摘要
奇异摄动系统的固有特性是两时间尺度(Two-Time-Scale),这个特性通常会导致数学模型是“刚性”(Stiff)的。而为了消除微分方程的刚性问题而提出的奇异摄动方法(也叫快慢分解方法),已经发展的相当成熟,并且也取得了很多有意义的成果。快慢分解方法是在两个时间尺度内分别独立完成设计任务,实际上是一种时标的分解。与时域上的两时间尺度性质相对应的是频域上的两频域尺度(Two-Frequency-Scale),即低频和高频。其中慢子系统对低频信号敏感,快子系统对高频信号敏感。而与时标分解方法,即在两个时间尺度内分别独立完成设计任务对应的是,在每个频域尺度上分别考虑相关设计任务,这种方法到目前为止还未见文献涉及。
鉴于此,本论文在不同的频域尺度上考虑了奇异摄动系统的鲁棒控制问题,其中主要讨论了H_∞控制和正实性能问题。至于时滞奇异摄动的鲁棒控制问题,非线性奇异摄动系统的鲁棒控制和离散奇异摄动系统的广义H_2等问题,其分频设计的结果还有待进一步研究,目前本论文只在不进行快慢动态分解意义上的统一时域上,讨论了其鲁棒控制问题。
综上,本论文完成了以下工作。
(1)从频域分解角度研究了奇异摄动系统的H_∞控制,相对于以前的对H_∞范数在全频段上优化,本论文在经典快慢动态分解和广义KYP引理方法的基础上,分别在奇异摄动系统慢动态的低频段,和快动态的高频段上设计H_∞控制器,(此处的H_∞范数是限制在局部频段上的)最后复合得到原系统的H_∞控制器,并且为了说明本文方法的有效性,以跟踪和H_∞模型匹配问题为例,对在全频段考虑H_∞范数优化和在部分频段上考虑所带来的设计效果的不同进行了比较。
(2)从频域分解角度研究了奇异摄动系统的正实性问题,在广义KYP引理方法的基础上,研究了奇异摄动系统低阶子系统的传递函数在其对应频段上的正实性和原系统传递函数正实性之间的关系。对奇异摄动系统的正实性能分析来说,本文的结论是已有结论的推广。
(3)在统一时间域内,研究了不确定时滞奇异摄动系统的鲁棒稳定及镇定的问题。通过建立改进的Lyapunov-Krasovskii泛函,得到了时滞奇异摄动系统鲁棒稳定的线性矩阵不等式形式的充分条件。并在此基础上设计了鲁棒镇定控制器,其分频设计的结果还有待进一步研究。
(4)在统一时间域内,研究了不确定模糊奇异摄动系统的鲁棒控制,包括离散和连续两种情况。特别是对离散情形,相对于已有文献给出的控制器存在的非线性矩阵不等式判别条件,本文通过引入矩阵变量,给出了判定上述系统鲁棒镇定的线性矩阵不等式条件。另外,还研究了基于快采样的离散模糊奇异摄动系统的l_2—l_∞控制。同样用线性矩阵不等式给出了控制器存在的充分条件,同样,其分频设计的结果还有待进一步研究。
另外,我们通过数值算例说明了本论文方法的有效性。
The inherited property of singularly perturbed systems (SPS) is two-time scale (TTS) property, which always leads to mathematical models that are "stiff". The "curse" of dimensionality coupled with stiffness poses formidable computational complexities for the analysis and control of the systems. To overcome these problems, the singular perturbation and time scale methods were put forward. These techniques had attained a certain level of maturity in theory of continuous control systems described by ordinary differential equations. In fact, the idea of time scale technique is to carry out the design procedure in each two different time scale. The two-frequency scale (TFS) property in frequency domain, which is in according to the property of TTS in time domain, poses high- and low-frequency categories. Here the slow subsystem of singular perturbed systems is sensitive to the low-frequency signals, while the fast subsystem is sensitive to the high-frequency signals. However, so far as we know, seldom research starts from the frequency decomposition points to deal with the related problem of SPS.
The aim of this dissertation was to develop a frequency decomposition technique for the positive realness performance analysis and H_∞control design problem of SPS based on the generalized KYP lemma. Besides this, we made further study on singularly perturbed systems in unified time scale, which involves time delay singularly perturbed systems, T-S fuzzy singularly perturbed systems and discrete-time singularly perturbed systems, the frequency decomposition results need to be further studied.
To sum up, the major works in this dissertation are as follows.
Firstly, H_∞control of the singularly perturbed systems was studied from a new perspective. Unlike the traditional H_∞norm optimization in all frequencies, here we considered the optimization in finite frequencies. Based on generalized KYP lemma, a low frequency controller and a high frequency controller was designed, respectively, for the slow and fast subsystems. A composite controller was constructed by the solutions of the above reduced order systems. A tracking problem and a H_∞model matching problem were provided to show the priority to the traditional H_∞control of SPS.
Secondly, the positive realness property of singularly perturbed systems was studied from a new perspective. Based on generalized KYP lemma, the positive realness property of the reduced subsystems in the corresponding frequency domain was studied. It was shown in this dissertation that the positive realness property in finite frequency ranges for the full-order system can be inferred by the reduced subsystems.
Thirdly, in the unified time scale, the robust stability and robust stabilization of time delay singularly perturbed systems with uncertainty was studied. By using improved Lyapunov-Krasovskii functional, a sufficient condition in terms of LMI was proposed to ensure the singularly perturbed systems to be robustly stable. On the basis of which, a stabilizing controller was designed.
Fourthly, in the unified time scale, the problem of robust fuzzy control for nonlinear singularly perturbed systems described by a T-S fuzzy model was studied. Two cases of the T-S fuzzy systems with parametric uncertainties, both continuous-time and discrete-time cases were considered. LMI-based sufficient conditions were proposed for stabilization of the controlled systems via fuzzy state feedback controllers. Especially for the discrete case, unlike the existing results, by introducing new matrix variables, LMI-based sufficient condition was proposed to ensure the fuzzy discrete-time systems robustly stable. Furthermore, generalized H_2 control for the fast sampling discrete-time nonlinear singularly perturbed systems described by T-S fuzzy model was studied.
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