Abstract In this dissertation, the eigenvalue analysis and decentralized robust controller design of uncertain multi-time-scale system with parametrical perturbations are considered. Because the eigenvalues of the multi-time-scale systems cluster in some difference regions of the complex plane, we can use the singular perturbation method to separate the systems into some subsystems. These subsystems are independent to each other. We can discuss the properties of eigenvalues and design controller for these subsystem respectively, then we composite these controllers to a decentralized controller. The eigenvalue positions dominate the stability and the performance of the dynamic system. However, we cannot obtain the precise position of the eigenvalues from the influence of parametrical perturbations. The sufficient conditions of the eigenvalues clustering for the multi-time-scale systems will be discussed. The uncertainties consider as unstructured and structured perturbations are taken into considerations. The design algorithm provides for designing a decentralized controller that can assign the poles to our respect regions. The specified regions are half-plane and circular disk. Furthermore, the concepts of decentralized control and optimal control are used to design the linear quadratic regulator (LQR) controller and linear quadratic Gaussian (LQG) controller for the perturbed multi-time-scale systems. That is, the system can get the optimal robust performance. The bound of the singular perturbation parameter would influence the robust stability of the multi-time-scale systems. Finally, the sufficient condition to obtain the upper bound of the singular perturbation parameter presented by the Lyapunov method and matrix norm. The condition also extends for the pole assignment in the specified regions of each subsystem respectively. The illustrative examples are presented behind each topic. They show the applicability of the proposed theorems, and the results are satisfactory.

<p>Problems in structural dynamics that involve rapid evolution of the material at multiple scales of length and time are challenging to solve numerically. One such problem is that of a structure un- dergoing fracture, where the material in the vicinity of a crack front may experience high stresses and strains while the remainder of the structure may be unaffected by it. Usually, such problems are solved using numerical methods based on a finite element discretization in space and a finite difference time-stepping scheme to capture dynamic response. Regions of interest within the struc- ture, where high transients are expected, are usually modeled with a fine discretization in space and time for better accuracy. In other regions of the model where the response does not change rapidly, a coarser discretization suffices and helps keep the computational cost down. This variation in spatial and temporal discretization is achieved through domain decomposition and multi-time-step coupling methods which allow the use of different levels of mesh discretization and time-steps in different regions of the mesh.</p>

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The present work introduces a methodology to obtain a discrete time state space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. This is the first time the connection between the EMTP nodal analysis solution and a corresponding state-space formulation is presented. Compared to conventional state space solutions, the nodal EMTP solution is computationally much more efficient. Compared to the phasor solutions used in transient stability analysis, the proposed approach captures a much wider range of eigenvalues and system operating states. A fundamental advantage of extracting the system eigenvalues directly from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. The system's trajectory can be accurately traced and the calculated eigenvalues and eigenvectors correctly represent the system's instantaneous dynamics. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time scale solutions. The proposed technique can be implemented as an extension to any EMTP-based simulator. Within our UBC research group, it is aimed at extending the capabilities of our real-time PC-cluster Object Virtual Network Integrator (OVNI) simulator.

The present work introduces a methodology to obtain a discrete time state space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. This is the first time the connection between the EMTP nodal analysis solution and a corresponding state-space formulation is presented. Compared to conventional state space solutions, the nodal EMTP solution is computationally much more efficient. Compared to the phasor solutions used in transient stability analysis, the proposed approach captures a much wider range of eigenvalues and system operating states. A fundamental advantage of extracting the system eigenvalues directly from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. The system's trajectory can be accurately traced and the calculated eigenvalues and eigenvectors correctly represent the system's instantaneous dynamics. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time scale solutions. The proposed technique can be implemented as an extension to any EMTP-based simulator. Within our UBC research group, it is aimed at extending the capabilities of our real-time PC-cluster Object Virtual Network Integrator (OVNI) simulator.

The present work introduces a methodology to obtain a discrete time state space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. This is the first time the connection between the EMTP nodal analysis solution and a corresponding state-space formulation is presented. Compared to conventional state space solutions, the nodal EMTP solution is computationally much more efficient. Compared to the phasor solutions used in transient stability analysis, the proposed approach captures a much wider range of eigenvalues and system operating states. A fundamental advantage of extracting the system eigenvalues directly from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. The system's trajectory can be accurately traced and the calculated eigenvalues and eigenvectors correctly represent the system's instantaneous dynamics. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time scale solutions. The proposed technique can be implemented as an extension to any EMTP-based simulator. Within our UBC research group, it is aimed at extending the capabilities of our real-time PC-cluster Object Virtual Network Integrator (OVNI) simulator. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

