Optimization of sampled-data systems with finite-dimensional linear continuousperiodic process and delay

该项目涉及解决线性采样数据系统的H2和L2优化问题的方法的发展。该系统由一个具有任意纯延迟的线性连续周期多输入多输出（MIMO）过程和一个在数字处理器上以程序形式实现的MIMO控制器组成。连续和数字部分之间的连接是由一个理想的模数转换器和一个零阶保持器完成的，脉冲的形状可以自由选择。采样周期T与连续过程周期T同步，使两者相称。设N是一个自然数。然后应该研究以下情况：Ts=T（等时钟），Ts=T/N（插值）和Ts=NT（抽取）。这些程序应该产生精确的解，并且还必须考虑样本间行为。尽管这类问题具有重要的理论和实践意义，但目前还没有系统的文献。其原因可能是缺乏合适的数学工具。所选择的研究方法是基于参数传递矩阵（PTM）的概念。在之前的研究项目中，申请人在理论方向上对该概念进行了完善，并与合作伙伴成功地将其应用于各种采样数据控制问题。当连续周期过程+延迟+采样这三个困难同时存在时，尚不知道能够提供精确解的替代方法。在求解上述H2和L2优化问题的过程中，需要进行以下步骤：*用Floquet-Lyapunov变换提取动力学并将其托管在定常部分*建立开链的PTM；*各稳定控制器参数化集的构造* H2范数和代价泛函的构造*解析解的Wiener-Hopf法的推广与应用*软件工具的编程及其与Matlab-Toolbox DirectSD的集成

The project concerns the development of methods for the solution of H2 and L2 optimization problems for linear sampled-data systems. The systems consist of a linear continuous periodic multi-input multi-output (MIMO) process with arbitrary pure delays, and a MIMO controller, which is realized as a program on a digital processor. The connection between the continuous and digital parts is accomplished by an ideal analogue-to-digital converter and a zero-order hold with free selectable shape of the pulses. The sampling period Ts is synchronized with the period of the continuous process T, such that both commensurate. Let N>1 be a natural number. Then the following cases should be investigated: Ts=T (equally clocked), Ts=T/N (interpolating), and Ts=NT (decimating).The procedures should yield exact solutions and they also have to consider the intersample behavior. Despite of the theoretical and practical importance of the named class of problems, there does not exist systematic literature about it. The reason for that could be the absence of an appropriate mathematical tool.The chosen approach for the investigations is based on the parametric transfer matrix (PTM) concept. In preceding research projects, the applicant has refined this concept in theoretical directions, and in cooperation with his partners he also successfully applied it to various sampled-data control problems. Alternative approaches are not known, which are able to provide exact solutions, when all three difficulties are present:continuous periodic process + delay + sampling.On the way to the solution of the above formulated H2 and L2 optimization problems, the following steps have to be taken:* Floquet-Lyapunov transformation for extracting the dynamics and hosting them in the time-invariant part* Establishing the PTM for the open chain, and then for the closed loop* Construction of the parametrized set of all stabilizing controllers* Construction of the H2 norm and of the cost functional* Extension and application of the Wiener-Hopf method for the analytical solution* Programming of software tools and their integration into the Matlab-Toolbox DirectSD