This research investigates the harmonic and narrowband disturbance rejection problem for linear time-periodic (LTP) systems. The consequence of disturbances on LTP systems is similar to their linear time-invariant (LTI) counterparts, but is complicated by the interaction of the disturbance and plant acting at different frequencies, which manifests itself in the modulation of the disturbance signal. The result, for an m-periodic plant and disturbance containing a single tone, is that the output contains m tones. Using various disturbance rejection architectures, harmonic and narrowband disturbance rejection is investigated for linear time-periodic plants. Included are classical and multivariable feedback controllers, fixed-gain feedforward designs using finite impulse response (FIR) filters and H-infinity synthesis tools, and adaptive feedforward controllers. The objective of time-periodic, narrowband, disturbance rejection seeks to place a zero in the controlled system's disturbance path and align the zero direction, defined by the null space of the controlled system at the disturbance frequency, with the disturbance. In this research, constraints on controlled system infinity-norms specify nominal performance and robust stability objectives. Periodic controllers are found using existing LTI H-infinity control theory, and causality is satisfied using two techniques which can be added easily to H-infinity solvers: loop-shifting and Q-parameterization. The resulting controllers are high-gain, narrowband-pass, periodic filters; the closed-loop sensitivity has a zero at the disturbance frequency, and the disturbance is in the sensitivity's null space. It is also shown that classical designs do not achieve the same performance levels as periodic controllers. Similar developments are made using the feedforward disturbance rejection architecture. Objectives are given which minimize the weighted infinity-norm of the controlled system. Such feedforward controllers achieve perfect disturbance rejection. A multivariable equivalent of the tapped-delay line is used in the description of periodic FIR filters. In addition, periodic FIR filters are made adaptive using an algorithm similar to filtered-X least mean square (LMS) but modified for periodic systems. / Ph. D.

Input-output systems with time periodic parameters are commonly found in nature (e.g., oceanic movements) and engineered systems (e.g., vibrations due to gyroscopic forces in vehicles). In a broader sense, periodic behaviors can arise when there is a dynamic equi- librium between inertia and various balancing forces. A classic example is a structure in a steady wind or current that undergoes large oscillations due to vortex shedding or flutter. Such phenomena can have either positive or negative outcomes, like the efficient operation of wind turbines or the collapse of the Tacoma Narrows Bridge. While the systems mentioned here are typically all modeled as systems of nonlinear partial differential equations, the pe- riodic behaviors of interest typically form part of a stable "center manifold," the analysis of which prompts linearization around periodic solutions. The linearization produces linear, time periodic partial differential equations. Discretization in the spatial dimension typically produces large scale linear time-periodic systems of ordinary differential equations. The need to simulate responses to a variety of inputs motivates the development of effective model re- duction tools. We seek to address this need by investigating partial Floquet transformations, which serve to simultaneously remove the time dependence of the system and produce effec- tive reduced order models. In this thesis we describe the time-periodic analogs of important concepts for time invariant model reduction such as the transfer function and the H2 norm. Building on these concepts we present an algorithm which converges to the dominant poles of an infinite dimensional operator. These poles may then be used to produce the partial Floquet transform. / Master of Science / Systems that exhibit time periodic behavior are commonly found both in nature and in human-made structures. Often, these system behaviors are a result of periodic forces, such as the Earth's rotation, which leads to tidal forces and daily temperature changes affecting atmospheric and oceanic movements. Similarly, gyroscopic forces in vehicles can cause no- ticeable vibrations and noise. In a broader sense, periodic behaviors can arise when there's a dynamic equilibrium between inertia and various balancing forces. A classic example is a structure in a steady wind or current that undergoes large oscillations due to vortex shedding or flutter. Such phenomena can have either positive or negative outcomes, like the efficient operation of wind turbines or the collapse of the Tacoma Narrows Bridge. Linear Time-Periodic (LTP) systems are crucial in understanding, simulating, and control- ling such phenomena, even in situations where the fundamental dynamics are non-linear. This importance stems from the fact that the periodic behaviors of interest typically form part of a stable "center manifold," especially under minor disturbances. In natural systems, the absence of this stability would mean these oscillatory patterns would not be commonly observed, and in engineered systems, they would not be desirable. Additionally, the process of deriving periodic solutions from nonlinear systems often involves solving large scale linear periodic systems, raising the question of how to effectively reduce the complexity of these models, a question we address in this thesis.

