Journal articles: 'Parametric polynomial systems'

1

Lazard, Daniel, and Fabrice Rouillier. "Solving parametric polynomial systems." Journal of Symbolic Computation 42, no. 6 (June 2007): 636–67. http://dx.doi.org/10.1016/j.jsc.2007.01.007.

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Franco-Medrano, Fermin, and Francisco J. Solis. "Stability of Real Parametric Polynomial Discrete Dynamical Systems." Discrete Dynamics in Nature and Society 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/680970.

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We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameterλand generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to realmth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept ofcanonical polynomial mapswhich are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termedProduct Position Functionfor a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termedstability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

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Lai, Yisheng, Renhong Wang, and Jinming Wu. "Solving parametric piecewise polynomial systems." Journal of Computational and Applied Mathematics 236, no. 5 (October 2011): 924–36. http://dx.doi.org/10.1016/j.cam.2011.05.008.

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4

Moreno Maza, Marc, Bican Xia, and Rong Xiao. "On Solving Parametric Polynomial Systems." Mathematics in Computer Science 6, no. 4 (November 28, 2012): 457–73. http://dx.doi.org/10.1007/s11786-012-0136-3.

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5

Ayad, A. "Complexity of solving parametric polynomial systems." Journal of Mathematical Sciences 179, no. 6 (December 2011): 635–61. http://dx.doi.org/10.1007/s10958-011-0616-z.

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Hashemi, Amir, Benyamin M.-Alizadeh, and Mahdi Dehghani Darmian. "Minimal polynomial systems for parametric matrices." Linear and Multilinear Algebra 61, no. 2 (April 2, 2012): 265–72. http://dx.doi.org/10.1080/03081087.2012.670235.

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Chen, Zhenghong, Xiaoxian Tang, and Bican Xia. "Generic regular decompositions for parametric polynomial systems." Journal of Systems Science and Complexity 28, no. 5 (July 30, 2015): 1194–211. http://dx.doi.org/10.1007/s11424-015-3015-6.

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8

Gerhard, Jürgen, D. J. Jeffrey, and Guillaume Moroz. "A package for solving parametric polynomial systems." ACM Communications in Computer Algebra 43, no. 3/4 (June 24, 2010): 61–72. http://dx.doi.org/10.1145/1823931.1823933.

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9

A.A., Nesenchuk. "Investigation and robust synthesis of polynomials under perturbations based on the root locus parameter distribution diagram." Artificial Intelligence 24, no. 1-2 (November 15, 2019): 25–33. http://dx.doi.org/10.15407/jai2019.01-02.025.

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Investigation of the 4 th order dynamic systems characteristic polynomials behavior in conditions of the interval parametric uncertainties is carried out on the basis of root locus portraits. The roots behavior regularities and corresponding diagrams for the root locus parameter distribution along the asymptotic stability bound are specified for the root locus portraits of the systems. On this basis the stability conditions are derived, graphic-analytical method is worked out for calculating intervals of variation for the polynomial family parameters ensuring its robust stability. The discovered regularities of the system root locus portrait behavior allow to extract hurwitz sub-families from the non-hurwitz families of interval polynomials and to determine whether there exists at least one stable polynomial in the unstable polynomial family.

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10

Šebek, Michael, Martin Hromčik, and Jan Ježek. "Polynomial Toolbox 2.5 and Systems with Parametric Uncertainties 1." IFAC Proceedings Volumes 33, no. 14 (September 2000): 757–61. http://dx.doi.org/10.1016/s1474-6670(17)36321-8.

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11

HOANG, VIET HA, and CHRISTOPH SCHWAB. "REGULARITY AND GENERALIZED POLYNOMIAL CHAOS APPROXIMATION OF PARAMETRIC AND RANDOM SECOND-ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS." Analysis and Applications 10, no. 03 (July 2012): 295–326. http://dx.doi.org/10.1142/s0219530512500145.

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Initial boundary value problems of linear second-order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform stability with respect to the support of the resulting coupled hyperbolic systems, and provide sufficient smoothness and compatibility conditions on the data for the solution to exhibit analytic, respectively, Gevrey regularity with respect to the countably many parameters. Sufficient conditions for the p-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best N-term polynomial chaos type approximations of the parametric solution are given. In addition, regularity both in space and time for the parametric family of solutions is proved for data satisfying certain compatibility conditions. The results allow obtaining convergence rates and stability of sparse space-time tensor product Galerkin discretizations in the parameter space.

