Controllability and stabilisability are two fundamental properties of control systems and it is intuitively appealing to conjecture that the former should imply the latter; especially so when the state of a control system is assumed to be known at every time instant. Such an implication can, indeed, be proven for certain types of controllability and stabilisability, and certain classes of control systems. In the present thesis, we consider real analytic control systems of the form $\Sgr:\dot{x}=f(x,u)$, with $x$ in a real analytic manifold and $u$ in a separable metric space, and we show that, under mild technical assumptions, small-time local controllability from an equilibrium $p$ of \Sgr\ implies the existence of a piecewise analytic feedback \Fscr\ that asymptotically stabilises \Sgr\ at $p$. As a corollary to this result, we show that nonlinear control systems with controllable unstable dynamics and stable uncontrollable dynamics are feedback stabilisable, extending, thus, a classical result of linear control theory. Next, we modify the proof of the existence of \Fscr\ to show stabilisability of small-time locally controllable systems in finite time, at the expense of obtaining a closed-loop system that may not be Lyapunov stable. Having established stabilisability in finite time, we proceed to prove a converse-Lyapunov theorem. If \Fscr\ is a piecewise analytic feedback that stabilises a small-time locally controllable system \mbox{$\Sgr:\dot{x}=f(x,u)$} in finite time, then the Lyapunov function we construct has the interesting property of being differentiable along every trajectory of the closed-loop system obtained by ``applying" \Fscr\ to \Sgr. We conclude this thesis with a number of open problems related to the stabilisability of nonlinear control systems, along with a number of examples from the literature that hint at potentially fruitful lines of future research in the area. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-09-24 10:24:22.51

The subject of this thesis is the motion planning algorithm known as the continuation method. To solve motion planning problems, the continuation method proceeds by lifting curves in state space to curves in control space; the lifted curves are the solutions of special initial value problems called path-lifting equations. To validate this procedure, three distinct obstructions must be overcome. The first obstruction is that the endpoint maps of the control system under study must be twice continuously differentiable. By extending a result of A. Margheri, we show that this differentiability property is satisfied by an inclusive class of time-varying fully nonlinear control systems. The second obstruction is the existence of singular controls, which are simply the singular points of a fixed endpoint map. Rather than attempting to completely characterize such controls, we demonstrate how to isolate control systems for which no controls are singular. To this end, we build on the work of S. A. Vakhrameev to obtain a necessary and sufficient condition. In particular, this result accommodates time-varying fully nonlinear control systems. The final obstruction is that the solutions of path-lifting equations may not exist globally. To study this problem, we work under the standing assumption that the control system under study is control-affine. By extending a result of Y. Chitour, we show that the question of global existence can be resolved by examining Lie bracket configurations and momentum functions. Finally, we show that if the control system under study is completely unobstructed with respect to a fixed motion planning problem, then its corresponding endpoint map is a fiber bundle. In this sense, we obtain a necessary condition for unobstructed motion planning by the continuation method. / Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2012-12-18 20:53:43.272

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.

Neste trabalho aprimorou-se um método para aproximar soluções de uma classe de equações diferenciais algébricas (DAEs), conhecida como sistemas semi-implícitos quadrados. O método, chamado aqui de MII, fundamenta-se na teoria geométrica de desacoplamento para sistemas não lineares, aliada a técnicas eficientes de análise numérica. Ele usa uma estratégia mista com cálculos simbólicos e numéricos para construir um sistema explícito, cujas soluções convergem exponencialmente para as soluções do sistema implícito original. Duas versões do método são apresentadas. Com a primeira, chamada de MIIcond, procura-se obter matrizes numericamente estáveis, através de balanceamentos. E a segunda, MIIproj, aproveita uma interpretação geométrica para o campo vetorial obtido. As implementações foram desenvolvidas em Matlab/simulink com o pacote de computação simbólica. Através dos benchmarks, realizando inclusive comparações com outros métodos atualmente disponíveis, constatou-se que o MIIcond foi inviável em alguns casos, devido ao tempo de processamento muito extenso. Por outro lado, o MIIproj mostrou-se uma boa alternativa para esta classe de problemas, em especial para sistemas de alto índex. / This work improves a method to approximate solutions for a class of differential algebraic equations (DAEs), known as systems semi-implicit square. The method, called here MII, is based on geometric theory of decoupling for nonlinear systems combined with efficient techniques numerical analysis. It uses an algorithum that mixes symbolic and numerical calculations to build an explicit system, whose solutions converge exponentially to solutions of the original implicit system. Two versions of the method are given. The first one is called MIIcond, trying to obtain numerically stable matrices through balancing. The second one is the MIIproj, taking advantage of a geometricinterpretation of the vector field there obtained. The implementations were developed in Matlab/Simulink with the symbolic toolbox. Through benchmarks, including performing comparisons with other methods currently available, it was found that the MIIcond was not feasible in some cases, due to processing time too long. On the other hand, the MIIproj presented itself as good alternative to this class of problems, especially for systems of high index.

