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## Motion on lie groups and its applications in control theory

Cariñena, José F.; Clemente-Gallardo, Jesús; Ramos, Arturo
Tipo: Artigo de Revista Científica Formato: aplication/PDF
ENG
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The usefulness in control theory of the geometric theory of motion on Lie groups and homogeneous spaces will be shown. We quickly review some recent results concerning two methods to deal with these systems, namely, a generalization of the method proposed by Wei and Norman for linear systems, and a reduction procedure. This last method allows us to reduce the equation on a Lie group G to that on a subgroup H, provided a particular solution of an associated problem in G/H is known. These methods are shown to be very appropriate to deal with control systems on Lie groups and homogeneous spaces, through the specific examples of the planar rigid body with two oscillators and the front-wheel driven kinematic car.; http://www.sciencedirect.com/science/article/B6VN0-49F836Y-3/1/3e135eb33cca05a026b85455b5544308

## Fenômeno Fuller em problemas de controle ótimo: trajetórias em tempo mínino de veículos autônomos subaquáticos; Fuller Phenomenon in optimal control problems: minimum time path of autonomous underwater vehicles.

Oda, Eduardo
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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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...

## Formas triangulares para sistemas não-lineares com duas entradas e controle de sistemas sem arrasto em SU(n) com aplicações em mecânica quântica.; Triangular forms for nonlinear systems with two inputs and control of driftless systems on SU(n) with applications in quantum mechanics.

Silveira, Hector Bessa
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
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A presente tese aborda dois problemas distintos e independentes: triangularização de sistemas não-lineares com duas entradas e controle de sistemas sem arrasto que evoluem no grupo especial unitário SU(n). Em relação ao primeiro, estabeleceu-se, através da generalização de resultados bem conhecidos, condições geométricas para que um sistema com duas entradas seja descrito por uma forma triangular específica após uma mudança de coordenadas e uma realimentação de estado estática regular. Para o segundo problema, desenvolveu-se uma estratégia de controle que força o estado do sistema a rastrear assintoticamente uma trajetória de referência periódica que passa por um estado objetivo arbitrário. O método de controle proposto utiliza os resultados de convergência de tipo- Lyapunov que foram estabelecidos pela presente pesquisa e que tiveram como inspiração uma versão periódica do princípio da invariância de LaSalle. Apresentou-se, ainda, os resultados de simulação obtidos com a aplicação da técnica de controle desenvolvida a um sistema quântico consistindo de duas partículas de spin-1/2, com o objetivo de gerar a porta lógica quântica C-NOT.; This thesis treats two distinct and independent problems: triangularization of nonlinear systems with two inputs and control of driftless systems which evolve on the special unitary group SU(n). Concerning the first...

## Integração numérica de sistemas não lineares semi-implícitos via teoria de controle geométrico; Numerical integration of non-linear semi-implicit square systems via geometric control theory.

Freitas, Celso Bernardo da Nobrega de
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Relevância na Pesquisa
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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)...

## Feedback Stabilisation of Locally Controllable Systems

Isaiah, Pantelis
Fonte: Quens University Publicador: Quens University
EN; EN
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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...

## Applications of Lie systems in Quantum Mechanics and Control Theory

Cariñena, José F.; Ramos, Arturo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems. We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the involved Lie group by a central extension of it. The geometric techniques developed for dealing with Lie systems are also used in problems of control theory. Specifically, we will study some examples of control systems on Lie groups and homogeneous spaces.; Comment: LaTeX, 28 pages

## Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces

Jouan, Philippe
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which have its own interest. The present proof makes use of geometric control theory arguments.

## Uniformly hyperbolic control theory

Kawan, Christoph
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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This paper gives a summary of a body of work at the intersection of control theory and smooth nonlinear dynamics. The main idea is to transfer the concept of uniform hyperbolicity, central to the theory of smooth dynamical systems, to control-affine systems. Combining the strength of geometric control theory and the hyperbolic theory of dynamical systems, it is possible to deduce control-theoretic results of non-local nature that reveal remarkable analogies to the classical hyperbolic theory of dynamical systems.

## Mathematical models for geometric control theory

Jafarpour, Saber; Lewis, Andrew D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
75.89%
Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory, leading to definitions, methodologies, and results that depend in unexpected ways on control parameterisation. In this paper a framework is presented for modelling systems in geometric control theory in a manner that does not make any choice of parameterisation by control; the systems are called "tautological control systems." For the framework to be coherent, it relies in a fundamental way on topologies for spaces of vector fields. As such, classes of systems are considered possessing a variety of degrees of regularity: finitely differentiable; Lipschitz; smooth; real analytic. In each case, explicit geometric seminorms are provided for the topologies of spaces of vector fields that enable straightforward descriptions of time-varying vector fields and control systems. As part of the development, theorems are proved for regular (including real analytic) dependence on initial conditions of flows of vector fields depending measurably on time. Classes of "ordinary" control systems are characterised that interact with the regularity under consideration in a comprehensive way. In this framework...

