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When the disturbance input matrix is nonlinear, existing disturbance observerdesign methods rely on the solvability of a partial differential equation orthe existence of an output function with a uniformly well-defined disturbancerelative degree, which can pose significant limitations. This note introduces asystematic approach for designing an Immersion and Invariance-based DisturbanceObserver (IIDOB) that circumvents these strong assumptions. The proposed IIDOBensures the disturbance estimation error is globally uniformly ultimatelybounded by approximately solving a partial differential equation whilecompensating for the approximation error. Furthermore, by integrating IIDOBinto the framework of control barrier functions, a filter-based safe controldesign method for control-affine systems with disturbances is established wherethe filter is used to generate an alternative disturbance estimation signalwith a known derivative. Sufficient conditions are established to guarantee thesafety of the disturbed systems. Simulation results demonstrate theeffectiveness of the proposed method.

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