Partial-state observers for nonlinear systems

This note deals with the design of reduced-order observers for a class of nonlinear systems. The order reduction of the observer is achieved by only estimating a required partial set of the state vector. Necessary and sufficient conditions are derived for the existence of reduced-order observers. An observer design procedure based on linear matrix inequalities is given. A numerical example is given to illustrate the design method.

Functional observer design with application to pre-compensated multi-variable systems

Partial state estimation of dynamical systems provides significant advantages in practical applications. Likewise, pre-compensator design for multi variable systems invokes considerable increase in the order of the original system. Hence, applying functional observer to pre-compensated systems can result in lower computational costs and more practicability in some applications such as fault diagnosis and output feedback control of these systems. In this note, functional observer design is investigated for pre-compensated systems. A lower order pre-compensator is designed based on a H2 norm optimization that is designed as the solution of a set of linear matrix inequalities (LMIs). Next, a minimum order functional observer is designed for the pre-compensated system. An LTI model of an irreversible chemical reactor is used to demonstrate our design algorithm, and to highlight the benefits of the proposed schemes.

Redução de ordem de modelos matemáticos lineares usando algoritmos genéticos

Nesta dissertação apresenta-se o problema de redução de ordem de modelos dinâmicos lineares, sob o ponto de vista de otimização via Algoritmos Genéticos. Uma função custo, obtida a partir da norma dos coeficientes do numerador da função de transferência do erro entre o modelo original e o reduzido, e minimizada por meio de um algoritmo genético, com consequente calculo dos parâmetros do modelo reduzido. O procedimento e aplicado em alguns exemplos que demonstram a validade da abordagem.

Controladores robustos do tipo LQG/LTR de ordem reduzida para sistemas MIMO com saídas independentes de seus modos não dominantes

O objetivo principal desta dissertação é apresentar uma solução eficiente, prática e de simples implementação para um problema recorrente em projetos de controladores robustos multivariáveis do tipo LQG/LTR: a elevada ordem que estes controladores podem obter dependendo das complicações apresentadas pelo sistema dificultando para que este possa ser controlado de maneira satisfatória. Para que esta meta seja alcançada, é apresentada uma técnica de redução do modelo de sistemas com metodologia bastante descomplicada, dispensando qualquer necessidade de complexas programações para a sua utilização. Esta metodologia porém, é somente aplicável a uma classe bastante específica de sistema. Em suma, o sistema deve possuir variáveis de estado desacopladas do restante do sistema, ou seja, variáveis que não sofram influências de outras e que também não provoquem grande efeito nas saídas do sistema. Foi escolhido um sistema multivariável de sexta ordem, com duas entradas e duas saídas para que a técnica de redução de ordem de modelo seja testada. Este sistema possui as características especiais mencionadas anteriormente bem como exige o projeto de compensador dinâmico e a adição de integradores às suas saídas para que seja controlado adequadamente. Este trabalho pretende apresentar o procedimento de todo o projeto mencionado, desde a obtenção de um modelo de ordem reduzida até a implementação do controlador LQG/LTR. Em seguida, o controlador obtido é testado através de diversas simulações e os resultados encontrados são discutidos para a avaliação da eficácia e da praticidade do método proposto para obtenção de controladores de ordem reduzida.

Nonlinear/electromagnetic co-simulation of microwaves and millimeter-waves links

The Ph.D. thesis describes the simulations of different microwave links from the transmitter to the receiver intermediate-frequency ports, by means of a rigorous circuit-level nonlinear analysis approach coupled with the electromagnetic characterization of the transmitter and receiver front ends. This includes a full electromagnetic computation of the radiated far field which is used to establish the connection between transmitter and receiver. Digitally modulated radio-frequency drive is treated by a modulation-oriented harmonic-balance method based on Krylov-subspace model-order reduction to allow the handling of large-size front ends. Different examples of links have been presented: an End-to-End link simulated by making use of an artificial neural network model; the latter allows a fast computation of the link itself when driven by long sequences of the order of millions of samples. In this way a meaningful evaluation of such link performance aspects as the bit error rate becomes possible at the circuit level. Subsequently, a work focused on the co-simulation an entire link including a realistic simulation of the radio channel has been presented. The channel has been characterized by means of a deterministic approach, such as Ray Tracing technique. Then, a 2x2 multiple-input multiple-output antenna link has been simulated; in this work near-field and far-field coupling between radiating elements, as well as the environment factors, has been rigorously taken into account. Finally, within the scope to simulate an entire ultra-wideband link, the transmitting side of an ultrawideband link has been designed, and an interesting Front-End co-design technique application has been setup.

