Polynomial Dynamics Research Articles

Measures and LMIs for Impulsive Nonlinear Optimal Control

This note shows how to use semi-definite programming to find lower bounds on a large class of nonlinear optimal control problems with polynomial dynamics and convex semialgebraic state constraints and an affine dependence on the control. This is done by relaxing an optimal control problem into a linear programming problem on measures, also known as a generalized moment problem. The handling of measures by their moments reduces the problem to a convergent series of standard linear matrix inequality relaxations. When the optimal control consists of a finite number of impulses, we can recover simultaneously the actual impulse times and amplitudes by simple linear algebra. Finally, our approach can be readily implemented with standard software, as illustrated by a numerical example.

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. The approach is demonstrated on several numerical examples.

Algebraic roots of Newtonian mechanics: correlated dynamics of particles on a unique worldline

In the development of the old ideas of Stueckelberg–Wheeler–Feynman on the ‘one-electron Universe’, we study the purely algebraic dynamics of the ensemble of (two kinds of) identical point-like particles. These are represented by the (real and complex conjugate) roots of a generic polynomial system of equations that implicitly defines a single ‘worldline’. The dynamics includes events of ‘merging’ of a pair of particles modelling the annihilation/creation processes. Correlations in the location and motion of the particles-roots relate, in particular, to the Vieta formulas. After a special choice of the inertial-like reference frame, the linear Vieta formulas guarantee that, for any worldline, the law of (non-relativistic) momentum conservation is identically satisfied. Thus, the general structure of Newtonian mechanics follows from the algebraic properties of a worldline alone. A simple example of, unexpectedly rich, ‘polynomial dynamics’ is retraced in detail and illustrated via an animation (available from stacks.iop.org/JPhysA/46/175206/mmedia).

Convex Computation of the Maximum Controlled Invariant Set For Polynomial Control Systems

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

천정형 크레인의 흔들림 억제제어에 관한 SOS 접근법

This paper proposes anti-swing control of overhead crane system using sum of squares method. The dynamic equations of overhead crane include nonlinear terms, which are transformed into polynomials by using Taylor series expansion. Therefore the dynamic equation of overhead crane can be changed to the system of polynomial equation. On the basis of polynomial dynamics of crane system, we propose the Sum of Squares (SOS) conditions considering the input constraints. In addition, control gains are obtained by numerical tool which is called by SOSTOOL. The effectiveness of the proposed method is demonstrated by numerical simulation.

Polynomial Dynamics and a Proof of the Fermat Little Theorem

We present a dynamical proof of the Fermat little theorem, based on the Chebyshev polynomials and their extraordinary composition property.

Optimization on linear matrix inequalities for polynomial systems control

Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate approximate solutions in floating point arithmetic.

Arboreal Galois representations and uniformization of polynomial dynamics

Given a polynomial f defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z↦ f(z) to the multiplicative action z↦ zdeg(f). The relation between this construction and the Böttcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory. As a consequence, we obtain some corollaries regarding Galois irreducibility of preimages of certain points under certain polynomials, as well as the rationality of preimages in one-parameter families.

Conditional Tests on Basins of Attraction with Finite Fields

An iterative method is given for computing the polynomials that vanish on the basin of attraction of a steady state in discrete polynomial dynamics with finite field coefficients. The algorithm is applied to dynamics of a T cell survival network where it is used to compare transition maps conditional on a basin of attraction.

Time-Delay Margin Analysis for an Adaptive Controller

robustness analysis tools are applied to adaptive flight control in this paper. A standard model reference adaptive controller with sigma modification is designed for a linear short-period aircraft model. The resulting nonlinear closed-loop system is governed by polynomial dynamics. The nonlinear analysis algorithms rely on sum-of-squares polynomial optimization to assess the robustness of the adaptive closed-loop system with respect to a time delay. Time-delaymarginsarecomputedforvariouscombinationsofdesignparametersintheadaptivecontrollaw,aswell as in the presence of parametric model uncertainty. Advantages and limitations of the proposed sum-of-squaresbased robustness analysis are presented. Analysis results show a significant promise in the context of recent development in nonlinear robustness analysis. I. Introduction A DAPTIVEcontrolhasthepotentialtoimprovetheperformance and reliability of aircraft systems. Typical adaptive control architectures are inherently nonlinear, which presents a number of challenges. There is currently a lack of tools available to rigorously analyze the robustness and performance of nonlinear control architectures. The inability toverifyrobustnessand performance is a significant roadblock to the implementation of adaptive control architectures on civilian andmilitary aircraft. Thus, the flight control community would greatly benefit from advances in the area of

Analysis of parametric biological models with non-linear dynamics

In this paper we present recent results on parametric analysis of biological models. The underlying method is based on the algorithms for computing trajectory sets of hybrid systems with polynomial dynamics. The method is then applied to two case studies of biological systems: one is a cardiac cell model for studying the conditions for cardiac abnormalities, and the second is a model of insect nest-site choice.

Measures and LMI for space launcher robust control validation

We describe a new temporal verification framework for safety and robustness analysis of nonlinear control laws, our target application being a space launcher vehicle. Robustness analysis, formulated as a nonconvex nonlinear optimization problem on admissible trajectories corresponding to piecewise polynomial dynamics, is relaxed into a convex linear programming problem on measures. This infinite-dimensional problem is then formulated as a generalized moment problem, which allows for a numerical solution via a hierarchy of linear matrix inequality relaxations solved by semidefinite programming. The approach is illustrated on space launcher vehicle benchmark problems, in the presence of closed-loop nonlinearities (saturations and dead-zones) and axis coupling.

