With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero. The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions. Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations. While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly. One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution. One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

Transformers are very important parts of any electrical network, and their size increase so does their price. Protecting these important devices is a daunting task due to the wide variety of operating conditions. This thesis develops a new protection scheme based on state estimation.The foundation upon which our protection scheme is built is the modeling of the single phase transformer system of equations. The transformer equations are composed of polynomial and differential equations and this system of equations involving the transformer's electrical quantities are modeled into a system of equations such that highest degree of each of the system's equations is quadratic―in a process named Quadratization and then integrated using a technique called Quadratic integration to give a set of algebraic companion equations that can be solved numerically to determine the health of the transformer.

We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

This thesis addressed non-smooth dynamics of lumped parameter systems, and was restricted to Filippov-type systems. The main objective of this thesis was twofold. Firstly, modeling aspects of Filippov-type non-smooth dynamical systems were addressed with an emphasis on the constitutive assumptions and mathematical structure behind these models. Secondly, robust algorithms were presented to obtain numerical solutions for various Filippov-type lumped parameter systems. Governing equations were written using two different mathematical approaches. The first approach was based on differential inclusions and the second approach was based on differential-algebraic equations. The differential inclusions approach is more amenable to mathematical analysis using existing mathematical tools. On the other hand, the approach based on differential-algebraic equations gives more insight into the constitutive assumptions of a chosen model and easier to obtain numerical solutions. Bingham-type models in which the force cannot be expressed in terms of kinematic variables but the kinematic variables can be expressed in terms of force were considered. Further, Coulomb friction was considered in which neither the force can be expressed in terms of kinematic variables nor the kinematic variables in terms of force. However, one can write implicit constitutive equations in terms of kinematic quantities and force. A numerical framework was set up to study such systems and the algorithm was devised. Towards the end, representative dynamical systems of practical significance were considered. The devised algorithm was implemented on these systems and the results were obtained. The results show that the setting offered by differential-algebraic equations is appropriate for studying dynamics of lumped parameter systems under implicit constitutive models.

Thesis (Ph. D.)--Aerospace Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Olivier A. Bauchau; Committee Member: Andrew Makeev; Committee Member: Carlo L. Bottasso; Committee Member: Dewey H. Hodges; Committee Member: Massimo Ruzzene. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Systems of differential-algebraic equations (DAEs) arise in many areas including chemical engineering, electrical circuit simulation, and robotics. Such systems are routinely generated by simulation and modeling environments, like MapleSim, Matlab/Simulink, and those based on the Modelica language. Before a simulation starts and a numerical solution method is applied, some kind of structural analysis (SA) is performed to determine the structure and the index of a DAE system. Structural analysis methods serve as a necessary preprocessing stage, and among them, Pantelides's graph-theory-based algorithm is widely used in industry. Recently, Pryce's Σ-method is becoming increasingly popular, owing to its straightforward approach and capability of analyzing high-order systems. Both methods are equivalent in the sense that (a) when one succeeds, producing a nonsingular Jacobian, the other also succeeds, and that (b) the two give the same structural index in the case of either success or failure. When SA succeeds, the structural results can be used to perform an index reduction process, or to devise a stage-by-stage solution scheme for computing derivatives or Taylor coefficients up to some order. Although such a success occurs on fairly many problems of interest, SA can fail on some simple, solvable DAEs with an identically singular Jacobian, and give incorrect structural information that usually includes the index. In this thesis, we focus on the Σ-method and investigate its failures. Aiming at making this SA more reliable, we develop two conversion methods for fixing SA failures. These methods reformulate a DAE on which the Σ-method fails into an equivalent problem on which SA is more likely to succeed with a nonsingular Jacobian. The implementation of our methods requires symbolic computations. We also combine our conversion methods with block triangularization of a DAE. Using a block triangular form of a Jacobian sparsity pattern, we identify which diagonal block(s) of the Jacobian is identically singular, and then perform a conversion on each singular block. This approach can reduce the computational cost and improve the efficiency of finding a suitable conversion for fixing SA's failures. / Thesis / Doctor of Philosophy (PhD)

Many critical systems are based on the combination of components from different physical domains (e.g. mechanical, electrical, hydraulic), and are mathematically modeled as Switched Multi-Domain Kirchhoff Networks (SMDKN). In this thesis, we tackle a major obstacle to formal verification of SMDKN, namely devising a global model amenable to verification in the form of a Hybrid Automaton. This requires the combination of the local dynamics of the components, expressed as Differential Algebraic Equations, according to Kirchhoff's laws, depending on the (exponentially many) operation modes of the network. We propose an automated SMT-based method to analyze networks from multiple physical domains, detecting which modes induce invalid (i.e. inconsistent) constraints, and to produce a Hybrid Automaton model that accurately describes, in terms of Ordinary Differential Equations, the system evolution in the valid modes, catching also the possible non-deterministic behaviors. The experimental evaluation demonstrates that the proposed approach allows several complex multi-domain systems to be formally analyzed and model checked against various system requirements.

