Artykuły w czasopismach: „Trajectory convergence”

1

Ning, Tan, Zhang Yun-Jun, Ouyang Qi i Geng Zhi. "Dynamical Convergence Trajectory in Networks". Chinese Physics Letters 22, nr 9 (25.08.2005): 2447–50. http://dx.doi.org/10.1088/0256-307x/22/9/085.

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Deng, Jiao-Jiao, Hsiao-Dong Chiang i Tian-Qi Zhao. "Newton Method and Trajectory-Based Method for Solving Power Flow Problems: Nonlinear Studies". International Journal of Bifurcation and Chaos 25, nr 06 (15.06.2015): 1530018. http://dx.doi.org/10.1142/s0218127415300189.

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This paper analyzes the convergence properties and convergence region of a class of trajectory-based power flow methods. The convergence region of the trajectory-based method is a connected set and possesses the near-by property. The convergence region of Newton method and trajectory-based method in solving power flow problems are numerically investigated. Since the convergence region of trajectory-based method corresponds to the stability region of the nonlinear dynamic system, the stability regions of two dynamic systems are computed. The numerical results indicate that the stability region of the dynamic system possesses better geometry features than the convergence region of Newton method. These properties make the trajectory-based power flow method robust, especially on heavy loading conditions.

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Cui, Hong, i Chongwen Yu. "A Mathematical Model for Self-Twist Yarn Spinning Process". Research Journal of Textile and Apparel 18, nr 1 (1.02.2014): 1–5. http://dx.doi.org/10.1108/rjta-18-01-2014-b001.

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A mathematical model is established for self-twist spinning process by applying laws of fluid mechanics (i.e. mass conservation and momentum conservation). The motion curve of the convergent point was determined and the convergence point was decided by densities and velocities of two fed-in strands and the resultant self-twist yarn, independent of their mechanical properties. The experimental verification proves that the motion trajectory of the convergence point reaches a better agreement with the theoretical trajectory. The results provide general way to study the self-twist yarn spinning process and optimize the process parameters.

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Xia, Youshen, i Gang Feng. "On Convergence Conditions of an Extended Projection Neural Network". Neural Computation 17, nr 3 (1.03.2005): 515–25. http://dx.doi.org/10.1162/0899766053019926.

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The output trajectory convergence of an extended projection neural network was developed under the positive definiteness condition of the Jacobian matrix of nonlinear mapping. This note offers several new convergence results. The state trajectory convergence and the output trajectory convergence of the extended projection neural network are obtained under the positive semidefiniteness condition of the Jacobian matrix. Comparison and illustrative examples demonstrate applied significance of these new results.

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Purchase, Sharon, Christina Kum i Doina Olaru. "An analysis of technical and commercialization paths for an innovation trajectory". Journal of Business & Industrial Marketing 32, nr 6 (3.07.2017): 848–63. http://dx.doi.org/10.1108/jbim-06-2015-0111.

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Purpose The purpose of this study is to investigate sequences of event and the resulting innovation paths and trajectories followed by a university spin-off organization. Design/methodology/approach A single longitudinal case study methodology was applied to analyze innovation events and paths along the trajectory. Narrative methods were used to analyze actor perceptions on innovation processes/events. Findings The study categorizes events and paths in two categories, technical and commercialization, and finds that lock-in events matter for convergence of an innovation trajectory. The results indicate that understanding critical events may assist timely interventions in the innovation paths, thus potentially avoiding disruptions of the development of an innovation trajectory. The temporal processes reveal contrasting convergence–divergence patterns in the trajectory, depending on the types of events that occur. Research limitations/implications Using a single case data may limit the applicability of the findings, which calls for future research. Practical implications Industries could monitor the technical and commercialization paths as a strategy to reduce “vulnerability” of the innovation trajectory and possible negative impacts. Knowledge about the role of the CEO is key for a university spin-off organization. Originality/value This study presents a new typology of events and paths, identifies and characterizes lock-in events and shows the relatively fragile dexterity between convergent and divergent paths along an innovation trajectory.

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Xu, Liu, Wei Min Li, Lin Zhang i An Tang Zhang. "The Applications of Improved Genetic Algorithm on Hypersonic Vehicle Trajectory Optimization". Advanced Materials Research 466-467 (luty 2012): 1095–99. http://dx.doi.org/10.4028/www.scientific.net/amr.466-467.1095.

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The Optimal trajectory design for hypersonic cruise missile is an optimal control problem with strict terminal constraints and variable final time. The classical algorithms always encounter the problems of high sensitivity to initial guess and local convergence in solving this problem. Aiming at these problems, genetic algorithm (GA) which is of good global convergence is applied to designing the optimal trajectory for hypersonic cruise missile. In order to improve the convergence rate of GA and overcome its premature problems, this text introduces a predatory search (PS) strategy to speed the convergence of genetic algorithms, robust and closer to the optimal solution. This text compares the original genetic algorithm (GA) and improved genetic algorithm by the emulate experiments, and the results show that the PSGA is a more effective method to design the Optimal trajectory for hypersonic cruise missile than the original genetic algorithm.

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Gan, Wenyang, Tianxing Xia i Zhenzhong Chu. "A Prognosis Technique Based on Improved GWO-NMPC to Improve the Trajectory Tracking Control System Reliability of Unmanned Underwater Vehicles". Electronics 12, nr 4 (12.02.2023): 921. http://dx.doi.org/10.3390/electronics12040921.

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The dynamics model of the unmanned underwater vehicle (UUV) system is highly nonlinear, multi-degree-of-freedom, strongly coupled, and time-varying. Its motion control has been a complex problem due to the unknown information about and the uncertainty of the working environment. To improve the performance and reliability of UUV trajectory tracking control, a trajectory tracking method based on nonlinear model predictive control is designed, and an improved gray wolf optimization (IGWO) is proposed for the optimization of nonlinear model predictive control. The convergence factor of IGWO is designed as a nonlinear attenuation function, and the memory function is added to the position update equation to enhance the effect of trajectory tracking control. Through the simulation in the ROS environment, the influence of the convergence factor on the convergence rate of trajectory tracking error and tracking control performance is obtained. By comparing the tracking effects of several groups of reference trajectories, it is shown that the proposed method is universally applicable and effective to the trajectory tracking control of UUV. Compared with traditional gray wolf optimization (GWO), SQP, and other optimization algorithms, the reliability of the proposed method for UUV trajectory tracking control is demonstrated.

