Orientador: Douglas Eduardo Zampieri / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-07-22T15:51:05Z (GMT). No. of bitstreams: 1 Alves_PauloSergioLima_M.pdf: 7176382 bytes, checksum: 382a907d22963bc53b27c051d42f780f (MD5) Previous issue date: 1997 / Resumo: É feito um estudo sobre a dinâmica de um modelo plano de um trator no qual é introduzido um eixo dianteiro. Considerando os modelos com e sem implemento agricola são introduzidos elementos ativos na suspensão e no elemento de ligação do implemento com o corpo do veículo. Verificando a controlabilidade e observabilidade do sistema e utilizando a teoria de controle ótimo, é aplicada a equação de Riccati para obtenção da lei de controle dos elementos ativos baseada na realimentação dos estados do sistema. As matrizes de ponderação, utilizadas na função custo do tipo integral quadrática, são obtidas a partir de parâmetros que representem o comportamento desejado do sistema. Considerando o conforto e a segurança do veículo, as matrizes de ponderação são obtidas representando-se a aceleração vertical do corpo do trator e as forças de contato dos pneus dianteiro e traseiro. São também consideradas as ponderações do espaço de trabalho da suspensão e o movimento do implemento. A influência destas ponderações sobre os autovalores e autovetores do sistema é, também, analisada. Finalmente, compara-se a resposta do sistema ao degrau considerando-se a suspensão e o elemento de ligação implemento/trator totalmente passivos e totalmente ativos / Abstract: The dynamics of a two dimensional tractor model with a frontal axle is studied. Active elements are introduced in the suspension and in the tractor body-implement linking element considering the model with and without implement. Verifying the system controllability and observability and using optimal control theory, Riccati equation is applied in order to obtain the control law of the active elements. The weighting matrices, used in the quadratic integral cost function, are obtained from the parameters that represent the desired behaviour of the system. Considering comfort and safety, the weighting matrices are obtained representing the verticalaccelerationof the tractor bodyand the contactforces of the ITontand rear tires as a vector multiplied by the state vector. The weighting matrices of the suspension working space and the movement of the implement are also considered. The effect of each weighting matrix on the eigenvalues and eigenvectors of the system is analyzed. The response of the active system to external excitation is compared to that obtained considering passive elements in the suspension and in the tractor body-implement linking element / Mestrado / Mecanica dos Solidos / Mestre em Engenharia Mecânica

The existence and uniqueness of Lagrangian invariant subspaces of Hamiltonian matrices is studied. Necessary and sufficient conditions are given in terms of the Jordan structure and certain sign characteristics that give uniqueness of these subspaces even in the presence of purely imaginary eigenvalues. These results are applied to obtain in special cases existence and uniqueness results for Hermitian solutions of continuous time algebraic Riccati equations.

We present a linear-quadratic regulator (LQR) design for a heat transfer model describing the cooling process of steel profiles in a rolling mill. Primarily we consider a feedback control approach for a linearization of the nonlinear model given there, but we will also present first ideas how to use local (in time) linearizations to treat the nonlinear equation with a regulator approach. Numerical results based on a spatial finite element discretization and a numerical algorithm for solving large-scale algebraic Riccati equations are presented both for the linear and nonlinear models.

We discuss a parallel library of efficient algorithms for the solution of linear-quadratic optimal control problems involving largescale systems with state-space dimension up to $O(10^4)$. We survey the numerical algorithms underlying the implementation of the chosen optimal control methods. The approaches considered here are based on invariant and deflating subspace techniques, and avoid the explicit solution of the associated algebraic Riccati equations in case of possible ill-conditioning. Still, our algorithms can also optionally compute the Riccati solution. The major computational task of finding spectral projectors onto the required invariant or deflating subspaces is implemented using iterative schemes for the sign and disk functions. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Itanium-2 processors.

We discuss numerical methods for the stabilization of large linear multi-input control systems of the form x=Ax + Bu via a feedback of the form u=Fx. The method discussed in this paper is a stabilization algorithm that is based on subspace splitting. This splitting is done via the matrix sign-function method. Then a projection into the unstable subspace is performed followed by a stabilization technique via the solution of an appropriate algebraic Riccati equation. There are several possibilities to deal with the freedom in the choice of the feedback as well as in the cost functional used in the Riccati equation. We discuss several optimality criteria and show that in special cases the feedback matrix F of minimal spectral norm is obtained via the Riccati equation with the zero constant term. A theoretical analysis about the distance to instability of the closed loop system is given and furthermore numerical examples are presented that support the practical experience with this method.

An algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes, but also dampens the closed-loop system dynamics. In other words, a feedback gain vector is computed such that the eigenvalues of the closed-loop state matrix are within the region of the left half-plane where the magnitude of the real part of each eigenvalue is greater than the imaginary part. This may be accomplished by solving one periodic algebraic Riccati equation and one degenerate Riccati equation. The solution to these equations are computed using numerically robust algorithms. Finally, the periodic Riccati equation is unusual in that it produces one symmetric and one skew symmetric solution, and as a result two different state feedbacks. Both feedbacks dampen the system dynamics, but produce different closed-loop eigenvalues, giving the controller designer greater freedom in choosing a desired feedback.

