## 3 Inverse problem: constrained sigma-point Kalman filter  
In our study the inverse problem has to provide real-time forecasts of the delamination processes, along with estimates of uncertain model parameters. Allowing for the above described explicit time integration scheme, a state vector \(\mathbf{x}\) gathering all the information on the evolving system is obtained by joining the nodal displacement, velocity and acceleration vectors to a vector \(\boldsymbol{\vartheta}\) gathering model parameters in need of calibration:  
\[\mathbf{x}= \begin{Bmatrix}\mathbf{u}^{h}\\ \dot{\mathbf{u}}^{h}\\ \boldsymbol{\vartheta}\end{Bmatrix}  \tag{21}\]  
Since laminate failure due to delamination is represented by displacement discontinuities arising along the inter-laminar surfaces, vector \(\mathbf{x}\) in (21) also provides the requested details of the failure mode. State vectors shaped in this way, with model parameters explicitly accounted for, are typically adopted in extended Kalman filtering [5; 40]. While the current structural state (here \(\mathbf{u}^{h}\), \(\dot{\mathbf{u}}^{h}\) and \(\ddot{\mathbf{u}}^{h}\)) is always at least partially observed, model parameters to be calibrated can not be directly measured; by joining structural and parameter vectors in \(\mathbf{x}\), state tracking can consistently enhance model calibration. Allowing for modeling and measurement errors in a stochastic framework, the discretized state-space model of the system within the time interval \([t_{i}\quad t_{i+1}]\) reads:  
\[ \begin{cases}\mathbf{x}_{i+1}&=\mathbf{f}_{i}\left(\mathbf{x}_{i}\right)+ \mathbf{v}_{i}\\ \mathbf{y}_{i}&=\mathbf{H}\mathbf{x}_{i}+\mathbf{w}_{i}\end{cases}  \tag{22}\]  
where: \(\mathbf{y}_{i}\) is the observation vector, which collects measurables; \(\mathbf{v}_{i}\) and \(\mathbf{w}_{i}\) are the process and measurement noises. These noises are assumed to be additive, uncorrelated white and Gaussian processes, with zero mean and time-invariant covariance matrices \(\mathbf{V}\) and \(\mathbf{W}\)[6; 40]. Because of inter-laminar strength degradation, mapping \(\mathbf{f}_{i}\) turns out to be highly nonlinear; on the other hand, since either surface displacements or velocities are measured during testing, the observation equation (22\({}_{2}\)) shows up as a linear relation between \(\mathbf{y}_{i}\) and \(\mathbf{x}_{i}\) (see Eq. 21). The mentioned nonlinearities of mapping \(\mathbf{f}_{i}\) cause estimates obtained through a standard EKF to be affected by biases [41; 42]; to get accurate results, the recently proposed SPKF [8; 9; 43; 44] is instead here adopted. While the EKF linearizes the mapping \(\mathbf{f}_{i}\), the SPKF deterministically samples the probability distribution of \(\mathbf{x}\) at time \(t_{i}\) (i.e. at the beginning of the time step) through a set of properly chosen sigma-points \(\mathbf{\hat{\chi}}_{i,\text{j}}\), \(\text{j}=0,...,N_{\chi}\). These sigma-points are allowed to evolve according to the nonlinear mapping \(\mathbf{f}_{i}\); the  
a weighted averaging procedure performed on the evolved sigma-points [8]. The filtering procedure is detailed in Table 1, where \(\mathbb{E}[\Box]\) represents the expected value of \(\Box\). In this scheme, the number \(N_{\chi}\) of sigma-points, terms \(\mathbf{\Delta}\mathbf{\chi}_{i,\mathrm{j}}\), and weights \(\omega_{\mathrm{j}}\) and \(\omega_{\mathrm{j}}^{\star}\) need to be set so as to achieve maximum accuracy; such issue is treated in Appendix A, where fundamental results are summarized. In Kalman filtering it is usually assumed that all the state vector components are Gaussian variables; hence, the relevant probability distributions cannot be compliant with physical side constraints (like, e.g. positiveness of inter-laminar strength and toughness).