Robust Stability Analysis of Sparsely Interconnected Uncertain Systems

In this paper, we consider robust stability analysis of large-scale sparsely interconnected uncertain systems. By modeling the interconnections among the subsystems with integral quadratic constraints, we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities. We also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness. The sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques, and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem.Comment: Provisionally accepted to appear in IEEE Transactions on Automatic Contro

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201

COSMO: A conic operator splitting method for convex conic problems

This paper describes the Conic Operator Splitting Method (COSMO) solver, an operator splitting algorithm for convex optimisation problems with quadratic objective function and conic constraints. At each step the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. A number of benchmarks against other state-of-the-art solvers for a variety of problems show the effectiveness of our approach. Our Julia implementation is open-source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.Comment: 45 pages, 11 figure

Passivity enforcement via chordal methods

Orientador: Prof. Dr. Gustavo Henrique da Costa OliveiraTese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Elétrica. Defesa : Curitiba, 27/08/2019Inclui referências: p. 164-175Resumo: Neste documento são propostos três algoritmos inéditos associados aos problemas subsequentes de aferição e imposição da passividade, a qual é uma propriedade qualitativa, geral e fundamental na modelagem matemática de transitórios eletromagnéticos de sistemas elétricos passivos, como transformadores. Esses algoritmos baseiam-se numa combinação de teoria dos grafos e otimização convexa. O primeiro deles consiste na aferição de subsistemas passivos contidos num sistema não passivo, intuitivamente busca-se partes passivas contidas num todo não passivo. Já na etapa de imposição de passividade, o segundo algoritmo é consequência natural do primeiro: retendo apenas os parâmetros associados às partes passivas e descartando os demais, parte-se de um sistema passivo parcialmente especificado para se determinar novos parâmetros em substituição àqueles descartados de modo que o sistema como um todo seja passivo. A possibilidade de determinação dos novos parâmetros depende de uma propriedade topológica de um grafo associado às matrizes de parâmetros do modelo, tal propriedade é denominada cordalidade. O terceiro algoritmo aborda novamente a questão de imposição da passividade e também faz uso da cordalidade, não mais como condição de existência de solução, mas sim como uma forma de explorar a esparsidade das matrizes de parâmetros. O problema de imposição da passividade encerra dois desafios no seu processo de solução, a saber: (i) compensação de parâmetros resultando na degradação do modelo bem como (ii) longos tempos de solução. Os algoritmos ora propostos são uma resposta a essas questões e os resultados obtidos demonstraram-se comparáveis àqueles já existentes na literatura especializada, em alguns casos apresentando melhorias, seja em termos de aproximação ou tempo computacionais. Os algoritmos foram testados a partir de dados de medição de um Transformador de Potencial Indutivo bem como de um Transformador de Potência. Palavras-chave: Macro-modelagem Passiva. Teoria de Sistemas. Álgebra Linear Aplicada. Análise de Transitórios. Transformadores.Abstract: Three novel algorithms are herein proposed to solve passivity assessment and enforcement problems. Passivity is a general, qualitative and fundamental property pertaining to the modeling associated with electromagnetic transients in passive power systems, such as transformers. These algorithms make combined use of Graph Theory and Convex Optimization. The first algorithm is concerned with passivity assesment. In particular, it searches for passive subsystems embedded into a larger nonpassive system and eventually specifies a partially specified passive system. Focusing on the subsequent step, algorithm two is a natural consequence of the preceeding one: retaining only the parameter set associated with passive subsystems as determined before, this partially specified passive system is used to further determine the remaining parameters so that the entire system be fully specified and passive. The existence condition for finding a fully specified system hinges on the fulfillment of a topological property of the graph associated the parameter matrices, namely chordality. The third algorithm also solves the passivity enforcement problem by making use of chordality, not as an existence condition, but rather by exploiting chordal sparsity patterns obtained with the parameter matrices. Solving passivity enforcement problems entails two persisting challenges, namely: (i) passivity compensations to parameters prompting increased model degradation as well as (ii) large computation times. The algorithms herein proposed tackle these issues and yield results comparable to those already in use, sometimes resulting in improved performance in terms of either approximation accuracy or runtime. These results herein reported entail data from actual measurements of an Inductive Voltage Transformer and a Power Transformer. Keywords: Passive Macromodeling. System Theory. Applied Linear Algebra. Transient Analysis. Transformers

Practical polynomial optimization through positivity certificates with and without denominators

Les certificats de positivité ou Positivstellens"atze fournissent des représentations de polynômes positifs sur des ensembles semialgébriques de basiques, c'est-à-dire des ensembles définis par un nombre fini d'inégalités polynomiales. Le célèbre Positivstellensatz de Putinar stipule que tout polynôme positif sur un ensemble semialgébrique basique fermé $S$ peut être écrit comme une combinaison pondérée linéaire des polynômes décrivant $S$, sous une certaine condition sur $S$ légèrement plus forte que la compacité. Lorsqu'il est écrit comme ceci, il devient évident que le polynôme est positif sur $S$, et donc cette description alternative fournit un certificat de positivité sur $S$. De plus, comme les poids polynomiaux impliqués dans le Positivstellensatz de Putinar sont des sommes de carrés (SOS), de tels certificats de positivité permettent de concevoir des relaxations convexes basées sur la programmation semidéfinie pour résoudre des problèmes d'optimisation polynomiale (POP) qui surviennent dans diverses applications réelles, par exemple dans la gestion des réseaux d'énergie et l'apprentissage automatique pour n'en citer que quelques unes. Développée à l'origine par Lasserre, la hiérarchie des relaxations semidéfinies basée sur le Positivstellensatz de Putinar est appelée la emph{hiérarchie Moment-SOS}. Dans cette thèse, nous proposons des méthodes d'optimisation polynomiale basées sur des certificats de positivité impliquant des poids SOS spécifiques, sans ou avec dénominateurs.Positivity certificates or Positivstellens"atze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set $S$ can be written as a linear weighted combination of the polynomials describing $S$, under a certain condition on $S$ slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on $S$, and therefore this alternative description provides a certificate of positivity on $S$. Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite programming to solve polynomial optimization problems (POPs) that arise in various real-life applications, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the emph{Moment-SOS hierarchy}. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators