Saturated finite-time stabilization of uncertain nonholonomic systems in feedforward-like form and its application

Abstract

This paper investigates the problem of finite-time stabilization by state feedback for a class of uncertain nonholonomic systems with inputs saturation. Comparing with the existing relevant literature, a distinguishing feature of the systems under investigation is that the x-subsystem is in feedforward-like form. Rigorous design procedure for saturated finite-time state feedback control is presented by using the adding a power integrator and the nested saturation methods. The development of saturated finite-time controller is also presented briefly for a class of dynamic nonholonomic systems in feedforward-like form. An application example for a kinematic hopping robot is provided to illustrate the effectiveness of the proposed approach.

Appendix

Proof of Theorem 1

The proof is proceeded in two steps. In the first step, it is proved that the control law with coefficients $\beta _{i}$’s preset in (33) ensures that all states will converge to a region determined by the saturation function $\sigma (\cdot )$. Then, the saturated controller (32) reduces to the unsaturated controller (23). As a result, the global finite-time stability for the closed-loop system (15) with (32) can be guaranteed. Since the proof is quite similar to that of Theorem 4.1 in [40], we only briefly present the first step of the proof.

Step 1 At this stage, we will find a time instant $t_{1}$ in such a way that for $t\ge t_{1}$

$$\begin{aligned} X_{n}(t)\in Q_{n}=\left\{ X_{n}:|x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t))|<\varepsilon \right\} \nonumber \\ \end{aligned}$$

(53)

By contradiction, it can be shown that there is a time instant $t_{1}$ such that

$$\begin{aligned} \displaystyle |x_{n}^{1/r_{n}}(t_{1})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}))|\le \frac{\varepsilon }{2} \end{aligned}$$

(54)

Otherwise, we assume that $|x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1})(t)|>\varepsilon /2$ for all $t\ge 0$. Now, the case when

$$\begin{aligned} \displaystyle x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t)) >\frac{\varepsilon }{2},\quad \forall t\ge 0 \end{aligned}$$

(55)

is considered. In this case, for all $t\ge 0$, by (15) and (32), we have

$$\begin{aligned} \displaystyle \dot{x}_{n}(t)= & {} -d_{n} \beta _{n}\sigma ^{r_{n+1}}\Big (x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t))\Big )\nonumber \\\le & {} -c_{n1} \beta _{n}(\varepsilon /2)^{r_{n+1}}\nonumber \\:= & {} -\mu _{n}\varepsilon ^{r_{n+1}} \end{aligned}$$

(56)

with $\mu _{n}=c_{n1} \beta _{n}(1/2)^{r_{n+1}}>0$, which implies that $x_{n}(t)<x_{n}(0)-\mu _{n}\varepsilon ^{r_{n+1}}t$,$\quad \forall t\ge 0$. Consequently, as time goes to infinity, $x_{n}(t)\rightarrow -\infty$, which leads to a contradiction disavowing (55) by noticing the fact $|v_{n-1}^{1/r_{n}}(X_{n-1}(t))|\le \beta _{n-1}^{1/r_{n}}\varepsilon$. Similarly, we can show the case $\displaystyle x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t)) <-\varepsilon /2,\quad \forall t\ge 0$, is also impossible. In conclusion, there must exist a time instant $t_{1}$ such that (54) holds.

Next, we will prove that the following holds after the time instant $t_{1}$

$$\begin{aligned} \displaystyle |x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t))| <\varepsilon ,\quad \forall t\ge t_{1} \end{aligned}$$

(57)

If (57) is not true, there exists at least one time instant $t_{1}^{*}$ such that $|x_{n}^{1/r_{n}}(t_{1}^{*})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))|=\varepsilon$. Specifically, there are $t_{1}^{'}<\infty$ and $t_{1}^{*}<\infty$ such that either

$$\begin{aligned}&x_{n}^{1/r_{n}}(t_{1}^{'})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))=\varepsilon /2 \end{aligned}$$

(58)

$$\begin{aligned}&x_{n}^{1/r_{n}}(t_{1}^{*})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))=\varepsilon \end{aligned}$$

(59)

$$\begin{aligned}&\displaystyle \frac{\varepsilon }{2}\le x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t))\le \varepsilon ,\quad t\in [t_{1}^{'},t_{1}^{*}]\nonumber \\ \end{aligned}$$

(60)

in the positive region, or $x_{n}^{1/r_{n}}(t_{1}^{'})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))=-\varepsilon /2$, $x_{n}^{1/r_{n}}(t_{1}^{*})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))=-\varepsilon$, $-\varepsilon \le x_{n}^{1/r_{n}}(t)-v_{n-1}^{1/r_{n}}(X_{n-1}(t))\le -\varepsilon /2$, $\forall t\in [t_{1}^{'},t_{1}^{*}]$ as the negative case.

