Fixed-time Input-to-state Stability via Generalized Homogeneous Approximations

In order to extend the existing fixed-time control robustness results for double-integrator systems to general nonlinear systems, the fixed-time input-to-state stability (FT-ISS) of a class of nonlinear system with bounded exogenous disturbances is studied through generalized homogeneous approximations in this paper. To this end, the definitions of finite-time and fixed-time class \({\cal K}{\cal L}\) $$\mathcal{K}ℒ$$ functions are given. Based on the definitions, the sufficient conditions of the FT-ISS property with respect to exogenous disturbances are proposed. Then, a class of fixed-time filters is designed under the principle of generalized homogeneous approximations. As a particular example, the stability and frequency characteristics of the low-pass fixed-time filter are analyzed by the presented sufficient conditions and describing function method, respectively. Finally, a simulation example is presented to illustrate the efficiency of the fixed-time filter.