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.

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. / Ph. D.

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.

Orientador: Prof. Dr. André Fenili / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia Mecânica, 2016. / A medição da espessura de cascos de navios e outras estruturas submarinas por ultra-som criou uma nova e desafiante aplicação para robótica e apresenta vários desafios de controle gerados pela não linearidade, acoplamento de movimentos, incertezas nos parâmetros hidrodinâmicos e forças de perturbação. Este trabalho trata do controle de posição e atitude do Veículo Híbrido Submersível Operado de forma Remota (VHSOR) Proteo da Universidade Federal do ABC (São Paulo, Brasil). Neste trabalho aplicou-se a metodologia de control não linear State-Dependent Riccati Equation (SDRE) num VHSOR e obteve-se um controlador único que leva em consideração todos os estados da dinâmica do veículo, a não linearidade, o acoplamento e incertezas ao mesmo tempo. Também foram modeladas as perturbações externas com a intenção de verificar a capacidade do controlador para superar os problemas que aparecem em ambientes reais (forças e momentos externos gerados pelas correntes marinhas). Neste trabalho mostram-se os resultados obtidos das simulações numéricas do controlador desenvolvido e também os resultados obtidos dos experimentos em tanque de testes da plataforma robótica Proteo. Como parte do projeto de controle, neste trabalho realizou-se a programação da plataforma robótica Proteo provendo um sistema embarcado que faz aquisição de dados dos sensores, estima as variáveis de estado, calcula as ações de controle, ativa os atuadores, comunica a posição e velocidade (de traslação e rotação) à tripulação e recebe as referências em posição e velocidade (de traslação e rotação) enviadas desde um computador externo. Para este computador externo também realizou-se um programa que recebe sinais por meio de um Joystick e envia para o sistema embarcado na plataforma robótica Proteo. / The ultrasonic thickness measurements of ship hulls and other underwater structures has created a new and challenging application for robotics and poses several control challenges generated by non-linearity, coupling movements, uncertainties in the hydrodynamic parameters and disturbance forces. This work is on the development of position and attitude control of Hybrid Remotely Operated Vehicle (HROV) Proteo of the Universidade Federal do ABC (São Paulo, Brazil). In this work we used the nonlinear control methodology State-Dependent Riccati Equation (SDRE) as a unique control system, that takes into account all the states of the vehicle dynamic, non-linearities, coupling and uncertainty at the same time. Also, was modeled external disturbances with the intention to verify the ability of the control system to overcome a problem which is always encountered in real environments (external forces and moments generated by sea currents). This work shows the results obtained from numerical simulations of the developed controller and also the results of the tests of the robotic platform Proteo in a water tank. As part of the control design, in this work was carried out the programming of robotic platform Proteo providing an embedded system on it that makes data acquisition, estimate state variables, calculates control actions, active actuators, communicates position and velocity (of translation and rotation) to the crew and receives position and velocity references (of translation and rotation) from an external computer. For this external computer also was made a program that receives signals from a Joystick and sends to the embedded system in the robotic platform Proteo.

Orientadora: Profª. Drª. Elvira Rafikova / Coorientador: Prof. Dr. Magno Enrique Mendonza Meza / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia Mecânica, Santo André, 2018.

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.

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.

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.

L'objectif de cette thèse est de développer une nouvelle méthodologie pour la détection active de défaillances, basée sur approche multimodèle et robuste des fautes. Ce travail prolonge des recherches effectuées dans le projet Metalau de l'Inria. L'apport essentiel de cette thèse est la prise en compte de modèles évoluant en boucle fermée. On utilise une approche multi-modèle pour modéliser le modèle en fonctionnement normal et le modèle défaillant. Les avantages potentiels de l'utilisation d'un feedback dynamique linéaire et ses propriétés de robustesse sont analysés dans la construction de signaux de détection auxiliaires. On compare les résultats obtenus avec ceux du cas boucle ouverte. La formulation du problème de détection active dans le cas d'un modèle en boucle fermée est nouvelle et repose sur la prise en considération de la norme du signal de détection auxiliaire comme critère d'optimisation. On considère aussi des fonctions coût plus générales, telles celles qui sont utilisées pour mesurer la performance de feedbacks dans des problèmes de la théorie de la commande linéaire robuste. La solution complète repose sur la résolution de plusieurs problèmes d'optimisation non standards / The aim is to develop a novel theory of robust active failure detection based on multi-model formulation of faults. The original method was already proposed by the Metalau group of INRIA. We have continued to work on the extension of this approach to more general cases. The focus is on the effects of feedback on the previous approach. The multi-model approach is still used to model the normal and the failed systems; however the possible advantages of using linear dynamic feedback in the construction of the auxiliary signal for robust fault detection is considered and the results are compared to the previously developed open-loop setup. An original formulation of the active fault detection problem using feedback is developed. The norm of the auxiliary signal is considered as a possible cost criterion. Also, we have considered a more general cost function that has already been used for measuring the performance of feedback configurations in Linear Control Theory. We have given a complete solution to this problem. In order to find a complete solution, several mathematical problems are solved

Esta dissertação apresenta um estudo dos filtros de Kalman para sistemas singulares em tempo discreto. Novos algoritmos são formulados para as estimativas filtradas, preditoras e suavizadas com as correspondentes equações de Riccati para sistemas singulares variantes no tempo. Nesta dissertação considera-se também uma aproximação do problema de filtragem de Kalman como um problema determinístico de ajuste ótimo de trajetória. A formulação proposta permite considerar um atraso no sinal de medida, sendo permitida a correlação entre os estados e os ruídos da medida. Apresentam-se também as provas da estabilidade e da convergência destes filtros. / This dissertation presents a study of Kalman filters for singular systems in discrete time. New algorithms are developed for the Kalman filtered, predicted and smoothed estimate recursions with the corresponding Riccati equations for time-variant singular systems. This dissertation addresses the Kalman filtering problem as a deterministic optimal trajectory fitting problem. The problem is formulated taking into account one delay in the measured signals and correlations between state and measurement noises. In the final, this work presents the stability and convergence proofs of these filters.