Research Initiation Award: Robust Stabilization Using Phase Information

在鲁棒控制理论中，相位信息在很大程度上被忽略了，但在控制复杂多变量系统时，相位信息对于最大化可实现的性能是必不可少的。在这里，相位信息是指建模不确定性在频域中的相位的表征。在时间域中的真实参数不确定性建模提供了频率域中的相位信息。其目的是发展有效的方法来利用相位信息来分析和设计鲁棒控制器。建议研究与动力系统，特别是反馈控制系统的相位特性有关的许多相互关联的概念。虽然二次Lyapunov函数为许多鲁棒控制理论提供了基础，但与小增益(H-无穷)理论的联系说明了该方法在存在相位信息的情况下的局限性。二次Lyapunov函数的扩展，如参数依赖的Lyapunov函数和结构化Lyapunov函数是克服这些限制的一些想法。并提出了建立具有正实不确定性的鲁棒镇定的理论基础。作者最近得到的关于正实不确定性的基于Riccati的稳健性结果是这些研究的起点。最后，将经典的频域稳定性判据，如Popov判据和Circle判据推广到基于Riccati方程的状态空间控制器综合中。这一发展将为扇区有界非线性的控制器综合提供必要的工具。这项研究将提供一种新的稳健稳定方法--一种涉及相位信息的方法，与H无穷小增益方法相比，这种方法可以显著降低保守性。

Phase information has largely been neglected in robust control theory but is essential for maximizing achievable performance in controlling complex multivariable systems. Phase information, here, refers to the characterization of the phase of the modeling uncertainty in the frequency domain. Real parameter uncertainty modeling in the time domain provides phase information in the frequency domain. The objective is to develop effective methods for utilizing phase information in the analysis and design of robust controllers. A proposal is made to study numerous interrelated ideas that are relevant to phase properties of dynamical systems in general and feedback control systems in particular. Although quadratic Lyapunov functions provide the foundation for much robust control theory, connections with small-gain (H-infinity) theory illustrate limitations of this approach in the presence of phase information. Extensions of quadratic Lyapunov functions, such as parameter-dependent Lyapunov functions, and structured Lyapunov functions are some of the ideas being proposed to overcome these limitations. Also proposed, is to establish a theoretical foundation of robust stabilization with positive real uncertainty. Recent Riccati-based robustness results for positive real uncertainty obtained by the author form the starting point for these investigations. Finally, it is proposed to generalize classical frequency-domain stability criteria, such as the Popov and circle criteria, to state-space controller synthesis using a Riccati equation approach. This development will provide the necessary tools for controller synthesis for sector-bounded nonlinearities. This research will provide a new approach to robust stabilization--one that involves phase information that can be significantly less conservative than an H-infinity small-gain approach.