Robust state-derivative feedback LMI-based designs for multivariable linear systems

In some practical problems, for instance in the control systems for the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. New necessary and sufficient linear matrix inequalities (LMI) conditions for the design of state-derivative feedback for multi-input (MI) linear systems are proposed. For multi-input/multi-output (MIMO) linear time-invariant or time-varying plants, with or without uncertainties in their parameters, the proposed methods can include in the LMI-based control designs the specifications of the decay rate, bounds on the output peak, and bounds on the state-derivative feedback matrix K. These design procedures allow new specifications and also, they consider a broader class of plants than the related results available in the literature. The LMIs, when feasible, can be efficiently solved using convex programming techniques. Practical applications illustrate the efficiency of the proposed methods.

Novos resultados sobre a estabilidade e controle de sistemas nao-lineares utilizando modelos fuzzy e LMI

Relaxed conditions for the stability study of nonlinear, continuous and discrete-time systems given by fuzzy models are presented. A theoretical analysis shows that the proposed method provides better or at least the same results of the methods presented in the literature. Digital simulations exemplify this fact. These results are also used for the fuzzy regulators design. The nonlinear systems are represented by the fuzzy models proposed by Takagi and Sugeno. The stability analysis and the design of controllers are described by LMIs (Linear Matrix Inequalities), that can be solved efficiently by convex programming techniques. The specification of the decay rate, constraints on control input and output are also described by LMIs. Finally, the proposed design method is applied in the control of an inverted pendulum.

Estabilização de sistemas fuzzy T-S incertos usando realimenta̧ão derivativa

In some practical problems, for instance, in the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. Thus, a method for state-derivative feedback design applied to uncertain nonlinear systems is proposed in this work. The nonlinear systems are represented by Takagi-Sugeno fuzzy models during the modeling of the problem, allowing to use Linear Matrix Inequalities (LMIs) in the controller design. This type of modeling ease the control design, because, LMIs are easily solved using convex programming technicals. The control design aimed at system stabilisation, with or without bounds on decay rate. The efficiency of design procedure is illustrated through a numerical example.

Controle robusto com realimentação derivativa de sistemas não lineares via LMI

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Controle robusto de sistemas não-lineares sujeitos a falhas estruturais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Controle robusto h-infinito chaveado para sistemas lineares

Pós-graduação em Engenharia Elétrica - FEIS

Projeto de reguladores fuzzy Takagi-Sugeno utilizando as condições iniciais da planta

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

New methods for the synthesis of radiation patterns of coupled antenna arrays = Nuevos métodos de síntesis de diagramas de radiación de agrupaciones de antenas acopladas

