and stochastic optimal semi-active control. It is seen that for a different intensity of excitation, the stochastic optimal semi-active control can achieve higher control effectiveness and efficiency than the most effective bang-bang control. When the maximum control force is selected for the bang-bang control, the control effectiveness and efficiency reduce and the stochastic optimal semi-active control strategy is superior to the bang-bang control strategy in the sense of higher control effectiveness and efficiency. Samples of the bang-bang control force and the stochastic optimal semi-active control force are given in Figure 8 and Figure 9 respectively for $2 D = 7.6 \times 10^{- 7}$ . It is seen that the stochastic optimal semi-active control force is less discontinuous than the bang-bang control force. Thus, the chattering in the bang-bang control can be attenuated by using the stochastic optimal semi-active control.

Figure 2.

Control effectiveness K_bc of bang-bang control as a function of control bound u_a for different excitation intensity 2D values.

Figure 3.

Control efficiency μ_bc of bang-bang control as a function of control bound u_a for different excitation intensity 2D values.

Figure 4.

Control effectiveness K_oc of the stochastic optimal semi-active control as a function of weight coefficient s₁₂ (s₃₂ = 1.05, R = 1.0) for different excitation intensity 2D values.

Figure 5.

Control efficiency μ_oc of the stochastic optimal semi-active control as a function of weight coefficient s₁₂ (s₃₂ = 1.05, R = 1.0) for different excitation intensity 2D values.

Figure 6.

Control effectiveness K_oc of the stochastic optimal semi-active control as a function of weight coefficient s₃₂ (s₁₂ = 1.055, R = 1.0) for different excitation intensity 2D values.

Figure 7.

Control efficiency μ_oc of the stochastic optimal semi-active control as a function of weight coefficient s₁₂ (s₁₂ = 0.0155, ; R = 1.0) for different excitation intensity 2D values.

Figure 8.

A sample of bang-bang control force (u_a = 0.002).

Figure 9.

A sample of stochastic optimal semi-active control force (s₁₂ = 0.0155, ; s₃₂ = 3.85, ; R = 1.0).

Table 1.

A comparison of the bang-bang control and the stochastic optimal semi-active control

The intensity of excitation $2 D$	Bang-bang control			Stochastic optimal semi-active control
The intensity of excitation $2 D$	$u_{a}$	$K_{bc}$	$μ_{bc}$	$s_{12}$	$s_{32}$	$K_{oc}$	$μ_{oc}$
$2.6 \times 10^{- 7}$	0.0006	0.8563	0.7283	0.0188	1.0500	0.8658	0.8175
	0.002	0.8073	0.2057	0.0155	0.7000	0.8645	0.8820
$5.1 \times 10^{- 7}$	0.001	0.8002	0.4078	0.0188	1.0500	0.8113	0.5336
	0.002	0.7655	0.1951	0.0155	0.7000	0.8100	0.5878
$7.6 \times 10^{- 7}$	0.0012	0.7561	0.3216	0.0188	1.0500	0.7685	0.4185
	0.002	0.7334	0.1869	0.0224	1.4000	0.7671	0.4599

6. Conclusions

In this paper, a stochastic optimal semi-active control strategy for a stay cable using the MR damper has been developed based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. It has been shown that the MR damper can completely implement the optimal control force. The case study demonstrated that compared with the bang-bang control using the stochastic optimal semi-active control strategy, the MR damper can achieve higher control effectiveness and efficiency and attenuates the chattering.

Footnotes

Acknowledgments

The work proposed in this paper is supported by the National Science Foundation of China under Grant Nos. 11072212 and 10932009 and the Zhejiang Natural Science Foundation of China under Grant No.7080070.

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