Structural controllability of driftless bilinear control systems

Structural control;Bilinear systems

Tsopelakos, Aristomenis ; Belabbas ; Mohamed Ali

关键词: Structural control;    Bilinear systems;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/92774/TSOPELAKOS-THESIS-2016.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis addresses the structural controllability of driftless bilinear systems with sparse matrices. We begin with a rigorous introduction to the controllability of nonholonomic nonlinear systems. We present the notion of structural controllability and the fact that the controllability of linear systems is a generic property. We give a detailed presentation of the structural controllability of linear systems, based on Lin (1974). Afterwards, we proceed to the analysis of the structural controllability of driftless bilinear systems. We examine two cases; in the first case the matrices of the driftless bilinear system belong to a single vector space of matrices (single pattern case); in the second case the matrices belong to more than one vector spaces (multiple pattern case). After a rigorous presentation of the preliminaries of the theory of Lie algebras, we provide a theorem which claims that in the single pattern case, the driftless bilinear systems with more than two matrices can have a realization consisting of two matrices. This important result extends the theorem of Boothby (1975) about the realization of driftless bilinear systems. We prove that the controllability of driftless bilinear systems in both single and multiple pattern cases is a generic property. We define the notion of the graph which corresponds to a vector space of matrices and we establish necessary and sufficient conditions that relate the connectivity of this graph with the structural controllability of the driftless bilinear system in both cases. For the two patterns case, we provide a theorem which states that driftless bilinear systems with more than four matrices can have a realization with four matrices and we prove that similar propositions can be stated for more than two patterns.