Polynomial Constraints Research Articles

Quadrotor Flight Envelope Protection with Trajectory and Yaw Tracking

This study proposes a reference governor control strategy to ensure that a quadrotor drone follows a planar trajectory while respecting the prescribed safety constraints. These constraints, including polynomial constraints accounting for aerodynamic and gyroscopic effects, were transformed into linear constraints using novel state-space augmentation. This allows for the design of a reference governor, which is designed to safeguard the drone’s flight envelope by adjusting the reference trajectory during trajectory tracking while also ensuring yaw tracking with lower priority than position tracking, especially when constraints are at risk of being violated. This study also presents simulations that illustrate the effectiveness of the proposed algorithms.

Principal Landau determinants

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/. Program summaryProgram title:PLD.jlCPC Library link to program files:https://doi.org/10.17632/7h5644mm4n.1Developer's repository link:https://mathrepo.mis.mpg.de/PLD/Licensing provisions: Creative Commons by 4.0Programming language:JuliaSupplementary material: The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).Nature of problem: A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.Solution method: We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.

PolyARBerNN: A Neural Network Guided Solver and Optimizer for Bounded Polynomial Inequalities

Constraints solvers play a significant role in the analysis, synthesis, and formal verification of complex cyber-physical systems. In this article, we study the problem of designing a scalable constraints solver for an important class of constraints named polynomial constraint inequalities (also known as nonlinear real arithmetic theory). In this article, we introduce a solver named PolyARBerNN that uses convex polynomials as abstractions for highly nonlinears polynomials. Such abstractions were previously shown to be powerful to prune the search space and restrict the usage of sound and complete solvers to small search spaces. Compared with the previous efforts on using convex abstractions, PolyARBerNN provides three main contributions namely (i) a neural network guided abstraction refinement procedure that helps selecting the right abstraction out of a set of pre-defined abstractions, (ii) a Bernstein polynomial-based search space pruning mechanism that can be used to compute tight estimates of the polynomial maximum and minimum values which can be used as an additional abstraction of the polynomials, and (iii) an optimizer that transforms polynomial objective functions into polynomial constraints (on the gradient of the objective function) whose solutions are guaranteed to be close to the global optima. These enhancements together allowed the PolyARBerNN solver to solve complex instances and scales more favorably compared to the state-of-the-art nonlinear real arithmetic solvers while maintaining the soundness and completeness of the resulting solver. In particular, our test benches show that PolyARBerNN achieved 100X speedup compared with Z3 8.9, Yices 2.6, and PVS (a solver that uses Bernstein expansion to solve multivariate polynomial constraints) on a variety of standard test benches. Finally, we implemented an optimizer called PolyAROpt that uses PolyARBerNN to solve constrained polynomial optimization problems. Numerical results show that PolyAROpt is able to solve high-dimensional and high order polynomial optimization problems with higher speed compared to the built-in optimizer in the Z3 8.9 solver.

Electrical Resistivity Inversion Based on Symmetric Polynomial Constraint

As we seek to make geophysical models more consistent with geological models, more and more a priori information is incorporated into geophysical inverse problems. How we make the most use of a priori information is still a key issue in geophysical inversion, particularly for effectively representing such information in inversion problems. To address this problem, we propose a symmetric polynomial constrained inversion algorithm that can represent multiple known physical properties for the lithology. The electrical resistivity synthetic model tests show that the inversion results constrained with symmetric polynomials can make the model resistivity be in line with a priori information and obtain more realistic sharp boundaries. Finally, we give a practical verification of our inversion algorithm with field data.

Calculating Global Minimum Points to Binary Polynomial Optimization Problem: Optimizing the Optimal PMU Localization Problem as a Case-Study