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.

There are a lot of interesting problems in surface chemistry where quantum chemistry could give great insight, like reaction mechanisms in heterogeneous catalysis, the effect of surface functionalization on semiconductors, or the influence of defects on the reactivity of crystal surfaces. Plane wave based methods applied to crystals cannot handle problems that are localized in nature like surface defects and adsorbates. On the other hand, molecular electronic structure techniques, which describe these effects and the locality of the electronic correlation well, are too computationally expensive to use on these systems. In this work, we introduce translationally-transformed coupled-cluster (TT-CC) theory, a new electronic structure method that incorporates the periodicity of crystals and the locality of electronic correlation. This is accomplished by encoding the periodicity into the amplitudes, instead of using plane waves, in order to be able to use a local basis to reflect the decay of the electronic correlation at sufficiently large distances. This avoids the calculation of redundant amplitudes. Perfectly periodic surfaces are envisioned as reference wavefunctions for localized defects and chemical reactions. The working equations in one dimension are derived starting from the amplitude equations of conventional coupled cluster singles and doubles (CCSD) on an infinite system and rearranging them such that the distance to an anonymous cell is an explicit degree of freedom, L. The formally infinite summations can be truncated by systematically neglecting numerically insignificant amplitudes. The generalization of the amplitude equations to higher dimensions is straightforward, albeit laborious. We show a general strategy to incorporate defects. These will be subjects of future dissertations. We present a proof of principle for 1-dimensional chemical systems of increasing size (He, H2, Be, Ne and N2) using the 6-31G basis set. We compute the energies, with TT-CCSD, at different distances and compared them against the perfectly periodic intensive energy (PPIE) using conventional CCSD. All results, up to L=3, show that the energies of TT-CCSD converge to the PPIE. For neon, TT-CCSD shows an error of -6.2x10-6 Eh per cell against the PPIE at the bonding distance with the potential computational cost of 7 cells using CCSD, as an upper bound. For nitrogen, TT-CCSD shows an error of -2.2x10-9 Eh at 7.5 Å per cell with the same potential cost as upper bound.

As a perturbation method, averaging is a mathematical tool for dynamic analysis of time-periodic and space-periodic dynamical systems, including those governed by partial differential equations. The control design procedure presented in this work uses averaging techniques, the well-developed linear control strategies, and finite element methods. The controller is designed based on the linear averaged dynamics of a time- or space-periodic system. The controller is then used for trajectory tracking or stabilization of the periodic system. The applicability and performance of the suggested method depend on different physical parameters of the periodic system and the control parameters of the controller. The effects of these parameters are discussed in this work. Numerical simulations show acceptable performance of the proposed control design strategy for two linear and nonlinear time- and space-periodic systems, namely, the one-dimensional heat equation and the Chafee-Infante equation with periodic coefficients. / M.S. / Dynamic analysis and control of dynamical systems with varying parameters is a challenging task. It is always of great help if one can perform the analyses for an approximate system with constant parameters and use the results to study and control the original system with varying parameters. Averaging is a mathematical tool that is used to approximate a system with periodic parameters with a ``simpler'' system with constant parameters. In this research averaging is used for design of controllers for systems with periodic parameters. First, an approximate system with constant parameters, called the averaged system, is determined. The averaged system is used for design of a controller which can be then be used for the original system with periodic parameters.

