Journal articles: 'Système polynomiaux invariants'

1

Zakharov, Victor G. "Reproducing solutions to PDEs by scaling functions." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 03 (February 7, 2020): 2050017. http://dx.doi.org/10.1142/s0219691320500174.

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A generalization of the multivariate Strang–Fix conditions to no scale-invariant (only shift-invariant) polynomial spaces multiplied by exponents is introduced. A method to construct nonstationary compactly supported interpolating scaling functions that the scaling functions reproduce polynomials multiplied by exponents is presented. The polynomials (multiplied by exponents) are solutions to systems of linear constant coefficient PDEs, where the symbols of the differential operators that define PDEs can be no scale-invariant and can contain constant terms. Analytically calculated graphs of the scaling functions, including nonstationary, are presented. A concept of the so-called [Formula: see text]-separate MRAs is considered; and it is shown that, in the case of isotropic dilation matrices, the [Formula: see text]-separate scaling functions appear naturally.

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Wang, Maw-Ling, Shwu-Yien Yang, and Rong-Yeu Chang. "Application of Generalized Orthogonal Polynomials to Parameter Estimation of Time-Invariant and Bilinear Systems." Journal of Dynamic Systems, Measurement, and Control 109, no. 1 (March 1, 1987): 7–13. http://dx.doi.org/10.1115/1.3143824.

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Generalized orthogonal polynomials (GOP) which can represent all types of orthogonal polynomials and nonorthogonal Taylor series are first introduced to estimate the parameters of a dynamic state equation. The integration operation matrix (IOP) and the differentiation operation matrix (DOP) of the GOP are derived. The key idea of deriving IOP and DOP of these polynomials is that any orthogonal polynomial can be expressed by a power series and vice versa. By employing the IOP approach to the identification of time invariant systems, algorithms for computation which are effective, simple and straightforward compared to other orthogonal polynomial approximations can be obtained. The main advantage of using the differentiation operation matrix is that the parameter estimation can be carried out starting at an arbitrary time of interest. In addition, the computational algorithm is even simpler than that of the integral operation matrix. Illustrative examples for using IOP and DOP approaches are given. Very satisfactory results are obtained.

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Kiritsis, Konstadinos H. "Pole Assignment by Proportional-plus-derivative State Feedback for Multivariable Linear Time-invariant Systems." WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL 17 (June 16, 2022): 262–68. http://dx.doi.org/10.37394/23203.2022.17.30.

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In this paper the pole assignment problem by proportional-plus-derivative state feedback for multivariable linear-time invariant systems is studied. In particular, explicit necessary and sufficient conditions are established for a given polynomial with real coefficients to be characteristic polynomial of closed-loop system obtained by proportional-plus-derivative state feedback from the given multivariable linear time-invariant system. A procedure is given for the calculation of proportional-plus-derivative state feedback which places the poles of closed-loop system at desired locations. Our approach is based on properties of polynomial matrices.

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Vajda, S. "Deterministic identifiability and algebraic invariants for polynomial systems." IEEE Transactions on Automatic Control 32, no. 2 (February 1987): 182–84. http://dx.doi.org/10.1109/tac.1987.1104546.

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TAN, SHAOHUA, and JOOS VANDEWALLE. "Generalized invariant polynomials and the generalized companion form." International Journal of Control 45, no. 3 (March 1987): 811–16. http://dx.doi.org/10.1080/00207178708933771.

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Kiritsis, Konstadinos H. "Stabilization of Linear Time-Invariant Systems by State-Derivative Feedback." WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL 18 (March 24, 2023): 65–72. http://dx.doi.org/10.37394/23203.2023.18.7.

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In this paper is studied the stabilization problem by state-derivative feedback for linear time-invariant continuous-time systems. In particular, explicit necessary and sufficient conditions are established for the stability of a closed-loop system, obtained by state-derivative feedback from the given linear time-invariant continuous-time system. Furthermore a procedure is given for the computation of stabilizing state-derivative feedback. Our approach is based on properties of real and polynomial matrices.

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7

WEI, KEHUl, and B. ROSS BARMLSH. "Making a polynomial Hurwitz-invariant by choice of feedback gains†." International Journal of Control 50, no. 4 (October 1989): 1025–38. http://dx.doi.org/10.1080/00207178908953414.

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8

Mohan, B. M., and K. B. Datta. "Lumped and Distributed Parameter System Identification Via Shifted Legendre Polynomials." Journal of Dynamic Systems, Measurement, and Control 110, no. 4 (December 1, 1988): 436–40. http://dx.doi.org/10.1115/1.3152709.

