Wind disturbance rejection for unmanned aerial vehicles using acceleration feedback enhanced $H_\infty$ method

Abstract

One of the most critical issues for unmanned aerial vehicle (UAV) safety and precision flight is wind disturbance. To this end, this paper presents an acceleration feedback (AF) enhanced $H_\infty$ method for UAV flight control against wind disturbance and its application on a hex-rotor platform. The dynamics of the UAV system are decoupled into attitude control and position control loops. A hierarchical $H_\infty$ controller is then designed for the decoupled system. Finally, an AF-enhanced method is introduced into the decoupled system without altering the original control structure. The stability of the AF-enhanced $H_\infty$ method for the UAV system is analyzed and verified using the $H_\infty$ theory. Two types of wind disturbances—continuous and gusty winds—are considered and analyzed for guiding the AF-enhanced controller design. The results of an experimental comparison between the $H_\infty$ controller and the AF-enhanced $H_\infty$ controller against wind disturbances demonstrate the robustness and effectiveness of the proposed method for wind disturbance rejection.

Appendices

A Proof of Theorem 1

Similar to the proof in (Fantoni et al. , 2008), we provide the following proof for Theorem 1. Defining $h(R, R_d) = [h_x, h_y,$$h_z]^T = (R_d - R)\mathbf {e}_3$, and combining it with (4) and attitude error $e_\Phi$, we obtain

$$\begin{aligned} \left\{ \begin{array}{ll} h_x &{} = c\phi s\theta c\psi + s\phi s\psi - (c\phi _{d}s\theta _{d}c\phi _{d} + s\phi _d s\psi _d) \\ h_y &{} = c\phi s\theta s\psi - s\phi c\psi - (c\phi _{d}s\theta _{d}s\phi _{d} + s\phi _d c\psi _d) \\ h_z &{} = c\phi c\theta - c\phi _d c\theta _d \end{array} \right. \end{aligned}$$

(46)

For $h_x$ in (46), replacing $[\phi , \theta , \psi ] ^T$ by $[\phi _d + e_\phi , \theta _d + e_\theta , \phi _d + e_\psi ]^T$ and combine the following trigonometric function

$$\begin{aligned} \left\{ \begin{array}{ll} \sin (a+b) &{} = \sin (a) + \sin (\frac{b}{2})\cos (a+\frac{b}{2}) \\ \cos (a+b) &{} = \cos (a) - \sin (\frac{b}{2})\sin (a+\frac{b}{2}) \\ \begin{array}{l} \sin (a)| \le 1 \\ |\cos (a)| \le 1 \end{array}&\end{array} \right. \end{aligned}$$

(47)

Thus, we can obtain,

$$\begin{aligned} |h_x|\le & {} 2\left| s\frac{e_\phi }{2}\right| + \left| s\frac{e_\theta }{2}\right| + 2\left| s\frac{e_\psi }{2}\right| + \left| s\frac{e_\phi }{2}\right| \cdot \left| s\frac{e_\theta }{2}\right| \nonumber \\+ & {} \left| s\frac{e_\theta }{2}\right| \cdot \left| s\frac{e_\psi }{2}\right| + 2\left| s\frac{e_\phi }{2}\right| \cdot \left| s\frac{e_\psi }{2}\right| \nonumber \\+ & {} \left| s\frac{e_\phi }{2}\right| \cdot \left| s\frac{e_\theta }{2}\right| \cdot \left| s\frac{e_\psi }{2}\right| \end{aligned}$$

(48)

Combining the following inequalities into (48)

$$\begin{aligned} \left\{ \begin{array}{l} |a|\cdot |b| \le \frac{1}{2}(|a| + |b|),~for~|a|,~|b| \le 1 \\ |a|\cdot |b|\cdot |c| \le \frac{1}{3}(|a| + |b| + |c|),~for~|a|,~|b|,~|c| \le 1 \\ |\sin (a)| \le |a| \end{array} \right. \end{aligned}$$

(49)

Thus, we can obtain

$$\begin{aligned} |h_x| \le \frac{10}{3}\left| s\frac{e_\phi }{2}\right| + \frac{7}{3}\left| s\frac{e_\theta }{2}\right| + \frac{10}{3}\left| s\frac{e_\psi }{2}\right| \le \frac{5}{3}(|e_\phi | + |e_\theta | + |e_\psi |) \end{aligned}$$

(50)

Similarly, the property of $h_y$ and $h_z$ is

$$\begin{aligned} \left\{ \begin{array}{l} |h_y| \le \frac{5}{3}(|e_\phi | + |e_\theta | + |e_\psi |) \\ |h_z| \le \frac{3}{4}(|e_\phi | + |e_\theta |) \end{array} \right. \end{aligned}$$