O princípio de invariância de LaSalle estuda o comportamento assintótico das soluções sem conhecer as soluções das equações diferenciais.Para isto,utiliza uma função auxiliar V usualmente chamada de função de Lyapunov. Este trabalho apresenta um princípio de invariância fuzzy e sua versão global para a classe de sistemas dinâmicos fuzzy descrito, via extensão de Zadeh,por equações diferenciais autônomas com incertezas na condição inicial.Ainda, apresentamos um princípio de invariância uniforme, no qual não se exige que a derivada da função de Lyapunov seja sempre definida negativa, para a classe de sistemas dinâmicos não lineares não autônomos que são descritos por um conjunto de equações diferenciais ordinárias periódicas. Aplicações para as duas classes de sistemas foram desenvolvidas. / The LaSalle\'s invariance principle studies the asymptotic behavior of the solutions without requiring the knowledge of the solutions of differential equations. For this, it uses an auxiliary function V usually called Lyapunov function. This work proposes a fuzzy invariance principle and its global version for the class of fuzzy dynamic systems described, via Zadeh\'s extension, by autonomous ordinary differential equation with uncertainties in the initial condition. Moreover, we develop an uniform invariance principle, in which the derivative of the Lyapunov function is not required to be always negative definite, for the class of non autonomous non linear dynamical system described by a set of periodic ordinary differential equations. Applications for the two classes of systems are also developed.

O princípio de invariância de LaSalle estuda o comportamento assintótico das soluções sem conhecer as soluções das equações diferenciais.Para isto,utiliza uma função auxiliar V usualmente chamada de função de Lyapunov. Este trabalho apresenta um princípio de invariância fuzzy e sua versão global para a classe de sistemas dinâmicos fuzzy descrito, via extensão de Zadeh,por equações diferenciais autônomas com incertezas na condição inicial.Ainda, apresentamos um princípio de invariância uniforme, no qual não se exige que a derivada da função de Lyapunov seja sempre definida negativa, para a classe de sistemas dinâmicos não lineares não autônomos que são descritos por um conjunto de equações diferenciais ordinárias periódicas. Aplicações para as duas classes de sistemas foram desenvolvidas. / The LaSalle\'s invariance principle studies the asymptotic behavior of the solutions without requiring the knowledge of the solutions of differential equations. For this, it uses an auxiliary function V usually called Lyapunov function. This work proposes a fuzzy invariance principle and its global version for the class of fuzzy dynamic systems described, via Zadeh\'s extension, by autonomous ordinary differential equation with uncertainties in the initial condition. Moreover, we develop an uniform invariance principle, in which the derivative of the Lyapunov function is not required to be always negative definite, for the class of non autonomous non linear dynamical system described by a set of periodic ordinary differential equations. Applications for the two classes of systems are also developed.

La validation des systèmes Mécatroniques tels que la supervision d'une chaînede traction hybride utilise de plus en plus la simulation Hardware-in-the-Loop. Cela consiste à interconnecter des composants réels du système et des composantssimulés. On parle alors de simulation temps réel car les composants simulés doivent avoir le même comportement temporel que les réels. En d'autres termes, la simulation temps réel d'un modèle nécessite le maillage de l'évolution du temps simulé sur celle du temps réel. Sur les outils existants, l'intégration de modèles physiques représentatifs se heurte à des modèles de calculs et des contraintes temporelles pessimistes. Cette thèse propose des solutions, analytiques ou tirées d'expérimentations au sein d'IFP Energies nouvelles, pour l'implantation adéquate de la simulation temps réel de modèles physiques. Des métriques ont été introduites pourqualifier et quantifier la validité d'une simulation temps réel. Une définition des contraintes temporelles propres à la simulation temps réel a été proposée, accompagnée des règles régissant leur propagation aux calculs sous-jacents. Ces méthodes ont ensuite été déclinées en étude d'ordonnançabilité pour deux systèmes au comportement pseudo périodique : un simulateur de moteur à combustion et un contrôle moteur. Des expérimentations sur la simulation temps réel distribuée d'un moteur, intégrant des modèles phénoménologiques de combustion, ont permis de justifier et de validerles méthodes proposées. Les dégradations dues à la simulation distribuée ont été corrigées par un mécanisme d'extrapolation paramétrable dont le coût d'exécution a été étudié / Validation of Mechatronics systems such as hybrid automotive powertrains isincreasingly relying on Harware-in-the-Loop simulation. It consists in interconnecting real components to the real-time simulation of physical models, involving their timely behavior to match their real counterpart. In other words, the evolution of simulated and real time have to be meshed together. Involving representative physical models is currently hindered by both pessimistic models of computation and temporal constraints.This thesis proposes several analytical and experimental answers, carried out at IFP Energies nouvelles, toward the proper implantation of real-time simulation of physical models. Several metrics able to qualify and quantify the success of real-time simulation were proposed, as well as the definition of its dedicated timed constraints, along with the rules for their propagation toward the underlying computations involved.Then, we showed how to take advantage of the pseudo periodic behavior of two systems to reach better schedulability bounds for the real-time simulation of : a combustion engine and an engine control. The methods discussed were then accounted for and validated by several experiments, involving the distributed real-time simulation of an engine including phenomenological combustion models. Also, the perturbations induced by the distributed simulation were addressed by proposing a configurable extrapolation mechanism, taking into account its execution time

