Local null controllability of the control-affine nonlinear systems with time-varying disturbances. Direct calculation of the null controllable region

TL;DR

This paper investigates local null controllability of control-affine nonlinear systems with time-varying disturbances, providing algebraic criteria and explicit calculations of the null controllable region, including conditions for global null controllability.

Contribution

It introduces a new algebraic approach and a Gronwall-type inequality to determine the null controllable region for systems with disturbances.

Findings

Explicit calculation of null controllable regions.

Criteria involving algebraic manipulations of system matrices.

Conditions for global null controllability when disturbances are zero.

Abstract

The problem of local null controllability for the control-affine nonlinear systems $\dot x(t)=f(x(t))+Bu(t)+w(t),$ $t\in[0,T]$ is considered in this paper. The principal requirements on the system are that the LTI pair $\left((\partial f/\partial x)(0), B\right)$ is controllable and the disturbance is limited by the constraint $|f(0)+w(t)|\leq M_d\left(1-\frac{t}{T}\right)^\eta,$ $M_d\geq0$ and $\eta>0.$ These properties together with one technical assumption yield a complete answer to the problem of deciding when the null controllable region have a nonempty interior. The criteria obtained involve purely algebraic manipulations of vector field $f,$ input matrix $B$ and bound on the disturbance $w(t).$ To prove the main result we have derived a new Gronwall-type inequality allowing the fine estimates of the closed-loop solutions. The theory is illustrated and the efficacy of proposed controller is demonstrated by the examples where the null controllable region is explicitly calculated. Finally we established the sufficient conditions to be the system under consideration (with $w(t)\equiv 0$) globally null controllable.