## Damage Estimation

In the model-based paradigm, damage estimation reduces to joint state-parameter estimation. The typical approach to state estimation is to use a state observer. Joint state-parameter estimation is typically performed using a state observer by augmenting the state vector with the parameter vector. In this way, both states and parameters are simultaneously estimated.

For nonlinear systems with non-Gaussian noise terms, particle filters are best suited, and offer approximate (suboptimal) solutions to the state estimation problem for such systems where optimal solutions are unavailable or intractable. In particle filters, the state distribution is approximated by a set of discrete weighted samples, called particles. As the number of particles is increased, performance increases and the optimal solution is approached.

With particle filters, the particle approximation to the state distribution is given by

where *N* denotes the number of particles, and for particle *i*, **x**_{k}^{i} denotes the state estimates, **θ**_{k}^{i} denotes the parameter estimates, and *w*_{k}^{i}denotes the weight. The posterior density is approximated by

with δ denoting the Dirac delta function.

We employ the sampling importance resampling (SIR) particle filter, and implement the resampling step using systematic resampling. The pseudocode for a single step of the SIR filter is shown as the algorithm below. Each particle is propagated forward to time *k* by first sampling new parameter values and sampling new states. The particle weight is assigned using the outputs at *k*. The weights are then normalized, followed by the resampling step.