Interpolation-Based Numerical Algorithms in Robust Control

在鲁棒控制中，控制器的设计考虑了实际过程与其描述模型之间的差异。这在实践中具有重要意义，因为数学模型只能大致描述一个真实的过程。因此，不仅对于标称模型，而且对于一族模型，都需要保证期望的性能要求，如抑制外部干扰和良好的参考跟踪，以及闭环系统的稳定性。因此，例如可以解决建模和逼近误差，在这个意义上，必须理解对模型不确定性的鲁棒性。本项目的目标是开发用于具有大状态空间维度和/或时滞的动态系统的(鲁棒)H_无穷控制器的新设计技术。对于这样的系统，许多经典的方法已经不再有效，例如，不能利用所涉及的系统矩阵的稀疏结构，这导致对内存和计算时间的高要求。因此，在本项目中，需要开发和分析基于内插的控制器设计算法。插值法用于生成降阶的线性模型，从而可以有效地进行控制器设计和鲁棒性分析。因此，计算适当的降阶模型是至关重要的，因为这些模型必须包含关于不受控制系统的容易激励的频率的信息，以便能够通过控制器有效地抑制具有这些频率的干扰。因此，提出在迭代过程中设计降阶模型和控制器。这样，就可以利用当前控制器的信息来更新降维模型，从而改进控制器。在这个项目中应该实现以下目标：*为大规模广义系统设计H-无穷控制器，*为时滞系统设计H-无穷控制器，*开发包含不确定性的鲁棒控制器，*为大系统设计有效的H-无穷回路成形，*在积分二次约束下的控制器设计。为了实现这些目标，首先应该从理论上分析这些问题，然后在一个有良好文档和测试的软件包中实现，该软件包应该公开可用。与已有的方法相比，预计将具有以下独特的优势：*通用性和广泛的适用性，*高效率，*适应性，*适合数据驱动的控制器设计。

In robust control, the discrepancy between a real process and the model chosen for its description, is taken into account for controller design. This is of essential significance in practice, since mathematical models can only describe a real process approximately. Therefore, it is necessary that desired performance requirements such as the suppression of external disturbances and a good reference tracking, as well as the stability of the closed-loop system are not only guaranteed for the nominal model but for a family of models. Thereby, modeling and approximation errors can for example be addressed and in this sense, robustness against model uncertainties has to be understood.The goal of this project is the development of novel design techniques for (robust) H-infinity controllers for the case of dynamical systems with a large state-space dimension and/or with delays. For such systems, many classical methods are no longer efficient, since, e. g., the sparsity structure of the involved system matrices cannot be exploited which leads to high requirements in memory and computational time. Therefore, in this project interpolation-based algorithms for controller design should be developed and analyzed. The interpolation is used to generate reduced linear models for which controller design and robustness analysis can be carried out efficiently. Thereby, the computation of appropriate reduced-order models is of essential importance, since these must contain information about the easily excitable frequencies of the uncontrolled system in order to be able to attenuate disturbances with these frequencies effectively by the controller. Thus, it is proposed to design the reduced model and the controller in an iterative process. In this way, information of the current controller can be used to update the reduced-order model and thus to improve the controller. The following goals should be achieved within this project:* design of H-infinity controllers for large-scale descriptor systems,* design of H-infinity controllers for delay systems,* development of robust controllers including uncertainties,* efficient H-infinity loop shaping for large-scale systems,* controller design under integral quadratic constraints.In order to achieve these goals, the problems should first be theoretically analyzed and then implemented in a well documented and tested software package which shall be made publicly available. In comparison to established methods, the following particular advantages are expected:* generality and broad applicability,* high efficiency,* adaptivity,* suitability for data-driven controller design.