## Infinite horizon control and minimax observer design for linear DAEs

Zhuk, Sergiy; Petreczky, Mihaly
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.65%
In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, in fact, the worst-case estimation error. Using geometric control theory, we construct an optimal control in the feed-back form and represent it as an output of a stable LTI system. The latter gives the minimax state estimator. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAEs.; Comment: This is an extended version of the paper which is to appear in the proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10-13, 2013

## A geometric control proof of linear Franks' lemma for geodesic flows

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We provide an elementary proof of the Franks lemma for geodesic flows that uses basic tools of geometric control theory.; Comment: 14 pages, 2 figures

## Discrete Control Systems

Lee, Taeyoung; Leok, Melvin; McClamroch, N. Harris
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.96%
Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control system aims to address this deficiency by restricting the approximation to choice of a discrete-time model...

## Solutions of differential-algebraic equations as outputs of LTI systems: application to LQ control problem

Petreczky, Mihaly; Zhuk, Sergiy
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper we synthesize behavioral ideas with geometric control theory and propose a unified geometric framework for representing all solutions of a Linear Time Invariant Differential-Algebraic Equation (DAE-LTI) as outputs of classical Linear Time Invariant systems (ODE-LTI). An algorithm for computing an ODE-LTI that generates solutions of a given DAE-LTI is described. It is shown that two different ODE-LTIs which represent the same DAE-LTI are feedback equivalent. The proposed framework is then used to solve an LQ optimal control problem for DAE-LTIs with rectangular matrices.; Comment: The main difference with respect to the previous version is that the supplementary files were included which were missing from the previous version. Note that part of the material of this report appeared in arXiv:1309.1235. A version of this paper was submitted to Automatica, first in January 2014 and then in November 2014, and then in November 2015

## Motion planning and control problems for underactuated robots

Martinez, Sonia; Cortes, Jorge; Bullo, Francesco
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Motion planning and control are key problems in a collection of robotic applications including the design of autonomous agile vehicles and of minimalist manipulators. These problems can be accurately formalized within the language of affine connections and of geometric control theory. In this paper we overview recent results on kinematic controllability and on oscillatory controls. Furthermore, we discuss theoretical and practical open problems as well as we suggest control theoretical approaches to them.; Comment: 16 pages, 5 figures, to appear as a book chapter in the Advanced Robotics Series, Springer-Verlag

## Geometric control theory I: mathematical foundations

Massa, Enrico; Bruno, Danilo; Pagani, Enrico
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
65.79%
A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on geometrical grounds, via the introduction of the concept of infinitesimal control . On the same basis, the usual classification of the extremals of a variational problem into normal and abnormal ones is also rationalized, showing the existence of a purely kinematical algorithm assigning to each admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. The whole machinery is then applied to constrained variational calculus. The argument provides an interesting revisitation of Pontryagin maximum principle and of the Erdmann-Weierstrass corner conditions, as well as a proof of the classical Lagrange multipliers method and a local interpretation of Pontryagin's equations as dynamical equations for a free (singular) Hamiltonian system. As a final, highly non-trivial topic, a sufficient condition for the existence of finite deformations with fixed endpoints is explicitly stated and proved.; Comment: replaced by the more recent article arXiv:1503.08808

## Geometric reduction in optimal control theory with symmetries

Echeverría-Enríquez, A.; Marín-Solano, J.; Muñoz-Lecanda, M. C.; Román-Roy, N.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in the regular and singular cases, which are previously described.; Comment: 24 pages. LaTeX file. The paper has been reorganized. Additional comments have been included in Section 3. The example in Section 5.2 has been revisited. Some references have been added

## On the locomotion and control of a self-propelled shape-changing body in a fluid

Chambrion, Thomas; Munnier, Alexandre
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper we study the locomotion of a shape-changing body swimming in a two-dimensional perfect fluid of infinite extent. The shape-changes are prescribed as functions of time satisfying constraints ensuring that they result from the work of internal forces only: conditions necessary for the locomotion to be termed self-propelled. The net rigid motion of the body results from the exchange of momentum between these shape-changes and the surrounding fluid. The aim of this paper is several folds: First, it contains a rigorous frame- work for the study of animal locomotion in fluid. Our model differs from previous ones mostly in that the number of degrees of freedom related to the shape-changes is infinite. . Second, we are interested in making clear the connection between shape- changes and internal forces. We prove that, when the number of degrees of freedom relating to the shape-changes is finite, both choices are actually equivalent in the sense that there is a one-to-one relation between shape-changes and internal forces. Third, we show how the control problem consisting in associating to each shape-change the resulting trajectory of the swimming body can be suitably treated in the frame of geometric control theory. For any given shape-changes producing a net displacement in the fluid (say...

## Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze's and Helmke's compactification arises naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze's compactification as a Quot scheme is given, whereas Helmke's compactification is shown to be an algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric description of both compactifications. As an application, we determine the cohomology ring of Helmke's compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive.; Comment: 24 pages, based on my Diplomarbeit completed in February 2005, to appear in Linear Algebra and its Applications (LAA)

## Geometric Control Methods for Quantum Computations

Giunashvili, Zakaria
Tipo: Artigo de Revista Científica