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12

Guo, Long-Chuan. "Adaptive Output Feedback Tracking Controller Design of Stochastic Nonlinear Systems with Parameter Uncertainty for Polynomial Function Growth Conditions." Mathematical Problems in Engineering 2018 (June 19, 2018): 1–10. http://dx.doi.org/10.1155/2018/7659536.

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This paper mainly focuses on output feedback practical tracking controller design for stochastic nonlinear systems with polynomial function growth conditions. Mostly, there are some studies on output feedback tracking control problem for general nonlinear systems with parametric certainty in existing achievements. Moreover, we extend it to stochastic nonlinear systems with parametric uncertainty and system nonlinear terms are assumed to satisfy polynomial function growth conditions which are more relaxed than linear growth conditions or power growth conditions. Due to the presence of unknown parametric uncertainty, an output feedback practical tracking controller with dynamically updated gains is constructed explicitly so that all the states of the closed-loop systems are globally bounded and the tracking error belongs to arbitrarily small interval after some positive finite time. An example illustrates the efficiency of the theoretical results.

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13

Aoki, Miho, and Yasuhiro Kishi. "On systems of fundamental units of certain quartic fields." International Journal of Number Theory 11, no. 07 (October 21, 2015): 2019–35. http://dx.doi.org/10.1142/s1793042115500864.

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14

ZHOU, Yongzhi, Hao WU, Chenghong GU, and Yonghua SONG. "Global optimal polynomial approximation for parametric problems in power systems." Journal of Modern Power Systems and Clean Energy 7, no. 3 (December 21, 2018): 500–511. http://dx.doi.org/10.1007/s40565-018-0469-2.

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15

Lewis, Robert H., Béla Paláncz, and Joseph Awange. "Solving geoinformatics parametric polynomial systems using the improved Dixon resultant." Earth Science Informatics 12, no. 2 (November 15, 2018): 229–39. http://dx.doi.org/10.1007/s12145-018-0366-2.

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16

Xia, Bingqing, Hao Wu, Wenbin Yang, Lu Cao, and Yonghua Song. "Parametric Transient Stability Constrained Optimal Power Flow Solved by Polynomial Approximation Based on the Stochastic Collocation Method." Energies 15, no. 11 (June 3, 2022): 4127. http://dx.doi.org/10.3390/en15114127.

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To better respond to the impact of power system-uncertain parameters on transient stability, a novel model named the parametric transient stability constrained optimal power flow (parametric TSCOPF) is proposed. It seeks the optimal control scheme of transient stability constrained optimal power flow (TSCOPF) expressed by the function of uncertain parameters in power systems. The key difficulty to solve this model lies in that the relationship between the parametric TSCOPF solution and uncertain parameters is implicit, which is hard to derive generally. To this end, this paper approximates the optimal solution of parametric TSCOPF by polynomial expressions of uncertain parameters based on the stochastic collocation method. First, the parametric TSCOPF model includes both uncertain parameters and transient stability constraints, in which the transient stability constraint is constructed as a set of polynomial expressions using the SCM. Then, to derive the relationship between the parametric TSCOPF solution and uncertain parameters, the SCM is applied to the parametric Karush–Kuhn–Tucker (KKT) conditions of the parametric TSCOPF model, so that the optimal solution of the parametric TSCOPF is approximated by using polynomial expressions with respect to uncertain parameters. The proposed parametric TSCOPF model has been tested on a 3-machine, 9-bus system, and the IEEE 145-bus system, which verifies the effectiveness of the proposed method.

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17

Barrowclough, Oliver J. D., and Tor Dokken. "Approximate Implicitization Using Linear Algebra." Journal of Applied Mathematics 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/293746.

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We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.

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18

Duff, Timothy, Cvetelina Hill, Anders Jensen, Kisun Lee, Anton Leykin, and Jeff Sommars. "Solving polynomial systems via homotopy continuation and monodromy." IMA Journal of Numerical Analysis 39, no. 3 (April 13, 2018): 1421–46. http://dx.doi.org/10.1093/imanum/dry017.

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Abstract We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. Under the theoretical . that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.

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19

Son, Jeongeun, and Yuncheng Du. "An Efficient Polynomial Chaos Expansion Method for Uncertainty Quantification in Dynamic Systems." Applied Mechanics 2, no. 3 (July 12, 2021): 460–81. http://dx.doi.org/10.3390/applmech2030026.