(51)

With (50) and (51), the norm of $h(R,R_d)$ satisfies

$$\begin{aligned} \Vert h(R,R_d)\Vert = \sqrt{h_x^{2} + h_y^{2} + h_z^{2}} \le k_1\Vert e_\Phi \Vert \end{aligned}$$

(52)

where $k_1 \le \sqrt{13}$. For the hex-rotor, we can assume the maximum thrust is $k_2$; thus,

$$\begin{aligned} \left\| \delta (R,R_d) \right\|\le & {} \left\| \frac{f}{m}\right\| \cdot \Vert h(R,R_d)\Vert \le \frac{k_1 k_2}{m} \Vert e_\Phi \Vert \nonumber \\= & {} \left\| {\left[ O_{3\times 3}~O_{3\times 3}~\sigma I_{3\times 3}~O_{3\times 3} \right] x} \right\| \end{aligned}$$

(53)

where $\sigma = \frac{k_1 k_2}{m}$.

B Proof of Theorem 2

First, a candidate Lyapunov function of the closed-loop system is chosen as

$$\begin{aligned} V(x) = x^{TPx} \end{aligned}$$

(54)

Then the differential of V(x) is

$$\begin{aligned} \dot{V}(x)= & {} x^T((A+BK)^T)P + P(A+BK))x + \Delta ^TD^{TPx} \nonumber \\+ & {} x^TPD\Delta + \delta ^T(R,R_d)E^{TPx} + x^TPE\delta (R,R_d) \nonumber \\= & {} x^T((A+BK)^TP + P(A+BK))x + x^TC^{TCx} \nonumber \\+ & {} \frac{1}{\gamma ^{2}}x^TPDD^{TPx} + \frac{1}{\lambda }x^T PEE^{TPx} + \lambda \left\| \delta (R,R_d)\right\| ^{2} \nonumber \\- & {} \gamma ^{2}\left\| \Delta - \frac{1}{\gamma ^{2}}D^{TPx}\right\| ^{2} - \lambda \left\| \delta (R,R_d) - \frac{1}{\lambda }E^{TPx}\right\| ^{2} \nonumber \\+ & {} \gamma ^{2}\left\| \Delta \right\| ^{2} - \left\| y\right\| ^{2} \end{aligned}$$

(55)

Combining with (17), we can get

$$\begin{aligned} \dot{V}(x)\le & {} x^T((A+BK)^TP + P(A+BK))x + x^TC^{TCx} \nonumber \\+ & {} \frac{1}{\gamma ^{2}}x^TPDD^{TPx} + \frac{1}{\lambda }x^T PEE^{TPx} + \lambda \left\| \delta (R,R_d)\right\| ^{2} \nonumber \\+ & {} \gamma ^{2}\left\| \Delta \right\| ^{2} - \left\| y\right\| ^{2} \nonumber \\\le & {} x^T((A+BK)^TP + P(A+BK))x + x^TC^{TCx} \nonumber \\+ & {} \frac{1}{\gamma ^{2}}x^TPDD^{TPx} + \frac{1}{\lambda }x^T PEE^{TPx} + \lambda x^TF^TFx \nonumber \\+ & {} \gamma ^{2}\left\| \Delta \right\| ^{2} - \left\| y\right\| ^{2} \end{aligned}$$

(56)

Thus, if (18) is satisfied, then

$$\begin{aligned} \dot{V}(x) \le \gamma ^{2}\left\| \Delta \right\| ^{2} - \left\| y\right\| ^{2} \end{aligned}$$

(57)

which means, for any $T > 0$, we have

$$\begin{aligned}&\int _0^T\left( \left\| y(t)\right\| ^{2} - \gamma ^{2}\left\| \Delta (t)\right\| ^{2}\right) dt \\&= \int _0^T \left( \left\| y(t)\right\| ^{2} - \gamma ^{2}\left\| \Delta (t)\right\| ^{2} + \dot{V}(x(t))\right) dt + V\left( x(T)\right) \\&\le V\left( x(T)\right) \end{aligned}$$

That is,

$$\begin{aligned} \int _0^T \left\| y(t)\right\| ^{2} \le \int _0^T\gamma ^{2} \left\| \Delta (t)\right\| ^{2} dt + V\left( x(T)\right) \end{aligned}$$

(58)

That means, the system is finite gain $L_2$-stable from disturbance $\Delta$ to outputs y, and the $L_2$ gain is less than or equal to $\gamma$.