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ER dampers
Probabilistic Engineering Mechanics 19 (2004) 279–285 www.elsevier.com/locate/probengmech
Semi-active control of wind excited building structures using MR/ER dampers W.Q. Zhu*, M. Luo, L. Dong Department of Mechanics, Zhejiang University, Hangzhou 310027, China Received 15 November 2003; revised 15 December 2003; accepted 2 February 2004
Abstract A semi-active control strategy for building structures subject to wind loading and controlled by MR/ER dampers is proposed. The power spectral density (PSD) matrix of the fluctuating part of wind velocity vector is diagonalized in the eigenvector space. Each element of the diagonalized PSD matrix is modeled as a set of second-order linear filter driven by white noise. A Bingham model for MR/ER dampers is used. The forces produced by MR/ER dampers are split into passive and active parts and the passive part is combined with structural damping forces. A set of partially averaged Itoˆ equations for controlled modal energies are derived by applying the stochastic averaging method for quasi-integrable-Hamiltonian systems. The optimal control law is then determined by using the stochastic dynamical programming principle and the cost function is so selected that the optimal control law can be implemented by the MR/ER dampers. The response of semiactive controlled structures is predicted by using the reduced Fokker – Planck– Kolmogorov equation associated with fully averaged Itoˆ equations of the controlled structures. A comparison with clipped linear quadratic Gaussian (LQG) control strategy, for an example, shows that the proposed semi-active control strategy for MR/ER dampers is superior to clipped LQG control strategy. q 2004 Elsevier Ltd. All rights reserved. Keywords: Tall building; Wind excitation; MR/ER damper; Bingham model; Stochastic optimal control; Stochastic averaging; Stochastic dynamical programming
1. Introduction Structural control of large civil engineering structures has been studied for more than two decades. It evolves from passive control, active control to semi-active control. Recently, semi-active control systems attract much attention for their low-energy requirement and cost, and having best features of both passive and active controls, offering the reliability of passive devices yet maintaining the versatility and adaptability of active systems. A number of semi-active control devices have been developed, such as variableorifice dampers, variable-friction dampers, controllable tuned liquid dampers, semi-active impact dampers and controllable-fluid dampers, etc. [1]. Because of the intrinsically nonlinear nature of semi-active control devices, the feedback control law must be nonlinear. The development * Corresponding author. Tel.: þ 86-571-8799-1150; fax: þ 86-571-87952651. E-mail address: [email protected] (W.Q. Zhu). 0266-8920/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2004.02.011
of such control strategy that is practically implementable and can fully utilize the capabilities of these unique devices is an important and challenging task. Some semi-active control strategies have been developed. Leitmann applied Lyapunov’s direct approach for the design of a semi-active controller [2]; McClamroch and Gavin used a similar approach to develop the decentralized bang-bang control law for using an ER damper [3]; Inaudi developed the modulated homogeneous friction controller for a variablefriction damper [4]; Sun and Goto used the fuzzy control method [5] and Dyke et al. presented the clipped linear optimal control law that has been shown effective for MR damper [6]. Recently, a stochastic optimal semi-active control strategy for MR/ER damper was proposed [7]. It has been shown that if parameters were properly selected, then the MR/ER damper can fully generate the optimal control force and the control effectiveness and efficiency are better than those of clipped linear quadratic Gaussian (LQG). In this paper, this stochastic optimal semi-active control strategy is extended to wind excited tall building structures.
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where wðtÞ is a l-dimensional vector of independent Gaussian white noises with unit intensity
2. Model of wind excitation The fluctuating part vðtÞ of a wind velocity vector is assumed to be a zero-mean Gaussian random process vector. It is characterized by power spectral density (PSD) matrix Sv ðvÞ; which can be decomposed in the eigenvector space [8], i.e. cT ðvÞSv ðvÞcðvÞ ¼ LðvÞ
ð1Þ
where cðvÞ is the normalized eigenmatrix of Sv ðvÞ; i.e. cT ðvÞcðvÞ ¼ I
ð2Þ
where I is a unit matrix. LðvÞ is a diagonal matrix whose ith element Li ðvÞ is the eigenvalue associated with ith column ci ðvÞ of cðvÞ: Since the eigenvectors vary slowly with frequency v; it is possible to take the following approximation Sv ðvÞ ø cðv~ÞLðvÞcT ðv~Þ
ð3Þ
where v~ is an appropriately selected fixed value of circular frequency v: Then Li ðvÞ can be regarded as the PSD of the output velocity y_ i1 ðtÞ governed by the following equations: y€ i ðtÞ þ Jif y_ i ðtÞ þ V2if yi ðtÞ ¼ Gi wi ðtÞ where 2 6 6 6 6 yi ¼ 6 6 6 4
ð4Þ
3
yi1
7 yi2 7 7 7 ; .. 7 7 . 7 5 yimi
2 6 6 6 6 6 6 Jif ¼ 6 6 6 6 6 4 2 6 6 6 6 2 Vif ¼ 6 6 6 4
y ¼ ½y1 ; y2 ; …; yl T ; Jf ¼ diagðJ1f ; J2f ; …; Jlf Þ
ð7Þ
V2f ¼ diagðV21f ; V22f ; …; V2lf Þ; G ¼ diagðG1 ; G2 ; …; Gl Þ Once the filter differential equation (6) has been determined, the fluctuating part of wind velocity vector, vðtÞ; is given in the form ~ yðtÞ ~ v~ÞA_ vðtÞ ø cð
ð8Þ
~ v~Þ is the sub-eigenmatrix consisting of the first l where cð wind-eigenvectors evaluated at v ¼ v~; y_ ðtÞ is obtained by ~ is a Boolean matrix whose non-zero solving Eq. (6) and A elements are those corresponding to the components y_ i1 ðtÞ; i ¼ 1; …; l of the filter response velocity vector y_ ðtÞ: The fluctuating part of the wind force vector can then be written in the form FðtÞ ¼ Tðv~Þ_yðtÞ
ð9Þ
~ r is air density; CD is drag ~ v~ÞA; where Tðv~Þ ¼ rCD Avcð coefficient; A is a diagonal matrix whose iith element is the ith nodal area Ai ; v is a diagonal matrix with the mean wind velocities v ðzi Þ as its elements.
3. Model of MR/ER damper
2zi1 vi1
21
0
···
0
0
2zi2 vi2
21
···
0
.. .
0
2zi3 vi3
···
0
.. .
.. .
.. .
..