The port-Hamiltonian framework as a comprehensive approach to model and simulate complex systems

This thesis describes modelling tools and methods suited for complex systems (systems that typically are represented by a plurality of models). The basic idea is that all models representing the system should be linked by well-defined model operations in order to build a structured repository of information, a hierarchy of models. The port-Hamiltonian framework is a good candidate to solve this kind of problems as it supports the most important model operations natively. The thesis in particular addresses the problem of integrating distributed parameter systems in a model hierarchy, and shows two possible mechanisms to do that: a finite-element discretization in port-Hamiltonian form, and a structure-preserving model order reduction for discretized models obtainable from commercial finite-element packages.

Comparación de un nuevo método de aceleración de análisis dinámicos con métodos de reducción de modelos

La dinámica estructural estudia la respuesta de una estructura ante cargas o fenómenos variables en el tiempo. En muchos casos, estos fenómenos requieren realizar análisis paramétricos de la estructura considerando una gran cantidad de configuraciones de diseño o modificaciones de la estructura. Estos cambios, ya sean en fases iniciales de diseño o en fases posteriores de rediseño, alteran las propiedades físicas de la estructura y por tanto del modelo empleado para su análisis, cuyo comportamiento dinámico se modifica en consecuencia. Un caso de estudio de este tipo de modificaciones es la supervisión de la integridad estructural, que trata de identificar la presencia de daño estructural y prever el comportamiento de la estructura tras ese daño, como puede ser la variación del comportamiento dinámico de la estructura debida a una delaminación, la aparición o crecimiento de grieta, la debida a la pérdida de pala sufrida por el motor de un avión en vuelo, o la respuesta dinámica de construcciones civiles como puentes o edificios frente a cargas sísmicas. Si a la complejidad de los análisis dinámicos requeridos en el caso de grandes estructuras se añade la variación de determinados parámetros en busca de una respuesta dinámica determinada o para simular la presencia de daños, resulta necesario la búsqueda de medios de simplificación o aceleración del conjunto de análisis que de otra forma parecen inabordables tanto desde el punto de vista del tiempo de computación, como de la capacidad requerida de almacenamiento y manejo de grandes volúmenes de archivos de datos. En la presente tesis doctoral se han revisado los métodos de reducción de elementos .nitos más habituales para análisis dinámicos de grandes estructuras. Se han comparado los resultados de casos de estudio de los métodos más aptos, para el tipo de estructuras y modificaciones descritas, con los resultados de aplicación de un método de reducción reciente. Entre los primeros están el método de condensación estática de Guyan extendido al caso con amortiguamiento no proporcional y posteriores implementaciones de condensaciones dinámicas en diferentes espacios vectoriales. El método de reducción recientemente presentado se denomina en esta tesis DACMAM (Dynamic Analysis in Complex Modal space Acceleration Method), y consiste en el análisis simplificado que proporciona una solución para la respuesta dinámica de una estructura, calculada en el espacio modal complejo y que admite modificaciones estructurales. El método DACMAM permite seleccionar un número reducido de grados de libertad significativos para la dinámica del fenómeno que se quiere estudiar como son los puntos de aplicación de la carga, localizaciones de los cambios estructurales o puntos donde se quiera conocer la respuesta, de forma que al implementar las modificaciones estructurales, se ejecutan los análisis necesarios sólo de dichos grados de libertad sin pérdida de precisión. El método permite considerar alteraciones de masa, rigidez, amortiguamiento y la adición de nuevos grados de libertad. Teniendo en cuenta la dimensión del conjunto de ecuaciones a resolver, la parametrización de los análisis no sólo resulta posible, sino que es también manejable y controlable gracias a la sencilla implementación del procedimiento para los códigos habituales de cálculo mediante elementos .nitos. En el presente trabajo se muestra la bondad y eficiencia del método en comparación con algunos de los métodos de reducción de grandes modelos estructurales, verificando las diferencias entre sí de los resultados obtenidos y respecto a la respuesta real de la estructura, y comprobando los medios empleados en ellos tanto en tiempo de ejecución como en tamaño de ficheros electrónicos. La influencia de los diversos factores que se tienen en cuenta permite identificar los límites y capacidades de aplicación del método y su exhaustiva comparación con los otros procedimientos. ABSTRACT Structural dynamics studies the response of a structure under loads or phenomena which vary over time. In many cases, these phenomena require the use of parametric analyses taking into consideration several design configurations or modifications of the structure. This is a typical need in an engineering o¢ ce, no matter the structural design is in early or final stages. These changes modify the physical properties of the structure, and therefore, the finite element model to analyse it. A case study, that exempli.es this circumstance, is the structural health monitoring to predict the variation of the dynamical behaviour after damage, such as a delaminated structure, a crack onset or growth, an aircraft that suffers a blade loss event or civil structures (buildings or bridges) under seismic loads. Not only large structures require complex analyses to appropriately acquire an accurate solution, but also the variation of certain parameters. There is a need to simplify the analytical process, in order to bring CPU time, data .les, management of solutions to a reasonable size. In the current doctoral thesis, the most common finite element reduction methods for large structures are reviewed. Results of case studies are compared between a recently proposed method, herein named DACMAM (Dynamic Analysis in Complex Modal space Acceleration Method), and different condensation methods, namely static or Guyan condensation and dynamic condensation in different vectorial spaces. All these methods are suitable for considering non-classical damping. The reduction method DACMAM consist of a structural modification in the complex modal domain which provides a dynamic response solution for the reduced models. This process allows the selection of a few degrees of freedom that are relevant for the dynamic response of the system. These d.o.f. are the load application points, relevant structural points or points in which it is important to know the response. Consequently, an analysis with structural modifications implies only the calculation of the dynamic response of the selected degrees of freedom added, but with no loss of information. Therefore, mass, stiffness or damping modifications are easily considered as well as new degrees of freedom. Taking into account the size of the equations to be solved, the parameterization of the dynamic solutions is not only possible, but also manageable and controllable due to the easy implementation of the procedure in the standard finite element solvers. In this thesis, the proposed reduction method for large structural models is compared with other published model order reduction methods. The comparison shows and underlines the efficiency of the new method, and veri.es the differences in the response when compared with the response of the full model. The CPU time, the data files and the scope of the parameterization are also addressed.

