Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric. In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization. We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study. / Master of Science / Mathematical modeling of dynamical systems provides a powerful means for studying physical phenomena. Due the complexities of real-world problems, many mathematical models face computational difficulties due to the costs of accurate modeling. Model-order reduction of large-scale dynamical systems circumvents this by approximating the large-scale model with a ``smaller'' one that still accurately describes the problem of interest. Balanced truncation model reduction for linear systems is one such example, yielding reduced-order models that satisfy a tractable upper bound on the approximation error. This work investigates conditions in which this bound is known to hold with equality, becoming an exact formula for the error in reduction. We additionally show how to determine these conditions for a special class of linear dynamical systems known as arrowhead systems, which arise in special applications of network modeling. We provide an example of one such system from power systems modeling that motivated our study.

In this thesis, we apply nonlinear Koopman mode analysis to decompose the swing dynamics of a power system into modes of oscillation, which are identified by analyzing the Koopman operator, a linear infinite-dimensional operator that may be defined for any nonlinear dynamical system. Specifically, power system modes of oscillation are identified through spectral analysis of the Koopman operator associated with a particular observable. This means that they can be determined directly from measurements. These modes, referred to as Koopman modes, are single-frequency oscillations, which may be extracted from nonlinear swing dynamics under small and large disturbances. They have an associated temporal frequency and growth rate. Consequently, they may be viewed as a nonlinear generalization of eigen-modes of a linearized system. Koopman mode analysis has been also applied to identify coherent swings and coherent groups of machines of a power system. This will allow us to carry out a model reduction of a large-scale system and to derive a precursor to monitor the loss of transient stability. / Master of Science

Due to the complexity, multi-scale, and multi-physics nature of the mathematical models for fires, current numerical models require too much computational effort to be useful in design and real-time decision making, especially when dealing with fires over large domains. To reduce the computational time while retaining the complexity of the domain and physics, our research has focused on several reduced-order modeling techniques. Our contributions are improving wildland fire reduced-order models (ROMs), creating new ROM techniques for nonlinear systems, and preserving optimality when discretizing a continuous-time ROM. Currently, proper orthogonal decomposition (POD) is being used to reduce wildland fire-spread models with limited success. We use a technique known as the discrete empirical interpolation method (DEIM) to address the slowness due to the nonlinearity. We create new methods to reduce nonlinear models, such as the Burgers' equation, that perform better than POD over a wider range of input conditions. Further, these ROMs can often be constructed without needing to capture full-order solutions a priori. This significantly reduces the off-line costs associated with creating the ROM. Finally, we investigate methods of time-discretization that preserve the optimality conditions in a certain norm associated with the input to output mapping of a dynamical system. In particular, we are able to show that the Crank-Nicholson method preserves the optimality conditions, but other single-step methods do not. We further clarify the need for these discrete-time ROMs to match at infinity in order to ensure local optimality. / Ph. D.

Generally, large-scale dynamical systems pose tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges we use several model reduction techniques. For stable linear models these techniques work very well and provide good approximations for the full model. However, large-scale unstable systems arise in many applications. Many of the known model reduction methods are not very robust, or in some cases, may not even work if we are dealing with unstable systems. When approximating an unstable system by a reduced order model, accuracy is not the only concern. We also need to consider the structure of the reduced order model. Often, it is important that the number of unstable poles in the reduced system is the same as the number of unstable poles in the original system. The Iterative Rational Krylov Algorithm (IRKA) is a robust model reduction technique which is used to locally reduce stable linear dynamical systems optimally in the ℋ₂-norm. While we cannot guarantee that IRKA reduces an unstable model optimally, there are no numerical obstacles to the reduction of an unstable model via IRKA. In this thesis, we investigate IRKA's behavior when it is used to reduce unstable models. We also consider systems for which we cannot obtain a first order realization of the transfer function. We can use Realization-independent IRKA to obtain a reduced order model which does not preserve the structure of the original model. In this paper, we implement a structure preserving algorithm for systems with nonlinear frequency dependency. / Master of Science

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.

