Global adaptive finite-time control for uncertain nonlinear systems with actuator faults and unknown control directions

Abstract

In this paper, the global adaptive finite-time stabilization problem is investigated for a class of uncertain nonlinear systems with actuator faults and unknown control directions. The lower bounds of the actuation effectiveness, the upper bounds of the disturbance and the stuck faults are not required to be known a prior. By adopting adding a power integrator technique, a switching-type adaptive finite-time controller is designed and a modified switching mechanism is also proposed. It is proven that the global finite-time stability can be guaranteed by the proposed controller. A simulation example is provided to verify the effectiveness of the proposed method.

Appendix

1.1 A.1. Proof of Proposition 1

By using Lemma 1, we have

$$\begin{aligned} |x_i-x_i^*|\le 2^{1-r_i}\left| x_i^\frac{1}{r_i}-{x_i^*}^\frac{1}{r_i}\right| ^{r_i}\le 2|\xi _i|^{r_i}. \end{aligned}$$

(54)

Then by using Lemma 3, we have

$$\begin{aligned} |b_{i-1}\xi _{i-1}^{2-r_{i-1}}(x_i-x_i^*)|\le \frac{1}{3}\sum \limits _{j=1}^{i-1}\xi _j^\alpha +\vartheta _{i1}\xi _i^\alpha , \end{aligned}$$

(55)

where $\vartheta _{i1}=\frac{2\bar{b}_{i-1}r_i}{\alpha }\left( \frac{\alpha }{6(2-r_{i-1})\bar{b}_{i-1}}\right) ^{-\frac{2-r_{i-1}}{r_i}}$ is a positive constant.

1.2 A.2. Proof of Proposition 2

According to Assumption 2, we have

$$\begin{aligned} |f_i(\bar{x}_i,\theta )|\le \vartheta \varphi _i(\bar{x}_i)\sum \limits _{j=1}^{i}|x_j|. \end{aligned}$$

(56)

For $1\le j\le i$, using Lemma 2 gets

$$\begin{aligned} |x_j|\le&\left| \xi _j+{x_j^*}^{\frac{1}{r_j}}\right| ^{r_j}\le |\xi _j|^{r_j}+|{x_j^*}|=|\xi _j|^{r_j}\nonumber \\&+|\xi _{j-1}|^{r_j}\beta _{j-1}(\bar{x}_{j-1},k), \end{aligned}$$

(57)

Since $r_j\ge r_{i+1}$, for $1\le j\le i$, substituting (57) into (56) yields

$$\begin{aligned} |f_i(\bar{x}_i,\theta )|\le \,&\vartheta \varphi _i(\bar{x}_i)\sum \limits _{j=1}^{i}(|\xi _j|^{r_j}+|\xi _{j-1}|^{r_j}\beta _{j-1}(\bar{x}_{j-1},k))\nonumber \\ \le&\,\vartheta \varphi _i'(\bar{x}_i,k)\sum \limits _{j=1}^{i}|\xi _j|^{r_{i+1}}, \end{aligned}$$

(58)

where $\varphi _i'(\bar{x}_i,k)$ is a $C^1$ positive function. Using Lemma 3 gets

$$\begin{aligned} |\xi _{i}^{2-r_{i}}f_i|\le&\,|\xi _{i}^{2-r_{i}}|\vartheta \varphi _i'(\bar{x}_i,k)\sum \limits _{j=1}^{i}|\xi _j|^{r_{i+1}}\nonumber \\ \le&\, \frac{1}{3}\sum \limits _{j=1}^{i-1}\xi _j^\alpha +\xi _i^\alpha \varphi _{i2}(\bar{x}_i,k)\vartheta _{i2}, \end{aligned}$$

(59)

where $\vartheta _{i2}=\frac{\vartheta r_{i+1}}{\alpha }\left( \frac{\alpha }{3(2-r_{i})\vartheta }\right) ^{-\frac{2-r_{i}}{r_{i+1}}}$, $\varphi _{i2}(\bar{x}_i,k)$ is a $C^1$ positive function.

