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This thesis deals with the control of a highly maneuverable miniature helicopter about trajectories, generated online, from a library of prespecified maneuvers. Linearizing the nonlinear equations describing the helicopter dynamics about the prespecified, library maneuvers results in a hybrid linear time-varying (LTV) model. Two control approaches are used to design controllers corresponding to each library maneuver: the standard L2-induced norm approach and an approach which also uses the L2-induced norm as a performance measure while accounting for uncertain initial states. Each control approach is evaluated in closed-loop simulation with a nonlinear helicopter model. The controllers are set to drive the helicopter model to track desired trajectories in the presence of disturbances such as wind gusts, turbulence, sensor noise, and uncertain initial conditions. For the specific plant formulations and trajectories presented, performance is comparable for both control approaches; however, it is possible to improve controller performance by exploiting some of the features of the approach accounting for uncertain initial states. These improvements in performance are topics for future work along with implementation of the presented approaches and results on a remote control helicopter. / Master of Science

Acumen is a language and tool for modeling and simulating cyber-physical systems. It allows the user to conduct simulations using a technique called rigorous simulation that produces results with explicit error bounds, expressed as intervals. This feature can be useful when designing and testing systems where the reliability of results or taking uncertainty into account is important. Unfortunately, analyzing these simulation results can be difficult, as Acumen supports only two ways of presenting them: raw data tables and 2D-plots. These views of the data make certain kinds of analysis cumbersome, such as understanding correlations between variables. This is especially true when the model in question is large. This project proposes a new way of visualising rigorous simulation results in Acumen. The goal of this project is to create a method for visualising intervallic values in 3D, and implement it in Acumen. To achieve that, every span of values is represented as a series of overlapping objects. This family of objects, which constitutes an under-approximation of the true simulation result, is then wrapped inside a semi-translucent box that is a conservative over-approximation of the simulation result. The resulting implementation makes for a combination of mathematical correctness (rigour), and mediation of intervals in question. It enables the user to explore the results of his rigorous simulations as conveniently as with the existing, non-rigorous simulation methods, using the 3D visualisation to simplify the study of real-life problems. To our knowledge, no existing software features visualisation of interval-based simulation results, nor is there any convention for doing this. Some ways in which the proposed solution could be improved are suggested at the end of this report

This dissertation advances tools for robust control and analysis of complex nonlinear dynamical systems. Specifically, it leverages standard synthesis and robustness analysis techniques developed for linear systems and provides additional results to design robust controllers for nonlinear systems over the considered operating envelopes. To facilitate the application of these linear techniques, nonlinear systems are represented as uncertain linear models. A significant contribution of this dissertation is the development of data-driven approaches to generate these uncertain linear models, which capture the behavior of nonlinear systems reasonably well over the considered operating envelopes without being overly conservative. We propose two approaches where a nominal linear time-invariant (LTI) approximation of a nonlinear system is first obtained using traditional linearization techniques, and data-driven methods are then applied to model the discrepancies arising from this simplification. In the first approach, the discrepancies are modeled using polynomials, resulting in an improved linear parameter-varying (LPV) approximation that can be expressed as a linear fractional transformation (LFT) on uncertainties. The second approach utilizes coprime factorization and a data-driven lifting technique to approximate the nonlinear discrepancy model with an LTI state-space system in a higher-dimensional state space. Additionally, a purely data-driven modeling approach is proposed for nonlinear systems with uncertain initial conditions. In this approach, a deep learning framework is developed to approximate nonlinear dynamical systems with LPV state-space models in higher-dimensional spaces while simultaneously characterizing the uncertain initial states within the lifted state space. Another contribution is the development of a systematic method for identifying critical attack points in cyber-physical systems using integral quadratic constraints (IQCs). IQC analysis is also used in developing a framework focused on the design and analysis of robust path-following controllers for an autonomous underwater vehicle (AUV). In this framework, the AUV is modeled as an LFT on uncertainties and is affected by exogenous inputs such as measurement noise and ocean currents. A tuning routine is developed for robust control design, using the robust performance level derived from IQC analysis to guide the tuning process. This framework is applied to design \( H_\infty \), \( H_2 \), and LPV controllers for the AUV, with the results validated through extensive nonlinear simulations and underwater experiments. Finally, this dissertation presents novel controller synthesis and IQC analysis techniques for LPV systems with uncertain initial conditions. These methods, combined with the lifting-based LPV modeling approach, enable the design of static, nonstationary LPV controllers for nonlinear systems in a higher-dimensional space. When interpreted in the original state space, these controllers become nonlinear with explicit dependence on both the scheduling parameters and time. Through examples, it is demonstrated that these controllers outperform those designed using nominal linearized models. / Doctor of Philosophy / This dissertation focuses on robust control design and analysis for complex nonlinear dynamical systems using well-established methods developed for linear systems. These methods are relatively easier to implement than their counterparts for nonlinear systems and can provide both stability and performance guarantees. A major contribution of this dissertation is the development of data-driven approaches to generate linear approximations of nonlinear systems that are valid over larger operating envelopes compared to those obtained through traditional linearization techniques. Another contribution is the development of a systematic method for identifying critical attack points in cyber-physical systems using robust control tools. Robust control methods are also used in developing a framework focused on the design and analysis of robust path-following controllers for an autonomous underwater vehicle (AUV). In this framework, the AUV is modeled as an uncertain linear system and is affected by external inputs such as measurement noise and ocean currents. A tuning routine is developed to automate the control design process, and the framework is validated through extensive nonlinear simulations and underwater experiments. Finally, this dissertation presents novel controller synthesis and analysis techniques for linear systems with uncertain initial conditions. These methods, combined with a data-driven modeling approach, enable the design of nonlinear controllers that are demonstrated to outperform those designed using nominal linearized models.

