Earthquake Engineering Research - HKU Libraries - The University ...

425
change continuously. In this paper, the EKF is employed to estimate the states of the nonlinear
structure in order to implement the structural control. Suppose the nonlinear dynamic equation of
structure with active control devices subjected to additive plant noise w may be
Z = f(z 9 U,X g ,w) (3.1)
And the measurement equation with noise v given by
y v ~C v Z + v (3.2)
where C v is the measurement matrix and y v are the observe structural states, w and v are
statistically independent zero mean Gaussian white noise with covariance O w and R v , respectively.
The EKF implements the state estimation through a recursion algorithm that is more suitable for online
simulation. Every calculating step is composed of time update and measurement update equations. The
recursion algorithm is given as follows:
(3-3)
(t M |O (3.4)
l
(3.5)
Z(t k \t k ) = Z(t t Kj+£(Oy-C,Z(f t Kj (3-6)
P(t k \t k } = [l-K(t k )C,]P(t k \t k J (3.7)
In which Z(t k \ t k ] is the state estimation at t k ,P(t k \ rj is covariance matrix of error 'mZ(t k \ t t ),
K(t k ) is Kalman gain at time t t , $(t M \t t ) is state transfer matrix from t t to t M , and
^fe+ilO is ^ discretirized plant noise input matrix. For the sampling time A, matrix