Nesta dissertação vamos estudar alguns importantes resultados acerca da resolubilidade local de operadores lineares de primeira ordem. Mais especificamente, seja o campo vetorial singular L em \'R POT. n\' e dado por: L = \'\\SIGMA SUP. m\' . INF. j=1\' a IND. j\' (x) \'SUP. \\PARTIAL\' INF. \\PARTIAL x INF. j\'. Esta trabalho dirige-se ao estudo da resolubilidade local de L, isto é, dada f \'PERTENCE A\' \' C POT. INFINITO\' (\'R POT. n\') e dado \'x IND. 0\' \'PERTENCE A\' \'R POT. n queremos encontrar u \'PERTENCE A\' D\'(\'R POT.n \') tal que Lu = f numa vizinhança de \'x INF. 0\'. Será dada atenção especial ao caso em que os coeficientes \'a IND. j\'(x) de L são função lineares. Também, serão apresentados resultados sobre a resolubilidade local da equação Lu = cu + f, sendo c \'PERTENCE A\' \'C POT. INFINITO\' (\'R POT. n\') / This dissertation aims to study some important results about local solvability of first order differential operators. Specifically, let L be a singular vector field on \'R POT. n\' given by L = \' \\SIGMA SUP. m INF.j=1\' \'a IND. j(x) \'\\PARTIAL SUP. INF. \\PARTIAL x INF. j\'. This work explore the local solvability of L, that is, given f \'IT BELONGS\' \'C POT. INFINITY\' (\'R POT. n\' and \'x INF. 0\' \'IT BELONGS\' \'R POT. n\' we want to find u \'IT BELONGS\' 2 D\'(\'R POT. n) such that Lu = f in a neighborhood of \'x INF. 0\'. We give special attention to the case where the coefficients \'a IND. j\'(x) are linear. We also present some results about local solvability of the equation Lu = cu + f for c \'IT BELONGS\' \'C POT. INFINITY\' (\'R POT. n\')

The control theory of nonlinear systems has been receiving increasing attention in recent years, both for its technical importance as well as for its impact in various fields of application. In several key areas, such as aerospace, chemical and petrochemical industries, bioengineering, and robotics, a new practical application for this tool appears every day. System nonlinearity is characterized when at least one component or subsystem is nonlinear. Classical methods used in the study of linear systems, particularly superposition, are not usually applied to the nonlinear systems. It is necessary to use other methods to study the control of these systems. For a wide class of nonlinear systems, a rather important structural feature comes from the strong nonlinearity appearing as coupling between spectrally decoupled parts of the system. Even in the case of low frequencies, where lumped models can still be employed the nonlinear coupling between parts of the system requires specific treatment, using advanced mathematical tools. In this context, an alternative, frequency domain approach is pursued here. In the rest of this work, a specific system form of linearly decoupled but nonlinearly coupled subsystems is examined. The mathematical toolbox of the Hilbert transform is appropriately introduced for obtaining two low-pass subsystems that form an equivalent description of the essential overall system dynamics. The nonlinear coupled dynamics is investigated systematically by partitioning the coupled system state vector in such a way as to fully exploit the low-pass and the band-pass intrinsic features of free dynamics. In particular, by employing the Hilbert Transform, a low-pass equivalent system is derived. Then, a typical case is investigated thoroughly by means of numerical simulation of the original coupled low and band-pass, real-state-variable system and the low-pass-equivalent, complex-state-variable derived one. The nonlinear model equations considered here pave the way for a systematic investigation of nonlinear feedback control options designed to operate mechatronic transducers in energy harvesting, sensing or actuation modes.

