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Feedback control optimization for seismically excited buildings
Acta Mechanica Solida Sinica, Vol. 20, No. 4, December, 2007 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-007-0740-y
ISSN 0894-9166
FEEDBACK CONTROL OPTIMIZATION FOR SEISMICALLY EXCITED BUILDINGS Xueping Li
Zuguang Ying
( Department of Mechanics, School of Aeronautics and Astronautics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China)
Received 8 December 2006, revision received 2 August 2007
ABSTRACT A feedback control optimization method of partially observable linear structures via stationary response is proposed and analyzed with linear building structures equipped with control devices and sensors. First, the partially observable control problem of the structure under horizontal ground acceleration excitation is converted into a completely observable control problem. Then the Itˆ o stochastic differential equations of the system are derived based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution to the Fokker-Plank-Kolmogorov (FPK) equation associated with the Itˆ o equations is obtained. The performance index in terms of the mean system energy and mean square control force is established and the optimal control force is obtained by minimizing the performance index. Finally, the numerical results for a three-story building structure model under El Centro, Hachinohe, Northridge and Kobe earthquake excitations are given to illustrate the application and the effectiveness of the proposed method.
KEY WORDS feedback control optimization, partially observable structure, stochastic averaging method, earthquake response, stationary probability density
I. INTRODUCTION Civil engineering structures, including existing and future buildings, towels and bridges, must be adequately protected from a variety of events such as earthquakes, winds and traffic. In order to protect the structures and reduce the destruction, the stochastic control of civil engineering structures is becoming a problem of public concern. The mathematical theory of stochastic optimal control has been quite well developed[1, 2] . A principal approach to the stochastic optimal control problem is based on the stochastic dynamical programming principle. So far, the theory of stochastic optimal control has been applied mainly to economics, especially finance. In the engineering field, only the linear quadratic Gaussian (LQG) control strategy has been widely used until now. In the past few years, a nonlinear stochastic optimal control strategy has been proposed for quasi-Hamiltonian systems with external and/or parametric stochastic excitations by the present second author and his co-workers[3, 4] based on the stochastic averaging method for quasi-Hamiltonian systems[5–7] and the stochastic dynamical programming principle[1, 2] . The strategy has been extended to the stochastic optimal control of partially observable linear and nonlinear systems[8, 9] . More effective and efficient[10] , this control strategy has several advantages over the LQG controller strategy[10].
Project supported by the National Natural Science Foundation of China under a key grant (No.10332030), the Research Fund for the Doctoral Program of Higher Education of China (No. 20060335125) and the Zhejiang Provincial Natural Science Foundation of China (No. Y607087).
Vol. 20, No. 4
Xueping Li et al.: Feedback Control Optimization for Seismically Excited Building
· 343 ·
In the control strategy developed in Refs.[3] and [4], the dynamical programming equation must be solved in order to obtain the optimal control forces. However, in practice, there is no effective method of obtaining the analytical solution to the dynamical programming equation. Generally, it is a burdensome task to solve it numerically. In the present paper, a control optimization for partially observable linear structures with stochastic excitations is proposed based on the separability principle, stochastic averaging method and optimization method. The mean square control force and the mean system energy are combined as the performance index of the system optimization and the optimum control force is obtained without solving the dynamical programming equation. The method is illustrated with a linear building structure equipped with control devices and sensors. The earthquake records, El Centro, Hachinohe, Northridge and Kobe[11] are used to show the effectiveness and efficiency of the control method proposed.
