Nonlinear Stabilization via System Immersion and Manifold Invariance: Survey and New Results

A survey on a recently developed methodology for the (adaptive) stabilization of nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and is well suited in applications where a controller for a reduced-order model is known, and we would like to robustify it with respect to higher-order dynamics. This is achieved by immersing the full (closed-loop) system dynamics into the (closed-loop) reduced-order one. The applicability of the method is discussed through several examples. It is shown that for a class of systems in feedback form the method yields new adaptive control laws with advantageous properties. The method can be extended to output feedback stabilization problems, where the design of an observer is typically required. In this case, the proposed approach treats unmeasured states and unknown parameters in a uniform manner.