Small-controllability of two-dimensional state-affine nonlinear systems

Abstract

For most nonlinear systems, it is in general a difficult task to obtain algebraic controllability criteria. In this paper, we consider small-controllability of discrete-time state-affine nonlinear systems. We first focus on the systems in dimension two with single-input and improve a previous controllability criterion. That is, we derive a sufficient algebraic criterion for small-controllability of the systems, which is easier to apply than the previous one. We then show that if the state-affine nonlinear systems are bilinear, a necessary and sufficient algebraic criterion for small-controllability can be obtained using invariant sets. We also extend the derived controllability criteria to continuous-time state-affine nonlinear systems and to discrete-time multi-input state-affine nonlinear systems. Examples are given to illustrate the derived controllability criteria of this paper.

Keywords:

• State-affine nonlinear systems
• bilinear systems
• small-controllability
• algebraic conditions

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 Here, A can always be put in canonical form $\tilde{A}\triangleq PA{P}^{-1}$ by a nonsingular linear transformation P. Similarly, $\tilde{B}\triangleq PB{P}^{-1}$ and $\tilde{x}\triangleq Px$. From Definition 2.1, this transformation does not affect the small-controllability.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 61973014 and 61573044].