II. STOCHASTIC OPTIMAL CONTROL OF A PARTIALLY OBSERVABLE STRUCTURE For an n-story building structure equipped with control devices and sensors under horizontal ground acceleration excitation, the equation of motion of the system is of the form ¨ + CX ˙ + KX = −M Γ x MX ¨g + P U (1) where X = {X1 , X2 · · · Xn }T , Xi is the horizontal displacement of the ith floor relative to the ground; M , C, K are the n × n mass, damping and stiffness matrices of the system, respectively; Γ is an n-dimensional unit vector; x ¨g is the horizontal acceleration; U is a k-dimensional control force vector with k being the number of control devices; P is a n × k control device placement matrix. Introducing modal transformation X = ΦQ (2) where Φ is the n × m (m ≤ n) real modal matrix of the system which can be normalized to yield ΦT M Φ = I, and I is an m × m unit matrix, m is the number of the dominant modes among n modes to control, Eq.(1) becomes ¨ + 2ζΩ Q ˙ + Ω 2 Q = u − Gp x Q ¨g (3) where Q = {Q1 , Q2 · · · Qm }T , Qi is the generalized displacement of the ith mode; 2ζΩ = ΦT CΩ with ζ = diag(ζi ) and ζi is the damping ratio of ith mode; Ω 2 = diag(ωi2 ) = ΦT KΦ and ωi is the natural frequency of the ith mode; Gp = −ΦT M Γ ; u = ΦT P U and ui is the generalized control force of the ith mode. Suppose that the horizontal ground acceleration x ¨g is modeled as a uniformly modulated stationary random process with the Kanai-Tajimi power spectral density described in Appendix A. Equations (3), (27) and (28) can be combined and converted into the following Itˆ o stochastic differential equation dZ = (Ac Z + B c u)dt + C c dB(t)
(4)
˙ T , F T }T , F 1 = {f, f˙}T , B(t) is the unit Wiener process. where Z = {QT , Q 1 ⎫ ⎧ ⎤ ⎡ ⎧ ⎫ 0 ⎪ ⎪ 0 Im 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ ⎪ ⎪ ⎬ ⎨ Im ⎪ ⎨ 0 ⎪ ⎬ ⎢ −Ω 2 −2ζΩ −Gp d0 −Gp d1 ⎥ ⎥ ⎢ , Cc = A=⎢ ⎥ , Bc = 0 ⎪ ⎪ ⎪ ⎢ 0 0 ⎪ 0 1 0 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦ ⎣ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 1 ⎪ ⎭ ⎩ σ(t) ⎪ 0 c1 /c2 0 0 −c0 /c2 c2
(5)
T
To derive the observation equation, suppose that l absolute accelerations a = {a1 , a2 , ...al } at the ˙ and u as follows: different floors are measured with noises on. They can be expressed in terms of Q, Q ¨ + Γx a = P 0 (X ¨g ) + σ 0 w1 (t) (6) where P 0 is an l × n sensor placement matrix, w 1 (t) is an m-dimensional vector of independent unit Gaussian white noises; σ 0 is an l ×m measurement noise intensity matrix. Equation (6) can be converted into the following Itˆ o stochastic differential equation dV = (D 0 Z + C 0 u)dt + σ 0 dB 1 (t)
(7)
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where dV /dt = a; B 1 (t) is an m-dimensional vector of the independent unit Wiener processes, which are independent of B(t). D 0 = P P 0 −ΦΩ 2 − Φ2ζΩ 0 0 , C 0 = P P 0 Φ (8) are measurement matrices and Eq.(7) is the observation equation of the system.
III. OPTIMAL CONTROL OF A COMPLETELY OBSERVABLE SYSTEM According to the separation principle[12, 13] , the partially observed stochastic optimal control problem formulated in the last section can be converted into a completely observed stochastic optimal control ˆ ˆ problem in the following way. Take the conditional mean Z(t) of Z(t) as the state of the system. Z(t) is governed by the following Itˆ o equation ˆ = (Ac Z ˆ 1 (t) ˆ + B c u)dt + F e σ 0 dB dZ (9) ˆ where B 1 (t) is the m-dimensional vector of the independent unit Wiener processes. −1 F e = Re (t)D T (10) 0 S0 ¯ = Z − Z, ˆ which satisfies the following Riccati-type matrix Re (t) is the covariance matrix of error Z differential equation. ˙ e = Ac Re + Re AT − Re D T S −1 D 0 Re + S c (11) R c 0 0 T T where S 0 = σ 0 σ 0 ; S c = C c C c . A linear optimal control can be applied.
IV. AVERAGED EQUATION AND STATIONARY SOLUTIONS The system (4) can be regarded as a controlled, stochastically excited and dissipated integrable Hamiltonian system. Suppose the damping forces, control forces and the intensity of ground acceleration are of the same order of ε and ε is a small constant. By applying the stochastic averaging method for quasiintegrable Hamiltonian systems[7] , the averaged Itˆ o equations can be obtained, in the non-resonance case dHi = [mi (Hi )+ < (∂Hi /∂ Q˙ i )ui >]dt + σi (Hi )dBi (t) (i = 1, 2, ..., n) (12) 2 2 2 ˙ where Hi = (Qi + ωi Qi )/2; Bi (t) are independent unit Wiener processes: mi (Hi ) = −2ζi ωi Hi + σi2 (Hi )/(2Hi ), σi2 (Hi ) = 2πHi [Gp ]2i Sx¨g (ωi , τ ) (13) Sx¨g (ωi , τ ) is the evolutionary spectral density of x¨g which can be expressed as 2 r−1 i di (jω) √ Sx¨g (ωi , τ ) = 2πσ 2 (τ ) i=0 (14) 2 , j = −1 r ci (jω)i i=0
The linear feedback control force for system (9) is ˆ˙ u = −kQ (15) where u = {u1 , u2 , ..., um }T , k = diag{k1 , k2 , ...km }. By substituting Eq.(15) into Eq.(12) and averaging the control force terms, the corresponding FPK equation is established as m m ∂p −∂ −(2ζi ωi + ki )Hi + σi2 (Hi )/(2Hi ) p ∂ 2 [σi2 (Hi )p] = + (16) ∂t ∂Hi 2∂Hi2 i=1 i=1 A stationary solution to FPK Eq.(16) is m p= pi , pi (Hi ) = Ci exp −(2ζi ωi + ki )Hi /[π[Gp ]2i Sx¨g (ωi , τ )]
(17)
i=1
where Ci are normalization constants. The stationary probability of the modal displacements and velocities of the controlled structure is then pi (Hi ) pi (Qi , Q˙ i ) = T (Hi ) 2 2 ˙2 Hi =(Qi +ωi Qi )/2
=
Ci ωi exp −(2ζi ωi + ki )(Q˙ 2i + ωi2 Q2i )/[2π[Gp ]2i Sx¨g (ωi , τ )] 2π
(18)
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V. DETERMINING THE OPTIMAL CONTROL FORCE The form of the performance index depends upon the objective of the control, for example, minimization of response, stabilization or maximization of the reliability. Here the objective is to minimize the response and the performance index is Ji = Ji1 + Ji2 , Ji1 = E(Hi ), Ji2 = Ri E(u2i )
where E(Hi ) =
∞
0
Hi pi (Hi )dHi ,
E(u2i ) =
0
∞
(−ki Q˙ i )2
0
∞
pi (Qi , Q˙ i )dQi dQ˙ i
(19) (20)
Substituting Eq.(20) into Eq.(19) yields the ith performance index Ji . The ki can be determined by minimizing Ji . Ri is selected as the balanced parameter which adjusts the amplitude of the control force.