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Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE).

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20

Duff, G. F. D., R. B. Leipnik, and C. E. M. Pearce. "Guide expansions for the recursive parametric solution of polynomial dynamical systems." ANZIAM Journal 47, no. 3 (January 2006): 387–96. http://dx.doi.org/10.1017/s1446181100009901.

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AbstractRecursive parametric series solutions are developed for polynomial ODE systems, based on expanding the system components in series of a form studied by Weiss. Individual terms involve first-order driven linear ODE systems with variable coefficients. We consider Lotka-Volterra systems as an example.

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21

Solis, Francisco. "Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems." Mathematical and Computational Applications 24, no. 1 (January 18, 2019): 13. http://dx.doi.org/10.3390/mca24010013.

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In this paper, we introduce and analyze a family of exponential polynomial discrete dynamical systems that can be considered as functional perturbations of a linear dynamical system. The stability analysis of equilibria of this family is performed by considering three different parametric scenarios, from which we show the intricate and complex dynamical behavior of their orbits.

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22

Shen, Danfeng, Hao Wu, Bingqing Xia, and Deqiang Gan. "Polynomial Chaos Expansion for Parametric Problems in Engineering Systems: A Review." IEEE Systems Journal 14, no. 3 (September 2020): 4500–4514. http://dx.doi.org/10.1109/jsyst.2019.2957664.

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23

Zheng, Qian, and Fen Wu. "Adaptive control design for uncertain polynomial nonlinear systems with parametric uncertainties." International Journal of Adaptive Control and Signal Processing 25, no. 6 (November 5, 2010): 502–18. http://dx.doi.org/10.1002/acs.1215.

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24

Trinchero, Riccardo, Paolo Manfredi, and Igor S. Stievano. "TMsim: An Algorithmic Tool for the Parametric and Worst-Case Simulation of Systems with Uncertainties." Mathematical Problems in Engineering 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/6739857.

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This paper presents a general purpose, algebraic tool—named TMsim—for the combined parametric and worst-case analysis of systems with bounded uncertain parameters. The tool is based on the theory of Taylor models and represents uncertain variables on a bounded domain in terms of a Taylor polynomial plus an interval remainder accounting for truncation and round-off errors. This representation is propagated from inputs to outputs by means of a suitable redefinition of the involved calculations, in both scalar and matrix form. The polynomial provides a parametric approximation of the variable, while the remainder gives a conservative bound of the associated error. The combination between the bound of the polynomial and the interval remainder provides an estimation of the overall (worst-case) bound of the variable. After a preliminary theoretical background, the tool (freely available online) is introduced step by step along with the necessary theoretical notions. As a validation, it is applied to illustrative examples as well as to real-life problems of relevance in electrical engineering applications, specifically a quarter-car model and a continuous-time linear equalizer.

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25

Ezangina, Tatiana Al, and Sergey An Gayvoronskiy. "Ensuring Maximum Stability Degree in the Systems with Interval Parameters." Applied Mechanics and Materials 752-753 (April 2015): 955–60. http://dx.doi.org/10.4028/www.scientific.net/amm.752-753.955.

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The robust control system objects with interval-undermined parameters is considers in this paper. Initial information about the system is its characteristic polynomial with interval coefficients. On the basis of coefficient estimations of quality indices and criterion of the maximum stability degree, the methods of synthesis of a robust regulator parametric is developed.

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26

Le, Huu Phuoc, and Mohab Safey El Din. "Solving parametric systems of polynomial equations over the reals through Hermite matrices." Journal of Symbolic Computation 112 (September 2022): 25–61. http://dx.doi.org/10.1016/j.jsc.2021.12.002.

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27

Ayad, Ali, Ali Fares, and Youssef Ayyad. "An algorithm for solving zero-dimensional parametric systems of polynomial homogeneous equations." Journal of Nonlinear Sciences and Applications 05, no. 06 (December 12, 2012): 426–38. http://dx.doi.org/10.22436/jnsa.005.06.03.

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28

Zhou, Yongzhi, Hao Wu, Chenghong Gu, and Yonghua Song. "A Novel Method of Polynomial Approximation for Parametric Problems in Power Systems." IEEE Transactions on Power Systems 32, no. 4 (July 2017): 3298–307. http://dx.doi.org/10.1109/tpwrs.2016.2623820.