The phenomenon of x-ray beam hardening (BH) has significant impact on preclinical micro-CT imaging systems. The causal factors are the polyenergetic nature of x-ray beam used for imaging and the energy dependence of linear attenuation coefficient of the imaged material. With increase in length of propagation of beam in the imaged object, lower energy photons in the projected beam become preferentially absorbed. The beam "hardens" (as average energy increases) and progressively becomes more penetrating, causing underestimation of the attenuation coefficient. When this phenomenon is not accounted for during CT reconstruction, it results in images with nonuniform CT number values across regions of uniform density. It leads to severe errors in quantitative applications of micro-CT and degradation in diagnostic quality of images. Hence, correction for beam hardening effect is of foremost importance and has been an active area of research since the advent of micro -CT. The Siemens Inveon micro-CT system uses a common linearization approach for BH correction. It provides a set of standard default coefficients to be applied during CT reconstruction. However, our initial experiments with uniform water phantoms of varying diameters indicated that the correction coefficients provided by default in the Inveon system are applicable for imaging mouse-size (~28 mm) objects only. For larger objects the correction factors yielded incorrect CT values along with characteristic 'cupping' observed in the uniform region in the center of the phantom. This study provides an insight into the nature and characteristics of beam hardening on the Inveon CT system using water phantoms of varying sizes. We develop and test a beam hardening correction scheme based on linearization using cylindrical water phantoms of two different diameters - 28 mm and 71 mm, selected to represent mouse and rat sizes respectively. The measured non-linear relationship between attenuation and length of propagation is fitted to a polynomial function, which is used to estimate the effective monoenergetic attenuation coefficient for water. The estimated effective linear attenuation coefficient value is used to generate the expected sum of attenuation coefficients along each x-ray path through the imaged object. The acquired poly-energetic data is then linearized to expected projections using a third order polynomial fit, which is consistent with the Inveon BH model and software. The coefficients of this trinomial are then applied for BH correction during CT reconstruction. Correction achieved with the proposed model demonstrates effective removal of the characteristic cupping artifact that was observed when default BHC coefficients were applied. In addition to water phantoms, we also test the effectiveness of the proposed scheme using solid cylindrical phantoms of three different densities and composition. The proposed method was also used to measure the BH effect for 12 different kVp/filtration combinations. By generating twelve distinct sets of BHC coefficients, for each setting, we achieve a significant expansion in the quantitative performance of the Inveon CT system.

A small satellite simulator with attitude determination and control was designed and implemented in hardware. The simulator consists of inertial sensors for attitude determination and a pyramidal four-wheel momentum exchange system as the control actuators. A linearized PV controller with trajectory generation and feedback linearization was implemented, with the focus on controlling yaw. The simulator was tested on a spherical air bearing platform to allow three-degree-of-freedom operation. The simulator software was developed to read measurements from the sensors, apply the control algorithm, and send commands to the actuators. A data processing routine was developed. Electromechanical testing for the system as well as test results are presented.

<p>Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models.</p><p>More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system.</p><p>In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented.</p><p>In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs.</p><p>It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.</p> / <p>Linjära tidsinvarianta approximationer av olinjära system har många användningsområden och kan tas fram på flera sätt. Om man har mätningar av in- och utsignalerna från ett olinjärt system kan man till exempel använda systemidentifiering och prediktionsfelsmetoden för att skatta en linjär modell utan att ta hänsyn till att systemet egentligen är olinjärt. Ett av huvudmålen med den här avhandlingen är att beskriva egenskaper för sådana approximativa modeller.</p><p>Framförallt studeras linjära tidsinvarianta modeller som är optimala approximationer i meningen att de minimerar ett kriterium baserat på medelkvadratfelet. Brusmodeller kan inkluderas i dessa modelltyper och både fallet med och utan brusmodell studeras här. Modeller som är optimala i medelkvadratfelsmening visar sig kunna uppvisa ett antal intressanta, men ibland oönskade, egenskaper. Bland annat visas det att en optimal linjär modell kan vara mycket känslig för små olinjäriteter. Denna känslighet är inte önskvärd i de flesta tillämpningar och innebär att en linjär approximation av ett nästan linjärt system kan vara oanvändbar för till exempel robust reglerdesign. Vidare visas det att en del valideringsmetoder som är framtagna för linjära system inte alltid kan användas för validering av linjära approximationer av olinjära system.</p><p>Man kan dock göra de optimala linjära modellerna mer användbara genom att välja lämpliga insignaler. Bland annat visas det att minfasfiltrerat vitt brus i många avseenden är ett bra val av insignal. Klassen av separabla signaler detaljstuderas också. Denna klass innehåller till exempel alla gaussiska signaler och just dessa signaler visar sig vara speciellt användbara för att ta fram approximationer av generaliserade wiener-hammerstein-system. Dessutom visas det att en viss typ av slumpmässiga multisinussignaler är separabel. Några teoretiska resultat om nästan linjära system presenteras också.</p><p>De flesta metoder för robust reglerdesign kan bara användas om storleken på modellfelet är känd för alla tänkbara insignaler. Detta är emellertid ofta inte realistiskt när ett olinjärt system approximeras med en linjär modell. I denna avhandling beskrivs därför ett alternativt sätt att göra en robust reglerdesign baserat på en tidsdiskret modell och en modellfelsmodell som bara är giltig för begränsade insignaler.</p><p>Ibland skulle det vara önskvärt om en linjär modell av ett system inte påverkades av förekomsten av små olinjäriteter i systemet. Denna oönskade påverkan kan i vissa fall reduceras om en liten olinjär term tas med i modellen. En identifieringsmetod för olinjära autoregressiva system med externa insignaler där denna möjlighet finns beskrivs här. Med hjälp av denna metod kan modeller som består av en parametrisk linjär del och en ickeparametrisk lipschitzkontinuerlig olinjär del skattas genom att man löser ett konvext optimeringsproblem.</p>

