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

Wall crossings for double Hurwitz numbers

AbstractDouble Hurwitz numbers count covers of P1 by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification (Goulden et al., 2005) [10], and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0 (Shadrin et al., 2008) [15]. This paper gives a unified approach to these results and strengthens them in several ways — the most important being the extension of the results of Shadrin et al. (2008) [15] to arbitrary genus.The main tool is the authorsʼ previous work (Cavalieri et al., 2010) [6] expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987) [17], and could have broader applications

The modular geometry of Random Regge Triangulations

We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio

Stabilising Model Predictive Control for Discrete-time Fractional-order Systems

In this paper we propose a model predictive control scheme for constrained fractional-order discrete-time systems. We prove that all constraints are satisfied at all time instants and we prescribe conditions for the origin to be an asymptotically stable equilibrium point of the controlled system. We employ a finite-dimensional approximation of the original infinite-dimensional dynamics for which the approximation error can become arbitrarily small. We use the approximate dynamics to design a tube-based model predictive controller which steers the system state to a neighbourhood of the origin of controlled size. We finally derive stability conditions for the MPC-controlled system which are computationally tractable and account for the infinite dimensional nature of the fractional-order system and the state and input constraints. The proposed control methodology guarantees asymptotic stability of the discrete-time fractional order system, satisfaction of the prescribed constraints and recursive feasibility

Stability robustness of linear systems: a field of values approach

Ankara : Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references leaves 48-51.One active area of research in stability robustness of linear time invariant systems is concerned with stability of matrix polytopes. Various structured real parametric uncertainties can be modeled by a family of matrices consisting of a convex hull of a finite number of known matrices, the matrix poly tope. An interval matrix family consisting of matrices whose entries can assume any values in given intervals are special types of matrix polytopes and it models a commonly encountered parametric uncertainty. Results that allow the inference of the stability of the whole polytope from stability of a finite number of elements of the polytope are of interest. Deriving such results is known to be difficult and few results of sufficient generality exist. In this thesis, a survey of results pertaining to robust Hurwitz and Schur stability of matrix polytopes and interval matrices are given. A seemingly new tool, the field of values, and its elementary properties are used to recover most results available in the literature and to obtain some new results. Some easily obtained facts through the field of values approach are as follows. Poly topes with normal vertex matrices turn out to be Hurwitz and Schur stable if and only if the vertex matrices are Hurwitz and Schur stable, respectively. If the polytope contains the transpose of each vertex matrix, Hurwitz stability of the symmetric part of the vertices is necessary and sufficient for the Hurwiz stability of the polytope. If the polytope is nonnegative and the symmetric part of each vertex matrix is Schur stable, then the polytope is also stable. For polytopes with spectral vertex matrices, Schur stability of vertices is necessary and sufficient for the Schur stability of the polytope.Saadaoui, KarimM.S

Investigation and Synthesis of Robust Polynomials in Uncertainty on the Basis of the Root Locus Theory

The root locus method is proposed in the chapter for searching intervals of uncertainty for coefficients of the given (source) polynomial with constant or interval coefficients under perturbations, which ensures its robust stability regardless of whether the given polynomial is Hurwitz or not. The method is based on introduction and application of the “extended root locus” notion. Polynomial adjustment is performed by setting up each one of its coefficients separately and sequentially and determining permissible values of coefficient variation intervals (intervals of uncertainty). The effect of each coefficient variation upon the polynomial root dynamics (behavior) is considered and analyzed separately, and this influence could be observed in the root locus portraits. Root locus method is thus generalized to the cases when the number of polynomial variable coefficients is arbitrary. The root locus parameter distribution diagram along the asymptotic stability bound is introduced and applied for observing the roots behavior regularities. On this basis, the stability conditions are derived, and analytical and graphic-analytical methods are worked out for calculating intervals of variation for the 4th order polynomial family parameters ensuring its robust stability. It also allows to extract Hurwitz subfamilies from the non-Hurwitz families of interval polynomials and to determine whether there exists at least one stable polynomial in the unstable polynomial family

Robust control of quasi-linear parameter-varying L2 point formation flying with uncertain parameters

Robust high precision control of spacecraft formation flying is one of the most important techniques required for high-resolution interferometry missions in the complex deep-space environment. The thesis is focussed on the design of an invariant stringent performance controller for the Sun-Earth L2 point formation flying system over a wide range of conditions while maintaining system robust stability in the presence of parametric uncertainties. A Quasi-Linear Parameter-Varying (QLPV) model, generated without approximation from the exact nonlinear model, is developed in this study. With this QLPV form, the model preserves the transparency of linear controller design while reflecting the nonlinearity of the system dynamics. The Polynomial Eigenstructure Assignment (PEA) approach used for Linear Time-Invariant (LTI) and Linear Parameter-Varying (LPV ) models is extended to use the QLPV model to perform a form of dynamic inversion for a broader class of nonlinear systems which guarantees specific system performance. The resulting approach is applied to the formation flying QLPV model to design a PEA controller which ensures that the closed-loop performance is independent of the operating point. Due to variation in system parameters, the performance of most closed-loop systems are subject to model uncertainties. This leads naturally to the need to assess the robust stability of nonlinear and uncertain systems. This thesis presents two approaches to this problem, in the first approach, a polynomial matrix method to analyse the robustness of Multiple-Input and Multiple-Output (MIMO) systems for an intersectingD-region,which can copewith time-invariant uncertain systems is developed. In the second approach, an affine parameterdependent Lyapunov function based Linear Matrix Inequality (LMI) condition is developed to check the robust D-stability of QLPV uncertain systems.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Multiple-Model Adaptive Control With Set-Valued Observers

This paper proposes a multiple-model adaptive control methodology, using set-valued observers (MMAC-SVO) for the identification subsystem, that is able to provide robust stability and performance guarantees for the closed-loop, when the plant, which can be open-loop stable or unstable, has significant parametric uncertainty. We illustrate, with an example, how set-valued observers (SVOs) can be used to select regions of uncertainty for the parameters of the plant. We also discuss some of the most problematic computational shortcomings and numerical issues that arise from the use of this kind of robust estimation methods. The behavior of the proposed control algorithm is demonstrated in simulation.Comment: Combined 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 200