时滞系统稳定性分析及其在网络控制中的应用
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摘要
时滞现象由于其深刻的实际背景已经引起广泛的关注.时滞通常是系统不稳定和性能恶化的主要根源之一.利用现有的时滞系统稳定性分析方法得到的往往是充分条件,降低结果的保守性是控制算法实用化、工业化的关键环节,目前还缺乏有关稳定性准则保守性的系统论述.
针对标称线性时滞系统,本论文提出稳定性准则的等价性定义,深入研究稳定性准则的保守性问题.基于Lyapunov-Krasovskii稳定性理论,采用线性矩阵不等式工具,对奇异时滞系统、Lur'e时滞系统、Markov跳变时滞系统以及时滞细胞神经网络系统进行稳定性分析,并结合系统各项性能指标给出控制器的设计方法.进而以网络控制系统为应用背景,深入研究网络控制系统特有的问题,如:时滞、丢包和量化等对系统性能的影响,设计控制器使得网络闭环系统渐近稳定.最后,以蒸馏塔为研究对象,将本论文提出的状态反馈迭代控制方法应用于蒸馏塔控制系统仿真.论文的主要工作如下:
(1)针对现有的线性时滞系统稳定性分析方法,从理论上证明了其中某些方法的等价性,改变了目前以数值例子来验证结果有效性的处理方法,进而给出判断自由变量是否是冗余变量的有效手段,得出自由变量法更适于多胞型不确定时滞系统的结论.
(2)应用积分不等式处理方法,分别导出了奇异时滞系统、Lur'e时滞系统、Markov跳变时滞系统以及时滞细胞神经网络系统的时滞相关稳定性条件,并提出了基于线性矩阵不等式求解的状态反馈稳定化控制器设计方法.从理论上证明了所导出的结果具有更小的保守性.其中奇异时滞系统的稳定性条件同时保证了系统正则、无脉冲.进一步,结合系统的二次性能指标,给出了奇异时滞系统和Lur'e时滞系统保性能控制器的设计方法,结合系统的增益性能和相位性能指标,给出Markov跳变时滞系统耗散性控制器的设计方法.
(3)应用积分不等式处理方法,导出具有时变时滞的网络控制系统稳定性条件,该条件克服了传统方法对时滞变化上界的依赖,允许时滞是快变的,更符合网络控制的实际情况.进一步研究丢包对状态估计的影响以及丢包的补偿策略,并结合系统的扰动抑制性能指标,给出使得时滞最大化的H_∞控制器设计方法.最后,给出保证闭环网络控制系统稳定的量化反馈控制器的设计方法,并揭示了数据丢包率、时滞和平均数率之间的量化关系.
(4)应用本论文给出的算法对蒸馏塔控制系统进行仿真,仿真结果验证了该方法的可行性和良好的性能.这一部分内容是时滞系统理论面向实际工程问题的尝试性应用.
Time-delay phenomenon has received much attention due to its profound practical background and time-delays are often the source of instability and degradation in control performance. By using the existing stability analysis methods, sufficient conditions are often obtained and reducing the conservatism is important for the practicality and industrialization of the control algorithm. However, there are few systematical discussions about the conservatism of the stability criteria.
For nominal linear time-delay systems, the equivalence of stability criteria is defined and the conservatism of the stability criteria is deeply analyzed. Based on Lyapunov-Krasovskii stability theory, the stability analysis of singular time-delay systems、Lur'e time-delay systems、time-delay systems with Markovian parameters、delayed cellular neural networks systems are proposed in this dissertation and the sufficient conditions for the existence of the controller are derived with respect to the performance. Some characteristics of network control systems are deeply studied, such as: time-delay, packet-drop out, quantization and the state feedback controller is designed which guarantees the closed-loop system stable. Finally, state feedback control algorithm developed in this dissertation is applied in the process of a distillation column control system simulation. The main contents of this dissertation are outlined as follows:
(1) For linear time-delay systems, some stability analysis methods are proved theoretically to be equivalent, which is different from the regular practice that most of the criteria are compared via numerical examples. Furthermore, a method is given to judge if a slack variable is redundant. Finally, we concluded that the slack variable method is more suitable to deal with polytopic uncertain systems.
(2) For singular time-delay systems、Lur'e time-delay systems、time-delay systems with Markovian parameters and delayed cellular neural networks systems, the integral inequality method is used to derive the delay-dependent stability criteria, based on which the controller design problem are formulated as the solvability of some iterative linear matrix inequalities. It is proved theoretically that the results obtained in this dissertation are less conservative than some existing results. The stability criteria obtained for singular time-delay systems also guarantee the system regular and impulse free. The controller is designed which satisfied the proposed guaranteed cost and dissipative performance.
(3) For the networked control systems with time-varying time-delay, the integral inequality method is used to derive the stability criteria. The criteria are independent of the delay derivative bound, e.g., the delay considered here is allowed to be fast-varying, which is more suitable for networked control systems. Furthermore, the analysis is focused on how the packet dropping affects state estimation and what we can do to compensate this unreliability. The H_∞controller is obtained which allows a maximum allowable delay size for a fixed disturbance rejection performance value. Finally, the quantized feedback controller is designed which guarantees the Lyapunov stability of closed-loop systems. Besides, the relationship among the packet-dropping rate, time delay and the average data rate is also obtained in the meanwhile.
