【作者】 陈武华；

【导师】 关治洪；

【作者基本信息】 华中科技大学 ， 控制理论与控制工程， 2004， 博士

【摘要】 现实系统中普遍存在时滞现象,常常是引起系统不稳定和性能变差的主要原因。时滞系统的稳定性分析和控制器综合是控制理论研究的一个重要课题。现有的时滞系统稳定性结果可以分为两类:与时滞无关的稳定性和与时滞相关的稳定性。与时滞无关的稳定性是指系统参数满足的条件能够保证对所有的有界时滞系统都是稳定的,与时滞相关的稳定性是指系统参数满足的条件只保证当系统时滞小于某个上界时系统都是稳定的。在实际应用中,系统的时滞往往是有限的,而且其上界是知道的。因此,基于时滞无关的分析和设计可能是保守的。本文运用M-矩阵方法或奇异系统方法,并结合不等式技巧,研究了几类时滞系统与时滞相关的稳定性问题和与时滞相关的控制器设计问题。基于M-矩阵理论的方法,主要是运用M-矩阵理论,将对系统解的估计转化为对解的一个分量的估计,再利用不等式的技巧,来获取系统与时滞相关稳定性的充分条件。基于奇异系统的方法,首先将系统变换为等价的奇异系统,再构造合适的Lyapunov泛函,并利用不等式放大向量积,就可以基于线性矩阵不等式,得到系统稳定性与时滞相关的充分条件。最后利用已得到的与时滞相关的稳定性结果,设计所考虑的控制器。虽然本文主要考虑时滞系统与时滞相关的分析和控制问题,但我们也得到了与时滞无关新的结果。本文的工作和研究成果具体体现在:1. 运用M-矩阵理论和不等式技巧,研究了变时滞Lur’e间接控制系统绝对稳定性问题,将现有的关于常时滞Lur’e间接控制系统与时滞无关绝对稳定性结果推广到变时滞系统,并首次得到了与时滞相关的绝对稳定性条件。此外,我们成功地将上述思想方法推广到变时滞神经网络,基于M-矩阵,得到了变时滞神经网络指数稳定性与时滞相关的准则。2. 对时滞Lur’e直接控制系统,通过奇异系统变换,构造出一个新型Lyapunov泛函,并由此研究时滞Lur’e直接控制系统的绝对稳定性。基于线性矩阵不等式,得到了常时滞和有界变时滞系统绝对稳定性与时滞相关的充分条件。此外,我们还基于线性矩阵不等式,得到了常时滞Lur’e直接控制系统绝对稳定性与时滞无关的充分条件。数值例子显示了我们结果的有效性。3. 运用奇异系统方法和不等式技巧,研究了变时滞神经网络指数稳定性问题,基于线性矩阵不等式,得到了神经网络指数稳定性与时滞相关的充分条件。此外,我们还基于线性矩阵不等式,得到了常时滞神经网络指数稳定性与时滞无关的充分条件。数值例子表明,相对于已有的结果,我们的方法具有较少的保守性。4. 将奇异系统方法推广到It型随机时滞系统,构造出一个新型Lyapunov泛函,<WP=5>并由此研究随机时滞系统与时滞相关的稳定性、镇定及H∞控制问题。基于线性矩阵不等式,得到了具有非线性随机扰动项的不确定随机时滞系统均方指数稳定性与时滞相关的充分条件。此外,基于线性矩阵不等式,我们还得到了线性随机时滞系统状态反馈随机H∞控制器存在的与时滞相关的充分条件。5. 运用奇异系统方法研究了时滞系统与时滞相关的保成本控制问题。首先研究了不确定离散时滞系统及控制存在滞后的离散时间系统与时滞相关的保成本控制问题,基于线性矩阵不等式,得到了系统状态反馈保成本控制器存在的与时滞相关的充分条件。我们还研究了不确定离散时滞系统H∞范数意义下干扰抑制保成本控制问题。然后,我们研究了不确定连续时间时滞系统与时滞相关的输出反馈保成本控制问题,得到了基于矩阵不等式的输出反馈保成本控制器存在的与时滞相关的充分条件,并提出了构造次优输出反馈保成本控制器的迭代算法。数值仿真表明,即使对可用时滞无关方法镇定的系统,用我们的方法得到的成本上界要小于用时滞无关设计方法得到的成本上界。6. 运用奇异系统方法研究了不确定具有模式依赖时滞的连续时间及离散时间马尔可夫切换系统与时滞相关的随机稳定性、镇定及随机H∞控制问题。通过构造新型的随机Lyapunov泛函,基于耦合的线性矩阵不等式,得到了系统随机稳定性及状态反馈随机H∞控制器存在的与时滞相关的充分条件。此外,基于耦合的线性矩阵不等式,我们还得到了系统状态反馈保成本控制器存在的与时滞相关及与时滞无关的充分条件。7. 运用奇异系统方法, 并结合新的变量变换方法,研究了马尔可夫切换时滞系统与时滞相关的输出反馈镇定问题,并给出了寻找系统可输出反馈镇定的最大时滞的迭代算法。更多还原

