Optimal design of two-degree-freedom control systems with preview feedforward based on noncausal inversion

本课题从基于双面拉普拉斯变换的模型匹配问题出发，研究了一种无限时间预览前馈控制器的设计问题。结果表明，基于预览的线性定常系统稳定反演等价于模型匹配问题的非calisal解。在已经建立了最优反馈控制器设计方法的情况下，可以在频域上方便地讨论前馈控制器。此外，采用与稳定反演相同的方法，我们推导了基于伴随迭代学习控制算法的传递函数表示。这表明该算法可以应用于无限维系统，如柔性结构。通过对柔性臂尖端控制的实验证明了这一点。其次，研究了采样数据系统的前馈控制。由于采样周期较短的采样数据系统的传递函数经常在单位圆外为零，因此人们认识到迭代学习控制（ILC）是一种获得系统反演的迭代方法，不能应用于这样的系统。为了克服这一困难，我们将非因果有界反演引入到采样数据系统中，并研究了该反演的极限性质。证明了第n个微分算子与包含在第n个积分元的脉冲传递函数中的n个欧拉算子之间的简单关系。基于这一关系，证明了采样数据系统的反演在非因果框架中近似于连续时间系统的反演。这一结果表明，ILC可以应用于该框架下的采样数据系统。

For this research project, it was investigated that the design problem of an infinite-time preview feedforward controller from the viewpoint of the model-matching problem based on the two-sided Laplace transform. It was shown that the so-mlled preview-based stable inversion for linear time-invariant systems is equivalent to the noncalisal solution to the model-matching problem. The preview feedforward controller can be easily discussed in the frequency domain where the design method for the optimal feedback controller has already been established. Moreover, with the same approach as stable inversion, we derived a transfer-function representation for the algorithm of adjoint-based iterative learning control. This indicates that the algorithm can be applied to infinite-dimensional systems such as flexible structures. This was demonstrated through an experiment on the tip control of a flexible arm.Next, preview feedforward control for sampled-data systems was investigated. Since the transfer function of sampled-data systems with a short sampling period frequently has zeros outside the unit circle, it is recognized that iterative learning control (ILC), which is an iterative approach to obtain an inversion of a system, cannot be applied to such systems. In order to overcome this difficulty, we introduced a non-causal bounded inversion to a sampled-data system and investigate the limiting property of the inversion. It was demonstrated that a simple relation between the n-th differential operator and n Euler operators that are included in the pulse transfer function of the n-th integral element. Based on this relation, it was proved that the inversion of the sampled-data system approximates its continuous-time counterpart in a non-causal framework. This result suggests that ILC can be applied to sampled-data systems in this framework.