The distance to uncontrollability of a controllable linear system is a measure of the degree of perturbation a system can undergo and remain controllable. The definition of the distance to uncontrollability leads to a non-convex optimization problem in two variables. In 2000 Gu proposed the first polynomial time algorithm to compute this distance. This algorithm relies heavily on efficient eigenvalue solvers. In this work we examine two alternative algorithms that result in linear matrix inequalities. For the first algorithm, proposed by Ebihara et. al., a semidefinite programming problem is derived via the Kalman-Yakubovich-Popov (KYP) lemma. The dual formulation is also considered and leads to rank conditions for exactness verification of the approximation. For the second algorithm, by Dumitrescu, Şicleru and Ştefan, a semidefinite programming problem is derived using a sum-of-squares relaxation of an associated matrix-polynomial and the associated Gram matrix parameterization. In both cases the optimization problems are solved using primal-dual-interior point methods that retain positive semidefiniteness at each iteration. Numerical results are presented to compare the three algorithms for a number of benchmark examples. In addition, we also consider a system that results from a finite element discretization of the one-dimensional advection-diffusion equation. Here our objective is to test these algorithms for larger problems that originate in PDE-control. / Master of Science

<p>Methodologies for the analysis and computer simulations of active optimal vibration control of complex elastic structures are considered. The structures, generally represented by a large number of degrees of freedom (DOF), are to be controlled by a comparatively small number of actuators.</p><p>Various techniques presently available to solve the optimal control problems are briefly discussed. A Parametric optimization technique that is versatile enough to solve almost any type of optimization problems is found to give poor accuracy and is time consuming. More promising is the optimality equations approach, which is based on Pontryagins principle. Several new numerical procedures are developed using this approach. Most of the problems in this thesis are analysed in the modal space. Even complex structures can be approximated accurately in the modal space by using only few modes. Different techniques have been first applied to the cases where the number of modes to control was the same as the number of actuators (determined optimal control problems), then to cases in which the number of modes to control is larger than the number of actuators (overdetermined optimal control problems). </p><p>The determined optimal control problems can be solved by applying the Independent Modal Space Control (IMSC) approach. Such an approach is implemented in the Beam Analogy (BA) method that solves the problem numerically by applying the Finite Element Method (FEM). The BA, which uses the ANSYS program, is numerically very efficient. The effects of particular optimization parameters involved in BA are discussed in detail. Unsuccessful attempts have been made to modify this method in order to make it applicable for solving overdetermined or underactuated problems. </p><p>Instead, a new methodology is proposed that uses modified optimality equations. The modifications are due to the extra constraints present in the overdetermined problems. These constraints are handled by time dependent Lagrange multipliers. The modified optimality equations are solved by using symbolic differential operators. The corresponding procedure uses the MAPLE programming, which solves overdetermined problems effectively despite of the high order of differential equations involved.</p><p>The new methodology is also applied to the closed loop control problems, in which constant optimal gains are determined without using Riccatis equations.</p>

<p>Methodologies for the analysis and computer simulations of active optimal vibration control of complex elastic structures are considered. The structures, generally represented by a large number of degrees of freedom (DOF), are to be controlled by a comparatively small number of actuators.</p><p>Various techniques presently available to solve the optimal control problems are briefly discussed. A Parametric optimization technique that is versatile enough to solve almost any type of optimization problems is found to give poor accuracy and is time consuming. More promising is the optimality equations approach, which is based on Pontryagins principle. Several new numerical procedures are developed using this approach. Most of the problems in this thesis are analysed in the modal space. Even complex structures can be approximated accurately in the modal space by using only few modes. Different techniques have been first applied to the cases where the number of modes to control was the same as the number of actuators (determined optimal control problems), then to cases in which the number of modes to control is larger than the number of actuators (overdetermined optimal control problems). </p><p>The determined optimal control problems can be solved by applying the Independent Modal Space Control (IMSC) approach. Such an approach is implemented in the Beam Analogy (BA) method that solves the problem numerically by applying the Finite Element Method (FEM). The BA, which uses the ANSYS program, is numerically very efficient. The effects of particular optimization parameters involved in BA are discussed in detail. Unsuccessful attempts have been made to modify this method in order to make it applicable for solving overdetermined or underactuated problems. </p><p>Instead, a new methodology is proposed that uses modified optimality equations. The modifications are due to the extra constraints present in the overdetermined problems. These constraints are handled by time dependent Lagrange multipliers. The modified optimality equations are solved by using symbolic differential operators. The corresponding procedure uses the MAPLE programming, which solves overdetermined problems effectively despite of the high order of differential equations involved.</p><p>The new methodology is also applied to the closed loop control problems, in which constant optimal gains are determined without using Riccatis equations.</p>

