Immersion and Invariance-Based Disturbance Observer and Its Application to Safe Control

When the disturbance input matrix is nonlinear, existing disturbance observer design methods rely on the solvability of a partial differential equation or the existence of an output function with a uniformly well-defined disturbance relative degree, which can pose significant limitations. This note introduces a systematic approach for designing an immersion and invariance-based disturbance observer (IIDOB) that circumvents these strong assumptions. The proposed IIDOB ensures the disturbance estimation error is globally uniformly ultimately bounded by approximately solving a partial differential equation while compensating for the approximation error. Furthermore, by integrating IIDOB into the framework of control barrier functions, a filter-based safe control design method for control-affine systems with disturbances is established, where the filter is used to generate an alternative disturbance estimation signal with a known derivative. Sufficient conditions are established to guarantee the safety of the disturbed systems. Simulation results demonstrate the effectiveness of the proposed method.