Matrix Convex Hulls of Free Semialgebraic Sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a Mathematica notebook) can be found at http://www.math.auckland.ac.nz/~igorklep/publ.htm

Filter design for linear systems with H-2, H-infinity and H-infinity in frequency interval criteria by means of matrix inequalities

Orientador: Pedro Luis Dias PeresDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Esta dissertação aborda o problema de filtragem para sistemas dinâmicos lineares utilizando metodologias baseadas em desigualdades matriciais lineares (do inglês, LMIs -- "Linear Matrix Inequalities"). Mais precisamente, são fornecidas condições para o projeto de filtros de ordem completa para sistemas lineares contínuos e discretos no tempo, usando como critérios de desempenho as normas H-2 e H-infinito, com extensões para tratar sistemas com incertezas politópicas. As condições possuem um parâmetro escalar, tornando-se LMIs para valores fixos do parâmetro. Como características principais, as condições propostas isolam a matriz de Lyapunov, usada para certificar a estabilidade com desempenho H-2 e H-infinito, das matrizes que produzem o filtro e contêm os resultados da literatura conhecidos como estabilidade quadrática para escolhas particulares do escalar. Adicionalmente, o problema de filtragem H-infinito com especificações em baixa, média e alta frequência é abordado a partir de uma extensão do Lema de Kalman-Yakubovich-Popov que relaciona desigualdades no domínio da frequência em intervalos de reta ou segmentos de círculo com desigualdades matriciais. São propostas condições baseadas em LMIs para o projeto de filtros H-infinito com especificações em intervalos de frequência que garantem uma realização estável com matrizes reais, nos casos contínuo e discreto no tempo, com extensões para tratar sistemas incertos. Exemplos numéricos ilustram os resultados do trabalhoAbstract: This thesis is devoted to the problem of filter design for linear dynamic systems using the Linear Matrix Inequality (LMI) framework. More precisely, LMI based conditions for the design of full-order filters for continuous- and discrete-time linear systems, using the H-2 and H-infinity norms as performance criteria, are proposed, with extensions to deal with polytopic uncertainty. The conditions have a scalar parameter and become LMIs for fixed values of the scalar. As attractive characteristics, the proposed conditions dissociate the Lyapunov matrix, that certifies the stability and the H-infinity or H-2 performance, from the matrices of the filter realization, encompassing the well-known quadratic stability based results from the literature for specific values of the scalar parameter. Additionally, the H-infinity filtering problem with low, middle and high frequency specifications is addressed through an extension of the Kalman-Yakubovich-Popov Lemma that relates frequency domain inequalities on line or circle segments with matrix inequalities. LMI based conditions for the design of H-infinity filters satisfying frequency range specifications with a stable realization and real matrices are proposed, for both continuous- and discrete-time cases, as well as extensions to cope with uncertain systems. Numerical examples illustrate the proposed resultsMestradoAutomaçãoMestre em Engenharia Elétrica2014/06408-4, 2015/13135-7FAPES