A backstepping-based procedure with saturation functions to control the PVTOL system

Abstract

In this study, we present a control strategy to solve the regulation problem for a simplified version of a PVTOL system strongly coupled. The strategy is split into two control actions that act simultaneously: one stabilizes asymptotically the vertical position; the other stabilizes both the horizontal and the angle positions. The first controller uses a simple feedback linearization procedure in conjunction with a saturation function. This controller assigns a quasilinear behavior to the vertical displacement. The second controller is based on a suitable backstepping procedure, and its task is to force the remaining variables converge asymptotically to the origin. In short, the resulting control is a nonlinear state feedback, whose performance is demonstrated by numerical simulations. The convergence analysis, based on the Lyapunov method, turned out to be quite simple if compared to other control methods found in the literature.

Appendix

Proof of Lemma 2

Notice that the assumption (ii) is a direct consequence of both the Lemma 1 and the definition of r given in (10). We first show that $\left| q_{3}(t)\right| \le \overline{\theta }$, for all $t>0$. From the definition of $q_{3}$, given in (12), and restrictions (i) and (iii), we have that following inequality:

$$\begin{aligned} \left| q_{3}\right| \le \left| \frac{1}{k_{2}}\left( s_{a}\left[ \widetilde{q}_{1}\right] +s_{b}\left[ p_{1}\right] +z_{1}\right) \right| \le \frac{a+b+\overline{z}}{k_{2}}\le \overline{\theta }, \end{aligned}$$

(28)

holds. Now, if a, b, and $k_{2}$ fulfill the condition (13), then we can assure that $\left| q_{3}\right| \le$ $\overline{\theta }$. This fact implies, both, that the term $\tan \left[ q_{3}\right]$ is well defined, and the system (12) is locally Lipschitz.

Boundedness of ($q_{1},p_{1}$):

Now, we show that $p_{1}$ is bounded. So, using the positive function $V_{1}=p_{1}^{2}/2$, we have that the time derivative of $V_{1}$, along the trajectories of the second equation of (12), leads to:

$$\begin{aligned} \dot{V}_{1}=-k_{r}(t)p_{1}\tan \left[ \frac{1}{k_{2}}\left( s_{a}\left[ \widetilde{q}_{1}\right] +s_{b}\left[ p_{1}\right] +z_{1}\right) \right] , \end{aligned}$$

where $k_{r}(t)>0$. Since $z_{1}$ is a decreasing function, there is a finite time $T_{0}>0$, where $\left| z_{1}(T_{0})\right| =\gamma$. Thus, selecting $b>a+\gamma$, we have that $\dot{V}_{1}<0$, for all $\left| p_{1}\right| >$ $b+\gamma$. So, there is a finite time $T_{1}>T_{0}$, afterward $\left| p_{1}(t)\right| <b$, for all $t>T_{1}$, that is, $p_{1}$ is uniformly bounded, and consequently, the system of Eq. (12) is globally Lipschitz. Therefore, the state $q_{1}$ remains bounded during a finite time, and a finite time of escape does not exists [11]. To analyze the boundedness of $q_{1}$, we must note that $s_{b}\left[ p_{1}\right] =p_{1}$, for all $t>T_{1}$ . Hence, the system (12) reads as:

$$\begin{aligned} \dot{q}_{1}= & {} p_{1}; \nonumber \\ \dot{p}_{1}= & {} -(1-r)\tan \left[ \frac{1}{k_{2}}(s_{a}[\widetilde{q} _{1}]+p_{1}+z_{1})\right] , \end{aligned}$$

(29)

where

$$\begin{aligned} \left| p_{1}(t)\right| \le b ; \left| z_{1}\right| \le \gamma , \end{aligned}$$

(30)

for all $t>T_{1}$. Using the following function

$$\begin{aligned} V_{(11)}=\int _{0}^{\widetilde{q}_{1}}\tan \left[ \frac{1}{k_{2}}s_{a}\left[ x \right] \right] \hbox {d}x+\frac{p_{1}^{2}}{2}. \end{aligned}$$

