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Dissertations / Theses on the topic 'Algebraic Riccati equations'
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1
Rajasingam, Prasanthan. "Numerical Solution of the coupled algebraic Riccati equations." OpenSIUC, 2013. https://opensiuc.lib.siu.edu/theses/1323.
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In this paper we develop new and improved results in the numerical solution of the coupled algebraic Riccati equations. First we provide improved matrix upper bounds on the positive semidefinite solution of the unified coupled algebraic Riccati equations. Our approach is largely inspired by recent results established by Liu and Zhang. Our main results tighten the estimates of the relevant dominant eigenvalues. Also by relaxing the key restriction our upper bound applies to a larger number of situations. We also present an iterative algorithm to refine the new upper bounds and the lower bounds and numerically compute the solutions of the unified coupled algebraic Riccati equations. This construction follows the approach of Gao, Xue and Sun but we use different bounds. This leads to different analysis on convergence. Besides, we provide new matrix upper bounds for the positive semidefinite solution of the continuous coupled algebraic Riccati equations. By using an alternative primary assumption we present a new upper bound. We follow the idea of Davies, Shi and Wiltshire for the non-coupled equation and extend their results to the coupled case. We also present an iterative algorithm to improve our upper bounds. Finally we improve the classical Newton's method by the line search technique to compute the solutions of the continuous coupled algebraic Riccati equations. The Newton's method for couple Riccati equations is attributed to Salama and Gourishanar, but we construct the algorithm in a different way using the Fr\'echet derivative and we include line search too. Our algorithm leads to a faster convergence compared with the classical scheme. Numerical evidence is also provided to illustrate the performance of our algorithm.
2
Guo, Chun-Hua. "Analysis and modification of Newton's method for algebraic Riccati equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0009/NQ31028.pdf.
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3
Rajasingam, Prasanthan. "On the numerical solution of continuous coupled algebraic Riccati equations." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/dissertations/1203.
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In this dissertation we first derive a new unified upper solution bound for the continuous coupled algebraic Riccati equation, which arises from the optimal control of a Markovian jump linear system. In particular, we address the issue of rank deficiency with the control matrices. In the case of rank deficiency the existing matrix upper bounds are inapplicable. Moreover, our new result is not restricted to rank deficiency cases only. It now contains the existing results as special cases. Next, an iterative refinement is presented to improve our new unified matrix upper solution bounds. In particular, this iterative refinement determines a monotonically decreasing sequence of upper bounds for the solution of the continuous coupled algebraic Riccati equation. We formulate a new iterative algorithm by modifying this iterative refinement. We also prove that this new algorithm generates a monotonically decreasing sequence of matrix upper solution bounds that converges to the maximal solution of the continuous coupled algebraic Riccati equation. Furthermore, we prove the convergence of an accelerated Riccati iteration which computes a positive semidefinite solution of the continuous coupled algebraic Riccati equation. In particular, we establish sufficient conditions for the convergence of this algorithm. We also prove that for particular initial values this algorithm determines a monotonically increasing sequence of positive semidefinite matrices that converge to the minimal solution of the continuous coupled algebraic Riccati equation. Additionally, we show that for specific initial values this algorithm generates a monotonically decreasing sequence that converges to the maximal solution of the continuous coupled algebraic Riccati equation. In addition, we prove that this accelerated Riccati iteration converges faster than the Riccati iteration. Finally, we formulate a weighted modified accelerated Riccati iteration which is a more generalized Riccati type iteration. All of the existing Riccati iterations are now the special cases of this algorithm. Furthermore, we establish sufficient conditions for the convergence of this algorithm and we prove the monotonic convergence of the sequence generated by this algorithm. We also discuss how the weight and other quantities affect the rate of convergence of this algorithm. Illustrative numerical examples are also presented.
4
Benner, P., A. J. Laub, and V. Mehrmann. "A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800758.
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A collection of benchmark examples is presented for the numerical solution of continuous-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.
5
Benner, P., A. J. Laub, and V. Mehrmann. "A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: Discrete-time case." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800765.
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This is the second part of a collection of benchmark examples for the numerical solution of algebraic Riccati equations. After presenting examples for the continuous-time case in Part I, our concern in this paper is discrete-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.
6
Mehrmann, V. "A step towards a unified treatment of continuous and discrete time control problems." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800581.
