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The contributions of this thesis lie in the area of global finite-time observer design for a class of nonlinear systems with bounded rational and mixed rational powers imposed on the incremental rate of the nonlinear terms whose solutions exist and are unique for all positive time. In the thesis, two different kinds of nonlinear global finite-time observers are designed by employing of finite-time theory and homogeneity properties with different methods. The global finite-time stability of both proposed observers is derived on the basis of Lyapunov theory. For a class of nonlinear systems with rational and mixed rational powers imposed on the nonlinearities, the first global finite-time observers are designed, where the global finite-time stability of the observation systems is achieved from two parts by combining asymptotic stability and local finitetime stability. The proposed observers can only be designed for the class of nonlinear systems with dimensions greater than 3. The observers have a dynamic high gain and two homogenous terms, one homogeneous of degree greater than 1 and the other of degree less than 1. In order to prove the global finite-time stability of the proposed results, two homogeneous Lyapunov functions are provided, corresponding with the two homogeneous items. One is homogeneous of degree greater than 1, which makes the observation error systems converging into a spherical area around the origin, and the other is of degree less than 1, which ensures local finite-time stability. The second global finite-time observers are also proposed based on the high-gain technique, which does not place any limitation on the dimension of the nonlinear systems. Compared with the first global finite-time observers, the newly designed observers have only one homogeneous term and a new gain update law where two new terms are introduced to dominate some terms in the nonlinearities and ensure global finite-time stability as well. The global finite-time stability is obtained directly based on a sufficient condition of finite-time stability and only one Lyapunov function is employed in the proof. The validity of the two kinds of global finite-time observers that have been designed is illustrated through some simulation results. Both of them can make the observation error systems converge to the origin in finite-time. The parameters, initial conditions as well as the high gain do have some impact on the convergence time, where the high gain plays a stronger role. The bigger the high gain is, the shorter the time it needs to converge. In order to show the performance of the two kinds of observers more clearly, two examples are provided and some comparisons are made between them. Through these, it can be seen that under the same parameters and initial conditions, although the amplitude of the observation error curve is slightly greater, the global finite-time observers with a new gain update law can make the observation error systems converge much more quickly than the global finite-time observers with two homogeneous terms. In the simulation results, one can see that, as a common drawback of high gain observers, they are noise-sensitive. Finding methods to improve their robustness and adaptiveness will be quite interesting, useful and challenging. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Electrical, Electronic and Computer Engineering / unrestricted

In this dissertation, we focus on two fundamental problems related to the navigation of ground robots and small Unmanned Aerial Vehicle (UAVs): cooperative localization and path planning. The theme running through in all of the work is the use of bearing only sensors, with a focus on monocular video cameras mounted on ground robots and UAVs. To begin with, we derive the conditions for the complete observability of the bearing-only cooperative localization problem. The key element of this analysis is the Relative Position Measurement Graph (RPMG). The nodes of an RPMG represent vehicle states and the edges represent bearing measurements between nodes. We show that graph theoretic properties like the connectivity and the existence of a path between two nodes can be used to explain the observability of the system. We obtain the maximum rank of the observability matrix without global information and derive conditions under which the maximum rank can be achieved. Furthermore, we show that for the complete observability, all of the nodes in the graph must have a path to at least two different landmarks of known location. The complete observability can also be obtained without landmarks if the RPMG is connected and at least one of the robots has a sensor which can measure its global pose, for example a GPS receiver. We validate these conditions by simulation and experimental results. The theoretical conditions to attain complete observability in a localization system is an important step towards reliable and efficient design of localization and path planning algorithms. With such conditions, a designer does not need to resort to exhaustive simulations and/or experimentation to verify whether a given selection of a control strategy, topology of the sensor network, and sensor measurements meets the observability requirements of the system. In turn, this leads to decreased requirements of time, cost, and effort for designing a localization algorithms. We use these observability conditions to develop a technique, for camera equipped UAVs, to cooperatively geo-localize a ground target in an urban terrain. We show that the bearing-only cooperative geo-localization technique overcomes the limitation of requiring a low-flying UAV to maintain line-of-sight while flying high enough to maintain GPS lock. We design a distributed path planning algorithm using receding horizon control that improves the localization accuracy of the target and of all of the UAVs while satisfying the observability conditions. Next, we use the observability analysis to explicitly design an active local path planning algorithm for UAVs. The algorithm minimizes the uncertainties in the time-to-collision (TTC) and bearing estimates while simultaneously avoiding obstacles. Using observability analysis we show that maximizing the observability and collision avoidance are complementary tasks. We provide sufficient conditions of the environment which maximizes the chances obstacle avoidance and UAV reaching the goal. Finally, we develop a reactive path planner for UAVs using sliding mode control such that it does not require range from the obstacle, and uses bearing to obstacle to avoid cylindrical obstacles and follow straight and curved walls. The reactive guidance strategy is fast, computationally inexpensive, and guarantees collision avoidance.

