Practical Stability: New Definitions and Innovative Results Part I: The Linear Systems Case

In this paper (Part I) new and useful definitions of finite time practical stability, exponential stability and practical convergence velocity, for an important class of nonlinear systems, including linear ones, subject to uncertainties and/or additional nonlinearities and/or bounded external signals, are proposed and illustrated. For the linear systems, using a time-domain approach, important theorems and efficient algorithms, which allow computing, for a generic instant belonging to an assigned limited interval, the minimum and maximum output values, for all the initial conditions and for all the external signals belonging to fixed polytopes or for all the external signals generated by an assigned linear system freely evolving from initial conditions belonging to a fixed polytope, are provided. For a class of second order linear and time invariant systems some theorems, also when there are parametric uncertainties, are stated. In Part II innovative results, obtained via Lyapunov and introducing the concept of majorant system, for an important class of nonlinear systems, including linear ones, subject to uncertainties, are reported. These definitions and theorems allow solving numerous analysis and synthesis practical problems.