A wavelet-based dynamic mode decomposition for modeling mechanical systems from partial observations

The paper defines the auxiliary state z by stacking the wavelet-coefficient observables and constructs C_w so that y(t_k) = C_w z(t_k); this is explicitly stated in eq. (31) and the surrounding definitions (wi, z, Cw) . It then fits [A_w B_w; C_w D_w] by least squares with the Moore–Penrose pseudoinverse, namely [Z1; Y0][Z0; U0]^† (eq. (36)) , and uses the predictor z(t_{k+1}) ≈ A_w z(t_k) + B_w u(t_k), y(t_k) ≈ C_w z(t_k) + D_w u(t_k) (eq. (34)) with state dimension d(J+1) (cf. z ∈ R^{d(J+1)} and the example text) . The model’s solution reproduces these steps and supplies the standard normal-equations/pseudoinverse argument for least squares. Aside from a minor typographical slip in Z1’s definition in the PDF, the two are aligned.