A unique robust controller for tracking and stabilisation of non-holonomic vehicles

ABSTRACT

It is known that for non-holonomic systems it is impossible to design a universal controller able to asymptotically stabilise any feasible reference trajectory. We present a smooth time-varying controller able to stabilise a wide class of reference trajectories that include converging (parking problem) and persistently exciting (tracking problem) ones, as well as set-points. For the first time in the literature we establish uniform global asymptotic stability for the origin of the closed-loop system in the kinematics state space. We also show that the kinematics controller renders the system robust to perturbations in the sense of integral-input-to-state stability. Then, we show that for the case in which the velocity dynamics equations are also considered (full model), any velocity-tracking controller with the property that the error velocities are square integrable may be used to ensure global tracking or stabilisation even under parametric uncertainty.

Keywords:

• Persistency of excitation
• autonomous vehicles
• Lyapunov's method
• input-to-state stability

Acknowledgments

The authors are grateful to H. Khalil for his keen technical remarks on the material presented in this paper. The work of Mohamed Maghenem was carried out while he was with Univ Paris-Saclay, Saclay, France

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Persistency of excitation in the sense defined in Loría et al. (1999) is also implicitly used in the earlier reference Samson (1993).

2 For a continuous function $t\mapsto \varphi$ we define $|\varphi \left(t\right){|}_{\mathrm{\infty }}:=\underset{t\ge 0}{sup}|\varphi \left(t\right)|$.

3 This proof of uniform stability replaces the corresponding one proposed in Maghenem, Loría, and Panteley (2017), which is incorrect.

Additional information

Funding

The work of E. Panteley is partially supported by the Government of the Russian Federation [grant number 074-U01] and Univ Paris Saclay [17IDEXCONSYNHE].