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In this paper, one shot operational matrix for repeated integration of the shifted Legendre polynomial basis vector is developed and double-shifted Legendre series is introduced to approximate functions of two independent variables. Then using these, systematic algorithms for the identification of linear time-invariant single input-single output continuous lumped and distributed parameter systems are presented. Illustrative examples are provided with satisfactory results.

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Augusta, Petr, and Zdeněk Hurák. "Distributed stabilisation of spatially invariant systems: positive polynomial approach." Multidimensional Systems and Signal Processing 24, no. 1 (April 21, 2011): 3–21. http://dx.doi.org/10.1007/s11045-011-0152-5.

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10

Barabanov, A. E. "Invariance and polynomial design of strategies in the linear-quadratic game." Automation and Remote Control 67, no. 10 (October 2006): 1547–72. http://dx.doi.org/10.1134/s000511790610002x.

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11

CABRELLI, CARLOS A., SIGRID B. HEINEKEN, and URSULA M. MOLTER. "REFINABLE SHIFT INVARIANT SPACES IN ℝd." International Journal of Wavelets, Multiresolution and Information Processing 03, no. 03 (September 2005): 321–45. http://dx.doi.org/10.1142/s0219691305000889.

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Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝd is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [cAi-j]i,j∈Γ and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

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12

Lu, Andong, and Barmak Honarvar Shakibaei Honarvar Shakibaei Asli. "Seismic Image Identification and Detection Based on Tchebichef Moment Invariant." Electronics 12, no. 17 (August 31, 2023): 3692. http://dx.doi.org/10.3390/electronics12173692.

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The research focuses on the analysis of seismic data, specifically targeting the detection, edge segmentation, and classification of seismic images. These processes are fundamental in image processing and are crucial in understanding the stratigraphic structure and identifying oil and natural gas resources. However, there is a lack of sufficient resources in the field of seismic image detection, and interpreting 2D seismic image slices based on 3D seismic data sets can be challenging. In this research, image segmentation involves image preprocessing and the use of a U-net network. Preprocessing techniques, such as Gaussian filter and anisotropic diffusion, are employed to reduce blur and noise in seismic images. The U-net network, based on the Canny descriptor is used for segmentation. For image classification, the ResNet-50 and Inception-v3 models are applied to classify different types of seismic images. In image detection, Tchebichef invariants are computed using the Tchebichef polynomials’ recurrence relation. These invariants are then used in an optimized multi-class SVM network for detecting and classifying various types of seismic images. The promising results of the SVM model based on Tchebichef invariants suggest its potential to replace Hu moment invariants (HMIs) and Zernike moment invariants (ZMIs) for seismic image detection. This approach offers a more efficient and dependable solution for seismic image analysis in the future.

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13

LIANG, YANYAN, ZHANCHUAN CAI, DONGXU QI, and ZESHENG TANG. "SCALE-INVARIANT V-TRANSFORM AND ITS APPLICATION TO SIGNAL DE-NOISING." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 05 (September 2013): 1350038. http://dx.doi.org/10.1142/s0219691313500380.

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An aporia of signal de-noising is that the local signal reconstruction at the singular points. Based on the analysis for the signal singular points, combining signal scaling and orthogonal transform, This paper present a novel method called Scale-Invariant V-Transform (SIVT) for signal de-noising based on V-System, which is polynomial multi-wavelets in invariant set. SIVT employs multiple redundant basis of various scale to suppress the artifacts appearing in the singular points of denoised signal. The test results reveal the SIVT reconstructions exhibit higher visual quality and numerical measurement of SNR than wavelet-based reconstructions. Existing theory of SIVT suggests that these new approaches can perform significantly better than wavelet methods in certain signal reconstruction problems.

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14

Zhou, Bin, and Huaiyuan Jiang. "On the invariant factors and the minimal polynomial assignments of linear systems." Automatica 164 (June 2024): 111632. http://dx.doi.org/10.1016/j.automatica.2024.111632.

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15

Singh, Sunil K., and Lin Shi. "Robust Adaptive Controllers for Interconnected Mechanical Systems: Influence of Types of Interconnections on Time-Invariant and Time-Varying Systems." Journal of Dynamic Systems, Measurement, and Control 116, no. 3 (September 1, 1994): 456–73. http://dx.doi.org/10.1115/1.2899241.

