This paper studies multistability and hidden oscillations in a four-dimensional Keldysh model of flutter suppression. The model is written in the Lurie form and is considered with discontinuous damper characteristics interpreted in the sense of Filippov. We analyze the stationary set, the sliding manifold, and the corresponding reduced sliding dynamics. To detect undesired periodic solutions, we apply the locus of a perturbed relay system (LPRS) method and extend it to a discontinuous piecewise-linear approximation of the original Keldysh nonlinearity. It allows to reveal symmetric and asymmetric stable periodic solutions, including periodic solutions with complicated switching configurations. Numerical continuation from the discontinuous piecewise-linear characteristic to the original discontinuous characteristic with a quadratic term detects five stable periodic solutions, four of which are hidden with respect to the rest segment. These results show that the four-dimensional Keldysh model has rich and complex nonsmooth dynamics and provide a constructive framework for estimating the boundary of global stability in discontinuous aircraft control systems.