This thesis is concerned with the efficient numerical simulation of finite element based flexible multibody systems. Scaling operations are systematically applied to the governing index-3 differential algebraic equations in order to solve the problem of ill conditioning for small time step sizes. The importance of augmented Lagrangian terms is demonstrated. The use of fast sparse solvers is justified for the solution of the linearized equations of motion resulting in significant savings of computational costs. Three time stepping schemes for the integration of the governing equations of flexible multibody systems are discussed in detail. These schemes are the two-stage Radau IIA scheme, the energy decaying scheme, and the generalized-α method. Their formulations are adapted to the specific structure of the governing equations of flexible multibody systems. The efficiency of the time integration schemes is comprehensively evaluated on a series of test problems. Formulations for structural and constraint elements are reviewed and the problem of interpolation of finite rotations in geometrically exact structural elements is revisited. This results in the development of a new improved interpolation algorithm, which preserves the objectivity of the strain field and guarantees stable simulations in the presence of arbitrarily large rotations. Finally, strategies for the spatial discretization of beams in the presence of steep variations in cross-sectional properties are developed. These strategies reduce the number of degrees of freedom needed to accurately analyze beams with discontinuous properties, resulting in improved computational efficiency.

Nesta pesquisa são analisados os mecanismos que levam um sistema elétrico de potência (SEP) à instabilidade proveniente de uma perturbação de grande porte e as formas de se avaliar diretamente a margem de estabilidade desse sistema quando o mesmo é modelado preservando a estrutura da rede de transmissão. O sistema foi matematicamente modelado por um conjunto de equações algébrico-diferenciais (EAD), que permite modelagens mais compreensivas da carga e da rede e possibilita melhor avaliação da estabilidade de um sistema quando comparado com o modelo de equações diferenciais ordinárias (EDO) utilizado tradicionalmente para o estudo de estabilidade transitória. A avaliação direta da margem de estabilidade do sistema de potência modelado por conjuntos de EAD foi realizada usando métodos diretos de análise com base no conceito de ponto de equilíbrio instável de controle (PEIC). Tais métodos permitem a obtenção da margem deforma local e rápida, sem requerer a integração numérica de equações diferenciais do modelo pós-falta. No entanto, existem alguns problemas abertos para se alcançar a completa fundamentação do método PEIC para modelos de EAD. Assim, neste estudo dá-se um passo nesta direção, mostrando que as definições existentes de PEIC e de outros pontos de interesse podem ser falhas, principalmente quando a trajetória de falta do sistema alcança superfícies singulares. Neste sentido, são propostos a correção destas definições e um método direto de detecção do PEIC. O método proposto é adequado para análise direta de estabilidade angular e de tensão de curto-termos devido a grandes perturbações e capaz de fornecer corretos tempos críticos de abertura e a identificação dos mecanismos de instabilização do sistema de EAD, mesmo quando as trajetórias do sistema alcançam superfícies singulares. / This thesis addresses to the mechanisms that lead an electric power system to instability due to large disturbances and to the methods to assess directly the stability margin when the system is modeled preserving the network structure. The system is modeled by a set of differential-algebraic equations (DAE) that permits more comprehensive models for the load and network and provides a better stability margin assessment when compared to the model of ordinary differential equations (ODE) traditionally used for transient stability analysis. The direct assessment of the stability margin was realized using direct methods based on the controlling unstable equilibrium point (CUEP) concept and permits to assess the margin in a local and fast manner, without requires the time integration of the post-fault system differential equations. Nevertheless, some open problems remain to be solved in order to provide a complete foundation of the CUEP method for DAE power system models. In this research a further step is given in this direction, showing that the existent definitions for the CUEP and other interest points may fail, mainly when the fault-on trajectory reaches singular surfaces. In this sense, it is proposed the correction of these definitions and a new CUEP method that is adequate to the angular and voltage short-term direct stability assessment due to large disturbances; capable to provide precise critical clearing times and the identification of the instability mechanisms for the DAE modeled power system, even in the presence of singular surfaces.

Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.