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We investigate robust adaptive controller designs for interconnected systems when no exact knowledge about the structure of the nonlinear interconnections between various subsystems is available. In this study, we concentrate on several different types of systems. We deal with both linear time-invariant (LTI) and linear time-varying (LTV) systems with nonlinear interconnections. For LTI systems, we examine the following types of interconnections: • interconnections that are bounded by first order polynomials in state space; • slowly time varying interconnections; • interconnections bounded by higher-order polynomials in state-space together with input channel interconnections. For LTV systems we deal with interconnections bounded by first-order polynomials in state space. We show that the nature of the nonlinear interactions influences the adaptation laws. We use the direct method of Lyapunov for the design of adaptive controllers for tracking in such systems. We investigate issues such as stability, transient performance and steady-state errors, and derive quantitative estimates and analytical bounds for various different adaptive controllers. For time-varying systems, we analyze the effect of the time variations of parameters and interactions and propose a modified adaptive control scheme with better performance. Simulation results are presented to validate our conclusions. We also investigate these results experimentally on a two-link robot manipulator. Experimental results validate theoretical conclusions and demonstrate the usefulness of such robust adaptive controllers for high-speed motions in uncertain systems.

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16

Wei, K., and R. Yedavalli. "Invariance of a strict Hurwitz property for uncertain polynomials with dependent coefficients." IEEE Transactions on Automatic Control 32, no. 10 (October 1987): 907–9. http://dx.doi.org/10.1109/tac.1987.1104466.

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17

Ichihara, Hiroyuki, Kenji Sawada, and Sophie Tarbouriech. "Invariant set analysis for SISO discrete-time polynomial systems with dynamic quantizers." International Journal of Robust and Nonlinear Control 28, no. 17 (September 6, 2018): 5495–508. http://dx.doi.org/10.1002/rnc.4326.

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18

Chyung, Dong H. "Tracking Controller for Linear Systems With Time Delays in Both State and Control Variables." Journal of Dynamic Systems, Measurement, and Control 115, no. 1 (March 1, 1993): 179–83. http://dx.doi.org/10.1115/1.2897394.

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Linear time invariant control systems with time delays in both state and control variables are studied. The problem is to design a controller so that the system response will follow a given reference input with zero steady-state error. The reference input is assumed to be a polynomial function of time. The controller is based on feeding forward the reference input and its delayed values. A method is derived for determining the feed-forward gain matrices. Necessary and sufficient condition for the existence of proposed tracking controller is also obtained. An example is given to illustrate the proposed method.

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19

Santos, Welington, and Marcelo Muniz Alves. "Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric." IEEE Transactions on Information Theory 66, no. 7 (July 2020): 4061–74. http://dx.doi.org/10.1109/tit.2020.2971989.

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20

Chen, Yiwen, Qijuan Jie, Yiqiao Zhang, Feng Shu, Xichao Zhan, Shihao Yan, Wenlong Cai, et al. "Two Rapid Power Iterative DOA Estimators for UAV Emitter Using Massive/Ultra-Massive Receive Array." Drones 7, no. 6 (May 30, 2023): 361. http://dx.doi.org/10.3390/drones7060361.

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To provide rapid direction finding (DF) for unmanned aerial vehicle (UAV) emitters in future wireless networks, a low-complexity direction of arrival (DOA) estimation architecture for massive multiple-input multiple-output (MIMO) receiver arrays is constructed. In this paper, we propose two strategies to address the extremely high complexity caused by eigenvalue decomposition of the received signal covariance matrix. Firstly, a rapid power iterative rotational invariance (RPI-RI) method is proposed, which adopts the signal subspace generated by power iteration to obtain the final direction estimation through rotational invariance between subarrays. RPI-RI causes a significant complexity reduction at the cost of a substantial performance loss. In order to further reduce the complexity and provide good directional measurement results, the rapid power iterative polynomial rooting (RPI-PR) method is proposed, which utilizes the noise subspace combined with the polynomial solution method to obtain the optimal direction estimation. In addition, the influence of initial vector selection on convergence in the power iteration is analyzed, especially when the initial vector is orthogonal to the incident wave. Simulation results show that the two proposed methods outperform the conventional DOA estimation methods in terms of computational complexity. In particular, the RPI-PR method achieves more than two orders of magnitude lower complexity than conventional methods and achieves performance close to the Cramér–Rao Lower Bound (CRLB). Moreover, it is verified that the initial vector and the relative error have a significant impact on the performance with respect to the computational complexity.

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21

KARCANIAS†, NICOS. "Invariance properties, and characterization of the greatest common divisor of a set of polynomials." International Journal of Control 46, no. 5 (November 1987): 1751–60. http://dx.doi.org/10.1080/00207178708934007.

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22

BACCOU, J., and J. LIANDRAT. "POSITION-DEPENDENT LAGRANGE INTERPOLATING MULTIRESOLUTIONS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 04 (July 2007): 513–39. http://dx.doi.org/10.1142/s0219691307001884.

