不确定时滞系统的稳定性分析与镇定
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摘要
在实际的工程问题中,时滞现象广泛存在。时滞的存在使得对系统进行分析和综合变得更加复杂,同时也会导致系统不稳定和性能变差。此外,在针对实际控制系统的研究中,存在着模型简化、环境变化和元器件的老化,使得系统的数学模型与实际系统之间存在误差。长期以来,时滞现象和模型不确定因素一直是控制界研究的热点。
本论文的主题是时滞系统的稳定性分析与镇定。应用线性矩阵不等式方法与数值方法研究了延迟型和中立型时滞系统的稳定性与镇定问题。具体的研究内容包括:
1.针对线性离散时滞系统,采用非奇异状态变换技术对系统的稳定性与镇定进行了研究,并且在此基础上研究了线性区间离散时滞系统的区域稳定与镇定问题,提出了线性矩阵不等式表示的充分条件。区域镇定保证了闭环系统的所有极点位于给定圆盘内,并给出了状态反馈镇定控制器的设计方法。数值实例验证了该方法的可行性。
2.针对具有区间时滞的中立型时滞系统的稳定性与镇定进行了分析,在自由权矩阵方法的基础上,通过构造合适的Lyapunov函数,不采用任何形式的不等式约束,得到了中立时滞独立和中立时滞依赖的鲁棒稳定性判据,并在此基础上研究了鲁棒镇定的问题,得到了线性矩阵不等式表示的充分条件。数值实例验证了结论的有效性以及计算上的方便性,与现有文献的结论相比具有更小的保守性。
3.应用频域分析中的数值方法,研究了连续时滞系统的稳定与镇定问题,针对延迟型时滞系统的渐近稳定、α-稳定性、D-稳定性、鲁棒稳定以及中立型时滞系统的渐近稳定和鲁棒稳定等问题进行了分析,并且在此基础上分析了时滞系统的镇定问题。数值实例验证了数值方法简单明了,与时域方法相比,具有更小的保守性。
The delay phenomenon is prevalent in the industrial and engineering problems. The existence of time-delay makes it more complex to analyze and synthesize of such system. Meanwhile, delay is frequently a source of instability and performance degradation in many dynamic systems. In addition, in the research of real control systems, there exist the uncertainties between the mathematic model and controlled plant, which reflects the existing of model reduction, changes in the environment, the aging of components. Time-delay systems have been an enduring theme for several decades. A great number of monographs have been devoted to this field.
The stability analysis and stabilization of time-delay system is the subject of this thesis. Based on linear matrix inequality and numerical methods, the stability and stabilization problems of time-delay systems are studied in this paper. The main contents and novelty are outlined as follows:
1. For the linear discrete time-delay systems, the stability and stabilization problems are studied. Based on nonsingular state transformation, sufficient conditions of stability and stabilization are derived, in terms of linear matrix inequality. Then the conditions are improved in the linear interval discrete time-delay systems. Sufficient conditions of D-stability and D-stabilization are proposed for the interval systems. D -stabilization criterion guarantees all poles of the resulting closed-loop system to be placed in a specified circular region. The robust D -stabilizing controller designing method is also proposed. Numerical example shows the feasibility of the proposed method.
2. A new stability and stabilization criteria for a class of neutral systems with interval delays are proposed. Based on the Lyapunov functional method, neutral-delay-independent and neutral-delay-dependent robust stability conditions are derived in terms of linear matrix inequalities. Some free weighting matrices are introduced to reduce the conservatism, avoiding any form of inequality restriction. The stabilization conditions are also obtained based on some assumption. Numerical examples are given to show the effectiveness of the proposed criteria finally.
3. The stability and stabilizing problem of time-delay systems are studied by the frequency domain stability theory. Based on the numerical methods, we solve the asymptotical stability,α-stability, D -stability problems and robust stability problems of retarded and neutral time-delay systems. Furthermore, based on the numerical methods, the stabilization problem for linear time-delay systems is studied. Finally, numerical examples demonstrate the effectiveness and improvement of our methods.
本论文的主题是时滞系统的稳定性分析与镇定。应用线性矩阵不等式方法与数值方法研究了延迟型和中立型时滞系统的稳定性与镇定问题。具体的研究内容包括:
1.针对线性离散时滞系统,采用非奇异状态变换技术对系统的稳定性与镇定进行了研究,并且在此基础上研究了线性区间离散时滞系统的区域稳定与镇定问题,提出了线性矩阵不等式表示的充分条件。区域镇定保证了闭环系统的所有极点位于给定圆盘内,并给出了状态反馈镇定控制器的设计方法。数值实例验证了该方法的可行性。
2.针对具有区间时滞的中立型时滞系统的稳定性与镇定进行了分析,在自由权矩阵方法的基础上,通过构造合适的Lyapunov函数,不采用任何形式的不等式约束,得到了中立时滞独立和中立时滞依赖的鲁棒稳定性判据,并在此基础上研究了鲁棒镇定的问题,得到了线性矩阵不等式表示的充分条件。数值实例验证了结论的有效性以及计算上的方便性,与现有文献的结论相比具有更小的保守性。
3.应用频域分析中的数值方法,研究了连续时滞系统的稳定与镇定问题,针对延迟型时滞系统的渐近稳定、α-稳定性、D-稳定性、鲁棒稳定以及中立型时滞系统的渐近稳定和鲁棒稳定等问题进行了分析,并且在此基础上分析了时滞系统的镇定问题。数值实例验证了数值方法简单明了,与时域方法相比,具有更小的保守性。
The delay phenomenon is prevalent in the industrial and engineering problems. The existence of time-delay makes it more complex to analyze and synthesize of such system. Meanwhile, delay is frequently a source of instability and performance degradation in many dynamic systems. In addition, in the research of real control systems, there exist the uncertainties between the mathematic model and controlled plant, which reflects the existing of model reduction, changes in the environment, the aging of components. Time-delay systems have been an enduring theme for several decades. A great number of monographs have been devoted to this field.
The stability analysis and stabilization of time-delay system is the subject of this thesis. Based on linear matrix inequality and numerical methods, the stability and stabilization problems of time-delay systems are studied in this paper. The main contents and novelty are outlined as follows:
1. For the linear discrete time-delay systems, the stability and stabilization problems are studied. Based on nonsingular state transformation, sufficient conditions of stability and stabilization are derived, in terms of linear matrix inequality. Then the conditions are improved in the linear interval discrete time-delay systems. Sufficient conditions of D-stability and D-stabilization are proposed for the interval systems. D -stabilization criterion guarantees all poles of the resulting closed-loop system to be placed in a specified circular region. The robust D -stabilizing controller designing method is also proposed. Numerical example shows the feasibility of the proposed method.
2. A new stability and stabilization criteria for a class of neutral systems with interval delays are proposed. Based on the Lyapunov functional method, neutral-delay-independent and neutral-delay-dependent robust stability conditions are derived in terms of linear matrix inequalities. Some free weighting matrices are introduced to reduce the conservatism, avoiding any form of inequality restriction. The stabilization conditions are also obtained based on some assumption. Numerical examples are given to show the effectiveness of the proposed criteria finally.
3. The stability and stabilizing problem of time-delay systems are studied by the frequency domain stability theory. Based on the numerical methods, we solve the asymptotical stability,α-stability, D -stability problems and robust stability problems of retarded and neutral time-delay systems. Furthermore, based on the numerical methods, the stabilization problem for linear time-delay systems is studied. Finally, numerical examples demonstrate the effectiveness and improvement of our methods.
引文
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