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In this paper introduce new approach for
unified theory for continuous and discrete time
(optimal) control problems based on the
generalized Cayley transformation. We also relate
the associated discrete and continuous generalized
algebraic Riccati equations. We demonstrate the
potential of this new approach proving new
result for discrete algebraic Riccati equations.
But we also discuss where this new approach as
well as all other approaches still is
non-satisfactory. We explain a discrepancy
observed between the discrete and continuous
cse and show that this discrepancy is partly due
to the consideration of the wrong analogues. We
also present an idea for a metatheorem that
relates general theorems for discrete and
continuous control problems.
7
Petkov, P. Hr, M. M. Konstantinov, and V. Mehrmann. "DGRSVX and DMSRIC: Fortran 77 subroutines for solving continuous-time matrix algebraic Riccati equations with condition and accuracy estimates." Universitätsbibliothek Chemnitz, 2005. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200501109.
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We present new Fortran 77 subroutines which implement the Schur method and the
matrix sign function method for the solution of the continuoustime matrix algebraic
Riccati equation on the basis of LAPACK subroutines. In order to avoid some of
the wellknown difficulties with these methods due to a loss of accuracy, we combine
the implementations with block scalings as well as condition estimates and forward
error estimates. Results of numerical experiments comparing the performance of both
methods for more than one hundred well and illconditioned Riccati equations of order
up to 150 are given. It is demonstrated that there exist several classes of examples for
which the matrix sign function approach performs more reliably and more accurately
than the Schur method. In all cases the forward error estimates allow to obtain a reliable
bound on the accuracy of the computed solution.
8
Hu, Weiwei. "Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/38664.
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In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory.<br>Ph. D.
9
Aziz, Waleed. "Analytic and algebraic aspects of integrability for first order partial differential equations." Thesis, University of Plymouth, 2013. http://hdl.handle.net/10026.1/1468.
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This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.
10
Kawamoto, Atsushi. "Studies on Generalized Algebraic Riccati Equation for Descriptor Systems and Its Applications." Kyoto University, 1999. http://hdl.handle.net/2433/181816.
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11
Benner, P., and R. Byers. "Newtons method with exact line search for solving the algebraic Riccati equation." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800775.
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This paper studies Newton's method for solving the algebraic Riccati equation combined with an exact line search. Based on these considerations we present a Newton{like method for solving algebraic Riccati equations. This method can improve the sometimes erratic convergence behavior of Newton's method.
12
Rance, Guillaume. "Commande H∞ paramétrique et application aux viseurs gyrostabilisés." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS152/document.
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Cette thèse porte sur la commande H∞ par loop-shaping pour les systèmes linéaires à temps invariant d'ordre faible avec ou sans retard et dépendant de paramètres inconnus. L'objectif est d'obtenir des correcteurs H∞ paramétriques, c'est-à-dire dépendant explicitement des paramètres inconnus, pour application à des viseurs gyrostabilisés.L'existence de ces paramètres inconnus ne permet plus l'utilisation des techniques numériques classiques pour la résolution du problème H∞ par loop-shaping. Nous avons alors développé une nouvelle méthodologie permettant de traiter les systèmes linéaires de dimension finie grâce à l'utilissation de techniques modernes de calcul formel dédiées à la résolution des systèmes polynomiaux (bases de Gröbner, variétés discriminantes, etc.).Une telle approche présente de multiples avantages: étude de sensibilités du critère H∞ par rapport aux paramètres, identification de valeurs de paramètres singulières ou remarquables, conception de correcteurs explicites optimaux/robustes, certification numérique des calculs, etc. De plus, nous montrons que cette approche peut s'étendre à une classe de systèmes à retard.Plus généralement, cette thèse s'appuie sur une étude symbolique des équations de Riccati algébriques. Les méthodologies génériques développées ici peuvent s'étendre à de nombreux problèmes de l'automatique, notamment la commande LQG, le filtrage de Kalman ou invariant<br>This PhD thesis deals with the H∞ loop-shaping design for low order linear time invariant systems depending on unknown parameters. The objective of the PhD thesis is to obtain parametric H∞ controllers, i.e. controllers which depend explicitly on the unknown model parameters, and to apply them to the stabilization of gyrostabilized sights.Due to the unknown parameters, no numerical algorithm can solve the robust control problem. Using modern symbolic techniques dedicated to the solving of polynomial systems (Gröbner bases, discriminant varieties, etc.), we develop a new methodology to solve this problem for finite-dimensional linear systems.This approach shows several advantages : we can study the sensibilities of the H∞ criterion to the parameter variations, identify singular or remarquable values of the parameters, compute controllers which depend explicitly on the parameters, certify the numerical computations, etc. Furthermore, we show that this approach can be extended to a class of linear time-delay systems.More generally, this PhD thesis develops an algebraic approach for the study of algebraic Riccati equations. Thus, the methodology obtained can be extended to many different problems such as LQG control and Kalman or invariant filtering
13
Kojima, Chiaki. "Studies on Lyapunov stability and algebraic Riccati equation for linear discrete-time systems based on behavioral approach." 京都大学 (Kyoto University), 2007. http://hdl.handle.net/2433/135968.