1.3 A.3. Proof of Proposition 3

To prove Proposition 3, a mathematical deduction is used. More details are shown as follows:

Step 1 For $i=2$, $j=1$, we have

$$\begin{aligned} \left| \frac{\partial {x_2^*}^{\frac{1}{r_2}}}{\partial x_j}\right| =\left| \frac{\partial {(x_1\beta _1^{\frac{1}{r_2}}(x_1,k)})}{\partial x_1}\right| \le \, \psi _{2,1}(x_{1},k). \end{aligned}$$

(60)

Step i Suppose that for $i-1$, $1\le j\le i-2$, there exists a $C^1$ positive function $\psi _{i-1,j}(\bar{x}_{i-2},k)$ such that

$$\begin{aligned} \left| \frac{\partial {x_{i-1}^*}^{\frac{1}{r_{i-1}}}}{\partial x_j}\right| \le \left( \sum \limits _{l=1}^{i-2}\xi _l^{1-r_j}\right) \psi _{i-1,j}(\bar{x}_{i-2},k). \end{aligned}$$

(61)

Next, we will prove that for i, $1\le j\le i-1$, there exists a $C^1$ positive function $\psi _{i,j}(\bar{x}_{i-1},k)$ such that

$$\begin{aligned} \left| \frac{\partial {x_{i}^*}^{\frac{1}{r_i}}}{\partial x_j}\right| \le \left( \sum \limits _{l=1}^{i-1}\xi _l^{1-r_j}\right) \psi _{i,j}(\bar{x}_{i-1},k). \end{aligned}$$

(62)

First of all, we will prove (62) holds for $1\le j\le i-2$. Noting the fact that ${x_{i}^*}^{\frac{1}{r_i}}=-\xi _{i-1}\beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k)$ and $\xi _{i-1}=x_{i-1}^{\frac{1}{r_{i-1}}}-{x_{i-1}^*}^{\frac{1}{r_{i-1}}}$, we have

$$\begin{aligned} \left| \frac{\partial {x_{i}^*}^{\frac{1}{r_i}}}{\partial x_j}\right|&\le \left| \xi _{i-1}\frac{\partial {\beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k})}{\partial x_j}\right| \nonumber \\&\quad +\left| \beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k)\frac{\partial {x_{i-1}^*}^{\frac{1}{r_{i-1}}}}{\partial x_j}\right| \nonumber \\&\le \left| \xi _{i-1}\frac{\partial {\beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k})}{\partial x_j}\right| \nonumber \\&\quad +\left| \beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k)\right| \left( \sum \limits _{l=1}^{i-2}\xi _l^{1-r_j}\right) \psi _{i-1,j}(\bar{x}_{i-2},k)\nonumber \\&\le \left( \sum \limits _{l=1}^{i-1}\xi _l^{1-r_j}\right) \psi _{i,j}(\bar{x}_{i-1},k). \end{aligned}$$

(63)

Then for $j=i-1$, we have

$$\begin{aligned} \left| \frac{\partial {x_{i}^*}^{\frac{1}{r_i}}}{\partial x_{i-1}}\right|&\le \left| \xi _{i-1}\frac{\partial {\beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k})}{\partial x_{i-1}}\right| +\left| \beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k)\frac{\partial x_{i-1}^{\frac{1}{r_{i-1}}}}{\partial x_{i-1}}\right| \nonumber \\&\le \left| \xi _{i-1}\frac{\partial {\beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k})}{\partial x_j}\right| \nonumber \\&\quad +\left| \frac{1}{r_{i-1}}\beta ^{\frac{1}{r_i}}(\bar{x}_{i-1},k)\right| \left( \xi _{i-1}^{1-r_{i-1}}+\xi _{i-2}^{1-r_{i-1}}\tilde{\beta }_{i-2}^{\frac{1}{r_{i-1}}-1}\right) \nonumber \\&\le \left( \sum \limits _{l=1}^{i-1}\xi _l^{1-r_j}\right) \psi _{i,i-1}(\bar{x}_{i-1},k). \end{aligned}$$

(64)

Thus, the proof is completed.

