Tropical geometry is algebraic geometry over the tropical semiring, or min-plus algebra. In this thesis, I discuss the basic geometry of plane tropical curves. By introducing the notion of abstract tropical curves, I am able to pass to a more abstract metric-topological setting. In this setting, I discuss divisors on tropical curves. I begin a study of $G$-invariant divisors and divisor classes.

One of the difficulties of human figure tracking is that humans move their bodies in complex, non-linear ways. An effective computational model of human motion could therefore be of great benefit in figure tracking. We are interested in the use of a class of dynamic models called switching linear dynamic systems for figure tracking. This thesis makes two contributions. First, we present an empirical analysis of some of the technical issues involved with applying linear dynamic systems to figure tracking. The lack of high-level theory in this area makes this type of empirical study valuable and necessary. We show that sensitivity of these models to perturbations in input is a central issue in their application to figure tracking. We also compare different types of LDS models and identification algorithms. Second, we describe 2-DAFT, a flexible software framework we have created for figure tracking. 2-DAFT encapsulates data and code involved in different parts of the tracking problem in a number of modules. This architecture leads to flexibility and makes it easy to implement new tracking algorithms.

Fuzzy set is a generalization of the classical set. A classical set is distinguished from another by a sharp boundary at some threshold value and therefore, they are also known as crisp set. In fuzzy theory, sharp boundary and crisp set are replaced by partial truth and fuzzy sets. The idea of partial truth facilitates information description especially those communicated through natural language whose transition between descriptive terms are not abrupt discontinuities. Instead, the transition is a smooth change over a range corresponding to the degree of fulfillment each intermediate elements has according to the operating definition of the concept. The shape of a fuzzy set is defined by its membership function. This, by far, has been the common extent of concern regarding the membership function. Different applications may use the membership function to describe different variables such as speed, position, temperature, dirtiness, traffic conditions etc. But the underlying application of fuzzy sets remains the same: to describe information whose membership function, created in an initial setting, preserve the same size and shape throughout its entire application. In other word, fuzzy sets are utilized as if they are static entities. Nothing has been said about how an initially defined membership function can develop over time with respect to a system. The current research proposes a new framework that concerns the evolution of membership functions. We introduce the concept of membership function propagation as a dynamic description of uncertainty. Given a dynamical system with a set of uncertain initial states which can be represented by membership functions, the membership function propagation describes how these membership functions evolve over time with respect to the system. The evolution produces a set of propagated membership functions that have different size and shape from their predecessors. They represent the uncertainty associated with the states of the system at a given time. This new description also confers new definitions for two important concepts in control theory, namely controllability and observability. These two concepts are re-introduced in a fuzzy sense, based on the concept of membership function propagation. By assuming convexity of the fuzzy set, criterions for controllability and observability can be derived. These criterions are illustrated by MATLAB and SIMULINK simulations of an inverted pendulum and a 2 degree of freedom mechanical manipulator.

Many dynamical systems in nature and technology involve delays in the interaction of variables forming the system. Furthermore, many of such systems involve external inputs or perturbations which might force the system to have arbitrary initial function. The conventional way to model these systems is using delay differential equations (DDE). However, DDEs with arbitrary initial functions has serious problems for finding analytical and computational solutions. This fact is a strong motivation for considering abstractions and approximations for dynamical systems involving delay. In this thesis, the piecewise linear systems with delay on piecewise constant part which is a useful subclass of hybrid dynamical systems is studied. We introduced various representations of these systems and studied the state transition conditions. We showed that there exists fixed point and periodic stable solutions. We modelled the genomic regulation of fission yeast cell cycle. We discussed various potential uses including approximating the DDEs and finally we concluded.

Regulatory processes and history dependent behavior appear in many dynamical systems in nature and technology. For modeling regulatory processes, hybrid systems offer several advances. From this point of view, to observe the capability of hybrid systems in a history dependent system is a strong motivation. In this thesis, we developed functional hybrid systems which exhibit memory dependent behavior such that the dynamics of the system is determined by both the location of the state vector and the memory. This property was explained by various examples. We used the hybrid system with memory in modeling the gene regulatory network of human immune response to Influenza A virus infection. We investigated the sensitivity of the piecewise linear model with memory. We introduced how the model can be developed in future.

When designing controllers with robust performance and stabilization requirements, H-infinity synthesis is a common tool to use. These controllers are often obtained by solving mathematical optimization problems. The controllers that result from these algorithms are typically of very high order, which complicates implementation. Low order controllers are usually desired, since they are considered more reliable than high order controllers. However, if a constraint on the maximum order of the controller is set that is lower than the order of the so-called augmented system, the optimization problem becomes nonconvex and it is relatively difficult to solve. This is true even when the order of the augmented system is low. In this thesis, optimization methods for solving these problems are considered. In contrast to other methods in the literature, the approach used in this thesis is based on formulating the constraint on the maximum order of the controller as a rational function in an equality constraint. Three methods are then suggested for solving this smooth nonconvex optimization problem. The first two methods use the fact that the rational function is nonnegative. The problem is then reformulated as an optimization problem where the rational function is to be minimized over a convex set defined by linear matrix inequalities (LMIs). This problem is then solved using two different interior point methods. In the third method the problem is solved by using a partially augmented Lagrangian formulation where the equality constraint is relaxed and incorporated into the objective function, but where the LMIs are kept as constraints. Again, the feasible set is convex and the objective function is nonconvex. The proposed methods are evaluated and compared with two well-known methods from the literature. The results indicate that the first two suggested methods perform well especially when the number of states in the augmented system is less than 10 and 20, respectively. The third method has comparable performance with two methods from literature when the number of states in the augmented system is less than 25.

