中立型跳跃系统的研究及其在复杂网络中的应用
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摘要
随着科学技术的迅猛发展,信息处理、计算机通讯、机器人控制、生产过程自动化等技术的完善和广泛应用,人们着眼的系统规模越来越大,所涵盖的现象也越来越复杂,出现了大量具有中立型时滞和切换特征的复杂系统,迫使人们开始关注这些系统。另一方面,随着计算机技术和网络理论高速发展,复杂动态网络已成为一个新的研究热点。在现实生活中存在着大量具有中立型时滞和切换特征的复杂网络,研究此类中立型复杂网络的动态行为,如稳定性和同步性等,对于人们深刻理解和应用复杂网络具有重要的理论意义。本文依托国家自然科学基金项目,研究了一类马尔可夫跳跃中立型系统的稳定性问题。在此基础上,进一步考虑了在马氏过程转移率矩阵部分可知情况下,马尔可夫跳跃中立型复杂网络及融合网络(复杂异构网络)的同步控制问题。主要工作如下：
1.针对带有马尔可夫跳跃参数的中立型时滞系统,在李雅普诺夫稳定性理论框架下,我们运用多李雅普诺夫泛函的方法,构造了一类带有三重积分项的李雅普诺夫泛函,并运用矩阵不等式,积分不等式,和凸组合的性质,推导出了时滞相关的随机稳定性准则。此外,将此结果推广到了外界非线性扰动的情况以及系统参数不确定的情况。通过数值仿真实例,与现有的一些文献相比在较大程度上降低了系统的保守性,从而验证所得结果的有效性和优越性。
2.针对模态转移率矩阵不完全确定的中立型马尔可夫跳跃系统,考虑其受到外界非线性扰动的情况以及系统跳跃参数不确定的情况,我们仍然在李雅普诺夫稳定性理论框架下,构造了一类带有三重积分项的随机李雅普诺夫泛函,运用矩阵不等式,积分不等式并结合往复凸引理(Reciprocally Convex Lemma)和自由权矩阵的方法,推导出了依赖模态和时滞相关的指数稳定性准则。通过一些数值仿真实例并与现有文献相比较,较大程度的提高了系统稳定的时滞上限,从而验证了本文所得结果的有效性和优越性。
3.针对模态转移率矩阵部分可知的一类带有分布时滞的马尔可夫跳跃中立型复杂网络,考虑其受到外界扇形有界的非线性扰动情况,我们利用采样数据控制技术,构造了一类带有三重积分项的随机李雅普诺夫泛函,运用Kronecker积的运算性质,矩阵不等式,积分不等式并结合往复凸引理和自由权矩阵的方法,推导出了依赖模态和时滞相关的指数同步准则。通过一些数值仿真实例并与最近有关的文献相比,在采样时间间隔方面有所提高,从而验证了本文给出的关于马氏跳跃中立型复杂网络同步准则的有效性和优越性。
4.针对模态转移率矩阵部分未知的一类马尔可夫跳跃中立型复杂网络,考虑其受到外界扇形有界的非线性扰动情况,我们采用牵制控制技术,在有限时间稳定性理论框架下,构造了一类带有权系数的随机李雅普诺夫泛函,运用Kronecker积的运算性质,矩阵不等式,积分不等式并结合自由权矩阵的方法,推导出了依赖模态和时滞相关的有限时间同步准则。通过一些数值仿真实例,与最近相关文献的同步时间相比较,验证了本文给出的关于马氏跳跃中立型复杂网络有限时间同步准则的有效性和优越性。
5.针对模态转移率矩阵部分未知的一类复杂融合网络(即按照马尔可夫过程切换规则由不同类型网络异构而成的复杂网络),考虑其异构的不同节点没有共同的平衡点,在此情况下我们定义所有节点状态的加权平均状态作为我们的虚拟同步目标状态,即引入拟同步(Quasi-synchronization)的定义,随后构造了一类带有权系数的随机李雅普诺夫泛函,运用Kronecker积的运算性质,矩阵不等式并结合自由权矩阵的方法,推导出了依赖模态和时滞相关的拟同步准则。通过一些数值仿真实例,验证了本文给出的关于此类复杂融合网络拟同步准则的有效性。
With the rapid development of science and technology, the system scale is becom-ing larger and the phenomena are becoming more and more complex. Due to the great application of neutral Markovian jumping systems, they have received close attention by scholars at home and abroad. On the other hand, with the rapid improvement of com-puter technology and network theory, complex dynamic network has become a new hot topic. Since neutral delay and Markovian switching are always existed in large number of networks, then it is of great importance and significance to investigate these neutral Markovian jump complex networks. The main results obtained in this dissertation can be concluded as follows.
