Orientador: Claudio Aguinaldo Buzzi / Banca: Marco antônio Teixeira / Banca: Paulo Ricardo Silva / Resumo: Neste trabalho estudamos os campos de vetores planares semi-homogênios quadráticos e também os campos de vetores planares com duas retas paralelas invariantes pelo fluxo. Para cada dessas classes, obtemos uma classificação dos retratos de fase global no disco de Poincaré e apresentamos as respectivas formas normais. Dentre as técnicas utilizadas no desenvolvimento do trabalho destacamos a Compactificação de Poincaré e o Método do Blow-up. / Mestre

Systems involving quantisation arise in many areas of engineering, especially when digital implementations are involved. In this thesis we consider different aspects of quantisation in feedback control systems. We study two topics of interest: (a) quantisers that quadratically stabilise a given system and are efficient in the use of their quantisation levels and (b) the derivation of ultimate bounds for perturbed systems, especially when the perturbations arise from the use of quantisers. In the first part of the thesis we address problem (a) above. We consider quadratic stabilisation of discrete-time multiple-input systems by means of quantised static feedback and we measure the efficiency of a quantiser via the concept of quantisation density. Intuitively, the lower the density of a quantiser is, the more separated its quantisation levels are. We thus deal with the problem of optimising density over all quantisers that quadratically stabilise a given system with respect to a given control Lyapunov function. Most of the available results on this problem treat single-input systems, and the ones that deal with the multiple-input case consider only two-input systems. In this thesis, we derive several new results for multiple-input systems and also provide an alternative approach to deal with the single-input case. Our new results for multiple-input systems include the derivation of the structure of optimal quantisers and the explicit design of multivariable quantisers with finite density that are able to quadratically stabilise systems having an arbitrary number of inputs. For single-input systems, we provide an alternative approach to the analysis and design of optimal quantisers by establishing a link between the separation of the quantisation levels of a quantiser and the size of its quantisation regions. In the second part of the thesis we address problem (b) above. In the presence of perturbations, asymptotic stabilisation may not be possible. However, there may exist a bounded region that contains the equilibrium point and has the property that the system trajectories converge to this bounded region. When this bounded region exists, we say that the system trajectories are ultimately bounded, and that this bounded region is an ultimate bound for the system. The size of the ultimate bound quantifies the performance of the system in steady state. Hence, it is important to derive ultimate bounds that are as tight as possible. This part of the thesis addresses the problem of ultimate bound computation in settings involving several scalar quantisers, each having different features. We consider each quantised variable in the system to be a perturbed copy of the corresponding unquantised variable. This turns the original quantised system into a perturbed system, where the perturbation has a natural \emph{componentwise} bound. Moreover, according to the type of quantiser employed, the perturbation bound may depend on the system state. Typical methods to estimate ultimate bounds are based on the use of Lyapunov functions and usually require a bound on the norm of the perturbation. Applying these methods in the setting considered here may disregard important information on the structure of the perturbation bound. We therefore derive ultimate bounds on the system states that explicitly take account of the componentwise structure of the perturbation bound. The ultimate bounds derived also have a componentwise form, and can be systematically computed without having to, e.g. select a suitable Lyapunov function for the system. The results of this part of the thesis, though motivated by quantised systems, apply to more general perturbations, not necessarily arising from quantisation. / PhD Doctorate

This paper examines the probability that a random polynomial of specific degree over a field has a specific number of distinct roots in that field. Probabilities are found for random quadratic polynomials with respect to various probability measures on the real numbers and p-adic numbers. In the process, some properties of the p-adic integer uniform random variable are explored. The measure Witt ring, a generalization of the canonical Witt ring, is introduced as a way to link quadratic forms and measures, and examples are found for various fields and measures. Special properties of the Haar measure in connection with the measure Witt ring are explored. Higher-degree polynomials are explored with the aid of numerical methods, and some conjectures are made regarding higher-degree p-adic polynomials. Other open questions about measure Witt rings are stated. / Graduation date: 1999