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This paper is devoted to the construction of interpolating multiresolutions using Lagrange polynomials and incorporating a position dependency. It uses the Harten's framework21 and its connection to subdivision schemes. Convergence is first emphasized. Then, plugging the various ingredients into the wavelet multiresolution analysis machinery, the construction leads to position-dependent interpolating bases and multi-scale decompositions that are useful in many instances where classical translation-invariant frameworks fail. A multivariate generalization is proposed and analyzed. We investigate applications to the reduction of the so-called Gibbs phenomenon for the approximation of locally discontinuous functions and to the improvement of the compression of locally discontinuous 1D signals. Some applications to image decomposition are finally presented.

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Yuno, Tsuyoshi, Eva Zerz, and Toshiyuki Ohtsuka. "Invariance of a class of semi-algebraic sets for polynomial systems with dynamic compensators." Automatica 122 (December 2020): 109243. http://dx.doi.org/10.1016/j.automatica.2020.109243.

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Vasu, G., M. Sivakumar, and M. Ramalingaraju. "A novel model reduction approach for linear time-invariant systems via enhanced PSO-DV algorithm and improved MPPA method." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 234, no. 2 (May 23, 2019): 240–56. http://dx.doi.org/10.1177/0959651819849372.

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In this article, the combination of stochastic search and conventional approaches are used to develop an optimal frequency-domain model order reduction method for determining the stable and accurate reduced-order model for the stable large-scale linear time-invariant systems. The method uses the enhanced particle swarm optimization with differentially perturbed velocity algorithm to determine the denominator polynomial coefficients of the reduced-order model, whereas the numerator polynomial coefficients of the reduced-order model are determined by using an improved multi-point Padé approximation method. The method generates an optimum reduced-order model by minimizing an objective function [Formula: see text], which is formulated using two functions. The first function, [Formula: see text], evaluates the measure of integral squared error between the step responses of the original system and the reduced-order model. And the second function evaluates the measure of retention of full impulse response energy of the original system in the reduced-order model. Therefore, by minimizing the objective function ‘ E’, the proposed method is guaranteed for preserving passivity, stability and the accuracy of the original higher order system in the reduced-order model. The proposed method is extended to the linear time-invariant multi-input multi-output system. In this case, an optimal reduced-order model is determined by minimizing a single objective function [Formula: see text], which is formulated by linear scalarizing of all the objective function [Formula: see text] components. The method is popular for preserving stability, passivity and accuracy of the original system in the reduced-order model. The validation of the method is shown by applying to a sixth-order single-input single-output hydropower system model as well as to the seventh-order two-area multi-input multi-output power system model. The comparison of the simulation results of integral squared error and impulse response energy values of the reduced-order models demonstrates the dominance of the proposed method than the existing reduction methods available in the literature.

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CHANG, RONG-YEU, KUN-CHOU CHEN, and MAW-LING WANG. "A new approach to the parameter estimation of linear time-invariant delayed systems via modified Laguerre polynomials." International Journal of Systems Science 16, no. 12 (December 1985): 1505–15. http://dx.doi.org/10.1080/00207728508926770.

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26

Niederwieser, Helmut, and Markus Reichhartinger. "On the characteristic polynomial of the dynamic matrix of linear time-invariant multivariable systems in Luenberger’s canonical forms." Automatica 162 (April 2024): 111532. http://dx.doi.org/10.1016/j.automatica.2024.111532.

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Coutinho, Daniel F., Daniel J. Pagano, and Alexandre Trofino. "On the estimation of robust stability regions for nonlinear systems with saturation." Sba: Controle & Automação Sociedade Brasileira de Automatica 15, no. 3 (September 2004): 269–78. http://dx.doi.org/10.1590/s0103-17592004000300003.

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This paper addresses the problem of determining robust stability regions for a class of nonlinear systems with time-invariant uncertainties subject to actuator saturation. The unforced nonlinear system is represented by differential-algebraic equations where the system matrices are allowed to be rational functions of the state and uncertain parameters, and the saturation nonlinearity is modelled by a sector bound condition. For this class of systems, local stability conditions in terms of linear matrix inequalities are derived based on polynomial Lyapunov functions in which the Lyapunov matrix is a quadratic function of the state and uncertain parameters. To estimate a robust stability region is considered the largest level surface of the Lyapunov function belonging to a given polytopic region of state. A numerical example is used to demonstrate the approach.

