212 R. Astroza et al. where, p∈Rnp ×1 = vector of model parameters, np =number of model parameters, M, C∈ Rn×n =mass and damping matrices, q, ˙q, ¨q ∈ Rn×1 = nodal displacement, velocity, and acceleration vectors, r(q(p), p) ∈Rn×1 =internal resisting force vector, L ∈ Rn×nu = influence matrix of the base excitation, n =number of degrees of freedom of the FE model, ¨u ∈Rnu×1 =input ground acceleration vector withnu =number of acceleration components of the base excitation, and the subscript denotes the time step. Aleatory and epistemic uncertainties are present when selecting the model parameters, because they are defined based on the information contained in material specifications, characterization of external loads, blueprints, etc., therefore, in real structures it is not possible to have a precise estimation of the model parameters. In this paper, the vector of model parameters is written as p =[θ T ϕ T]T, with θ ∈ Rnθ×1 = vector of unknown model parameters to be estimated, and ϕ∈Rnϕ×1 =vector of modeling uncertainty parameters. Then, Eq. (24.1) can be written as M(θ, ϕ) ¨qk+1 (θ, ϕ) +C(θ, ϕ) ˙qk+1 (θ, ϕ) +rk+1 (qk+1 (θ, ϕ), θ, ϕ) =−M(θ, ϕ) L(θ, ϕ) ¨uk+1 (24.2) From Eq. (24.2), the FE-predicted response, yk+1 ∈ Rny×1, can be expressed as a nonlinear function of θ and ϕ, the time-history of earthquake excitation (¨u1:k+1 = ¨uT 1 , . . . , ¨uT k+1 T), and the initial conditions (q0 and ˙q0), i.e., yk+1 =hk+1 (θ, ϕ, ¨u1:k+1, q0, ˙q0) (24.3) where hk +1(·) =nonlinear response function of the FE model. Measured response, yk+1 ∈ Rny×1, can be used to calibrate θsince it is related to the FE-predicted response by means of: vk+1 =yk+1 − yk+1 (24.4) where vk+1 ∈ Rny×1 =simulation error vector assumed Gaussian white with zero mean and covariance matrix Rk+1 ∈ Rny×ny, i.e., v k +1 ∼N(0, Rk +1). Effects of measurement noise and modeling errors are included in the simulation error vector. In this paper, modeling uncertainty is considered by defining different model classes for response simulation (M0) and model updating (Mj) phases [11], i.e., Mj =M0. Measured response data (y) is obtained by using a set of pre-defined model parameters denoted by θ true and polluting the associated response (ytrue) with additive noise. Unknown model parameters defining the FE-predicted response, y, will be estimated based on a model class Mj. Different cases and levels of modeling uncertainty are analyzed, then, different model classes Mj (j =1, 2, . . . ) are investigated. Since the unknown model parameters are time invariant, they can be estimated assuming a random walk model. Then, the following nonlinear state-space model is defined θk+1 =θk +wk yk+1 =hk+1 (θk+1, ϕ, ¨u1:k+1) +vk+1 (24.5) where the initial conditions have been omitted for notational convenience andwk ∈ Rnθ×1 denotes the process noise assumed to be uncorrelated withvk +1, and Gaussian white with zero mean and covariance matrixQk ∈ Rnθ×nθ , i.e., wk ∼N(0, Qk). 24.2.1 Parameter-Only Estimation Based on the UKF The UKF can be used to recursively estimate the mean vector and covariance matrix of the θ, denoted by θ and Pθθ, respectively. This parameter-only estimation approach has been investigated in detail by Astroza et al. [9, 11, 17], proving its robustness to input and output measurement noises and also to minor modeling uncertainty. Figure 24.1 summarizes the parameter-only estimation approach for nonlinear FE model updating based on the UKF. More details about this approach can be found in [17, 18].
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