21
0
0
0 3
0
v2i1
···
···
0
v2i2
···
.. .
.. .
..
0
0
0
.
.
Several mechanical models such as Bingham viscoplastic model and Bouc –Wen model have been proposed for MR/ER dampers. For a steady and fully developed flow, the shear resistance of MR/ER fluids may be modeled as
3 7 7 7 7 7 7 7; 7 7 7 7 5
ð5Þ
2zimi vimi
in which wi ðtÞ is a Gaussian white noise with unit intensity; mi is the number of the filters required for appropriately modeling eigenvalue Li ðvÞ; vij ; zij and gi are parameters of the filters. By collecting the first l equations corresponding to the first l significant eigenvalues, the following compact form of the filter differential equations is obtained y€ ðtÞ þ Jf y_ ðtÞ þ V2f yðtÞ ¼ GwðtÞ
ð10Þ
which is called Bingham model [9]. In Eq. (10), t is the shear stress in fluid, h is the Newtonian viscosity independent of the applied magnetic/electric filed, g_ is the shear strain rate and ty ðEÞ is the yielding shear stress controlled by the applied magnetic/electric field. For a Bingham material in a rectangular duct, an approximate relationship between force Pd and relative velocity u_ of the piston relative to the damper cylinder is given as follows [10]
2 3 0 7 7 6 607 0 7 7 6 7 7 7 7; Gi ¼ 6 6 .. 7 7 7 6 7 . 0 5 4 5 gi v2imi 0
t ¼ hg_ þ ty ðEÞsgnðg_Þ
ð6Þ
Pd ðtÞ ¼ Cd u_ þ Fd ðEÞsgnð_uÞ
ð11Þ
where Cd ¼ C1
12hLAp Ap ; bh3
Lty ðEÞ Ap þ Py Fd ðEÞ ¼ C2 ðEÞ h
ð12Þ
W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
_ Eq. (15) can be rewritten as By combining Up with CX;
For the flow-type damper C1 ¼ 1:0;
C2 ðEÞ ¼ 2:07 þ
1:0 ; 1:0 þ 0:4TðEÞ
_ þ KX ¼ PUa þ FðtÞ; € þ Cp X MX ð13Þ
bh2 ty ðEÞ TðEÞ ¼ 12Ap hu_
V¼
ð19Þ
where Cp ¼ C þ PCd P1 : Introducing model transform ð20Þ
where Q ¼ ½Q1 ; Q2 ; …; Qn T ; F is an N £ n modal submatrix consisting of the first n ðn # NÞ columns of the modal matrix. Assuming that Cp satisfies the classical orthogonality condition, then equation for the n dominating modes can be written as
bh ; 2Ap
C2 ðEÞ ¼ 2:07 þ
Xð0Þ ¼ X0
X ¼ FQ
For the mixed type damper C1 ¼ 1:0 2
281
1:0 1:5V 2 2 ; 1:0 þ 0:4TðEÞ 1:0 þ 0:4T 2 ðEÞ
ð14Þ
€ þ 2zVQ _ þ V2 Q ¼ FT FðtÞ þ FT PUa ; Q
ð21Þ
Qð0Þ ¼ Q0
bh 2Ap
where L is the effective axial pole length; Ap is the effective cross-sectional area of the piston; Py is the mechanical friction force in the damper; h is the gap between two magnetic/ electric poles; and b is the circumference of the inner magnetic/electric pole. Clearly, Fd is a function of the yielding shear stress and can be controlled through change in the applied field intensity but Cd is independent of the applied magnetic/electric field.
4. Equation of control system Considering an N-storey building structure subject to wind loading and installed with m MR/ER dampers. The equation of the system is € þ CX _ þ KX ¼ PU þ FðtÞ; MX
Xð0Þ ¼ X0
ð15Þ
where M; C; K are N £ N mass, damping and stiffness matrix, respectively; X ¼ ½X1 ; X2 ; …; XN T ; FðtÞ is wind force vector; P is N £ m placement matrix of MR/ER dampers and U ¼ ½U1 ; U2 ; …; Um T is control force vector generated by MR/ER dampers. Suppose that the rth MR/ER damper is installed between the ðr 2 1Þth and rth floor. Then, Ur is of the form Ur ¼ 2Cdr ðX_ r 2 X_ r21 Þ 2 Fdr sgnðX_ r 2 X_ r21 Þ
where FT KF ¼ V2 ¼ diagðv2i Þ; FT Cp F ¼ 2zV ¼ diagð2zi vi Þ; i ¼ 1; 2; …; n: Eqs. (6), (9) and (21) can be combined and converted into the following Itoˆ stochastic differential equation dZ ¼ ðAZ þ BUa Þ þ C dBðtÞ;
Zð0Þ ¼ Z0
ð22Þ
_ T ; yT ; y_ T T ; BðtÞ is a vector of standard where Z ¼ ½QT ; Q wiener processes; Z0 is a Gaussian random vector representing the initial state of the system, which is independent of BðtÞ; 3 2 0 I 0 0 7 6 6 2V2 22zV 0 FT Tðv~Þ 7 7 6 7; A¼6 7 6 7 6 0 0 0 I 5 4 2 0 0 2Vf 2Jf ð23Þ 2 3 3 2 0 0 6 7 6 T 7 607 6F P7 6 7 7 6 7 7 6 B¼6 s¼6 6 7 7; 607 6 0 7 4 5 5 4 G 0
5. Stochastic averaging
ð16Þ
ð17Þ
To simplify the equation of controlled system and reduce its dimension, the stochastic averaging method for quasiintegrable-Hamiltonian system [11] is applied to equations _ in Eq. (22). Then the following partially for Q and Q averaged Itoˆ stochastic differential equations for modal energies are obtained
where Up ¼ ½Up1 ; Up2 ; …; Upm T and Ua ¼ ½Ua1 ; Ua2 ; …; Uam T
_ i ÞFTik Pkr Uar ldt þ si ðHi ÞdB i ðtÞ; dHi ¼ ½mi ðHi Þ þ kð›Hi =›Q i ¼ 1; 2; …; n ð24Þ
_ Up ¼ 2Cd P1 X;
where
where X_ r ; X_ r21 is the velocity of the rth and ðr 2 1Þth floor, respectively. Obviously, U can be split into a passive subvector Up independent of the external voltage and an active sub-vector Ua depending on the external voltage, i.e. U ¼ Up þ Ua
Uar ¼ 2Fdr sgnðX_ r 2 X_ r21 Þ
ð18Þ
where Cd is a diagonal matrix of viscous damping coefficients and P1 is a m £ N matrix depending on the location of MR/ER dampers.