(4) The state feedback control strategy developed in this dissertation is applied for the distillation column control system simulation. Simulation results demonstrate the validity and excellent performance of this method. This part constitutes an attempt of applying the Lyapunov stability idea to practical engineering problems.
针对标称线性时滞系统,本论文提出稳定性准则的等价性定义,深入研究稳定性准则的保守性问题.基于Lyapunov-Krasovskii稳定性理论,采用线性矩阵不等式工具,对奇异时滞系统、Lur'e时滞系统、Markov跳变时滞系统以及时滞细胞神经网络系统进行稳定性分析,并结合系统各项性能指标给出控制器的设计方法.进而以网络控制系统为应用背景,深入研究网络控制系统特有的问题,如:时滞、丢包和量化等对系统性能的影响,设计控制器使得网络闭环系统渐近稳定.最后,以蒸馏塔为研究对象,将本论文提出的状态反馈迭代控制方法应用于蒸馏塔控制系统仿真.论文的主要工作如下:
(1)针对现有的线性时滞系统稳定性分析方法,从理论上证明了其中某些方法的等价性,改变了目前以数值例子来验证结果有效性的处理方法,进而给出判断自由变量是否是冗余变量的有效手段,得出自由变量法更适于多胞型不确定时滞系统的结论.
(2)应用积分不等式处理方法,分别导出了奇异时滞系统、Lur'e时滞系统、Markov跳变时滞系统以及时滞细胞神经网络系统的时滞相关稳定性条件,并提出了基于线性矩阵不等式求解的状态反馈稳定化控制器设计方法.从理论上证明了所导出的结果具有更小的保守性.其中奇异时滞系统的稳定性条件同时保证了系统正则、无脉冲.进一步,结合系统的二次性能指标,给出了奇异时滞系统和Lur'e时滞系统保性能控制器的设计方法,结合系统的增益性能和相位性能指标,给出Markov跳变时滞系统耗散性控制器的设计方法.
(3)应用积分不等式处理方法,导出具有时变时滞的网络控制系统稳定性条件,该条件克服了传统方法对时滞变化上界的依赖,允许时滞是快变的,更符合网络控制的实际情况.进一步研究丢包对状态估计的影响以及丢包的补偿策略,并结合系统的扰动抑制性能指标,给出使得时滞最大化的H_∞控制器设计方法.最后,给出保证闭环网络控制系统稳定的量化反馈控制器的设计方法,并揭示了数据丢包率、时滞和平均数率之间的量化关系.
(4)应用本论文给出的算法对蒸馏塔控制系统进行仿真,仿真结果验证了该方法的可行性和良好的性能.这一部分内容是时滞系统理论面向实际工程问题的尝试性应用.
Time-delay phenomenon has received much attention due to its profound practical background and time-delays are often the source of instability and degradation in control performance. By using the existing stability analysis methods, sufficient conditions are often obtained and reducing the conservatism is important for the practicality and industrialization of the control algorithm. However, there are few systematical discussions about the conservatism of the stability criteria.
For nominal linear time-delay systems, the equivalence of stability criteria is defined and the conservatism of the stability criteria is deeply analyzed. Based on Lyapunov-Krasovskii stability theory, the stability analysis of singular time-delay systems、Lur'e time-delay systems、time-delay systems with Markovian parameters、delayed cellular neural networks systems are proposed in this dissertation and the sufficient conditions for the existence of the controller are derived with respect to the performance. Some characteristics of network control systems are deeply studied, such as: time-delay, packet-drop out, quantization and the state feedback controller is designed which guarantees the closed-loop system stable. Finally, state feedback control algorithm developed in this dissertation is applied in the process of a distillation column control system simulation. The main contents of this dissertation are outlined as follows:
(1) For linear time-delay systems, some stability analysis methods are proved theoretically to be equivalent, which is different from the regular practice that most of the criteria are compared via numerical examples. Furthermore, a method is given to judge if a slack variable is redundant. Finally, we concluded that the slack variable method is more suitable to deal with polytopic uncertain systems.
(2) For singular time-delay systems、Lur'e time-delay systems、time-delay systems with Markovian parameters and delayed cellular neural networks systems, the integral inequality method is used to derive the delay-dependent stability criteria, based on which the controller design problem are formulated as the solvability of some iterative linear matrix inequalities. It is proved theoretically that the results obtained in this dissertation are less conservative than some existing results. The stability criteria obtained for singular time-delay systems also guarantee the system regular and impulse free. The controller is designed which satisfied the proposed guaranteed cost and dissipative performance.
(3) For the networked control systems with time-varying time-delay, the integral inequality method is used to derive the stability criteria. The criteria are independent of the delay derivative bound, e.g., the delay considered here is allowed to be fast-varying, which is more suitable for networked control systems. Furthermore, the analysis is focused on how the packet dropping affects state estimation and what we can do to compensate this unreliability. The H_∞controller is obtained which allows a maximum allowable delay size for a fixed disturbance rejection performance value. Finally, the quantized feedback controller is designed which guarantees the Lyapunov stability of closed-loop systems. Besides, the relationship among the packet-dropping rate, time delay and the average data rate is also obtained in the meanwhile.
(4) The state feedback control strategy developed in this dissertation is applied for the distillation column control system simulation. Simulation results demonstrate the validity and excellent performance of this method. This part constitutes an attempt of applying the Lyapunov stability idea to practical engineering problems.
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