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Tran, Duong Dinh, Kazuhiro Ogata, Santiago Escobar, Sedat Akleylek, and Ayoub Otmani. "Kyber, Saber, and SK-MLWR Lattice-Based Key Encapsulation Mechanisms Model Checking with Maude." IET Information Security 2023 (October 30, 2023): 1–17. http://dx.doi.org/10.1049/2023/9399887.

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Facing the potential threat raised by quantum computing, a great deal of research from many groups and industrial giants has gone into building public-key post-quantum cryptographic primitives that are resistant to the quantum attackers. Among them, there is a large number of post-quantum key encapsulation mechanisms (KEMs), whose purpose is to provide a secure key exchange, which is a very crucial component in public-key cryptography. This paper presents a formal security analysis of three lattice-based KEMs including Kyber, Saber, and SK-MLWR. We use Maude, a specification language supporting equational and rewriting logic and a high-performance tool equipped with many advanced features, such as a reachability analyzer that can be used as a model checker for invariant properties, to model the three KEMs as state machines. Because they all belong to the class of lattice-based KEMs, they share many common parts in their designs, such as polynomials, vectors, and message exchange patterns. We first model these common parts and combine them into a specification, called base specification. After that, for each of the three KEMs, by extending the base specification, we just need to model some additional parts and the mechanism execution. Once completing the three specifications, we conduct invariant model checkings with the Maude search command, pointing out a similar man-in-the-middle attack. The occurrence of this attack is due to the fact that authentication is not part of the KEMs, and therefore an active attacker can modify all communication between two honest parties.

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Kramar, Vadim, Aleksey Kabanov, and Vasiliy Alchakov. "Multiloop Multirate Continuous-Discrete Drone Stabilization System: An Equivalent Single-Rate Model." Drones 5, no. 4 (November 1, 2021): 129. http://dx.doi.org/10.3390/drones5040129.

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The article discusses the UAV lateral motion stabilization system, as a MIMO multiloop multirate continuous-discrete system, specified in the form of an input–output model in the domain of discrete Laplace transform or in the form of a structural diagram. Approaches to the construction of equivalent T and NT single-rate models for MIMO multiloop multirate continuous-discrete systems are considered. Here, T is the largest common divisor of the sampling periods of the system, N is a natural number that is the smallest common multiple of the numbers characterizing the sampling periods of the system. The resulting impulse representations of the outputs of equivalent models are in the form of rational functions. The basis for the construction of these models is a matrix of sampling densities—a structural invariant of sampling chains. An example of the construction of the indicated matrix and an equivalent single-rate model are given. Obtaining equivalent single-rate models for MIMO multiloop multirate systems allows us to extend the methods of research and synthesis of MIMO continuous and continuous-discrete systems to a common theoretical base—the theory of polynomials and rational functions, which are typical elements of the description of these classes of systems.

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Rescio, Gabriele, Alessandro Leone, and Pietro Siciliano. "Supervised Expert System for Wearable MEMS Accelerometer-Based Fall Detector." Journal of Sensors 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/254629.

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Falling is one of the main causes of trauma, disability, and death among older people. Inertial sensors-based devices are able to detect falls in controlled environments. Often this kind of solution presents poor performances in real conditions. The aim of this work is the development of a computationally low-cost algorithm for feature extraction and the implementation of a machine-learning scheme for people fall detection, by using a triaxial MEMS wearable wireless accelerometer. The proposed approach allows to generalize the detection of fall events in several practical conditions. It appears invariant to the age, weight, height of people, and to the relative positioning area (even in the upper part of the waist), overcoming the drawbacks of well-known threshold-based approaches in which several parameters need to be manually estimated according to the specific features of the end user. In order to limit the workload, the specific study on posture analysis has been avoided, and a polynomial kernel function is used while maintaining high performances in terms of specificity and sensitivity. The supervised clustering step is achieved by implementing an one-class support vector machine classifier in a stand-alone PC.

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Bonnet, Édouard, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. "Twin-width I: Tractable FO Model Checking." Journal of the ACM 69, no. 1 (February 28, 2022): 1–46. http://dx.doi.org/10.1145/3486655.

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Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

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Dudarenko, Natalia, and Nina Vunder. "A Priori Estimation of Potential Degeneration of Continuous Multichannel Dynamic Systems." SPIIRAS Proceedings 19, no. 1 (February 7, 2020): 104–27. http://dx.doi.org/10.15622/10.15622/sp.2020.19.1.4.

