Rhythmic motion is ubiquitous in nature and technology. Various motions of organisms like the heart beating and walking require stable periodic execution. The stability of the rhythmic execution of human movement can be altered by neurological or orthopedic impairment. In robotics, successful development of legged robots heavily depends on the stability of the controlled limit-cycle. An accurate measure of the stability of rhythmic execution is critical to the diagnosis of several performed tasks like walking in human locomotion. Floquet multipliers have been widely used to assess the stability of a periodic motion. The conventional approach to extract the Floquet multipliers from actual data depends on the least squares method. We devise a new way to measure the Floquet multipliers with reduced bias and estimate orbital stability more accurately. We show that the conventional measure of the orbital stability has bias in the presence of noise, which is inevitable in every experiment and observation. Compared with previous method, the new method substantially reduces the bias, providing acceptable estimate of the orbital stability with fewer cycles even with different noise distributions or higher or lower noise levels. The new method can provide an unbiased estimate of orbital stability within a reasonably small number of cycles. This is important for experiments with human subjects or clinical evaluation of patients that require effective assessment of locomotor stability in planning rehabilitation programs. / Graduate / 2018-11-22

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The main problem evaluated in this manuscript is the stabilization of periodic orbits of non-linear dynamical systems by use of feedback control. The goal of the control methods proposed in this work is to achieve a stable periodic oscillation. These control methods are applied to systems that present unstable periodic orbits in the state space, and the latter are the orbits to be stabilized.The methods proposed here are such that the resulting stable oscillation is obtained with low control effort, and the control signal is designed to converge to zero when the trajectory tends to the stabilized orbit. Local stability of the periodic orbits is analyzed by studying the stability of some linear time-periodic systems, using the Floquet stability theory. These linear systems are obtained by linearizing the trajectories in the vicinity of the periodic orbits.The control methods used for stabilization of periodic orbits here are the proportional feedback control, the delayed feedback control and the prediction-based feedback control. These methods are applied to discrete and continuous-time systems with the necessary modifications. The main contributions of the thesis are related to these methods, proposing an alternative control gain design, a new control law and related results.

Le problème principalement étudié dans ce manuscrit est la stabilisation d’orbites périodiques de systèmes dynamiques non linéaires à l’aide d’une commande de rétroaction (feedback). Le but des méthodes de contrôle proposées ici est d’obtenir une oscillation périodique stable. Ces méthodes de contrôle sont appliquées à des systèmes présentant des orbites périodiques instables dans l’espace d’état, et ces dernières sont les orbites destinées à être stabilisées.Les méthodes proposées ici sont telles que l’oscillation stable qui en résulte est obtenue avec un effort de contrôle faible, et que la valeur de la commande tend vers zéro lorsque la trajectoire tend vers l’orbite stabilisée. La stabilité locale des orbites périodiques est analysée par l’étude de la stabilité des systèmes linéaires périodiques à l’aide de la théorie de Floquet. Ces systèmes linéaires sont obtenus par linéarisation des trajectoires au voisinage de l’orbite périodique.Les méthodes de contrôle utilisées ici pour la stabilisation des orbites périodiques sont une loi de commande proportionnelle, une loi de commande de rétroaction retardée et une loi de commande de rétroaction basée sur une prédiction. Ces méthodes sont appliquées aux systèmes en temps discret et aux systèmes en temps continu avec les modifications nécessaires. Les contributions principales de cette thèse sont associées à ces méthodes, proposant une méthode alternative de design de gain, une nouvelle loi de commande et des résultats associés. / The main problem evaluated in this manuscript is the stabilization of periodic orbits of non-linear dynamical systems by use of feedback control. The goal of the control methods proposed in this work is to achieve a stable periodic oscillation. These control methods are applied to systems that present unstable periodic orbits in the state space, and the latter are the orbits to be stabilized.The methods proposed here are such that the resulting stable oscillation is obtained with low control effort, and the control signal is designed to converge to zero when the trajectory tends to the stabilized orbit. Local stability of the periodic orbits is analyzed by studying the stability of some linear time-periodic systems, using the Floquet stability theory. These linear systems are obtained by linearizing the trajectories in the vicinity of the periodic orbits.The control methods used for stabilization of periodic orbits here are the proportional feedback control, the delayed feedback control and the prediction-based feedback control. These methods are applied to discrete and continuous-time systems with the necessary modifications. The main contributions of the thesis are related to these methods, proposing an alternative control gain design, a new control law and related results. / O principal problema avaliado neste manuscrito é a estabilização de órbitas periódicas em sistemas dinâmicos não-lineares utilizando controle por realimentação. O objetivo dos métodos de controle propostos neste trabalho é obter uma oscilação periódica estável. Estes métodos de controle são aplicados a sistemas que apresentam órbitas periódicas instáveis no espaço de estados, estas são as órbitas a serem estabilizadas.Os métodos propostos aqui são tais que a oscilação periódica estável resultante é obtida utilizando um baixo esforço de controle, e o sinal de controle é projetado de forma a convergir para zero quanto a trajetória tende à órbita estabilizada. A estabilidade local de órbitas periódicas é analisada através do estudo da estabilidade de alguns sistemas lineares periódicos no tempo, utilizando a teoria de estabilidade de Floquet. Estes sistemas lineares são obtidos por linearização das trajetórias na vizinhança da órbita periódica.Os métodos de controle utilizados aqui para estabilização de órbitas periódicas são o proportional feedback control, o delayed feedback control e o prediction-based feedback control (controle por realimentação baseado em predição). Estes métodos são aplicados a sistemas de tempo discreto e de tempo contínuo, com as modificações necessárias. As principais contribuições da tese são relacionadas a esses métodos, propondo um projeto de ganho de controle alternativo, uma nova lei de controle e resultados relacionados.