Plane geometry and convexity of polynomial stability regions

The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible

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

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

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 $\Omega$, 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 ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-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

Global time estimates for solutions to equations of dissipative type

Global time estimates of Lp-Lq norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.Comment: Journees "Equations aux Derivees Partielles

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology