非线性系统的稳定性与同步控制研究
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摘要
非线性系统在国防、工业、人类社会和生物界等领域有着广泛的应用背景,一直是控制界研究的热点问题.本文主要研究了非线性系统的稳定性和同步控制问题,主要工作可以归结为以下几方面:
1.针对含有时变时滞的Lurie系统,利用时滞划分的方法分别讨论了直接控制系统和中立型间接控制系统的鲁棒绝对稳定性问题.通过选择适当的Lyapunov函数,以及采用不同的时滞划分方法,给出系统绝对稳定的时滞相关判据.同时,通过数值仿真比较,可以发现本文所得结果具有较小的保守性.
2.针对非线性函数属于扇形区域和满足斜率条件的混沌Lurie系统,首次利用收缩分析的方法研究了带有反馈控制器的Lurie系统的同步问题.不需要讨论误差系统的稳定性,给出了一般Lurie系统同步的判定条件,并把结论推广到Chua’s电路中非线性函数不可微的情形.同时,数值仿真也表明了理论结果的正确性.
3.针对Lurie型复杂动态网络系统,首次利用收缩分析的方法,研究了两种含有不同耦合时滞的复杂网络的同步问题.基于收缩理论以及矩阵测度的相关性质,先讨论了仅含有连续耦合的同步问题,然后考虑了连续与离散时滞耦合同时存在的同步问题,给出了系统同步的时滞无关判据.仿真结果也说明了结论的有效性.
4.针对含有时变时滞的二阶多智能体系统,研究了系统的一致性问题.指出二阶多智能体系统的一致性性问题可以转化为线性多智能体误差系统的稳定性问题.利用时滞划分的方法得出系统实现一致可取的最大时滞上界,所得结果以线性矩阵不等式的形式给出,便于求解.
5.针对含有耦合变量的二阶多智能体系统,首先利用线性多智能体系统的相关结论给出了系统在不含有领导者的情形下实现一致的充要条件.然后将其推广到有领导者的情形,并进行了数值仿真.
6.针对一般线性多智能体系统以及含有非线性项的多智能体系统,首次利用收缩分析的方法研究了智能体系统的一致性问题.给出了系统在不含有领导者和带有领导者的情形下实现一致的判定条件.另外,通过数值仿真说明了结论的有效性.
Nonlinear system has broad application background in national defense, industry,human society and biosphere, and so on. And it has always been the hot topic of researchin the control feld. This dissertation mainly studies the stability and synchronizationcontrol problems of the nonlinear systems. The main works of this dissertation can besummarized into the following aspects:
1. For the Lurie systems with time-varying delay, a delay decomposition method isemployed to solve the problems of robust absolute stability of direct system and neutralindirect system, respectively. By selecting suitable Lyapunov functions and diferent delaydivision methods, some delay-dependent conditions for absolute stability of these systemsare given. Through simulations and comparison, it can be found that the obtained resultsof this paper has less conservativeness.
2. For the chaotic Lurie systems with sector and slope restricted nonlinearities,contraction analysis is used for the frst time to study the synchronization problem ofLurie systems with feedback controller. Without considering the stability of the errorsystem, synchronization condition for general Lurie systems is given. Furthermore, theresult is extended to the Chua’s system whose nonlinear function is not diferentiable.Meanwhile, numerical simulations are also demonstrated to show the correctness of thetheoretical results.
3. For Lurie complex dynamical networks, contraction analysis is utilized for thefrst time to study two synchronization problems of complex networks with diferentdelay-couplings. Based on contraction theory and matrix measure properties, the syn-chronization problem of system with constant coupling is frstly discussed, and then thesynchronization problem of system with constant coupling and discrete-delay coupling isalso considered. Some delay-independent synchronization conditions are proposed. Andsimulation results are illustrated to show the efectiveness of these conclusions.
4. For the second-order multi-agent system with time-varying delay, the consen-sus problem of this system is studied. It is pointed out that the consensus problemof the second-order multi-agent system can be changed into the stability of the errorsystem about the linear multi-agent system. Then, the maximal advisable time-delaybound about the system achieving consensus is obtained through time-delay decomposi-tion method. The obtained results are presented in the form of linear matrix inequalities.This can help us to get a solution.
5. For the second-order multi-agent systems with coupling strengths, the necessaryand sufcient conditions of these systems reaching leaderless consensus are frstly given by using some conclusions originated from the linear multi-agent systems. Then, the resultsare extended to the leader-following case. Numerical simulations are also given.
6. For the multi-agent systems with general linear and nonlinear dynamics, con-traction analysis is used to consider their consensus problems for the frst time. Someconditions about these systems reaching consensus in leaderless and leader-following caseare presented. Moreover, simulation results are also given to validate the efectiveness ofthe conclusions.
