On Near‐controllability of Discrete‐time Upper‐triangular Bilinear Systems with Applications to Controllability

Abstract

Near-controllability is defined for those systems that are uncontrollable but have a large controllable region. It is a property of nonlinear control systems introduced recently, and it has been demonstrated on two classes of discrete-time bilinear systems. This paper studies near-controllability of discrete-time upper-triangular bilinear systems, which are uncontrollable and are more general than the two classes mentioned. A necessary and sufficient condition for the systems in dimension two to be nearly controllable is presented, which covers the existing results. For the systems with high dimensions, necessary conditions and sufficient conditions of near-controllability are provided, which generalize the existing results. In particular, the obtained near-controllability results are applied to controllability of discrete-time bilinear systems. An example also is given to demonstrate the effectiveness, which shows that the controllability problems of discrete-time bilinear systems can be solved by near-controllability.In this subsection, we consider near-controllability of two-dimensional discrete-time upper-triangular bilinear systems, i.e. system (2), and obtain a necessary and sufficient condition. We first show that system (1) can always be transformed into system (2) if A, B in (1) have a real eigenvector in common. Let n = 2 and ξ be the common real eigenvector of A, B. Denote P −1 by

On Controllability and Near-controllability of Multi-input Discrete-time Bilinear Systems in Dimension Two

“…From Remark 2 A has two real eigenvectors 1 −1 T , 3 −4 T of which the former is also an eigenvector of B 1 , B 2 . In particular, one can verify that span 1 −1 T is invariant for system (19). By x (k) = P −1 x (k) where…”

“…This property was first defined and was demonstrated on two classes of discrete-time bilinear systems [16,17], and it was then generalized to continuoustime bilinear systems and to continuous-time and discrete-time nonlinear systems that are not necessarily bilinear in [18]. Recently, the near-controllability problems were raised in [19] for discrete-time uppertriangular bilinear systems which are more general than those considered in [16,17], and necessary conditions and sufficient conditions for near-controllability were derived. However, the results in [16,17,19] are for single-input systems only and the study on the topic of near-controllability is just at the beginning.…”

“…Recently, the near-controllability problems were raised in [19] for discrete-time uppertriangular bilinear systems which are more general than those considered in [16,17], and necessary conditions and sufficient conditions for near-controllability were derived. However, the results in [16,17,19] are for single-input systems only and the study on the topic of near-controllability is just at the beginning.…”