_ 2i þ v2i Q2i Þ=2; H i ¼ ðQ
ði ¼ 1; 2; …; nÞ
ð25Þ
denotes the energy of the ith mode; B i ðtÞ are standard
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W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
obtained by substituting the resultant ›V=›H into the Eq. (31).
Wiener processes, mi ðHi Þ ¼ 22zi vi Hi þ FTil Fki Slk ðvi Þ;
ð26Þ
s2i ðHi Þ ¼ 2Hi FTil Fki Slk ðvi Þ
7. Semi-active control strategy
Sls ðvi Þ is the lsth elements of spectral density matrix SðvÞ of FðtÞ evaluated at vi :
p ¼2 Uar
6. Optimal control law Eq. (24) implies that Hi are controlled diffusion processes. The objective of stochastic optimal control is to seek the optimal feedback control law Upa for minimizing a semi-infinite time-interval performance index 1 ðtf JðUa Þ ¼ lim kLðHðtÞ; Ua ðtÞÞldt ð27Þ tf !1 tf 0 Based on the stochastic optimal dynamical programming principle, the following dynamical programming equation is established # ( " ›V T ›H T l ¼ min kLðHðtÞ; Ua ðtÞÞl þ mðHÞ þ k F PUa l _ Ua ›H ›Q " #) 1 ›2 V T sðHÞs ðHÞ ð28Þ þ tr 2 ›H 2 where
l ¼ lim
tf !1
1 ð tf kLðHðtÞ; Upa ðtÞÞldt tf 0
ð29Þ
ð30Þ
where R is an m £ m positive-define diagonal matrix. Then Upa is of the following form 1 _ T ð›V=›HÞ Upa ¼ 2 R21 PT Fð›H=›QÞ 2
n 1X _ ›V=›Hj R21 PT F Q 2 j r rk kj j
ð33Þ
where Ri is the rth diagonal element of R: On the other hand, the active control force can be generated by the ith MR/ER damper is Uar ¼ 2Fdi sgnðX_ r 2 X_ r21 Þ
ð34Þ
MR/ER dampers can implement the optimal control law, if p ¼ Fdr
n 1X _ ›V=›Hj sgnðX_ r 2 X_ r21 Þ $ 0 ð35Þ R21 PT F Q 2 j r rk kj j
Otherwise, clipping is necessary. If the rth MR/ER damper is installed between lth and rth floor, then the rth column of device displacement matrix P is 0; · · ·; 0 T Pr ¼ ½ 0; · · ·; 0; 1;l 0; · · ·; 0; 21; r
ð36Þ
and PTrk Fkj ¼ Flj 2 Frj ;
is the optimal average cost function and V is the value function. The expression of Upa can be obtained from minimizing the right hand side of Eq. (28). Suppose that the cost function L is of the form [12] L ¼ gðHÞ þ UTa RUa
The rth component of the optimal control forces Upa requested to be produced by the rth MR/ER damper is
ð31Þ
Substituting Eq. (31) into Eq. (28), and completing _ yields the following final averaging with respect to Q and Q dynamical programming equation:
l ¼ gðHÞ " # n X ›V 1 ›V 2 1 2 ›2 V mi ðHi Þ 2 Dii Hi þ si ðHi Þ 2 þ 4 2 ›H i ›H i ›H i i¼1 ð32Þ where Dii ¼ ½FT PR21 PT Fii : The value function VðHÞ can be obtained by solving the final dynamical programming equation (32). Then the optimal control law Upa can be
p Uar ¼2
n 1X _ j ›V=›Hj R21 ðFlj 2 Frj ÞQ 2 j r
ð37Þ
Since R is taken to be positive-definite diagonal matrix, if gðHÞ and l are selected so that ›V=›Hj ¼ ›V=›H $ 0; then 1 p _ j ›V=›H Uar ¼ 2 R21 ðFlj 2 Frj ÞQ 2 r 1 ðX_ l 2 X_ r Þ›V=›H ¼ 2 R21 2 r 1 lX_ l 2 X_ r l›V=›H sgnðX_ l 2 X_ r Þ ¼ 2 R21 2 r
ð38Þ
then p _ _ Fdr ¼ R21 r ›V=›HlXl 2 Xr l $ 0
ð39Þ
Obviously, the damper may always produce exactly the optimal control forces all the time. If one end of the rth MR/ER dampers is fixed, and the other end is attached to the rth floor, then the rth column of placement matrix P is Pr ¼ ½ 0;
· · ·;
0;
1; r
0;
···
0 T
ð40Þ
W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
283
and PTrk Fkj ¼ Frj ;
p Uar ¼2
n 1X _ j ›V=›Hj R21 Frj Q 2 j r
ð41Þ
In the same way, if gðHÞ and l are selected so that ›V=›Hj ¼ ›V=›H $ 0; then 1 p _ j ›V=›H Uar ¼ 2 R21 Frj Q 2 r 1 ¼ 2 R21 X_ ›V=›H 2 r r 1 lX_ r l›V=›H sgnðX_ r Þ ¼ 2 R21 2 r
ð42Þ
Obviously, the damper may also always produce exactly the optimal control forces all the time.
8. Clipped linear quadratic Gaussian control Some results by using clipped LQG control are also obtained. Rewriting system Eq. (21) as system state equation _ ¼ Al Zl ðtÞ þ Bl Ua þ Cl F ZðtÞ _ T T where Zl ðtÞ ¼ ½QT ; Q " # 0 I Al ¼ ; 2V2 22zV " # 0 Cl ¼ FT
ð43Þ " Bl ¼
0 FT P
Fig. 1. RMS displacements of passively controlled, clipped LQG controlled and proposed semi-actively controlled structures. The parameters are R ¼ 5 £ 1027 £ I20£20 ; s ¼ ½1; 1; …; 1T1£10 for proposed semi-active control strategy and Ql ¼ I20£20 ; Rl ¼ 1:3 £ 1025 £ I20£20 for clipped LQG control strategy.
with Frp ¼ ðKgain Þrk Zk sgnðX_ r 2 X_ r21 Þ
ð49Þ
Obviously, the MR/ER dampers may generate these control force only if Frp $ 0: In general, it is impossible and clipping is necessary.
