Nonsmooth dynamical systems : on stability of hybrid trajectories and bifurcations of discontinuous systems

Abstract

Nonsmooth and hybrid dynamical systems can represent complex behaviour occurring in engineer- ing, physics, biology, and economy. The benets of nonsmooth and hybrid systems are two-fold. Firstly, certain phenomena are inherently nonsmooth, such as switching control actions. Secondly, nonsmooth systems can represent dynamical behaviour at desirable abstraction levels, for example, rigid body models with impulsive contact laws can describe the motion of unilaterally constrained mechanical systems in an ecient manner. The goal of this research is to make contributions to the theory of stability and robustness for nonsmooth systems and therewith support the analysis, design and control of such systems. The contributions are structured in two parts. In the rst part of this thesis, the robustness of dynamical systems is studied for the class of non-dierentiable and discontinuous dierential equations. Small parameter variations can induce complex bifurcations in two-dimensional dierential equations with non-dierentiable right-hand sides. In this thesis, bifurcations are studied where multiple equilibria and periodic solutions are created from an isolated equilibrium point. We present a procedure to identify all limit sets that can be created or destroyed by a bifurcation. This procedure is an important tool to assess what behaviour can occur in planar non-dierentiable dynamical models. In addition to non-dierentiable systems, a class of discontinuous systems is studied, which exhibits even more diverse behaviour. We focus on bifurcations of discontinuous systems describing mechanical systems with dry friction acting in one interface. We show that equilibria of these systems are non-isolated and form a line interval in the state space. Parameter variations can only induce bifurcations by changing the dynamics near the endpoints of this line interval. Using this insight, we present sucient conditions for structural stability of the vector eld near the equilibrium set and identify various bifurcations. These results make it possible to assess the robustness to parameter variation in the modelling and control of mechanical systems with friction. Furthermore, time-varying force perturbations are shown to induce unexpected behaviour in this type of systems. A new type of chaotic limit set is discovered that is created by the perturbation of a discontinuous system that has a homoclinic solution. The forward dynamics of trajectories in this limit set is shown to be qualitatively dierent from the reversed-time dynamics. Since many trajectories spend a long transient time in the neighbourhood of this limit set, the properties of this limit set will strongly aect the behaviour of the dynamical system under study. In the second part of this thesis, the stability of jumping trajectories is studied for hybrid systems, in which solutions display both continuous-time evolution and jumps (i.e. discrete events). This research is motivated by tracking control problems for hybrid systems, e.g. for mechanical systems with unilateral contacts and impacts. A major complication in the analysis of the stability of trajectories in hybrid systems is that, in general, the jump times of two trajectories do not coincide. Consequently, the conventional Euclidean distance between two trajectories does not converge to zero, even if the trajectories converge to each other in between jumps, and the jump time mismatch tends to zero. We propose a novel stability formulation that overcomes this problem by comparing the jumping trajectories using a new distance function. This leads to sucient conditions for asymptotic stability of hybrid system trajectories. In two examples, we show that these conditions support the design of tracking controllers that achieve desired behaviour in hybrid systems. The contributions of this thesis lead to a better understanding of nonsmooth dynamics, such that the advantages of nonsmooth system models can be more eectively exploited.