VI. NUMERICAL RESULTS AND DISCUSSION A three-story experimental building structure model and its parameters are given in Appendix B. The control parameters, R1 = 0.01, R2 = 0.01, R3 = 0.01, and k can be obtained by using the proposed method as k = diag{4.8 2.8 1.8} (21) The performance indices Ji , Ji1 and Ji2 for varying ki are shown in Figs.1, 2 and 3. According to Eqs.(7), (9) and (15), the converted completely observable control system is ˆ˙ = (A Z ˆ + B c u) + F e (V˙ − D 0 Z ˆ − C 0 u) Z c
(22)
¯Z ˆ u = −k
(23)
Fig. 1. The performance indices J1 , J11 , J12 v.s. k1 .
Fig. 2. The performance indices J2 , J21 , J22 v.s. k2 .
Fig. 3. The performance indices J3 , J31 , J32 v.s. k3 .
Fig. 4. Simulink block diagram for vibration control simulator.
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Substitution of Eq.(23) into Eq.(22) leads to the controlled system equation ˆ˙ = (A − B k ¯ ¯ ˆ ˙ Z c c − F e D 0 + F e C 0 k)Z + F e V
(24)
The controller can be obtained using the bilinear transformation[14] to yield the following compensator X ck+1 = Ac X ck + B c Y sk ,
uk = C c X ck + Dc Y sk
(25)
The calculation for determining Ac , B c , C c and D is performed in MATLAB using the c2dm.m routine with the control toolbox. The control algorithm is implemented in simulation with the block illustrated in Fig.4. Figures 5-7 show the analytical probability densities (solid line) and the simulation results (dots • and ). It is seen that the analytical results and simulation results are consistent. Figures 8-13 show the response and control force of the structure subjected to the El Centro earthquake excitation. It is seen that the response of the system is greatly reduced and the control forces are bounded. The uncontrolled and controlled maximal re- Fig. 5 The uncontrolled and controlled probability density sponse and control force for the three-story building of H1 . c
Fig. 6. The uncontrolled and controlled probability density of H2 .
Fig. 7. The uncontrolled and controlled probability density of H3 .
Fig. 8. The uncontrolled (dashed) and controlled (solid) displacement of the roof.
Fig. 9. The uncontrolled (dashed) and controlled (solid) velocity of the roof.
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Fig. 10. The uncontrolled (dashed) and controlled (solid) acceleration of the roof.
Fig. 11. The control force of the first story.
Fig. 12. The control force of the second story.
Fig. 13. The control force of the third story.
structure under El Centro, Hachinohe, Northridge and Kobe earthquake excitations are given in Table 1, where xmax , x˙ max , x ¨max are the uncontrolled maximal displacement, velocity, acceleration, xmax , u u u c max max x˙ max , x ¨ are the controlled maximal displacement, velocity, acceleration, respectively and u is c c the maximal control force.
VII. CONCLUSIONS In the present paper a feedback control optimization method of partially observable linear structures via stationary response is proposed and analyzed with linear building structures equipped with control devices and sensors. The method consists of four steps: converting the partially observable control problem into a completely observable control problem; performing stochastic averaging and establishing the corresponding FPK equation; solving the FPK equation to obtain a stationary solution; combining the mean system energy and the control force as the performance index of the system and minimizing the performance index to obtain the optimal control force. The control effectiveness of the proposed procedure is illustrated by the application to a three-story experimental building structure model. The main advantage of the proposed procedure is that the optimum control force is obtained without solving the dynamical programming equation. It is shown that the approach presented may be a practical control strategy for seismic-excited building structures.
References [1] Fleming,W.H. and Soner,H.M., Controlled Markov Processes and Viscosity Solutions. New York: Springer, 1993. [2] Yong,J.M. and Zhou,X.Y., Stochastic Control, Hamiltonian Systems and HJB Equations. New York: Springer, 1999.
