摘要
现代控制理论自上世纪50年代诞生以来,得到了迅速发展,并且在许多领域取得了成功的应用.但是,现代控制理论在当前工业应用中也遇到了较大的困难.主要是因为在这些众多被控对象中,其动态特性一般都难以用精确的数学模型来描述,甚至有时能够得到精确的模型,但由于模型过于复杂,为了有效的进行分析和综合,我们要对系统进行简化处理.而最优控制和现代控制理论都是以精确的模型作为研究对象的,因此由简化后的系统进行分析和综合得到的控制器将难以达到预期的性能指标.因此,在系统建模时,需要考虑系统中经常遇到的时滞现象,不确定参数扰动以及随机因素等对系统的影响,即随机时滞系统的鲁棒H∞控制,也即设计相应的控制器使得随机闭环系统保持内部稳定和理想的性能要求.
本文主要研究了中立型随机时滞系统的鲁棒H∞控制和滤波器设计问题.针对几类中立型随机时滞系统,利用Lyapunov-Krasovskii泛函,时滞分割,自由权矩阵等时滞处理技巧,结合线性(非线性)矩阵不等式技术,建立了具有Markov跳跃的随机时滞系统的有界实引理,分别设计了随机动态输出反馈控制器和H∞滤波器,非脆弱鲁棒H∞控制器以及基于观测器的非脆弱鲁棒H∞控制器,时滞反馈鲁棒H∞控制器,得到了一些研究成果.本文的主要工作和贡献如下：
针对一类具有变时滞的中立型随机不确定系统,通过构造Lyapunov泛函,研究了随机闭环系统在无干扰输入和不确定满足可容许条件下的随机鲁棒镇定问题,以及在非零的外部干扰输入下,以半线性矩阵不等式(BMI)的形式给出了系统满足鲁棒H∞性能指标的随机动态输出反馈控制器的设计方法,并利用分支切割算法给出了半线性矩阵不等式的可解方法.最后针对该类系统,用类似的方法,以线性矩阵不等式的形式,给出了全阶鲁棒H∞滤波器的设计方法.
给出了具有Markov跳跃的随机系统鲁棒H∞控制的定义,以及相应性能指标的一些数学描述.然后综合应用Doob鞅不等式,随机积分不等式,Markov链的遍历性,建立了随机Markov跳跃的有界实引理.利用该引理结合线性矩阵不等式(LMI)技术对一类随机Markov跳跃系统的鲁棒H∞控制进行了研究,设计出了状态反馈鲁棒H∞控制器.
针对一类具有Markov跳跃的中立型随机时滞系统,通过构造一个Lyapunov-Krasovskii泛函,首先建立了时滞依赖的随机Markov跳跃有界实引理,然后利用该引理讨论了在加性和乘性控制器不确定增益下,非脆弱鲁棒H∞控制器的设计.并在此基础上研究了随机闭环系统在无干扰输入下基于观测器的随机鲁棒镇定问题,以及在非零的外部干扰输入下,设计出了系统满足鲁棒H∞性能指标的基于观测器的非脆弱鲁棒H∞控制器.该类控制器和观测器对它的增益误差是不敏感或非脆弱的.
针对一类非线性中立型随机时滞系统,通过对时滞进行分割,构造相应的Lyapunov-Krasovskii泛函,结合随机积分不等式,非线性随机分析等数学工具,基于时滞反馈的思想,以线性矩阵不等式的形式,给出了使得随机闭环系统在无干扰输入下随机鲁棒镇定的时滞相关的充分条件,以及在非零的外部干扰输入下,系统满足鲁棒H∞性能指标的鲁棒H∞控制器的设计方法.在推导过程中,没有使用模型变换以及交叉项有界等可能产生保守性的方法.并引入了合适的自由权矩阵,减小了控制器的保守性和算法的复杂度.
最后对全文工作进行了总结,并指出了下一步的研究方向.总之,本文关于中立型随机时滞系统的鲁棒H∞和滤波器设计的研究,不仅丰富了随机系统鲁棒H∞控制理论,而且拓广了随机系统鲁棒H∞控制的研究方法,数值仿真例子也说明了文中结论的正确性和方法的有效性.