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29

Gavalec, Martin, and Zuzana Němcová. "Solvability of a Bounded Parametric System in Max-Łukasiewicz Algebra." Mathematics 8, no. 6 (June 23, 2020): 1026. http://dx.doi.org/10.3390/math8061026.

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The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system.

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Buldaev, Alexander, and Dmitry Trunin. "On a Method for Optimizing Controlled Polynomial Systems with Constraints." Mathematics 11, no. 7 (April 2, 2023): 1695. http://dx.doi.org/10.3390/math11071695.

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A new optimization approach is considered in the class of polynomial in-state optimal control problems with constraints based on nonlocal control improvement conditions, which are constructed in the form of special fixed-point problems in the control space. The proposed method of successive approximations of control retains all constraints at each iteration and does not use the operation of parametric variation of control at each iteration, in contrast to known gradient methods. In addition, the initial approximation of the iterative process may not satisfy the constraints, which is a significant factor in increasing the efficiency of the approach. The comparative efficiency of the proposed method of fixed points in the considered class of problems is illustrated in a model example.

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31

LI, JIBIN, and FENGJUAN CHEN. "KNOTTED PERIODIC ORBITS AND CHAOTIC BEHAVIOR OF A CLASS OF THREE-DIMENSIONAL FLOWS." International Journal of Bifurcation and Chaos 21, no. 09 (September 2011): 2505–23. http://dx.doi.org/10.1142/s0218127411029896.

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This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

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32

Wan, Yiming, Dongying Erin Shen, Sergio Lucia, Rolf Findeisen, and Richard D. Braatz. "Polynomial chaos-based H2 output-feedback control of systems with probabilistic parametric uncertainties." Automatica 131 (September 2021): 109743. http://dx.doi.org/10.1016/j.automatica.2021.109743.

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33

DU, WeiPing, YiSheng LAI, DeXin DUAN, and XiaoKe FANG. "Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial systems." SCIENTIA SINICA Mathematica 45, no. 9 (September 1, 2015): 1423–40. http://dx.doi.org/10.1360/n012015-00063.

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34

Srinivasa Rao, Danaboyina, Mangipudi Siva Kumar, and Manyala Ramalinga Raju. "New algorithm for the design of robust PI controller for plants with parametric uncertainty." Transactions of the Institute of Measurement and Control 40, no. 5 (February 1, 2017): 1481–89. http://dx.doi.org/10.1177/0142331216685393.

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This paper proposes a new algorithm for the design of robust PI controller for plants with parametric uncertainty using new necessary and sufficient stability conditions. Most of the control systems operate under large uncertainty causing degradation of system performance and destabilization. In order to compensate these shortcomings, a robust PI controller is designed based on new necessary and sufficient conditions for stability of a plant with parametric uncertainty, a class of interval polynomial. New necessary and sufficient conditions for the determination of robust stability of interval polynomials have been developed using the results of Routhe’s theorem and Karitonov theorem. A set of inequalities are derived based on these developed new necessary and sufficient conditions to obtain robust controller parameters. The proposed method is simple and involves less computational complexity compared with the available methods in the literature. The efficacy of the proposed methodology is demonstrated with a numerical example for successful implementation.

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35

Gayvoronskiy, S. A., T. A. Ezangina, I. V. Khozhaev, and A. A. Nesenchuk. "Analyzing Robust Stability of an Interval Control System on the Basis of Vertex Polynomials." Mekhatronika, Avtomatizatsiya, Upravlenie 20, no. 5 (May 25, 2019): 266–73. http://dx.doi.org/10.17587/mau.20.266-273.