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[3] Zhu,W.Q. and Ying,Z.G., Optimal nonlinear feedback control of quasi Hamiltonian systems. Science in China, Series A, 1999, 42: 1213-1219. [4] Zhu,W.Q., Ying,Z.G. and Soong,T.T., An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dynamics, 2001, 24: 31-51. [5] Zhu,W.Q. and Yang,Y.Q., Stochastic averaging of quasi non-integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64: 157-164. [6] Zhu,W.Q., Huang,Z.L. and Yang,Y.Q., Stochastic averaging of quasi integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64: 975-984. [7] Zhu,W.Q., Huang,Z.L. and Suzuki,Y., Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. International Journal of Non-linear Mechanics, 2002, 37: 419-437. [8] Zhu,W.Q. and Ying,Z.G., Nonlinear stochastic optimal control of partially observable linear structures. Engineering Structures, 2002, 24: 333-342. [9] Zhu,W.Q. and Ying,Z.G., stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems. Journal of Zhejiang University Science, 2004, 5: 1313-1317. [10] Zhu,W.Q., Nonlinear Stochastic Dynamics and Control—Hamiltonian Framework. Beijing: Science Press, 2003 (in Chinese). [11] Ou,J.P., Structure Vibration Control—Active, Semi-active and Intelligent Control. Beijing: Science Press, 2003 (in Chinese). [12] Fleming,W.H. and Rishel R.W., Deterministic and Stochastic Optimal Control. Berlin: Spring-Verlag, 1975. [13] Bensoussan,A., Stochastic Control of Partially Observable Systems. Cambridge: Cambridge University Press, 1992. [14] Antoniou,A., Digital Filters: Analysis, Design and Application. New York: McGraw-Hill, Inc., 1993: 444446.
Table 1. The uncontrolled and controlled maximal response and control force for the three-story building structure
Earthquakes Story xmax (m) u xmax (m) c x˙ max (m/s) u x˙ max (m/s) c (m/s2 ) x ¨max u x ¨max (m/s2 ) c max u (kN) Earthquakes Story xmax (m) u xmax (m) c x˙ max (m/s) u x˙ max (m/s) c x ¨max (m/s2 ) u max x ¨c (m/s2 ) max (kN) u
El Centro Earthquake 1 2 3 0.0397 0.0682 0.0830 0.0154 0.0268 0.0334 0.3693 0.6720 0.9004 0.154 0.3416 0.4582 6.2580 9.7587 12.098 6.0053 7.5415 8.2645 1000 1000 707.91 Northridge Earthquake 1 2 3 0.0384 0.0651 0.0849 0.0344 0.0583 0.0705 0.5750 1.2311 1.6790 0.4329 0.9589 1.3105 11.541 18.155 22.905 10.761 17.033 21.177 1000 1000 1000
Hachinohe Earthquake 1 2 3 0.0183 0.0317 0.0378 0.01 0.0172 0.0209 0.1713 0.2885 0.4061 0.121 0.1969 0.2538 2.6896 3.6619 4.5607 2.228 2.3028 3.4654 1000 925.90 513.27 Kobe Earthquake 1 2 3 0.0886 0.1642 0.2069 0.0659 0.114 0.138 0.9327 1.6218 1.9512 0.6008 1.0811 1.4023 8.7386 13.977 17.541 8.8495 14.815 14.269 1000 1000 1000
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APPENDIX A. THE KANAI-TAJIMI POWER SPECTRAL DENSITY OF THE EARTHQUAKE EXCITATION 2
Sx¨g (ω, τ ) = σ (τ )
r−1
bj ω
2j
j=0
r
aj ω 2j
(26)
j=0
df dt d2 f 1 c0 c1 df + σ (τ ) W (t) =− f− 2 dt c2 c2 dt c2 a0 = ωg4 , a1 = 4ζg2 ωg2 − 2ωg2 , a2 = 1, b0 = ωg4 , b1 = 4ζg2 ωg2
(29)
c20
(30)
x ¨g = d0 f + d1
=
a0 , c21
− 2c0 c2 = 0,
c22
=
a2 , d20
=
b0 , d21
= b1
(27) (28)
According to the four records of the El Centro, Hachinohe, Kobe and Northridge earthquakes, the power spectral density can be obtained as shown in Fig.14, where c0 = 625, c1 = 12.5, c2 = 1, d0 = 625, d1 = 12.5, σ(τ ) = 0.015.
Fig. 14. The simulative power spectral density (PSD) of the earthquake excitation.
Fig. 15. The three-story experimental building structure model.