Modern control theory, which came into being in the 1950s, has been quickly devel-oped, and has been successfully applied in all kinds of engineering area. But, it has subjected to plenty of difficulty in the modern industrial applications. The main reason is that the dy-namics character is difficult to describe by the accurate mathematics model for the plenty of control objects. Even sometimes the precise mathematics model can be obtained, but the given model is so complex that we have to simplify the systems for effective analysis and synthesis. However, the accurate model is the research object in optimal control theory and modern control theory. So, it is difficult to satisfy the expected performance index for the controller, which is analyzed and synthesized for the reduced systems. Therefore, in the system modeling, we have to consider the effectiveness of time-delay, which is frequently appeared in the systems, uncertain parameter disturbance and stochastic factor etc. That is stochastic robust H∞control theory, which is to design corresponding controller such that the systems are inner stochastic stable and satisfy expected performance index.
The dissertation focuses on the investigation of the robust H∞control and filter design for neutral stochastic time-delay systems. For several different classes of neutral stochas-tic time-delay systems, in virtue of the approach of Lyapunov-Krasovskii functional, delay partition, free weighting matrices etc. time-delay methods, together with linear (nonlinear) matrix inequality technique, stochastic Markov jump bounded real lemma is obtained, and the stochastic dynamic output feedback controller, H∞filter, non-fragile robust H∞, con-troller, based on observer non-fragile robust H∞controller, delay feedback H∞controller are designed respectively. Some results are also presented. The main research work and contribution of this thesis are listed as follows:
Through constructing a Lyapunov functional, without the disturbance input and the un-certainties satisfy admissible conditions, the stochastic stabilization problems are studied. Under the nonzero disturbance input, a full-order stochastic dynamic output feedback con-troller is designed by solving a bilinear matrix inequality (BMI). At last a full-order filter is designed for all admissible uncertainties and time-varying delay by similar approach, which is expressed in the form of linear matrix inequality (LMI).
The conception of stochastic robust H∞control for stochastic Markov jump systems is given, and the mathematical description of corresponding performance index is presented. Then, together with Doob martingale inequality, Borel-Cantelli lemma, stochastic integral inequality, ergodic theory of Markov chain, the stochastic Markov jump bounded real lemma is presented. Combing the lemma with linear matrix inequality technique, the robust H∞control problem for a class of stochastic system with Markov jump parameter is discussed, and the state feedback robust H∞controller is designed.
For a class of neutral stochastic delay system with Markov jump parameter, by em-ploying a Lyapunov-Krasovskii functional, the delay-dependent stochastic Markov jump bounded real lemma is presented at first. Then, non-fragile robust H∞controller is de-signed by using the lemma, the controller gain are variable in additive form or multiplying form. The non-fragile observer-based stabilization and H∞control problems for the neutral stochastic hybrid systems with time-varying delay are studied. The delay-dependent suffi-cient conditions for the existence of the non-fragile H∞controller and observer are given, which is non-fragile or resilient with respect to errors in the controller coefficients and ob-server coefficients. Under the control of the non-fragile observer-based H∞controller, the resulting closed-loop system not only is robust stochastic exponential stable in mean square but also satisfies the H∞performance index.
Dividing the delay interval into multiple segments, different weighting matrices in the new Lyapunov-Krasovskii functional are chosen. Based on the ideal of delay feedback, combining the stochastic integral inequality with nonlinear stochastic analysis, the delay-dependent sufficient criteria of stabilization and robust H∞control problems for a class of neutral nonlinear stochastic delay system are discussed in term of linear matrix inequality form. In the derivative process, neither model transformations nor boundedness techniques for cross terms are employed, which possibly produced the conservatism. A appropriate free weight matrix is also introduced, which reduced the conservatism of the controller and the complexity of the algorithm.
Finally, the concluding remarks are summarized, and the future works which may be further investigated are presented. Overall, the study on robust H∞and filter design for neutral stochastic delay systems in this dissertation not only enriches the stochastic robust H∞control theory, but also extends the approach for stochastic robust H∞control. Numerical examples illustrate the validity of the results and the effectiveness of the proposed methods.
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