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In the paper, a characteristic polynomial of an interval control system, whose coefficients are unknown or may vary within certain ranges of values, is considered. Parametric variations cause migration of interval characteristic polynomial roots within their allocation areas, whose borders determine robust stability degree of the interval control system. To estimate a robust stability degree, a projection of a polytope of interval characteristic polynomial coefficients on a complex plane must be examined. However, in order to find a robust stability degree it is enough to examine some vertices of a coefficient polytope and not the whole polytope. To find these vertices, which fully determine a robust stability degree, it is proposed to use a basic phase equation of a root locus method. Considering the requirements to placing allocation areas of system poles an interval extension of expressions for angles included to the phase equation. The set of statements, allowing to find a sum of pole angles intervals in the case of degree of oscillating robust stability, were formulated and proved. From these statements, a set of double interval angular inequalities was derived. The inequalities determine ranges of angles of all root locus edge branches departure from every pole. Considered research resulted in a procedure of finding coordinates of verifying vertices of a coefficients polytope and vertex polynomials according to these vertices. Such polynomials were found for oscillating robust stability degree analysis of interval control systems of the second, the third and the forth order. Also, similar statements were derived for aperiodical robust stability degree analysis. Numerical examples of vertex analysis of oscillating and aperiodical robust stability degree were provided for interval control systems of the second, the third and the fourth order. Obtained results were proved by examining root allocation areas of interval characteristic polynomials examined in application examples of proposed methods.

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36

Sadchikov, Pavel, Tatyana Khomenko, and Galina Ternovaya. "Numerical optimization of the transfer function of the intelligent building management system." E3S Web of Conferences 97 (2019): 01015. http://dx.doi.org/10.1051/e3sconf/20199701015.

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The paper deals with structural-parametric models for describing dynamic processes of technical systems of an intelligent building. The task of searching for the transfer function of the synthesized elements and devices of its information-measuring and control systems based on the Mason method is formalized. The components of the transfer function are presented in the form of characteristic polynomials in the structural scheme of the energy-information model of the circuit. The results of a comparative analysis of search methods for multiple real and complex conjugate polynomial roots are presented. To organize their search, an iterative method of unconditional optimization of Fletcher-Reeves was chosen. This conjugate gradient method allows to solve the problem of numerical optimization in a finite number of steps and shows the best convergence in comparison with the methods of the fastest descent, with the same order of difficulty of performing the steps of the algorithm.

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Allgower, Eugene L., and Kurt Georg. "Continuation and path following." Acta Numerica 2 (January 1993): 1–64. http://dx.doi.org/10.1017/s0962492900002336.

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The main ideas of path following by predictor–corrector and piecewise-linear methods, and their application in the direction of homotopy methods and nonlinear eigenvalue problems are reviewed. Further new applications to areas such as polynomial systems of equations, linear eigenvalue problems, interior methods for linear programming, parametric programming and complex bifurcation are surveyed. Complexity issues and available software are also discussed.

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Son, Jeongeun, Dongping Du, and Yuncheng Du. "Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems." Applied Mechanics 1, no. 3 (August 22, 2020): 153–73. http://dx.doi.org/10.3390/applmech1030011.

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Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive because it often involves high-dimensional integrals. To address this, a generalized dimension reduction method (gDRM) is coupled with quadrature rules to convert a high-dimensional integral in the SG into a few lower dimensional ones that can be rapidly solved. The performance of the algorithm is validated with two examples describing the dynamic behavior of cells. Compared to other UQ techniques (e.g., nonintrusive PCE), the results show the potential of the algorithm to tackle UQ in more complicated biological systems.

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COHEN, ALBERT, RONALD DEVORE, and CHRISTOPH SCHWAB. "ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S." Analysis and Applications 09, no. 01 (January 2011): 11–47. http://dx.doi.org/10.1142/s0219530511001728.

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Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space [Formula: see text] of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number N dof of degrees of freedom is the minimum of the convergence rates afforded by the best N-term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.

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40

Sánchez, Marcelino, and Miguel Bernal. "LMI–Based Robust Control of Uncertain Nonlinear Systems via Polytopes of Polynomials." International Journal of Applied Mathematics and Computer Science 29, no. 2 (June 1, 2019): 275–83. http://dx.doi.org/10.2478/amcs-2019-0020.

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Abstract This investigation is concerned with robust analysis and control of uncertain nonlinear systems with parametric uncertainties. In contrast to the methodologies from the field of linear parameter varying systems, which employ convex structures of the state space representation in order to perform analysis and design, the proposed approach makes use of a polytopic form of a generalisation of the characteristic polynomial, which proves to outperform former results on the subject. Moreover, the derived conditions have the advantage of being cast as linear matrix inequalities under mild assumptions.

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41

Kopchak, Bohdan, Andrii Kushnir, Inna Onoshko, and Sergiy Vovk. "Synthesis of two-mass electromechanical systems with cascade connection of fractional-order controllers." Eastern-European Journal of Enterprise Technologies 6, no. 5 (126) (December 28, 2023): 26–35. http://dx.doi.org/10.15587/1729-4061.2023.293206.

