372 research outputs found
On convexity of the frequency response of a stable polynomial
In the complex plane, the frequency response of a univariate polynomial is
the set of values taken by the polynomial when evaluated along the imaginary
axis. This is an algebraic curve partitioning the plane into several connected
components. In this note it is shown that the component including the origin is
exactly representable by a linear matrix inequality if and only if the
polynomial is stable, in the sense that all its roots have negative real parts
Some control design experiments with HIFOO
A new MATLAB package called HIFOO was recently proposed for H-infinity
fixed-order controller design. This document illustrates how some standard
controller design examples can be solved with this software
Semidefinite representation of convex hulls of rational varieties
Using elementary duality properties of positive semidefinite moment matrices
and polynomial sum-of-squares decompositions, we prove that the convex hull of
rationally parameterized algebraic varieties is semidefinite representable
(that is, it can be represented as a projection of an affine section of the
cone of positive semidefinite matrices) in the case of (a) curves; (b)
hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized
by bivariate quartics; all in an ambient space of arbitrary dimension
Semidefinite geometry of the numerical range
The numerical range of a matrix is studied geometrically via the cone of
positive semidefinite matrices (or semidefinite cone for short). In particular
it is shown that the feasible set of a two-dimensional linear matrix inequality
(LMI), an affine section of the semidefinite cone, is always dual to the
numerical range of a matrix, which is therefore an affine projection of the
semidefinite cone. Both primal and dual sets can also be viewed as convex hulls
of explicit algebraic plane curve components. Several numerical examples
illustrate this interplay between algebra, geometry and semidefinite
programming duality. Finally, these techniques are used to revisit a theorem in
statistics on the independence of quadratic forms in a normally distributed
vector
Semidefinite geometry of the numerical range
The numerical range of a matrix is studied geometrically via the cone of
positive semidefinite matrices (or semidefinite cone for short). In particular
it is shown that the feasible set of a two-dimensional linear matrix inequality
(LMI), an affine section of the semidefinite cone, is always dual to the
numerical range of a matrix, which is therefore an affine projection of the
semidefinite cone. Both primal and dual sets can also be viewed as convex hulls
of explicit algebraic plane curve components. Several numerical examples
illustrate this interplay between algebra, geometry and semidefinite
programming duality. Finally, these techniques are used to revisit a theorem in
statistics on the independence of quadratic forms in a normally distributed
vector
On semidefinite representations of plane quartics
This note focuses on the problem of representing convex sets as projections
of the cone of positive semidefinite matrices, in the particular case of sets
generated by bivariate polynomials of degree four. Conditions are given for the
convex hull of a plane quartic to be exactly semidefinite representable with at
most 12 lifting variables. If the quartic is rationally parametrizable, an
exact semidefinite representation with 2 lifting variables can be obtained.
Various numerical examples illustrate the techniques and suggest further
research directions
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
Convex computation of the region of attraction of polynomial control systems
We address the long-standing problem of computing the region of attraction
(ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a
controlled nonlinear system with polynomial dynamics and semialgebraic state
and input constraints. We show that the ROA can be computed by solving an
infinite-dimensional convex linear programming (LP) problem over the space of
measures. In turn, this problem can be solved approximately via a classical
converging hierarchy of convex finite-dimensional linear matrix inequalities
(LMIs). Our approach is genuinely primal in the sense that convexity of the
problem of computing the ROA is an outcome of optimizing directly over system
trajectories. The dual infinite-dimensional LP on nonnegative continuous
functions (approximated by polynomial sum-of-squares) allows us to generate a
hierarchy of semialgebraic outer approximations of the ROA at the price of
solving a sequence of LMI problems with asymptotically vanishing conservatism.
This sharply contrasts with the existing literature which follows an
exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix
inequalities or conservative LMI conditions. The approach is simple and readily
applicable as the outer approximations are the outcome of a single semidefinite
program with no additional data required besides the problem description
SDLS: a Matlab package for solving conic least-squares problems
This document is an introduction to the Matlab package SDLS (Semi-Definite
Least-Squares) for solving least-squares problems over convex symmetric cones.
The package is shortly presented through the addressed problem, a sketch of the
implemented algorithm, the syntax and calling sequences, a simple numerical
example and some more advanced features. The implemented method consists in
solving the dual problem with a quasi-Newton algorithm. We note that SDLS is
not the most competitive implementation of this algorithm: efficient, robust,
commercial implementations are available (contact the authors). Our main goal
with this Matlab SDLS package is to provide a simple, user-friendly software
for solving and experimenting with semidefinite least-squares problems. Up to
our knowledge, no such freeware exists at this date
- âŠ