State estimation (SE) is an algorithmic function of an energy management system (EMS). SE provides an actual-time monitoring and control of modern electrical power grids. State Estimation can be worked with sufficiency using Phasor Measurement Units optimally placed within a power grid. This paper concerns the implementation of proper algorithms embedded in optimization solvers to the optimal PMU localization problem solving globally. The optimization model is formulated as a 0 - 1 nonlinear minimization problem. The problem is transformed to a polyhedron using linearization methods and B&B tree. In this model, we use a linear cost function under polynomial constraints and binary restrictions on the design variables in a symbolic format. This mathematical model is programmed in the YALMIP environment which is fully compatible with MATLAB. The 0 - 1 Nonlinear Programming (NLP) model is suitable for getting concisely global optimal solutions. The optimal solution is given by a wrapped optimization engine including a local optimizer routine performing together with a mixed-Integer-Linear Programming routine. The solution is achieved within a zero-gap precisely encountered during the iterative process. This tolerance criterion is a necessity for a successful implementation of the B&B tree because it ensures global optimality with an acceptance relative gap. The minimization model is implemented in a YALMIP code fully compatible with MATLAB in two stages. Initially, an objective function with one term is minimized to discover a number of sensors for wide-area monitoring, control and state estimator applications. Then, an extra product is considered in the objective to suffice maximum reliability for observing the network buses. The numerical minimization models are applied to standard power networks in the direction to be solved globally.

Impacts of g-g-v Constraints Formulations on Online Minimum-Time Vehicle Trajectory Planning

The g-g-v diagram is a popular representation of a vehicle’s maximum longitudinal and lateral accelerations, as a function of its speed. Recently, g-g-v diagrams have been used as performance constraints for online minimum-time vehicle trajectory planning with economic nonlinear model predictive control (E-NMPC). However, in the related literature, the modeling accuracy of the g-g-v formulations was often compromised in favor of computational efficiency for online E-NMPC. This paper compares various formulations of the g-g-v constraints, evaluating their accuracy, computational efficiency and impact on the lap times, when applied by an artificial race driver (ARD) to control a sports car model on a circuit. Also, we propose a new g-g-v diagram formulation, based on convex polytopic and non-convex polynomial constraints. Our formulation yields improved modeling accuracy while preserving a low computational burden for online E-NMPC. Finally, we analyze the vehicle trajectories, to understand how ARD achieves lower lap times through an improved knowledge of the maximum performance.

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

Fast angles-only relative navigation using polynomial dynamics

A fast and efficient method for initial relative orbit determination from bearing-angle measurements is introduced. The range ambiguity problem for angles-only relative navigation is addressed by modeling nonlinear effects with a novel second-order mapping from relative orbit elements (ROE) to relative position coordinates. This model is used to form a system of polynomial constraint equations linking the line-of-sight measurements to the ROE. An efficient method for solving this system is developed around the insight that the ROE scale with the ratio of the inter-spacecraft separation to the orbit radius and are therefore small for most applications of interest. The method uses a truncated expansion of the quadratic formula to recursively eliminate unknowns, reduce the dimension of the system, and ultimately acquire an approximate solution. Strategies for improving robustness, efficiency, and accuracy are developed and the method is applied to general second-order systems as well as to a broad range of intial relative orbit determination scenarios. Modifications to the constraint equations and a solution algorithm are introduced to address the challenge of bias in the bearing-angle measurements.

Midpoint projection algorithm for stochastic differential equationson manifolds.

Stochastic differential equationsprojected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology, and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equationson the manifold are sometimes computationally impractical, and numerical projections are therefore useful in many cases. In this paper a combined midpoint projection algorithm is proposed that uses a midpoint projection onto a tangent space, combined with a subsequent normal projection to satisfy the constraints. We also show that the Stratonovich form of stochastic calculus is generally obtained with finite bandwidth noise in the presence of a strong enough external potential that constrains the resulting physical motion to a manifold. Numerical examples are given for a wide range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, higher-order polynomial constraints that give a quasicubical surface, and a ten-dimensional hypersphere. In all cases the combined midpoint method has greatly reduced errors compared to other methods used for comparison, namely, a combined Euler projection approach and a tangential projection algorithm. We derive intrinsic stochastic equationsfor spheroidal and hyperboloidal surfaces for comparison purposes to verify the results. Our technique can handle multiple constraints, which allows manifolds that embody several conserved quantities. The algorithm is accurate, simple, and efficient. A reduction of an order of magnitude in the diffusion distance error is found compared to the other methods and an up to several orders of magnitude reduction in constraint function errors.

Computation of maximal output admissible sets for linear systems with polynomial constraints

AbstractIn this technical note we study the computation of the Maximal Output Admissible Set for linear systems subject to polynomial constraints. The computation of an inner approximation of the Maximal Output Admissible Sets requires the determination of constraint redundancy. We use a procedure to determine polynomial constraint redundancy based on a consequence of Putinar's Positivstellensatz. Further, we present a modification of the algorithm to compute the Maximal Output Admissible Set with improved performance. Lastly, demonstrate the potential for practical applications in two case studies of spacecraft rendezvous and control of an electromagnetic actuator.