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Dubanov, A. A. "Kinematic model of the parallel convergence method in space". Journal of Physics: Conference Series 2182, nr 1 (1.03.2022): 012004. http://dx.doi.org/10.1088/1742-6596/2182/1/012004.

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Abstract In this article, the implementation of the method of parallel convergence in space in a computer mathematics system is considered and discussed. In this method, the pursuer’s velocity vector is directed arbitrarily. The pursuer’s trajectory gradually approaches movement in the plane formed by the line connecting the initial positions of the pursuer and the target, and the velocity vector. In this task, the target moves uniformly and rectilinearly. The pursuer moves evenly. The points of the pursuer’s trajectory are calculated sequentially. They are being the result of the intersection of the plane containing the line of sight, sphere and cone. As we approach the plane where the target is moving, the algorithm for calculating the trajectory points changes. Now the point of the pursuer’s trajectory is the result of the intersection of the sphere, the plane of movement of the target and the plane containing the line of sight.

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Ling, Guo, Du Weiwei, Yang Jiancheng, Wang Liping, Wan Ping i Liu Ling. "A Thermal Load Forecasting Algorithm Based on Trajectory Tracking". Mathematical Problems in Engineering 2020 (16.11.2020): 1–12. http://dx.doi.org/10.1155/2020/5919238.

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The heating load forecast provides a basis for saving heating energy. Considering the nonstationary, nonlinear, and large time-delay characteristics of thermal load, this paper introduces the trajectory tracking stability theory into the field of load forecasting and proposes a heuristic correction that can ensure the convergence of forecast errors and does not depend on the system prediction model algorithm. The Lyapunov method is used to derive an error convergence criterion that has nothing to do with the prediction model, and a heuristic correction algorithm is designed for the predicted value with error divergence trend to ensure the error convergence of the load forecast sequence.

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Cvetkovic, Bosko, i Mihailo Lazarevic. "Fractional-order iterative learning control for robotic Arm-PD2Dα type". Filomat 35, nr 1 (2021): 1–10. http://dx.doi.org/10.2298/fil2101001c.

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In this paper, a new open-loop PD2D? type a fractional order iterative learning control (ILC) is studied for joint space trajectory tracking control of a linearized uncertain robotic arm. The robust convergent analysis of the tracking errors has been done in time domain where it is theoretically proven that the boundednesses of the tracking error are guaranteed in the presence of model uncertainty. The convergence of the proposed open-loop ILC law is proven mathematically using Gronwall integral inequality for a linearized robotic system and sufficient conditions for convergence and robustness are obtained.

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Zhironkin, Sergey A., Vladimir V. Guzyr i Magerram A. Gasanov. "Economic convergence in the methodology of structural shifts". Vestnik Tomskogo gosudarstvennogo universiteta. Ekonomika, nr 58 (2022): 24–41. http://dx.doi.org/10.17223/19988648/58/2.

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Throughout the entire period of market transformations, the trend towards the distance of the Russian economy from technologically advanced countries has persisted, in terms of both the dynamics of the main macroeconomic indicators and structural proportions. This is due to the high sensitivity of the Russian economy to external shocks, including technological ones, as well as a number of successive negative structural shifts. At the same time, the commonality of endogenous processes, which affect the structure of factors of production and distribution of national income, in the economies of advanced and catching up countries is not fully associated with economic convergence. The studies are dominated by exogenous sources of convergence - sociopolitical, cultural. Therefore, the study of the role and place of structural shifts in the system of economic convergence is a methodological approach that allows one to come close to the theoretical foundations for stabilizing macroeconomic dynamics, reducing the susceptibility of the national economy to external shocks, and entering a long-term growth trajectory above world rates. In this article, the authors consider the problem of launching convergent structural shifts - the basis for the transition of the Russian economy from economic divergence to convergence with advanced countries, which was the basis for the aim of this study. Within the framework of this article, a methodology for analyzing structural shifts in their mutual influence on technological, structural and economic convergence is presented. The authors give a typology of convergent and divergent structural shifts that bring the national economy closer or further away from technologically advanced market systems. Such a typology of structural shifts implies the endogenous nature of economic convergence, due to a change in the proportions of accumulation and consumption, the involvement of production factors in the creation of national income and its distribution, contrary to the prevailing exogenous approach to the study of economic convergence. Based on the presented typology, the article highlights the factors of structural convergence -neo-industrial and fundamental investment, associated with a priority increase in funding for fundamental research, the creation of new innovation and investment partnerships between the state and business, and the modernization of basic industries based on convergent technologies. The authors show that these factors should be institutionalized in a system of convergence-oriented structural policy, which is a key condition for the transition of the Russian economy from structural divergence to convergence, with access to a sustainable growth trajectory and a reduction in exposure to external shocks.

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Rzezuchowski, Tadeusz. "Strong convergence of selections implied by weak". Bulletin of the Australian Mathematical Society 39, nr 2 (kwiecień 1989): 201–14. http://dx.doi.org/10.1017/s0004972700002677.

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In some situations weak convergence in L1, implies strong convergence. Let P, L: T → C∘(ℝd) be measurable multifunctions (C∘(ℝd) being the set of closed, convex subsets of ℝd) the values L(t) affine sets and W(t) = P(t) ∩ L(t) extremal faces of P(t). Let pk be integrable selections of P, the projection of pk,(t) on L(t) and pk(t) on W(t). We prove that if converges weakly to zero then pk − k converges to zero in measure. We give also some extensions of this theorem. As applications to differential inclusions we investigate convergence of derivatives of convergent sequences of solutions and we describe solutions which are in some sense isolated. Finally we discuss what can be said about control functions u when the corresponding trajectories of ẋ = f(t, x, u) are convergent to some trajectory.

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Huang, Hangjie, i Jinfeng Gao. "Backstepping and Novel Sliding Mode Trajectory Tracking Controller for Wheeled Mobile Robots". Mathematics 12, nr 10 (8.05.2024): 1458. http://dx.doi.org/10.3390/math12101458.

