截断误差有界的鲁棒$\mathcal{H}_\infty$静态输出反馈控制的多项式混沌方法
A Polynomial Chaos Approach to Robust $\mathcal{H}_\infty$ Static Output-Feedback Control with Bounded Truncation Error
摘要：
研究了线性时不变不确定系统的$\mathcal{H}_\infty$静态输出反馈控制问题，该系统对概率时不变参数不确定性具有多项式依赖性。应用多项式混沌理论，利用一个具有随机状态不确定性传播特性的高维扩展系统解决了控制综合问题。提出了一种闭环多项式混沌变换，导出了闭环扩展系统。该方法明确占闭环动力学，并保留了$\mathcal{L}_2$诱导的增益，这导致在较小的变换误差相比，现有的多项式混沌变换。使用有限次多项式混沌展开的效果首先捕获的范数有界的线性微分包含，然后通过制定一个强大的多项式混沌控制综合问题。该方法避免了使用高次多项式混沌展开来减轻截断误差的不稳定影响，从而显著降低了计算复杂度。此外，还对扩展系统的鲁棒稳定蕴含原系统鲁棒稳定的条件进行了分析。数值算例说明了该方法的有效性。
英文摘要：
This article considers the $\mathcal{H}_\infty$ static output-feedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the $\mathcal{L}_2$-induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. In addition, some analysis is given for the condition under which the robustly stabilized expanded system implies the robust stability of the original system. A numerical example illustrates the effectiveness of the proposed approach.
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