El principal objetivo de esta tesis es el desarrollo de métodos de síntesis de diagramas de radiación de agrupaciones de antenas, en donde se realiza una caracterización electromagnética rigurosa de los elementos radiantes y de los acoplos mutuos existentes. Esta caracterización no se realiza habitualmente en la gran mayoría de métodos de síntesis encontrados en la literatura, debido fundamentalmente a dos razones. Por un lado, se considera que el diagrama de radiación de un array de antenas se puede aproximar con el factor de array que únicamente tiene en cuenta la posición de los elementos y las excitaciones aplicadas a los mismos. Sin embargo, como se mostrará en esta tesis, en múltiples ocasiones un riguroso análisis de los elementos radiantes y del acoplo mutuo entre ellos es importante ya que los resultados obtenidos pueden ser notablemente diferentes. Por otro lado, no es sencillo combinar un método de análisis electromagnético con un proceso de síntesis de diagramas de radiación. Los métodos de análisis de agrupaciones de antenas suelen ser costosos computacionalmente, ya que son estructuras grandes en términos de longitudes de onda. Generalmente, un diseño de un problema electromagnético suele comprender varios análisis de la estructura, dependiendo de las variaciones de las características, lo que hace este proceso muy costoso. Dos métodos se utilizan en esta tesis para el análisis de los arrays acoplados. Ambos están basados en el método de los elementos finitos, la descomposición de dominio y el análisis modal para analizar la estructura radiante y han sido desarrollados en el grupo de investigación donde se engloba esta tesis. El primero de ellos es una técnica de análisis de arrays finitos basado en la aproximación de array infinito. Su uso es indicado para arrays planos de grandes dimensiones con elementos equiespaciados. El segundo caracteriza el array y el acoplo mutuo entre elementos a partir de una expansión en modos esféricos del campo radiado por cada uno de los elementos. Este método calcula los acoplos entre los diferentes elementos del array usando las propiedades de traslación y rotación de los modos esféricos. Es capaz de analizar agrupaciones de elementos distribuidos de forma arbitraria. Ambas técnicas utilizan una formulación matricial que caracteriza de forma rigurosa el campo radiado por el array. Esto las hace muy apropiadas para su posterior uso en una herramienta de diseño, como los métodos de síntesis desarrollados en esta tesis. Los resultados obtenidos por estas técnicas de síntesis, que incluyen métodos rigurosos de análisis, son consecuentemente más precisos. La síntesis de arrays consiste en modificar uno o varios parámetros de las agrupaciones de antenas buscando unas determinadas especificaciones de las características de radiación. Los parámetros utilizados como variables de optimización pueden ser varios. Los más utilizados son las excitaciones aplicadas a los elementos, pero también es posible modificar otros parámetros de diseño como son las posiciones de los elementos o las rotaciones de estos. Los objetivos de las síntesis pueden ser dirigir el haz o haces en una determinada dirección o conformar el haz con formas arbitrarias. Además, es posible minimizar el nivel de los lóbulos secundarios o del rizado en las regiones deseadas, imponer nulos que evitan posibles interferencias o reducir el nivel de la componente contrapolar. El método para el análisis de arrays finitos basado en la aproximación de array infinito considera un array finito como un array infinito con un número finito de elementos excitados. Los elementos no excitados están físicamente presentes y pueden presentar tres diferentes terminaciones, corto-circuito, circuito abierto y adaptados. Cada una de estas terminaciones simulará mejor el entorno real en el que el array se encuentre. Este método de análisis se integra en la tesis con dos métodos diferentes de síntesis de diagramas de radiación. En el primero de ellos se presenta un método basado en programación lineal en donde es posible dirigir el haz o haces, en la dirección deseada, además de ejercer un control sobre los lóbulos secundarios o imponer nulos. Este método es muy eficiente y obtiene soluciones óptimas. El mismo método de análisis es también aplicado a un método de conformación de haz, en donde un problema originalmente no convexo (y de difícil solución) es transformado en un problema convexo imponiendo restricciones de simetría, resolviendo de este modo eficientemente un problema complejo. Con este método es posible diseñar diagramas de radiación con haces de forma arbitraria, ejerciendo un control en el rizado del lóbulo principal, así como en el nivel de los lóbulos secundarios. El método de análisis de arrays basado en la expansión en modos esféricos se integra en la tesis con tres técnicas de síntesis de diagramas de radiación. Se propone inicialmente una síntesis de conformación del haz basado en el método de la recuperación de fase resuelta de forma iterativa mediante métodos convexos, en donde relajando las restricciones del problema original se consiguen unas soluciones cercanas a las óptimas de manera eficiente. Dos métodos de síntesis se han propuesto, donde las variables de optimización son las posiciones y las rotaciones de los elementos respectivamente. Se define una función de coste basada en la intensidad de radiación, la cual es minimizada de forma iterativa con el método del gradiente. Ambos métodos reducen el nivel de los lóbulos secundarios minimizando una función de coste. El gradiente de la función de coste es obtenido en términos de la variable de optimización en cada método. Esta función de coste está formada por la expresión rigurosa de la intensidad de radiación y por una función de peso definida por el usuario para imponer prioridades sobre las diferentes regiones de radiación, si así se desea. Por último, se presenta un método en el cual, mediante técnicas de programación entera, se buscan las fases discretas que generan un diagrama de radiación lo más cercano posible al deseado. Con este método se obtienen diseños que minimizan el coste de fabricación. En cada uno de las diferentes técnicas propuestas en la tesis, se presentan resultados con elementos reales que muestran las capacidades y posibilidades que los métodos ofrecen. Se comparan los resultados con otros métodos disponibles en la literatura. Se muestra la importancia de tener en cuenta los diagramas de los elementos reales y los acoplos mutuos en el proceso de síntesis y se comparan los resultados obtenidos con herramientas de software comerciales. ABSTRACT The main objective of this thesis is the development of optimization methods for the radiation pattern synthesis of array antennas in which a rigorous electromagnetic characterization of the radiators and the mutual coupling between them is performed. The electromagnetic characterization is usually overlooked in most of the available synthesis methods in the literature, this is mainly due to two reasons. On the one hand, it is argued that the radiation pattern of an array is mainly influenced by the array factor and that the mutual coupling plays a minor role. As it is shown in this thesis, the mutual coupling and the rigorous characterization of the array antenna influences significantly in the array performance and its computation leads to differences in the results obtained. On the other hand, it is difficult to introduce an analysis procedure into a synthesis technique. The analysis of array antennas is generally expensive computationally as the structure to analyze is large in terms of wavelengths. A synthesis method requires to carry out a large number of analysis, this makes the synthesis problem very expensive computationally or intractable in some cases. Two methods have been used in this thesis for the analysis of coupled antenna arrays, both of them have been developed in the research group in which this thesis is involved. They are based on the finite element method (FEM), the domain decomposition and the modal analysis. The first one obtains a finite array characterization with the results obtained from the infinite array approach. It is specially indicated for the analysis of large arrays with equispaced elements. The second one characterizes the array elements and the mutual coupling between them with a spherical wave expansion of the radiated field by each element. The mutual coupling is computed using the properties of translation and rotation of spherical waves. This method is able to analyze arrays with elements placed on an arbitrary distribution. Both techniques provide a matrix formulation that makes them very suitable for being integrated in synthesis techniques, the results obtained from these synthesis methods will be very accurate. The array synthesis stands for the modification of one or several array parameters looking for some desired specifications of the radiation pattern. The array parameters used as optimization variables are usually the excitation weights applied to the array elements, but some other array characteristics can be used as well, such as the array elements positions or rotations. The desired specifications may be to steer the beam towards any specific direction or to generate shaped beams with arbitrary geometry. Further characteristics can be handled as well, such as minimize the side lobe level in some other radiating regions, to minimize the ripple of the shaped beam, to take control over the cross-polar component or to impose nulls on the radiation pattern to avoid possible interferences from specific directions. The analysis method based on the infinite array approach considers an infinite array with a finite number of excited elements. The infinite non-excited elements are physically present and may have three different terminations, short-circuit, open circuit and match terminated. Each of this terminations is a better simulation for the real environment of the array. This method is used in this thesis for the development of two synthesis methods. In the first one, a multi-objective radiation pattern synthesis is presented, in which it is possible to steer the beam or beams in desired directions, minimizing the side lobe level and with the possibility of imposing nulls in the radiation pattern. This method is very efficient and obtains optimal solutions as it is based on convex programming. The same analysis method is used in a shaped beam technique in which an originally non-convex problem is transformed into a convex one applying symmetry restrictions, thus solving a complex problem in an efficient way. This method allows the synthesis of shaped beam radiation patterns controlling the ripple in the mainlobe and the side lobe level. The analysis method based on the spherical wave expansion is applied for different synthesis techniques of the radiation pattern of coupled arrays. A shaped beam synthesis is presented, in which a convex formulation is proposed based on the phase retrieval method. In this technique, an originally non-convex problem is solved using a relaxation and solving a convex problems iteratively. Two methods are proposed based on the gradient method. A cost function is defined involving the radiation intensity of the coupled array and a weighting function that provides more degrees of freedom to the designer. The gradient of the cost function is computed with respect to the positions in one of them and the rotations of the elements in the second one. The elements are moved or rotated iteratively following the results of the gradient. A highly non-convex problem is solved very efficiently, obtaining very good results that are dependent on the starting point. Finally, an optimization method is presented where discrete digital phases are synthesized providing a radiation pattern as close as possible to the desired one. The problem is solved using linear integer programming procedures obtaining array designs that greatly reduce the fabrication costs. Results are provided for every method showing the capabilities that the above mentioned methods offer. The results obtained are compared with available methods in the literature. The importance of introducing a rigorous analysis into the synthesis method is emphasized and the results obtained are compared with a commercial software, showing good agreement.