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Benner, Peter, and Heike Faßbender. "On the solution of the radical matrix equation $X=Q+LX^{-1}L^T$." Universitätsbibliothek Chemnitz, 2007. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200701929.
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We study numerical methods for finding the maximal
symmetric positive definite solution of the nonlinear matrix equation
$X = Q + LX^{-1}L^T$, where Q is symmetric positive definite and L is
nonsingular. Such equations arise for instance in the analysis of
stationary Gaussian reciprocal processes over a finite interval.
Its unique largest positive definite solution coincides with the unique
positive definite solution of a related discrete-time algebraic
Riccati equation (DARE). We discuss how to use the butterfly
SZ algorithm to solve the DARE. This approach is compared to
several fixed point type iterative methods suggested in the
literature.
15
SILVA, Fabio Nogueira da. "Métodos neuronais para a solução da equação algébrica de Riccati e o LQR." Universidade Federal do Maranhão, 2008. http://tedebc.ufma.br:8080/jspui/handle/tede/1817.
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Previous issue date: 2008-06-20<br>Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPQ)<br>Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e Tecnológico do Maranhão (FAPEMA)<br>We present in this work the results about two neural networks methods to solve the algebraic Riccati(ARE), what are used in many applications, mainly in the Linear Quadratic Regulator (LQR), H2 and H1 controls. First is showed the real symmetric form of the ARE and two methods based on neural computation. One feedforward neural network (FNN), that de¯nes an error as function of the ARE and a recurrent neural network (RNN), which converts a constrain optimization problem, restricted to the state space model, into an unconstrained convex optimization problem de¯ning an energy as function of the ARE and Cholesky factor. A proposal to chose the learning parameters of the RNN used to solve the ARE, by making a surface of the parameters variations, thus we can tune the neural network for a better performance. Computational experiments related with the plant matrices perturbations of the tested systems in order to perform an analysis of the behavior of the presented methodologies, that are based on homotopies methods, where we chose a good initial condition and compare the results to the Schur method. Two 6th order systems were used, a Doubly Fed Induction Generator(DFIG) and an aircraft plant. The results showed the RNN a good alternative compared with the FNN and Schur methods.<br>Apresenta-se nesta dissertação os resultados a respeito de dois métodos neuronais para a resolução da equação algébrica de Riccati(EAR), que tem varias aplicações, sendo principalmente usada pelos Regulador Linear Quadrático(LQR), controle H2 e controle H1. É apresentado a EAR real e simétrica e dois métodos baseados em uma rede neuronal direta (RND) que tem a função de erro associada a EAR e uma rede neuronal recorrente (RNR) que converte um problema de otimização restrita ao modelo de espaço de estados em outro de otimização convexa em função da EAR e do fator de Cholesky de modo a usufruir das propriedades de convexidade e condições de otimalidade. Uma proposta para a escolha dos parâmetros da RNR usada para solucionar a EAR por meio da geração de superfícies com a variação paramétrica da RNR, podendo assim melhor sintonizar a rede neuronal para um melhor desempenho. Experimentos computacionais relacionados a perturbações nos sistemas foram realizados para analisar o comportamento das metodologias apresentadas, tendo como base o princípio dos métodos homotópicos, com uma boa condição inicial, a partir de uma ponto de operação estável e comparamos os resultados com o método de Schur. Foram usadas as plantas de dois sistemas: uma representando a dinâmica de uma aeronave e outra de um motor de indução eólico duplamente alimentado(DFIG), ambos sistemas de 6a ordem. Os resultados mostram que a RNR é uma boa alternativa se comparado com a RND e com o método de Schur.
