23 Optimal Sensor Placement for Response Reconstruction in Structural Dynamics 207 with observation and prediction equation y(t) =C δ,ϕ x(t) +D δ,ϕ u(t) +e(t) (23.5) z(t) =Cp ϕ x(t) +Dp ϕ u(t) +ε(t) (23.6) where the state matrices A, B, CandDdepend on the stiffness, mass and damping matrices of the structure, es(t), e(t) andε(t) are respectively the zero-mean state, measurement and prediction errors with covariance matricesQs, Qe andQε, respectively, while Cp and Dp are system matrices that connect the output QoI to the state and input vectors. Similar description in the discrete state space form is also available. Also, a description can also be obtained in the modal space to simplify the formulation for complex linear systems in the case where only a fraction of the modes contribute to the response. For known input characteristics that are given by a linear filtering technique (for example, Kanai-Tajimi filter for earthquake excitations), the state vector can be augmented to include the states of the input filter, while the parameter set ϕ is augmented to include the parameters defining the input. Using Eq. (23.6), the uncertainty in the prediction of output QoI z(t) is given by Pz δ,ϕ = Cp ϕ Dp ϕ P δ,ϕ Cp ϕ Dp ϕ T +Qε (23.7) and depends on the covariance matrixP δ,ϕ of the error in the state and input estimates. Various techniques exist to estimate the covariance of the state and input in the case of output-only vibration measurements. For example, consider the case of white noise input. This case arises also for non-white excitations modelled by a set of stochastic differential equations with parameters that captures the characteristics of the excitation. The filter parameters are usually uncertain and are included in the parameter set ϕ. The stationary error covariance of the state estimate for displacement and velocity measurements is provided by the steady-state Ricatti equation in the form A ϕ Px δ,ϕ +Px δ,ϕ A ϕ −Px δ,ϕ C T δ,ϕ Q−1 e C δ,ϕ Px δ,ϕ +B ϕ QsB T ϕ =0 (23.8) Substituting the covariance matrix Px δ,ϕ into Eq. (23.7), one obtains the covariance of the state and input vector as a function of the sensor configuration δ. Similarly, for the case where the input in not white noise but unknown, one can use existing input-state estimation techniques to estimate the error covariance for the predictions of both the state and the input. In this case the Ricatti Eq. (23.8) is replaced by similar equation(s) that can be solved to estimate the joint covariance for state and input. The resulting formulation depends on the method used. One can find such formulation in references [8, 9]. The end result is that the covariance matrixP δ,ϕ depends on the characteristics of the system and the input contained in the parameter vector ϕ. It is clear that the covariance of the error estimate for the state and input is independent of the measurements and depends only on the structural model parameters, as well as the state and measurement error covariances. The parameters that define the state and measurement error covariances can be included in the parameter set ϕ. The covariance matrix P δ,ϕ , described in terms of the parameters ϕ and the sensor locations δ, is the main quantity used in the next section to solve the optimal sensor location problem for response reconstruction. To account for uncertain model and input characteristics ϕ, a prior probability distribution p ϕ can be used to quantify such uncertainties in the values of the model parameters. The data can also be used to learn a partition of the model parameter set ϕ, with the rest of the parameters to be nuisance parameters. Using Bayesian inference for the parameters that are learned using the data, the prior probability distribution of these parameters can be replaced by the posterior probability distribution. Herein we will denote such distribution byp ϕ , without making the distinction between prior and posterior.
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