scholarly journals Adaptive Fixed-Time Tracking Consensus Control for Multiagent Nonlinear Pure-Feedback Systems with Performance Constraints

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Pinwei Li ◽  
Jiyang Dai ◽  
Jin Ying
Keyword(s):  
Fixed Time ◽  
Mean Value ◽  
High Order ◽  
Mean Value Theorem ◽  
Feedback Systems ◽  
Feedback Form ◽  
Consensus Control ◽  
Performance Constraints ◽  
Whole Process ◽  
Constraint Variable

This paper investigates adaptive fixed-time tracking consensus control problems for multiagent nonlinear pure-feedback systems with performance constraints. Compared with existing results of first/second/high-order multiple agent systems, the studied systems have more complex nonlinear dynamics with each agent being modeled as a high-order pure-feedback form. The mean value theorem is introduced to address the problem of nonaffine structure in nonlinear pure-feedback systems. Meanwhile, radial basis function neural networks (RBFNNs) are employed to approximate unknown functions. Furthermore, a constraint variable is used to guarantee that all local tracking errors are within the prescribed boundaries. It is shown that, by utilizing the proposed consensus control protocol, each tracking consensus error can converge into a neighborhood around zero within designed fixed time, the tracking consensus performance can be ensured during the whole process, and all signals in the investigated systems are bounded. Finally, two simulations are performed and the results demonstrate the effectiveness of the proposed control strategy.

scholarly journals Approximation-Based Fixed-Time Adaptive Tracking Control for a Class of Uncertain Nonlinear Pure-Feedback Systems

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Cheng He ◽  
Jian Wu ◽  
Jiyang Dai ◽  
Zhe Zhang ◽  
Libin Xu ◽  
...  
Keyword(s):  
Tracking Control ◽  
Tracking Error ◽  
Fixed Time ◽  
Mean Value ◽  
Design Parameters ◽  
Mean Value Theorem ◽  
Feedback Systems ◽  
Adaptive Tracking Control ◽  
Adaptive Tracking ◽  
Time Control

This paper examines approximation-based fixed-time adaptive tracking control for a class of uncertain nonlinear pure-feedback systems. Novel virtual and actual controllers are designed that resolve the meaninglessness of virtual and actual controllers at the origin and in the negative domain, and the sufficient condition for the system to have semiglobal fixed-time stability is also provided. Radial basis function neural networks are introduced to approximate unknown functions for solving the fixed-time control problem of unknown nonlinear pure-feedback systems, and the mean value theorem is used to solve the problem of nonaffine structure in nonlinear pure-feedback systems. The controllers designed in this paper ensure that all signals in the closed-loop system are semiglobally uniform and ultimately bounded in a fixed time. Two simulation results show that appropriate design parameters can limit the tracking error within a region of the origin in a fixed time.

scholarly journals Distributed Adaptive Fixed-Time Tracking Consensus Control for Multiple Uncertain Nonlinear Strict-Feedback Systems under a Directed Graph

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Pinwei Li ◽  
Jiyang Dai ◽  
Jin Ying ◽  
Zhe Zhang ◽  
Cheng He
Keyword(s):  
Directed Graph ◽  
Control Strategy ◽  
Design Procedure ◽  
Fixed Time ◽  
Convergence Time ◽  
Approximation Technique ◽  
Feedback Systems ◽  
Consensus Control ◽  
Consensus Tracking ◽  
Strict Feedback

In this brief, we study the distributed adaptive fixed-time tracking consensus control problem for multiple strict-feedback systems with uncertain nonlinearities under a directed graph topology. It is assumed that the leader’s output is time varying and has been accessed by only a small fraction of followers in a group. The distributed fixed-time tracking consensus control is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader and ensure the error convergence time is independent of the systems’ initial state. The function approximation technique using radial basis function neural networks (RBFNNs) is employed to compensate for unknown nonlinear terms induced from the controller design procedure. From the Lyapunov stability theorem and graph theory, it is shown that, by using the proposed fixed-time control strategy, all signals in the closed-loop system and the consensus tracking errors are cooperatively semiglobally uniformly bounded and the errors converge to a neighborhood of the origin within a fixed time. Finally, the effectiveness of the proposed control strategy has been proved by rigorous stability analysis and two simulation examples.

