Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions

Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In the first part of the paper the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result

Contraction analysis of switched systems with application to control and observer design

In many control problems, such as tracking and regulation, observer design, coordination and synchronization, it is more natural to describe the stability problem in terms of the asymptotic convergence of trajectories with respect to one another, a property known as incremental stability. Contraction analysis exploits the stability properties of the linearized dynamics to infer incremental stability properties of nonlinear systems. However, results available in the literature do not fully encompass the case of switched dynamical systems. To overcome these limitations, in this thesis we present a novel extension of contraction analysis to such systems based on matrix measures and differential Lyapunov functions. The analysis is conducted first regularizing the system, i.e. approximating it with a smooth dynamical system, and then applying standard contraction results. Based on our new conditions, we present design procedures to synthesize switching control inputs to incrementally stabilize a class of smooth nonlinear systems, and to design state observers for a large class of nonlinear switched systems including those exhibiting sliding motion. In addition, as further work, we present new conditions for the onset of synchronization and consensus patterns in complex networks. Specifically, we show that if network nodes exhibit some symmetry and if the network topology is properly balanced by an appropriate designed communication protocol, then symmetry of the nodes can be exploited to achieve a synchronization/consensus pattern

Controle híbrido para estabilização de pose usando quaternions duais

Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2018.Motivado tanto pelas vantagens da representação em dual quatérnios duais e por problemas relativos à obstrução topológica de se ter um equilíbrio assintótico globalmente estável, esse trabalho visa usar o formalismo de quaternion dual e as ferramentas de sistemas dinâmicos híbridos para tratar o problema de estabilização de pose de corpos rígidos. O grupo de Lie dos quatérnios duais proporciona um modo eficiente de representar a cinemática linear e rotacional de um corpo rígido sem singularidades. Algumas estratégias híbridas são propostas para lidar com o problema de “chattering” presente em todos os controladores por realimentação descontínuos enquanto ao mesmo tempo garantindo atratividade global da pose de estabilização do corpo rígido.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Fundação de Apoio à Pesquisa do Distrito Federal (FAP-DF).Motivated both by the advantages of the dual quaternion representation and by the problems concerning the topological obstruction to global asymptotic stability, this work addresses the rigid body pose stabilization problem using dual quaternion formalism and dynamic hybrid systems tools. The Lie group of unit dual quaternions provides a computationally efficient way to represent coupled linear and rotational kinematics without singularities. Some hybrid control strategies are proposed to overcome the chattering problem present in all discontinuous-based feedback controllers while at same time also guaranteeing global attractivity of the stabilization pose of the rigid body