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Silva, Fabio Nogueira da. "Métodos Neuronais para a Solução da Equação Algébrica de Riccati e o LQR." Universidade Federal do Maranhão, 2008. http://tedebc.ufma.br:8080/jspui/handle/tede/401.
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Previous issue date: 2008-06-20<br>FUNDAÇÃO DE AMPARO À PESQUISA E AO DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO DO MARANHÃO<br>We present in this work the results about two neural networks methods to solve
the algebraic Riccati(ARE), what are used in many applications, mainly in the
Linear Quadratic Regulator (LQR), H2 and H1 controls. First is showed the
real symmetric form of the ARE and two methods based on neural computation.
One feedforward neural network (FNN), that de¯nes an error as function of
the ARE and a recurrent neural network (RNN), which converts a constrain
optimization problem, restricted to the state space model, into an unconstrained
convex optimization problem de¯ning an energy as function of the ARE and
Cholesky factor. A proposal to chose the learning parameters of the RNN used
to solve the ARE, by making a surface of the parameters variations, thus we can
tune the neural network for a better performance.
Computational experiments related with the plant matrices perturbations of
the tested systems in order to perform an analysis of the behavior of the presented
methodologies, that are based on homotopies methods, where we chose a good
initial condition and compare the results to the Schur method. Two 6th order
systems were used, a Doubly Fed Induction Generator(DFIG) and an aircraft
plant. The results showed the RNN a good alternative compared with the FNN
and Schur methods.<br>Apresenta-se nesta dissertação os resultados a respeito de dois métodos neuronais
para a resolução da equação algébrica de Riccati(EAR), que tem varias aplicações,
sendo principalmente usada pelos Regulador Linear Quadrático(LQR), controle
H2 e controle H1. É apresentado a EAR real e simétrica e dois métodos baseados
em uma rede neuronal direta (RND) que tem a função de erro associada a EAR
e uma rede neuronal recorrente (RNR) que converte um problema de otimização
restrita ao modelo de espaço de estados em outro de otimização convexa em
função da EAR e do fator de Cholesky de modo a usufruir das propriedades de
convexidade e condições de otimalidade.
Uma proposta para a escolha dos parâmetros da RNR usada para solucionar
a EAR por meio da geração de superfícies com a variação paramétrica da RNR,
podendo assim melhor sintonizar a rede neuronal para um melhor desempenho.
Experimentos computacionais relacionados a perturbações nos sistemas foram
realizados para analisar o comportamento das metodologias apresentadas, tendo
como base o princípio dos métodos homotópicos, com uma boa condição inicial,
a partir de uma ponto de operação estável e comparamos os resultados com o
método de Schur. Foram usadas as plantas de dois sistemas: uma representando
a dinâmica de uma aeronave e outra de um motor de indução eólico duplamente
alimentado(DFIG), ambos sistemas de 6a ordem. Os resultados mostram que
a RNR é uma boa alternativa se comparado com a RND e com o método de
Schur.
17
Benner, P., R. Byers, and E. Barth. "HAMEV and SQRED: Fortran 77 Subroutines for Computing the Eigenvalues of Hamiltonian Matrices Using Van Loanss Square Reduced Method." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800926.
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This paper describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamilto- nian matrix to a square-reduced Hamiltonian uses only orthogonal symplectic similarity transformations. The eigenvalues can then be determined by applying the Hessenberg QR iteration to a matrix of half the order of the Hamiltonian matrix and taking the square roots of the computed values. Using scaling strategies similar to those suggested for algebraic Riccati equations can in some cases improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.
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Benner, P., and H. Faßbender. "A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800797.
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A restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. Breakdowns and near-breakdowns are overcome by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors and invariant subspaces of large and sparse Hamiltonian matrices and low rank approximations to the solution of continuous-time algebraic Riccati equations with large and sparse coefficient matrices.
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Alencar, Andrà Luiz Sampaio de. "Uma nova metodologia de jogos dinÃmicos lineares quadrÃticos." Universidade Federal do CearÃ, 2011. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=6934.
