The spectral radius of a matrix is a widely used concept in linear algebra. It expresses the asymptotic growth rate of successive powers of the matrix. This concept can be extended to sets of matrices, leading to the notion of "joint spectral radius". The joint spectral radius of a set of matrices was defined in the 1960's, but has only been used extensively since the 1990's. This concept is useful to study the behavior of multi-agent systems, to determine the continuity of wavelet basis functions or for expressing the capacity of binary codes. Although the joint spectral radius shares some properties with the usual spectral radius, it is much harder to compute, and the problem of approximating it is NP-hard. In this thesis, we first review theoretical results that lead to basic approximations of the joint spectral radius. Then, we list various specific cases where it is effectively computable, before presenting a specific type of sets of matrices, for which we solve the problem of computing it with a polynomial computational cost.

Advances in mechatronics have renewed interest in the harvesting and storage of ambient vibration energy. This work documents recent efforts to model a novel electromagnetic transducer design that is intended for use in energy harvesting. The thesis details methods of experimental characterization as well as model validation. Also presented are methods of state space and parametric modelling eforts. In addition, this thesis presents equivalent electrical circuit models with a focus on switched pulse-width-modulated topologies that seek to maximize harvested energy. / Master of Science

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

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Un grand nombre de phénomènes nous entourant peuvent être décrit par des modèles hybrides, c'est-à-dire, mettant en jeu simultanément une dynamique continu et une dynamique discrète. Également, il n'est pas rare que ces dynamiques puissent évoluer dans des échelles de temps différentes. Dans cette thèse, nous nous intéressons à l'analyse de stabilité des systèmes à commutations singulièrement perturbés à temps continu. En présence de commutations, l'analyse de stabilité des systèmes singulièrement perturbés dite "classique" (séparation des échelles de temps) n'est plus valable. En nous plaçant en dimension deux et en considérant deux modes, nous donnons une caractérisation complète du comportement asymptotique de tels systèmes lorsque le paramètre de perturbation tend vers zéro. Ensuite, nous étudions la discrétisation des systèmes à commutations singulièrement perturbés, en portant un intérêt particulier aux méthodes de discrétisation permettant de préserver la stabilité et les fonctions de Lyapunov quadratiques communes / Many phenomena we encounter can be described by hybrid models, namely, consisting of one continuous dynamic and one discret dynamic at the same time. Moreover, these dynamics often evolves in different time scales. In this thesis, we deal with the stability analysis of singularly perturbed switched systems in continuous time. When we consider switchings, the "classical" approach (decoupling fast and slow dynamics) allowing to analyse stability of singularly perturbed systems doesn't hold anymore. Considering second order singularly perturbed switched systems woth two modes, we completely characterize de stability behavior of such systems when the perturbation parameter goes to zero. Then, we study the discretization of singularly perturbed switched systems. In particular, we focus on methods allowing to preserve stability and common quadratic Lyapunov functions

Following results over recent years, this thesis enhances the problem of minimizing a cost functional defined on a state trajectory of an autonomous switched dynamical system. The cost functional traditionally used, is augmented with explicit costs on the switching times and the final time is set by a constraint as opposed to being given. An equation for the gradient of the cost functional is derived and an algorithm is proposed for computing local minima. The algorithm is based on existing steepest descent methods including the Armijo procedure and gradient projection. A matlab implementation of the algorithm is developed in order to solve optimal problems that can be modelled with costs on or between the switching times. An existing problem, the motivation for this research, where repairs on a bridge is to be optimized, is provided and solved.

<p>Sensor path planning and control refer to the problems of determining the trajectory and feedback control law that best support sensing objectives, such as monitoring, detection, classification, and tracking. Many autonomous systems developed, for example, to conduct environmental monitoring, search-and-rescue operations, demining, or surveillance, consist of a mobile vehicle instrumented with a suite of proprioceptive and exteroceptive sensors characterized by a bounded field-of-view (FOV) and a performance that is highly dependent on target and environmental conditions and, thus, on the vehicle position and orientation relative to the target and the environment. As a result, the sensor performance can be significantly improved by planning the vehicle motion and attitude in concert with the measurement sequence. This dissertation develops a general and systematic approach for deriving information-driven path planning and control methods that maximize the expected utility of the sensor measurements subject to the vehicle kinodynamic constraints.</p><p>The approach is used to develop three path planning and control methods: the information potential method (IP) for integrated path planning and control, the optimized coverage planning based on the Dirichlet process-Gaussian process (DP-GP) expected Kullback-Leibler (KL) divergence, and the optimized visibility planning for simultaneous target tracking and localization. The IP method is demonstrated on a benchmark problem, referred to as treasure hunt, in which an active vision sensor is mounted on a mobile unicycle platform and is deployed to classify stationary targets characterized by discrete random variables, in an obstacle-populated environment. In the IP method, an artificial potential function is generated from the expected conditional mutual information of the targets and is used to design a closed-loop switched controller. The information potential is also used to construct an information roadmap for escaping local minima. Theoretical analysis shows that the closed-loop robotic system is asymptotically stable and that an escaping path can be found when the robotic sensor is trapped in a local minimum. Numerical simulation results show that this method outperforms rapidly-exploring random trees and classical potential methods. The optimized coverage planning method maximizes the DP-GP expected KL divergence approximated by Monte Carlo integration in order to optimize the information value of a vision sensor deployed to track and model multiple moving targets. The variance of the KL approximation error is proven to decrease linearly with the inverse of the number of samples. This approach is demonstrated through a camera-intruder problem, in which the camera pan, tilt, and zoom variables are controlled to model multiple moving targets with unknown kinematics by nonparametric DP-GP mixture models. Numerical simulations as well as physical experiments show that the optimized coverage planning approach outperforms other applicable algorithms, such as methods based on mutual information, rule-based systems, and randomized planning. The third approach developed in this dissertation, referred to as optimized visibility motion planning, uses the output of an extended Kalman filter (EKF) algorithm to optimize the simultaneous tracking and localization performance of a robot equipped with proprioceptive and exteroceptive sensors, that is deployed to track a moving target in a global positioning system (GPS) denied environment.</p><p>Because active sensors with multiple modes can be modeled as a switched hierarchical system, the sensor path planning problem can be viewed as a hybrid optimal control problem involving both discrete and continuous state and control variables. For example, several authors have shown that a sensor with multiple modalities is a switched hybrid system that can be modeled by a hierarchical control architecture with components of mission planning, trajectory planning, and robot control. Then, the sensor performance can be represented by two Lagrangian functions, one function of the discrete state and control variables, and one function of the continuous state and control variables. Because information value functions are typically nonlinear, this dissertation also presents an adaptive dynamic programming approach for the model-free control of nonlinear switched systems (hybrid ADP), which is capable of learning the optimal continuous and discrete controllers online. The hybrid ADP approach is based on new recursive relationships derived in this dissertation and is proven to converge to the solution of the hybrid optimal control problem. Simulation results show that the hybrid ADP approach is capable of converging to the optimal controllers by minimizing the cost-to-go online based on a fully observable state vector.</p> / Dissertation