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A novel variable structure controller based on sliding mode is developed for addressing the trajectory tracking challenge encountered by wheeled mobile robots. Firstly, the trajectory tracking error model under the global coordinate system is established according to the kinematic model of the wheeled mobile robot. Secondly, the novel sliding mode algorithm and backstepping method are introduced to design the motion controller of the system, respectively. Different sliding mode surfaces are formulated to guarantee rapid and stable convergence of the system’s trajectory tracking error to zero. Ultimately, comparative simulation trials validate the controller’s ability to swiftly and consistently follow the reference trajectory. In contrast to traditional controllers, this controller shows rapid convergence, minimal error, and robustness.

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Sun, Hanqing, Shijie Zhang, Qingle Quan, Zheng Liu, Guizhi Wang i Weimin Lian. "Trajectory Tracking Robust Control Method Based on Finite-Time Convergence of Manipulator with Nonsingular Fast Terminal Sliding Mode Surface". Journal of Control Science and Engineering 2022 (7.04.2022): 1–9. http://dx.doi.org/10.1155/2022/2271804.

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In order to make the manipulator track the desired trajectory in a finite time, a new control method based on fast nonsingular terminal sliding mode has been designed. This method combines the traditional fast terminal sliding mode with the nonsingular terminal sliding mode, has rapidity, nonsingularity, finite-time convergence, and strong robustness, and can effectively suppress the inherent chattering phenomenon of the sliding mode controller. The nonsingular fast terminal sliding mode surface is used to accelerate the convergence speed of manipulator trajectory tracking error, and the singularity problem in terminal sliding mode is solved. The finite-time convergence of the algorithm is proved by the Lyapunov function. The simulation results demonstrate that the proposed method can achieve accurate finite-time trajectory tracking characteristics and has robustness against external disturbances.

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Liu, Junjie, Hui Wang, Xue Li, Kai Chen i Chaoyu Li. "Robotic arm trajectory optimization based on multiverse algorithm". Mathematical Biosciences and Engineering 20, nr 2 (2022): 2776–92. http://dx.doi.org/10.3934/mbe.2023130.

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<abstract><p>For inefficient trajectory planning of six-degree-of-freedom industrial manipulators, a trajectory planning algorithm based on an improved multiverse algorithm (IMVO) for time, energy, and impact optimization are proposed. The multi-universe algorithm has better robustness and convergence accuracy in solving single-objective constrained optimization problems than other algorithms. In contrast, it has the disadvantage of slow convergence and quickly falls into local optimum. This paper proposes a method to improve the wormhole probability curve, adaptive parameter adjustment, and population mutation fusion to improve the convergence speed and global search capability. In this paper, we modify MVO for multi-objective optimization to derive the Pareto solution set. We then construct the objective function by a weighted approach and optimize it using IMVO. The results show that the algorithm improves the timeliness of the six-degree-of-freedom manipulator trajectory operation within a specific constraint and improves the optimal time, energy consumption, and impact problems in the manipulator trajectory planning.</p></abstract>

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Bigras, Pascal, Maarouf Saad i Jules O'shea. "Convergence Analysis of an Inverse Flexible Manipulator Model Algorithm". Journal of Vibration and Control 9, nr 10 (październik 2003): 1141–58. http://dx.doi.org/10.1177/107754603030705.

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In this paper we present a convergence analysis of an iterative method to obtain the inverse model in the virtual joint space of a class of flexible manipulators. Virtual joints are defined so as to be related kinematically to the workspace. Thus, when the desired trajectory can be transformed from the workspace to the virtual joint space, the inverse model in the virtual joint space can be used to obtain a trajectory tip-tracking controller. Because the robot model in the virtual joint space is a non-minimum phase system, its inverse is unstable. The inverse method is then based on a causal-anticausal integration approach used in an iterative manner. The convergence analysis of this inverse algorithm is based on the contraction-mapping theorem and provides a very simple way to expand the desired trajectory to ensure its convergence. Finally, the inverse algorithm and its convergence analysis are validated by simulating a controller, based on the inverse model, that drives a two-link manipulator made up of one rigid and one flexible link. The simulation results demonstrate the good performance of the inverse algorithm and its convergence analysis presented here.

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Qiu, Shi Ying, Peng Yi, Rui Bo Yuan, Huan Yang, Sen Hui i Rong Ping Xue. "Trajectory Control for Cartesian Pneumatic Manipulator Using the Terminal Sliding Mode Control Method". Applied Mechanics and Materials 444-445 (październik 2013): 1354–59. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.1354.

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This paper applies the terminal sliding mode control method to control the trajectory of a three axises Cartesian Pneumatic Manipulator. A mathematical model of the pneumatic servo control system was established at first, then the terminal sliding mode control method was used for trajectory control. The simulation results shows that the tracking error of the terminal sliding mode control method become large only in the time period of not fully reaching the convergence point in time when the manipulator tracks the space straight line, whereas it can fully track the target trajectory after reaching the convergence point.

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Zhai, Guanyu, Jundong Zhang, Shuyun Wu i Yongkang Wang. "Predefined-Time Tracking Control of Unmanned Surface Vehicle under Complex Time-Varying Disturbances". Electronics 13, nr 8 (16.04.2024): 1510. http://dx.doi.org/10.3390/electronics13081510.

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Aiming at the unmanned surface vehicle (USV) trajectory tracking control under complex time-varying environment, a predefined-time convergence sliding mode disturbance observer (PTC-SMO) is introduced to effectively handle the internal parameter uncertainties and external environmental disturbances, thereby guaranteeing precise compensation of the lumped disturbance term within a set time. Then, in order to achieve precise tracking of the desired trajectory using USV under a predetermined time constraint, a novel fast trajectory tracking control strategy with predefined-time convergence (PTC-FTTCS) is established to improve tracking performance and ensure that the trajectory tracking error converges quickly in the predefined time. Through rigorous comparative simulation under ideal conditions and time-varying disturbances, the results demonstrate reliable trajectory tracking and disturbance handling effects, and the tracking performance and disturbance observation performance are significantly better than state-of-the-art methods.

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Eshmetova, S. D., i O. N. Khakimov. "On dynamics of infinite dimensional Volterra QSOs". UZBEK MATHEMATICAL JOURNAL 68, nr 2 (30.08.2024): 69–75. http://dx.doi.org/10.29229/uzmj.2024-2-8.

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In this paper, we investigate the behavior of infinite-dimensional Volterra quadratic stochastic operators corresponding to the negative upper triangular skew-symmetric matrix. It is proved that the trajectory of such operators cannot consist of convergent subsequences. This property immediately implies non-ergodicity of the operator. Moreover, it is shown that the operator's dynamics with respect to pointwise convergence are regular.