APPENDIX B. THE THREE-STORY EXPERIMENTAL BUILDING STRUCTURE MODEL The mass, ⎡ 4 M = ⎣0 0
stiffness and damping matrices of the system shown in Fig.15 are ⎤ ⎡ ⎤ 0 0 4 −2 0 4 0 ⎦ × 105 kg, K = ⎣ −2 4 −2 ⎦ × 108 N/m, C = 0.733M + 0.0026K 0 4 0 −2 2 ⎡ ⎤ 1 −1 0 1 −1 ⎦. The control force device placement matrix is P = ⎣ 0 0 0 1
ISSN 0894-9166
FEEDBACK CONTROL OPTIMIZATION FOR SEISMICALLY EXCITED BUILDINGS Xueping Li
Zuguang Ying
( Department of Mechanics, School of Aeronautics and Astronautics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China)
Received 8 December 2006, revision received 2 August 2007
ABSTRACT A feedback control optimization method of partially observable linear structures via stationary response is proposed and analyzed with linear building structures equipped with control devices and sensors. First, the partially observable control problem of the structure under horizontal ground acceleration excitation is converted into a completely observable control problem. Then the Itˆ o stochastic differential equations of the system are derived based on the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution to the Fokker-Plank-Kolmogorov (FPK) equation associated with the Itˆ o equations is obtained. The performance index in terms of the mean system energy and mean square control force is established and the optimal control force is obtained by minimizing the performance index. Finally, the numerical results for a three-story building structure model under El Centro, Hachinohe, Northridge and Kobe earthquake excitations are given to illustrate the application and the effectiveness of the proposed method.
KEY WORDS feedback control optimization, partially observable structure, stochastic averaging method, earthquake response, stationary probability density
I. INTRODUCTION Civil engineering structures, including existing and future buildings, towels and bridges, must be adequately protected from a variety of events such as earthquakes, winds and traffic. In order to protect the structures and reduce the destruction, the stochastic control of civil engineering structures is becoming a problem of public concern. The mathematical theory of stochastic optimal control has been quite well developed[1, 2] . A principal approach to the stochastic optimal control problem is based on the stochastic dynamical programming principle. So far, the theory of stochastic optimal control has been applied mainly to economics, especially finance. In the engineering field, only the linear quadratic Gaussian (LQG) control strategy has been widely used until now. In the past few years, a nonlinear stochastic optimal control strategy has been proposed for quasi-Hamiltonian systems with external and/or parametric stochastic excitations by the present second author and his co-workers[3, 4] based on the stochastic averaging method for quasi-Hamiltonian systems[5–7] and the stochastic dynamical programming principle[1, 2] . The strategy has been extended to the stochastic optimal control of partially observable linear and nonlinear systems[8, 9] . More effective and efficient[10] , this control strategy has several advantages over the LQG controller strategy[10].
Project supported by the National Natural Science Foundation of China under a key grant (No.10332030), the Research Fund for the Doctoral Program of Higher Education of China (No. 20060335125) and the Zhejiang Provincial Natural Science Foundation of China (No. Y607087).
Vol. 20, No. 4
Xueping Li et al.: Feedback Control Optimization for Seismically Excited Building
· 343 ·
In the control strategy developed in Refs.[3] and [4], the dynamical programming equation must be solved in order to obtain the optimal control forces. However, in practice, there is no effective method of obtaining the analytical solution to the dynamical programming equation. Generally, it is a burdensome task to solve it numerically. In the present paper, a control optimization for partially observable linear structures with stochastic excitations is proposed based on the separability principle, stochastic averaging method and optimization method. The mean square control force and the mean system energy are combined as the performance index of the system optimization and the optimum control force is obtained without solving the dynamical programming equation. The method is illustrated with a linear building structure equipped with control devices and sensors. The earthquake records, El Centro, Hachinohe, Northridge and Kobe[11] are used to show the effectiveness and efficiency of the control method proposed.