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The research focused on two-mass electromechanical systems widely utilized in industry. The challenge addressed in this work was to improve the synthesis of controllers for such systems to simplify it and enhance the quality of transition processes. Traditionally, the synthesis of control system loops for these systems was carried out using integer controllers and standard forms. However, this approach led to the synthesis of complex integer controllers that are difficult to implement. To overcome this issue, an original approach to the synthesis of control system loops based on the fractional characteristic polynomial is proposed. The fractional characteristic polynomial ensures the desired quality of the transition process given the implementation of a specified structure of the fractional controller. A new method of structural-parametric synthesis of fractional-order controllers is developed for the case of their cascade connection in multi-loop two-mass electromechanical systems. Additionally, an algorithm for synthesizing fractional-order controllers for the corresponding control loops is presented. This enabled the structural-parametric synthesis of fractional-order controllers for a two-mass electromechanical system with the cascade connection of controllers. Such an approach provides better quality of transition processes compared to classical integer controllers, simplifies the synthesis, and thereby enhances the quality of the synthesized systems. The impact of the synthesized fractional-order controllers using the proposed approach on the dynamic properties of the two-mass «thyristor converter – motor» system was investigated. The research results demonstrated the practical applicability of fractional controllers designed using the proposed method for the synthesis of automatic control systems of two-mass electromechanical systems in the industry.

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Peruzzi, NJ, FR Chavarette, JM Balthazar, AM Tusset, ALPM Perticarrari, and RMFL Brasil. "The dynamic behavior of a parametrically excited time-periodic MEMS taking into account parametric errors." Journal of Vibration and Control 22, no. 20 (August 8, 2016): 4101–10. http://dx.doi.org/10.1177/1077546315573913.

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Micro-electromechanical systems (MEMS) are micro scale devices that are able to convert electrical energy into mechanical energy or vice versa. In this paper, the mathematical model of an electronic circuit of a resonant MEMS mass sensor, with time-periodic parametric excitation, was analyzed and controlled by Chebyshev polynomial expansion of the Picard interaction and Lyapunov-Floquet transformation, and by Optimal Linear Feedback Control (OLFC). Both controls consider the union of feedback and feedforward controls. The feedback control obtained by Picard interaction and Lyapunov-Floquet transformation is the first strategy and the optimal control theory the second strategy. Numerical simulations show the efficiency of the two control methods, as well as the sensitivity of each control strategy to parametric errors. Without parametric errors, both control strategies were effective in maintaining the system in the desired orbit. On the other hand, in the presence of parametric errors, the OLFC technique was more robust.

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Xing, Wei W., Ming Cheng, Kaiming Cheng, Wei Zhang, and Peng Wang. "InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients." Materials 14, no. 13 (June 29, 2021): 3635. http://dx.doi.org/10.3390/ma14133635.

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Composition-dependent interdiffusion coefficients are key parameters in many physical processes. However, finding such coefficients for a system with few components is challenging due to the underdetermination of the governing diffusion equations, the lack of data in practice, and the unknown parametric form of the interdiffusion coefficients. In this work, we propose InfPolyn, Infinite Polynomial, a novel statistical framework to characterize the component-dependent interdiffusion coefficients. Our model is a generalization of the commonly used polynomial fitting method with extended model capacity and flexibility and it is combined with the numerical inversion-based Boltzmann–Matano method for the interdiffusion coefficient estimations. We assess InfPolyn on ternary and quaternary systems with predefined polynomial, exponential, and sinusoidal interdiffusion coefficients. The experiments show that InfPolyn outperforms the competitors, the SOTA numerical inversion-based Boltzmann–Matano methods, with a large margin in terms of relative error (10× more accurate). Its performance is also consistent and stable, whereas the number of samples required remains small.

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Maillet, Denis, Célien Zacharie, and Benjamin Rémy. "Identification of an impulse response through a model of ARX structure." Journal of Physics: Conference Series 2444, no. 1 (February 1, 2023): 012002. http://dx.doi.org/10.1088/1742-6596/2444/1/012002.

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Abstract Polynomial parametric models of ARX structure are becoming increasingly popular for characterizing heat transfer for linear thermal systems with time invariant coefficients. This stems from their robustness when applied to inverse problems, either for model reduction, for experimental model identification or for inverse input problems. Their parsimonious character allows to get residuals of very low levels with a limited number of coefficients. This paper shows, on a theoretical algebraic basis, that ARX models can be deduced from convolutive models.