# ;
9. Performance criteria ð44Þ
Let the cost function LðZl ; Ua Þ is of the following form LðZl ; Ua Þ ¼ ZTl Ql Zl þ UTa Rl Ua
ð45Þ
where Ql is a 2n £ 2n semi-definite symmetric matrix and Rl is a m £ m positive-definite symmetric matrix. In case of semi-infinite time-interval control with performance index 1 ðT Jl ¼ lim LðZl ; Ua Þdt ð46Þ T!1 T 0
The performance criteria used to evaluate the proposed optimal control strategy are the percentage reduction of root-mean-square (RMS) response, Kresponse ; and the percentage RMS optimal damping force Kus relative to the structure total weight [13], i.e. Kresponse ¼
RMSðresponseÞp 2 RMSðresponseÞs £ 100% RMSðresponseÞp ð50Þ
the optimal control force requested is of the form Upa ¼ 2Kgain Zl
ð47Þ
where Kgain is the control gain matrix. Eq. (47) can be rewritten as p Uar ¼ Frp sgnðX_ r 2 X_ r21 Þ
ð48Þ
Table 1 Basic parameters of MR dampers and materials used in the example L (m) 1.0
Ap (m) 0.10
b (m) 2.51
h (m) 0.001
Py (kN) 0.05
h (kPa s) 0.00001
ðty Þmin (kPa) 0.05
ðty Þmax (kPa) 20.00
Fig. 2. RMS inter-storey drifts of passively controlled, clipped LQG controlled and proposed semi-actively controlled structures. The parameters are the same as those in Fig. 1.
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W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
Fig. 3. RMS absolute accelerations of passively controlled, clipped LQG controlled and proposed semi-actively controlled structures. The parameters are the same as those in Fig. 1.
Kus ¼
pffiffiffiffiffiffiffiffi E½up2 s £ 100% trðMÞg
ð51Þ
where RMS stands for the root-mean-square; subscripts p and s denote passive and semi-active control, respectively; and trð·Þ is the trace operator of square matrix. Higher values of the percentage reduction Kresponse and the percentage relative optimal damping force Kus indicate more effective and efficient control strategy.
10. Numerical example A 40-storey shear building subject to along-wind loading is taken as an example. The structural data for the building are: individual storey height is 4 m; lumped mass at individual floor is 1.29 £ 106 kg; elastic shear stiffness between any two storeys is 10 9 N/m; the damping coefficient is 2.155 £ 104 N s/m. The natural frequencies
Fig. 4. Percentage reduction of RMS displacements of proposed semiactive controlled and clipped LQG controlled structures. The parameters are the same as those in Fig. 1.
Fig. 5. Percentage reduction of RMS inter-storey drifts of proposed semiactive controlled and clipped LQG controlled structures. The parameters are the same as those Fig. 1.
for the first ten modes are 0.17, 0.52, 0.86, 1.20, 1.54, 1.88, 2.21, 2.54, 2.87, 3.19 (Hz). The aerodynamic data are: the wind-load tributary area Ai for each storey is 192 m2; zref is 10 m; v ðzref Þ is 30.00 m/s; a is 0.4; k0 is 0.03; the drag coefficient CD is 1.2; C1z is 7.7, the air density r is 1.23 kg/m3. A spectral decomposition method is adopted to establish the state representation of wind excitation. Only the first wind mode is taken into account. All the parameters are the same as those in the paper [8]. There are 20 MR dampers installed at the lower 20 floors. Each damper is installed between two adjacent floors. The parameters for MR/ER dampers are listed in Table 1. The first ten modes of the building structure are controlled. An approximately analytical solution to equation (32) is obtained [14] with gðHÞ ¼ s0 þ sT1 H, where H ¼ ½H1 ; · · ·; H10 . Some numerical results are shown in Figs. 1– 6. Figs. 1– 3 show the RMS displacements, RMS inter-storey drifts and RMS accelerations for passively controlled, clipped LQG controlled and proposed semiactively controlled structures, respectively; Figs. 4 –6 show
Fig. 6. Percentage reduction of RMS accelerations of proposed semi-active controlled and clipped LQG controlled structures. The parameters are the same as those in Fig. 1.
W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
the percentage reduction of RMS displacements, RMS interstorey drifts and RMS accelerations of proposed semi-active control strategy and clipped LQG control strategy; The RMS optimal damping force of the proposed semi-active control strategy is equal to 0.341% of the structural total weight, while that of the clipped LQG control strategy is equal to 0.290%. Thus, the proposed semi-active control strategy is better than clipped LQG control strategy.
11. Conclusions In this paper, a semi-active optimal control method for wind excited building structures using MR/ER dampers has been developed based on the stochastic averaging method and the stochastic dynamic programming principle. If certain conditions are satisfied, the MR/ER dampers can completely generate the optimal control force without clipping. The numerical study of a 40-storey tall building structure demonstrates that using the proposed semi-active optimal control strategy, the MR/ER dampers can achieve significant reduction of displacement, inter-storey drift and, specially, acceleration, and the proposed control strategy is better than clipped LQG control strategy.
Acknowledgements The work reported in this paper was supported by the National Natural Science Foundation of China under Key Grant No. 10332030 and the special Fund for Doctor Programs in the Institutions of Higher Learning of China under Grant No. 20020225092.