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The problem of a priory control of potential degeneration of continuous multichannel dynamic systems is considered in the paper. Degeneracy is a property of a system describing operability of a multichannel dynamic system together with the basic properties of stability, reliability and invariance to the changing conditions. An assessment of potential generation of a system and its configuration together with the interconnections and polynomial exogenous signal is proposed. Degeneration process of a multichannel dynamic systems is a process of the rank reducing of the linear operator of the system. This statement is a basic concept of the degeneration factors approach. Algebraic properties of the matrix of the system’s operator is considered, and the matrix is named as the criterion matrix. Degeneration factor is calculated with the singular values of the criterion matrix. The global degeneration factor is conditional number of the criterion matrix of a system. In contrast to previous solutions it is proposed to form the criterion matrix of a system with the resolvent of its state matrix. Deparameterization of the linear algebraic problem is realized by additive decomposition of the output vector of the system by derivatives of the exogenous signal, and the steady-state mode of the system is considered. The procedure of a priori estimation of degeneration of continuous multichannel dynamic systems is proposed. The ways to achieve the required value of degeneration of the criterion matrix of the system with the modal control methods are discussed. The paper is supported with examples.

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Gourary, Mark, Sergey Rusakov, Mikhail Zharov, and Sergey Ulyanov. "Computational Algorithms for Reducing Rational Transfer Functions’ Order." SPIIRAS Proceedings 19, no. 2 (April 23, 2020): 330–56. http://dx.doi.org/10.15622/sp.2020.19.2.4.

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A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at a given tolerance on the approximation error. Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained. To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function. The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed. Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.

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Hrushovski, Ehud, Joël Ouaknine, Amaury Pouly, and James Worrell. "On Strongest Algebraic Program Invariants." Journal of the ACM, August 8, 2023. http://dx.doi.org/10.1145/3614319.

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A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location. Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this paper we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

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He, Bing, Jun Liu, Tengfei Yang, Bin Xiao, and Yanguo Peng. "Quaternion fractional-order color orthogonal moment-based image representation and recognition." EURASIP Journal on Image and Video Processing 2021, no. 1 (May 17, 2021). http://dx.doi.org/10.1186/s13640-021-00553-7.

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AbstractInspired by quaternion algebra and the idea of fractional-order transformation, we propose a new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractional-order transformations, which extract only the global features from color images, our proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their geometric invariants in the representation and recognition of color images.

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van Iersel, Leo, Vincent Moulton, and Yukihiro Murakami. "Polynomial invariants for cactuses." Information Processing Letters, March 2023, 106394. http://dx.doi.org/10.1016/j.ipl.2023.106394.

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Saha, Premjit, Tarunraj Singh, and Gary F. Dargush. "Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos." Journal of Computational and Nonlinear Dynamics, July 17, 2021. http://dx.doi.org/10.1115/1.4051821.

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Abstract The focus of this paper is on the use of Polynomial Chaos for developing surrogate models for Differential Algebraic Equations with time-invariant uncertainties. Intrusive and non-intrusive approaches to synthesize Polynomial Chaos surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations, if the algebraic constraints are a function of the stochastic variable, some initial conditions of the differential algebraic equations are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a Polynomial Chaos based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the Polynomial Chaos models are validated using Monte Carlo simulations in addition to estimating the evolving PDFs of the states of the pendulum.

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Nandi, Souransu, and Tarunraj Singh. "Nonintrusive Global Sensitivity Analysis for Linear Systems With Process Noise." Journal of Computational and Nonlinear Dynamics 14, no. 2 (January 7, 2019). http://dx.doi.org/10.1115/1.4041622.

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The focus of this paper is on the global sensitivity analysis (GSA) of linear systems with time-invariant model parameter uncertainties and driven by stochastic inputs. The Sobol' indices of the evolving mean and variance estimates of states are used to assess the impact of the time-invariant uncertain model parameters and the statistics of the stochastic input on the uncertainty of the output. Numerical results on two benchmark problems help illustrate that it is conceivable that parameters, which are not so significant in contributing to the uncertainty of the mean, can be extremely significant in contributing to the uncertainty of the variances. The paper uses a polynomial chaos (PC) approach to synthesize a surrogate probabilistic model of the stochastic system after using Lagrange interpolation polynomials (LIPs) as PC bases. The Sobol' indices are then directly evaluated from the PC coefficients. Although this concept is not new, a novel interpretation of stochastic collocation-based PC and intrusive PC is presented where they are shown to represent identical probabilistic models when the system under consideration is linear. This result now permits treating linear models as black boxes to develop intrusive PC surrogates.

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Tapia-Flores, Tonatiuh, and Ernesto López-Mellado. "Discovering workflow nets of concurrent iterative processes." Acta Informatica, September 14, 2023. http://dx.doi.org/10.1007/s00236-023-00445-5.