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CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior<br>A teoria dos jogos à um ramo da matemÃtica dedicado ao estudo de situaÃÃes que surgem quando mÃltiplos agentes de decisÃo buscam atingir seus objetivos individuais, possivelmente conflitantes entre si. Em sua formulaÃÃo dinÃmica linear quadrÃtica (LQ), as soluÃÃes de equilÃbrio de Nash dos jogadores podem ser obtidas em termos das equaÃÃes algÃbricas de Riccati acopladas, que, a depender do mÃtodo numÃrico utilizado para seu cÃlculo, podem gerar resultados insatisfatÃrios sob o ponto de vista da estabilidade e precisÃo numÃrica. Neste sentido, esta dissertaÃÃo propÃe um novo algoritmo para uma soluÃÃo alternativa das equaÃÃes algÃbricas de Riccati acopladas associadas aos jogos dinÃmicos (LQ), com estrutura de informaÃÃo em malha aberta, utilizando, para isso, conceitos da teoria da dualidade e otimizaÃÃo estÃtica convexa. Em adiÃÃo, obtÃm-se uma nova metodologia para a sÃntese de uma famÃlia de controladores Ãtimos.
A teoria dos jogos tambÃm revela um enorme potencial de aplicaÃÃo em problemas de controle multiobjetivo, no qual està incluÃdo o controle Hinf, que pode ser formulado como um jogo dinÃmico de soma-zero. Considerando essa formulaÃÃo, as novas metodologias propostas neste trabalho sÃo estendidas aos problemas de controle Hinf com rejeiÃÃo de perturbaÃÃo, gerando resultados com melhores propriedades de desempenho e estabilidade que os obtidos via equaÃÃo algÃbrica de Riccati modificada.
Por fim, atravÃs de exemplos numÃricos e simulaÃÃes computacionais, as novas metodologias sÃo confrontadas com as metodologias tradicionais, evidenciando-se os aspectos mais relevantes de cada abordagem.
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Queiroz, Jonathan Araujo. "Solução geral da equação algébrica de Riccati Discreta utilizando estimador não quadrático e decomposição matricial aplicado no modelo em espaço de estado de um gerador eólico." Universidade Federal do Maranhão, 2016. http://tedebc.ufma.br:8080/jspui/handle/tede/1691.
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Previous issue date: 2016-03-08<br>Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPQ)<br>The discrete Riccati algebraic equation has played an increasingly important role in optimal control theory and adaptive ltering. For this reason, various techniques have been developed to solve the DARE, for example the approach based on self vectors or approaches related to invariant subspaces [1], which require mathematical rigor and precision. However, these approaches present a number of problems, among them the fact that they can not be implemented in real-time due to its high computational cost to estimate the solution of DARE in many systems, especially systems with higher order three. In order to overcomes this problem, we propose to solve the DARE using as an estimator based on the sum of potential error pairs. The estimator is similar to the Recursive Least Squares (RLS), but with a better performance in terms of convergence speed and estimation accuracy without a signi- cant increase in computational complexity. The estimator is called Recursive Least Non-Squares (RLNS). One other aspect in unraveling the general DARE is to ensure that DARE is numerically well conditioned. To perform the numerical conditioning of DARE, a matrix decomposition technique known as Moore-Penrose inverse or generalized inverse is used. The proposed method is evaluated in a multivariate system 6th order corresponding to the wind generator. The method is evaluated under the numerical stability point of view and speed of convergence.<br>A equação algébrica Riccati discreta (discrete algebraic Riccati equation (DARE)) tem desempenhado uma papel cada vez mais importante na teoria de controle ótimo. Por esse motivo, varias técnicas tem sido desenvolvidas para solucionar a DARE, por exemplo a abordagem baseada em auto vetores ou ainda abordagens relacionadas a subespaços invariantes, as quais requerem rigor e precisão matemáticas. No entanto, estas abordagens apresentam uma serie de problemas, dentre eles, o fato de não poderem ser implementadas em tempo real devido ao seu alto custo computacional para estimar a solução da DARE em diversos sistemas, sobretudo sistemas com ordem superior a três. Com o intuito de contorna este problema, propomos solucionar a DARE utilizando um estimador baseado na soma das potencias pares do erro. O estimador e similar ao Recursive least squares (RLS), mas com um desempenho melhor em termos de velocidade de convergência e precisão de estimativa, sem aumento significativo da complexidade computacional. O estimador é denominado Recursive Least Non-Squares (RLNS). Um outra aspecto para que possamos solucionar a DARE de forma geral, e garantir que a DARE seja numericamente bem condicionada. Para efetuar o condicionamento numérico da DARE, será utilizada uma técnica de decomposição matricial conhecida como inversa de Moore-Penrose ou inversa generalizada. A metodologia proposta e avaliada em um sistema multivariavel de 6th ordem correspondente ao gerador eólico.