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References [1] Housner GW, Bergman LA, Caughey TK, Chassiakos AG, Claus RO, Masri SF, Skelton RE, Soong TT, Spencer BF, Yao JTP. Structural control: past, present, and future. ASCE J Engng Mech 1997;123(9): 897–971. [2] Leitmann G. Semiactive control for vibration attenuation. J Intell Mater Syst Struct 1994;5:841–6. [3] McClamroch NH, Gavin HP. Closed loop structural control using electrorheological dampers. Proceeding of the American Control Conference, Seattle, Washington; 1995. p. 4173–7. [4] Inaudi JA. Modulated homogeneous friction: a semi-active damping strategy. Earthquake Engng Struct Dyn 1997;26(3):361. [5] Sun L, Goto Y. Applications of fuzzy theory to variable damper for bridge vibration control. Proceeding of First World Conference on Structural Control WPI, WPI.; 1994. p. 31– 40. [6] Dyke SJ, Spencer BF, Sain MK, Carlson JD. Modeling and control of magnetorheological dampers for seismic response reduction. Smart Mater Struct 1996;5:565–75. [7] Ying ZG, Zhu WQ, Soong TT. A stochastic optimal semi-active control strategy for ER/MR dampers. J Sound Vib 2003;259(1): 45–62. [8] Benfratello S, Muscolino G. Filter approach to the stochastic analysis of MDOF wind-excited structures. Probab Engng Mech 1999;14: 311–21. [9] Spencer BF, Dyke SJ, Sain MK, Carson JD. Phenomenological model for magnetorheological dampers. J Engng Mech 1997;123(3):230 –8. [10] Gavin GP, Hanson RD, Filisko FE. Electrorheological dampers. Part II. Testing and modeling. ASME J Appl Mech 1996;63:676 –82. [11] Zhu WQ, Huang ZL, Yang YQ. Stochastic averaging of quasiintegrable Hamiltonian systems. ASME J Appl Mech 1997;64(9): 975–84. [12] Zhu WQ, Ying ZG, Soong TT. An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dyn 2001; 24(1):31– 51. [13] Ni YQ, Ying ZG, Wang JY, Ko JM, Spencer BF. Semi-active control of randomly wind-excited tall building structures using MR-TLCDs. Advances in stochastic structural dynamics, Proceedings of the Fifth International Conference on Stochastic Structural Dynamics, Zhu WQ et al. (eds.), CRC Press, p. 377 –84. [14] Zhu WQ, Ying ZG. Nonlinear stochastic optimal control of partially observable linear structures. Engng Struct 2002;24:333–42.
Semi-active control of wind excited building structures using MR/ER dampers W.Q. Zhu*, M. Luo, L. Dong Department of Mechanics, Zhejiang University, Hangzhou 310027, China Received 15 November 2003; revised 15 December 2003; accepted 2 February 2004
Abstract A semi-active control strategy for building structures subject to wind loading and controlled by MR/ER dampers is proposed. The power spectral density (PSD) matrix of the fluctuating part of wind velocity vector is diagonalized in the eigenvector space. Each element of the diagonalized PSD matrix is modeled as a set of second-order linear filter driven by white noise. A Bingham model for MR/ER dampers is used. The forces produced by MR/ER dampers are split into passive and active parts and the passive part is combined with structural damping forces. A set of partially averaged Itoˆ equations for controlled modal energies are derived by applying the stochastic averaging method for quasi-integrable-Hamiltonian systems. The optimal control law is then determined by using the stochastic dynamical programming principle and the cost function is so selected that the optimal control law can be implemented by the MR/ER dampers. The response of semiactive controlled structures is predicted by using the reduced Fokker – Planck– Kolmogorov equation associated with fully averaged Itoˆ equations of the controlled structures. A comparison with clipped linear quadratic Gaussian (LQG) control strategy, for an example, shows that the proposed semi-active control strategy for MR/ER dampers is superior to clipped LQG control strategy. q 2004 Elsevier Ltd. All rights reserved. Keywords: Tall building; Wind excitation; MR/ER damper; Bingham model; Stochastic optimal control; Stochastic averaging; Stochastic dynamical programming
1. Introduction Structural control of large civil engineering structures has been studied for more than two decades. It evolves from passive control, active control to semi-active control. Recently, semi-active control systems attract much attention for their low-energy requirement and cost, and having best features of both passive and active controls, offering the reliability of passive devices yet maintaining the versatility and adaptability of active systems. A number of semi-active control devices have been developed, such as variableorifice dampers, variable-friction dampers, controllable tuned liquid dampers, semi-active impact dampers and controllable-fluid dampers, etc. [1]. Because of the intrinsically nonlinear nature of semi-active control devices, the feedback control law must be nonlinear. The development * Corresponding author. Tel.: þ 86-571-8799-1150; fax: þ 86-571-87952651. E-mail address: [email protected] (W.Q. Zhu). 0266-8920/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2004.02.011
of such control strategy that is practically implementable and can fully utilize the capabilities of these unique devices is an important and challenging task. Some semi-active control strategies have been developed. Leitmann applied Lyapunov’s direct approach for the design of a semi-active controller [2]; McClamroch and Gavin used a similar approach to develop the decentralized bang-bang control law for using an ER damper [3]; Inaudi developed the modulated homogeneous friction controller for a variablefriction damper [4]; Sun and Goto used the fuzzy control method [5] and Dyke et al. presented the clipped linear optimal control law that has been shown effective for MR damper [6]. Recently, a stochastic optimal semi-active control strategy for MR/ER damper was proposed [7]. It has been shown that if parameters were properly selected, then the MR/ER damper can fully generate the optimal control force and the control effectiveness and efficiency are better than those of clipped linear quadratic Gaussian (LQG). In this paper, this stochastic optimal semi-active control strategy is extended to wind excited tall building structures.
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where wðtÞ is a l-dimensional vector of independent Gaussian white noises with unit intensity
2. Model of wind excitation The fluctuating part vðtÞ of a wind velocity vector is assumed to be a zero-mean Gaussian random process vector. It is characterized by power spectral density (PSD) matrix Sv ðvÞ; which can be decomposed in the eigenvector space [8], i.e. cT ðvÞSv ðvÞcðvÞ ¼ LðvÞ
ð1Þ
where cðvÞ is the normalized eigenmatrix of Sv ðvÞ; i.e. cT ðvÞcðvÞ ¼ I
ð2Þ
where I is a unit matrix. LðvÞ is a diagonal matrix whose ith element Li ðvÞ is the eigenvalue associated with ith column ci ðvÞ of cðvÞ: Since the eigenvectors vary slowly with frequency v; it is possible to take the following approximation Sv ðvÞ ø cðv~ÞLðvÞcT ðv~Þ
ð3Þ
where v~ is an appropriately selected fixed value of circular frequency v: Then Li ðvÞ can be regarded as the PSD of the output velocity y_ i1 ðtÞ governed by the following equations: y€ i ðtÞ þ Jif y_ i ðtÞ þ V2if yi ðtÞ ¼ Gi wi ðtÞ where 2 6 6 6 6 yi ¼ 6 6 6 4
ð4Þ
3
yi1
7 yi2 7 7 7 ; .. 7 7 . 7 5 yimi
2 6 6 6 6 6 6 Jif ¼ 6 6 6 6 6 4 2 6 6 6 6 2 Vif ¼ 6 6 6 4
y ¼ ½y1 ; y2 ; …; yl T ; Jf ¼ diagðJ1f ; J2f ; …; Jlf Þ
ð7Þ
V2f ¼ diagðV21f ; V22f ; …; V2lf Þ; G ¼ diagðG1 ; G2 ; …; Gl Þ Once the filter differential equation (6) has been determined, the fluctuating part of wind velocity vector, vðtÞ; is given in the form ~ yðtÞ ~ v~ÞA_ vðtÞ ø cð
ð8Þ
~ v~Þ is the sub-eigenmatrix consisting of the first l where cð wind-eigenvectors evaluated at v ¼ v~; y_ ðtÞ is obtained by ~ is a Boolean matrix whose non-zero solving Eq. (6) and A elements are those corresponding to the components y_ i1 ðtÞ; i ¼ 1; …; l of the filter response velocity vector y_ ðtÞ: The fluctuating part of the wind force vector can then be written in the form FðtÞ ¼ Tðv~Þ_yðtÞ
ð9Þ
~ r is air density; CD is drag ~ v~ÞA; where Tðv~Þ ¼ rCD Avcð coefficient; A is a diagonal matrix whose iith element is the ith nodal area Ai ; v is a diagonal matrix with the mean wind velocities v ðzi Þ as its elements.