II. STOCHASTIC OPTIMAL CONTROL OF A PARTIALLY OBSERVABLE STRUCTURE For an n-story building structure equipped with control devices and sensors under horizontal ground acceleration excitation, the equation of motion of the system is of the form ¨ + CX ˙ + KX = −M Γ x MX ¨g + P U (1) where X = {X1 , X2 · · · Xn }T , Xi is the horizontal displacement of the ith floor relative to the ground; M , C, K are the n × n mass, damping and stiffness matrices of the system, respectively; Γ is an n-dimensional unit vector; x ¨g is the horizontal acceleration; U is a k-dimensional control force vector with k being the number of control devices; P is a n × k control device placement matrix. Introducing modal transformation X = ΦQ (2) where Φ is the n × m (m ≤ n) real modal matrix of the system which can be normalized to yield ΦT M Φ = I, and I is an m × m unit matrix, m is the number of the dominant modes among n modes to control, Eq.(1) becomes ¨ + 2ζΩ Q ˙ + Ω 2 Q = u − Gp x Q ¨g (3) where Q = {Q1 , Q2 · · · Qm }T , Qi is the generalized displacement of the ith mode; 2ζΩ = ΦT CΩ with ζ = diag(ζi ) and ζi is the damping ratio of ith mode; Ω 2 = diag(ωi2 ) = ΦT KΦ and ωi is the natural frequency of the ith mode; Gp = −ΦT M Γ ; u = ΦT P U and ui is the generalized control force of the ith mode. Suppose that the horizontal ground acceleration x ¨g is modeled as a uniformly modulated stationary random process with the Kanai-Tajimi power spectral density described in Appendix A. Equations (3), (27) and (28) can be combined and converted into the following Itˆ o stochastic differential equation dZ = (Ac Z + B c u)dt + C c dB(t)
(4)
˙ T , F T }T , F 1 = {f, f˙}T , B(t) is the unit Wiener process. where Z = {QT , Q 1 ⎫ ⎧ ⎤ ⎡ ⎧ ⎫ 0 ⎪ ⎪ 0 Im 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ ⎪ ⎪ ⎬ ⎨ Im ⎪ ⎨ 0 ⎪ ⎬ ⎢ −Ω 2 −2ζΩ −Gp d0 −Gp d1 ⎥ ⎥ ⎢ , Cc = A=⎢ ⎥ , Bc = 0 ⎪ ⎪ ⎪ ⎢ 0 0 ⎪ 0 1 0 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦ ⎣ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ 1 ⎪ ⎭ ⎩ σ(t) ⎪ 0 c1 /c2 0 0 −c0 /c2 c2
(5)
T
To derive the observation equation, suppose that l absolute accelerations a = {a1 , a2 , ...al } at the ˙ and u as follows: different floors are measured with noises on. They can be expressed in terms of Q, Q ¨ + Γx a = P 0 (X ¨g ) + σ 0 w1 (t) (6) where P 0 is an l × n sensor placement matrix, w 1 (t) is an m-dimensional vector of independent unit Gaussian white noises; σ 0 is an l ×m measurement noise intensity matrix. Equation (6) can be converted into the following Itˆ o stochastic differential equation dV = (D 0 Z + C 0 u)dt + σ 0 dB 1 (t)
(7)
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where dV /dt = a; B 1 (t) is an m-dimensional vector of the independent unit Wiener processes, which are independent of B(t). D 0 = P P 0 −ΦΩ 2 − Φ2ζΩ 0 0 , C 0 = P P 0 Φ (8) are measurement matrices and Eq.(7) is the observation equation of the system.
III. OPTIMAL CONTROL OF A COMPLETELY OBSERVABLE SYSTEM According to the separation principle[12, 13] , the partially observed stochastic optimal control problem formulated in the last section can be converted into a completely observed stochastic optimal control ˆ ˆ problem in the following way. Take the conditional mean Z(t) of Z(t) as the state of the system. Z(t) is governed by the following Itˆ o equation ˆ = (Ac Z ˆ 1 (t) ˆ + B c u)dt + F e σ 0 dB dZ (9) ˆ where B 1 (t) is the m-dimensional vector of the independent unit Wiener processes. −1 F e = Re (t)D T (10) 0 S0 ¯ = Z − Z, ˆ which satisfies the following Riccati-type matrix Re (t) is the covariance matrix of error Z differential equation. ˙ e = Ac Re + Re AT − Re D T S −1 D 0 Re + S c (11) R c 0 0 T T where S 0 = σ 0 σ 0 ; S c = C c C c . A linear optimal control can be applied.
IV. AVERAGED EQUATION AND STATIONARY SOLUTIONS The system (4) can be regarded as a controlled, stochastically excited and dissipated integrable Hamiltonian system. Suppose the damping forces, control forces and the intensity of ground acceleration are of the same order of ε and ε is a small constant. By applying the stochastic averaging method for quasiintegrable Hamiltonian systems[7] , the averaged Itˆ o equations can be obtained, in the non-resonance case dHi = [mi (Hi )+ < (∂Hi /∂ Q˙ i )ui >]dt + σi (Hi )dBi (t) (i = 1, 2, ..., n) (12) 2 2 2 ˙ where Hi = (Qi + ωi Qi )/2; Bi (t) are independent unit Wiener processes: mi (Hi ) = −2ζi ωi Hi + σi2 (Hi )/(2Hi ), σi2 (Hi ) = 2πHi [Gp ]2i Sx¨g (ωi , τ ) (13) Sx¨g (ωi , τ ) is the evolutionary spectral density of x¨g which can be expressed as 2 r−1 i di (jω) √ Sx¨g (ωi , τ ) = 2πσ 2 (τ ) i=0 (14) 2 , j = −1 r ci (jω)i i=0
The linear feedback control force for system (9) is ˆ˙ u = −kQ (15) where u = {u1 , u2 , ..., um }T , k = diag{k1 , k2 , ...km }. By substituting Eq.(15) into Eq.(12) and averaging the control force terms, the corresponding FPK equation is established as m m ∂p −∂ −(2ζi ωi + ki )Hi + σi2 (Hi )/(2Hi ) p ∂ 2 [σi2 (Hi )p] = + (16) ∂t ∂Hi 2∂Hi2 i=1 i=1 A stationary solution to FPK Eq.(16) is m p= pi , pi (Hi ) = Ci exp −(2ζi ωi + ki )Hi /[π[Gp ]2i Sx¨g (ωi , τ )]
(17)
i=1
where Ci are normalization constants. The stationary probability of the modal displacements and velocities of the controlled structure is then pi (Hi ) pi (Qi , Q˙ i ) = T (Hi ) 2 2 ˙2 Hi =(Qi +ωi Qi )/2
=
Ci ωi exp −(2ζi ωi + ki )(Q˙ 2i + ωi2 Q2i )/[2π[Gp ]2i Sx¨g (ωi , τ )] 2π
(18)
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V. DETERMINING THE OPTIMAL CONTROL FORCE The form of the performance index depends upon the objective of the control, for example, minimization of response, stabilization or maximization of the reliability. Here the objective is to minimize the response and the performance index is Ji = Ji1 + Ji2 , Ji1 = E(Hi ), Ji2 = Ri E(u2i )
where E(Hi ) =
∞
0
Hi pi (Hi )dHi ,
E(u2i ) =
0
∞
(−ki Q˙ i )2
0
∞
pi (Qi , Q˙ i )dQi dQ˙ i
(19) (20)
Substituting Eq.(20) into Eq.(19) yields the ith performance index Ji . The ki can be determined by minimizing Ji . Ri is selected as the balanced parameter which adjusts the amplitude of the control force.