Mathematical models of physical systems are pervasive in engineering. These models can be used to analyze properties of the system, to simulate the system, or synthesize controllers. However, many of these models are too complex or too large for standard analysis and synthesis methods to be applicable. Hence, there is a need to reduce the complexity of models. In this thesis, techniques for reducing complexity of large linear time-invariant (lti) state-space models and linear parameter-varying (lpv) models are presented. Additionally, a method for synthesizing controllers is also presented. The methods in this thesis all revolve around a system theoretical measure called the H2-norm, and the minimization of this norm using nonlinear optimization. Since the optimization problems rapidly grow large, significant effort is spent on understanding and exploiting the inherent structures available in the problems to reduce the computational complexity when performing the optimization. The first part of the thesis addresses the classical model-reduction problem of lti state-space models. Various H2 problems are formulated and solved using the proposed structure-exploiting nonlinear optimization technique. The standard problem formulation is extended to incorporate also frequency-weighted problems and norms defined on finite frequency intervals, both for continuous and discrete-time models. Additionally, a regularization-based method to account for uncertainty in data is explored. Several examples reveal that the method is highly competitive with alternative approaches. Techniques for finding lpv models from data, and reducing the complexity of lpv models are presented. The basic ideas introduced in the first part of the thesis are extended to the lpv case, once again covering a range of different setups. lpv models are commonly used for analysis and synthesis of controllers, but the efficiency of these methods depends highly on a particular algebraic structure in the lpv models. A method to account for and derive models suitable for controller synthesis is proposed. Many of the methods are thoroughly tested on a realistic modeling problem arising in the design and flight clearance of an Airbus aircraft model. Finally, output-feedback H2 controller synthesis for lpv models is addressed by generalizing the ideas and methods used for modeling. One of the ideas here is to skip the lpv modeling phase before creating the controller, and instead synthesize the controller directly from the data, which classically would have been used to generate a model to be used in the controller synthesis problem. The method specializes to standard output-feedback H2 controller synthesis in the lti case, and favorable comparisons with alternative state-of-the-art implementations are presented.

High Redundancy Actuation (HRA) is a novel type of fault tolerant actuator. By comprising a relatively large number of actuation elements, faults in the elements can be inherently accommodated without resulting in a failure of the complete actuation system. By removing the possibility of faults detection and reconfiguration, HRA can provide high reliability and availability. The idea is motivated by the composition of human musculature. Our musculature can sustain damage and still function, sometimes with reduced performance, and even complete loss of a muscle group can be accommodated through kinematics redundancy, e.g. the use of just one leg. Electro-mechanical actuation is used as single element inside HRA. This thesis is started with modelling and simulation of individual actuation element and two basic structures to connect elements, in series and in parallel. A relatively simple HRA is then modelled which engages a two-by-two series-in-parallel configuration. Based on this HRA, position feedback controllers are designed using both classical and optimal algorithms under two control structures. All controllers are tested under both healthy and faults injected situations. Finally, a hardware demonstrator is set up based simulation studies. The demonstrator is controlled in real time using an xPC Target system. Experimental results show that the HRA can continuously work when one element fails, although performance degradation can be expected.

This dissertation deals with some computational and analytic challenges for dynamic process operations using first-principles models. For processes with significant spatial variations, spatially distributed first-principles models can provide accurate physical descriptions, which are crucial for offline dynamic simulation and optimization. However, the large amount of time required to solve these detailed models limits their use for online applications such as nonlinear model predictive control (NMPC). To cope with the computational challenge, we develop computationally efficient and accurate dynamic reduced order models which are tractable for NMPC using temporal and spatial model reduction techniques. Then we introduce an input and state blocking strategy for NMPC to further enhance computational efficiency. To improve the overall economic performance of process systems, one promising solution is to use economic NMPC which directly optimizes the economic performance based on first-principles dynamic models. However, complex process models bring challenges for the analysis and design of stable economic NMPC controllers. To solve this issue, we develop a simple and less conservative regularization strategy with focuses on a reduced set of states to design stable economic NMPC controllers. In this thesis, we study the operation problems of a solid sorbent-based CO2 capture system with bubbling fluidized bed (BFB) reactors as key components, which are described by a large-scale nonlinear system of partial-differential algebraic equations. By integrating dynamic reduced models and blocking strategy, the computational cost of NMPC can be reduced by an order of magnitude, with almost no compromise in control performance. In addition, a sensitivity based fast NMPC algorithm is utilized to enable the online control of the BFB reactor. For economic NMPC study, compared with full space regularization, the reduced regularization strategy is simpler to implement and lead to less conservative regularization weights. We analyze the stability properties of the reduced regularization strategy and demonstrate its performance in the economic NMPC case study for the CO2 capture system.