This thesis investigates the frequency weighted balanced model reduction problem for linear time invariant systems. Both continuous and discrete time systems are considered, in one and two-dimensions. First the frequency weighted balanced model reduction problem is formulated, then a novel frequency weighted, balanced, model reduction method for continuous time systems is proposed. This method is based on the retention of frequency weighted Hankel singular values of the original system, and yields stable reduced order models even when two sided weightings are employed. An alternative frequency weighted balanced model reduction technique (applicable for controller reduction applications) is then developed. This is based on a parametrized combination of the frequency weighted partial fraction expansion technique with balanced truncation and the singular perturbation approximation techniques. This method yields stable models even when two sided weightings are employed. An a priori error bound for the model reduction method is derived. Lower frequency response errors and error bounds are obtained using free parameters and equivalent anti-stable weightings. Based on the same idea, a novel parameterized frequency weighted optimal Hankel norm model reduction method with a tighter a priori error bound is proposed. The proposed methods are extended to include discrete time systems. A frequency interval Gramians based stability preserving model reduction scheme with error bounds is also presented. In this case, frequency weights are not explicitly predefined. Discrete time system related results are also included. Several frequency weighted model reduction results for two-dimensional (2-D) systems are also proposed. The advantages of these schemes include error bounds, guaranteed stability and applicability to general stable (non-separable denominator) weighting functions. Finally, a novel 2-D identification based frequency weighted model reduction scheme is outlined. Numerically robust algorithms based on square root and balancing free techniques are proposed for frequency weighted balanced truncation techniques. Several practical examples are included to illustrate the effectiveness of the algorithms.

Στην παρούσα εργασία ασχοληθήκαμε με την παρατηρησιμότητα σε μη-γραμμικά συστήματα, αφού πρώτα εξετάσαμε τα γραμμικά συστήματα μέσω μίας γεωμετρικής προσέγγισης. Αρχικώς, ασχοληθήκαμε με την παρατηρησιμότητα affine συστημάτων υπό τμηματικά σταθερές εισόδους. Στη συνέχεια μελετήσαμε τα ίδια συστήματα υπό διαφορίσιμες εισόδους. Κατόπιν, αποδείξαμε ένα θεώρημα που θέτει ένα άνω φράγμα στο πλήθος των Lie παραγωγίσεων της συνάρτησης εξόδου. Τέλος, εξετάσαμε τη μη-γραμμική μη-παρατηρησιμότητα σε γενικότερες κατηγορίες μη-γραμμικών συστημάτων με χρήση γραμμικών μεθόδων και logic dynamic approach. / This thesis is focused on the problem of observability in non-linear systems. A brief examination of linear systems via a geometric approach is initially given. First, observability of affine systems under piecewise constant controls is examined. Second, the same examination is repeated for the same systems under differentiable inputs. Moreover, a theorem that sets an upper bound on the number of Lie derivatives of the output functions is proved and explained. Finally, the non-linear unobservability of non-linear systems in general is presented under the scope of linear methods and logic dynamic approach.

Orientador: Paulo Ricardo da Silva / Banca: Weber Flavio Pereira / Banca: Marcelo Messias / Resumo: Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle. / Abstract: We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems. / Mestre

Internal Models (IMs) play a significant role in autonomous robotics. They are mechanisms able to represent the input-output characteristics of the sensorimotor loop. In developmental robotics, open-ended learning of skills and knowledge serves the purpose of reaction to unexpected inputs, to explore the environment and to acquire new behaviours. The development of the robot includes self-exploration of the state-action space and learning of the environmental dynamics. In this dissertation, we explore the properties and benefits of the self-organisation of robot behaviour based on the homeokinetic learning paradigm. A homeokinetic robot explores the environment in a coherent way without prior knowledge of its configuration or the environment itself. First, we propose a novel approach to self-organisation of behaviour by artificial curiosity in the sensorimotor loop. Second, we study how different forward models settings alter the behaviour of both exploratory and goal-oriented robots. Diverse complexity, size and learning rules are compared to assess the importance in the robot’s exploratory behaviour. We define the self-organised behaviour performance in terms of simultaneous environment coverage and best prediction of future sensori inputs. Among the findings, we have encountered that models with a fast response and a minimisation of the prediction error by local gradients achieve the best performance. Third, we study how self-organisation of behaviour can be exploited to learn IMs for goal-oriented tasks. An IM acquires coherent self-organised behaviours that are then used to achieve high-level goals by reinforcement learning (RL). Our results demonstrate that learning of an inverse model in this context yields faster reward maximisation and a higher final reward. We show that an initial exploration of the environment in a goal-less yet coherent way improves learning. In the same context, we analyse the self-organisation of central pattern generators (CPG) by reward maximisation. Our results show that CPGs can learn favourable reward behaviour on high-dimensional robots using the self-organised interaction between degrees of freedom. Finally, we examine an on-line dual control architecture where we combine an Actor-Critic RL and the homeokinetic controller. With this configuration, the probing signal is generated by the exertion of the embodied robot experience with the environment. This set-up solves the problem of designing task-dependant probing signals by the emergence of intrinsically motivated comprehensible behaviour. Faster improvement of the reward signal compared to classic RL is achievable with this configuration.