Nonlinear robust output feedback tracking control of a quadrotor UAV using quaternion representation

Abstract

In this paper, a new quaternion-based nonlinear robust output feedback tracking controller is developed to address the attitude and altitude tracking problem of a quadrotor unmanned aerial vehicle which is subject to structural uncertainties and unknown external disturbances. By using the unit quaternion representation, the singularity associated with orientation representations can be avoided. A set of non-model-based filters are introduced to provide estimations for the unmeasurable angular velocities and translational velocity in the altitude direction of the quadrotor in the case that velocity feedback is unavailable. Approximation components based on neural network (NN) are introduced to estimate the modeling uncertainties, and robust feedback components are designed to compensate for external disturbances and NN reconstruction errors. The Lyapunov-based stability analysis is employed to prove that a semiglobally asymptotic tracking result is achieved and all the closed-loop states remain bounded. Numerical simulation results are provided to illustrate the good tracking performance of the proposed control methodologies.

Appendix

1.1 Proof of Lemma 1

Based on (29) and (30), the auxiliary signal $\eta (t)$ can be rewritten as

$$\begin{aligned} \eta =\dot{\mu }+\mu \end{aligned}$$

(95)

where $\mu =e_{v}+e_{f}$. After substituting (95) into (47), it can be obtained that

$$\begin{aligned} \int _{0}^{t}L_{1}(s)\hbox {d}s&= \int _{0}^{t}\mu ^{T}(\bar{\tau }_{d}-K_{1}\hbox {sgn}(s))\hbox {d}s\nonumber \\&\quad +\,\int _{0}^{t}\dot{\mu }^{T}\bar{\tau }_{d}\hbox {d}s-\int _{0}^{t}\dot{\mu }^{T}K_{1} \hbox {sgn}(s)\hbox {d}s\text {.}\nonumber \\ \end{aligned}$$

(96)

After integrating the second and third integrals in (96) by parts, it can be obtained that

$$\begin{aligned} \int _{0}^{t}L_{1}(s)\hbox {d}s&= \int _{0}^{t}\left( \mu ^{T}\bar{\tau }_{d}-\frac{d\bar{\tau }_{d}}{\hbox {d}s}-K_{1}\hbox {sgn}(\mu )\right) \hbox {d}s\nonumber \\&\quad +\,\mu ^{T}\bar{\tau }_{d}-\mu ^{T}(0)\bar{\tau }_{d}(0)- {\sum \limits _{i=1}^{3}} K_{1i}\left| \mu _{i}\right| \nonumber \\&\quad +\,{\sum \limits _{i=1}^{3}} K_{1i}\left| e_{vi}(0)\right| \text {.} \end{aligned}$$

(97)

The right side of (97) can be upper bounded by

$$\begin{aligned} \int _{0}^{t}L_{1}(s)\hbox {d}s&\le \int _{0}^{t} {\sum \limits _{i=1}^{3}} \left| \mu _{i}\right| \left( \left| \bar{\tau }_{di}\right| +\left| \frac{d\bar{\tau }_{di}}{\hbox {d}s}\right| -K_{1i}\!\right) \hbox {d}s\nonumber \\&\quad +\,{\sum \limits _{i=1}^{3}} \left| \mu _{i}\right| (\left| \bar{\tau }_{di}\right| -K_{1i})\nonumber \\&\quad +\,{\sum \limits _{i=1}^{3}} K_{1i}\left| e_{vi}(0)\right| -\mu ^{T}(0)D_{1}^{\prime }(0)\text {.} \end{aligned}$$

(98)

If the control gain matrix $K_{1}$ satisfies the condition in (46), the result in Lemma 1 can be proved. $\square$

1.2 Proof of Theorem 2

Let the auxiliary function $Q_{z}(t)\in {\mathbb {R}}$ be defined as follows

$$\begin{aligned} Q_{z}=\zeta _{bz}-\int _{0}^{t}L_{z}(s)\hbox {d}s \end{aligned}$$

(99)

where the $\zeta _{bz}$ and $L_{z}(t)$ have been introduced in Lemma 2. To prove the above theorem, an nonnegative function $V_{z}(t)\in {\mathbb {R}}$ is defined as follows

$$\begin{aligned} V_{2}=\frac{1}{2}m\eta _{z}^{2}+\frac{1}{2}e_{fz}^{T}+\frac{1}{2}e_{z} ^{2}+\frac{1}{2}r_{fz}^{2}+Q_{z}. \end{aligned}$$

