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

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

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

Bounds for Input- and State-to-Output Properties of Uncertain Linear Systems

We consider the effect of parametric uncertainty on properties of Linear Time Invariant systems. Traditional approaches to this problem determine the worst-case gains of the system over the uncertainty set. Whilst such approaches are computationally tractable, the upper bound obtained is not necessarily informative in terms of assessing the influence of the parameters on the system performance. We present theoretical results that lead to simple, convex algorithms producing parametric bounds on the $\mathcal{L}_2$-induced input-to-output and state-to-output gains as a function of the uncertain parameters. These bounds provide quantitative information about how the uncertainty affects the system.Comment: To appear in the proceedings of the 8th IFAC Symposium on Robust Control Design - ROCOND'1

LMI techniques for optimization over polynomials in control: A survey

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Plya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers. © 2006 IEEE.published_or_final_versio

Optimal bounds with semidefinite programming: an application to stress driven shear flows

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler--Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times, and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.Comment: 17 pages; typos removed; extended discussion of linear matrix inequalities in Section III; revised argument in Section IVC, results unchanged; extended discussion of computational setup and limitations in Sectios IVE-IVF. Submitted to Phys. Rev.

Convex dwell-time characterizations for uncertain linear impulsive systems

New sufficient conditions for the characterization of dwell-times for linear impulsive systems are proposed and shown to coincide with continuous decrease conditions of a certain class of looped-functionals, a recently introduced type of functionals suitable for the analysis of hybrid systems. This approach allows to consider Lyapunov functions that evolve non-monotonically along the flow of the system in a new way, broadening then the admissible class of systems which may be analyzed. As a byproduct, the particular structure of the obtained conditions makes the method is easily extendable to uncertain systems by exploiting some convexity properties. Several examples illustrate the approach.Comment: Accepted at IEEE Transactions on Automatic Contro