Practical Stability: New Definitions and Innovative Results Part II: The Nonlinear Systems Case

In the paper, on the basis of the new formulation of the finite-time practical stability problem and of the new and useful definitions provided in Part I, this issue is approached for the nonlinear systems with parametric uncertainties. In detail, for an important class of nonlinear systems, including linear ones, subject to uncertainties, the concept of majorant system is introduced and, via Lyapunov, new important theorems and very efficient algorithms, which allow determining a good conservative estimate of practical convergence velocity and a good excess estimate of each output value, for all the initial conditions belonging to assigned compact sets and for all the external signals and/or the additional nonlinearities and/or parametric uncertainties belonging to assigned polytopes, are provided. These theorems allow solving numerous analysis and synthesis practical problems, in particular these results allow designing very simple control laws in order to force a quite general linear or nonlinear system subject to parametric uncertainties to track sufficiently regular trajectories with prefixed errors and prefixed convergence velocity.