A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities

In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning theory, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chervonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of these problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality. The effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters.Comment: 19 pages, 2 figures, Accepted for Publication in Automatic

Robust Fixed-Order Controller Design with Common Lyapunov Strictly Positive Realness Characterization

This paper investigates the design of a robust fixed-order controller for a polytopic system with interval uncertainties, with the aim that the closed-loop stability is appropriately ensured and the performance specifications on sensitivity shaping are conformed in a specific finite frequency range. Utilizing the notion of common Lyapunov strictly positive realness (CL-SPRness), the equivalence between strictly positive realness (SPRness) and strictly bounded realness (SBRness) is elegantly established; and then the specifications on robust stability and performance are transformed into the SPRness of newly constructed systems and further characterized in the framework of linear matrix inequality (LMI) conditions. Compared with the traditional robust controller synthesis approaches, the proposed methodology here avoids the tedious yet mandatory evaluations of the specifications on all vertices of the polytopic system; only a one-time checking of matrix existence is needed exclusively, and thus the typically heavy computational burden involved (in such robust controller design problems) is considerably alleviated. Moreover, it is noteworthy that the proposed methodology additionally provides essential necessary and sufficient conditions for this robust controller design with the consideration of a prescribed finite frequency range; and therefore significantly less conservatism is attained in the system performance.Comment: 10 pages, 6 figure

Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper proposes a set-membership state estimator and a zonotopic Kalman observer for discrete-time descriptor systems. Both approaches are developed in a set-based context considering system disturbances, measurement noise, and unknown inputs. This set-membership state estimation approach determines the set of consistent states with the model and measurements by constructing a parameterized intersection zonotope. Two methods to minimize the size of this intersection zonotope are provided: one inspired by Kalman filtering and the other based on solving an optimization problem involving a series of linear matrix inequalities. Additionally, we propose a zonotopic Kalman observer for discrete-time descriptor systems. Moreover, the relationship between both approaches is discussed. In particular, it is proved that the zonotopic Kalman observer in the current estimation type is equivalent to the set-membership approach. Finally, a numerical example is used to illustrate and compare the effectiveness of the proposed approaches.Peer ReviewedPostprint (author's final draft

A new vertex result for robustness problems with interval matrix uncertainty

Copia personal con permiso para subir en otro formatoThis paper1 addresses a family of robustness problems in which the system under consideration is affected by interval matrix uncertainty. The main contribution of the paper is a new vertex result that drastically reduces the number of extreme realizations required to check robust feasibility. This vertex result allows one to solve, in a deterministic way and without introducing conservatism, the corresponding robustness problem for small and medium size problems. For example, consider quadratic stability of an autonomous dimensional system. In this case, instead of checking vertices, we show that it suffices to check specially constructed systems. This solution is still exponential, but this is not surprising because the problem is NP-hard. Finally, vertex extensions to multiaffine interval families and some sufficient conditions (in LMI form) for robust feasibility are presented. Some illustrative examples are also given.Ministerio de Ciencia y Tecnología DPI2005-0456

Sobre o controle para uma classe de sistemas não-lineares com atuadores saturantes

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2009.Apresentamos uma metodologia sistemática visando estudar e computar leis de controle para uma classe de sistemas não-lineares, em tempo contínuo e em tempo discreto, sujeitos à saturação de atuadores. Para modelar o sistema saturado não-linear, utiliza-se uma não-linearidade de tipo zona-morta satisfazendo uma condição de setor modificada, a qual engloba a representação clássica da saturação como uma não-linearidade de setor. Para proposição dos resultados teóricos e algoritmos, consideramos um sistema não-linear tipo Lur'e e, baseados em ferramentas de estabilidade absoluta, uma condição de setor modificada para levar em conta os efeitos da saturação nas entradas de controle. A estrutura do controlador é composta por uma parte linear, um termo associado a saída da não-linearidade dinâmica e, no caso do compensador dinâmico, de uma malha anti-windup. Abordamos, para uma classe de sistemas não-lineares em tempo discreto e com parâmetros variantes, o problema de estabilização sob saturação via uma lei de controle dependente de parâmetros e uma lei de controle a ganhos fixos, ambas sob a forma de uma realimentação de estados mais uma realimentação da não-linearidade. Com base nos resultados obtidos, desenvolvemos também um compensador dinâmico não-linear parcialmente dependente de parâmetros e analisamos a influência da realimentação das não-linearidades consideradas. Para os problemas de controle considerados ao longo deste trabalho, são propostos problemas de otimização convexa com restrições de tipo LMI para o projeto dos controladores, com o objetivo de determinar a maximização da região de atração ou melhoramento do desempenho com a garantia de estabilidade. Exemplos numéricos foram desenvolvidos para ilustrar as potencialidades dos algoritmos propostos