Ultimate Boundedness Analysis

Abstract

Physically-motivated systems are generally subject to nonlinearities and uncertainties; for these systems, classical stability definitions (like asymptotic or exponential stability in the sense of Lyapunov) can be too restrictive. Namely, the state of a system may be mathematically unstable in some classical sense, but the response oscillates close enough to the equilibrium, to be considered as acceptable. In many stabilization problems, the aim is to bring states close to certain sets rather than to a particular state. In these situations, appropriate performance specifications are given by the concept of ultimate boundedness with a fixed bound, also referred to as practical stability which not only provides information on the stability of the system, but also characterizes its transient behavior with estimates of the bounds on the system trajectories. In this chapter, the ultimate boundedness analysis of the drilling system described by the wave equation with nonlinear boundary conditions is investigated. A proposal of Lyapunov functional allows determining (Linear Matrix Inequalities) LMI-type conditions to establish ultimate bounds on the system response. It is proven that under certain conditions, the nongrowth of the system energy is guaranteed.