Finite-time observer-based output-feedback control for the global stabilisation of the PVTOL aircraft with bounded inputs

Abstract

In this work, an output-feedback scheme for the global stabilisation of the planar vertical take-off and landing aircraft with bounded inputs is developed taking into account the positive nature of the thrust. The global stabilisation objective is proven to be achieved avoiding input saturation and by exclusively considering the system positions in the feedback. To cope with the lack of velocity measurements, the proposed algorithm involves a finite-time observer. The generalised versions of the involved finite-time stabilisers have not only permitted to solve the output-feedback stabilisation problem avoiding input saturation, but also provide additional flexibility in the control design that may be used in aid of performance improvements. With respect to previous approaches, the developed finite-time observer-based scheme guarantees the global stabilisation objective disregarding velocity measurements in a bounded input context. Simulation tests corroborate the analytical developments. The study includes further experimental results on an actual flying device.

Keywords:

• nonlinear control
• global stabilisation
• output feedback
• finite-time observers
• bounded inputs
• PVTOL aircraft

Notes

1. As pointed out in Zavala-Río et al. (2003) and Lopez-Araujo et al. (2010), notice from the vertical motion equation (1b(1b) $$\begin{array}{cc}\hfill \ddot{\zeta }& =u_{1}cos\theta +ϵu_{2}sin\theta -1\end{array}$$(1b) ) that U₁ > 1 is a necessary condition for the system in Equations (4) to be stabilisable, since any steady-state condition implies that the aircraft weight be compensated.

2. Let us note that their strictly increasing character renders σ₃₂ and σ₃₃ invertible functions mapping $\mathbb{R}$ onto ${\sigma }_{32}\left(\mathbb{R}\right)$ and ${\sigma }_{33}\left(\mathbb{R}\right)$, respectively, and consequently, σ^{− 1}₃₂ and σ^{− 1}₃₃ are well-defined functions, respectively, mapping ${\sigma }_{32}\left(\mathbb{R}\right)$ and ${\sigma }_{33}\left(\mathbb{R}\right)$ onto $\mathbb{R}$. In particular, observe that, by Equation (22c(22c) $$\underset{s\to -\infty }{lim}{\sigma }_{33}\left(s\right)<-M_{30}^{-},\underset{s\to \infty }{lim}{\sigma }_{33}\left(s\right)>M_{30}^{+}$$(22c) ), we have that $M_{30}^{+}\in {\sigma }_{33}\left(\mathbb{R}\right)$ and $-M_{30}^{-}\in {\sigma }_{33}\left(\mathbb{R}\right)$.

3. Let us note that the dotted ( $\dot{}$ ) and double-dotted ( $\ddot{}$ ) variables defined through Equations (23)–(27) do not a priori correspond to the first-order and second-order change rate of the referred variables. However, such a correspondence will be later on proven to (ultimately) hold from some finite time t₁ on.

4. Let us note that the satisfaction of inequalities (22c(22c) $$\underset{s\to -\infty }{lim}{\sigma }_{33}\left(s\right)<-M_{30}^{-},\underset{s\to \infty }{lim}{\sigma }_{33}\left(s\right)>M_{30}^{+}$$(22c) ) and (22d(22d) $$\begin{array}{c}\underset{s\to -\infty }{lim}{\sigma }_{32}\left(s\right)<-M_{31}^{-}+{\sigma }_{33}^{-1}(-M_{30}^{-}),\hfill \\ \underset{s\to \infty }{lim}{\sigma }_{32}\left(s\right)>M_{31}^{+}+{\sigma }_{33}^{-1}\left(M_{30}^{+}\right)\hfill \end{array}$$(22d) ) ensures positivity of the right-hand side expressions of inequalities (32).

5. Recall that, in view of the strictly increasing character of σ₃₂ and σ₃₃, σ^{− 1}₃₂ and σ^{− 1}₃₃ are well-defined functions, respectively, mapping ${\sigma }_{32}\left(\mathbb{R}\right)$ and ${\sigma }_{33}\left(\mathbb{R}\right)$ onto $\mathbb{R}$. In particular, by inequalities (32), we have that $M_{30}^{+}+M_{34}^{+}\in {\sigma }_{33}\left(\mathbb{R}\right)$, $-M_{30}^{-}-M_{34}^{-}\in {\sigma }_{33}\left(\mathbb{R}\right)$, $M_{31}^{+}+{\sigma }_{33}^{-1}(M_{30}^{+}+M_{34}^{+})\in {\sigma }_{32}\left(\mathbb{R}\right)$, and $-M_{31}^{-}+{\sigma }_{33}^{-1}(-M_{30}^{-}-M_{34}^{-})\in {\sigma }_{32}\left(\mathbb{R}\right)$.

6. Theorem 4.18 of Khalil (2002) is being applied by considering the closed-loop rotational motion dynamics a first-order subsystem with respect to $\dot{\theta }$, i.e. $\frac{d}{dt}\dot{\theta }=u_2(t,\dot{\theta })$ where (along the closed-loop trajectories) the rest of the system variables, involved in u₂, are considered time-varying functions.

7. Let us notice that generalised saturation functions with sufficiently low slope in v₁ and v₂ and small enough bounds in u₂ would also be helpful to get $B_{\dot{\theta }}\le L_0$.

Additional information

Funding

This work was carried out in the framework of the Labex MS2T and the Equipex ROBOTEX, which were funded by the French Government, through the programme ‘Investments for the future’ managed by the National Agency for Research (reference ANR-11-IDEX-0004-02 and ANR-10-EQPX-44-01). The third author is financed by the European Regional Development Fund. European Union is investing in your future.

Notes on contributors

A. Zavala-Río

Arturo Zavala-Río received his BS degree in electronic systems engineering and MS degree in control engineering from the Instituto Tecnológico y de Estudios Superiores de Monterrey, Mexico, in 1989 and 1992, respectively, and his DEA and PhD degrees in automatic control from the Institut National Politechnique de Grenoble, France, in 1994 and 1997, respectively. He held professor–researcher positions at Universidad Autonoma de Queretaro (1999–2000), Mexico, and Universidad Autonoma de San Luis Potosi (2001), Mexico. He has been visiting researcher at Mechanical Engineering Laboratory (1998), Japan, and Université de Technologie de Compiègne (2001–2002), France. Since 2002, he is a full-time researcher at the Instituto Potosino de Investigacion Cientifica y Tecnologica, Mexico. His research topics focus on the modelling, analysis, and control of nonlinear systems.

I. Fantoni

Isabelle Fantoni received her PhD degree, with the European label, in nonlinear control for underactuated mechanical systems, in 2000, from the ‘Université de Technologie de Compiègne’, in France. Since October 2001, she is a permanent researcher at Heudiasyc laboratory, UTC, in Compiègne, France, employed by the French National Foundation for Scientific Research (CNRS) and CNRS Research Director since October 2013. Her research interests include nonlinear control, modelling and control for UAVs, fault-tolerant control for UAVs, vision for navigation of aerial vehicles, cooperation of UAVs, and heterogeneous robotic systems in cooperation.

G. Sanahuja

Guillaume Sanahuja has done his PhD thesis at Heudiasyc Laboratory, Université de Technologie de Compiègne, in 2010, on the topic of vision for UAVs. He is now working in the same laboratory as a research engineer. His work is related to the French project Equipex (equipment of excellence) ROBOTEX, whose aim is to equip Heudiasyc and its partners with robotic platforms. His function is to design and develop embedded architectures at both software and hardware levels. His topics of interests are embedded systems, embedded vision, and nonlinear control.