Cette thèse se situe dans le cadre de l’analyse et de la synthèse des systèmes périodiques. Les contributions présentées dans ce mémoire portent sur la commande sous contraintes des systèmes linéaires discrets périodiques. Ces contraintes, portant sur l’état du système et/ ou sur la commande, peuvent être des contraintes de positivité ou de bornitude. Dans ce travail, des conditions d’analyse en stabilité et positivité des systèmes périodiques en termes de LMI (Inégalité Matricielle Linéaire) strictes, sont présentées. Ces outils d’analyse ont ensuite permis d’élaborer une loi de commande par retour d’état périodique. Les résultats obtenus sont exploités par la suite pour développer une commande par retour d’état périodique robuste pour les systèmes périodiques incertains. Des conditions de stabilisation robuste sont élaborées en utilisant la S-procédure. En outre, des conditions de stabilité et stabilisation par retour d’état périodique des systèmes périodiques avec retards sont établies. Le problème de stabilisation de ce type de systèmes sous un certain nombre de contraintes est résolu en suivant deux approches, la première est basée sur les techniques de Lyapunov la seconde fait appel à la programmation linéaire. Outre la notion de stabilité, la notion de performance des systèmes en boucle fermée est traitée. Pour cela, nous proposons une commande de type H∞ pour résoudre le problème de rejet de perturbations. / This thesis deals with the analysis and the control problem of periodic linear discrete systems (PLDS). The contributions presented in this work focuses on the constrained control of PLDS. Conditions for stability analysis and positivity are established in terms of strict LMI (Linear Matrix Inequalities). The stabilization of PLDS under the condition that the closed-loop system is positive and stable is addressed as well as the case of bounded state and/ or control variables. The obtained results are then extended to the synthesis of robust state feedback controllers, where some of which are based on the S − procedure technique. Furthermore, some conditions of stability and stabilization of PLDS with delays are established. The problem of stabilization of constrained PLDS is addressed based on the Lyapunov techniques or the Linear Programming techniques. The robust H∞ state feedback control in which both robust stability and a prescribed H∞ performance are required is investigated.

