Functional $\mathscr {H}_{\infty }$ Filtering for Descriptor Systems With Incrementally Quadratic Nonlinearities Under Disturbances

This paper develops a functional $\mathscr {H}_{\infty }$ filter for nonlinear descriptor systems subject to external disturbances. Conventional $\mathscr {H}_{\infty }$ filtering approaches for descriptor systems impose restrictive regularity assumptions and employ implicit descriptor-form filters, leading to practical implementation difficulties. Moreover, existing approaches mainly target full-or reduced-order state estimation, which is computationally inefficient when only a specific functional of the state is required. To address these limitations, the filter is formulated directly in an explicit state-space framework and can be initialized with arbitrary initial values. The filter order is chosen to be less than or equal to the dimension of the functional vector to be estimated, thereby reducing computational complexity. The considered nonlinearities are characterized using incremental quadratic constraints parameterized by appropriate multiplier matrices, which encompass Lipschitz, one-sided Lipschitz, monotone, and many other nonlinearities. Sufficient criteria for the existence of the proposed filter are established through a rank condition imposed on the system matrices together with a set of linear matrix inequalities (LMIs). Under these conditions, asymptotic stability of the estimation error dynamics is guaranteed, while the influence of external disturbances on the error is bounded within a prescribed $\mathcal {L}_{2}$ -performance framework. Finally, numerical simulations demonstrate and validate the effectiveness of our theoretical results.