Control for Nonlinear Chemical System

In this paper, we propose a new design of the feedback control of state vector for the plants with polynomial dynamics. A genetic algorithm is employed to find suitable gain, and algebraic geometric concept is used to simplify the design. Finally, an example is given to illustrate the effectiveness of the proposed method.

Trends and Developments in Complex Dynamics

Holomorphic dynamics is a thriving field which has experienced tremendous progress over the last 25 years, involving mathematicians such as Douady, Hubbard, McMullen, Milnor, Sullivan, Thurston, and Yoccoz. Holomorphic dynamics is gifted with the tools of conformal and hyperbolic geometry that allow a deep penetration into its nature, with many further applications to real dynamics. Conformal dynamics is currently in a phase of transition. Previously, most of the successful and deep work has been focused on one-parameter model families: families of quadratic or unimodal polynomials and families of meromorphic transcendental functions with one singular orbit. The theory is particularly advanced in the case of the quadratic family; in fact, a complete measure-theoretic picture of the dynamics was obtained in the real quadratic family (Lyubich, Avila, de Melo, Moreira, and others). During the past two years, significant breakthroughs have appeared in two important directions: rigidity in polynomial dynamics, and Julia sets of positive measure. These results answer major long-standing open questions and simultaneously open up new perspectives. Both problems have served as landmarks in the field, and both have a long history going back to Fatou (and more recently, to Ahlfors, Smale and Mostow). The recent results have brought us very close to a full understanding of the model families, and in particular of the real unimodal dynamics. Time is now ripe for holomorphic dynamics to move on from the study of the special model cases to a general theory of holomorphic dynamics, delivering on the promise that the model cases have impact in greater generality. Holomorphic dynamics is a vast area of research giving impetus to this transition from many directions. Given the time and number of participants we can not cover all of them within one workshop. We have thus chosen to focus on 5 such aspects of holomorphic dynamics with recent and substantial progress. The organising principle of the workshop is to present the state of the art of each aspect, to evaluate the progress achieved and to formulate questions which may guide the transition.

Modeling and analysis of mass-action kinetics

Mass-action kinetics are used in chemistry and chemical engineering to describe the dynamics of systems of chemical reactions, that is, reaction networks. These models are a special form of compartmental systems, which involve mass- and energy-balance relations. Aside from their role in chemical engineering applications, mass-action kinetics have numerous analytical properties that are of inherent interest from a dynamical systems perspective. Because of physical considerations, however, mass- action kinetics have special properties, such as nonnegative solutions, that are useful for analyzing their behavior. With this motivation in mind, this article has several objectives. First, a general construction of the kinetic equations based on the reaction laws is provided in a state-space form. Next, the nonnegativity of solutions to the kinetic equations is considered. The realizability problem, which is concerned with the inverse problem of constructing a reaction network having specified essentially nonnegative dynamics, is also considered. In particular, an explicit construction of a reaction network for essentially nonnegative polynomial dynamics involving a scalar state is provided. Next, the reducibility of the kinetic equations is considered as well as the stability of the equilibria of the kinetic equations. Lyapunov methods are applied to the kinetic equations, and semistability is guaranteed through the convergence to a Lyapunov- stable equilibrium that depends on the initial concentrations. Semistability is the appropriate notion of stability for compartmental systems in general, and reaction networks in particular, where the limiting concentration maybe nonzero and may depend on the initial concentrations. Finally, the zero deficiency result for mass-action kinetics in standard matrix terminology is presented and semistability is proven.

Local Stability Analysis for Uncertain Nonlinear Systems

We propose a method to compute provably invariant subsets of the region-of-attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems. We consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field. This uncertainty description is motivated by both incapabilities in modeling, as well as bilinearity and dimension of the sum-of-squares programming problems whose solutions provide invariant subsets of the region-of-attraction. We demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example.

Minimal polynomial dynamics on the set of 3-adic integers

In this paper, the polynomials that have all of their orbits dense in the set of 3-adic integers ℤ3 are characterized in terms of their coefficients.

Gene expression from polynomial dynamics in the 2-adic information space

We perform geometrization of genetics by representing genetic information by points of the 4-adic information space. By well known theorem of number theory this space can also be represented as the 2-adic space. The process of DNA-reproduction is described by the action of a 4-adic (or equivalently 2-adic) dynamical system. As we know, the genes contain information for production of proteins. The genetic code is a degenerate map of codons to proteins. We model this map as functioning of a polynomial dynamical system. The purely mathematical problem under consideration is to find a dynamical system reproducing the degenerate structure of the genetic code. We present one of possible solutions of this problem.

Intertwined internal rays in Julia sets of rational maps

We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of t

Forecasting polynomial dynamics

The reverse engineering method of gene regulatory networks, recently implemented by Laubenbacher and Stigler, is used to generate a multivariate time series with a finite set of levels. A formalism of fuzzy control allows one to forecast the evolutionary behavior of a network. When the multivariate time series is converted to a single-component series, an excellent agreement is obtained between the actual and forecasted values. The tests include both simulated and realistic models. For a binary model, with random bit changes simulating mutations, this approach can be used to detect anomalies in the network evolution.