Comparative Study on Multi-Objective Genetic Algorithms for Seismic Response Controls of Structures - Design Optimization of Active and Passive Structural Control Systems - page 349

Figure 10. Stiffness of two-way asymmetric system (Chopra 2000)

where M is the seismic mass matrix of the
20-story building and b and d are the distances of
structural plan of the NS- and EW-directions,
respectively. Finally, the governing equation of
motion is

output, y r is the vector of regulated responses, v
is a measurement noise vector, and C r , D r , and F r
depend on the sensor locations and the number of
actuators and locations, and

0 I

,
(23)

ndof ndof × ndof ndof ×

A = −

M x +C  x + K 



x =

M

− 1

K − M

− 1

C

3D20s 3D20s 3D20s

3D20s 3D20s ndof ndof × 3D20s 3 D20s ndof ndof ×

(19)

- M G x g + P f

3D20s 3D20s 3D20s 3D20 s

0

where K 3D20s (60x60) is the global stiffness ma-
trix, M 3D20s (60x60) is the global mass matrix,
C 3D20s (60x60) is the global damping matrix that
is defined by 2% proportional Rayleigh damping,
G 3D20s is ground motion matrix, P 3D20s is a loca-
tion vector of the control device forces, and f 3D20s
is the control force input. The 2 nd order differen-
tial equations can be transformed into the 1 st order
state space equations

ndof ndof ×

B = , (24)

-1

M f ”

3D20s a ndof ndof ×

I 0

ndof ndof × ndof ndof ×

,

C = 0 I

m 3 3 × ndof ndof ×

− 1 − 1

− M K − M C

3D20s 3D20s ndof × ndof 3D20s 3D20s ndof ndof ×

(25)

0
0

ndof ndof ×

D = , (26)

m ndof ndof ×

x =  Ax Bf + + E x g  (20)

− 1 ”

f M ( )

a 3D20s ndof ndof ×

y = C x D f + + F  x g + v (21)

0
G 3D20s

m m m m

ndof × 1

E = − , (27)

ndof 1

×

  x g

y = x +C x D f + + F (22)

r r r r

where x is the state vector, f is control force input,
y m is the vector corresponding to the measured