VI. NUMERICAL RESULTS AND DISCUSSION A three-story experimental building structure model and its parameters are given in Appendix B. The control parameters, R1 = 0.01, R2 = 0.01, R3 = 0.01, and k can be obtained by using the proposed method as k = diag{4.8 2.8 1.8} (21) The performance indices Ji , Ji1 and Ji2 for varying ki are shown in Figs.1, 2 and 3. According to Eqs.(7), (9) and (15), the converted completely observable control system is ˆ˙ = (A Z ˆ + B c u) + F e (V˙ − D 0 Z ˆ − C 0 u) Z c
(22)
¯Z ˆ u = −k
(23)
Fig. 1. The performance indices J1 , J11 , J12 v.s. k1 .
Fig. 2. The performance indices J2 , J21 , J22 v.s. k2 .
Fig. 3. The performance indices J3 , J31 , J32 v.s. k3 .
Fig. 4. Simulink block diagram for vibration control simulator.
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Substitution of Eq.(23) into Eq.(22) leads to the controlled system equation ˆ˙ = (A − B k ¯ ¯ ˆ ˙ Z c c − F e D 0 + F e C 0 k)Z + F e V
(24)
The controller can be obtained using the bilinear transformation[14] to yield the following compensator X ck+1 = Ac X ck + B c Y sk ,
uk = C c X ck + Dc Y sk
(25)
The calculation for determining Ac , B c , C c and D is performed in MATLAB using the c2dm.m routine with the control toolbox. The control algorithm is implemented in simulation with the block illustrated in Fig.4. Figures 5-7 show the analytical probability densities (solid line) and the simulation results (dots • and ). It is seen that the analytical results and simulation results are consistent. Figures 8-13 show the response and control force of the structure subjected to the El Centro earthquake excitation. It is seen that the response of the system is greatly reduced and the control forces are bounded. The uncontrolled and controlled maximal re- Fig. 5 The uncontrolled and controlled probability density sponse and control force for the three-story building of H1 . c
Fig. 6. The uncontrolled and controlled probability density of H2 .
Fig. 7. The uncontrolled and controlled probability density of H3 .
Fig. 8. The uncontrolled (dashed) and controlled (solid) displacement of the roof.
Fig. 9. The uncontrolled (dashed) and controlled (solid) velocity of the roof.
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Fig. 10. The uncontrolled (dashed) and controlled (solid) acceleration of the roof.
Fig. 11. The control force of the first story.
Fig. 12. The control force of the second story.
Fig. 13. The control force of the third story.
structure under El Centro, Hachinohe, Northridge and Kobe earthquake excitations are given in Table 1, where xmax , x˙ max , x ¨max are the uncontrolled maximal displacement, velocity, acceleration, xmax , u u u c max max x˙ max , x ¨ are the controlled maximal displacement, velocity, acceleration, respectively and u is c c the maximal control force.
VII. CONCLUSIONS In the present paper a feedback control optimization method of partially observable linear structures via stationary response is proposed and analyzed with linear building structures equipped with control devices and sensors. The method consists of four steps: converting the partially observable control problem into a completely observable control problem; performing stochastic averaging and establishing the corresponding FPK equation; solving the FPK equation to obtain a stationary solution; combining the mean system energy and the control force as the performance index of the system and minimizing the performance index to obtain the optimal control force. The control effectiveness of the proposed procedure is illustrated by the application to a three-story experimental building structure model. The main advantage of the proposed procedure is that the optimum control force is obtained without solving the dynamical programming equation. It is shown that the approach presented may be a practical control strategy for seismic-excited building structures.
References [1] Fleming,W.H. and Soner,H.M., Controlled Markov Processes and Viscosity Solutions. New York: Springer, 1993. [2] Yong,J.M. and Zhou,X.Y., Stochastic Control, Hamiltonian Systems and HJB Equations. New York: Springer, 1999.
