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。控制理论研究的是部分由用户规定其动力学的系统。用户控制的动态可以设计成各种各样的结果。与任何设计方法相关的控制系统的两个重要特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。可控性很重要的一个应用例子是卫星控制。如果希望在空间中重新定位卫星，首先要了解从给定的初始方向可以实现卫星的哪些方向。稳定，粗略地说，是使用控制使一个可能不稳定的工作点稳定，通常是通过使用反馈，即通过使用状态的测量来指定控制。稳定的一个简单例子是汽车的巡航控制系统，它控制油门以保持几乎恒定的速度。上面的简单例子说明了可控性和稳定性的效用。然而，这些也是基本的系统属性，因此理解系统的这些属性通常是理解系统本身的第一步。虽然对简单类系统的可控性和稳定性理解得很好，但对许多大型和有趣类系统的可控性和稳定性理解得很少。在本研究中，将研究可控性、稳定性问题以及它们之间的联系。所研究的系统是一般的非线性系统，其中微分几何的工具是有价值的。事实上，这些工具的新用途将是拟议研究的重要组成部分。该研究将推动控制理论的前沿理解和应用。与拟议的研究有关的研究生所接受的培训将使他们能够为加拿大的科学界和高科技经济作出贡献。

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.