Composite Learning Control of Uncertain Fractional-Order Nonlinear Systems with Actuator Faults Based on Command Filtering and Fuzzy Approximation

Abstract

The paper devotes to investigate the tracking problem for a class of unknown fractional-order nonlinear systems suffering from actuator faults, and proposes an adaptive composite learning control approach by using approximation principle of fuzzy logic systems and backstepping layout. It is well known that the standard backstepping control has inherent computational complexity. Therefore, a type of fractional-order command filters are introduced to overcome such a shortcoming, that is, by means of the proposed command filters, the virtual input signals and their own fractional derivatives can be estimated properly as anticipated. To enable that more better performance is realized while one copes with the filtering error, an error-compensation mechanism is concerned. Besides, in the fuzzy composite learning, fuzzy logic system is exploited to approximate each of the unknown functions in the dynamic system. Meanwhile, the online historical data and instantaneous data are applied to induce the prediction errors, and the tracking error and the prediction errors are used to update the parameters of fuzzy logic systems. The stability analysis is established on the basis of fractional-order Lyapunov method, and the boundedness of all signals is guaranteed. Finally, the effectiveness of the proposed configuration is authenticated by numerical simulation.

Appendix

Lemma 9

Let $\phi _{i,1}(0)=\varpi _{i}(0)$ and $\phi _{i,2}(0)=0$ be the initial conditions of the fractional-order CF determined by (10) and (11). Then the following assertions hold:

• (1) $\phi _{i,1}(t)$ and $\phi _{i,2}(t)$ converge to 0 in finite time.

• (2) Suppose that the input $\varpi _{i}(t)$ determined by (10) and (11) satisfies $|{\mathcal {D}}^{\alpha }\varpi _{i}(t)|\le d_{1i}$ with $d_{1i}\in {\mathbb {R}}^+$. Then $|\phi _{i,1}(t)-\varpi _{i}(t)|\le b_{i}$ for some $b_{i}>0$.

Proof

Consider the fractional-order CF determined by (10) and (11) in the next vector form:

$$\begin{aligned} {\mathcal {D}}^{\gamma }\phi _{i}(t)=f(t,\phi _{i}(t)), \end{aligned}$$

(69)

where $\phi _{i}(t)=[\phi _{i,1}(t),\phi _{i,2}(t)]^T$, $f=[f_1,f_2]^T$ with

$$\begin{aligned} \begin{aligned} f_1(t,\phi _{i}(t))&=k\phi _{i,2}(t),\\ f_2(t,\phi _{i}(t))&=-2k\xi \phi _{i,2}(t)-k(\phi _{i,1}(t)-\varpi _{i}(t)), \end{aligned} \end{aligned}$$

(70)

By Remark 4, $\Vert f(t,\phi _{i}(t))\Vert \le M$ holds for some constant $M>0$. Let

$$\begin{aligned} g(\nu ,\phi _{i}^*(\nu ))=f(t-(t^{\gamma }-\nu \Gamma (\gamma +1))^{\frac{1}{\gamma }}, \phi _{i}(t-(t^{\gamma }-\nu \Gamma (\gamma +1)^{\frac{1}{\gamma }})). \end{aligned}$$

(71)

Based on Lemma 6, the fractional-order CF determined by (10) and (11) can be equivalently converted into its integer-order analogue ${\dot{\phi }}^*_{i}(\nu )=g(\nu ,\phi ^*_{i}(\nu ))$.

According to the related results in [44], it can be examined that ${\dot{\phi }}^*_{i}(\nu )=g(\nu ,\phi ^*_{i}(\nu ))$ is stable. Hence by Lemma 6, $\phi _{i}(t)=\phi ^*_{i}\left( \frac{t^{\alpha }}{\Gamma (\alpha +1)}\right)$. Assume that $\nu =\nu _1$ is the stabilization time of system ${\dot{\phi }}^*_{i}(\nu )=g(\nu ,\phi ^*_{i}(\nu ))$. Since $\phi _{i}(t)=\phi ^*_{i}\left( \frac{t^{\alpha }}{\Gamma (\alpha +1)}\right)$, there is a bijective correspondence between f and g, so the solution of the problem $f(t,\phi _{i}(t))=g(\nu _1,\phi _{i}^*(\nu _1))$, denoted by $t=t_1$, is the stabilization time of system (69). From (71), $t_1=t_1-(t_1^{\gamma }-\nu _1\Gamma (\gamma +1))^{\frac{1}{\gamma }}$. Therefore, the stabilization time of system (69) is $t_{1}=(\nu _{1}\Gamma (\alpha +1))^{\frac{1}{\alpha }}$.