Cette thèse se concentrera exclusivement sur l’application de méthodes non paramétriques basées sur le noyau à des problèmes d’identification non-linéaires. Comme pour les autres méthodes non-linéaires, deux questions clés dans l’identification basée sur le noyau sont les questions de comment définir un modèle non-linéaire (sélection du noyau) et comment ajuster la complexité du modèle (régularisation). La contribution principale de cette thèse est la présentation et l’étude de deux critères d’optimisation (un existant dans la littérature et une nouvelle proposition) pour l’approximation structurale et l’accord de complexité dans l’identification de systèmes non-linéaires basés sur le noyau. Les deux méthodes sont basées sur l’idée d’intégrer des contraintes de complexité basées sur des caractéristiques dans le critère d’optimisation, en pénalisant les dérivées de fonctions. Essentiellement, de telles méthodes offrent à l’utilisateur une certaine souplesse dans la définition d’une fonction noyau et dans le choix du terme de régularisation, ce qui ouvre de nouvelles possibilités quant à la facon dont les modèles non-linéaires peuvent être estimés dans la pratique. Les deux méthodes ont des liens étroits avec d’autres méthodes de la littérature, qui seront examinées en détail dans les chapitres 2 et 3 et formeront la base des développements ultérieurs de la thèse. Alors que l’analogie sera faite avec des cadres parallèles, la discussion sera ancrée dans le cadre de Reproducing Kernel Hilbert Spaces (RKHS). L’utilisation des méthodes RKHS permettra d’analyser les méthodes présentées d’un point de vue à la fois théorique et pratique. De plus, les méthodes développées seront appliquées à plusieurs «études de cas» d’identification, comprenant à la fois des exemples de simulation et de données réelles, notamment : • Détection structurelle dans les systèmes statiques non-linéaires. • Contrôle de la fluidité dans les modèles LPV. • Ajustement de la complexité à l’aide de pénalités structurelles dans les systèmes NARX. • Modelisation de trafic internet par l’utilisation des méthodes à noyau / This thesis will focus exclusively on the application of kernel-based nonparametric methods to nonlinear identification problems. As for other nonlinear methods, two key questions in kernel-based identification are the questions of how to define a nonlinear model (kernel selection) and how to tune the complexity of the model (regularisation). The following chapter will discuss how these questions are usually dealt with in the literature. The principal contribution of this thesis is the presentation and investigation of two optimisation criteria (one existing in the literature and one novel proposition) for structural approximation and complexity tuning in kernel-based nonlinear system identification. Both methods are based on the idea of incorporating feature-based complexity constraints into the optimisation criterion, by penalising derivatives of functions. Essentially, such methods offer the user flexibility in the definition of a kernel function and the choice of regularisation term, which opens new possibilities with respect to how nonlinear models can be estimated in practice. Both methods bear strong links with other methods from the literature, which will be examined in detail in Chapters 2 and 3 and will form the basis of the subsequent developments of the thesis. Whilst analogy will be made with parallel frameworks, the discussion will be rooted in the framework of Reproducing Kernel Hilbert Spaces (RKHS). Using RKHS methods will allow analysis of the methods presented from both a theoretical and a practical point-of-view. Furthermore, the methods developed will be applied to several identification ‘case studies’, comprising of both simulation and real-data examples, notably: • Structural detection in static nonlinear systems. • Controlling smoothness in LPV models. • Complexity tuning using structural penalties in NARX systems. • Internet traffic modelling using kernel methods

Le diagnostic et la prise en charge des infections ostéo-articulaires (IOA) sont souvent complexes occasionnant une perte osseuse irréversible. La variabilité intra et inter-patient en terme de présentation clinique rend impossible le recours à une description systématique ou à une analyse statistique pour le diagnostic et l'étude de cette pathologie. Le développement d'IOA résulte d'interactions complexes entre les mécanismes cellulaires et moléculaires du tissu osseux et les bactéries. L'objectif de cette thèse est de modéliser l'IOA afin de simuler le comportement du système suite à des interactions au niveau cellulaire et moléculaire en utilisant l'approche de modélisation à base d'agents. Nous avons utilisé une méthode basée sur l'analyse bibliographique pour extraire les caractéristiques du modèle et les utiliser pour deux aspects. Le premier consiste en l'élaboration de la structure du modèle en identifiant les agents et les interactions, et le deuxième concerne l'estimation quantitative des différents paramètres du modèle. La réponse du système BJI aux différentes