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AbstractA novel and efficient method for discovering concurrent workflow processes is presented. It allows building a suitable workflow net (WFN) from a large event log $$\lambda $$ λ , which represents the behaviour of complex iterative processes involving concurrency. First, the t-invariants are determined from $$\lambda $$ λ ; this allows computing the causal and concurrent relations between the events and the implicit causal relations between events that do not appear consecutively in $$\lambda $$ λ . Then a 1-bounded WFN is built, which could be eventually adjusted if its t-invariants do not match with those computed from $$\lambda $$ λ . The discovered model allows firing all the traces in $$\lambda $$ λ . The procedures derived from the method are polynomial time on $$|\lambda |$$ | λ | ; they have been implemented and tested on artificial logs.

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Fradkov, Alexander, Irina Pchelkina, Mikhail Ananyevskiy, and Anatolii Tomchin. "Control of oscillations by control of invariants in quasi-polynomial nonlinear systems." Nonlinear Dynamics, June 18, 2023. http://dx.doi.org/10.1007/s11071-023-08566-9.

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Nandi, Souransu, Victor Migeon, Tarunraj Singh, and Puneet Singla. "Polynomial Chaos-Based Controller Design for Uncertain Linear Systems With State and Control Constraints." Journal of Dynamic Systems, Measurement, and Control 140, no. 7 (January 24, 2018). http://dx.doi.org/10.1115/1.4038800.

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For linear dynamic systems with uncertain parameters, design of controllers which drive a system from an initial condition to a desired final state, limited by state constraints during the transition is a nontrivial problem. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system.

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Leventides, J., N. Karcanias, and N. C. Poulios. "The notion of almost zeros and randomness." IMA Journal of Mathematical Control and Information, April 22, 2021. http://dx.doi.org/10.1093/imamci/dnab008.

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Abstract We investigate the problem of almost zeros of polynomial matrices as used in system theory. It is related to the controllability and observability notion of systems as well as the determination of Macmillan degree and complexity of systems. We also present some new results on this important invariant in the light of randomness and we prove an uncertainty type relation appearing in such ensembles of operators.

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Kumar, Manoj, Aman Jhinga, and J. T. Majithia. "Solutions of Time-Space Fractional PDEs Using Picard's Iterative Method." Journal of Computational and Nonlinear Dynamics, January 25, 2024, 1–23. http://dx.doi.org/10.1115/1.4064553.

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Abstract In this paper, we present Picard's iterative method for solving time-space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The present method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.

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Wei, Cui, Antonis Vardulakis, and Tianyou Chai. "Stabilization of the Pendubot: a polynomial matrix approach." IMA Journal of Mathematical Control and Information, May 10, 2023. http://dx.doi.org/10.1093/imamci/dnad011.

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Abstract This paper concerns the stabilization problem for an underactuated robot called the Pendubot. Relying on a computational algorithm which is based on various results of the ‘polynomial matrix approach’, we propose an output-feedback-based internally stabilizing controller to stabilize the Pendubot at the unstable vertical upright position. The algorithm utilizes results for the solution of polynomial matrix Diophantine equations required for the computation and parameterization of proper ‘denominator assigning’ and internally stabilizing controllers for linear time invariant multivariable systems and reduces the problem to that of the solution of a set of numerical linear equations. The controller presented uses only the measured output which consists of the angles of the two links and does not require knowledge of the angular velocities which are usually not directly measurable. Comparative simulations are carried out to verify the good performance of the proposed controller. Finally, experimental results are provided to demonstrate the validity and feasibility of the proposed method.

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Gerbet, Daniel, and Klaus Roebenack. "Application of LaSalle's Invariance Principle on Polynomial Differential Equations using Quantifier Elimination." IEEE Transactions on Automatic Control, 2021, 1. http://dx.doi.org/10.1109/tac.2021.3103887.

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Dabrowski-Tumanski, Pawel, Pawel Rubach, Wanda Niemyska, Bartosz Ambrozy Gren, and Joanna Ida Sulkowska. "Topoly: Python package to analyze topology of polymers." Briefings in Bioinformatics, September 16, 2020. http://dx.doi.org/10.1093/bib/bbaa196.

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Abstract The increasing role of topology in (bio)physical properties of matter creates a need for an efficient method of detecting the topology of a (bio)polymer. However, the existing tools allow one to classify only the simplest knots and cannot be used in automated sample analysis. To answer this need, we created the Topoly Python package. This package enables the distinguishing of knots, slipknots, links and spatial graphs through the calculation of different topological polynomial invariants. It also enables one to create the minimal spanning surface on a given loop, e.g. to detect a lasso motif or to generate random closed polymers. It is capable of reading various file formats, including PDB. The extensive documentation along with test cases and the simplicity of the Python programming language make it a very simple to use yet powerful tool, suitable even for inexperienced users. Topoly can be obtained from https://topoly.cent.uw.edu.pl.