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Boţ, Radu Ioan, i Dang-Khoa Nguyen. "Improved convergence rates and trajectory convergence for primal-dual dynamical systems with vanishing damping". Journal of Differential Equations 303 (grudzień 2021): 369–406. http://dx.doi.org/10.1016/j.jde.2021.09.021.

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Xu, Huiyan, Guoqing Zhai i Xiaofan Li. "Precipitation Efficiency and Water Budget of Typhoon Fitow (2013): A Particle Trajectory Study". Journal of Hydrometeorology 18, nr 9 (15.08.2017): 2331–54. http://dx.doi.org/10.1175/jhm-d-16-0273.1.

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Abstract In this study, the WRF Model is used to simulate the torrential rainfall of Typhoon Fitow (2013) over coastal areas of east China during its landfall. Data from the innermost model domain are used to trace trajectories of particles in three major 24-h accumulated rainfall centers using the Lagrangian flexible particle dispersion model (FLEXPART). Surface rainfall budgets and cloud microphysical budgets as well as precipitation efficiency are analyzed along the particles’ trajectories. The rainfall centers with high precipitation efficiency are associated with water vapor convergence, condensation, accretion of cloud water by raindrops, and raindrop loss/convergence. The raindrop loss/convergence over rainfall centers is supported by the raindrop gain/divergence over the areas adjacent to rainfall centers. Precipitation efficiency is mainly determined by hydrometeor loss/convergence. Hydrometeor loss/convergence corresponds to the hydrometeor flux convergence, which may be related to the increased vertical advection of hydrometeors in response to the upward motions and upward decrease of hydrometeors, whereas hydrometeor gain/divergence corresponds to the reduction in hydrometeor flux convergence, which may be associated with the decreased horizontal advection of hydrometeors in response to the zonal decrease in hydrometeors and easterly winds and the meridional increase in hydrometeors and southerly winds. The water vapor convergence and associated condensation do not show consistent relationships with orographic lifting all the time.

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Hota, Sikha, i Debasish Ghose. "Optimal Trajectory Generation for Convergence to a Rectilinear Path". Journal of Intelligent & Robotic Systems 75, nr 2 (26.07.2013): 223–42. http://dx.doi.org/10.1007/s10846-013-9863-1.

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Seibert, Petra. "Convergence and Accuracy of Numerical Methods for Trajectory Calculations". Journal of Applied Meteorology 32, nr 3 (marzec 1993): 558–66. http://dx.doi.org/10.1175/1520-0450(1993)032<0558:caaonm>2.0.co;2.

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Ljubich, Yu I. "On trajectory convergence of dissipative flows in Banach spaces". Integral Equations and Operator Theory 13, nr 1 (styczeń 1990): 138–44. http://dx.doi.org/10.1007/bf01195297.

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Liang, Haizhao, Yunhao Luo, Haohui Che, Jingxian Zhu i Jianying Wang. "A Reentry Trajectory Planning Algorithm via Pseudo-Spectral Convexification and Method of Multipliers". Mathematics 12, nr 9 (25.04.2024): 1306. http://dx.doi.org/10.3390/math12091306.

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The reentry trajectory planning problem of hypersonic vehicles is generally a continuous and nonconvex optimization problem, and it constitutes a critical challenge within the field of aerospace engineering. In this paper, an improved sequential convexification algorithm is proposed to solve it and achieve online trajectory planning. In the proposed algorithm, the Chebyshev pseudo-spectral method with high-accuracy approximation performance is first employed to discretize the continuous dynamic equations. Subsequently, based on the multipliers and linearization methods, the original nonconvex trajectory planning problem is transformed into a series of relaxed convex subproblems in the form of an augmented Lagrange function. Then, the interior point method is utilized to iteratively solve the relaxed convex subproblem until the expected convergence precision is achieved. The convex-optimization-based and multipliers methods guarantee the promotion of fast convergence precision, making it suitable for online trajectory planning applications. Finally, numerical simulations are conducted to verify the performance of the proposed algorithm. The simulation results show that the algorithm possesses better convergence performance, and the solution time can reach the level of seconds, which is more than 97% less than nonlinear programming algorithms, such as the sequential quadratic programming algorithm.

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Li, Ningbo, Humin Lei, Lei Shao, Tao Liu i Bin Wang. "Trajectory Optimization Based on Multi-Interval Mesh Refinement Method". Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/8521368.

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In order to improve the optimization accuracy and convergence rate for trajectory optimization of the air-to-air missile, a multi-interval mesh refinement Radau pseudospectral method was introduced. This method made the mesh endpoints converge to the practical nonsmooth points and decreased the overall collocation points to improve convergence rate and computational efficiency. The trajectory was divided into four phases according to the working time of engine and handover of midcourse and terminal guidance, and then the optimization model was built. The multi-interval mesh refinement Radau pseudospectral method with different collocation points in each mesh interval was used to solve the trajectory optimization model. Moreover, this method was compared with traditional h method. Simulation results show that this method can decrease the dimensionality of nonlinear programming (NLP) problem and therefore improve the efficiency of pseudospectral methods for solving trajectory optimization problems.

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Zhu, Wenxing, Lihui Wang, Yuan Ren i Yong Li. "Trajectory Tracking of UAVs Using Sigmoid Tracking Differentiator and Variable Gain Finite-Time Extended State Observer". Drones 6, nr 11 (12.11.2022): 350. http://dx.doi.org/10.3390/drones6110350.

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The problem of quadrotor attitude and position control is considered in the presence of generally lumped disturbances: external disturbances and model uncertainty. The improved active disturbance rejection controller (ADRC) for quadrotor trajectory tracking is proposed for compensating the lumped disturbances. Firstly, the improved sigmoid tracking differentiator (ISTD), combining improved Sigmoid function and sliding mode terminal attractor is proposed, which can accelerate the global convergence rate and effectively reduce the chattering. Secondly, a novel variable gain finite-time extended state observer (VGFESO) approach is proposed to effectively estimate the lumped disturbances, while the observation errors are convergent to zero in finite time. Then, a super-twisting sliding model controller (STWSMC) is utilized for tracking control of the desired position and attitude. Finally, the convergence of VGFESO and the closed-loop stability of the control system are proved. The results show that the convergence time of the proposed control scheme is the shortest, and the integral absolute error of improved ADRC is reduced from 2.64 to 0.91. The anti-disturbance capability of the proposed controller is fully illustrated when compared with ADRC and robust adaptive nonsingular fast terminal sliding-mode controller (RANFTSMC).

