Stabilization of Uncertain Nonlinear Systems via Immersion and Invariance

This paper presents a method for designing asymptotically stabilising and adaptive control laws for uncertain nonlinear systems. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilising control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output-zeroing manifold and its friend, and is well suited, for instance, in situations where we know a stabilising controller of a nominal reduced-order model, which we would like to robustify with respect to higher-order dynamics.In adaptive control problems the method yields stabilising schemes that counter the effect of the uncertain parameters adopting a robustness perspective. This is in contrast with most existing adaptive designs that (relying on certain matching conditions) treat these terms as disturbances to be rejected. The construction does not invoke certainty equivalence, nor requires a linear parameterisation. Furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states.To illustrate the concepts presented in this paper we introduce several academic and physical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics, and an adaptive, nonlinearly parameterised, visual servoing problem.