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Blicha, Martin, Antti E. J. Hyvärinen, Jan Kofroň, and Natasha Sharygina. "Using linear algebra in decomposition of Farkas interpolants." International Journal on Software Tools for Technology Transfer, August 5, 2021. http://dx.doi.org/10.1007/s10009-021-00641-z.

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AbstractThe use of propositional logic and systems of linear inequalities over reals is a common means to model software for formal verification. Craig interpolants constitute a central building block in this setting for over-approximating reachable states, e.g. as candidates for inductive loop invariants. Interpolants for a linear system can be efficiently computed from a Simplex refutation by applying the Farkas’ lemma. However, these interpolants do not always suit the verification task—in the worst case, they can even prevent the verification algorithm from converging. This work introduces the decomposed interpolants, a fundamental extension of the Farkas interpolants, obtained by identifying and separating independent components from the interpolant structure, using methods from linear algebra. We also present an efficient polynomial algorithm to compute decomposed interpolants and analyse its properties. We experimentally show that the use of decomposed interpolants in model checking results in immediate convergence on instances where state-of-the-art approaches diverge. Moreover, since being based on the efficient Simplex method, the approach is very competitive in general.

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Pakkhesal, Sajjad, and Saeed Shamaghdari. "Policy iteration for H_∞ control of polynomial time‐varying systems." IET Control Theory & Applications, April 12, 2024. http://dx.doi.org/10.1049/cth2.12661.

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AbstractThis paper studies the H∞ control problem for polynomial time‐varying systems. The H∞ control problem has been much less investigated for time‐varying systems in comparison to the time‐invariant systems. Approximate dynamic programming (ADP) is an optimal method to solve the control problems. Therefore, it is valuable to solve the polynomial time‐varying H∞ control problem with the ADP approach. Considering the time as an independent variable for sum‐of‐squares (SOS) optimization problems, an SOS‐based ADP method is proposed to solve this problem. A policy iteration algorithm is presented, where in its policy evaluation step it is sufficient to solve an optimization problem. Some constraints are added to this optimization problem to guarantee the closed‐loop exponential stability. The convergence and stability properties of the proposed algorithm are stated and proven. Moreover, in order to design an H∞ controller with a smaller disturbance attenuation coefficient, a two‐loop algorithm is suggested. Finally, the effectiveness of the proposed method is demonstrated by simulation examples.

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Jun, Wang, Yang Zi-Jian, Zhang Yun-Hao, and Zhang Jian-Chao. "Energy Transfer and Dissipation in Combined-Stiffness Nonlinear Energy Sink Systems." Journal of Computational and Nonlinear Dynamics 19, no. 3 (December 26, 2023). http://dx.doi.org/10.1115/1.4064271.

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Abstract Nonlinear energy sinks (NES) are highly efficient vibration energy absorption and dissipation devices, and play an important vibration-suppression role in many types of structures. In this study, the influence of parameters on the combined stiffness nonlinear energy sink system is revealed from the perspective of energy, in which combined-stiffness terms are composed of piecewise linear stiffness and cubic stiffness. First, the slow-varying derivative of the combined-stiffness nonlinear energy sink system is calculated based on the complexification-averaging and multiscale methods. Second, an approximate expression for the extreme points on the slow-invariant manifold (SIM) of the system is derived by polynomial approximation, and the energy dissipation equation of the combined-stiffness nonlinear energy sink system is derived. The impacts of the stiffness gap, piecewise linear stiffness coefficient, and cubic stiffness coefficient on the system are analyzed by studying the energy transfer efficiency equation. Additionally, the relationship between the damping ratio of the primary structure and the dissipation time is analyzed.

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Grant Kirkland, W., and S. C. Sinha. "Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1." Journal of Computational and Nonlinear Dynamics 11, no. 4 (May 13, 2016). http://dx.doi.org/10.1115/1.4033382.

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Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) Φ(t,α), associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix Q(t,α) and a time-invariant matrix R(α) containing a set of symbolic system parameters α. Computation of Q(t,α) and R(α) in symbolic form as a function of α is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the Q(t,α) matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of Q(t,α) and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, R(α) is computed using a Gaussian quadrature integral formula. Finally, Q(t,α) is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

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Journal articles on the topic 'Système polynomiaux invariants'

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