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Pan, Huihui, i Guangming Zhang. "Adaptive Fast Nonsingular Fixed-Time Tracking Control for Robot Manipulators". Complexity 2021 (8.05.2021): 1–16. http://dx.doi.org/10.1155/2021/6629993.

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This paper studies the fixed-time trajectory tracking control problem of robot manipulators in the presence of uncertain dynamics and external disturbances. First, a novel nonsingular fixed-time sliding mode surface is presented, which can ensure that the convergence time of the suggested surface is bounded regardless of the initial states. Subsequently, a novel fast nonsingular fixed-time sliding mode control (NFNFSMC) is developed so that the closed-loop system is fixed-time convergent to the equilibrium. By applying the proposed NFNFSMC method and the adaptive technique, a novel adaptive nonsingular fixed-time control scheme is proposed, which can guarantee fast fixed-time convergence of the tracking errors to small regions around the origin. With the proposed control method, the lumped disturbance is compensated by the adaptive technique, whose prior information about the upper bound is not needed. The fixed-time stability of the trajectory tracking control under the proposed controller is proved by the Lyapunov stability theory. Finally, corresponding simulations are given to illustrate the validity and superiority of the proposed control approach.

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K. Wilburn, Brenton, Mario G. Perhinschi i Jennifer N. Wilburn. "A modified genetic algorithm for UAV trajectory tracking control laws optimization". International Journal of Intelligent Unmanned Systems 2, nr 2 (6.05.2014): 58–90. http://dx.doi.org/10.1108/ijius-03-2014-0002.

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Purpose – The purpose of this paper is to gain trajectory-tracking controllers for autonomous aircraft are optimized using a modified evolutionary, or genetic algorithm (GA). Design/methodology/approach – The GA design utilizes real representation for the individual consisting of the collection of all controller gains subject to tuning. The initial population is generated randomly over pre-specified ranges. Alternatively, initial individuals are produced as random variations from a heuristically tuned set of gains to increase convergence time. A two-point crossover mechanism and a probabilistic mutation mechanism represent the genetic alterations performed on the population. The environment is represented by a performance index (PI) composed of a set of metrics based on tracking error and control activity in response to a commanded trajectory. Roulette-wheel selection with elitist strategy are implemented. A PI normalization scheme is also implemented to increase the speed of convergence. A flexible control laws design environment is developed, which can be used to easily optimize the gains for a variety of unmanned aerial vehicle (UAV) control laws architectures. Findings – The performance of the aircraft trajectory-tracking controllers was shown to improve significantly through the GA optimization. Additionally, the novel normalization modification was shown to encourage more rapid convergence to an optimal solution. Research limitations/implications – The GA paradigm shows much promise in the optimization of highly non-linear aircraft trajectory-tracking controllers. The proposed optimization tool facilitates the investigation of novel control architectures regardless of complexity and dimensionality. Practical implications – The addition of the evolutionary optimization to the WVU UAV simulation environment enhances significantly its capabilities for autonomous flight algorithm development, testing, and evaluation. The normalization methodology proposed in this paper has been shown to appreciably speed up the convergence of GAs. Originality/value – The paper provides a flexible generalized framework for UAV control system evolutionary optimization. It includes specific novel structural elements and mechanisms for improved convergence as well as a comprehensive PI for trajectory tracking.

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Wang, Feng Bo, i Chang Hong Dong. "Coevolutionary Algorithm Applied to Skip Reentry Trajectory Optimization Design". Applied Mechanics and Materials 427-429 (wrzesień 2013): 1424–31. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.1424.

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This paper proposed a coevolutionary algorithm combining improved particle swarm optimization algorithm with differential evolution method and its application was provided. Adaptive position escapable mechanism is introduced in the particle swarm optimization to improve the diversity of population and guarantee to achieve the global optima. The differential algorithm is employed in a cooperative manner to maintain the characteristic of fast convergence speed in the later convergence phase. The coevolutionary algorithm is then applied to skip trajectory optimization design for crew exploration vehicle with low-lift-to-drag and several comparative cases are conducted, Results show that coevolutionary algorithm is quite effective in finding the global optimal solution with great accuracy.

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Xie, Jian, Yongquan Zhou i Huan Chen. "A Novel Bat Algorithm Based on Differential Operator and Lévy Flights Trajectory". Computational Intelligence and Neuroscience 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/453812.

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Aiming at the phenomenon of slow convergence rate and low accuracy of bat algorithm, a novel bat algorithm based on differential operator and Lévy flights trajectory is proposed. In this paper, a differential operator is introduced to accelerate the convergence speed of proposed algorithm, which is similar to mutation strategy “DE/best/2” in differential algorithm. Lévy flights trajectory can ensure the diversity of the population against premature convergence and make the algorithm effectively jump out of local minima. 14 typical benchmark functions and an instance of nonlinear equations are tested; the simulation results not only show that the proposed algorithm is feasible and effective, but also demonstrate that this proposed algorithm has superior approximation capabilities in high-dimensional space.

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An, Zhenzhou, Xiaoyan Wang i Xinling Shi. "A Study on the Convergence of Family Particle Swarm Optimization". Mathematical Problems in Engineering 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/6129865.

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The sociological concept of family has been introduced in the particle swarm optimization (PSO) and the family PSO (FPSO) has been proposed, in which the particle swarm consisted of different families, each family consisted of different members, and there were different constraint relationships between family members. To further study the sensitivity of FPSO to the control parameters, this paper proposed a special model of FPSO and analyzed the convergence of FPSO theoretically. This model offered a new view to research the particle trajectory and divided the position sequence of particle into the even and odd subsequences. By mathematical analysis, the condition of two subsequences convergence was obtained and the related convergent theories and corollaries were proved. Simulations for benchmark functions showed that the convergence behavior of model and experimental results provided a valuable guideline for selecting control parameters.

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Chen, Shuo, Jiaying Peng, Xiaolong Li i Yao Zhao. "On the Unstable Convergence Regime of Gradient Descent". Proceedings of the AAAI Conference on Artificial Intelligence 38, nr 10 (24.03.2024): 11373–80. http://dx.doi.org/10.1609/aaai.v38i10.29017.

