By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. This article surveys recent developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.

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An existing hybrid control strategy is extended and applied to the control of a novel courier robot based on planar (Sawyer) motor technology augmented with integral sensing. The controller is designed to accomodate actuator limits and the desired motion characteristics, eliminating the need for careful matching of trajectory parameters with controller gains. Controller extensions allow for the unusual actuator constraints of the planar motors and independent adjustment of the parameters for each axis. Simulation results are presented for several cases to demonstrate the diversity of situations within the capability of the controller. 1

This thesis considers the control of multiple model systems. These are systems for which only one model out of some finite set of models gives the system dynamics at any given time. In particular, the model that gives the system dynamics can change over time. This thesis covers some of the theoretical aspects of these systems, including controllability and stabilizability. As an application, "overconstrained " mechanical systems are modeled as multiple model systems. Examples of such systems include distributed manipulation problems such as microelectromechanical systems and many wheeled vehicles such as the Sojourner vehicle of the Mars Pathfinder mission. Such systems are typified by having more Pfa#an constraints than degrees of freedom. Conventional classical motion planning and control theories do not directly apply to overconstrained systems. Control issues for two examples are specifically addressed. The first example is distributed manipulation. Distributed manipulation systems control an object's motion through contact with a high number of actuators. Stability results are shown for such systems and control schemes based on these results are implemented on a distributed manipulation test-bed. The second example is that of overconstrained vehicles, of which the Mars rover is an example. The nonlinear controllability test for multiple model systems is used to answer whether a kinematic model of the rover is or is not controllable.

ii Animal legs are much more complex than necessary for running on horizontal surfaces. Past research shows that, despite their complexity, they act essentially like simple pogo sticks or “Spring Loaded Inverted Pendulums ” (SLIP). Only recently was it discovered that SLIP possesses some degree of passive stability. In this research, the possibility of the realization of the SLIP model is investigated first in numerical simulation. Next, a small one-legged hopping robot with only one actuator is designed and built, and a simple hopping controller is implemented, which results in running with approximately 0.80 m/s (6.7 leg lengths per second). Comparison between simulation and experimental data is undertaken, and the stability of the resulting motion is investigated. ii Résumé Les pattes animales sont beaucoup plus complexes que ce qui est nécessaire pour courir sur des surfaces horizontales. En effet, des recherches passées montrent que malgré leur complexité, elles agissent essentiellement comme de simples « pogo sticks » ou « Spring Loaded Inverted Pendulums » (SLIP). Il a été découvert récemment que le SLIP possède un certain degré de stabilité passive. La présente recherche étudie tout d’abord la possibilité de réalisation du SLIP à l’aide de simulations numériques. Ensuite, un petit robot à une patte et un actuateur est développé et conçu, et un contrôleur de bonds est implémenté. Avec ce contrôleur, le robot peut atteindre une vitesse de 0.80 m/s, ce qui correspond à 6.7 longueurs de patte par seconde. Finalement, les résultats simulés et expérimentaux sont comparés et la stabilité du mouvement du robot est étudiée. iii Acknowledgments I’m happy that I have studied in the Ambulatory Robotics Laboratory and studied at McGill University and lived in Montreal. First of all, I thank the people who gave me the chance: � My family: I thank for their understanding and support. � Prof. Ken Tomiyama, ex-supervisor: He encouraged my plan to study abroad.

This paper develops a local controllability result for Multiple Model Driftless Affine (MMDA) control systems. The controllability result can be interpreted as a non-smooth extension of Chow's theorem, and uses a set-valued Lie Bracket. These results are interpreted in terms of an illustrative example involving an overconstrained wheeled vehicle.

Hybrid and switched dynamic systems are of major research interest nowadays due to their use as models in many applications in computer science and systems control. One of the major problems in hybrid and switched dynamic systems is establishing their key property of stability, which also is important in controller design. Stability may prove also critical for real-time systems, embedded systems, and hybrid systems in general that arise in computer science problems where verification tests are undecidable. Relaxing demands in the search for a stability proof may be necessary for a specific problem. In this paper, we present a brief survey of stability results for hybrid and switched systems while taking into consideration the used model and the technique used for establishing stability. Emphasis is also given in results that take into consideration the interaction of continuous-time systems with the discrete-event part.

We analyze a class of mechanisms that locomote by switching between constraints. Because of the hybrid nature of such systems, most of the existing analysis tools, developed primarily for smooth systems, can not be directly applied. Our aim is to exploit the special structure provided by Lagrangian mechanics to study the controllability of this class of mechanisms. We base the analysis on a series representation of the evolution of the system. Our main result is a description of trajectories involving switches between constraints at nonzero velocity (impacts) in the presence of large inertial forces (drift). The analysis provides a basis for local motion planning. The results are applied to an example of a two-link planar mechanism that can locomote by clamping one of the links.

This paper develops a local controllability result for Multiple Model Driftless Affine (MMDA) control systems. The controllability result can be interpreted as a non-smooth extension of Chow's theorem, and uses a set-valued Lie Bracket. These results are interpreted in terms of an illustrative example involving an overconstrained wheeled vehicle.

This paper extends recent results in local controllability analysis for Multiple Model Driftless A#ne (MMDA) control systems. Such controllability results can be interpreted as non-smooth extensions of Chow's theorem, and use a set-valued Lie Bracket. In particular, we formulate controllability in terms of generalized di#erential quotients (GDQs). Additionally, we present an extensive example in order to illustrate how these results can provide insight into the control of some specific physical systems. Moreover, this paper indicates that a multiple model system consisting of individually controllable models is not necessarily controllable.