tailles d’inoculum bactérien a été simulée par la variation de différents paramètres. L'évolution des agents simulés a ensuite été analysée en utilisant une modélisant par des systèmes dynamiques non linéaires et une méthodologie "Datadriven", grâce auxquelles nous avons décrit le système d'IOA et identifié des relations plausibles entre les agents. Le modèle a réussi à présenter la dynamique des bactéries, des cellules immunitaires innées et des cellules osseuses au cours de la première étape de l'IOA et pour différentes tailles d'inoculum bactérien. La simulation a mis en évidence les conséquences sur le tissu osseux résultant du processus de remodelage osseux au cours de l'IOA. Ces résultats peuvent être considérés comme une base pour une analyse plus approfondie et pour la proposition de différentes hypothèses et scénarios de simulation qui pourraient être étudiés dans ce laboratoire virtuel. / Bone and joint infections are one of the most challenging bone pathologies that associated with irreversible bone loss and long costly treatment. The high intra and inter patient's variability in terms of clinical presentation makes it impossible to rely on the systematic description or classical statistical analysis for its diagnosis or studying. The development of BJI encompasses a complex interplay between the cellular and molecular mechanisms of the host bone tissue and the infecting bacteria. The objective of this thesis is to provide a novel computational modeling framework that simulates the behavior resulting from the interactions on the cellular and molecular levels to explore the BJI dynamics qualitatively and comprehensively, using an agent-based modeling approach. We relied on a meta-analysis-like method to extract the quantitative and qualitative data from the literature and used it for two aspects. First, elaborating the structure of the model by identifying the agents and the interactions, and second estimating quantitatively the different parameters of the model. The BJI system’s response to different microbial inoculum sizes was simulated with respect to the variation of several critical parameters. The simulation output data was then analyzed using a data-driven methodology and system dynamics approach, through which we summarized the BJI complex system and identified plausible relationships between the agents using differential equations. The BJI model succeeded in imitating the dynamics of bacteria, the innate immune cells, and the bone cells during the first stage of BJI and for different inoculum size in a compatible way. The simulation displayed the damage in bone tissue as a result of the variation in bone remodeling process during BJI. These findings can be considered as a foundation for further analysis and for the proposition of different hypotheses and simulation scenarios that could be investigated through this BJI model as a virtual lab.

The advances in computational and experimental modeling in the area of structural mechanics have stimulated research in a class of hybrid problems that require both of these modeling capabilities to be combined to achieve certain objectives. Real-time substructure (RTS) testing, structural system identification (SSI) and active control techniques fall in the category of hybrid problems that need efficient tools in both computational and experimental phases for their successful implementation. RTS is a hybrid testing method, which aims to overcome the scaling problems associated with the conventional dynamic testing methods (such as shake table test, effective force test and pseudo dynamic test) by testing the critical part of the structure experimentally with minimum compromise on spatio-temporal scaling, while modeling the remaining part numerically. The problem of SSI constitutes an important component within the broader framework of problems of structural health monitoring where, based on the in-situ measurements on the loading and a subset of critical responses of the structure, the system parameters are estimated with a view to detecting damage/degradation. Active control techniques are employed to maintain the functionality of important structures, especially under extreme dynamic loading. The work reported in the present thesis contributes to the areas of RTS, SSI and active control of nonlinear systems, the main focus being the computational aspects, i.e., in developing numerical strategies to address some of the unsolved issues, although limited efforts have also been made to undertake laboratory experimental investigations in the area of nonlinear SSI. The thesis is organized into seven chapters and five appendices. The first chapter contains an overview of the state of the art techniques in dynamic testing, SSI and structural control. The topics covered include effective force test, pseudo dynamic test, RTS test, time and frequency domain methods of nonlinear system identification, dynamic state estimation techniques with emphasis on particle filters, Rao-Blackwellization, structural control methods, control algorithms and active control of nonlinear systems. The review