1. The delay-range-dependent stochastic stability and exponential stabili-ty are investigated for the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations. A novel augment-ed Lyapunov functional which contains some triple-integral terms is intro-duced. Then by employing some integral inequalities and the nature of convex combination, some less conservative stochastic stability condition-s are presented in terms of linear matrix inequalities. Finally, numerical examples are provided to demonstrate the effectiveness and to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.
2. The exponential stability is investigated for neutral Markovian jump sys-tems with interval mode-dependent time-varying delays, nonlinear pertur-bations and partially known transition rates. A novel augmented stochastic Lyapunov functional is constructed, which employs the improved bounding technique and contains triple-integral terms to reduce conservatism, then the exponential stability criteria are developed by Lyapunov stability theo-ry, reciprocally convex lemma and free-weighting matrices. Moreover, the corresponding results are extended to the uncertain case. Finally, numeri-cal examples are given to illustrate the effectiveness and superiority of the proposed criteria.
3. The synchronization problem is investigated for a neutral complex dy-namical network with distributed delay, Markovian switching and partially unknown transition rates via sampled-data controller. A new augmented stochastic Lyapunov functional is constructed, which contains some triple-integral terms to reduce the conservativeness. Then the exponential stabili-ty conditions for the closed-loop error system are obtained by the Lyapunov stability theory, integral matrix inequalities and reciprocally convex lem-ma. Based on these new stability conditions, the sampled-data exponential synchronization controllers are found in terms of the solutions to linear ma-trix inequalities. Finally, numerical examples are given to demonstrate the feasibility and superiority of the proposed theoretic result.
4. The finite-time synchronization problem is investigated for a class of neutral Markovian jumping complex networks with partly known transi-tion rates and mode-dependent delays. By utilizing the pinning control technique and constructing the appropriate stochastic Lyapunov functional, several sufficient conditions are proposed to ensure the finite-time synchro-nization for the neutral Markovian jumping complex dynamical networks, based on the Kronecker product, inequality techniques and finite-time sta-bility theorem. Finally, numerical examples and simulations are given to illustrate the feasibility and superiority of the proposed results.
5. It is concerned with the synchronization problem for a class of Marko-vian jump complex heterogeneous networks with partly unknown transition rates and time-varying delay. Based on the concept of quasi-synchronization, a novel stochastic Lyapunov functional is constructed to solve the problem. Then two sufficient quasi-synchronization conditions are presented, and ex-plicit expressions of error levels are proposed to estimate the synchroniza-tion error. Finally, numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed theoretic results.
1.针对带有马尔可夫跳跃参数的中立型时滞系统,在李雅普诺夫稳定性理论框架下,我们运用多李雅普诺夫泛函的方法,构造了一类带有三重积分项的李雅普诺夫泛函,并运用矩阵不等式,积分不等式,和凸组合的性质,推导出了时滞相关的随机稳定性准则。此外,将此结果推广到了外界非线性扰动的情况以及系统参数不确定的情况。通过数值仿真实例,与现有的一些文献相比在较大程度上降低了系统的保守性,从而验证所得结果的有效性和优越性。
2.针对模态转移率矩阵不完全确定的中立型马尔可夫跳跃系统,考虑其受到外界非线性扰动的情况以及系统跳跃参数不确定的情况,我们仍然在李雅普诺夫稳定性理论框架下,构造了一类带有三重积分项的随机李雅普诺夫泛函,运用矩阵不等式,积分不等式并结合往复凸引理(Reciprocally Convex Lemma)和自由权矩阵的方法,推导出了依赖模态和时滞相关的指数稳定性准则。通过一些数值仿真实例并与现有文献相比较,较大程度的提高了系统稳定的时滞上限,从而验证了本文所得结果的有效性和优越性。
3.针对模态转移率矩阵部分可知的一类带有分布时滞的马尔可夫跳跃中立型复杂网络,考虑其受到外界扇形有界的非线性扰动情况,我们利用采样数据控制技术,构造了一类带有三重积分项的随机李雅普诺夫泛函,运用Kronecker积的运算性质,矩阵不等式,积分不等式并结合往复凸引理和自由权矩阵的方法,推导出了依赖模态和时滞相关的指数同步准则。通过一些数值仿真实例并与最近有关的文献相比,在采样时间间隔方面有所提高,从而验证了本文给出的关于马氏跳跃中立型复杂网络同步准则的有效性和优越性。
4.针对模态转移率矩阵部分未知的一类马尔可夫跳跃中立型复杂网络,考虑其受到外界扇形有界的非线性扰动情况,我们采用牵制控制技术,在有限时间稳定性理论框架下,构造了一类带有权系数的随机李雅普诺夫泛函,运用Kronecker积的运算性质,矩阵不等式,积分不等式并结合自由权矩阵的方法,推导出了依赖模态和时滞相关的有限时间同步准则。通过一些数值仿真实例,与最近相关文献的同步时间相比较,验证了本文给出的关于马氏跳跃中立型复杂网络有限时间同步准则的有效性和优越性。
