Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximization for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimization problems with the help of convex semidefinite programming (optimization over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach

Влияние нулей передаточной функции объекта на свойства регулятора

The paper is devoted to the influence of transfer function zeros on state feedback controller parameters. Nowadays, during the modal control system design process, the most attention is paid to the analysis of transfer function poles. If transfer function zeros are located closely to the poles, the control object tends to singularity, when the influence of the input control signals to the output signals becomes weaker. This fact leads to the designing of state-space controllers with extremely high values of control coefficients. Available methods for selecting the desired poles of closed loop systems are represented by the following variants: quantitative evaluation of the controllability and observability of object models and object model reduction. These methods have some disadvantages: dependence on the state-space representation of the control object model, ignoring some parts and characteristics of the control object model. In this paper, to analyze the state feedback controller parameters the invariant of the state-space representation characteristic is used – catalecticant (Hankel) matrix. As a result, it was found that control coefficients of the state feedback controller are inversely proportional to the determinant of the catalecticant matrix of the control object, and the determinant of the catalecticant matrix is equal to the resultant of transfer function polynomials. A method of converting the block diagram of the control object model using the residues of the transfer function was suggested. As a result of the conversion, poles, which cause high values of control coefficients, are isolated as a multiplicative uncertainty in the object model structure. The robust control theory can be used for control system designing with such organization of the control object.Настоящая работа посвящена исследованию влияния нулей передаточной функции на свойства регуляторов состояния. В настоящее время при проектировании регуляторов с использованием алгоритмов модального управления наибольшее внимание уделяется выбору желаемого распределения полюсов объекта управления. При наличии нулей передаточной функции, близких к полюсам, объект управления стремится к вырождению, что проявляется в ослаблении влияния входных управляющих сигналов на выходные сигналы. При расчете регуляторов состояния это приводит к появлению чрезмерно больших коэффициентов регулятора, чувствительных к изменению параметров объекта и снижению параметрической робастности системы управления. Существующие методы анализа математической модели объекта сводятся к количественной оценке характеристик управляемости и наблюдаемости или редукции объекта управления. Перечисленные методы обладают рядом недостатков, таких как зависимость от базиса в пространстве состояний, игнорирование части модели объекта управления. В настоящей работе для проведения анализа свойств математической модели используется инвариантная по отношению к базису характеристика объекта — матрица вырожденности. В результате исследования установлено, что коэффициенты регулятора состояния обратно пропорциональны определителю матрицы вырожденности объекта управления, определитель матрицы вырожденности равен результанту полиномов передаточной функции и его величина зависит от расположения нулей передаточной функции. Предложен способ декомпозиции модели объекта управления с использованием вычетов передаточной функции. В результате преобразования полюса, которые вызывают появление больших коэффициентов регуляторов, выделяются в виде структурной помехи в составе объекта. Проектирование систем управления для подобного представления объекта может быть реализовано с использованием теории робастного управления

Robust analysis of stability and control of interval systems and multi-incident parameters via interval analysis

Orientador: Paulo Augusto Valente FerreiraTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Esta tese apresenta uma metodologia para a análise de estabilidade e controle de sistemas dinâmicos lineares com parâmetros intervalares. A fim de reduzir o conservadorismo de soluções devido ao chamado fenômeno de dependência, presente na análise intervalar clássica, introduz-se a análise intervalar parametrizada, a qual considera a existência de parâmetros intervalares multi-incidentes. A solução apresentada é baseada em uma metodologia para solução de sistemas lineares intervalares que considera o tratamento das multi-incidências de parâmetros. Diversos aspectos relacionados à estabilidade e controle de sistemas dinâmicos lineares intervalares são considerados, como por exemplo, regularidade e positividade de matrizes intervalares. A análise de estabilidade de sistemas lineares intervalares é realizada através da solução da equação de Lyapunov intervalar, utilizando-se transformações a fim de obter um sistema intervalar de equações. Os resultados obtidos são comparados a métodos de solução por LMIs (Desigualdades Matriciais Lineares) e métodos baseados em análise intervalar clássica. A alocação robusta de polos é obtida através da solução da equação Diofantina intervalar e os resultados obtidos são comparados com a análise intervalar clássica e com um método não intervalar recentemente proposto na literaturaAbstract: This thesis presents a methodology for stability analysis and control of linear dynamic systems with interval parameters. In order to reduce the conservativeness due to the so-called dependency phenomenon present in classical interval analysis methods, a parameterized interval analysis approach is proposed. The approach proposed is based on a solution strategy for linear interval systems that takes into account multi-incident parameters. Several aspects related to the stability and control of interval linear systems, as, for example, nonsingularity and positivity of interval matrices, are considered. The stability analysis of linear interval systems is carried out through the solution of the interval Lyapunov equation transformed into an ordinary interval system of linear equations. The results obtained are compared with LMI (Linear Matrix Inequality) and classical interval analysis methods. The robust pole placement problem is tackled by solving the interval Diophantine equation. The results obtained by the parametrized interval analysis method are compared with those of the classic interval analysis and a non-interval method recently proposed in the literatureDoutoradoAutomaçãoDoutora em Engenharia Elétrica159829/2013-5CNP