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Traditional gradient descent (GD) has been fully investigated for convex or L-smoothness functions, and it is widely utilized in current neural network optimization. The classical descent lemma ensures that for a function with L-smoothness, the GD trajectory converges stably towards the minimum when the learning rate is below 2 / L. This convergence is marked by a consistent reduction in the loss function throughout the iterations. However, recent experimental studies have demonstrated that even when the L-smoothness condition is not met, or if the learning rate is increased leading to oscillations in the loss function during iterations, the GD trajectory still exhibits convergence over the long run. This phenomenon is referred to as the unstable convergence regime of GD. In this paper, we present a theoretical perspective to offer a qualitative analysis of this phenomenon. The unstable convergence is in fact an inherent property of GD for general twice differentiable functions. Specifically, the forwardinvariance of GD is established, i.e., it ensures that any point within a local region will always remain within this region under GD iteration. Then, based on the forward-invariance, for the initialization outside an open set containing the local minimum, the loss function will oscillate at the first several iterations and then become monotonely decreasing after the GD trajectory jumped into the open set. This work theoretically clarifies the unstable convergence phenomenon of GD discussed in previous experimental works. The unstable convergence of GD mainly depends on the selection of the initialization, and it is actually inevitable due to the complex nature of loss function.

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Lee, Jonathan N., Michael Laskey, Ajay Kumar Tanwani, Anil Aswani i Ken Goldberg. "Dynamic regret convergence analysis and an adaptive regularization algorithm for on-policy robot imitation learning". International Journal of Robotics Research 40, nr 10-11 (24.01.2021): 1284–305. http://dx.doi.org/10.1177/0278364920985879.

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On-policy imitation learning algorithms such as DAgger evolve a robot control policy by executing it, measuring performance (loss), obtaining corrective feedback from a supervisor, and generating the next policy. As the loss between iterations can vary unpredictably, a fundamental question is under what conditions this process will eventually achieve a converged policy. If one assumes the underlying trajectory distribution is static (stationary), it is possible to prove convergence for DAgger. However, in more realistic models for robotics, the underlying trajectory distribution is dynamic because it is a function of the policy. Recent results show it is possible to prove convergence of DAgger when a regularity condition on the rate of change of the trajectory distributions is satisfied. In this article, we reframe this result using dynamic regret theory from the field of online optimization and show that dynamic regret can be applied to any on-policy algorithm to analyze its convergence and optimality. These results inspire a new algorithm, Adaptive On-Policy Regularization (Aor), that ensures the conditions for convergence. We present simulation results with cart–pole balancing and locomotion benchmarks that suggest Aor can significantly decrease dynamic regret and chattering as the robot learns. To the best of the authors’ knowledge, this is the first application of dynamic regret theory to imitation learning.

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Liu, Lei, Yongji Wang, Fuqiang Xie i Jiashi Gao. "Legendre Cooperative PSO Strategies for Trajectory Optimization". Complexity 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/5036791.

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Particle swarm optimization (PSO) is a population-based stochastic optimization technique in a smooth search space. However, in a category of trajectory optimization problem with arbitrary final time and multiple control variables, the smoothness of variables cannot be satisfied since the linear interpolation is widely used. In the paper, a novel Legendre cooperative PSO (LCPSO) is proposed by introducing Legendre orthogonal polynomials instead of the linear interpolation. An additional control variable is introduced to transcribe the original optimal problem with arbitrary final time to the fixed one. Then, a practical fast one-dimensional interval search algorithm is designed to optimize the additional control variable. Furthermore, to improve the convergence and prevent explosion of the LCPSO, a theorem on how to determine the boundaries of the coefficient of polynomials is given and proven. Finally, in the numeral simulations, compared with the ordinary PSO and other typical intelligent optimization algorithms GA and DE, the proposed LCPSO has traits of lower dimension, faster speed of convergence, and higher accuracy, while providing smoother control variables.

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Velasco–Villa, M., A. Rodriguez–Angeles, I. Z. Maruri–López, J. A. Báez-Hernández i R. D. Cruz Morales. "Leader–follower formation control based on non-inertial frames for non–holonomic mobile robots". PLOS ONE 19, nr 1 (29.01.2024): e0297061. http://dx.doi.org/10.1371/journal.pone.0297061.

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A chain formation strategy based on mobile frames for a set of n differential drive mobile robots is presented. Considering two consecutive robots in the formation, robots Ri and Ri+1. It is intended that robot Ri+1 follows the delayed trajectory, τ units of time, of the leader robot Ri. In this way, the follower robot Ri+1 becomes the leader robot for robot Ri+ 2 in the formation and so on. With this formation policy, the trailing distance between two consecutive robots varies accordingly to the velocity of the Ri leader robot. Mobile frames are located on the body of the vehicles, in such a way that the position of robot Ri is determined with respect to the frame located on Ri+1 robot. The strategy relies on the fact that the general leader robot R1 describes any trajectory generated by bounded linear v1(t) and angular ω1(t) velocities. For the remaining vehicles in the string, the strategy considers a desired trajectory for the follower robot Ri+1 obtained by an estimation of the delayed trajectory of the leader robot Ri. This desired estimated trajectory is obtained under the knowledge of the actual and past input velocities of the Ri robot. To formally prove the convergence of the formation strategy, the equations describing the time variation of the relative posture between any pair of consecutive vehicles in the formation are obtained, and a feedback law based on local measurements is proposed to get the convergence of robot Ri+1 to the delayed trajectory, τ units of time, of the trajectory previously described by robot Ri. Lyapunov techniques are considered for this fact. The effectiveness of the chain formation solution is evaluated by means of numerical simulations and real time experiments showing an adequate convergence.

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Panahandeh, Pouya, Khalil Alipour, Bahram Tarvirdizadeh i Alireza Hadi. "A self-tuning trajectory tracking controller for wheeled mobile robots". Industrial Robot: the international journal of robotics research and application 46, nr 6 (21.10.2019): 828–38. http://dx.doi.org/10.1108/ir-02-2019-0032.