Robust fault-tolerant control design for polynomial fuzzy systems

This paper focuses on the robust H∞ fault-tolerant control issue for a family of polynomial fuzzy systems subject to actuator faults and external disturbances. Especially, asynchronous fuzzy membership function technique is considered to minimize the expenses of controller implementation. To derive the robust stability conditions, actuator fault factors are taken into the account in the closed-loop system formulation. Moreover, to utilize the exact shape information of fuzzy membership functions in the stability criterion, they are approximated as sum-of-square polynomials with the aid of polynomial curve fitting method. In addition, to reduce the conservatism, a generalized polynomial Lyapunov function is considered. However, there are some non-convex terms in the stability conditions due to the construction of the generalized Lyapunov function that consists of all state-dependent polynomial matrix. Thus, some existing convex methods for general polynomial fuzzy systems are rendered inapplicable. To deal with this issue, the actual non-convex stability conditions converts into a convex optimization problem by introducing some additional convex constraints. Moreover, the membership-function-dependent robust stability conditions of the closed-loop system are provided as sum-of-square convex polynomial constraints. Lastly, the usefulness and superiority of the designed controller are illustrated by the simulation examples.

Toward Efficient Direct Dynamics Studies of Chemical Reactions: A Novel Matrix Completion Algorithm.

This paper describes the development and testing of a polynomial variety-based matrix completion (PVMC) algorithm. Our goal is to reduce computational effort associated with reaction rate coefficient calculations using variational transition state theory with multidimensional tunneling (VTST-MT). The algorithm recovers eigenvalues of quantum mechanical Hessians constituting the minimum energy path (MEP) of a reaction using only a small sample of the information, by leveraging underlying properties of these eigenvalues. In addition to the low-rank property that constitutes the basis for most matrix completion (MC) algorithms, this work introduces a polynomial constraint in the objective function. This enables us to sample matrix columns unlike most conventional MC methods that can only sample elements, which makes PVMC readily compatible with quantum chemistry calculations as sampling a single column requires one Hessian calculation. For various types of reactions─SN2, hydrogen atom transfer, metal-ligand cooperative catalysis, and enzyme chemistry─we demonstrate that PVMC on average requires only six to seven Hessian calculations to accurately predict both quantum and variational effects.

Problèmes de plongement finis sur les corps non commutatifs

We extend the notion of finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of centrally finite division rings. We first show that solving a finite embedding problem over such a division ring H is equivalent to finding a solution to some finite embedding problem over the center of H involving a polynomial constraint. We then show that every constant finite split embedding problem over the skew field of rational fractions H(t) with central indeterminate t has a solution, if the center of H is an ample field, thus providing with a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over skew fields of rational fractions H(t, σ), where σ denotes an arbitrary automorphism of H with finite order. Our results generalize previous works on the inverse Galois problem over skew fields.

A reference governor for linear systems with polynomial constraints

The paper considers reference governor design for a class of linear discrete-time systems with constraints given by polynomial inequalities and constant reference commands. For such systems, we propose a novel algorithm to compute the maximal output admissible set. The reference governor solves a constrained nonlinear minimization problem at the initialization and then uses a bisection algorithm at the subsequent time steps. The effectiveness of the method is demonstrated by two numerical examples.

Algebra-Based Reasoning for Loop Synthesis

Provably correct software is one of the key challenges of our software-driven society. Program synthesis—the task of constructing a program satisfying a given specification—is one strategy for achieving this. The result of this task is then a program that is correct by design. As in the domain of program verification, handling loops is one of the main ingredients to a successful synthesis procedure. We present an algorithm for synthesizing loops satisfying a given polynomial loop invariant. The class of loops we are considering can be modeled by a system of algebraic recurrence equations with constant coefficients, thus encoding program loops with affine operations among program variables. We turn the task of loop synthesis into a polynomial constraint problem by precisely characterizing the set of all loops satisfying the given invariant. We prove soundness of our approach, as well as its completeness with respect to an a priori fixed upper bound on the number of program variables. Our work has applications toward synthesizing loops satisfying a given polynomial loop invariant—program verification—as well as generating number sequences from algebraic relations. To understand viability of the methodology and heuristics for synthesizing loops, we implement and evaluate the method using the Absynth tool.