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[3] Zhu,W.Q. and Ying,Z.G., Optimal nonlinear feedback control of quasi Hamiltonian systems. Science in China, Series A, 1999, 42: 1213-1219. [4] Zhu,W.Q., Ying,Z.G. and Soong,T.T., An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dynamics, 2001, 24: 31-51. [5] Zhu,W.Q. and Yang,Y.Q., Stochastic averaging of quasi non-integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64: 157-164. [6] Zhu,W.Q., Huang,Z.L. and Yang,Y.Q., Stochastic averaging of quasi integrable Hamiltonian systems. ASME Journal of Applied Mechanics, 1997, 64: 975-984. [7] Zhu,W.Q., Huang,Z.L. and Suzuki,Y., Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. International Journal of Non-linear Mechanics, 2002, 37: 419-437. [8] Zhu,W.Q. and Ying,Z.G., Nonlinear stochastic optimal control of partially observable linear structures. Engineering Structures, 2002, 24: 333-342. [9] Zhu,W.Q. and Ying,Z.G., stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems. Journal of Zhejiang University Science, 2004, 5: 1313-1317. [10] Zhu,W.Q., Nonlinear Stochastic Dynamics and Control—Hamiltonian Framework. Beijing: Science Press, 2003 (in Chinese). [11] Ou,J.P., Structure Vibration Control—Active, Semi-active and Intelligent Control. Beijing: Science Press, 2003 (in Chinese). [12] Fleming,W.H. and Rishel R.W., Deterministic and Stochastic Optimal Control. Berlin: Spring-Verlag, 1975. [13] Bensoussan,A., Stochastic Control of Partially Observable Systems. Cambridge: Cambridge University Press, 1992. [14] Antoniou,A., Digital Filters: Analysis, Design and Application. New York: McGraw-Hill, Inc., 1993: 444446.
Table 1. The uncontrolled and controlled maximal response and control force for the three-story building structure
Earthquakes Story xmax (m) u xmax (m) c x˙ max (m/s) u x˙ max (m/s) c (m/s2 ) x ¨max u x ¨max (m/s2 ) c max u (kN) Earthquakes Story xmax (m) u xmax (m) c x˙ max (m/s) u x˙ max (m/s) c x ¨max (m/s2 ) u max x ¨c (m/s2 ) max (kN) u
El Centro Earthquake 1 2 3 0.0397 0.0682 0.0830 0.0154 0.0268 0.0334 0.3693 0.6720 0.9004 0.154 0.3416 0.4582 6.2580 9.7587 12.098 6.0053 7.5415 8.2645 1000 1000 707.91 Northridge Earthquake 1 2 3 0.0384 0.0651 0.0849 0.0344 0.0583 0.0705 0.5750 1.2311 1.6790 0.4329 0.9589 1.3105 11.541 18.155 22.905 10.761 17.033 21.177 1000 1000 1000
Hachinohe Earthquake 1 2 3 0.0183 0.0317 0.0378 0.01 0.0172 0.0209 0.1713 0.2885 0.4061 0.121 0.1969 0.2538 2.6896 3.6619 4.5607 2.228 2.3028 3.4654 1000 925.90 513.27 Kobe Earthquake 1 2 3 0.0886 0.1642 0.2069 0.0659 0.114 0.138 0.9327 1.6218 1.9512 0.6008 1.0811 1.4023 8.7386 13.977 17.541 8.8495 14.815 14.269 1000 1000 1000
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APPENDIX A. THE KANAI-TAJIMI POWER SPECTRAL DENSITY OF THE EARTHQUAKE EXCITATION 2
Sx¨g (ω, τ ) = σ (τ )
r−1
bj ω
2j
j=0
r
aj ω 2j
(26)
j=0
df dt d2 f 1 c0 c1 df + σ (τ ) W (t) =− f− 2 dt c2 c2 dt c2 a0 = ωg4 , a1 = 4ζg2 ωg2 − 2ωg2 , a2 = 1, b0 = ωg4 , b1 = 4ζg2 ωg2
(29)
c20
(30)
x ¨g = d0 f + d1
=
a0 , c21
− 2c0 c2 = 0,
c22
=
a2 , d20
=
b0 , d21
= b1
(27) (28)
According to the four records of the El Centro, Hachinohe, Kobe and Northridge earthquakes, the power spectral density can be obtained as shown in Fig.14, where c0 = 625, c1 = 12.5, c2 = 1, d0 = 625, d1 = 12.5, σ(τ ) = 0.015.
Fig. 14. The simulative power spectral density (PSD) of the earthquake excitation.
Fig. 15. The three-story experimental building structure model.
APPENDIX B. THE THREE-STORY EXPERIMENTAL BUILDING STRUCTURE MODEL The mass, ⎡ 4 M = ⎣0 0
stiffness and damping matrices of the system shown in Fig.15 are ⎤ ⎡ ⎤ 0 0 4 −2 0 4 0 ⎦ × 105 kg, K = ⎣ −2 4 −2 ⎦ × 108 N/m, C = 0.733M + 0.0026K 0 4 0 −2 2 ⎡ ⎤ 1 −1 0 1 −1 ⎦. The control force device placement matrix is P = ⎣ 0 0 0 1