Método computacional para la aceleración de análisis paramétricos de modificaciones locales en estructuras complejas sometidas a cargas dinámicas

Dentro del análisis y diseño estructural surgen frecuentemente problemas de ingeniería donde se requiere el análisis dinámico de grandes modelos de elementos finitos que llegan a millones de grados de libertad y emplean volúmenes de datos de gran tamaño. La complejidad y dimensión de los análisis se dispara cuando se requiere realizar análisis paramétricos. Este problema se ha abordado tradicionalmente desde diversas perspectivas: en primer lugar, aumentando la capacidad tanto de cálculo como de memoria de los sistemas informáticos empleados en los análisis. En segundo lugar, se pueden simplificar los análisis paramétricos reduciendo su número o detalle y por último se puede recurrir a métodos complementarios a los elementos .nitos para la reducción de sus variables y la simplificación de su ejecución manteniendo los resultados obtenidos próximos al comportamiento real de la estructura. Se propone el empleo de un método de reducción que encaja en la tercera de las opciones y consiste en un análisis simplificado que proporciona una solución para la respuesta dinámica de una estructura en el subespacio modal complejo empleando un volumen de datos muy reducido. De este modo se pueden realizar análisis paramétricos variando múltiples parámetros, para obtener una solución muy aproximada al objetivo buscado. Se propone no solo la variación de propiedades locales de masa, rigidez y amortiguamiento sino la adición de grados de libertad a la estructura original para el cálculo de la respuesta tanto permanente como transitoria. Adicionalmente, su facilidad de implementación permite un control exhaustivo sobre las variables del problema y la implementación de mejoras como diferentes formas de obtención de los autovalores o la eliminación de las limitaciones de amortiguamiento en la estructura original. El objetivo del método se puede considerar similar a los que se obtienen al aplicar el método de Guyan u otras técnicas de reducción de modelos empleados en dinámica estructural. Sin embargo, aunque el método permite ser empleado en conjunción con otros para obtener las ventajas de ambos, el presente procedimiento no realiza la condensación del sistema de ecuaciones, sino que emplea la información del sistema de ecuaciones completa estudiando tan solo la respuesta en las variables apropiadas de los puntos de interés para el analista. Dicho interés puede surgir de la necesidad de obtener la respuesta de las grandes estructuras en unos puntos determinados o de la necesidad de modificar la estructura en zonas determinadas para cambiar su comportamiento (respuesta en aceleraciones, velocidades o desplazamientos) ante cargas dinámicas. Por lo tanto, el procedimiento está particularmente indicado para la selección del valor óptimo de varios parámetros en grandes estructuras (del orden de cientos de miles de modos) como pueden ser la localización de elementos introducidos, rigideces, masas o valores de amortiguamientos viscosos en estudios previos en los que diversas soluciones son planteadas y optimizadas, y que en el caso de grandes estructuras, pueden conllevar un número de simulaciones extremadamente elevado para alcanzar la solución óptima. Tras plantear las herramientas necesarias y desarrollar el procedimiento, se propone un caso de estudio para su aplicación al modelo de elementos .nitos del UAV MILANO desarrollado por el Instituto Nacional de Técnica Aeroespacial. A dicha estructura se le imponen ciertos requisitos al incorporar un equipo en aceleraciones en punta de ala izquierda y desplazamientos en punta de ala derecha en presencia de la sustentación producida por una ráfaga continua de viento de forma sinusoidal. La modificación propuesta consiste en la adición de un equipo en la punta de ala izquierda, bien mediante un anclaje rígido, bien unido mediante un sistema de reducción de la respuesta dinámica con propiedades de masa, rigidez y amortiguamiento variables. El estudio de los resultados obtenidos permite determinar la optimización de los parámetros del sistema de atenuación por medio de múltiples análisis dinámicos de forma que se cumplan de la mejor forma posible los requisitos impuestos con la modificación. Se comparan los resultados con los obtenidos mediante el uso de un programa comercial de análisis por el método de los elementos .nitos lográndose soluciones muy aproximadas entre el modelo completo y el reducido. La influencia de diversos factores como son el amortiguamiento modal de la estructura original, el número de modos retenidos en la