Orthogonal anti-jamming waveform design with extended Doppler tolerance based on the LFM-PC signal

Designing orthogonal waveforms based on linear frequency modulation and phase-coded (LFM-PC) signals with extended Doppler tolerance plays a vital role in multiple-input multiple-output (MIMO) radar systems. However, it is difficult to address the major problem of the mixed nonconvex unimodulus and nonlinear fourth-order polynomial constraints in sequence set design. In this paper, an orthogonal waveform based on the LFM-PC signal is designed for the discrete phase-coded sequence (DPCS) and continuous phase-coded sequence (CPCS). The hybrid alternating direction method of multipliers (ADMM) and coordinate descending (CD) algorithm are applied to the DPCS design. A novel method is designed with the approach of ADMM and the Lagrange programming neural network (LPNN) algorithm for the CPCS. The specific objective of this study is to propose new chirplike phase codes and a new algorithm framework that can separate the complex nonconvex and nonlinear constraints by ADMM, and we transfer the fourth-order polynomials from the nonlinear constraints into the Lagrange function. As a result, the optimized variables are updated in the ADMM-CD and ADMM-LPNN frameworks. Numerical simulation examples indicate the good performance of the proposed methods, and we compare these two effective methods.

Polynomial Constraint Generalized Maximum Correntropy Normalized Subband Adaptive Filter Algorithm

Polynomial Constraint Generalized Maximum Correntropy Normalized Subband Adaptive Filter Algorithm

On Radius of Robust Feasibility for Convex Conic Programs with Data Uncertainty

The radius of robust feasibility is the maximal size of uncertain set in which the robust feasible set for an uncertain program is nonempty. In this paper, we employ robust optimization technique to study a class of uncertain convex conic program, and give its formulas for radius of robust feasibility under several data uncertain sets. First, with aid of the distance from the origin to the so-called epigraphcal set, we provide computable upper and lower bounds of the radius of robust feasibility for convex conic program in face of ball uncertainty. Second, a formula is presented for the radius of robust feasibility for robust convex optimization problem with SOS-convex polynomial constraints under ball uncertain sets. Finally, some exact formulas of radius of robust feasibility are given for convex conic program with piecewise linear function constraints under boxes or polytopes uncertain sets.

Membership-function-dependent model predictive control for nonlinear systems in a piecewise-fuzzy framework

This paper is concerned with the membership-function-dependent model predictive control (MPC) problem for a class of Takagi-Sugeno (T-S) fuzzy systems with hard constraints. In order to design a set of membership-function-dependent controllers in the framework of MPC, the original T-S fuzzy systems are equivalently converted into a piecewise-fuzzy model by means of the partition method of the premise variable space. Then, instead of calculating vertices to construct the fuzzy MPC controller as adopted in the traditional T-S fuzzy control strategies, some staircase membership functions are employed to approximate the continuous membership functions, such that the information of membership functions can be adequately considered in the control synthesis to reduce the conservatism of the controller design. Furthermore, associated with the piecewise-fuzzy model, an online optimization problem based on the membership-function-dependent terminal constraint set is formulated in terms of the piecewise-Lyapunov-function-based MPC to calculate the feedback gains. Besides, by employing polynomial constraints based on the shape information of the membership functions, errors between the staircase membership functions and the original continuous membership functions are taken into account for facilitating the feasibility investigation and the stability analysis. Finally, two illustrative examples are used to illustrate the validity of the proposed membership-function-dependent MPC strategy.

Tube-model predictive control based on sum of squares for hypersonic vehicle with state-dependent input constraints

In this paper, tube-model predictive control based on the sum of squares technique is developed for hypersonic vehicles with state-dependent input constraints. Firstly, the longitudinal non-linear model in the presence of uncertain parameters is transformed into the polytopic linear parameter varying model with bounded disturbance by feedback linearization. Then the actual input constraints are converted to the virtual state-dependent input constraints in linear multivariable polynomial. A composite feedback control law based on tube-model predictive control is designed into a convex combination of unconstrained and constrained control. The real control law can be obtained by inversion. The sum of squares technique is used to transform the polynomial constraints into the convex matrix sum of squares condition via linear matrix inequality. Finally, simulation results verify the effectiveness of the proposed controller.