Nous utilisons la méthode d'équivalence de Cartan pour réaliser une étude géométrique des équations différentielles ordinaires du second ordre et du premier ordre, sous l'action des transformations ponctuelles préservant les aires dans le cas du second ordre et de certaines autres transformations dans le cas du premier. Cela nous permet de caractériser de manière invariante toutes les équations différentielles du second ordre se ramenant à y"=0. De plus nous associons à toute telle équation, une connexion de Cartan affine normale dont la courbure contient tous ses invariants. Dans le cas du premier ordre nous apportons un regard nouveau sur une étude de R. Liouville concernant l'équation différentielle d'Abel. Enfin dans un autre ordre d'idées nous réalisons une étude de certaines solutions algébriques de l'équation de Riccati. / Abstract

Nous nous intéressons dans cette thèse, à l’étude des méthodes itératives pour la résolutiond’équations matricielles de grande taille : Lyapunov, Sylvester, Riccati et Riccatinon symétrique.L’objectif est de chercher des méthodes itératives plus efficaces et plus rapides pour résoudreles équations matricielles de grande taille. Nous proposons des méthodes itérativesde type projection sur des sous espaces de Krylov par blocs Km(A, V ) = Image{V,AV, . . . ,Am−1V }, ou des sous espaces de Krylov étendus par blocs Kem(A, V ) = Image{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V } . Ces méthodes sont généralement plus efficaces et rapides pour les problèmes de grande dimension. Nous avons traité d'abord la résolution numérique des équations matricielles linéaires : Lyapunov, Sylvester, Stein. Nous avons proposé une nouvelle méthode itérative basée sur la minimisation de résidu MR et la projection sur des sous espaces de Krylov étendus par blocs Kem(A, V ). L'algorithme d'Arnoldi étendu par blocs permet de donner un problème de minimisation projeté de petite taille. Le problème de minimisation de taille réduit est résolu par différentes méthodes directes ou itératives. Nous avons présenté ainsi la méthode de minimisation de résidu basée sur l'approche global à la place de l'approche bloc. Nous projetons sur des sous espaces de Krylov étendus Global Kem(A, V ) = sev{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. Nous nous sommes intéressés en deuxième lieu à des équations matricielles non linéaires, et tout particulièrement l'équation matricielle de Riccati dans le cas continu et dans le cas non symétrique appliquée dans les problèmes de transport. Nous avons utilisé la méthode de Newtown et l'algorithme MINRES pour résoudre le problème de minimisation projeté. Enfin, nous avons proposé deux nouvelles méthodes itératives pour résoudre les équations de Riccati non symétriques de grande taille : la première basée sur l'algorithme d'Arnoldi étendu par bloc et la condition d'orthogonalité de Galerkin, la deuxième est de type Newton-Krylov, basée sur la méthode de Newton et la résolution d'une équation de Sylvester de grande taille par une méthode de type Krylov par blocs. Pour toutes ces méthodes, les approximations sont données sous la forme factorisée, ce qui nous permet d'économiser la place mémoire en programmation. Nous avons donné des exemples numériques qui montrent bien l'efficacité des méthodes proposées dans le cas de grandes tailles. / In this thesis, we focus in the studying of some iterative methods for solving large matrix equations such as Lyapunov, Sylvester, Riccati and nonsymmetric algebraic Riccati equation. We look for the most efficient and faster iterative methods for solving large matrix equations. We propose iterative methods such as projection on block Krylov subspaces Km(A, V ) = Range{V,AV, . . . ,Am−1V }, or block extended Krylov subspaces Kem(A, V ) = Range{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. These methods are generally most efficient and faster for large problems. We first treat the numerical solution of the following linear matrix equations : Lyapunov, Sylvester and Stein matrix equations. We have proposed a new iterative method based on Minimal Residual MR and projection on block extended Krylov subspaces Kem(A, V ). The extended block Arnoldi algorithm gives a projected minimization problem of small size. The reduced size of the minimization problem is solved by direct or iterative methods. We also introduced the Minimal Residual method based on the global approach instead of the block approach. We projected on the global extended Krylov subspace Kem(A, V ) = Span{V,A−1V,AV,A−2V,A2V, · · · ,Am−1V,A−m+1V }. Secondly, we focus on nonlinear matrix equations, especially the matrix Riccati equation in the continuous case and the nonsymmetric case applied in transportation problems. We used the Newton method and MINRES algorithm to solve the projected minimization problem. Finally, we proposed two new iterative methods for solving large nonsymmetric Riccati equation : the first based on the algorithm of extended block Arnoldi and Galerkin condition, the second type is Newton-Krylov, based on Newton’s method and the resolution of the large matrix Sylvester equation by using block Krylov method. For all these methods, approximations are given in low rank form, wich allow us to save memory space. We have given numerical examples that show the effectiveness of the methods proposed in the case of large sizes.

Can we capture the explosive nature of volatility skew observed in the market, without resorting to non-Markovian models? We show that, in terms of skew, the Heston model cannot match the market at both long and short maturities simultaneously. We introduce Abi Jaber (2019)'s Lifted Heston model and explain how to price options with it using both the cosine method and standard Monte-Carlo techniques. This allows us to back out implied volatilities and compute skew for both models, confirming that the Lifted Heston nests the standard Heston model. We then produce and analyze the skew for Lifted Heston models with a varying number N of mean reverting terms, and give an empirical study into the time complexity of increasing N. We observe a weak increase in convergence speed in the cosine method for increased N, and comment on the number of factors to implement for practical use.

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.