1.针对含有时变时滞的Lurie系统,利用时滞划分的方法分别讨论了直接控制系统和中立型间接控制系统的鲁棒绝对稳定性问题.通过选择适当的Lyapunov函数,以及采用不同的时滞划分方法,给出系统绝对稳定的时滞相关判据.同时,通过数值仿真比较,可以发现本文所得结果具有较小的保守性.
2.针对非线性函数属于扇形区域和满足斜率条件的混沌Lurie系统,首次利用收缩分析的方法研究了带有反馈控制器的Lurie系统的同步问题.不需要讨论误差系统的稳定性,给出了一般Lurie系统同步的判定条件,并把结论推广到Chua’s电路中非线性函数不可微的情形.同时,数值仿真也表明了理论结果的正确性.
3.针对Lurie型复杂动态网络系统,首次利用收缩分析的方法,研究了两种含有不同耦合时滞的复杂网络的同步问题.基于收缩理论以及矩阵测度的相关性质,先讨论了仅含有连续耦合的同步问题,然后考虑了连续与离散时滞耦合同时存在的同步问题,给出了系统同步的时滞无关判据.仿真结果也说明了结论的有效性.
4.针对含有时变时滞的二阶多智能体系统,研究了系统的一致性问题.指出二阶多智能体系统的一致性性问题可以转化为线性多智能体误差系统的稳定性问题.利用时滞划分的方法得出系统实现一致可取的最大时滞上界,所得结果以线性矩阵不等式的形式给出,便于求解.
5.针对含有耦合变量的二阶多智能体系统,首先利用线性多智能体系统的相关结论给出了系统在不含有领导者的情形下实现一致的充要条件.然后将其推广到有领导者的情形,并进行了数值仿真.
6.针对一般线性多智能体系统以及含有非线性项的多智能体系统,首次利用收缩分析的方法研究了智能体系统的一致性问题.给出了系统在不含有领导者和带有领导者的情形下实现一致的判定条件.另外,通过数值仿真说明了结论的有效性.
Nonlinear system has broad application background in national defense, industry,human society and biosphere, and so on. And it has always been the hot topic of researchin the control feld. This dissertation mainly studies the stability and synchronizationcontrol problems of the nonlinear systems. The main works of this dissertation can besummarized into the following aspects:
1. For the Lurie systems with time-varying delay, a delay decomposition method isemployed to solve the problems of robust absolute stability of direct system and neutralindirect system, respectively. By selecting suitable Lyapunov functions and diferent delaydivision methods, some delay-dependent conditions for absolute stability of these systemsare given. Through simulations and comparison, it can be found that the obtained resultsof this paper has less conservativeness.
2. For the chaotic Lurie systems with sector and slope restricted nonlinearities,contraction analysis is used for the frst time to study the synchronization problem ofLurie systems with feedback controller. Without considering the stability of the errorsystem, synchronization condition for general Lurie systems is given. Furthermore, theresult is extended to the Chua’s system whose nonlinear function is not diferentiable.Meanwhile, numerical simulations are also demonstrated to show the correctness of thetheoretical results.
3. For Lurie complex dynamical networks, contraction analysis is utilized for thefrst time to study two synchronization problems of complex networks with diferentdelay-couplings. Based on contraction theory and matrix measure properties, the syn-chronization problem of system with constant coupling is frstly discussed, and then thesynchronization problem of system with constant coupling and discrete-delay coupling isalso considered. Some delay-independent synchronization conditions are proposed. Andsimulation results are illustrated to show the efectiveness of these conclusions.
4. For the second-order multi-agent system with time-varying delay, the consen-sus problem of this system is studied. It is pointed out that the consensus problemof the second-order multi-agent system can be changed into the stability of the errorsystem about the linear multi-agent system. Then, the maximal advisable time-delaybound about the system achieving consensus is obtained through time-delay decomposi-tion method. The obtained results are presented in the form of linear matrix inequalities.This can help us to get a solution.
5. For the second-order multi-agent systems with coupling strengths, the necessaryand sufcient conditions of these systems reaching leaderless consensus are frstly given by using some conclusions originated from the linear multi-agent systems. Then, the resultsare extended to the leader-following case. Numerical simulations are also given.
6. For the multi-agent systems with general linear and nonlinear dynamics, con-traction analysis is used to consider their consensus problems for the frst time. Someconditions about these systems reaching consensus in leaderless and leader-following caseare presented. Moreover, simulation results are also given to validate the efectiveness ofthe conclusions.
引文
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