An interval approach for stability analysis

The stability problem plays an important role not only in industry but also in our everyday life. Unstable systems are dangerous and can cause financial losses or can lead to injuries and loss of human life. Starting with the 19th century, many methods were proposed for analysing the stability of linear dynamical systems, nonlinear dynamical systems, and for systems with uncertainties. This thesis proposes several methods to study the stability of nonlinear dynamic systems with uncertainties. The majority of existing methods can only be applied to nonlinear dynamical systems under some particular circumstances. New approaches are developed in this thesis by combining the existing theory with interval analysis. The new methods are used to analyse the stability of Sliding Mode Control (SMC) strategy, for generating automatic Lyapunov functions and for solving Ordinary Differential Equations (ODE) and ODE with uncertainties. New types of sliding mode control structures, able to cope with faults, are developed and tested on a differential drive robot. Suitable tuning parameters for the SMC when dealing with multiple uncertainties are found by using novel methods which employ interval analysis techniques. To the best of my knowledge, there is no analytical method for tuning and estimating the robustness for an SMC in the presence of uncertainties. Moreover, the stability of uncertain nonlinear dynamical systems was studied from a Lyapunov perspective. Lyapunov functions can be automatically generated for mechanical systems based on their energy. In the majority of the cases, the energy function does not guarantee an asymptotic convergence to the equilibrium point. LaSalle theorem solved this problem for systems without uncertainties. However, when the systems have uncertainty it is not possible to apply LaSalle invariant principle. In this thesis, this was solved by adding perturbation functions to the energy function of the system, and by using interval analysis, the asymptotic convergence for systems with uncertainties was proved. New guaranteed integration methods are proposed for solving nonlinear ODEs in the last part of the thesis. It was shown, that by using interval methods, we are able to bracket the boundary of the solution manifold. Two methods for generating a boundary approximation were proposed: the first method used radial basis function networks and the second method used linear interpolation for a mesh triangulation.