<p>Today’s diesel engines are complex with systems like VGT and EGR to be able to fulfil the stricter emission legislations and the demands on the fuel consumption. Controlling a system like this demands a sophisticated control system. Furthermore, the authorities demand on self diagnosis requires an equal sophisticated diagnosis system. These systems require good knowledge about the signals present in the system and how they affect each other.</p><p>One way to achieve this is to have a good model of the system and based on this calculate an observer. The observer is then used to estimate signals used for control and diagnosis. Advantages with an observer instead of using just sensors are that the sensor signals often are noisy and need to be filtered before they can be used. This causes time delay which further complicates the control and diagnosis systems. Other advantages are that sensors are expensive and that some engine quantities are hard to measure.</p><p>In this Master’s thesis a model of a Scania diesel engine is developed and an observer is calculated. Due to the non-linearities in the model the observer is based on a constant gain extended Kalman filter.</p>

Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models. More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system. In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented. In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs. It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem. / Linjära tidsinvarianta approximationer av olinjära system har många användningsområden och kan tas fram på flera sätt. Om man har mätningar av in- och utsignalerna från ett olinjärt system kan man till exempel använda systemidentifiering och prediktionsfelsmetoden för att skatta en linjär modell utan att ta hänsyn till att systemet egentligen är olinjärt. Ett av huvudmålen med den här avhandlingen är att beskriva egenskaper för sådana approximativa modeller. Framförallt studeras linjära tidsinvarianta modeller som är optimala approximationer i meningen att de minimerar ett kriterium baserat på medelkvadratfelet. Brusmodeller kan inkluderas i dessa modelltyper och både fallet med och utan brusmodell studeras här. Modeller som är optimala i medelkvadratfelsmening visar sig kunna uppvisa ett antal intressanta, men ibland oönskade, egenskaper. Bland annat visas det att en optimal linjär modell kan vara mycket känslig för små olinjäriteter. Denna känslighet är inte önskvärd i de flesta tillämpningar och innebär att en linjär approximation av ett nästan linjärt system kan vara oanvändbar för till exempel robust reglerdesign. Vidare visas det att en del valideringsmetoder som är framtagna för linjära system inte alltid kan användas för validering av linjära approximationer av olinjära system. Man kan dock göra de optimala linjära modellerna mer användbara genom att välja lämpliga insignaler. Bland annat visas det att minfasfiltrerat vitt brus i många avseenden är ett bra val av insignal. Klassen av separabla signaler detaljstuderas också. Denna klass innehåller till exempel alla gaussiska signaler och just dessa signaler visar sig vara speciellt användbara för att ta fram approximationer av generaliserade wiener-hammerstein-system. Dessutom visas det att en viss typ av slumpmässiga multisinussignaler är separabel. Några teoretiska resultat om nästan linjära system presenteras också. De flesta metoder för robust reglerdesign kan bara användas om storleken på modellfelet är känd för alla tänkbara insignaler. Detta är emellertid ofta inte realistiskt när ett olinjärt system approximeras med en linjär modell. I denna avhandling beskrivs därför ett alternativt sätt att göra en robust reglerdesign baserat på en tidsdiskret modell och en modellfelsmodell som bara är giltig för begränsade insignaler. Ibland skulle det vara önskvärt om en linjär modell av ett system inte påverkades av förekomsten av små olinjäriteter i systemet. Denna oönskade påverkan kan i vissa fall reduceras om en liten olinjär term tas med i modellen. En identifieringsmetod för olinjära autoregressiva system med externa insignaler där denna möjlighet finns beskrivs här. Med hjälp av denna metod kan modeller som består av en parametrisk linjär del och en ickeparametrisk lipschitzkontinuerlig olinjär del skattas genom att man löser ett konvext optimeringsproblem.

