On the Nonlinear Stabilization Problem via Quadratic Immersion. Part I: Sigma/Pi-systems and biased solutions of driver-type systems

Based on our recently issued paper on Quadratic Immersion (QI), (also said 'exact quadratization') for nonlinear control systems, the present work aims to define the main features of a possible new approach in nonlinear control, able to exploit the structural properties of a quadratized system in order to design global, statefeedback- regulators, with an exponential, and tunable, performance, for a meaningful class of nonlinear systems. The paper is divided into two parts. In the Part I we go through the properties of QI and explore the possibility of writing explicitly the solution for the class of the, so called, Sigma/Pi-systems. The main result consists in showing that, under certain conditions, Sigma/Pi-systems are always forward complete, and the solution can be calculated at the steady-state.