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Wang, Rui. "Distributed Cooperative Communications and Wireless Power Transfer." Digital WPI, 2018. https://digitalcommons.wpi.edu/etd-dissertations/62.
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In telecommunications, distributed cooperative communications refer to techniques which allow different users in a wireless network to share or combine their information in order to increase diversity gain or power gain. Unlike conventional point-to-point communications maximizing the performance of the individual link, distributed cooperative communications enable multiple users to collaborate with each other to achieve an overall improvement in performance, e.g., improved range and data rates.
The first part of this dissertation focuses the problem of jointly decoding binary messages from a single distant transmitter to a cooperative receive cluster. The outage probability of distributed reception with binary hard decision exchanges is compared with the outage probability of ideal receive beamforming with unquantized observation exchanges. Low- dimensional analysis and numerical results show, via two simple but surprisingly good approximations, that the outage probability performance of distributed reception with hard decision exchanges is well-predicted by the SNR of ideal receive beamforming after subtracting a hard decision penalty of slightly less than 2 dB. These results, developed in non-asymptotic regimes, are consistent with prior asymptotic results (for a large number of nodes and low per-node SNR) on hard decisions in binary communication systems.
We next consider the problem of estimating and tracking channels in a distributed transmission system with multiple transmitters and multiple receivers. In order to track and predict the effective channel between each transmit node and each receive node to facilitate coherent transmission, a linear time-invariant state- space model is developed and is shown to be observable but nonstabilizable. To quantify the steady-state performance of a Kalman filter channel tracker, two methods are developed to efficiently compute the steady-state prediction covariance. An asymptotic analysis is also presented for the homogenous oscillator case for systems with a large number of transmit and receive nodes with closed-form results for all of the elements in the asymptotic prediction covariance as a function of the carrier frequency, oscillator parameters, and channel measurement period. Numeric results confirm the analysis and demonstrate the effect of the oscillator parameters on the ability of the distributed transmission system to achieve coherent transmission.
In recent years, the development of efficient radio frequency (RF) radiation wireless power transfer (WPT) systems has become an active research area, motivated by the widespread use of low-power devices that can be charged wirelessly. In this dissertation, we next consider a time division multiple access scenario where a wireless access point transmits to a group of users which harvest the energy and then use this energy to transmit back to the access point. Past approaches have found the optimal time allocation to maximize sum throughput under the assumption that the users must use all of their harvested power in each block of the "harvest-then-transmit" protocol. This dissertation considers optimal time and energy allocation to maximize the sum throughput for the case when the nodes can save energy for later blocks. To maximize the sum throughput over a finite horizon, the initial optimization problem is separated into two sub-problems and finally can be formulated into a standard box- constrained optimization problem, which can be solved efficiently. A tight upper bound is derived by relaxing the energy harvesting causality.
A disadvantage of RF-radiation based WPT is that path loss effects can significantly reduce the amount of power received by energy harvesting devices. To overcome this problem, recent investigations have considered the use of distributed transmit beamforming (DTB) in wireless communication systems where two or more individual transmit nodes pool their antenna resources to emulate a virtual antenna array. In order to take the advantages of the DTB in the WPT, in this dissertation, we study the optimization of the feedback rate to maximize the energy efficiency in the WPT system. Since periodic feedback improves the beamforming gain but requires the receivers to expend energy, there is a fundamental tradeoff between the feedback period and the efficiency of the WPT system. We develop a new model to combine WPT and DTB and explicitly account for independent oscillator dynamics and the cost of feedback energy from the receive nodes. We then formulate a "Normalized Weighted Mean Energy Harvesting Rate" (NWMEHR) maximization problem to select the feedback period to maximize the weighted averaged amount of net energy harvested by the receive nodes per unit of time as a function of the oscillator parameters. We develop an explicit method to numerically calculate the globally optimal feedback period.