This is the first study to identify and analyse postmodern romances as a new development of the romance and to relate this late twentieth-century subgenre to its tradition. Based on a selection of works published between 1969 and 1994, by A. S. Byatt, Lindsay Clarice, Michael Dorris and Louise Erdrich, John Fowles, Iris Murdoch, Susan Sontag and Jeanette Winterson, it seeks to demonstrate how this new orientation of the romance produces meaning in dialogue with generic conventions and traditional works and, in doing so, both criticises and rehabilitates the genre.A 'postmodern romance' is a double-natured or hybrid text influenced both by inherited romance strategies and experimental postmodern techniques, such as those specified in Linda Hutcheon's study of the "poetics* of postmodernism: ambiguity, parody, paradox, contradiction and self-reflexivity. Hutcheon's theories, as well as theories of the romance, of intertextuality, of feminism, of New Historicism and of popular culture provide the theoretical framework for my argument.Intertextuality is an important manifestation of literary postmodernism, and I isolate three kinds of intertextual relationships which 1 see as characteristic of postmodern romances. Taking as its starting point Julia Kristeva's view that intertextuality includes social, political and cultural, as well as literary, contexts, 1 argue that feminist ideologies appear as cultural intertexts in postmodern romances, thereby challenging the association between the romance genre and a patriarchal world-view. The connections between postmodern and chivalric, historical and women's popular romances are instances of generic intertextuality, where particularly postmodern literary strategies are fused with more conventional attributes of the romance. The links between the postmodern works and the various subgenres of romance affect both the former and the latter, making the postmodern texts accessible to a larger audience, but also revealing forgotten or overlooked complexities in earlier examples of the romance. The return to individual texts is an instance of specific intertextuality, where postmodern romances reinterpret and rewrite particular, earlier romances. Since the relationship between the texts involved is dialogic and, hence, unpredictable, the modern works are also reinterpreted by their intertexts.Postmodern romances transcend the boundaries between real and unreal, male and female, "high" and "low" literature, and in the process they show that this might be equally characteristic of traditional romances. As a result of the fusion of postmodern and romantic literary modes, the inherent duality of the romance genre as such is brought to the fore at the same time as the genre is revived. / <p>Swedish Science Press, Uppsala (distribution).</p> / digitalisering@umu

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences.

