On adaptive control of nonlinearly parameterized nonlinear systems: Towards a constructive procedure

A new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized, nonlinear systems was recently proposed by the authors. The key step is the construction of a monotone mapping, via a suitable selection of a controller tuning function, which has to satisfy some integrability conditions—this translates into the need to solve a partial differential equation (PDE). In this paper this result is extended providing some answers to the questions of characterization of “monotonizable” systems and solvability of the PDE. First, adding to the design a nonlinear dynamic scaling, we obviate the need to solve the PDE. Second, for the case of factorizable nonlinearities, the following results are established. (i) It is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible. (ii) Directly verifiable involutivity conditions that ensure the solution of the PDE are presented. (iii) An explicit formula for the required tuning function is given, provided the regressor matrix satisfies some rank conditions. Hence, adding a dynamic scaling, this yields a constructive solution to the problem.