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Dobson, William Keith. "Method for Improving the Efficiency of Image Super-Resolution Algorithms Based on Kalman Filters." Digital Archive @ GSU, 2009. http://digitalarchive.gsu.edu/math_theses/82.
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The Kalman Filter has many applications in control and signal processing but may also be used to reconstruct a higher resolution image from a sequence of lower resolution images (or frames). If the sequence of low resolution frames is recorded by a moving camera or sensor, where the motion can be accurately modeled, then the Kalman filter may be used to update pixels within a higher resolution frame to achieve a more detailed result. This thesis outlines current methods of implementing this algorithm on a scene of interest and introduces possible improvements for the speed and efficiency of this method by use of block operations on the low resolution frames. The effects of noise on camera motion and various blur models are examined using experimental data to illustrate the differences between the methods discussed.
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Hsu, Yung-an, and 徐永諳. "On Generalized Algebraic Riccati Equations for Descriptor Systems." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/94490747309184067368.
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碩士<br>國立海洋大學<br>電機工程學系<br>91<br>In this thesis, the generalized continuous-time algebraic Riccati equation (GCARE) for descriptor systems is studied.
The necessary and sufficient condition for the existence of
stabilizing solutions to a nonsymmetric generalized algebraic Riccati equations is derived based on a Hamiltonian matrix pencil approach. Some of the generalizations of the well known properties of the algebraic Riccati equation to the GCARE case are also proposed.
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CHEN, YI-DE, and 陳禕德. "Iterative solution of lyapunov matrix equations and algebraic matrix riccati equations." Thesis, 1991. http://ndltd.ncl.edu.tw/handle/83051584972001685077.
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Chao, Yi-ru, and 趙怡茹. "A Survey on Numerical Solutions for Algebraic Riccati Equations." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/84611187575213308157.
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碩士<br>國立成功大學<br>數學系應用數學碩博士班<br>95<br>In the last decades, a number of numerical methods for solving algebraic Riccati equations are proposed. According to that the kernel of the optimal control problems is used to be the stabilizing solutions of corresponding AREs, the thesis hence aims to survey several popular numerical methods for solving AREs, for
instance, Newton's method, matrix sign function method, Schur method and SDA method.
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chang, chih-lee, and 張志禮. "Matrix Bounds of the Solution for the Algebraic Riccati Equations." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/96431388976389889215.
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碩士<br>國立中央大學<br>電機工程研究所<br>89<br>Abstract
In this thesis, we will presents new matrix bounds of the solution for the discrete and continuous Riccati equations. Based on the solution of certain continuous and discrete Lyapunov equations, the improved upper and lower matrix bounds are obtained. In the discrete systems, the upper and lower matrix bounds always exist if the DLE solution exists. Then, further improvements on the bounds are presented. On other hand, these upper and lower matrix bounds of solution for the continuous Riccati equation are always exist if the system is stabilizable. Numerical examples are given to show that our methods are less conservative and less restrictive than some recent results.
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Wu, Ying-Han, and 吳盈漢. "Solutions of the Generalized Discrete –Time Algebraic Riccati Equations in H2 Control." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/72520582094945479113.
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碩士<br>國立臺灣海洋大學<br>電機工程學系<br>97<br>In this thesis, we study the generalized discrete-time algebraic Riccati
equation (GDARE). In particular we express a sufficient condition for solvability associated with (GDARE) and derive a set of solutions using some algorithms. These results may play a essential role in H2 control problem for discrete-time descriptor systems.
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Chien-YingLi and 李建穎. "Numerical Study on Low-Rank Approximate Solutions to Large-Scale Algebraic Riccati Equations." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/02361218488708007957.
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碩士<br>國立成功大學<br>數學系應用數學碩博士班<br>102<br>In recent years, large-scale computing has become an important research topic. Algebraic Riccati equations is a control problem comes from the quadratic optimization. In this paper, we study the relationship between the large-scale sparse algebraic Riccati equations low-rank approximate solutions and the control systems, the control system controllability and observability that can be used to obtain low-rank approximate solution of large sparse algebriac Riccati equations; we use Newton's method solving the algebriac Riccati equations, the convergence rate of Newton's method is quadratic, but each iteration requires solving the Lyapunov equation, so that the convergence rate significantly lower, for solving the Lyapunov equations Cholesky Factor Alternating Direction Implicit iterative method can be kept low-rank structure of the solution, thereby reducing its computation; further use of two strategies: Guess initial value, Relaxed CFADI, reducing the total number of inner iteration to accelerate the convergence rate of Newton's method, and finally provide some numerical results.