Radio frequency (RF) communications are limited to a number of frequency bands scattered over the radio spectrum. Applications over such bands increasingly require more versatile, data extensive wireless communications that leads to the necessity of high bandwidth efficient interfaces, operating over wideband frequency ranges. Whether for a base station or mobile device, the regulations and adequate transmission of such schemes place stringent requirements on the design of transmitter front-ends. Increasingly strenuous and challenging hardware design criteria are to be met, especially so in the design of power amplifiers (PA), the bottle neck of the transmitter’s design tradeoff between linearity and power efficiency. The power amplifier exhibits a nonideal behavior, characterized by both nonlinearity and memory effects, heavily affecting that tradeoff, and therefore requiring an effective linearization technique, namely Digital Predistortion (DPD). The effectiveness of the DPD is highly dependent on the modeling scheme used to compensate for the PA’s nonideal behavior. In fact, its viability is determined by the scheme’s accuracy and implementation complexity. Generic behavioral models for nonlinear systems with memory have been used, considering the PA as a black box, and requiring RF designers to perform extensive testing to determine the minimal complexity structure that achieves satisfactory results. This thesis first proposes a direct systematic approach based on the parallel Hammerstein structure to determine the exact number of coefficients needed in a DPD. Then a physical explanation of memory effects is detailed, which leads to a close-form expression for the characteristic behavior of the PA entirely based on circuit properties. The physical expression is implemented and tested as a modeling scheme. Moreover, a link between this formulation and the proven behavioral models is explored, namely the Volterra series and Memory Polynomial. The formulation shows the correlation between parameters of generic behavioral modeling schemes when applied to RF PAs and demonstrates redundancy based on the physical existence or absence of modeling terms, detailed for the proven Memory polynomial modeling and linearization scheme.

Today’s diesel engines are complex with systems like VGT and EGR to be able to fulfil the stricter emission legislations and the demands on the fuel consumption. Controlling a system like this demands a sophisticated control system. Furthermore, the authorities demand on self diagnosis requires an equal sophisticated diagnosis system. These systems require good knowledge about the signals present in the system and how they affect each other. One way to achieve this is to have a good model of the system and based on this calculate an observer. The observer is then used to estimate signals used for control and diagnosis. Advantages with an observer instead of using just sensors are that the sensor signals often are noisy and need to be filtered before they can be used. This causes time delay which further complicates the control and diagnosis systems. Other advantages are that sensors are expensive and that some engine quantities are hard to measure. In this Master’s thesis a model of a Scania diesel engine is developed and an observer is calculated. Due to the non-linearities in the model the observer is based on a constant gain extended Kalman filter.

Radio frequency (RF) communications are limited to a number of frequency bands scattered over the radio spectrum. Applications over such bands increasingly require more versatile, data extensive wireless communications that leads to the necessity of high bandwidth efficient interfaces, operating over wideband frequency ranges. Whether for a base station or mobile device, the regulations and adequate transmission of such schemes place stringent requirements on the design of transmitter front-ends. Increasingly strenuous and challenging hardware design criteria are to be met, especially so in the design of power amplifiers (PA), the bottle neck of the transmitter’s design tradeoff between linearity and power efficiency. The power amplifier exhibits a nonideal behavior, characterized by both nonlinearity and memory effects, heavily affecting that tradeoff, and therefore requiring an effective linearization technique, namely Digital Predistortion (DPD). The effectiveness of the DPD is highly dependent on the modeling scheme used to compensate for the PA’s nonideal behavior. In fact, its viability is determined by the scheme’s accuracy and implementation complexity. Generic behavioral models for nonlinear systems with memory have been used, considering the PA as a black box, and requiring RF designers to perform extensive testing to determine the minimal complexity structure that achieves satisfactory results. This thesis first proposes a direct systematic approach based on the parallel Hammerstein structure to determine the exact number of coefficients needed in a DPD. Then a physical explanation of memory effects is detailed, which leads to a close-form expression for the characteristic behavior of the PA entirely based on circuit properties. The physical expression is implemented and tested as a modeling scheme. Moreover, a link between this formulation and the proven behavioral models is explored, namely the Volterra series and Memory Polynomial. The formulation shows the correlation between parameters of generic behavioral modeling schemes when applied to RF PAs and demonstrates redundancy based on the physical existence or absence of modeling terms, detailed for the proven Memory polynomial modeling and linearization scheme.