2D系统的有限频域滤波方法研究

2-D systems are widely applied in various engineering fields such as digital image processing, iterative learning control, seismic data analysis and so on. At present, most of the existing filtering theory of 2-D systems assumes that the noise inputs uniformly distribute in the entire frequency domain, which nevertheless, is inconsistent with the fact that the energy of many actual noise signals usually lies in finite frequeny ranges. Hence, this discrepency leads to remarkable conservatism for these filtering results when applied to practice. In view of the above problem, this application project will investigate filtering theory of 2-D systems in finite frequency domain. Firstly, we research the generalized KYP lemmas for 2-D systems, and obtain finite frequency performance analysis conditions for several 2-D system models; then by taking the finite frequency property of practical noises into account, we study the finite frequency filtering problem of 2-D systems and present full- and reduced-order design methods of finite frequency filters; in the next, based on the obtained filtering results, the problem of fault detection of 2-D systems will be solved in the finite frequency framework for several typical fault cases; finally, by virtue of the obtained generalized KYP lemmas for 2-D systems, new design methods of 2-D digital filters are proposed, which will be applied to design digital image filters for the space rendezvous-and-docking ground simulation system. The goal of this project is to develop systematic methods for finite frequency filtering analysis and design for 2-D systems, improving and completing the existing 2-D system filtering theory.

2D系统具有很强的工程背景，在数字图像处理、迭代学习控制、地震数据分析等工程领域具有广泛的应用。目前，2D系统的滤波理论大都假设噪声的分布是全频范围的，与噪声信号的能量往往只集中在某些有限频率段的实际情况不符，因而在实际应用时存在很大的保守性。针对上述关键问题，本项目即在2D系统的有限频域滤波理论方面展开研究。首先研究2D系统的广义KYP引理，获得多种2D系统模型的有限频域性能分析准则；其次针对实际噪声的有限频域特性，研究2D系统的有限频域滤波问题，给出有限频域滤波器的全阶和降阶设计方法；在此基础上，针对多种类型故障信号，进一步解决2D系统有限频域故障检测问题；最后，利用得到的2D系统广义KYP引理，研究2D数字滤波器设计新方法，并将该方法应用于空间交会对接地面仿真系统的数字图像滤波器设计。本项目的宗旨是提出一套完整的2D系统有限频域滤波器分析与设计方法，完善现有2D系统滤波理论。

本项目系统深入地研究了二维系统的有限频域滤波分析与综合。首先建立了二维FM模型的广义KYP引理，给出了二维系统的一般二次性能的分析方法，成功统一了已有的正实引理和有界实引理，充分考虑了FM模型的结构特征以降低结果保守性。提出了研究二维系统稳定性和时滞系统时滞无关稳定性的频率分割方法，得到的稳定性充要条件便于进一步拓展的复杂系统综合问题。构建了基于广义KYP引理的系统综合问题的研究框架，提出了新颖的分离系统矩阵和Lyapunov矩阵的方法。在此基础上，研究了时滞系统的全阶有限频域H无穷滤波和二维系统的鲁棒全阶有限频域H无穷滤波，提出了保守性更低的线性矩阵不等式设计方法；考察了线性系统的降阶有限频域H无穷滤波和二维数字滤波器的有限频域H无穷模型降阶，给出了计算滤波器和降阶模型的凸优化方法。此外，解决了有理传递函数的有限频域逼近问题，并成功用于线性相位IIR滤波器的设计和DS调制器的噪声传递函数优化。