Orientador: Sandra Augusta Santos / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T08:38:37Z (GMT). No. of bitstreams: 1 Pereira_RicardoAlexandreAlves_M.pdf: 1562416 bytes, checksum: 5b626a8b83a97556be648d6eef873195 (MD5) Previous issue date: 2010 / Resumo: Este trabalho teve por objetivo produzir um texto didático que auxiliasse no estudo e na compreensão dos Problemas de Programação Linear (PPL). Procuramos diminuir o "degrau" que existe entre o Cálculo, a Geometria e a Álgebra Linear no tratamento desses problemas, utilizando uma linguagem clara e objetiva. Dessa forma, fizemos apenas as demonstrações dos resultados que julgamos essenciais. Trabalhamos com os principais conceitos e definições que envolvem os PPL (otimização, vetor gradiente, derivada direcional, máximos e mínimos sobre conjunto compacto, Multiplicadores de Lagrange, espaço de exigência, solução ótima, dualidade entre outros) fazendo sempre que possível contextualizações através de diversas aplicações. Finalizamos este texto com o Lema de Farkas, utilizando argumentos simples e lógicos para a sua demonstração, com o uso de cálculo e da álgebra linear / Abstract: This study aimed to produce a didactic text which would help in the study and understanding of Linear Programming Problems (LPP). We seek to reduce the "gap" that exists between the Calculus, Geometry and Linear Algebra in the treatment of such problem using a clear and objective language. Thus, we have included only the proofs of the results that we consider essential. We work with key concepts and definitions involving PPL (optimization, gradient vector, directional derivative, maximum and minimum on a compact set, Lagrange multipliers, space requirement, optimal solution, duality, among others) including wherever possible a contextualization through various applications. We finish this text with the Farkas' Lemma, using simple and logical arguments for their demonstration with the use of calculus and linear algebra / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

No description available.

Redundantly actuated manipulators have only recently aroused significant scientific interest. Their advantages in terms of enlarged workspace, higher payload ratio and better manipulability with respect to non-redundantly actuated systems explain the appearance of numerous applications in various fields: high-precision machining, fault-tolerant manipulators, transport and outer-space applications, surgical operation assistance, etc. The present Ph.D. research proposes a unified approach for modeling and actuation of redundantly actuated parallel manipulators. The approach takes advantage of the actuator redundancy principles and thus allows for following trajectories that contain parallel (force) singularities, and for eliminating the negative effect of the latter. As a first step of the approach, parallel manipulator kinematic and dynamic models are generated and treated in such a way that they do not suffer from kinematic loop closure numeric problems. Using symbolic models based on the multibody formalism and a Newton-Euler recursive computation scheme, faster-than-real-time computer simulations can thus be achieved. Further, an original piecewise actuation strategy is applied to the manipulators in order to eliminate singularity effects during their motion. Depending on the manipulator and the trajectories to be followed, this strategy results in non-redundant or redundant actuation solutions that satisfy actuator performance limits and additional optimality criteria. Finally, a validation of the theoretical results and the redundant actuation benefits is performed on the basis of well-known control algorithms applied on two parallel manipulators of different complexity. This is done both by means of computer simulations and experimental runs on a prototype designed at the Center for Research in Mechatronics of the UCL. The advantages of the actuator redundancy of parallel manipulators with respect to the elimination of singularity effects during motion and the actuator load optimization are thus confirmed (virtually and experimentally) and highlighted thanks to the proposed approach for modeling, simulation and control.

It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas. In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM). Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula. It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using only a single variational step. Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations too / indeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented. Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green&rsquo / s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well. In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-05-03T18:30:03Z No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-05-04T14:06:15Z (GMT) No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-05-04T14:06:23Z (GMT) No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) / Made available in DSpace on 2017-05-04T14:10:34Z (GMT). No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) Previous issue date: 2016-10-21 / Não recebi financiamento / In this work we explain an elegant and accessible proof of the Fundamental Theorem of Algebra using the Lagrange Multipliers method. We believe this will be a valuable resource not only to Mathematics students, but also to students in related areas, as the Lagrange Multipliers method that lies at the heart of the proof is widely taught. / Neste trabalho expomos uma demonstração acessível e elegante do Teorema Fundamental da Álgebra utilizando o método dos multiplicadores de Lagrange. Acreditamos que este trabalho seria uma fonte valiosa não são para estudantes de Matemática, mas também para estudantes de áreas relacionadas, uma vez que o método dos multiplicadores de Lagrange é amplamente ensinado em cursos de exatas.

High-amplitude freestream turbulence and surface roughness elements can excite a laminar boundary-layer flow sufficiently enough to cause streamwise-oriented vortices to develop. These vortices resemble elongated streaks having alternate spanwise variations of the streamwise velocity. Following the transient growth phase, the fully developed vortex structures downstream undergo an inviscid secondary instability mechanism and, ultimately, transition to turbulence. This mechanism becomes much more complicated in high-speed boundary layer flows due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this research, we formulate and test an optimal control algorithm to suppress the growth rate of the aforementioned streamwise vortex system. The derivation of the optimal control algorithm follows two stages. In the first stage, to optimize the computational cost of the analysis, the study develops an efficient numerical algorithm based on the nonlinear boundary region equations (NBREs), a reduced form of the compressible Navier-Stokes equations in a high-Reynolds-number asymptotic framework. The NBREs algorithm results agree well with direct numerical simulation (DNS) results. The numerical simulations are substantially less computationally costly than a full DNS and have a more rigorous theoretical foundation than parabolized stability equation (PSE) based models. The substantial reduction in computational time required to predict the full development of a G\"{o}rtler vortex system in high-speed flows allows investigation into feedback control in reasonable total computational time, which is the focus of the second part of the study. In the second stage, the method of Lagrange multipliers is utilized -- via an appropriate transformation of the original constrained optimization problem into an unconstrained form -- to obtain the adjoint compressible boundary-region equations (ACBREs) and corresponding optimality conditions, which constitute the basis of the optimal control approach. Numerical solutions for high-supersonic and hypersonic flows reveal a significant decrease in the kinetic energy and wall shear stress for all configurations considered. Streamwise velocity contour plots illustrate the qualitative effect of the optimal control iterations, demonstrating a significant decrease in the amplitude of the primary vortex instabilities.

No description available.