3. Model of MR/ER damper
2zi1 vi1
21
0
···
0
0
2zi2 vi2
21
···
0
.. .
0
2zi3 vi3
···
0
.. .
.. .
.. .
..
21
0
0
0 3
0
v2i1
···
···
0
v2i2
···
.. .
.. .
..
0
0
0
.
.
Several mechanical models such as Bingham viscoplastic model and Bouc –Wen model have been proposed for MR/ER dampers. For a steady and fully developed flow, the shear resistance of MR/ER fluids may be modeled as
3 7 7 7 7 7 7 7; 7 7 7 7 5
ð5Þ
2zimi vimi
in which wi ðtÞ is a Gaussian white noise with unit intensity; mi is the number of the filters required for appropriately modeling eigenvalue Li ðvÞ; vij ; zij and gi are parameters of the filters. By collecting the first l equations corresponding to the first l significant eigenvalues, the following compact form of the filter differential equations is obtained y€ ðtÞ þ Jf y_ ðtÞ þ V2f yðtÞ ¼ GwðtÞ
ð10Þ
which is called Bingham model [9]. In Eq. (10), t is the shear stress in fluid, h is the Newtonian viscosity independent of the applied magnetic/electric filed, g_ is the shear strain rate and ty ðEÞ is the yielding shear stress controlled by the applied magnetic/electric field. For a Bingham material in a rectangular duct, an approximate relationship between force Pd and relative velocity u_ of the piston relative to the damper cylinder is given as follows [10]
2 3 0 7 7 6 607 0 7 7 6 7 7 7 7; Gi ¼ 6 6 .. 7 7 7 6 7 . 0 5 4 5 gi v2imi 0
t ¼ hg_ þ ty ðEÞsgnðg_Þ
ð6Þ
Pd ðtÞ ¼ Cd u_ þ Fd ðEÞsgnð_uÞ
ð11Þ
where Cd ¼ C1
12hLAp Ap ; bh3
Lty ðEÞ Ap þ Py Fd ðEÞ ¼ C2 ðEÞ h
ð12Þ
W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
_ Eq. (15) can be rewritten as By combining Up with CX;
For the flow-type damper C1 ¼ 1:0;
C2 ðEÞ ¼ 2:07 þ
1:0 ; 1:0 þ 0:4TðEÞ
_ þ KX ¼ PUa þ FðtÞ; € þ Cp X MX ð13Þ
bh2 ty ðEÞ TðEÞ ¼ 12Ap hu_
V¼
ð19Þ
where Cp ¼ C þ PCd P1 : Introducing model transform ð20Þ
where Q ¼ ½Q1 ; Q2 ; …; Qn T ; F is an N £ n modal submatrix consisting of the first n ðn # NÞ columns of the modal matrix. Assuming that Cp satisfies the classical orthogonality condition, then equation for the n dominating modes can be written as
bh ; 2Ap
C2 ðEÞ ¼ 2:07 þ
Xð0Þ ¼ X0
X ¼ FQ
For the mixed type damper C1 ¼ 1:0 2
281
1:0 1:5V 2 2 ; 1:0 þ 0:4TðEÞ 1:0 þ 0:4T 2 ðEÞ
ð14Þ
€ þ 2zVQ _ þ V2 Q ¼ FT FðtÞ þ FT PUa ; Q
ð21Þ
Qð0Þ ¼ Q0
bh 2Ap
where L is the effective axial pole length; Ap is the effective cross-sectional area of the piston; Py is the mechanical friction force in the damper; h is the gap between two magnetic/ electric poles; and b is the circumference of the inner magnetic/electric pole. Clearly, Fd is a function of the yielding shear stress and can be controlled through change in the applied field intensity but Cd is independent of the applied magnetic/electric field.
4. Equation of control system Considering an N-storey building structure subject to wind loading and installed with m MR/ER dampers. The equation of the system is € þ CX _ þ KX ¼ PU þ FðtÞ; MX
Xð0Þ ¼ X0
ð15Þ
where M; C; K are N £ N mass, damping and stiffness matrix, respectively; X ¼ ½X1 ; X2 ; …; XN T ; FðtÞ is wind force vector; P is N £ m placement matrix of MR/ER dampers and U ¼ ½U1 ; U2 ; …; Um T is control force vector generated by MR/ER dampers. Suppose that the rth MR/ER damper is installed between the ðr 2 1Þth and rth floor. Then, Ur is of the form Ur ¼ 2Cdr ðX_ r 2 X_ r21 Þ 2 Fdr sgnðX_ r 2 X_ r21 Þ
where FT KF ¼ V2 ¼ diagðv2i Þ; FT Cp F ¼ 2zV ¼ diagð2zi vi Þ; i ¼ 1; 2; …; n: Eqs. (6), (9) and (21) can be combined and converted into the following Itoˆ stochastic differential equation dZ ¼ ðAZ þ BUa Þ þ C dBðtÞ;
Zð0Þ ¼ Z0
ð22Þ
_ T ; yT ; y_ T T ; BðtÞ is a vector of standard where Z ¼ ½QT ; Q wiener processes; Z0 is a Gaussian random vector representing the initial state of the system, which is independent of BðtÞ; 3 2 0 I 0 0 7 6 6 2V2 22zV 0 FT Tðv~Þ 7 7 6 7; A¼6 7 6 7 6 0 0 0 I 5 4 2 0 0 2Vf 2Jf ð23Þ 2 3 3 2 0 0 6 7 6 T 7 607 6F P7 6 7 7 6 7 7 6 B¼6 s¼6 6 7 7; 607 6 0 7 4 5 5 4 G 0
5. Stochastic averaging
ð16Þ
ð17Þ
To simplify the equation of controlled system and reduce its dimension, the stochastic averaging method for quasiintegrable-Hamiltonian system [11] is applied to equations _ in Eq. (22). Then the following partially for Q and Q averaged Itoˆ stochastic differential equations for modal energies are obtained
where Up ¼ ½Up1 ; Up2 ; …; Upm T and Ua ¼ ½Ua1 ; Ua2 ; …; Uam T
_ i ÞFTik Pkr Uar ldt þ si ðHi ÞdB i ðtÞ; dHi ¼ ½mi ðHi Þ þ kð›Hi =›Q i ¼ 1; 2; …; n ð24Þ
_ Up ¼ 2Cd P1 X;
where
where X_ r ; X_ r21 is the velocity of the rth and ðr 2 1Þth floor, respectively. Obviously, U can be split into a passive subvector Up independent of the external voltage and an active sub-vector Ua depending on the external voltage, i.e. U ¼ Up þ Ua
Uar ¼ 2Fdr sgnðX_ r 2 X_ r21 Þ
ð18Þ
where Cd is a diagonal matrix of viscous damping coefficients and P1 is a m £ N matrix depending on the location of MR/ER dampers.
