Adaptive neural control for a tilting quadcopter with finite-time convergence

Abstract

This paper addresses the tracking control problem of the tilting quadcopter with unknown nonlinearities. A novel tilting quadcopter conception is proposed with a fully actuated version, which suggests that the translational and rotational movements can be controlled independently. Based on the Euler-Lagrange equations, the dynamics of tilting quadcopter is developed with uncertainties, where Neural Networks (NNs) are utilized to approximate the unknown nonlinearities in systems. We construct a novel auxiliary filter to obtain the estimation errors explicitly to achieve better approximation ability of NNs. By introducing new leakage terms in the adaptive scheme, the weights of identifier of NNs can converge to their optimal values. And a simple online verification is provided to test the parameter estimation convergence, which relaxes the requirement of persistent excitation condition. Moreover, we propose an Adaptive Finite-time Neural Control for the tilting quadcopter, where all the tracking errors can converge to a small neighborhood around zero in finite time as well as the estimation errors. Finally, comparative simulation results are presented to illustrate the effectiveness and superiority of our proposed control.

Appendices

Appendix A

The matrices $J(\eta )$ and $C(\eta ,{\dot{\eta }})$ are given as follows:

$$\begin{aligned} J(\eta )=[j_{ij}]\in {\mathbb {R}}^{3\times 3},\quad C(\eta ,{\dot{\eta }})=[c_{ij}]\in {\mathbb {R}}^{3\times 3}, \end{aligned}$$

(51)

with

$$\begin{aligned}&j_{11}=I_{xx};~ j_{12}=0; ~ j_{13}=-I_{xx}\mathrm{s}\theta ;j_{21}=0;~ j_{22}=I_{yy}{\mathrm{c}^2\phi }+I_{zz}\mathrm{s}^2\phi \nonumber \\&j_{23}=(\textit{I}_{yy}-\textit{I}_{zz})\mathrm{c}\phi \mathrm{s}\phi \mathrm{c}\theta ;~j_{31}=-I_{xx}\mathrm{s}\theta ;\ j_{32}=(\textit{I}_{yy}-\textit{I}_{zz})\mathrm{c}\phi \mathrm{s}\phi \mathrm{c}\theta ;\nonumber \\&j_{33}=I_{xx}\mathrm{s}^2\theta +I _{yy}\mathrm{s}^2\phi \mathrm{c}^2\theta +I _{zz}\mathrm{c}^2\phi \mathrm{c}^2\theta ;\nonumber \\&c_{11}=0;\ c_{12}=\frac{1}{2}(I_{yy}-I_{zz})({\dot{\theta }}\mathrm{s}2\phi -{\dot{\psi }}\mathrm{c}2\phi \mathrm{c}\theta ) \nonumber \\&c_{13}=-I_{xx}{\dot{\theta }}\mathrm{c}\theta -\frac{1}{2}(I _{yy}-I _{zz})({\dot{\theta }}\mathrm{c}2\phi \mathrm{c}\theta +{\dot{\psi }}\mathrm{s}2\phi \mathrm{c}^2\theta ); \nonumber \\&c_{21}=\frac{1}{2}I_{xx}{\dot{\psi }}\mathrm{c}\theta ;~c_{31}=-I_{xx}{\dot{\theta }}\mathrm{c}\theta ;\nonumber \\&c_{22}=-(I_{yy}-I_{zz}){\dot{\phi }}\mathrm{s}2\phi +\frac{1}{2}(I _{yy}-I _{zz}){\dot{\psi }}\mathrm{c}\phi \mathrm{s}\phi \mathrm{s}\theta ; \nonumber \\&c_{23}=(I_{yy}-I_{zz})({\dot{\phi }}\mathrm{c}2\phi \mathrm{c}\theta -\frac{1}{2}{\dot{\theta }}\mathrm{s}\phi \mathrm{c}\phi \mathrm{s}\theta )+\frac{1}{2}{} I _{xx}{\dot{\phi }}\mathrm{c}\theta -(I _{xx}\nonumber \\&\quad \quad -I _{yy}\mathrm{s}^2\phi -I _{zz}\mathrm{c}^2\phi ){\dot{\psi }}\mathrm{c}\theta \mathrm{s}\theta ;\nonumber \\&c_{32}=(I_{yy}-I_{zz})({\dot{\phi }}\mathrm{c}2\phi \mathrm{c}\theta -{\dot{\theta }}\mathrm{s}\phi \mathrm{c}\phi \mathrm{s}\theta ); \nonumber \\&c_{33}=I_{xx}{\dot{\theta }}\mathrm{s}2\theta +(I _{yy}-I _{zz}){\dot{\phi }}\mathrm{s}2\phi \mathrm{c}^2\theta -(I _{yy}\mathrm{s}^2\phi +I _{zz}\mathrm{c}^2\phi ){\dot{\theta }}\mathrm{s}2\theta . \end{aligned}$$

(52)

Appendix B

The matrix P is positive-definite such that $P=\int _0^te^{-\delta (t-r)}{S}_f{S}_f^{\text {T}}\text {dr}\ge \lambda _f I$ where $\lambda _f>0$. Then we have,

$$\begin{aligned} \int _0^te^{-\delta (t-r)}{S}_f{S}_f^{\text {T}}\text {dr}&=\int _0^{t-T}e^{-\delta (t-r)}{S}_f{S}_f^{\text {T}}\text {dr}\nonumber \\&\quad +\int _{t-T}^te^{-\delta (t-r)}{S}_f{S}_f^{\text {T}}\text {dr}\nonumber \\&\le \frac{e^{-\delta T}}{\delta }\Vert {S}_f\Vert _\infty ^2+\int _{t-T}^t{S}_f{S}_f^{\text {T}}\text {dr}. \end{aligned}$$

(53)

We can further have,

$$\begin{aligned} \int _{t-T}^t{S}_f{S}_f^{\text {T}}\text {dr}\ge \left( \lambda _f-\frac{e^{-\delta T}}{\delta }\Vert {S}_f\Vert _\infty ^2\right) I. \end{aligned}$$

(54)

If we set T and $\delta$ in reason, $\lambda _f-\frac{e^{-\delta T}}{\delta }\Vert {S}_f\Vert _\infty ^2$ can be positive. Since the filter in (13) is stable, then the regressor S is PE.