(31)

Notice that $V_{(11)}$ is a well-defined Lyapunov function because, from inequality (28), $a/k_{2}<\overline{\theta }$ holds. Furthermore, it is easy to show that the time derivative of $V_{(11)}$, around the trajectories of (29), is given by:

$$\begin{aligned} \dot{V}_{(11)}=p_{1}q_{11}-rp_{1}\tan \left[ \frac{1}{k_{2}}(s_{a}[ \widetilde{q}_{1}]+p_{1}+z_{1})\right] -p_{1}q_{12} \end{aligned}$$

(32)

where

$$\begin{aligned} q_{11}= & {} \tan \left[ \frac{1}{k_{2}}s_{a}\left[ \widetilde{q}_{1}\right] \right] -\tan \left[ \frac{1}{k_{2}}(s_{a}[\widetilde{q}_{1}]+p_{1})\right] \\ q_{12}= & {} \tan \left[ \frac{1}{k_{2}}(s_{a}[\widetilde{q}_{1}]+p_{1}+z_{1}) \right] \\&+\,\tan \left[ \frac{1}{k_{2}}(s_{a}[\widetilde{q}_{1}]+p_{1})\right] \end{aligned}$$

From relation (13), we have that $\tan q_{3}$ is a Lipschitz function, for all $\left| q_{3}\right| \le$ $\overline{\theta }$. Consequently, there is a $L>0$, such that:

$$\begin{aligned} \left| q_{12}\right| \le L\left| z_{1}\right| . \end{aligned}$$

On the other hand, from p4, we have that:

$$\begin{aligned} p_{1}q_{11}<0;\forall \,p_{1}\ne 0. \end{aligned}$$

From the above two relations, we have that (32) can be upper bounded by:

$$\begin{aligned} \dot{V}_{(11)}\le p_{1}q_{11}+\left| r\right| b\tan \overline{ \theta }+bL\left| z_{1}\right| \end{aligned}$$

where $p_{1}q_{11}\le 0$. Integrating the inequality above, we have that:

$$\begin{aligned} V_{(11)}(t)\le & {} b\tan \overline{\theta }\int _{0}^{t}\left| r(s)\right| ds+bL\int _{0}^{t}\left| z_{1}(s)\right| ds\nonumber \\&+V_{(11)}(0). \end{aligned}$$

Because the signals r and $z_{1}$ satisfy (i) and (ii), we can assure that the left side of the last inequality is bounded, that is, $V_{(11)}(t)\le M<\infty$, for all $t>0$. Now, as $V_{(11)}$ is radially bounded, then $q_{1}$ and $p_{1}$ are also bounded.

Convergence at the origin: From the last discussion, we can conclude that all solutions of (29) are well defined and are bounded. Hence, when $t\rightarrow \infty$, the system (29) becomes to:

$$\begin{aligned} \dot{q}_{1}= & {} p_{1}; \nonumber \\ \dot{p}_{1}= & {} -\tan \left[ \frac{1}{k_{2}}\left( s_{a}\left[ \widetilde{q}_{1} \right] +p_{1}\right) \right] , \end{aligned}$$

(33)

with $\left| p_{1}\right| \le b$. Using once again the corresponding Lyapunov function (31), we have, after using some simple algebra, the following equality:

$$\begin{aligned} \dot{V}_{2}=p_{1}\left( \tan \left[ \frac{1}{k_{2}}s_{a}\left[ \widetilde{q} _{1}\right] \right] -\tan \left[ \frac{1}{k_{2}}\left( s_{a}\left[ \widetilde{q}_{1}\right] +p_{1}\right) \right] \right) . \end{aligned}$$

(34)

From p3, we have that $\dot{V}_{2}\le 0$. Invoking the theorem of LaSalle and using similar arguments as in the proof of the Lemma 1 (a), we conclude that the whole state of the system (33) asymptotically converges to the origin. This concludes the proof of the Lemma 2.