identifies a set of open problems that are subsequently addressed, to an extent, in the thesis. Chapter 2 focuses on the development of a time domain coupling technique, involving combined computational and experimental modeling, for vibration analysis of structures built-up of linear/nonlinear substructures. The numerical and experimental substructures are allowed to interact in real-time. The equation of motion of the numerical substructure is integrated using a step-by-step procedure that is formulated in the state space. For systems with nonlinear substructures, a multi-step transversal linearization method is used to integrate the equations of motion; and, a multi-step extrapolation scheme, based on the reproducing kernel particle method, is employed to handle the time delays that arise while accounting for the interaction between the substructures. Numerical illustrations on a few low dimensional vibrating structures are presented and these examples are fashioned after problems of seismic qualification testing of engineering structures using RTS testing techniques. The concept of substructuring is extended in Chapter 3 for implementing Rao-Blackwellization, a technique of combining particle filters with analytical computation through Kalman filters, for state and parameter estimations of a class of nonlinear dynamical systems with additive Gaussian process/observation noises. The strategy is based on decomposing the system to be estimated into mutually coupled linear and nonlinear substructures and then putting in place a rational framework to account for coupling between the substructures. While particle filters are applied to the nonlinear substructures, estimation of linear substructures proceeds using a bank of Kalman filters. Numerical illustrations for state/parameter estimations of a few linear and nonlinear oscillators with noise in both the process and measurements are provided to demonstrate the potential of the Rao-Blackwellized particle filter (RBPF) with substructuring. In Chapter 4, the concept of Rao-Blackwellization is extended to handle more general nonlinear systems, using two different schemes of linearization. A semi-analytical filter and a conditionally linearized filter, within the framework of Monte Carlo simulations, are proposed for state and parameter estimations of nonlinear dynamical systems with additively Gaussian process/observation noises. The first filter uses a local linearization of the nonlinear drift fields in the process/observation equations based on explicit Ito-Taylor expansions to transform the given nonlinear system into a family of locally linearized systems. Using the most recent observation, conditionally Gaussian posterior density functions of the linearized systems are analytically obtained through the Kalman filter. In the second filter, the marginalized posterior distribution of an appropriately chosen subset of the state vector is obtained using a particle filter. Samples of these marginalized states are then used to construct a family of conditionally linearized system of equations to obtain the posterior distribution of the states using a bank of Kalman filters. The potential of the proposed filters in state/parameter estimations is demonstrated through numerical illustrations on a few nonlinear oscillators. The problem of active control of nonlinear structural dynamical systems, in the presence of both process and measurement noises, is considered in Chapter 5. The focus of the study is on the exploitability of particle filters for state estimation in feedback control algorithms for nonlinear structures, when a limited number of noisy output measurements are available. The control design is done using the state dependent Riccati equation (SDRE) method. The Bayesian bootstrap filter and another based on sequential importance sampling are employed for state estimation. Numerical illustrations are provided for a few typically nonlinear oscillators of interest in structural engineering. The experimental validation of the RBPF using substructuring (developed in Chapter 3) and the conditionally linearized Monte Carlo filter (developed in Chapter 4), for parameter estimation, is reported in Chapter 6. Measured data available through laboratory experiments on simple building frame models subjected to harmonic base motions is processed using the proposed algorithms to identify the unknown parameters of the model. A brief summary of the contributions made in this thesis, together with a few suggestions for future research, are presented in Chapter 7. Appendix A provides an account of the multi-step transversal linearization method. The derivation of the reproducing kernel shape functions are presented in Appendix B. Appendix C provides the details of the stochastic Taylor expansion and derivation of the covariance structure of Gaussian MSI-s. The performance of a particle filtering algorithm (bootstrap filter) and Kalman filter in the state estimation of a linear system is provided in Appendix D and Appendix E contains the theoretical details of the Rao-Blackwellized particle filter.

