On small-controllability and controllability of a class of nonlinear systems

Abstract

In this paper, small-controllability and controllability of a class of nonlinear systems are studied. Both the discrete-time case and continuous-time case are considered. By proposing an implicit function approach, sufficient conditions for the systems respectively to be small-controllable and controllable are obtained. Examples are also provided to illustrate the results of the paper.

Keywords:

• nonlinear systems
• discrete-time systems
• bilinear systems
• controllability
• small-controllability
• implicit function approach

Acknowledgment

The author wishes to thank the anonymous reviewer for his/her constructive comments and suggestions.

Notes

1. From the definition, small-controllable systems can be controllable with arbitrarily small control inputs since μ can be chosen arbitrarily small. It is a stronger property than the general constrained controllability, and it would be useful in practice since in real systems the control inputs are often constrained.

2. Since v ∈ O(0^{m(M + 1)}, μ) and ‖v_i‖₂ ⩽ ‖v‖₂, we have ‖v_i‖₂ < μ and hence ‖v_i‖₂ ≤ μ for i = 0, 1, … , M.

3. If i = 0, then for j = 1, … , m; if i = L − 1, then for  j = 1, … , m.

Additional information

Funding

This work was supported by the China Postdoctoral Science Foundation funded project [grant number 2012T50035] and the National Natural Science Foundation of China [grant number 61203231], [grant number 61203022].