This article presents graph theoretic conditions for the controllability and accessibility of bilinear systems over the special orthogonal group, the special linear group and the general linear group, respectively, in the presence of drift terms. The controlled terms are assumed to take place between pairwise states. Such bilinear systems naturally induce two interaction graphs: one graph from the drift, and another from the controlled dynamics. As a result, the system controllability or accessibility becomes a property of the two graphs in view of the classical Lie algebra rank condition. We establish a systemic way of transforming the Lie bracket operations in the underlying Lie algebra, into specific operations of removing or creating links over the drift and controlled interaction graphs. As a result, we establish a series of graphical conditions for the controllability and accessibility of such bilinear systems, which rely only on the connectivity of the union of the drift and controlled interaction graphs. We present examples to illustrate the validity of the established results, and show that the proposed conditions are in fact considerably tight.

　　Publication:

　　IEEE Transactions on Automatic Control (Volume: 68, Issue: 4, April 2023)

　　Author:

　　Xing Wang

　　Key Laboratory of Mathematics Mechanization, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

　　Bo Li

　　Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

　　Email: [email protected]

　　Jr-Shin Li

　　Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130, USA

　　Ian R. Petersen

　　Research School of Electrical, Energy and Materials Engineering, Australian National University, Canberra, ACT 0200, Australia.

　　Guodong Shi

　　Australian Center for Field Robotics, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia