750 research outputs found
Polynomials in Control Theory Parametrized by Their Roots
The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem
Continuity argument revisited: geometry of root clustering via symmetric products
We study the spaces of polynomials stratified into the sets of polynomial
with fixed number of roots inside certain semialgebraic region , on its
border, and at the complement to its closure. Presented approach is a
generalisation, unification and development of several classical approaches to
stability problems in control theory: root clustering (-stability) developed
by R.E. Kalman, B.R. Barmish, S. Gutman et al., -decomposition(Yu.I.
Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A.
Fam, J. Meditch, J.Ackermann).
Our approach is based on the interpretation of correspondence between roots
and coefficients of a polynomial as a symmetric product morphism.
We describe the topology of strata up to homotopy equivalence and, for many
important cases, up to homeomorphism. Adjacencies between strata are also
described. Moreover, we provide an explanation for the special position of
classical stability problems: Hurwitz stability, Schur stability,
hyperbolicity.Comment: 45 pages, 4 figure
Polynomials in Control Theory Parametrized by Their Roots
The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem
A robust implementation of the Carathéodory-Fejér method
Best rational approximations are notoriously difficult to compute. However, the difference between the best rational approximation to a function and its Carathéodory-Fejér (CF) approximation is often so small as to be negligible in practice, while CF approximations are far easier to compute. We present a robust and fast implementation of this method in the chebfun software system and illustrate its use with several examples. Our implementation handles both polynomial and rational approximation and substantially improves upon earlier published software
Simple Approximations of Semialgebraic Sets and their Applications to Control
Many uncertainty sets encountered in control systems analysis and design can
be expressed in terms of semialgebraic sets, that is as the intersection of
sets described by means of polynomial inequalities. Important examples are for
instance the solution set of linear matrix inequalities or the Schur/Hurwitz
stability domains. These sets often have very complicated shapes (non-convex,
and even non-connected), which renders very difficult their manipulation. It is
therefore of considerable importance to find simple-enough approximations of
these sets, able to capture their main characteristics while maintaining a low
level of complexity. For these reasons, in the past years several convex
approximations, based for instance on hyperrect-angles, polytopes, or
ellipsoids have been proposed. In this work, we move a step further, and
propose possibly non-convex approximations , based on a small volume polynomial
superlevel set of a single positive polynomial of given degree. We show how
these sets can be easily approximated by minimizing the L1 norm of the
polynomial over the semialgebraic set, subject to positivity constraints.
Intuitively, this corresponds to the trace minimization heuristic commonly
encounter in minimum volume ellipsoid problems. From a computational viewpoint,
we design a hierarchy of linear matrix inequality problems to generate these
approximations, and we provide theoretically rigorous convergence results, in
the sense that the hierarchy of outer approximations converges in volume (or,
equivalently, almost everywhere and almost uniformly) to the original set. Two
main applications of the proposed approach are considered. The first one aims
at reconstruction/approximation of sets from a finite number of samples. In the
second one, we show how the concept of polynomial superlevel set can be used to
generate samples uniformly distributed on a given semialgebraic set. The
efficiency of the proposed approach is demonstrated by different numerical
examples
Polynomial Schur's theorem
We resolve the Ramsey problem for for all polynomials
over .Comment: 21 page
Systems Structure and Control
The title of the book System, Structure and Control encompasses broad field of theory and applications of many different control approaches applied on different classes of dynamic systems. Output and state feedback control include among others robust control, optimal control or intelligent control methods such as fuzzy or neural network approach, dynamic systems are e.g. linear or nonlinear with or without time delay, fixed or uncertain, onedimensional or multidimensional. The applications cover all branches of human activities including any kind of industry, economics, biology, social sciences etc
Multivariate Polya-Schur classification problems in the Weyl algebra
A multivariate polynomial is {\em stable} if it is nonvanishing whenever all
variables have positive imaginary parts. We classify all linear partial
differential operators in the Weyl algebra \A_n that preserve stability. An
important tool that we develop in the process is the higher dimensional
generalization of P\'olya-Schur's notion of multiplier sequence. We
characterize all multivariate multiplier sequences as well as those of finite
order. Next, we establish a multivariate extension of the Cauchy-Poincar\'e
interlacing theorem and prove a natural analog of the Lax conjecture for real
stable polynomials in two variables. Using the latter we describe all operators
in \A_1 that preserve univariate hyperbolic polynomials by means of
determinants and homogenized symbols. Our methods also yield homotopical
properties for symbols of linear stability preservers and a duality theorem
showing that an operator in \A_n preserves stability if and only if its
Fischer-Fock adjoint does. These are powerful multivariate extensions of the
classical Hermite-Poulain-Jensen theorem, P\'olya's curve theorem and
Schur-Mal\'o-Szeg\H{o} composition theorems. Examples and applications to
strict stability preservers are also discussed.Comment: To appear in Proc. London Math. Soc; 33 pages, 4 figures, LaTeX2
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