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Li, Xiaoxue, and Xiaorong Hou. "Design of Parametric Controller for Two-Dimensional Polynomial Systems Described by the Fornasini-Marchesini Second Model." IEEE Access 7 (2019): 44070–79. http://dx.doi.org/10.1109/access.2019.2906272.

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Li, G., B. S. Mordukhovich, T. T. A. Nghia, and T. S. Phạm. "Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates." Mathematical Programming 168, no. 1-2 (April 23, 2016): 313–46. http://dx.doi.org/10.1007/s10107-016-1014-6.

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Shen, Dongying E., Sergio Lucia, Yiming Wan, Rolf Findeisen, and Richard D. Braatz. "Polynomial Chaos-Based H 2 -optimal Static Output Feedback Control of Systems with Probabilistic Parametric Uncertainties." IFAC-PapersOnLine 50, no. 1 (July 2017): 3536–41. http://dx.doi.org/10.1016/j.ifacol.2017.08.949.

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Mykhailenko, Oleksii. "Modeling and simulating dynamics of lithium-ion batteries using block-oriented models with piecewise linear static nonlinearity." E3S Web of Conferences 280 (2021): 05004. http://dx.doi.org/10.1051/e3sconf/202128005004.

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The article deals with the research of the efficiency of modelling the dynamics of voltage change in lithium-ion rechargeable batteries in charging/discharging modes using nonlinear block-oriented systems. Drawing on experimental data, a structural and parametric identification of the Hammerstein, Wiener and Hammerstein-Wiener models with a polynomial structure of the linear dynamic block and piecewise linear static nonlinearities was performed. It has been established that the best modelling accuracy was ensured by using the Hammerstein-Wiener system with a linear model having the 6th order of the numerator and denominator polynomials and an input delay of 3 samples. It showed 15.67% and 6.2% higher accuracy compared to the Wiener and Hammerstein systems, respectively. The application of those models in battery management systems will make it possible to improve the control quality for battery assemblies of solar and wind power plants in the context of the variable nature of the charging/discharging processes due to the variability of weather conditions and fluctuations in power consumption during a 24-hour period. This will ensure a wider introduction of renewable power generation into existing power systems, which is currently the leading way to ensure sustainable development of the energy sector.

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Kantarakias, Kyriakos Dimitrios, and George Papadakis. "Application of Generalized Polynomial Chaos for Quantification of Uncertainties of Time Averages and Their Sensitivities in Chaotic Systems." Algorithms 13, no. 4 (April 13, 2020): 90. http://dx.doi.org/10.3390/a13040090.

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In this paper, we consider the effect of stochastic uncertainties on non-linear systems with chaotic behavior. More specifically, we quantify the effect of parametric uncertainties to time-averaged quantities and their sensitivities. Sampling methods for Uncertainty Quantification (UQ), such as the Monte–Carlo (MC), are very costly, while traditional methods for sensitivity analysis, such as the adjoint, fail in chaotic systems. In this work, we employ the non-intrusive generalized Polynomial Chaos (gPC) for UQ, coupled with the Multiple-Shooting Shadowing (MSS) algorithm for sensitivity analysis of chaotic systems. It is shown that the gPC, coupled with MSS, is an appropriate method for conducting UQ in chaotic systems and produces results that match well with those from MC and Finite-Differences (FD).

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Kürkçü, Ömür Kıvanç, and Mehmet Sezer. "Charlier Series Solutions of Systems of First Order Delay Differential Equations with Proportional and Constant Arguments." Scientific Research Communications 2, no. 1 (January 30, 2022): 1–11. http://dx.doi.org/10.52460/src.2022.004.

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This study is devoted to obtaining the Charlier series solutions of first order delay differential equations involving proportional and constant arguments by employing an inventive numerical method dependent upon a collaboration of matrix structures derived from the parametric Charlier polynomial. The method essentially conducts the conversion of the unknown terms into a unique matrix equation at the collocation points, which yields a direct computation for these stiff equations. Two illustrative examples are included to test the accuracy and efficiency of the method. According to the investigation of the graphical and numerical results, the method holds fast, inventive and accurate computation, regularizing the matrix forms in compliance with the equations in question.

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Journal articles on the topic 'Parametric polynomial systems'

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