truncatura o la precisión proporcionada por el barrido en frecuencia se analiza en detalle para, por último, señalar la eficiencia en términos de tiempo y volumen de datos de computación que ofrece el método propuesto en comparación con otras aproximaciones. Por lo tanto, puede concluirse que el método propuesto se considera una opción útil y eficiente para el análisis paramétrico de modificaciones locales en grandes estructuras. ABSTRACT When developing structural design and analysis some projects require dynamic analysis of large finite element models with millions of degrees of freedom which use large size data .les. The analysis complexity and size grow if a parametric analysis is required. This problem has been approached traditionally in several ways: one way is increasing the power and the storage capacity of computer systems involved in the analysis. Other obvious way is reducing the total amount of analyses and their details. Finally, complementary methods to finite element analysis can also be employed in order to limit the number of variables and to reduce the execution time keeping the results as close as possible to the actual behaviour of the structure. Following this third option, we propose a model reduction method that is based in a simplified analysis that supplies a solution for the dynamic response of the structure in the complex modal space using few data. Thereby, parametric analysis can be done varying multiple parameters so as to obtain a solution which complies with the desired objetive. We propose not only mass, stiffness and damping variations, but also addition of degrees of freedom to the original structure in order to calculate the transient and steady-state response. Additionally, the simple implementation of the procedure allows an in-depth control of the problem variables. Furthermore, improvements such as different ways to obtain eigenvectors or to remove damping limitations of the original structure are also possible. The purpose of the procedure is similar to that of using the Guyan or similar model order reduction techniques. However, in our method we do not perform a true model order reduction in the traditional sense. Furthermore, additional gains, which we do not explore herein, can be obtained through the combination of this method with traditional model-order reduction procedures. In our procedure we use the information of the whole system of equations is used but only those nodes of interest to the analyst are processed. That interest comes from the need to obtain the response of the structure at specific locations or from the need to modify the structure at some suitable positions in order to change its behaviour (acceleration, velocity or displacement response) under dynamic loads. Therefore, the procedure is particularly suitable for parametric optimization in large structures with >100000 normal modes such as position of new elements, stiffness, mass and viscous dampings in previous studies where different solutions are devised and optimized, and in the case of large structures, can carry an extremely high number of simulations to get the optimum solution. After the introduction of the required tools and the development of the procedure, a study case is proposed with use the finite element model (FEM) of the MILANO UAV developed by Instituto Nacional de Técnica Aeroespacial. Due to an equipment addition, certain acceleration and displacement requirements on left wing tip and right wing tip, respectively, are imposed. The structure is under a continuous sinusoidal wind gust which produces lift. The proposed modification consists of the addition of an equipment in left wing tip clamped through a rigid attachment or through a dynamic response reduction system with variable properties of mass, stiffness and damping. The analysis of the obtained results allows us to determine the optimized parametric by means of multiple dynamic analyses in a way such that the imposed requirements have been accomplished in the best possible way. The results achieved are compared with results from a commercial finite element analysis software, showing a good correlation. Influence of several factors such as the modal damping of the original structure, the number of modes kept in the modal truncation or the precission given by the frequency sweep is analyzed. Finally, the efficiency of the proposed method is addressed in tems of computational time and data size compared with other approaches. From the analyses performed, we can conclude that the proposed method is a useful and efficient option to perform parametric analysis of possible local modifications in large structures.

