Gradient Methods for Large-Scale and Distributed Linear Quadratic Control

This thesis considers methods for synthesis of linear quadratic controllers for large-scale, interconnected systems. Conventional methods that solve the linear quadratic control problem are only applicable to systems with moderate size, due to the rapid increase in both computational time and memory requirements as the system size increases. The methods presented in this thesis show a much slower increase in these requirements when faced with system matrices with a sparse structure. Hence, they are useful for control design for systems of large order, since they usually have sparse systems matrices. An equally important feature of the methods is that the controllers are restricted to have a distributed nature, meaning that they respect a potential interconnection structure of the system. The controllers considered in the thesis have the same structure as the centralized LQG solution, that is, they are consisting of a state predictor and feedback from the estimated states. Strategies for determining the feedback matrix and predictor matrix separately, are suggested. The strategies use gradient directions of the cost function to iteratively approach a locally optimal solution in either problem. A scheme to determine bounds on the degree of suboptimality of the partial solution in every iteration, is presented. It is also shown that these bounds can be combined to give a bound on the degree of suboptimality of the full output feedback controller. Another method that treats the synthesis of the feedback matrix and predictor matrix simultaneously is also presented. The functionality of the developed methods is illustrated by an application, where the methods are used to compute controllers for a large deformable mirror, found in a telescope to compensate for atmospheric disturbances. The model of the mirror is obtained by discretizing a partial differential equation. This gives a linear, sparse representation of the mirror with a very large state space, which is suitable for the methods presented in the thesis. The performance of the controllers is evaluated using performance measures from the adaptive optics community

On some dynamic properties of electrical power systems : sobre algunas propiedades dinámicas de los sistemas eléctricos de potencia

This thesis treats some dynamic properties of power system models. An extension of the classical concept of dissipativity is formulated to deal with these systems described by differential-algebraic equations on phasor variables. A class of models of these systems the same that is known to admit an energy function is shown to be dissipative in the sense mentioned above, to later be extended to include realistic models of synchronous machines and other devices. The small signal models are shown to satisfy a convex constraint in the frequency domain that is later articulated with Integral Quadratic Constraints, a well-known stability analysis tool. Specific features of realistic power system models, as the effect of voltage regulation and damping injection, are precisely captured and incorporated into the analysis. It is shown that a trade-off between the mentioned control actions and the voltage sensitivity is a sufficient condition to establish the robustness of the electromechanical modes. This result and others mentioned above are validated through several examples.Esta tesis trata algunas propiedades dinámicas de los sistemas eléctricos de potencia. Se formula una extensión del concepto clásico de disipatividad compatible con estos sistemas, descritos por ecuaciones algebraico-diferenciales sobre variables fasoriales. Se muestra que una clase de modelos de estos sistemas satisface este concepto de disipatividad y se muestra que también lo hacen modelos detallados de máquinas síncronas y otros dispositivos de potencia. Se demuestra que los modelos en pequeña señal satisfacen una restricción convexa en el dominio de la frecuencia, capaz de ser articulada con herramientas bien conocidas de análisis de estabilidad. Características específicas de los sistemas eléctricos reales, tales como la acción de la regulación de tensión y las señales estabilizadoras, son precisamente definidas e incorporadas al análisis. Se demuestra que un adecuado balance entre las acciones de control mencionadas y la sensibilidad a variaciones de tensión es una condición suficiente para la robustez de los modos electromecánicos del sistema. Este resultado y otros mencionados anteriormente son validados mediante el análisis de varios ejemplos

Applied Dynamics and Geometric Mechanics

This one week workshop was organized around several central subjects in applied dynamics and geometric mechanics. The specific organization with afternoons free for discussion led to intense exchanges of ideas. Bridges were forged between researchers representing different fields. Links were established between pure mathematical ideas and applications. The meeting was not restricted to any particular application area. One of the main goals of the meeting, like most others in this series for the past twenty years, has been to facilitate cross fertilization between various areas of mathematics, physics, and engineering. New collaborative projects emerged due to this meeting. The workshop was well attended with participants from Europe, North America, and Asia. Young researchers (doctoral students, postdocs, junior faculty) formed about 30% of the participants

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Application of system theory to power processing problems

The work in power processing is reported. Input-output models, and Lie groups in control theory are discussed along with the methods of analysis for time invariant electrical networks