_ 2i þ v2i Q2i Þ=2; H i ¼ ðQ
ði ¼ 1; 2; …; nÞ
ð25Þ
denotes the energy of the ith mode; B i ðtÞ are standard
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obtained by substituting the resultant ›V=›H into the Eq. (31).
Wiener processes, mi ðHi Þ ¼ 22zi vi Hi þ FTil Fki Slk ðvi Þ;
ð26Þ
s2i ðHi Þ ¼ 2Hi FTil Fki Slk ðvi Þ
7. Semi-active control strategy
Sls ðvi Þ is the lsth elements of spectral density matrix SðvÞ of FðtÞ evaluated at vi :
p ¼2 Uar
6. Optimal control law Eq. (24) implies that Hi are controlled diffusion processes. The objective of stochastic optimal control is to seek the optimal feedback control law Upa for minimizing a semi-infinite time-interval performance index 1 ðtf JðUa Þ ¼ lim kLðHðtÞ; Ua ðtÞÞldt ð27Þ tf !1 tf 0 Based on the stochastic optimal dynamical programming principle, the following dynamical programming equation is established # ( " ›V T ›H T l ¼ min kLðHðtÞ; Ua ðtÞÞl þ mðHÞ þ k F PUa l _ Ua ›H ›Q " #) 1 ›2 V T sðHÞs ðHÞ ð28Þ þ tr 2 ›H 2 where
l ¼ lim
tf !1
1 ð tf kLðHðtÞ; Upa ðtÞÞldt tf 0
ð29Þ
ð30Þ
where R is an m £ m positive-define diagonal matrix. Then Upa is of the following form 1 _ T ð›V=›HÞ Upa ¼ 2 R21 PT Fð›H=›QÞ 2
n 1X _ ›V=›Hj R21 PT F Q 2 j r rk kj j
ð33Þ
where Ri is the rth diagonal element of R: On the other hand, the active control force can be generated by the ith MR/ER damper is Uar ¼ 2Fdi sgnðX_ r 2 X_ r21 Þ
ð34Þ
MR/ER dampers can implement the optimal control law, if p ¼ Fdr
n 1X _ ›V=›Hj sgnðX_ r 2 X_ r21 Þ $ 0 ð35Þ R21 PT F Q 2 j r rk kj j
Otherwise, clipping is necessary. If the rth MR/ER damper is installed between lth and rth floor, then the rth column of device displacement matrix P is 0; · · ·; 0 T Pr ¼ ½ 0; · · ·; 0; 1;l 0; · · ·; 0; 21; r
ð36Þ
and PTrk Fkj ¼ Flj 2 Frj ;
is the optimal average cost function and V is the value function. The expression of Upa can be obtained from minimizing the right hand side of Eq. (28). Suppose that the cost function L is of the form [12] L ¼ gðHÞ þ UTa RUa
The rth component of the optimal control forces Upa requested to be produced by the rth MR/ER damper is
ð31Þ
Substituting Eq. (31) into Eq. (28), and completing _ yields the following final averaging with respect to Q and Q dynamical programming equation:
l ¼ gðHÞ " # n X ›V 1 ›V 2 1 2 ›2 V mi ðHi Þ 2 Dii Hi þ si ðHi Þ 2 þ 4 2 ›H i ›H i ›H i i¼1 ð32Þ where Dii ¼ ½FT PR21 PT Fii : The value function VðHÞ can be obtained by solving the final dynamical programming equation (32). Then the optimal control law Upa can be
p Uar ¼2
n 1X _ j ›V=›Hj R21 ðFlj 2 Frj ÞQ 2 j r
ð37Þ
Since R is taken to be positive-definite diagonal matrix, if gðHÞ and l are selected so that ›V=›Hj ¼ ›V=›H $ 0; then 1 p _ j ›V=›H Uar ¼ 2 R21 ðFlj 2 Frj ÞQ 2 r 1 ðX_ l 2 X_ r Þ›V=›H ¼ 2 R21 2 r 1 lX_ l 2 X_ r l›V=›H sgnðX_ l 2 X_ r Þ ¼ 2 R21 2 r
ð38Þ
then p _ _ Fdr ¼ R21 r ›V=›HlXl 2 Xr l $ 0
ð39Þ
Obviously, the damper may always produce exactly the optimal control forces all the time. If one end of the rth MR/ER dampers is fixed, and the other end is attached to the rth floor, then the rth column of placement matrix P is Pr ¼ ½ 0;
· · ·;
0;
1; r
0;
···
0 T
ð40Þ
W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
283
and PTrk Fkj ¼ Frj ;
p Uar ¼2
n 1X _ j ›V=›Hj R21 Frj Q 2 j r
ð41Þ
In the same way, if gðHÞ and l are selected so that ›V=›Hj ¼ ›V=›H $ 0; then 1 p _ j ›V=›H Uar ¼ 2 R21 Frj Q 2 r 1 ¼ 2 R21 X_ ›V=›H 2 r r 1 lX_ r l›V=›H sgnðX_ r Þ ¼ 2 R21 2 r
ð42Þ
Obviously, the damper may also always produce exactly the optimal control forces all the time.