Dynamic-Data Driven Real-Time Identification for Electric Power Systems

Power system engineers face a double challenge: to operate electric power systems within narrow stability and security margins, and to maintain high reliability. There is an acute need to better understand the dynamic nature of power systems in order to be prepared for critical situations as they arise. Innovative measurement tools, such as phasor measurement units, can capture not only the slow variation of the voltages and currents but also the underlying oscillations in a power system. Such dynamic data accessibility provides us a strong motivation and a useful tool to explore dynamic-data driven applications in power systems. To fulfill this goal, this dissertation focuses on the following three areas: Developing accurate dynamic load models and updating variable parameters based on the measurement data, applying advanced nonlinear filtering concepts and technologies to real-time identification of power system models, and addressing computational issues by implementing the balanced truncation method. By obtaining more realistic system models, together with timely updated parameters and stochastic influence consideration, we can have an accurate portrait of the ongoing phenomena in an electrical power system. Hence we can further improve state estimation, stability analysis and real-time operation.

DYNAMIC MODEL BASED STATE ESTIMATION IN A HEAVY DUTY DIESEL AFTERTREATMENT SYSTEM FOR ONBOARD DIAGNOSTICS AND CONTROLS

Estimating un-measurable states is an important component for onboard diagnostics (OBD) and control strategy development in diesel exhaust aftertreatment systems. This research focuses on the development of an Extended Kalman Filter (EKF) based state estimator for two of the main components in a diesel engine aftertreatment system: the Diesel Oxidation Catalyst (DOC) and the Selective Catalytic Reduction (SCR) catalyst. One of the key areas of interest is the performance of these estimators when the catalyzed particulate filter (CPF) is being actively regenerated. In this study, model reduction techniques were developed and used to develop reduced order models from the 1D models used to simulate the DOC and SCR. As a result of order reduction, the number of states in the estimator is reduced from 12 to 1 per element for the DOC and 12 to 2 per element for the SCR. The reduced order models were simulated on the experimental data and compared to the high fidelity model and the experimental data. The results show that the effect of eliminating the heat transfer and mass transfer coefficients are not significant on the performance of the reduced order models. This is shown by an insignificant change in the kinetic parameters between the reduced order and 1D model for simulating the experimental data. An EKF based estimator to estimate the internal states of the DOC and SCR was developed. The DOC and SCR estimators were simulated on the experimental data to show that the estimator provides improved estimation of states compared to a reduced order model. The results showed that using the temperature measurement at the DOC outlet improved the estimates of the CO , NO , NO2 and HC concentrations from the DOC. The SCR estimator was used to evaluate the effect of NH3 and NOX sensors on state estimation quality. Three sensor combinations of NOX sensor only, NH3 sensor only and both NOX and NH3 sensors were evaluated. The NOX only configuration had the worst performance, the NH3 sensor only configuration was in the middle and both the NOX and NH3 sensor combination provided the best performance.

Empirical analysis of the effect of dimension reduction and word order on semantic vectors

The aim of this paper is to provide a comparison of various algorithms and parameters to build reduced semantic spaces. The effect of dimension reduction, the stability of the representation and the effect of word order are examined in the context of the five algorithms bearing on semantic vectors: Random projection (RP), singular value decom- position (SVD), non-negative matrix factorization (NMF), permutations and holographic reduced representations (HRR). The quality of semantic representation was tested by means of synonym finding task using the TOEFL test on the TASA corpus. Dimension reduction was found to improve the quality of semantic representation but it is hard to find the optimal parameter settings. Even though dimension reduction by RP was found to be more generally applicable than SVD, the semantic vectors produced by RP are somewhat unstable. The effect of encoding word order into the semantic vector representation via HRR did not lead to any increase in scores over vectors constructed from word co-occurrence in context information. In this regard, very small context windows resulted in better semantic vectors for the TOEFL test.

Optimal H2 order-one reduction by solving eigenproblems for polynomial equations

A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included.

A symmetry reduction technique for higher order Painleve systems

The symmetry reduction of higher order Painleve systems is formulated in terms of Dirac procedure. A set of canonical variables that admit Dirac reduction procedure is proposed for Hamiltonian structures governing the A(2M)((1)) and A(2M-1)((1)) Painleve systems for M = 2,3,.... (C) 2012 Elsevier B.V. All rights reserved.