5.针对模态转移率矩阵部分未知的一类复杂融合网络(即按照马尔可夫过程切换规则由不同类型网络异构而成的复杂网络),考虑其异构的不同节点没有共同的平衡点,在此情况下我们定义所有节点状态的加权平均状态作为我们的虚拟同步目标状态,即引入拟同步(Quasi-synchronization)的定义,随后构造了一类带有权系数的随机李雅普诺夫泛函,运用Kronecker积的运算性质,矩阵不等式并结合自由权矩阵的方法,推导出了依赖模态和时滞相关的拟同步准则。通过一些数值仿真实例,验证了本文给出的关于此类复杂融合网络拟同步准则的有效性。
With the rapid development of science and technology, the system scale is becom-ing larger and the phenomena are becoming more and more complex. Due to the great application of neutral Markovian jumping systems, they have received close attention by scholars at home and abroad. On the other hand, with the rapid improvement of com-puter technology and network theory, complex dynamic network has become a new hot topic. Since neutral delay and Markovian switching are always existed in large number of networks, then it is of great importance and significance to investigate these neutral Markovian jump complex networks. The main results obtained in this dissertation can be concluded as follows.
1. The delay-range-dependent stochastic stability and exponential stabili-ty are investigated for the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations. A novel augment-ed Lyapunov functional which contains some triple-integral terms is intro-duced. Then by employing some integral inequalities and the nature of convex combination, some less conservative stochastic stability condition-s are presented in terms of linear matrix inequalities. Finally, numerical examples are provided to demonstrate the effectiveness and to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.
2. The exponential stability is investigated for neutral Markovian jump sys-tems with interval mode-dependent time-varying delays, nonlinear pertur-bations and partially known transition rates. A novel augmented stochastic Lyapunov functional is constructed, which employs the improved bounding technique and contains triple-integral terms to reduce conservatism, then the exponential stability criteria are developed by Lyapunov stability theo-ry, reciprocally convex lemma and free-weighting matrices. Moreover, the corresponding results are extended to the uncertain case. Finally, numeri-cal examples are given to illustrate the effectiveness and superiority of the proposed criteria.
3. The synchronization problem is investigated for a neutral complex dy-namical network with distributed delay, Markovian switching and partially unknown transition rates via sampled-data controller. A new augmented stochastic Lyapunov functional is constructed, which contains some triple-integral terms to reduce the conservativeness. Then the exponential stabili-ty conditions for the closed-loop error system are obtained by the Lyapunov stability theory, integral matrix inequalities and reciprocally convex lem-ma. Based on these new stability conditions, the sampled-data exponential synchronization controllers are found in terms of the solutions to linear ma-trix inequalities. Finally, numerical examples are given to demonstrate the feasibility and superiority of the proposed theoretic result.
4. The finite-time synchronization problem is investigated for a class of neutral Markovian jumping complex networks with partly known transi-tion rates and mode-dependent delays. By utilizing the pinning control technique and constructing the appropriate stochastic Lyapunov functional, several sufficient conditions are proposed to ensure the finite-time synchro-nization for the neutral Markovian jumping complex dynamical networks, based on the Kronecker product, inequality techniques and finite-time sta-bility theorem. Finally, numerical examples and simulations are given to illustrate the feasibility and superiority of the proposed results.
5. It is concerned with the synchronization problem for a class of Marko-vian jump complex heterogeneous networks with partly unknown transition rates and time-varying delay. Based on the concept of quasi-synchronization, a novel stochastic Lyapunov functional is constructed to solve the problem. Then two sufficient quasi-synchronization conditions are presented, and ex-plicit expressions of error levels are proposed to estimate the synchroniza-tion error. Finally, numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed theoretic results.
引文
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