(100)

Note that the function $V_{z}(t)$ can be bounded as

$$\begin{aligned} \lambda _{3}\left\| y_{z}\right\| ^{2}\le V_{2}\le \lambda _{4}\left\| y_{z}\right\| ^{2} \end{aligned}$$

(101)

where $y_{z}=\left[ \begin{array}{cc} z_{h}^{T}&\sqrt{Q_{z}} \end{array} \right] ^{T}\in {\mathbb {R}}^{5}$, and $\lambda _{3}$, $\lambda _{4}\in {\mathbb {R}}$ are defined as

$$\begin{aligned} \begin{array}{cc} \lambda _{3}=\frac{1}{2}\min (1,m)&\lambda _{4}=\max (\frac{1}{2}m,1) \end{array} \text {.} \end{aligned}$$

(102)

After taking the time derivative of (100), and substituting (70), (71), (72), (80), and (99) into the resulting equation, the following expression can be obtained

$$\begin{aligned} \dot{V}_{2}&=-e_{fz}^{2}-e_{z}^{2}-r_{fz}^{2}-k_{2z}m\eta _{z}^{2}+\eta _{z}\tilde{N}_{z}\nonumber \\&=-\left\| z_{z}\right\| ^{2}+(1-k_{2z}m)\eta _{z}^{2}+\eta _{z}\tilde{N}_{z} \end{aligned}$$

(103)

upon the use of the definition of $z_{h}(t)$. After applying (78) to (103), it can be obtained

$$\begin{aligned} \dot{V}_{2}&\le -\left\| z_{z}\right\| ^{2}+\left[ \left\| \eta _{z}\right\| \rho _{z}(\left\| z_{z}\right\| )-k_{nz}\left\| \eta _{z}\right\| ^{2}\right] \nonumber \\&\le -\left( 1-\frac{\rho _{z}^{2}(\left\| z_{h}\right\| )}{4k_{nz} }\right) \left\| z_{z}\right\| ^{2} \end{aligned}$$

(104)

where $k_{nz}\in {\mathbb {R}}$ is a constant and satisfies the following inequality

$$\begin{aligned} k_{nz}<k_{2z}m-1. \end{aligned}$$

(105)

The inequality in (105) implies that

$$\begin{aligned} k_{2z}>\frac{1}{m}(k_{nz}+1). \end{aligned}$$

(106)

From (104), it can be obtained that

$$\begin{aligned} \dot{V}_{2}\le -\gamma _{z}\left\| z_{z}\right\| ^{2}\text { for } k_{nz}>\frac{1}{4}\rho _{z}^{2}(\left\| z_{z}\right\| ) \end{aligned}$$

(107)

where $\gamma _{z}$ is some positive constant. The attract region $\mathcal {G}$ can be determined similarly as the one in Theorem 1. It should be noted that even though the selection of $k_{nz}$ in (107) is related with the state $z_{z}(t)$, but it can be transfer to the following sufficient condition

$$\begin{aligned} k_{nz}>\frac{1}{4}\rho _{z}^{2}\left( \sqrt{\frac{\lambda _{1z}}{\lambda _{2z}} }\left\| y_{z}(0)\right\| \right) \end{aligned}$$

(108)

and the initial value $\vert y_{z}(0)\vert$ is set as

$$\begin{aligned} \left| y_{z}(0)\right| \!=\!\sqrt{\left| e_{z}^{2}(0)\right| \!+\!\left| \eta _{z}^{2}(0)\right| \!+\!k_{1z}\left| e_{z}(0)\right| \!-\!e_{z}(0)N_{zd}(0)}.\nonumber \\ \end{aligned}$$

(109)

The condition in (108) which is only dependent on the system’s initial value can be obtained by following the similar steps in the proof of Theorem 1. By utilizing (107) and following the similar steps in the proof of Theorem 1, it can be proved that all the closed-loop signals remain bounded and the attitude tracking is achieved, provided the control gains being selected to satisfy (84), (106), (108), and (102). $\blacksquare$