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Purpose Trajectory tracking is a common problem in the field of mobile robots which has attracted a lot of attention in the past two decades. Therefore, besides the search for new controllers to achieve a better performance, improvement and optimization of existing control rules are necessary. Trajectory tracking control laws usually contain constant gains which affect greatly the robot’s performance. Design/methodology/approach In this paper, a method based on neural networks is introduced to automatically upgrade the gains of a well-known trajectory tracking controller of wheeled mobile robots. The suggested method speeds up the convergence rate of the main controller. Findings Simulations and experiments are performed to assess the ability of the suggested scheme. The obtained results show the effectiveness of the proposed method. Originality/value In this paper, a method based on neural networks is introduced to automatically upgrade the gains of a well-known trajectory tracking controller of wheeled mobile robots. The suggested method speeds up the convergence rate of the main controller.

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Kim, Maksim, Olesya Glushchenko i Catherine Puck. "Convergent Journalism: The Main Vectors of Development and Prospects". Theoretical and Practical Issues of Journalism 12, nr 3 (30.09.2023): 399–417. http://dx.doi.org/10.17150/2308-6203.2023.12(3).399-417.

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Today, convergence is called both the business strategy for the development of media holdings, and the tactics of convergence of different media to promote services, and the merging of competencies in the training and professional activities of various media professionals, and the adaptation of public text to the realities of the latest technological revolution in the digital world, etc. The classical understanding of convergence as a mechanical merger is being revised. The generalized results of convergence in the media are new formats of information transmission and organizational and managerial structures optimized for this, hybrid content-genre forms, functional transformation of the media and much more. Convergence appears to be an evolutionary process that unfolds at different speeds across a variety of vectors in the instrumental-technological, organizational-managerial and text-centric spheres. With a large assumption, it is possible to construct a universal periodization of media convergence with an indication of the time range for each stage, since different media follow their own trajectory in convergent evolution. The authors propose to count the start of evolutionary processes in each area and, according to the configuration of new features and specific results of evolution, predict a development model with promising tracks for a particular media. Using structural and comparative analysis, the authors study in detail the approaches to the interpretation of the concept of "convergence", comprehend its true nature, and also explore the instrumental-technological, organizational-managerial and text-centric spheres of media convergence. This makes it possible to predict promising directions of convergent evolution in each area. Special attention is paid to the analysis of new selective features of journalistic text in the digital environment, the vector of their development is determined. The authors deal with the problems of professional training of journalists, analyzes changes in the professional skills of specialists for convergent journalism.

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Bekmaganbetov, Kuanysh A., Gregory A. Chechkin i Vladimir V. Chepyzhov. "Application of Fatou’s Lemma for Strong Homogenization of Attractors to Reaction–Diffusion Systems with Rapidly Oscillating Coefficients in Orthotropic Media with Periodic Obstacles". Mathematics 11, nr 6 (16.03.2023): 1448. http://dx.doi.org/10.3390/math11061448.

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We study reaction–diffusion systems with rapidly oscillating terms in the coefficients of equations and in the boundary conditions, in media with periodic obstacles. The non-linear terms of the equations only satisfy general dissipation conditions. We construct trajectory attractors for such systems in the strong topology of the corresponding trajectory dynamical systems. By means of generalized Fatou’s lemma we prove the strong convergence of the trajectory attractors of considered systems to the trajectory attractors of the corresponding homogenized reaction–diffusion systems which contain an additional potential.

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40

Zhang, Biao. "Quadrotor Convergence Trajectory Optimization Model Based on Minimum Energy Consumption". Advances in Applied Mathematics and Mechanics 10, nr 3 (czerwiec 2018): 673–89. http://dx.doi.org/10.4208/aamm.oa-2017-0059.

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Bansal, Jagdish Chand, Anshul Gopal i Atulya K. Nagar. "Analysing Convergence, Consistency, and Trajectory of Artificial Bee Colony Algorithm". IEEE Access 6 (2018): 73593–602. http://dx.doi.org/10.1109/access.2018.2884255.

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Hota, Sikha, i Debasish Ghose. "Optimal trajectory planning for path convergence in three-dimensional space". Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 228, nr 5 (20.03.2013): 766–80. http://dx.doi.org/10.1177/0954410013479714.

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Hendrickx, Julien M., i Balázs Gerencsér. "Trajectory convergence from coordinate-wise decrease of general energy functions". Automatica 154 (sierpień 2023): 111099. http://dx.doi.org/10.1016/j.automatica.2023.111099.

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Shi, Junren, Kexin Li, Changhao Piao, Jun Gao i Lizhi Chen. "Model-Based Predictive Control and Reinforcement Learning for Planning Vehicle-Parking Trajectories for Vertical Parking Spaces". Sensors 23, nr 16 (11.08.2023): 7124. http://dx.doi.org/10.3390/s23167124.

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This paper proposes a vehicle-parking trajectory planning method that addresses the issues of a long trajectory planning time and difficult training convergence during automatic parking. The process involves two stages: finding a parking space and parking planning. The first stage uses model predictive control (MPC) for trajectory tracking from the initial position of the vehicle to the starting point of the parking operation. The second stage employs the proximal policy optimization (PPO) algorithm to transform the parking behavior into a reinforcement learning process. A four-dimensional reward function is set to evaluate the strategy based on a formal reward, guiding the adjustment of neural network parameters and reducing the exploration of invalid actions. Finally, a simulation environment is built for the parking scene, and a network framework is designed. The proposed method is compared with the deep deterministic policy gradient and double-delay deep deterministic policy gradient algorithms in the same scene. Results confirm that the MPC controller accurately performs trajectory-tracking control with minimal steering wheel angle changes and smooth, continuous movement. The PPO-based reinforcement learning method achieves shorter learning times, totaling only 30% and 37.5% of the deep deterministic policy gradient (DDPG) and twin-delayed deep deterministic policy gradient (TD3), and the number of iterations to reach convergence for the PPO algorithm with the introduction of the four-dimensional evaluation metrics is 75% and 68% shorter compared to the DDPG and TD3 algorithms, respectively. This study demonstrates the effectiveness of the proposed method in addressing a slow convergence and long training times in parking trajectory planning, improving parking timeliness.

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Dai, Xiaoqiang, Hewei Xu, Hongchao Ma, Jianjun Ding i Qiang Lai. "Dual closed loop AUV trajectory tracking control based on finite time and state observer". Mathematical Biosciences and Engineering 19, nr 11 (2022): 11086–113. http://dx.doi.org/10.3934/mbe.2022517.