This master’s thesis describes equations of motion for dynamic model of nonholonomic constrained system, namely the trident robotic snakes. The model is studied in the form of Lagrange's equations and D’Alembert’s principle is applied. Actually this thesis is a continuation of the study going at VUT about the simulations of non-holonomic mechanisms, specifically robotic snakes. The kinematics model was well-examined in the work of of Byrtus, Roman and Vechetová, Jana. So here we provide equations of motion and address the motion planning problem regarding dynamics of the trident snake equipped with active joints through basic examples and propose a feedback linearization algorithm.

Tato práce se zabývá simulacemi neholonomních mechanismů, konkrétně robotických hadů. V práci jsou uvedeny základní poznatky geometrické teorie řízení. Tyto poznatky jsou využity k odvození řídících modelů robotických systémů a následně jsou tyto modely simulovány v prostředí V-REP.

<p>Cyber-physical systems (CPS) are a class of complicated systems integrating cyber components with physical components. Although such a cyber-physical interaction improves the system performance and intelligence, it increases the system complexity and makes the system vulnerable to various types of faults, failures, and cyber-attacks. To assure the security and improve the resiliency of CPS, it is found that the hybrid system model can be a powerful tool in the domain of fault detection and isolation, cyber-attack diagnosis and containment, as well as resilient control and reconfiguration. Several problems are concerned in this dissertation. For situational awareness, \textit{mode discernibility}, which stands for whether the discrete state of a hybrid system can be correctly identified, is characterized and discussed with potential applications to monitoring system design. For CPS vulnerability analysis, the problem of stealthy attack design for systems with switching structures is investigated, which is motivated by the recent literature. To further understand and remedy for the vulnerabilities, the detectability and identifiability for severe cyber-attacks are defined and characterized, which are followed by the discussions on the methodologies for cyber-attack detection and identification. Last but not least, based on the understanding of identifiability, a framework of resilient control design is proposed to mitigate the impact of cyber-attacks, which can be generalized in future to account for additional design criteria.</p>

Cette thèse est divisée en trois parties. Dans la première partie, nous commençons par décrire des résultats très connus en théorie du contrôle géométrique tels que le théorème de Chow-Rashevsky, la condition de rang de Kalman, l'application Entrée-Sortie et le test linéaire. De plus, nous définissons et nous étudions brièvement la contrôlabilité locale au voisinage d'un contrôle de référence au premier et au second ordre. Dans la deuxième partie, nous donnons une preuve élémentaire du lemme de Franks linéaire pour les flots géodésiques qui utilise des techniques basiques de théorie du contrôle géométrique. Dans la dernière partie, étant donnée une variété Riemanienne compacte, nous prouvons un lemme de Franks uniforme au second ordre pour les flots géodésiques et on applique le résultat à la théorie de la persistance. Dans cette partie, nous introduisons avec plus de détails les notions de contrôlabilité locale au premier et au second ordre. En effet, nous donnons un résultat de contrôlabilité au second ordre dont la preuve est longue et technique. / This thesis is devided into three parts. In the first part we begin by describing some well known results in geometric control theory such as the Chow Rashevsky Theorem, the Kalman rank condition, the End-Point Mapping and the linear test. Moreover, we define and study briefly local controllability around a reference control at first and second order. In the second part we provide an elementary proof of the Franks lemma for geodesic flows using basic tools of geometric control theory. In the last part, given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory. In this part we introduce with more details notions of local controllability at first and second order. In fact, we provide a second order controllability result whose proof is long and technical.

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático. / The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.

As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático. / The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin\'s Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon, providing criteria to detect its presence at junctions with an algebraic notation. We present a numerical approach to treat optimal control problems and we conclude with an application of the results in the bidimesional model of autonomous underwater vehicle.