On the interplay of controllability and stabilisation

该提案旨在支持首席研究员及其研究生在几何控制理论领域的研究。控制理论是对其动力学可以部分由用户指定的系统的研究。用户控制的动态可以被设计来实现各种结果。与任何设计方法相关的控制系统的两个重要属性是可控性和稳定性。粗略地说，可控性是研究系统从给定的初始状态可以达到哪些状态。可控性非常重要的一个应用示例是卫星控制。如果希望在太空中重新定位卫星，首先需要了解从给定的初始方向可以实现卫星的哪些方向。粗略地说，稳定是使用控制来使可能不稳定的工作点稳定，通常通过使用反馈，即通过使用状态测量来指定控制。稳定性的一个简单例子是汽车中的巡航控制系统，它控制油门以保持几乎恒定的速度。上面的简单例子表明了可控性和稳定性的实用性。然而，这些也是基本的系统属性，因此了解系统的这些属性通常是了解系统本身的第一步。虽然对于简单类型的系统，可控性和稳定性已得到很好的理解，但对于许多大型且有趣的系统类型，它们仍然知之甚少。在拟议的研究中，将研究可控性、稳定性问题以及它们之间的联系。研究的系统是一般的非线性系统，微分几何工具在其中很有价值。事实上，这些工具的新颖使用将是拟议研究的重要组成部分。该研究将推进控制理论的理解和应用的前沿。研究生接受的与拟议研究相关的培训将使他们能够为加拿大的科学界和高科技经济做出贡献。

The proposal is to support the research of the principal investigator and his graduate students in the area of geometric control theory. Control theory is the study of systems whose dynamics can be partially prescribed by the user. The user-controlled dynamics can be designed to achieve a variety of outcomes. Two important properties of a control system connected to any design methodology are controllability and stabilisation. Roughly speaking, controllability is the study of which states of the system can be reached from a given initial state. An example of an application where controllability is important is in satellite control. If one wishes to reorient a satellite in space, it is first useful to understand which orientations of the satellite can be achieved from a given initial orientation. Stabilisation, roughly speaking, is the use of control to make a possibly unstable operating point stable, typically by the use of feedback, i.e., by using measurements of the state to specify the control. A simple example of stabilisation is the cruise control in a car which controls the throttle to maintain a nearly constant speed. The simple examples above suggest the utility of controllability and stabilisation. However, these are also fundamental system attributes, and so understanding these for a system is often a first step towards understanding the system per se. While controllability and stabilisation are well understood for simple classes of systems, they remain poorly understood for many large and interesting classes of systems. In the proposed research, the problems of controllability, stabilisation, and the connections between them will be studied. The systems studied are general nonlinear systems, where the tools of differential geometry are valuable. Indeed, the novel use of these tools will be an essential part of the proposed research. The research will advance the cutting edge of the understanding and application of control theory. The training received by graduate students connected to the proposed research will enable them to contribute to Canada's scientific community and high-tech economy.