Denote ${\tilde{\varpi }}_{i}(t)=\phi _{i,1}(t)-\varpi _{i}(t)$, which means the approximation error of the CF with respect to the intermediate control function $\varpi _{i}$. It should be noted that ${\tilde{\varpi }}_{i}(0)=\phi _{i,1}(0)-\varpi _{i}(0)=0$.

Employing both (10) and (11), one can verify

$$\begin{aligned} \begin{aligned} {\mathcal {D}}^{\gamma }{\tilde{\varpi }}_{i}(t)=&{\mathcal {D}}^{\gamma }\phi _{i,1}(t)-{\mathcal {D}}^{\gamma }\varpi _{i}(t)\\ =&-\frac{1}{2\xi }{\mathcal {D}}^{\gamma }\phi _{i,2}(t)-\frac{k}{2\xi }{\tilde{\varpi }}_{i}(t)-{\mathcal {D}}^{\gamma }\varpi _{i}(t). \end{aligned} \end{aligned}$$

Define a function $z_{i}(t)$ such that $-2d_{i}\le z_{i}(t)\le 2d_{i}$ with $d_{i}=d_{1i}+d_{2i}$. Then, it shows that

$$\begin{aligned} {\mathcal {D}}^{\gamma }{\tilde{\varpi }}_{i}(t)+z_{i}(t)=-\frac{k}{2\xi }{\tilde{\varpi }}_{i}(t)+z_{i}. \end{aligned}$$

(72)

Due to ${\tilde{\varpi }}_{i}(0)=0$, exploiting the Laplace transformation on (72) arouses that

$$\begin{aligned} {\tilde{\varpi }}_{i}(s)=\frac{z_{i}}{s(s^{\gamma }+\frac{k}{2\xi })}-\frac{Z_{i}(s)}{s^{\gamma }+\frac{k}{2\xi }}, \end{aligned}$$

(73)

where ${\mathcal {L}}({\tilde{\varpi }}_{i}(t))={\tilde{\varpi }}_{i}(s)$ and ${\mathcal {L}}(z_{i}(t))=Z_{i}(s)$. From (3), the solution for (73) is figured out as below:

$$\begin{aligned} {\tilde{\varpi }}_{i}(t)=d_{i}t^{\gamma }E_{\gamma ,1+\gamma }\left( -\frac{k}{2\xi }t^{\gamma }\right) -z_{i}(t)*t^{-1}E_{\gamma ,0}\left( -\frac{k}{2\xi }t^{\gamma }\right) . \end{aligned}$$

Analogizing the proof of [17, Lemma. 2], one can find a $t_{1}>t$, ensuring that

$$\begin{aligned} d_{i}t^{\gamma }E_{\gamma ,1+\gamma }\left( -\frac{k}{2\xi }t^{\gamma }\right) \le \frac{2d_{i}C_{i}}{k}. \end{aligned}$$

Meanwhile, there is $t_{2}>t$ with

$$\begin{aligned} \begin{aligned} \left| z_{i}(t)*t^{-1}E_{\gamma ,0}\left( -\frac{k}{2\xi }t^{\gamma }\right) \right| \le \frac{4d_{i}C_{i}}{k}, \end{aligned} \end{aligned}$$

where $C_{i}\in {\mathbb {R}}^{+}$.

For any $b_{i}>0$, while the choice of k is limited to $k\ge \frac{6d_{i}C_{i}}{3b_{i}}$, it yields that $|{\tilde{\varpi }}_{i}(t)|\le b_{i}$, where $t_{0}=\max \{t_{1},t_{2}\}$. $\square$