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La réduction du temps de calcul lors de la résolution de problèmes d’évolution dans le cadre du calcul de structure constitue un enjeu majeur pour, par exemple, la mise en place de critères de rupture des pièces dans le secteur de l’aéronautique et de l’automobile. En particulier, la prédiction du cycle stabilisé des polymères sollicités sous chargement cyclique nécessite de résoudre un problème thermo-viscoélastique à grand nombre de cycles. La présence de différentes échelles de temps telles que le temps de relaxation (viscosité), le temps caractéristique associé au problème thermique et le temps du cycle de chargement conduit à un temps de calcul significatif lorsqu’un schéma incrémental est utilisé comme c’est le cas avec la méthode des éléments finis (MEF). De plus, un nombre important de données doit être stocké (au moins à chaque cycle). L’objectif de cette thèse est de proposer de nouvelles méthodes ainsi que d’étendre des méthodes existantes. Il est choisi de résoudre un problème thermique transitoire cyclique impliquant différentes échelles de temps avec l’objectif de réduire le temps de calcul réduit. Les méthodes proposées font partie des méthodes de réduction de modèles. Tout d’abord, la méthode de décomposition propre généralisée(PGD) a été étendue à un problème transitoire cyclique 3D non linéaire, la non-linéarité a été traitée en combinant la méthode PGD à la Méthode d’interpolation empirique discrète (DEIM), stratégie numérique déjà proposée dans la littérature. Les résultats ont montré l’efficacité de la PGD pour générer des résultats précis par rapport à la solution FEM avec une erreur relative inférieure à (1%). Ensuite, afin de réduire le temps de calcul, une autre approche alternative a été développée. Cette approche est basée sur l’utilisation d’une collection de modes, les modes les plus significatifs, issus de solutions PGD pour différentes échelles de temps et différentes valeurs de paramètres. Un dictionnaire regroupant ces modes est alors utilisé pour construire des solutions pour différents temps caractéristiques et différentes conditions aux limites, uniquement par projection de la solution sur les modes du dictionnaire. Cette approche a été adaptée pour traiter un problème faiblement couplé diffuso-thermique. La nouveauté de cette approche est de considérer un dictionnaire composé de bases spatio-temporelles et non pas uniquement de bases spatiales comme dans la fameuse méthode POD. Les résultats obtenus avec cette approche sont précis et permettent une réduction notable du temps de calcul on line. Néanmoins, lorsque différents temps de cycles sont considérés, le nombre de modes dans le dictionnaire augmente, ce qui en limite son utilisation. Afin de pallier cette limitation,une troisième stratégie numérique est proposée dans cette thèse. Elle consiste à considérer comme a priori connues des bases temporelles, elle est appelée stratégie mixte. L’originalité dans cette approche réside dans la construction de la base temporelle a prior basée sur l’analyse de Fourier de différentes simulations pour différents temps et différentes valeurs des paramètres. Une fois cette étude réalisée, une expression analytique des bases temporelles fonction des paramètres tels que le temps caractéristique et le temps du cycle est proposée. Les bases spatiales associées sont calculées à l’aide d’un algorithme type PGD. Cette méthode est ensuite testée pour la résolution de problèmes thermiques 3D sous chargement cyclique linéaires et non linéaires et un problème faiblement couplé thermo-diffusion. / In the framework of structural calculation, the reduction of computation time plays an important rolein the proposition of failure criteria in the aeronautic and automobile domains. Particularly, the prediction of the stabilized cycle of polymer under cyclic loading requires solving of a thermo-viscoelastic problem with a high number of cycles. The presence of different time scales, such as relaxation time (viscosity), thermal characteristic time (thermal), and the cycle time (loading) lead to a huge computation time when an incremental scheme is used such as with the Finite Element Method (FEM).In addition, an allocation of memory will be used for data storage. The objective of this thesis isto propose new techniques and to extend existent ones. A transient thermal problem with different time scales is considered in the aim of computation time reduction. The proposed methods are called model reduction methods. First, the Proper Generalized Decomposition method (PGD) was extended to a nonlinear transient cyclic 3D problems. The non-linearity was considered by combining the PGD method with the Discrete Empirical Interpolation Method (DEIM), a numerical strategy used in the literature. Results showed the efficiency of the PGD in generating accurate results compared to the FEM solution with a relative error less than 1%. Then, a second approach was developed in order to reduce the computation time. It was based on the collection of the significant modes calculated from the PGD method for different time scales. A dictionary assembling these modes is then used to calculate the solution for different characteristic times and different boundary conditions. This approach was adapted in the case of a weak coupled diffusion thermal problem. The novelty of this method is to consider a dictionary composed of spatio-temporal bases and not spatial only as usedin the POD. The results showed again an exact reproduction of the solution in addition to a huge time reduction. However, when different cycle times are considered, the number of modes increases which limits the usage of the approach. To overcome this limitation, a third numerical strategy is proposed in this thesis. It consists in considering a priori known time bases and is called the mixed strategy. The originality in this approach lies in the construction of a priori time basis based on the Fourier analysis of different simulations for different time scales and different values of parameters.Once this study is done, an analytical expression of time bases based on parameters such as the characteristic time and the cycle time is proposed. The related spatial bases are calculated using the PGD algorithm. This method is then tested for the resolution of 3D thermal problems under cyclic loading linear and nonlinear and a weak coupled diffusion thermal problem.