Polynomial Systems Research Articles

Parallelization of triangular decompositions: Techniques and implementation

We discuss the parallelization of algorithms for solving polynomial systems by way of triangular decomposition. The Triangularize algorithm proceeds through incremental intersections of polynomials to produce different components (points, curves, surfaces, etc.) of the solution set. Independent components imply the opportunity for concurrency. This “component-level” parallelization of triangular decompositions, our focus here, belongs to the class of dynamic irregular parallelism. Potential parallel speed-up depends only on geometrical properties of the solution set (number of components, their dimensions and degrees); these algorithms do not scale with the number of processors. To manage the irregularities of component-level parallelization we combine different concurrency patterns: map, workpile, producer-consumer, pipeline, and fork-join. We report on our implementation in the freely available BPAS library. Comprehensive experimentation with thousands of polynomial systems yields examples with up to 10.8× speed-up on a 12-core machine.

A certified iterative method for isolated singular roots

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f=(f1,…,fN)∈C[x1,…,xn]N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

Contact detection between a small ellipsoid and another quadric

We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given.

Data-Driven Stabilization of Nonlinear Polynomial Systems With Noisy Data

In a recent article, we have shown how to learn controllers for unknown linear systems using finite-length noisy data by solving linear matrix inequalities. In this article, we extend this approach to deal with unknown nonlinear polynomial systems by formulating stability certificates in the form of data-dependent sum of squares programs, whose solution directly provides a stabilizing controller and a Lyapunov function. We then derive variations of this result that lead to more advantageous controller designs. The results also reveal connections to the problem of designing a controller starting from a least-square estimate of the polynomial system.

Reduced-Order Nonlinear Observers Via Contraction Analysis and Convex Optimization

In this article, we propose a new approach to design globally convergent reduced-order observers for nonlinear control systems via contraction analysis and convex optimization. Despite the fact that contraction is a concept naturally suitable for state estimation, the existing solutions are either local or relatively conservative when applying to physical systems. To address this, we show that this problem can be translated into an offline search for a coordinate transformation after which the dynamics is (transversely) contracting. The obtained sufficient condition consists of some easily verifiable differential inequalities, which, on one hand, identify a very general class of “detectable” nonlinear systems, and on the other hand, can be expressed as computationally efficient convex optimization, making the design procedure more systematic. Connections with some well-established approaches and concepts are also clarified in this article. Finally, we illustrate the proposed method with several numerical and physical examples, including polynomial, mechanical, electromechanical, and biochemical systems.

On generalizing Descartes' rule of signs to hypersurfaces

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations.

Tracking control design for periodic piecewise polynomial systems with multiple disturbances

AbstractThe present study examines the state tracking problem of periodic piecewise polynomial systems subject to multiple disturbances and actuator faults. Specifically, the fundamental period is partitioned into several subintervals and the piecewise matrix polynomial functions can be put together to approximate the dynamics over each period, which are characterized by Bernstein polynomials. Moreover, the system dynamics contains both matched and mismatched disturbances, wherein the matched disturbance is unknown and resulted from some exogenous system, and mismatched case is norm‐bounded. The control law is configured by integrating the output of the disturbance observer with state‐feedback reliable control law. Particularly, a disturbance observer is designed to estimate the disturbance caused by an exogenous system. Besides, by considering time‐varying polynomial Lyapunov function and making use of Bernstein polynomial approach, a set of sufficient conditions is derived which are expressed in terms of linear matrix inequalities. Further, by virtue of MATLAB software the desired controller and observer gain matrices can be computed. In conclusion, the potential and significance of the theoretical findings are confirmed by presenting a numerical example with simulation results.

On Sobolev orthogonal polynomials on a triangle

We use the invariance of the triangle T 2 = { ( x , y ) ∈ R 2 : 0 ⩽ x , y , 1 − x − y } \mathbf {T}^2=\{(x,y)\in \mathbb {R}^2:\, 0\leqslant x,y,\, 1-x-y\} under the permutations of { x , y , 1 − x − y } \{x,y,1-x-y\} to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on T 2 \mathbf {T}^2 . These orthogonal polynomials can be constructed from two sequences of univariate orthogonal polynomials. In particular, one of the two univariate sequences of polynomials is orthogonal with respect to a Sobolev inner product and the other is a sequence of classical Jacobi polynomials.

Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations

We study linear transformations \(T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]\) of the form \(T[x^n]=P_n(x)\) where \(\{P_n(x)\}\) is a real orthogonal polynomial system. With \(T=\sum \tfrac{Q_k(x)}{k!}D^k\), we seek to understand the behavior of the transformation T by studying the roots of the \(Q_k(x)\). We prove four main things. First, we show that the only case where the \(Q_k(x)\) are constant and \(\{P_n(x)\}\) is an orthogonal system is when the \(P_n(x)\) form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials \(Q_k(x)\) have real roots when the \(P_n(x)\) are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems.

On the cyclicity of quasi-homogeneous polynomial systems

In this paper, we study the weakened 16th Hilbert problem for a class of quasi-homogeneous polynomial systems and obtain the upper bound as well as the lower one for the number of limit cycles bifurcating from the period annulus of such systems.

