Stabilization of two-dimensional nonlinear systems through barrier-function-based integral sliding-mode control: application to a magnetic levitation system

Abstract

The sliding-mode control (SMC) has been used successfully to stabilize nonlinear systems, yet a side effect of SMC is chattering, an undesired phenomenon. Researchers have then introduced the so-called integral-type approach to diminish chattering. Additional studies have shown that SMC with the so-called barrier function can drive the system state to a region close to the origin. The main contribution of this paper is to join both setups in the context of SMC, i.e., integral term and barrier function. We show that SMC with both integral term and barrier function can be used to track any given trajectory while ensuring both stability and chattering-free performance. The potential of this novel control for applications is illustrated through real-time experiments performed on a magnetic levitation system.

Appendix

1.1 Proof of Proposition 1

From (1) and (2), we have

$$\begin{aligned} {\dot{e}}(t)={\dot{x}}_{1r}(t)-x_2(t), \quad \forall t\ge 0. \end{aligned}$$

(36)

Taking the derivative with respect to the time on both sides of (36), and substituting (1) into the corresponding expression, we obtain (for all $t>0$)

$$\begin{aligned} \begin{array}{l} \ddot{e}(t)=\ddot{x}_{1r}(t)-f(t)-b(t)u(t)-w(t),\\ \dddot{e}(t)=\dddot{x}_{1r}(t)-{\dot{f}}(t)-{\dot{b}}(t)u(t)-b(t){\dot{u}}(t)-{\dot{w}}(t). \end{array} \end{aligned}$$

(37)

Now, recall from (3) that

$$\begin{aligned} s(t)=c_1e(t)+ {\dot{e}}(t),\quad \forall t > 0, \end{aligned}$$

(38)

and from (5) that

$$\begin{aligned} \sigma (t)={\dot{s}}(t)+\beta s(t)+\alpha \int _0^t s(\tau ) d\tau , \quad \forall t> 0. \end{aligned}$$

(39)

As a result,

$$\begin{aligned} {\dot{\sigma }}(t)&=\ddot{s}(t)+\beta {\dot{s}}(t)+\alpha s(t) \nonumber \\&=c_1\ddot{e}(t)+\dddot{e}(t)+\beta \left( c_1{\dot{e}}(t)+ \ddot{e}(t) \right) + \alpha s(t) . \end{aligned}$$

(40)

Substituting (37) into (40) yields (11). This argument completes the proof. $\square$

1.2 Summary of the SMC control approach given in [52]

This section presents the SMC control method, which is borrowed from [52] and adapted to our MagLev device. In [52], the uncertainty part of the system is bounded by a known positive function, which is not available in our approach. Therefore, we suppose that we know an upper bound of the un-modeled part of the MagLev system. It then follows that the result in [52, Thm. 1] can be written as

$$\begin{aligned} \left\{ \begin{array}{l} u(t)=G^{-1}\left( x_{2d}-x_{1r}-(K+\delta )|\alpha {\dot{e}}_I|\frac{s}{|s|}-\delta \right) \\ s=e(t)+\alpha e_I(t)\\ {\dot{e}}_I(t)=e^{\frac{q}{p}}(t), \end{array}\right. \end{aligned}$$

(41)

with $p>q>0$, $\delta >0$ and $K>0$ control parameters to be tuned. The function G depends on the disturbance upper bound.