La nécessité de simuler des systèmes complexes et multi-physiques est de plus en plus courante dans l’industrie, en particulier avec l’avènement de la conception mécatronique. Ce phénomène couplé à la pression économique poussant les industriels dans la voie de l’optimisation de leurs produits conduit à une augmentation forte des temps de simulation que les progrès techniques ne parviennent pas à compenser. Les travaux menés lors de cette thèse ont permis de proposer une approche hybride analytique/éléments finis pour la simulation temporelle de la dynamique des transmissions par engrenages en présence de non-linéarités de contact. Couplée à une réduction des modèles éléments finis, cette approche permet la résolution rapide de la dynamique d’un réducteur et offre ainsi la possibilité d’intégrer le réducteur comme composant d’un système complexe tel qu’un système mécatronique. La résolution de la dynamique du réducteur peut être menée en parallèle des autres physiques en prenant en compte des couplages forts. L’optimisation d’un tel système est abordée sur un exemple pour clore cette étude. / The need to simulate complex and multi-physics systems is increasingly common in the industry, especially with the advent of mechatronic design. This coupled with economic pressure pushing the industry towards optimizing their products led to a strong increase in simulation time that technological advances can not compensate. An hybrid method analytical/finite element has been developed for the time domain simulation of gear transmissions involving contact non-linearities. Coupled with a reduction of finite element models, this approach allows fast resolution of the dynamics of a gearbox. Consequently, it is possible to integrate a gearbox as a part of a more complex mechatronic system. All physical phenomena involved in such a complex product are solved at the same time allowing strong coupling to be considered. The optimization of such a system is discussed with an example to conclude this study.

La perturbation d’un système peut donner lieu à de la propagation d’onde. Une façon classique d’appréhender ce phénomène est de rechercher les modes propres de vibration du milieu. Mathématiquement, trouver ces modes consiste à rechercher les valeurs et fonctions propres de l’opérateur de propagation. Cependant, d’un point de vue numérique, l’opération peut s’avérer coûteuse car les matrices peuvent avoir de très grandes tailles. En outre, dans la plupart des applications, des incertitudes sont inévitablement associées à notre modèle. La question se pose alors de savoir s’il faut attribuer d’importantes ressources de calcul pour une simulation dont la précision du résultat n’est pas assurée. Nous proposons dans cette thèse une démarche qui permet à la fois de mieux comprendre l’influence des incertitudes sur la propagation et de réduire considérablement les coûts de calcul pour la propagation des infrasons dans l’atmosphère. L’idée principale est que tous les modes n’ont pas la même importance et souvent, seule une poignée d’entre eux suffit à décrire le phénomène sans perte notable de précision. Ces modes s’avèrent être ceux qui sont les plus sensibles aux perturbations atmosphériques. Plus précisément, l’analyse de sensibilité permet d’identifier les structures de l’atmosphère les plus influentes, les groupes de modes qui leur sont associés et les parties du signal infrasonore qui leur correspondent. Ces groupes de modes peuvent être spécifiquement ciblés dans un calcul de spectre au moyen de techniques de projection sur des sous-espace de Krylov, ce qui implique un gain important en coût de calcul. Cette méthode de réduction de modèle peut être appliquée dans un cadre statistique et l’estimation de l’espérance et de la variance du résultat s’effectue là aussi sans perte notable de précision et avec un coût réduit. / The perturbation of a system can give rise to wave propagation. A classical approach to understand this phenomenon is to look for natural modes of vibration of the medium. Mathematically, finding these modes requires to seek the eigenvalues and eigenfunctions of the propagation operator. However, from a numerical point of view, the operation can be costly because the matrices can be of very large size. Furthermore, in most applications, uncertainties are inevitably associated with our model. The question then arises as to whether we should allocate significant computational resources for simulation while the accuracy of the result is not guaranteed. We propose in this thesis an approach that allows both a better understanding of the influence of uncertainties on the propagation and a significant decrease of computational costs for infrasound propagation in the atmosphere. The main idea is that all modes do not have the same importance and only a few of them is often sufficient to account for the phenomenon without a significant loss of accuracy. These modes appear to be those which are most sensitive to atmospheric disturbances. Specifically, a sensitivity analysis is used to identify the most influential structures of the atmosphere, the associated groups of modes and their associated parts of the infrasound signal. These groups of modes can be specifically targeted in a spectrum calculation with the projection of the operator onto Krylov subspaces, that allows a significant decrease of the computational cost. This method of model reduction can be applied in a statistical framework as well and estimations of the expectation and the variance of the results are carried out without a significant loss of accuracy and still with a low cost.