Parameter synthesis of polynomial dynamical systems

Parametric dynamical systems emerge as a formalism for modeling natural and engineered systems ranging from biology, epidemiology, and medicine to cyber-physics. Parameter tuning is a complex task usually performed exploiting heavy simulations having high computational complexity and not ensuring the correctness of the synthesized systems. We consider the problem of parameter synthesis for discrete-time polynomial systems. We propose a formal method based on Bernstein coefficients that allows refining the set of parameters according to a temporal specification defined as a Signal Temporal Logic formula. The synthesized system is correct with respect to the specification and we demonstrate the scalability of the approach by implementing it in the C++ library Sapo, also available within a stand-alone application and as a web application. Finally, we illustrate the tool usage and the interface through a simple yet realistic epidemiological model and consider an intriguing application enhancing accuracy of the verification of neural networks.

Design of output feedback proportional–integral controller for polynomial fuzzy systems

AbstractIn this paper, the design of dynamic output feedback control law for polynomial fuzzy systems is studied. The main purpose is to design a tracker control law for fuzzy polynomial systems that are subject to external bounded disturbances. The structure of the control law is considered as fuzzy proportional–integral control. The conditions for deriving the control law are formulated in the form of a sum of square (SOS) feasibility problem. The designed control law will be able to force the system state vector to follow the state vector of a stable reference model in addition to guaranteeing the performance measure. The simulation results show the efficiency of the proposed method exposed to external disturbance.

Multigraded Sylvester forms, duality and elimination matrices

In this paper we study the equations of the elimination ideal associated with n+1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

The Anticanonical Complex for Non-degenerate Toric Complete Intersections

The anticanonical complex generalizes the Fano polytope from toric geometry and has been used to study Fano varieties with torus action so far. We provide an explicit description of the anticanonical complex for complete intersections in toric varieties defined by non-degenerate systems of Laurent polynomials. As an application, we classify the terminal Fano threefolds that are embedded into a fake weighted projective space via a general system of Laurent polynomials.

Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator

The Inverse Kinematics (IK) problem is concerned with finding robot control parameters to bring the robot into a desired position under the kinematics and joint limit constraints. We present a globally optimal solution to the IK problem for a general serial 7DOF manipulator with revolute joints and a polynomial objective function. We show that the kinematic constraints due to rotations can be all generated by the second-degree polynomials. This is an important result since it significantly simplifies the further step where we find the optimal solution by Lasserre relaxations of nonconvex polynomial systems. We demonstrate that the second relaxation is sufficient to solve a general 7DOF IK problem. Our approach is certifiably globally optimal. We demonstrate the method on the 7DOF KUKA LBR IIWA manipulator and show that we are, in practice, able to compute the optimal IK or certify infeasibility in 99.9% tested poses. We also demonstrate that by the same approach, we are able to solve the IK problem for any generic (random) manipulator with seven revolute joints.

On Stability of Motion of Polynomial Systems with Aftereffect*

On Stability of Motion of Polynomial Systems with Aftereffect*

Toric eigenvalue methods for solving sparse polynomial systems

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety X X . Our starting point is a homogeneous ideal I I in the Cox ring of X X , which in practice might arise from homogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical approach for solving this problem. Our method works in particular for systems having isolated solutions with arbitrary multiplicities. It depends on the multigraded regularity properties of I I . We study these properties and provide bounds on the size of the matrices in our approach when I I is a complete intersection.

Оптимальный гироскопический стабилизатор многомерной вибрационной системы

We study the stabilizing problem for a multidimensional vibration system using a gyroscopic stabilizer. A vibration system is given by a second order differential equation with symmetrical matrix coefficients: a positive definite stiffness matrix and an indefinite damping matrix; in general, such a system is unstable. We need find an optimal gyroscopic stabilizer for it represented by a skew-symmetric matrix coefficient in a speed term. The choice of this control type is dictated by the tendency to avoid additional vibrations caused by slip-slide friction. Its feature is a reduced order of the controller, regardless of the dimension of the system and the number of tunable parameter. Our goal is to elucidate the most important properties of the gyroscopic stabilizers variety. It is described by a polynomial equations system. The dimension of the general solution variety in the regular case is easily found and some of its points can be calculated numerically. We start with an example of dimension 3, which leads to a system of 6th order ordinary differential equations (ODE) and then a system of five polynomial equations with regard to six unknowns. Its general solution turns out to be a one-dimensional algebraic variety presented in a table form. The second example has dimension 5; it corresponds to a tenth-order system of differential equations and nine polynomial equations in fifteen unknowns. The dimension of the solution manifold is equal to six; we find a one-dimensional subvariety and some singular points. The main difficulty is the divergence of numerical calculations near the multiple poles of a closed system. One of the important properties, that manifested itself in both examples, was the presence of complex conjugate poles and occasionally multiples of real ones; thus, almost all solutions for an optimal gyroscopic stabilizer are made up of relatively rapidly decaying oscillations. In both examples, the variety of solutions consists mostly of simple poles and allows one to choose a stabilizer that does not create resonant effects.

Imitation learning of stabilizing policies for nonlinear systems

There has been a recent interest in imitation learning methods that are guaranteed to produce a stabilizing control law with respect to a known system. Work in this area has generally considered linear systems and controllers, for which stabilizing imitation learning takes the form of a biconvex optimization problem. In this paper it is demonstrated that the same methods developed for linear systems and controllers can be readily extended to polynomial systems and controllers using sum of squares techniques. A projected gradient descent algorithm and an alternating direction method of multipliers algorithm are proposed as heuristics for solving the stabilizing imitation learning problem, and their performance is illustrated through numerical experiments.

Simultaneous stoquasticity

Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.