<p>Entering the era of big data, human beings are faced with an unprecedented amount of geometric data today. Many computational challenges arise in processing the new deluge of geometric data. A critical one is data uncertainty: the data is inherently noisy and inaccuracy, and often lacks of completeness. The past few decades have witnessed the influence of geometric algorithms in various fields including GIS, spatial databases, and computer vision, etc. Yet most of the existing geometric algorithms are built on the assumption of the data being precise and are incapable of properly handling data in the presence of uncertainty. This thesis explores a few algorithmic challenges in what we call geometric computing over uncertain data.</p><p>We study the nearest-neighbor searching problem, which returns the nearest neighbor of a query point in a set of points, in a probabilistic framework. This thesis investigates two different nearest-neighbor formulations: expected nearest neighbor (ENN), where we consider the expected distance between each input point and a query point, and probabilistic nearest neighbor (PNN), where we estimate the probability of each input point being the nearest neighbor of a query point.</p><p>For the ENN problem, we consider a probabilistic framework in which the location of each input point and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance. We present methods for computing an exact ENN or an \\eps-approximate ENN, for a given error parameter 0 < \\eps < 1, under different distance functions. These methods build an index of near-linear size and answer ENN queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or \\eps-approximate ENN queries with provable performance guarantees. Moreover, we extend our results to answer exact or \\eps-approximate k-ENN queries. Notably, when only the query points are uncertain, we obtain state-of-the-art results for top-k aggregate (group) nearest-neighbor queries in the L1 metric using the weighted SUM operator.</p><p>For the PNN problem, we consider a probabilistic framework in which the location of each input point is specified as a probability distribution function. We present efficient algorithms for (i) computing all points that are nearest neighbors of a query point with nonzero probability; (ii) estimating, within a specified additive error, the probability of a point being the nearest neighbor of a query point; (iii) using it to return the point that maximizes the probability being the nearest neighbor, or all the points with probabilities greater than some threshold to be the nearest neighbor. We also present some experimental results to demonstrate the effectiveness of our approach.</p><p>We study the convex-hull problem, which asks for the smallest convex set that contains a given point set, in a probabilistic setting. In our framework, the uncertainty of each input point is described by a probability distribution over a finite number of possible locations including a null location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \\beta-hull that may be a useful representation of uncertain hulls.</p><p>We study contour trees of terrains, which encode the topological changes of the level set of the height value \\ell as we raise \\ell from -\\infty to +\\infty on the terrains, in a probabilistic setting. We consider a terrain that is defined by linearly interpolating each triangle of a triangulation. In our framework, the uncertainty lies in the height of each vertex in the triangulation, and we assume that it is described by a probability distribution. We first show that the probability of a vertex being a critical point, and the expected number of nodes (resp. edges) of the contour tree, can be computed exactly efficiently. Then we present efficient sampling-based methods for estimating, with high probability, (i) the probability that two points lie on an edge of the contour tree, within additive error; (ii) the expected distance of two points p, q and the probability that the distance of p, q is at least \\ell on the contour tree, within additive error and/or relative error, where the distance of p, q on a contour tree is defined to be the difference between the maximum height and the minimum height on the unique path from p to q on the contour tree.</p> / Dissertation

Le travail de la thèse traite le problème de suivi de trajectoires des systèmes non linéaires incertains,dont le modèle nominal est construit à l’aide d’un système flou TS (Takagi-Sugeno) de type-2. Cedernier, exploite les modèles locaux du système obtenus par linéarisation autour de certains pointsde fonctionnement. La commande développée est basée sur les modes glissants d’ordre deux avecSuper-Twisting. Nous avons proposé deux systèmes flous type-2 adaptatifs, qui ont comme uniqueentrée la surface de glissement, pour résoudre le problème du calcul de la valeur optimale des gainsde la commande. Des résultats de simulation ont permis de comparer les performances de l’approcheproposée avec la méthode classique. Ensuite, nous avons introduit le concept de l’intégral sliding modepour imposer à priori le temps d’arrivée sur la surface de glissement. Les approches proposées sontgénéralisées aux cas des systèmes multivariables. Plusieurs résultats par simulation et implémentationen temps réel sont présentés pour illustrer les performances des approches développées / This work deals with a fuzzy tracking control design for uncertain nonlinear dynamic system withexternal disturbances and using a TS (Takagi-Sugeno) fuzzy model description. The control is basedon the Super-Twisting algorithm, which is among of second order sliding mode control. Moreover, twoadaptive fuzzy type-2 systems have been introduced to generate the two Super-Twisting signals toavoid both the chattering and the constraint on the knowledge of disturbances and uncertainties upperbounds. These adaptive fuzzy type-2 systems has only one input : the sliding surface, and one output :the optimale values of the control gains, which are hard to compute with the original algorithm.Simulation results are obtained in order to compare the performances of the proposed method tothat given by Levant. Then, we have introduced the integral sliding mode concept to impose inadvance the convergence time and the arrival on the sliding surface. The proposed approaches aregeneralized to the case of multivariable systems. Several results in simulation and in real time usinga benchmark are obtained to validate and to confirm the performances of our contributions.

