Teoria de estabilidade de equações diferenciais ordinárias e aplicações: modelos presa-predador e competição entre espécies

Pós-graduação em Matemática Universitária - IGCE

Projeto de controladores baseados em LMIs: realimentação derivativa e sistemas chaveados utilizando estrutura variável

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Controle robusto chaveado de sistemas lineares variantes no tempo com aplicação em falhas estruturais

Pós-graduação em Engenharia Elétrica - FEIS

Characterization of Stability Region for General Autonomous Nonlinear Dynamical Systems

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Stability in Terms of Two Measures for Differential Equations with “Maxima”

Атанаска Георгиева, Стела Глухчева, Снежана Христова - Изследвана е устойчивостта на нелинейни диференциални уравнения с “максимуми” по отношение на две мерки. Приложени са две различни мерки за началните условия и за решението. Използван е методът на Разумихин, а също така и методът на сравнението на обикновени скаларни диференциални уравнения. Приложението на получените резултати и достатъчни условия за устойчивост е илюстрирано с пример.

On Convergence of Differential Evolution Over a Class of Continuous Functions With Unique Global Optimum

Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms of current interest. Since its inception in the mid 1990s, DE has been finding many successful applications in real-world optimization problems from diverse domains of science and engineering. This paper takes a first significant step toward the convergence analysis of a canonical DE (DE/rand/1/bin) algorithm. It first deduces a time-recursive relationship for the probability density function (PDF) of the trial solutions, taking into consideration the DE-type mutation, crossover, and selection mechanisms. Then, by applying the concepts of Lyapunov stability theorems, it shows that as time approaches infinity, the PDF of the trial solutions concentrates narrowly around the global optimum of the objective function, assuming the shape of a Dirac delta distribution. Asymptotic convergence behavior of the population PDF is established by constructing a Lyapunov functional based on the PDF and showing that it monotonically decreases with time. The analysis is applicable to a class of continuous and real-valued objective functions that possesses a unique global optimum (but may have multiple local optima). Theoretical results have been substantiated with relevant computer simulations.

Lyapunov's stability and operational margin of grid-connected photovoltaic modules

Analyses of photovoltaic power generation based on Lyapunov's theorems are presented. The characteristics of the photovoltaic module and the power conditioning unit are analyzed in order to establish energy functions that assess the stability of solutions and define safe regions of operation. Furthermore, it is shown that grid-connected photovoltaic modules driven at maximum power may become unstable under normal grid transients. In such cases, stability can be maintained by allowing an operational margin defined as the energy difference between the stable and the unstable solutions of the system. Simulations show that modules cope well with grid transients when a sufficiently large margin is used.

Lyapunov-like conditions for the existence of zeno behavior in hybrid and lagrangian hybrid systems

Lyapunov-like conditions that utilize generalizations of energy and barrier functions certifying Zeno behavior near Zeno equilibria are presented. To better illustrate these conditions, we will study them in the context of Lagrangian hybrid systems. Through the observation that Lagrangian hybrid systems with isolated Zeno equilibria must have a onedimensional configuration space, we utilize our Lyapunov-like conditions to obtain easily verifiable necessary and sufficient conditions for the existence of Zeno behavior in systems of this form. © 2007 IEEE.

Application of Lyapunov's Direct Method to the Existence of Integral Manifolds of Impulsive Differential Equations

The present paper investigates the existence of integral manifolds for impulsive differential equations with variable perturbations. By means of piecewise continuous functions which are generalizations of the classical Lyapunov’s functions, sufficient conditions for the existence of integral manifolds of such equations are found.

The Expanded Human Kallikrein (KLK) gene family : genomic organisation, tissue specific expression and potential functions

The tissue kallikreins are serine proteases encoded by highly conserved multigene families. The rodent kallikrein (KLK) families are particularly large, consisting of 13 26 genes clustered in one chromosomal locus. It has been recently recognised that the human KLK gene family is of a similar size (15 genes) with the identification of another 12 related genes (KLK4-KLK15) within and adjacent to the original human KLK locus (KLK1-3) on chromosome 19q13.4. The structural organisation and size of these new genes is similar to that of other KLK genes except for additional exons encoding 5 or 3 untranslated regions. Moreover, many of these genes have multiple mRNA transcripts, a trait not observed with rodent genes. Unlike all other kallikreins, the KLK4-KLK15 encoded proteases are less related (25–44%) and do not contain a conventional kallikrein loop. Clusters of genes exhibit high prostatic (KLK2-4, KLK15) or pancreatic (KLK6-13) expression, suggesting evolutionary conservation of elements conferring tissue specificity. These genes are also expressed, to varying degrees, in a wider range of tissues suggesting a functional involvement of these newer human kallikrein proteases in a diverse range of physiological processes.