Controllability of Continuous-Time Bilinear Systems Without Using the Lie-Algebraic Conditions: A New Perspective

• 成果类型: Article
• 署名作者: Tie, Lin
• 署名单位: Beihang University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3170374
• 发表日期: 2023
• 页码: 2094-2108
• 关键词: Controllability; Nonlinear systems; Algebra; Nonhomogeneous media; Linear systems; Task analysis; Transforms; Bilinear systems; continuous-time systems; CONTROLLABILITY; discrete-time systems; inhomogeneous systems; Lie-algebraic conditions; state-affine nonlinear systems

摘要

Lie algebra methods are powerful for studying controllability of continuous-time nonlinear systems and most controllability criteria of continuous-time nonlinear systems are Lie-algebraic conditions. However, the Lie-algebraic conditions are in general difficult to verify if the system dimension is high. In this article, we propose a new perspective to study controllability of continuous-time bilinear systems without using the Lie-algebraic conditions. Specifically, we first consider controllability of the bilinear systems in the single-input case under a commutativity condition. We show that, although the Lie algebra rank condition for the homogeneous counterparts, which is necessary for a classical controllability result to work, does not fit such bilinear systems, they can still be controllable. Our approach to proving controllability is using controllability of the discrete-time counterparts of the continuous-time systems, and we derive a necessary and sufficient controllability criterion without the Lie algebra rank condition, which is algebraically verifiable for any finite dimension. More importantly, through this controllability study, we propose a new perspective to deal with the controllability problems of continuous-time bilinear systems by changing the verification of the Lie algebra rank condition to solving two linear algebra problems. That is, we establish a new method beyond Lie algebra methods to controllability of continuous-time bilinear systems. We also generalize the proposed perspective to continuous-time state-affine nonlinear systems to obtain algebraic controllability criteria. Examples are given to illustrate the obtained controllability results of this article.