This thesis provides a survey of the attacks on multivariate cryptosystems. We begin by providing an outline of the general multivariate cryptosystem. Proceeding from there, we show that even with this level of detail, there are several attacks that are possible, including the method of Groebner bases, the XL method, and the recently announced method of Dixon resultants. Less general attack techniques also exist, such as MinRank attacks and differential analysis. These attacks lack the universality of the first three mentioned. In order to explore these less general attacks further, more details are required, so we present four different multivariate cryptosystems. Then, we attack them, using the less general attacks of MinRank, differential analysis and even an attack specific to one system. This concludes our study of the attacks themselves, and we move on to note that not all routes of attack are promising. Specifically, quantum computing does not seem to be helpful beyond the quadratic speed-up of Grover's algorithm. We also note that not all multivariate cryptosystems have been successfully attacked as of the writing of this thesis. We conclude with the fact that multivariate cryptography is gaining more and more active study.

This thesis provides a survey of the attacks on multivariate cryptosystems. We begin by providing an outline of the general multivariate cryptosystem. Proceeding from there, we show that even with this level of detail, there are several attacks that are possible, including the method of Groebner bases, the XL method, and the recently announced method of Dixon resultants. Less general attack techniques also exist, such as MinRank attacks and differential analysis. These attacks lack the universality of the first three mentioned. In order to explore these less general attacks further, more details are required, so we present four different multivariate cryptosystems. Then, we attack them, using the less general attacks of MinRank, differential analysis and even an attack specific to one system. This concludes our study of the attacks themselves, and we move on to note that not all routes of attack are promising. Specifically, quantum computing does not seem to be helpful beyond the quadratic speed-up of Grover's algorithm. We also note that not all multivariate cryptosystems have been successfully attacked as of the writing of this thesis. We conclude with the fact that multivariate cryptography is gaining more and more active study.

Clock mesh is widely used in microprocessor designs for achieving low clock skew and high process variation tolerance. Clock mesh optimization is a very diffcult problem to solve because it has a highly connected structure and requires accurate delay models which are computationally expensive. Existing methods on clock network optimization are either restricted to clock trees, which are easy to be separated into smaller problems, or naive heuristics based on crude delay models. A clock mesh sizing algorithm, which is aimed to minimize total mesh wire area with consideration of clock skew constraints, has been proposed in this research work. This algorithm is a systematic solution search through rigorous Sequential Quadratic Programming (SQP). The SQP is guided by an efficient adjoint sensitivity analysis which has near-SPICE(Simulation Program for Integrated Circuits Emphasis)-level accuracy and faster-than-SPICE speed. Experimental results on various benchmark circuits indicate that this algorithm leads to substantial wire area reduction while maintaining low clock skew in the clock mesh. The reduction in mesh area achieved is about 33%.

Thesis (doctoral)--Universitate Litteraria Friderica Guilelma, 1835. / Vita.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaf 98).

Small aerial vehicles such as quad-rotors have been widely used commercially, for research and for hobby for the last decade with use still growing. The high interest is mainly due to the vehicles being small, simple, cheap and versatile. Among rigid body dynamics fast dynamics exist cohering to motors and other fast actuators. A linear quadratic control design technique is here investigated. The design technique suggests that the linear quadratic controller can be designed with penalties on the slow states only. The fast dynamics are modeled but the states are not penalised in the linear quadratic design. The design technique is here applied and evaluated. The results show that this in several cases is a suitable design technique for linear quadratic control design. MATLAB and Simulink have been widely used for design and implementation of control systems. With additional toolboxes these control systems can be compiled to and run on remote computers. Small, lightweight computers with high computational capacity are now easily accessible. In this thesis an avionics solution based on a small, powerful computer is presented. Simulink models can be compiled and transferred to the computer from the Simulink environment. The result is a user friendly way of rapid prototyping and evaluation of control systems.

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