Stability Analysis Of Feedback Control Differential System

Energy is indispensable in our country's development,it provides an important guarantee for our country's economic and social development.The energy demand-supply system is a study on the relationship between energy demand and energy supply in the two regions.The study of energy system can provide a certain theoretical basis for the development of energy.Due to the complexity of the energy system,certain controls need to be applied to make the stability of the system reach the required goal.Based on the established energy demand-supply system model,this paper applies time-delayed feedback control and conducts dynamic analysis of the system.The research content of this article is mainly composed of the following two parts:The first part focuses on the four-dimensional energy demand-supply system of the relationship between the energy demand shortage,energy supply,energy import and renewable energy in the two regions,by designing the delayed feedback control,making the original energy demand-supply system change from an unstable state to asymptotically stable or periodic changes.For the system with the control term applied,we separately studied the stability of the system equilibrium point O at?(28)0and?(29)0.The results show that when different time delay?and feedback gain k are selected,the system will have different states.After the value of the feedback gain k is determined,as the value of?changes,it is found that the system is in a stable state when the time delay?is in a certain interval.By calculating^d(Re?)_{d??(28)?i?1,2}?0,it is obtained that when?(28)?₁ⁿ,₂(n(28),0,1,2(43)),the system has Hopf bifurcation and the calculation formula for the properties of Hopf bifurcation is derived.Given appropriate coefficients,numerical simulations verify the theoretical results and illustrate the effectiveness of the feedback control.In the second part,a five-dimensional energy demand-supply system with time-delayed feedback control is established and the stability of the system is analyzed.In order to realize the sustainable development of energy,the new energy industry plays an important role.However,the addition of the new energy variable will change the stable state of the original four-dimensional energy system,so delayed feedback control is imposed on the original system.In the controlled system,the Hurwitz criterion is used to obtain the condition that the system is asymptotically stable when?(28)0.For the case of?(29)0,firstly calculate the characteristic equation of the linear system corresponding to the controlled system;Secondly,because the characteristic equation is a transcendental equation,the method of introducing a polynomial to completely discriminate the system can obtain the distribution of the root of the characteristic equation;According to the change characteristics of the real part of the characteristic root,the conclusion of how the system stability changes is obtained.Finally,according to the Hopf bifurcation theory,the existence of the Hopf bifurcation is obtained,and the calculation formula for the Hopf bifurcation is obtained.Given appropriate coefficients,numerical simulations illustrate the theoretical results.The innovations of this thesis:1.Due to the influence of renewable energy and new energy,for the more realistic four-dimensional and five-dimensional energy system,feedback control is established to make the stability of the system reach the required state.2.Time delay phenomenon is common,and it is of practical significance to design time delayed feedback control with time delay.3.In the controlled four-dimensional and five-dimensional systems,the methods of discriminating the roots of cubic equations and polynomials to fully discriminate the system are introduced to obtain the value of the feedback gain.4.Popularize low-dimensional system stability analysis methods and apply them to high-dimensional systems.