【Abstract】 Time delay is commonly encountered in real systems, and its existence is frequently a source of instability and poor performance. The problem of stability analysis and controller synthesis for time-delay systems is an important topic in automatic control. The existing stability results for time-delay systems can be classified into two types: delay-independent stability and delay-dependent stability. In the first case, the stability condition that the system parameters satisfy is applicable for any bounded delay. In the second case, the stability condition that the system parameters satisfy only guarantee the system to be stable up to some maximum value for the time-delay. In applications, system delays are frequently limited, and their bounds are known. So, delay-independent stability analysis and controller synthesis may be conservative. In this paper, by combing M-matrix approach or descriptor system approach with inequality techniques, we study the problems of delay-dependent stability and delay-dependent controller design for several classes of time-delay systems. For the M-matrix approach, the idea is to transform the problem of estimating the solutions of the time-delay system into the problem of estimating one element of the solution by using M-matrix theory. Then using the inequality techniques, we can obtain delay-dependent conditions for stability of the time-delay system of concern. For the descriptor system approach, the idea is to transform the original system into a descriptor system with time-delay and construct an appropriate Lyapunov functional for the descriptor system. Then, applying the new Lyapunov functional and using inequality on product of two vectors, we can derive sufficient conditions for stability of the time-delay system of concern in terms of delay-dependent linear matrix inequalities. Finally based on the delay-dependent stability result, we can design the controllers to be considered. Although the purpose of this thesis is to study the problems of delay-dependent analysis and control for time-delay systems, we derive new results concerning delay-independent analysis and control. The main work and research results of this thesis lie in the following.1. By applying M-matrix theory and using inequality techniques, we study the problem of absolute stability for Lur’e indirect control systems with variable delays. We extend the existing delay-independent stability results for Lur’e indirect control systems with constant delays to the systems with variable delays. Moreover, we first obtain delay-dependent <WP=7>stability results for the above systems. In addition, we successfully apply this idea to study the problem of delay-dependent exponential stability for neural networks with variable delays, and derive delay-dependent criteria for exponential stability based on M-matrix.2. By using the descriptor system transformation, we construct a new type of Lyapunov functional to study the absolute stability for Lur’e direct control systems with delays. Based on linear matrix inequalities, we obtain sufficient conditions for delay-dependent absolute stability for the systems with constant or continuous bounded delays. We also derive delay-independent sufficient conditions for absolute stability for the systems with constant delays. Numerical examples illustrate the effectiveness of our methods. 3. By combining descriptor system approach with inequality techniques, we study the problem of exponential stability for neural networks with variable delays. Based on linear matrix inequalities, we obtain sufficient conditions for delay-dependent exponential stability for neural networks with constant or continuous bounded delays. We also derive delay-independent sufficient conditions for exponential stability for neural networks with constant delays in terms of linear matrix inequalities. Numerical examples are presented which show that our results are less conservative that the existing stability criteria.4. By extending the descriptor approach to the It-type stochastic system with delays,更多还原