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<abstract><p>The three-dimensional trajectory tracking of AUV is an important basis for it to complete its task. Due to many uncertain disturbances such as wind, wave and current on the sea, it is easy to cause problems such as slow convergence speed of the controller and saturation of the controller output in the three-dimensional trajectory tracking control of AUV. And the dynamic uncertainty of AUV's own model will have a great negative impact on AUV's trajectory tracking control. In order to solve the problem of slow convergence speed of the above controller, the finite time control method is introduced into the designed position controller. In order to solve the problem of AUV controller output saturation, an auxiliary dynamic system is designed to compensate the system control output saturation. In order to solve the uncertainty of AUV model, a reduced order extended observer is designed in the dynamic controller. It can observe the motion parameters of AUV at any time, and compensate the uncertainty of model uncertainty and external environment disturbance in real time. The control method in this paper is simulated in a three-dimensional model. The experimental results show that the convergence speed, control accuracy, robustness and tracking effect of AUV are higher than those of common trajectory tracker. The algorithm is loaded into the "sea exploration Ⅱ" AUV and verified by experiments in Suzhou lake. The effect of AUV navigation basically meets the task requirements, in which the mean value of pitch angle and heading angle error is less than 8 degrees and the mean value of depth error is less than 0.1M. The trajectory tracker can better meet the trajectory tracking control needs of the AUV.</p></abstract>

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Huang, Shurui, Yueneng Yang, Ye Yan, Shifeng Zhang i Zhiyang Liu. "Observer-Based Robust Finite-Time Trajectory Tracking Control for a Stratospheric Satellite Subject to External Disturbance and Actuator Saturation". International Journal of Aerospace Engineering 2022 (29.09.2022): 1–19. http://dx.doi.org/10.1155/2022/1601771.

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The stratospheric satellite is regarded as an ideal stratosphere flight platform and is able to accomplish various missions such as surveillance, earth observation, and remote sensing, which requires a robust and effective trajectory tracking control method to support these tasks. A novel observer-based robust finite-time control scheme is proposed to address the trajectory tracking control problem dedicated to a stratospheric satellite in the presence of external disturbance and actuator saturation. Firstly, an extended state observer (ESO) is adopted to observe the unavailable velocity states and unknown disturbances simultaneously, and the estimated data are utilized in the robust control law design. Then, an auxiliary system based on anti-windup compensator is developed to directly compensate for the actuator saturation difference. After that, a backstepping nonsingular fast terminal sliding mode control (BNFTSMC) strategy is designed to track the desired trajectory with high accuracy, fast convergence rate, and finite-time convergence. Then, a stability analysis using Lyapunov-based theory is performed, in which the stabilization of the stratospheric satellite system and finite-time convergence are proven. Furthermore, a number of simulations are conducted further to verify the excellent performance of the designed control strategy.

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Luo, Lan, Tongbin Guo, Kangkang Cui i Qi Zhang. "Trajectory Planning in Robot Joint Space Based on Improved Quantum Particle Swarm Optimization Algorithm". Applied Sciences 13, nr 12 (11.06.2023): 7031. http://dx.doi.org/10.3390/app13127031.

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Trajectory planning is a crucial step in controlling robot motion. The efficiency and accuracy of trajectory planning directly impact the real-time control and accuracy of robot motion. The robot’s trajectory is mapped to the joint space, and a mathematical model of trajectory planning is established to meet physical constraints during motion and avoid joint coupling problems. To enhance convergence speed and avoid local optima, an improved quantum particle swarm optimization algorithm is proposed and applied to solve the mathematical model of robot trajectory planning. The trajectory planning in robot joint space is then tested based on the improved quantum particle swarm optimization algorithm. The results demonstrate that this method can replace the traditional trajectory planning algorithms and offers higher accuracy and efficiency.

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KE, YIFEN, i CHANGFENG MA. "A NEURAL NETWORK FOR THE GENERALIZED NONLINEAR COMPLEMENTARITY PROBLEM OVER A POLYHEDRAL CONE". Journal of the Australian Mathematical Society 99, nr 3 (30.10.2015): 364–79. http://dx.doi.org/10.1017/s1446788715000300.

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In this paper, we consider a neural network model for solving the generalized nonlinear complementarity problem (denoted by GNCP) over a polyhedral cone. The neural network is derived from an equivalent unconstrained minimization reformulation of the GNCP, which is based on the penalized Fischer–Burmeister function ${\it\phi}_{{\it\mu}}(a,b)={\it\mu}{\it\phi}_{\mathit{FB}}(a,b)+(1-{\it\mu})a_{+}b_{+}$. We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability and exponential stability. It is found that a larger ${\it\mu}$ leads to a better convergence rate of the trajectory. Simulation results are also reported.

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Yessenbekova, U. M., A. A. Mamankul, A. A. Syzdykova i B. Kutym. "Media convergence in social communication and cyberculture". BULLETIN of L.N. Gumilyov Eurasian National University. Journalism Series 144, nr 3 (2023): 87–102. http://dx.doi.org/10.32523/2616-7174-2023-144-3-87-102.

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The article is devoted to a theoretical study of the role of media convergence in the formation of social communication and cyber culture. Comprehensive analysis showed that the convergence of mass media has become a defining factor in the interaction between people and the media. During the discussion of various aspects of media convergence and its place in public relations, connections with traditional culture, scientific conclusions were drawn. The study found how media convergence contributes to the development of cyberculture. People actively interact through social networks, forums, blogs and other online platforms, forming their own digital communities and groups. In the virtual space, they share their knowledge, interests and new ideas, thus creating modern forms of cyberculture. The authors showed that media convergence, driven by such active connections, increases the accessibility and convenience of information and entertainment.The final results of the article confirm that media convergence is an important determining factor in the formation of social communication and cyber culture. The influence of convergence on the relationship between people and the trajectory of the development of the information society as a whole is assessed.The findings of the study show that the study of convergent trends in the mass media helps to predict the development of the state and society, to develop strategies that direct social communication and virtual culture in a direction that is useful for society

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Ieong, Meng U. "Macao and Hong Kong, Convergence or Divergence?" Asian Survey 57, nr 3 (maj 2017): 504–27. http://dx.doi.org/10.1525/as.2017.57.3.504.

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By examining the origins of Macao’s 2014 protest against the Retirement Package Bill, this paper intends to answer whether Macao is encountering a governance crisis like Hong Kong’s. Through a careful comparison, it concludes that Macao and Hong Kong are unlikely to follow the same political development trajectory, as there are considerable differences in the level of social mobilization and governmental coercive capacity between the two cities.

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