scholarly journals Adaptive Fixed-Time Control of Strict-Feedback High-Order Nonlinear Systems

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 963
Author(s):  
Yang Li ◽  
Jianhua Zhang ◽  
Xiaoyun Ye ◽  
Cheng Siong Chin
Keyword(s):  
Nonlinear Systems ◽  
Fixed Time ◽  
High Order ◽  
Convergence Time ◽  
Feedback Form ◽  
Time Control ◽  
Control Scheme ◽  
Lyapunov Stability Criterion ◽  
The Stability ◽  
Strict Feedback

This paper examines the adaptive control of high-order nonlinear systems with strict-feedback form. An adaptive fixed-time control scheme is designed for nonlinear systems with unknown uncertainties. In the design process of a backstepping controller, the Lyapunov function, an effective controller, and adaptive law are constructed. Combined with the fixed-time Lyapunov stability criterion, it is proved that the proposed control scheme can ensure the stability of the error system in finite time, and the convergence time is independent of the initial condition. Finally, simulation results verify the effectiveness of the proposed control strategy.

scholarly journals Adaptive Differentiator-Based Predefined-Time Control for Nonlinear Systems Subject to Pure-Feedback Form and Unknown Disturbance

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Man Yang ◽  
Qiang Zhang ◽  
Ke Xu ◽  
Ming Chen
Keyword(s):  
Nonlinear Systems ◽  
Control Method ◽  
Tracking Error ◽  
Nonlinear Function ◽  
Original System ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Rbf Neural Networks ◽  
Feedback Form ◽  
Output Tracking Control

In this article, by utilizing the predefined-time stability theory, the predefined-time output tracking control problem for perturbed uncertain nonlinear systems with pure-feedback structure is addressed. The nonaffine structure of the original system is simplified as an affine form via the property of the mean value theorem. Furthermore, the design difficulty from the uncertain nonlinear function is overcome by the excellent approximation performance of RBF neural networks (NNs). An adaptive predefined-time controller is designed by introducing the finite-time differentiator which is used to decrease the computational complexity problem appeared in the traditional backstepping control. It is proved that the proposed control method guarantees all signals in the closed-loop system remain bound and the tracking error converges to zero within the predefined time. Based on the controller designed in this paper, the expected results can be obtained in predefined time, which can be illustrated by the simulation results.

GLOBAL EXPONENTIAL STABILITY OF HIGH-ORDER BAM NEURAL NETWORKS WITH REACTION–DIFFUSION TERMS

2010 ◽  
Vol 20 (10) ◽  
pp. 3209-3223 ◽  
Author(s):  
FENG-YAN ZHOU ◽  
CHENG-RONG MA
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Time Delays ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Mean Value ◽  
High Order ◽  
Mean Value Theorem ◽  
Bam Neural Networks

The global exponential stability is studied for a class of high-order bi-directional associative memory (BAM) neural networks with time delays and reaction–diffusion terms. By constructing suitable Lyapunov functional, using differential mean value theorem and homeomorphism, several sufficient conditions guaranteeing the existence, uniqueness and global exponential stability of high-order BAM neural networks with time delays and reaction–diffusion terms are given. Two illustrative examples are also given in the end to show the effectiveness of our results.

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

scholarly journals On a hybrid version of the Vinogradov mean value theorem

2021 ◽  
Vol 163 (1) ◽  
pp. 1-17
Author(s):  
C. Chen ◽  
I. E. Shparlinski
Keyword(s):  
Mean Value ◽  
Mean Value Theorem

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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