In what follows, it will be claimed that the positive case (58)–(60) is impossible. By (60) and (56), we have

$$\begin{aligned} \displaystyle \dot{x}_{n}(t)\le -\mu _{n}\varepsilon ^{r_{n+1}},\quad t\in [t_{1}^{'},t_{1}^{*}] \end{aligned}$$

(61)

which leads to

$$\begin{aligned} \mu _{n}\varepsilon ^{r_{n+1}}(t_{1}^{*}-t_{1}^{'})\le x_{n}(t_{1}^{'})-x_{n}(t_{1}^{*}) \end{aligned}$$

(62)

By (58), (59) and the fact that $|v_{n-1}^{1/r_{n}}(X_{n-1})|\le \beta _{n-1}^{1/r_{n}}\varepsilon$, using Lemma 3, we obtain

$$\begin{aligned} \displaystyle x_{n}(t_{1}^{'})= & {} \Big (\frac{\varepsilon }{2}+v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))\Big )^{r_{n}}\nonumber \\\le & {} (1+\beta _{n-1}^{1/r_{n}})^{r_{n}}\varepsilon ^{r_{n}}\le (1+\beta _{n-1})\varepsilon ^{r_{n}} \end{aligned}$$

(63)

and

$$\begin{aligned} \displaystyle x_{n}(t_{1}^{*})= & {} \Big (\varepsilon +v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))\Big )^{r_{n}}\nonumber \\\ge & {} -(1+\beta _{n-1}^{1/r_{n}})^{r_{n}}\varepsilon ^{r_{n}}\ge -(1+\beta _{n-1})\varepsilon ^{r_{n}} \end{aligned}$$

(64)

Combining (63) with (64), from (62), the following time estimate is obtained:

$$\begin{aligned} \displaystyle t_{1}^{*}-t_{1}^{'}=\frac{ x_{n}(t_{1}^{'})-x_{n}(t_{1}^{*})}{\mu _{n}\varepsilon ^{r_{n}+1}}\le \frac{ 2}{\mu _{n}}(1+\beta _{n-1})\varepsilon ^{-\omega _{1}} \end{aligned}$$

(65)

Further, in light of (62), it yields that $x_{n}(t_{1}^{*})\le x_{n}(t_{1}^{'})$ which implies

$$\begin{aligned}&x_{n}^{1/r_{n}}(t_{1}^{*})-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))\le x_{n}^{1/r_{n}}(t_{1}^{'})\nonumber \\&\quad -\,v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'})) +v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))\nonumber \\&\quad -\,v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*})) \end{aligned}$$

(66)

A direct substitution of (58) and (59) into (66) leads to

$$\begin{aligned} \varepsilon /2\le |v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))| \end{aligned}$$

(67)

By considering (60) and the fact that $|v_{n-1}^{1/r_{n}}(X_{n-1})|\le \beta _{n-1}^{1/r_{n}}\varepsilon$, one has

$$\begin{aligned}&\displaystyle |x_{n}(t)|\le (1+\beta _{n-1}^{1/r_{n}})^{r_{n}}\varepsilon ^{r_{n}}\nonumber \\&\quad \le (1+\beta _{n-1})\varepsilon ^{r_{n}},\quad t\in [t_{1}^{'},t_{1}^{*}] \end{aligned}$$

(68)

Under this condition, applying Lemma 6 and combining (65) yields

$$\begin{aligned}&\displaystyle |v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))|\nonumber \\&\quad \le \alpha _{n}(\cdot )\varepsilon ^{1+\omega _{1}} (t_{1}^{*}-t_{1}^{'})\nonumber \\&\quad \le \frac{ 2}{\mu _{n}}(1+\beta _{n-1})\alpha _{n}(\cdot )\varepsilon \end{aligned}$$

(69)

By the definition of $\mu _{n}$ and the choice of (33), we have

$$\begin{aligned} \displaystyle \mu _{n}= & {} c_{n1} \beta _{n}(1/2)^{r_{n+1}}\nonumber \\&\displaystyle \ge c_{n1} (1/2)^{r_{n+1}} 2^{r_{n+1}}\nonumber \\&\quad \Big (\frac{4(1+\beta _{n-1})\alpha _{n-1}(\cdot )+1+2c_{n2}}{c_{n1}}\Big )\nonumber \\&\displaystyle >4(1+\beta _{n-1})\alpha _{n-1}(\cdot ) \end{aligned}$$

(70)

Substituting (70) into (69) and noticing (67), we obtain

$$\begin{aligned} \varepsilon /2\le |v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{*}))-v_{n-1}^{1/r_{n}}(X_{n-1}(t_{1}^{'}))|<\varepsilon /2\nonumber \\ \end{aligned}$$

(71)

which obviously is a contradiction. Therefore the case of (58)–(60) will never happen. Similarly, it can be shown, using an almost same argument as the positive case, that $v_{n-1}^{1/r_{n}}(X_{n-1}(t))-v_{n-1}^{1/r_{n}}(X_{n-1}(t))$ will never cross $-\varepsilon$. Hence for $t\le t_{1}$ we have

$$\begin{aligned} |v_{n-1}^{1/r_{n}}(X_{n-1}(t))-v_{n-1}^{1/r_{n}}(X_{n-1}(t))|<\varepsilon \end{aligned}$$

(72)

Following the same line shown in the first step, at the final step, we can obtain that there exists a time instant $t_{n}$, such that when $t\ge t_{n}$, $X_{n}(t)$ will enter and stay in the set

$$\begin{aligned}&Q=\Big \{X_{n}:|x_{1}^{1/r_{1}}|<\varepsilon ,|x_{2}^{1/r_{2}}-v_{1}^{1/r_{2}}(x_{1}(t))|\nonumber \\&\quad \qquad <\varepsilon ,\ldots ,|x_{n}^{1/r_{n}}-v_{n-1}^{1/r_{i}}(X_{n-1}(t))|<\varepsilon \Big \} \end{aligned}$$

(73)

Therefore, when $t\ge t_{n}$, then by tuning parameter $\varepsilon$, we can assure that $Q\in \Lambda$ where $\Lambda$ is defined in Lemma 5. By Lemma 5, we have the closed-loop system (15) and (23) is locally finite-time stable. Therefore, it can be concluded that closed-loop systems (15) and (32) are globally finite-time stable.