8. Clipped linear quadratic Gaussian control Some results by using clipped LQG control are also obtained. Rewriting system Eq. (21) as system state equation _ ¼ Al Zl ðtÞ þ Bl Ua þ Cl F ZðtÞ _ T T where Zl ðtÞ ¼ ½QT ; Q " # 0 I Al ¼ ; 2V2 22zV " # 0 Cl ¼ FT
ð43Þ " Bl ¼
0 FT P
Fig. 1. RMS displacements of passively controlled, clipped LQG controlled and proposed semi-actively controlled structures. The parameters are R ¼ 5 £ 1027 £ I20£20 ; s ¼ ½1; 1; …; 1T1£10 for proposed semi-active control strategy and Ql ¼ I20£20 ; Rl ¼ 1:3 £ 1025 £ I20£20 for clipped LQG control strategy.
with Frp ¼ ðKgain Þrk Zk sgnðX_ r 2 X_ r21 Þ
ð49Þ
Obviously, the MR/ER dampers may generate these control force only if Frp $ 0: In general, it is impossible and clipping is necessary.
# ;
9. Performance criteria ð44Þ
Let the cost function LðZl ; Ua Þ is of the following form LðZl ; Ua Þ ¼ ZTl Ql Zl þ UTa Rl Ua
ð45Þ
where Ql is a 2n £ 2n semi-definite symmetric matrix and Rl is a m £ m positive-definite symmetric matrix. In case of semi-infinite time-interval control with performance index 1 ðT Jl ¼ lim LðZl ; Ua Þdt ð46Þ T!1 T 0
The performance criteria used to evaluate the proposed optimal control strategy are the percentage reduction of root-mean-square (RMS) response, Kresponse ; and the percentage RMS optimal damping force Kus relative to the structure total weight [13], i.e. Kresponse ¼
RMSðresponseÞp 2 RMSðresponseÞs £ 100% RMSðresponseÞp ð50Þ
the optimal control force requested is of the form Upa ¼ 2Kgain Zl
ð47Þ
where Kgain is the control gain matrix. Eq. (47) can be rewritten as p Uar ¼ Frp sgnðX_ r 2 X_ r21 Þ
ð48Þ
Table 1 Basic parameters of MR dampers and materials used in the example L (m) 1.0
Ap (m) 0.10
b (m) 2.51
h (m) 0.001
Py (kN) 0.05
h (kPa s) 0.00001
ðty Þmin (kPa) 0.05
ðty Þmax (kPa) 20.00
Fig. 2. RMS inter-storey drifts of passively controlled, clipped LQG controlled and proposed semi-actively controlled structures. The parameters are the same as those in Fig. 1.
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Fig. 3. RMS absolute accelerations of passively controlled, clipped LQG controlled and proposed semi-actively controlled structures. The parameters are the same as those in Fig. 1.
Kus ¼
pffiffiffiffiffiffiffiffi E½up2 s £ 100% trðMÞg
ð51Þ
where RMS stands for the root-mean-square; subscripts p and s denote passive and semi-active control, respectively; and trð·Þ is the trace operator of square matrix. Higher values of the percentage reduction Kresponse and the percentage relative optimal damping force Kus indicate more effective and efficient control strategy.
10. Numerical example A 40-storey shear building subject to along-wind loading is taken as an example. The structural data for the building are: individual storey height is 4 m; lumped mass at individual floor is 1.29 £ 106 kg; elastic shear stiffness between any two storeys is 10 9 N/m; the damping coefficient is 2.155 £ 104 N s/m. The natural frequencies
Fig. 4. Percentage reduction of RMS displacements of proposed semiactive controlled and clipped LQG controlled structures. The parameters are the same as those in Fig. 1.
Fig. 5. Percentage reduction of RMS inter-storey drifts of proposed semiactive controlled and clipped LQG controlled structures. The parameters are the same as those Fig. 1.
for the first ten modes are 0.17, 0.52, 0.86, 1.20, 1.54, 1.88, 2.21, 2.54, 2.87, 3.19 (Hz). The aerodynamic data are: the wind-load tributary area Ai for each storey is 192 m2; zref is 10 m; v ðzref Þ is 30.00 m/s; a is 0.4; k0 is 0.03; the drag coefficient CD is 1.2; C1z is 7.7, the air density r is 1.23 kg/m3. A spectral decomposition method is adopted to establish the state representation of wind excitation. Only the first wind mode is taken into account. All the parameters are the same as those in the paper [8]. There are 20 MR dampers installed at the lower 20 floors. Each damper is installed between two adjacent floors. The parameters for MR/ER dampers are listed in Table 1. The first ten modes of the building structure are controlled. An approximately analytical solution to equation (32) is obtained [14] with gðHÞ ¼ s0 þ sT1 H, where H ¼ ½H1 ; · · ·; H10 . Some numerical results are shown in Figs. 1– 6. Figs. 1– 3 show the RMS displacements, RMS inter-storey drifts and RMS accelerations for passively controlled, clipped LQG controlled and proposed semiactively controlled structures, respectively; Figs. 4 –6 show
Fig. 6. Percentage reduction of RMS accelerations of proposed semi-active controlled and clipped LQG controlled structures. The parameters are the same as those in Fig. 1.
W.Q. Zhu et al. / Probabilistic Engineering Mechanics 19 (2004) 279–285
the percentage reduction of RMS displacements, RMS interstorey drifts and RMS accelerations of proposed semi-active control strategy and clipped LQG control strategy; The RMS optimal damping force of the proposed semi-active control strategy is equal to 0.341% of the structural total weight, while that of the clipped LQG control strategy is equal to 0.290%. Thus, the proposed semi-active control strategy is better than clipped LQG control strategy.
11. Conclusions In this paper, a semi-active optimal control method for wind excited building structures using MR/ER dampers has been developed based on the stochastic averaging method and the stochastic dynamic programming principle. If certain conditions are satisfied, the MR/ER dampers can completely generate the optimal control force without clipping. The numerical study of a 40-storey tall building structure demonstrates that using the proposed semi-active optimal control strategy, the MR/ER dampers can achieve significant reduction of displacement, inter-storey drift and, specially, acceleration, and the proposed control strategy is better than clipped LQG control strategy.
Acknowledgements The work reported in this paper was supported by the National Natural Science Foundation of China under Key Grant No. 10332030 and the special Fund for Doctor Programs in the Institutions of Higher Learning of China under Grant No. 20020225092.
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