1.4 A.4. Proof of Proposition 4

Firstly, from (19), we have

$$\begin{aligned} \frac{\partial W_i}{\partial x_j}=-(2-r_i)\frac{\partial {x_{i}^*}^{\frac{1}{r_i}}}{\partial x_j}\int _{x_i^*}^{x_i}\left( s^{\frac{1}{r_i}}-{x_i^*}^{\frac{1}{r_i}}\right) ^{1-r_i}\text {d}s. \end{aligned}$$

(65)

And it is easy to prove that

$$\begin{aligned} \left| \int _{x_i^*}^{x_i}\left( s^{\frac{1}{r_i}}-{x_i^*}^{\frac{1}{r_i}}\right) ^{1-r_i}\text {d}s\right|&\le |x_i-x_i^*||\xi _i|^{1-r_i}\nonumber \\&\le 2^{1-r_i}|\xi _i|. \end{aligned}$$

(66)

Then, for $1\le j\le i-1$, we have

$$\begin{aligned} |\dot{x}_j|&=|b_jx_{j+1}+f_j|\le |b_jx_{j+1}|+\vartheta \varphi _j(\bar{x}_j)\sum \limits _{l=1}^{j}|x_l|\nonumber \\&\le \bar{b}_j|\xi _{j+1}|^{r_{j+1}}+\bar{b}_j|\xi _{j}|^{r_{j+1}}\beta _{j}(\bar{x}_{j},k)\nonumber \\&\quad +\vartheta \varphi _j'(\bar{x}_j,k) \sum \limits _{l=1}^{j}|\xi _l|^{r_{j+1}}\nonumber \\&\le \sum \limits _{l=1}^{j+1}|\xi _l|^{r_{j+1}}\vartheta '\varphi _j'(\bar{x}_j,k). \end{aligned}$$

(67)

where $\vartheta =\max \{\vartheta , \bar{b}_j\}$, $\varphi _j'(\bar{x}_j,k)$ is a nonnegative function. Then by using Proposition 3, we have

$$\begin{aligned} \left| \frac{\partial {x_{i}^*}^{\frac{1}{r_i}}}{\partial x_j}\dot{x}_j\right|&\le \left( \sum \limits _{l=1}^{i-1}\xi _l^{1-r_j}\right) \left( \sum \limits _{l=1}^{j+1}|\xi _l|^{r_{j+1}}\right) \nonumber \\&\quad \times \psi _{i,j}(\bar{x}_{i-1},k) \vartheta '\varphi _j'(\bar{x}_j,k)\nonumber \\&\le \left( \sum \limits _{l=1}^{i}|\xi _l|^{\frac{2n-1}{2n+1}}\right) \varphi _{i,j}''(\bar{x}_i,k)\vartheta '. \end{aligned}$$

(68)

where $\varphi _{i,j}''(\bar{x}_i,k)$ is a $C^1$ positive function. Then by using Lemma 3, we can prove

$$\begin{aligned} \left| \sum \limits _{j=1}^{i-1}\frac{\partial W_i}{\partial x_j}\dot{x}_j\right|&\le (2-r_i)2^{1-r_i}|\xi _i|\left( \sum \limits _{l=1}^{i}|\xi _l|^{\frac{2n-1}{2n+1}}\right) \nonumber \\&\quad \times \left( \sum \limits _{j=1}^{i-1}\varphi _{i,j}''\bar{x}_i,k\right) \vartheta '\nonumber \\&\le \frac{1}{3}\sum \limits _{j=1}^{i-1}\xi _j^\alpha +\xi _i^\alpha (\varphi _{i3}\bar{x}_i,k)\vartheta _{i3}. \end{aligned}$$

(69)

where $\vartheta _{i3}=\frac{(2-r_i)2^{1-r_i}\vartheta '}{\alpha }(\frac{\alpha }{3r_{2}(2-r_i)2^{1-r_i}\vartheta '})^{-\frac{2n-1}{2n+1}}$; $\varphi _{i3}(\bar{x}_i,k)$ is a $C^1$ positive function.