This thesis deals with the estimation of unobserved variables or states from a time series of noisy observations. Approximate minimum variance filters for a class of discrete time systems with both additive and multiplicative noise, where the measurement might be delayed randomly by one or more sample times, are investigated. The delayed observations are modelled by up to N sample times by using N Bernoulli random variables with values of 0 or 1. We seek to minimize variance over a class of filters which are linear in the current measurement (although potentially nonlinear in past measurements) and present a closed-form solution. An interpretation of the multiplicative noise in both transition and measurement equations in terms of filtering under additive noise and stochastic perturbations in the parameters of the state space system is also provided. This filtering algorithm extends to the case when the system has continuous time state dynamics and discrete time state measurements. The Euler scheme is used to transform the process into a discrete time state space system in which the state dynamics have a smaller sampling time than the measurement sampling time. The number of sample times by which the observation is delayed is considered to be uncertain and a fraction of the measurement sample time. The same problem is considered for nonlinear state space models of discrete time systems, where the measurement might be delayed randomly by one sample time. The linearisation error is modelled as an additional source of noise which is multiplicative in nature. The algorithms developed are demonstrated throughout with simulated examples.

La commande H∞ est de nos jours utilisée pour la régulation de nombreux systèmes. Cette technique de contrôle permet de synthétiser des lois de commande robustes, dans le sens où le comportement du système régulé est peu sensible aux perturbations externes. De plus, la commande H∞ permet de prendre en compte des incertitudes liés au modèle décrivant le système à réguler. Par conséquence, cette technique de contrôle est robuste vis-à-vis des perturbations et des incertitudes de modèle. Afin de synthétiser une loi de commande robuste, les spécifications des performances du système en boucle fermée sont traduites en critères H∞ à partir desquels est formulé un problème d'optimisation. La loi de commande est une solution de ce problème, qui est non convexe dans le cas général. Les deux principales approches pour la résolution de ce problème sont basées sur la reformulation convexe et les méthodes d'optimisations locales, mais ne garantissent pas l'optimalité de la loi de commande vis-à-vis des critères H∞. Cette thèse propose une approche de la commande H∞ par des méthodes d'optimisation globales, rarement considérées jusqu'à présent. Contrairement aux approches classiques, bien qu'au prix d'une complexité algorithmique supérieure, la convergence vers la loi de commande optimale est garantie par les méthodes globales. De plus, les incertitude de modèle sont prises en compte de manière garantie, ce qui n'est pas nécessairement le cas avec les approches convexes et locales. / H∞ control is nowadays used in many applications. This control technique enables to synthesize control laws which are robust with respect to external disturbances. Moreover, it allows to take model uncertainty into account in the synthesis process. As a consequence, H∞ control laws are robust with respect to both external disturbances and model uncertainty. A robust control law is a solution to an optimization problem, formulated from H∞ criteria. These criteria are the mathematical translations of the desired closed loop performance specifications. The two classical approaches to the optimization problem rely on the convex reformulation and local optimization methods. However, such approaches are unable to guarantee the optimality, with respect to the H∞ criteria, of the control law. This thesis proposes to investigate a global optimization approach to H∞ control. Contrary to convex and local approaches, global optimization methods enable to guarantee the optimality of the control, and also to take into account model uncertainty in a reliable way.