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Chen, Chiao-Wen, and 陳巧雯. "Generalized Structure-Preserving Doubling Algorithms for Generalized Continuous and Discrete Time Algebraic Riccati Equations." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/23004248494708789317.
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碩士<br>國立清華大學<br>數學系<br>93<br>In this paper we extend the structure-preserving doubling algorithm (SDA) to compute the symmetric positive semi-definite solutions of the generalized continuous as well as discrete algebraic Riccati equations. Our main idea is to relate continuous and discrete time control systems based on the generalized Caley transformation.
In the end, we select some examples to illustrate that the G-SDA performs better than the MATLAB commands in the control toolbox.
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Kasinathan, Dhanaraja. "H-∞ optimal actuator location." Thesis, 2012. http://hdl.handle.net/10012/6732.
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There is often freedom in choosing the location of actuators on systems governed by partial differential equations.
The actuator locations should be selected in order to optimize the performance criterion of interest. The main focus of this thesis is to consider H-∞-performance with state-feedback. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output of a full-information plant.
Optimal H-∞-disturbance attenuation as a function of actuator location is used as the cost function. It is shown that the corresponding actuator location problem is well-posed. In practice, approximations are used to determine the optimal actuator location. Conditions for the convergence of optimal performance and the corresponding actuator location to the exact performance and location are provided. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied.
Systems of large model order arise in a number of situations; including approximation of partial differential equation models and power systems. The system descriptions are sparse when given in descriptor form but not when converted to standard first-order form. Numerical calculation of H-∞-attenuation involves iteratively solving large H-∞-algebraic Riccati equations (H-∞-AREs) given in the descriptor form. An iterative algorithm that preserves the sparsity of the system description to calculate the solutions of large H-∞-AREs is proposed. It is shown that the performance of our proposed algorithm is similar to a Schur method in many cases. However, on several examples, our algorithm is both faster and more accurate than other methods.
The calculation of H-∞-optimal actuator locations is an additional layer of optimization over the calculation of optimal attenuation. An optimization algorithm to calculate H-∞-optimal actuator locations using a derivative-free method is proposed. The results are illustrated using several examples motivated by partial differential equation models that arise in control of vibration and diffusion.
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Han, Chuan Shiang, and 韓傳祥. "Homotopy Method for Soving Algebraic Riccati Equation." Thesis, 1994. http://ndltd.ncl.edu.tw/handle/94470085570502695876.
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Chang, Chih-Peng, and 張志鵬. "A Structure-Preserving Doubling Algorithm for Nonsymmetric Algebraic Riccati Equation." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/70367263068604887527.
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碩士<br>國立清華大學<br>數學系<br>93<br>In this paper we consider the nonsymmetric algebraic Riccati
equation (NARE) for which the four coefficient matrices form an M-matrix. Nonsymmetric algebraic Riccati equations of this type appear in applied probability and transport theory. The minimal nonnegative solution of these equations can be found by a structure-preserving doubling algorithm (SDA).
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Chen, Hsin-An, and 陳信安. "A Study On Perturbation Analysis Of The Stochastic Algebraic Riccati Equation." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/67395690432699647127.
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碩士<br>國立中正大學<br>應用數學研究所<br>100<br>In this work we study a class of stochastic algebraic Riccati equations
from the indenite stochastic linear quadratic control problems and stochastic
H∞ control problems. We obtain a perturbation bound for the stabilizing
solution of the perturbed and analyze the existence of stabilizing solution
to the perturbed SARE. A numerical example is presented to illustrate the
sharpness of the perturbation bound.
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CHEN, JIAN-WEN, and 陳建文. "A symplectic jacobi-like algorithm and a symplectic direct method for the solution of the algebraic Riccati equation on a parallel computer." Thesis, 1989. http://ndltd.ncl.edu.tw/handle/21105521038428967037.
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Bhattacharya, Atreyee. "On an ODE Associated to the Ricci Flow." Thesis, 2013. http://etd.iisc.ernet.in/2005/3427.
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We discuss two topics in this talk. First we study compact Ricci-flat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat K¨ahler surfaces with similar but weaker restrictions on holomorphic sectional curvature.
Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various(locally) symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.