scholarly journals Global Stability for a Discrete Space-Time Lotka–Volterra System with Feedback Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Li Xu ◽  
Ruiwen Han
Keyword(s):  
Feedback Control ◽  
Lyapunov Functions ◽  
Nonnegative Solution ◽  
Space Time ◽  
Discrete Space ◽  
Volterra Model ◽  
Discrete Version ◽  
Positive Equilibrium ◽  
Volterra System ◽  
Quadratic Lyapunov Functions

In this paper, a discrete space-time Lotka–Volterra model with the periodic boundary conditions and feedback control is proposed. By means of a discrete version of comparison theorem, the boundedness of the nonnegative solution of the system is proved. By the combination of the Volterra-type and quadratic Lyapunov functions, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.

Global dissipativity of stochastic Lotka–Volterra system with feedback controls

2017 ◽  
Vol 10 (02) ◽  
pp. 1750022 ◽  
Author(s):  
Qimin Zhang ◽  
Xinjing Zhang ◽  
Hongfu Yang
Keyword(s):  
Linear Matrix Inequalities ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Jensen Inequality ◽  
Linear Matrix ◽  
Mean Square ◽  
Matrix Inequalities ◽  
Volterra System ◽  
Feedback Controls ◽  
The Mean

In this paper, a class of stochastic Lotka–Volterra system with feedback controls is considered. The purpose is to establish some criteria to ensure the system is globally dissipative in the mean square. By constructing suitable Lyapunov functions as well as combining with Jensen inequality and It[Formula: see text] formula, the sufficient conditions are established and they are expressed in terms of the feasibility to a couple linear matrix inequalities (LMIs). Finally, the main results are illustrated by examples.

scholarly journals Synthesis of Guaranteed Stability Regions of a Nonstationary Nonlinear System with a Fuzzy Controller

2019 ◽  
Vol 5 (1) ◽  
pp. 107 ◽  
Author(s):  
Vasiliy Berdnikov ◽  
Valeriy Lokhin
Keyword(s):  
Lyapunov Functions ◽  
Pid Controller ◽  
Sufficient Conditions ◽  
Control Object ◽  
Fuzzy Pid ◽  
Nonlinear Characteristics ◽  
Stability Regions ◽  
Fuzzy Pid Controller ◽  
Quadratic Lyapunov Functions ◽  
Conditions Of Stability

The paper proposes a method for constructing guaranteed regions of stability of nonstationary nonlinear systems on the plane of parameters of a fuzzy PID controller. It is shown that this method allows to determine the full stability areas, which are significantly larger than the areas determined by classical methods (frequency circle criterion, quadratic Lyapunov functions). This improvement is achieved by using the algorithm for constructing spline Lyapunov functions. This type of Lyapunov functions is based on the necessary and sufficient conditions of stability, while the classical methods are only sufficient conditions of stability. In this regard, on the basis of the proposed method, it is possible to calculate the maximum sizes of the sectors in which the nonlinear characteristics in the channels of the fuzzy PID controller should be located. Examples of the synthesis of fuzzy P, PI, PID controllers for a nonstationary control object of the third order are given. Numerical experiments show that the expansion of the boundaries of nonlinear characteristics allows to improve the accuracy in the steady state, and also to almost double the speed of the automatic control system with a nonstationary object. The advantages over linear controllers are demonstrated. The proposed method guarantees the stability inside the calculated stability regions for any character of the change in the nonstationary parameter, as well as for any character of the change in the nonlinear characteristics in the corresponding sectors.

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