L'objectif principal de ce travail a été d'apporter une nouvelle contribution quant à l'approche de contrôle optimal pendant la phase d'identification. Il s'agissait de trouver la commande à appliquer pendant l'expérience qui permet d'optimiser un critère qui est fonction des sensibilités des sorties par rapport aux paramètres du modèle à identifier. Cette approche couplant contrôleur prédictif sous contraintes et estimateur a résolu en ligne le problème d'identification à chaque instant en utilisant l'observateur. En ce sens, c'est une approche permettant d'automatiser et d'optimiser le design d'expérience, tout en réalisant conjointement l'identification d'un paramètre du modèle spécifié. L'aspect temps réel a été pris en compte dans la formulation de la solution apportée. Dans ce contexte, nous avons introduit deux stratégies de commande pour l'identification optimale. La première était basée sur un modèle de prédiction non linéaire et la seconde sur un modèle linéaire temps variant. Si le temps devient un paramètre critique pour l'implémentation de l'approche, cette dernière vise à réduire le temps alloué à l'optimisation. L'approche d'identification optimale en ligne a été appliquée à deux problèmes concrets du Génie des Procédés (réaction de saponification et cuisson de peintures). Ces exemples ont permis de vérifier en simulation, l'efficacité et la faisabilité de cette approche. / The main aim of this work is to give a new approach of optimal control during the phase of identification. The question is how to tune the control action to be applied during the experiment optimize a criterion which is function of the sensitivities of the mesure with respect to the parameters of the model to be identified. This approach coupling constrained predictive controller and estimator solves on line the problem of identification by using the observer. In that sense, it is an approach allowing an optimal and automatic design of experiment, while performing at the same time the identification of one parameter of the specified model. The real time aspect was taken into account in the formulation of the solution. In this context, we introduced two strategies for optimal identification : the first one is based on a nonlinear model of prediction and the second one is based on a linear time varying model that may be used if the real time aspect becomes a critical parameter for the implementation of the approach. This approach of on line optimal identification was applied on two concrete problems in Chemical Engineering. These examples show the performance and the feasibility of this approach.

This thesis deals with a simulation of analogue audio effects using the nonlinear models that replace the analogue nonlinear devices in discrete domain. The thesis describes Volterra system model and simplified Volterra system model that can be realized in two ways, either Wiener model, or Hammerstein model. The method for the analysis and modeling of audio and acoustic nonlinear systems is presented in this thesis. This method allows through knowledge of the input swept-sine signal and the response of the analogue nonlinear system to the input signal to determine the coefficients of the discrete nonlinear system. This allows simulating the analogue nonlinear system in discrete domain. The method was first tested and then used successfully for simulation of the analogue nonlinear system in discrete domain. Concretely, it was simulated a musical guitar effect of the type of distortion. Last part of this thesis is devoted a description of VST technology and an implementation of VST plug-in module, which realizations Hammerstein model.

With the development of technology and availability of data, it is sometimes easier to learn the control policies directly from the data, rather than modeling a plant and designing a controller. Modeling a plant is not always possible due to the complexity of the plant. Data-enabled predictive control (DeePC) is a recently proposed approach that combines system identification, estimation, and control in a single optimization problem. DeePC is primarily designed for LTI systems. The purpose of this thesis is to extend the application of DeePC to nonlinear systems with a particular focus on a non-holonomic ground robot. To reach this goal, we decompose the system states into different working modes where each mode can be linearly approximated. Furthermore, the data collection policies were also evaluated to conclude how they affect the performance of the DeePC. We identified several key challenges in this direction, namely: data-demanding structure, high computational complexity, and performance deterioration with increased non-linearity. While these challenges prohibited the application of DeePC to the ground robot system; we successfully applied the method to a benchmark non-linear system, the inverted pendulum on cart problem, and studied the effect of various design choices on control performance. Our observations indicate potential areas of improvement toward enabling DeePC for highly nonlinear systems. / Med utvecklingen av teknik och tillgänglighet av data är det ibland enklare att lära sig styrpolicyerna direkt från data, snarare än att modellera ett system och designa en styrenhet. Att modellera ett system är inte alltid möjligt på grund av systemets komplexitet. Data aktiverad prediktiv kontroll (DeePC) är en nyligen föreslagen metod som kombinerar systemidentifiering, uppskattning och kontroll i ett enda optimeringsproblem. DeePC är främst designad för LTI-system. Syftet med denna avhandling är att utöka tillämpningen av DeePC till icke-linjära system med särskilt fokus på en icke-holonomisk markrobot. För att nå detta mål delar vi upp systemtillstånden i olika arbetslägen där varje läge kan approximeras linjärt. Dessutom utvärderades datainsamlingspolicyerna för att dra slutsatser om hur de påverkar DeePCs prestation. Vi identifierade ett antal nyckelutmaningar i denna riktning, nämligen: datakrävande struktur, hög beräkningskomplexitet och prestandaförsämring med ökad icke-linjäritet. Även om de utmaningerna hindrade tillämpningen av DeePC på markrobot systemet; har vi framgångsrikt tillämpat metoden på ett benchmark icke-linjärt system, problemet med inverterad pendel på vagn, och studerade effekten av olika designval på kontrollprestanda. Våra observationer indikerar potentiella förbättringsområden för att möjliggöra DeePC för mycket olinjära system.

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