基于时滞/时滞导数二维分解的时滞系统分析与设计

By analyzing the limitations of widely adopted delay-decomposition methods, a complete delay-decomposition approach for constructing delay-interval-dependent Lyapunov functional will be investigated. Furthermore, the idea of the complete delay-decomposition approach will be further extended to handle the information of delay-derivative. As a result, a newly delay/delay-derivative two-dimensional decomposition approach will be proposed to construct Lyapunov functional. In addition, the convex combination method will be applied to obtain delay-dependent stability conditions. Based on the framework of delay/delay-derivative two-dimensional decomposition, the problems of delay-dependent robust stabilization and robust performance design will be discussed. For both state feedback and output feedback control,the relationship among robust stability analysis, robust stabilization and robust performance design will be estabilished. Furthermore, delay-dependent robust stabilization and robust performance conditions will be investigated and the corresponding robust stabilization control and robust performance synthesis algorithms will be presented. The delay/delay-derivative two-dimensional decomposition approach will be able to overcome the limitation of the traditional method and its conservativeness. It is believed that the research results of this project will provide an efficient and feasible method for the analysis and design of time-delay control systems. They will be very important in the field of control theory research.

针对时变时滞系统中普遍采用的时滞分解方法存在的局限性，研究更加有效的基于完全时滞分解的时滞区间依赖Lyapunov泛函方法。在此基础上，将完全时滞分解的思想推广应用于处理时滞导数信息，从二维空间的角度，研究时滞/时滞导数二维分解的Lyapunov泛函方法，进一步应用凸组合方法获得时滞相关鲁棒稳定性条件；同时在时滞/时滞导数二维分解的总体框架下，讨论时滞相关鲁棒镇定和鲁棒性能设计问题。基于状态反馈和输出反馈两种情形，建立时滞相关鲁棒稳定性分析、鲁棒镇定和鲁棒性能设计之间的内在联系，寻找时滞相关鲁棒镇定和时滞相关鲁棒性能的条件，研究基于时滞/时滞导数二维分解的时滞相关鲁棒镇定和鲁棒性能设计方法，以及相应的鲁棒镇定控制算法和鲁棒性能综合算法；时滞/时滞导数二维分解方法将有效克服传统方法的局限性和由此带来的保守性。研究结果将为时滞系统分析和设计提供一种有效可行的新方法，在理论上具有重要的科学意义。

本课题研究时滞系统的稳定性分析与设计问题，针对普时滞分解方法存在的局限性，提出了更低保守性的完全时滞分解方法；针对时变时滞系统的稳定性问题，充分利用时滞和时滞导数信息，建立了更低保守性的时滞相关稳定条件；针对时滞系统分析中普通存在的积分项界定问题，提出了更低保守性的自由矩阵不等式方法，相比以往Jessen不等式和Wirtinger不等式，这一方法具有更低的保守性并且无需引入逆凸组合技术；针对控制器的设计问题，提出了改进的锥补迭代算法，这一方法减少了控制器求解过程的迭代次数。基于上述新的技术，讨论了时滞系统的时滞相关鲁棒镇定、无源性分析和耗散设计问题，实验结果验证了所提方法的有效性。项目实施期间，完成论文共24篇，15 篇为SCI收录论文，9篇为EI收录论文其中两篇论文发表在国际控制领域顶级权威刊物《Automatica》和《IEEE Transactions on Automatic Control》。这些研究成果丰富了时滞系统的理论研究及应用。