Adaptive neural network finite time control for quadrotor UAV with unknown input saturation

Abstract

This study presents a novel adaptive robust control strategy for the position and attitude tracking of quadrotor unmanned aerial vehicles (UAVs) in the presence of input saturation, unmodeled nonlinear dynamics, and external disturbances. To deal with the negative effects of completely unknown input saturation constraints, a nonsymmetric saturation nonlinearity approxiator constructed by the hyperbolic tangent function attached with a parameter adjustment mechanism is incorporated into the controller design. Then, a novel neural network (NN) finite time backstepping-based anti-saturation control approach is proposed by introducing the NN finite time backstepping, designing the new virtual control signals and the modified error compensation mechanism. The proposed approach not only holds the advantages of the NN finite time backstepping control, but also prevents the system from degradation or even instability caused by unknown nonsymmetric saturation nonlinearities in actuator. The finite time convergence of all signals in the closed-loop aircraft system is guaranteed via Lyapunov finite time methodology despite the input saturations, unmodeled dynamics, and external disturbances. Finally, numerical simulations are carried out to illustrate the effectiveness and robustness of the proposed controller.

Appendix

Proof of Theorem 3:

We construct the Lyapunov function as

$$\begin{aligned} V^{\prime }=V_T^{\prime }+V_R^{\prime }. \end{aligned}$$

(111)

Taking time derivative of V yields

$$\begin{aligned} \begin{aligned} {\dot{V}}^{\prime }&={\dot{V}}^{\prime }_T+{\dot{V}}_R^{\prime }\\&\le -a_0V_T^{\prime }-b_0V_T^{' \frac{2-\sigma }{2}}+c_0\\&\quad -a_2V_R^{\prime }-b_2V_R^{' \frac{2-\sigma }{2}}+c_2\\&\le -a^{\prime }V^{\prime }-b^{\prime }V^{' \frac{2-\sigma }{2}}+c^{\prime }, \end{aligned} \end{aligned}$$

(112)

where

$$\begin{aligned} \begin{aligned}&a^{\prime }=\mathrm{min}\left\{ a_0,a_2\right\} ,\\&b^{\prime }=\mathrm{min}\left\{ b_0,b_2\right\} ,\\&c^{\prime }=c_0+c_2. \end{aligned} \end{aligned}$$

(113)

Then,

$$\begin{aligned} {\dot{V}}^{\prime }\le -\left( a-\frac{c}{2V^{\prime }}\right) V^{\prime }-\left( b-\frac{c}{2V^{' \frac{2-\sigma }{2}}}\right) V^{' \frac{2-\sigma }{2}}.\nonumber \\ \end{aligned}$$

(114)

From (114), if $\gamma =a-c/(2V^{\prime })>0$ and $\beta =b-c/(2V^{' \frac{2-\sigma }{2}})>0$, then by applying Lemma 1, $e_1,e_2$$,e_3,e_4$, ${\tilde{\theta }}_0$, and ${\tilde{\theta }}_2$ will converge to the region in finite time

$$\begin{aligned} T=\frac{1}{\gamma (1-\frac{2-\sigma }{2})}\mathrm{ln}\frac{\gamma V^{' 1-\frac{2-\sigma }{2}}(\imath )+\beta }{\beta }, \end{aligned}$$

(115)

where $\imath =[e_{1,0},e_{2,0},e_{3,0},e_{4,0},{\tilde{\theta }}_{0,0},{\tilde{\theta }}_{2,0}]^T$ is the error vector $[e_{1},e_{2},e_{3},e_{4},{\tilde{\theta }}_{0},{\tilde{\theta }}_{2}]^{T}$ at $t=0$. $\square$

Proof of Theorem 4:

Considering Theorems 1, 2, we construct the Lyapunov function as

$$\begin{aligned} V=V_T+V_R. \end{aligned}$$

(116)

Taking time derivative of V yields

$$\begin{aligned} \begin{aligned} {\dot{V}}&={\dot{V}}_T+{\dot{V}}_R\\&\le -a_1V_T-b_1V_T^{\frac{2-\sigma }{2}}+c_1\\&\quad -a_3V_R-b_3V_R^{\frac{2-\sigma }{2}}+c_3\\&\le -aV-bV^{\frac{2-\sigma }{2}}+c, \end{aligned} \end{aligned}$$

(117)

where

$$\begin{aligned} \begin{aligned}&a=\mathrm{min}\left\{ a_1,a_3\right\} ,\\&b=\mathrm{min}\left\{ b_1,b_3\right\} ,\\&c=c_1+c_3. \end{aligned} \end{aligned}$$

(118)

Then,

$$\begin{aligned} {\dot{V}}\le -\left( a-\frac{c}{2V}\right) V-\left( b-\frac{c}{2V^{\frac{2-\sigma }{2}}}\right) V^{\frac{2-\sigma }{2}}. \end{aligned}$$

(119)

From (114), if $\gamma =a-c/(2V)>0$ and $\beta =b-c/(2V^{\frac{2-\sigma }{2}})>0$, then by applying Lemma 1, $e_1,e_2,$$e_3,e_4$, ${\tilde{\theta }}_1$, and ${\tilde{\theta }}_3$ will converge to the region in finite time

$$\begin{aligned} T=\frac{1}{\gamma (1-\frac{2-\sigma }{2})}\mathrm{ln}\frac{\gamma V^{1-\frac{2-\sigma }{2}}(\imath )+\beta }{\beta }, \end{aligned}$$

(120)

where $\imath =[e_{1,0},e_{2,0},e_{3,0},e_{4,0},{\tilde{\theta }}_{1,0},{\tilde{\theta }}_{3,0}]^T$ is the error vector $[e_{1},e_{2},e_{3},e_{4},{\tilde{\theta }}_{1},{\tilde{\theta }}_{3}]^{T}$ at $t=0$. $\square$