Discontinuous Integral Control for Systems with Arbitrary Relative Degree

Abstract

For systems with arbitrary relative degree, we propose a homogeneous controller capable of tracking a smooth but unknown reference signal, despite a Lipschitz continuous perturbation, and by means of a continuous control signal. The proposed control scheme consists of two terms: (i) a continuous and homogeneous state feedback and (ii) a discontinuous integral term. The state feedback term aims at stabilizing (in finite time) the closed-loop while the (discontinuous) integral term estimates the perturbation and the unknown reference signal in finite time and provides for perfect compensation in closed loop. By adding an exact and robust differentiator, we complete an output feedback scheme, when only the output is available for measurement. The global finite-time stability of the closed-loop system and its insensitivity with respect to the Lipschitz continuous perturbations are proved in detail using several (smooth) homogeneous Lyapunov functions for different versions of the algorithm.

Appendix: Some Technical Lemmas on Homogeneous Functions

We recall some useful lemmas needed for the development of our main results. Lemmas 2.6 and 2.7 are extensions of classical results for homogeneous continuous functions to semicontinuous ones [21, Theorems 4.4 and 4.1] (for their proofs see [12, 46]).

Lemma 2.5

([19] Young’s inequality) For any positive real numbers $a>0$, $b>0$, $c>0$, $p>1$, and $q>1$, with $\frac{1}{p}+\frac{1}{q}=1$, the following inequality is always satisfied:

$$ ab\le {\displaystyle \frac{c^{p}}{p}a^{p}+\frac{c^{-q}}{q}b^{q},} $$

and equality holds if and only if $a^{p}=b^{q}$.

And the lemma is given as follows.

Lemma 2.6

Let $\eta :\mathbb {R}^{n}\rightarrow \mathbb {R}$ and $\gamma :\mathbb {R}^{n}\rightarrow \mathbb {R}_{+}$ (resp. $\gamma : \mathbb {R}^{n} \rightarrow \mathbb {R}_{-}$) be two lower (upper) semicontinuous single-valued $\mathbf {r}$-homogeneous functions of degree $m>0$. Suppose that $\gamma \left( x\right) \ge 0$ $\left( \mathrm {resp.}\,\gamma \left( x\right) \le 0\right)$ on $\mathbb {R}^{n}$. If $\eta \left( x\right) >0$ $\left( \mathrm {resp.}\,\eta \left( x\right) <0\right)$ for all $x\ne 0$ such that $\gamma \left( x\right) =0$, then there is a constant $\lambda ^{*}\in \mathbb {R}$ and a constant $c>0$ such that for all $\lambda \ge \lambda ^{*}$ and for all $x\in \mathbb {R}^{n}\setminus \{0\}$,

$$ \eta (x)+\lambda \gamma (x)\ge c\left\| x\right\| _{\mathbf {r},p}^{m}\;,\;\left( \mathrm {resp.}\,\eta (x)+\lambda \gamma (x)\le -c\left\| x\right\| _{\mathbf {r},p}^{m}\right) \,. $$

Lemma 2.7

Let $\eta :\mathbb {R}^{n}\rightarrow \mathbb {R}$ be an upper semicontinuous, single-valued $\mathbf {r}$-homogeneous function, with weights $\mathbf{r} =[r_{1},\ldots ,r_{n}]^{\top }$ and degree $m>0$. Then there is a point $x_{2}$ in the unit homogeneous sphere $S=\{x\in \mathbb {R}^{n}:\Vert x\Vert _{\mathbf {r},p}=1\}$ such that the following inequality holds for all $x\in \mathbb {R}^{n}$:

$$\begin{aligned} \eta (x)\le \eta \left( x_{2}\right) \left\| x\right\| _{\mathbf {r},p}^{m}\,. \end{aligned}$$

(2.47)

Under the same conditions, if $\eta$ is lower semicontinuous, there is a point $x_{1}$ in the unit homogeneous sphere S such that the following inequality holds for all $x\in \mathbb {R}^{n}$:

$$\begin{aligned} \eta \left( x_{1}\right) \left\| x\right\| _{\mathbf {r},p}^{m}\le \eta (x)\,. \end{aligned}$$

(2.48)