Modern control theory is today an interdisciplinary area of research. Just as much as this can be problematic, it also provides a rich research environment where practice and theory meet. This Thesis is conducted in the borderline between computing science (numerical analysis) and applied control theory. The design and analysis of a modern control system is a complex problem that requires high qualitative software to accomplish. Ideally, such software should be based on robust methods and numerical stable algorithms that provide quantitative as well as qualitative information. The introduction of the Thesis is dedicated to the underlying control theory and to introduce the reader to the main subjects. Throughout the Thesis, the theory is illustrated with several examples, and similarities and differences between the terminology from mathematics, systems and control theory, and numerical linear algebra are highlighted. The main contributions of the Thesis are structured in two parts, dealing with two mainly unrelated subjects. Part I is devoted to the qualitative information which is provided by the stratification of orbits and bundles of matrices, matrix pencils and system pencils. Before the theory of stratification is established the reader is introduced to different canonical forms which reveal the system characteristics of the model under investigation. A stratification reveals which canonical structures of matrix (system) pencils are near each other in the sense of small perturbations of the data. Fundamental concepts in systems and control, like controllability and observability of linear continuous-time systems, are considered and it is shown how these system characteristics can be investigated using the stratification theory. New results are presented in the form of the cover relations (nearest neighbours) for controllability and observability pairs. Moreover, the permutation matrices which take a matrix pencil in the Kronecker canonical form to the corresponding system pencil in (generalized) Brunovsky canonical form are derived. Two novel algorithms for determining the permutation matrices are provided. Part II deals with numerical methods for solving periodic Riccati differential equations (PRDE:s). The PRDE:s under investigation arise when solving the linear quadratic regulator (LQR) problem for periodic linear time-varying (LTV) systems. These types of (periodic) LQR problems turn up for example in motion planning of underactuated mechanical systems, like a humanoid robot, the Furuta pendulum, and pendulums on carts. The constructions of the nonlinear controllers are based on linear versions found by stabilizing transverse dynamics of the systems along cycles. Three different methods explicitly designed for solving the PRDE are evaluated on both artificial systems and stabilizing problems originating from experimental control systems. The methods are the one-shot generator method and two recently proposed methods: the multi-shot method (two variants) and the SDP method. As these methods use different approaches to solve the PRDE, their numerical behavior and performance are dependent on the nature of the underlying control problem. Such method characteristics are investigated and summarized with respect to different user requirements (the need for accuracy and possible restrictions on the solution time). / Modern reglerteknik är idag i högsta grad ett interdisciplinärt forskningsområde. Lika mycket som detta kan vara problematiskt, resulterar det i en stimulerande forskningsmiljö där både praktik och teori knyts samman. Denna avhandling är utförd i gränsområdet mellan datavetenskap (numerisk analys) och tillämpad reglerteknik. Att designa och analysera ett modernt styrsystem är ett komplext problem som erfordrar högkvalitativ mjukvara. Det ideala är att mjukvaran består av robusta metoder och numeriskt stabila algoritmer som kan leverera både kvantitativ och kvalitativ information.Introduktionen till avhandlingen beskriver grundläggande styr- och reglerteori samt ger en introduktion till de huvudsakliga problemställningarna. Genom hela avhandlingen illustreras teori med exempel. Vidare belyses likheter och skillnader i terminologin som används inom matematik, styr- och reglerteori samt numerisk linjär algebra. Avhandlingen är uppdelade i två delar som behandlar två i huvudsak orelaterade problemklasser. Del I ägnas åt den kvalitativa informationen som ges av stratifiering av mångfalder (orbits och bundles) av matriser, matrisknippen och systemknippen. Innan teorin för stratifiering introduceras beskrivs olika kanoniska former, vilka var och en avslöjar olika systemegenskaper hos den undersökta modellen. En stratifiering ger information om bl.a. vilka kanoniska strukturer av matrisknippen (systemknippen) som är nära varandra med avseende på små störningar i datat. Fundamentala koncept i styr- och reglerteori behandlas, så som styrbarhet och observerbarhet av linjära tidskontinuerliga system, och hur dessa systemegenskaper kan undersökas med hjälp av stratifiering. Nya resultat presenteras i form av relationerna för täckande (närmsta grannar) styrbarhets- och observerbarhets-par. Dessutom härleds permutationsmatriserna som tar ett matrisknippe i Kroneckers kanoniska form till motsvarande systemknippe i (generaliserade) Brunovskys kanoniska form. Två algoritmer för att bestämma dessa permutationsmatriser presenteras. Del II avhandlar numeriska metoder för att lösa periodiska Riccati differentialekvationer (PRDE:er). De undersökta PRDE:erna uppkommer när ett linjärt kvadratiskt regulatorproblem för periodiska linjära tidsvariabla (LTV) system löses. Dessa typer av (periodiska) regulatorproblem dyker upp till exempel när man planerar rörelser för understyrda (underactuated) mekaniska system, så som en humanoid (mänsklig) robot, Furuta-pendeln och en vagn med en inverterad (stående) pendel. Konstruktionen av det icke-linjära styrsystemet är baserat på en linjär variant som bestäms via stabilisering av systemets transversella dynamik längs med cirkulära banor. Tre metoder explicit konstruerade för att lösa PRDE:er evalueras på både artificiella system och stabiliseringsproblem av experimentella styrsystem. Metoderna är sk. en- och flerskotts metoder (one-shot, multi-shot) och SDP-metoden. Då dessa metoder använder olika tillvägagångssätt för att lösa en PRDE, beror dess numeriska egenskaper och effektivitet på det underliggande styrproblemet. Sådana metodegenskaper undersöks och sammanfattas med avseende på olika användares behov, t.ex. önskad noggrannhet och tänkbar begränsning i hur lång tid det får ta att hitta en lösning.