Generalized methods and solvers for noise removal from piecewise constant signals. I. Background theory

Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.

First-Order Conditions for Optimization Problems with Quasiconvex Inequality Constraints

2000 Mathematics Subject Classification: 90C46, 90C26, 26B25, 49J52.

A QUASI-SIMPLEX METHOD FOR STRICTLY CONVEX QUADRATIC PROGRAMMING

本文提出一个不用 Kuhn- Tucker条件而直接搜索严格凸二次规划最优目标点的鲁棒方法 .在搜索过程中 ,目标点沿约束多面体边界上的一条折线移动 .这种移动目标点的思想可以被认为是线性规划单纯形法的自然推广 ,在单纯形法中 ,目标点从一个顶点移到另一个顶点。

Comparative study of RPSALG algorithm for convex semi-infinite programming

The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

Learning the kernel matrix with semidefinite programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space - classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

Learning the Kernel Matrix with Semi-Definite Programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space -- classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semi-definite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -- using the labelled part of the data one can learn an embedding also for the unlabelled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method to learn the 2-norm soft margin parameter in support vector machines, solving another important open problem. Finally, the novel approach presented in the paper is supported by positive empirical results.

Linear programming for large-scale Markov decision problems

We consider the problem of controlling a Markov decision process (MDP) with a large state space, so as to minimize average cost. Since it is intractable to compete with the optimal policy for large scale problems, we pursue the more modest goal of competing with a low-dimensional family of policies. We use the dual linear programming formulation of the MDP average cost problem, in which the variable is a stationary distribution over state-action pairs, and we consider a neighborhood of a low-dimensional subset of the set of stationary distributions (defined in terms of state-action features) as the comparison class. We propose a technique based on stochastic convex optimization and give bounds that show that the performance of our algorithm approaches the best achievable by any policy in the comparison class. Most importantly, this result depends on the size of the comparison class, but not on the size of the state space. Preliminary experiments show the effectiveness of the proposed algorithm in a queuing application.