Continuous finite-time regulation of Euler-Lagrange systems via energy shaping

Abstract

In this paper, we provide conditions on the energy-like functions employed in the classical energy shaping control technique in order to solve the finite-time regulation problem of a class of Euler-Lagrange systems. Inherited from the energy shaping design, the obtained control schemes are described by the gradient of suitable artificial potential energy and energy dissipation functions. In order to achieve finite-time convergence at a desired constant position, these energy functions are provided with some homogeneity properties and designed to have an appropriate homogeneity degree. Using different energy function definitions, this paper provides different novel controllers as an example of the proposed control design methodology.

Keywords:

• Energy shaping
• finite-time regulation
• Euler-Lagrange systems

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The vector field ${\mathbf{\text{f}}}_H\left(\mathbf{\text{x}}\right)$ is known as the $\mathbf{\text{r}}$-homogeneous approximation of $\mathbf{\text{f(x)}}$. Similarly, an $\mathbf{\text{r}}$-homogeneous function $V_H:{\mathbb{R}}^m\mapsto \mathbb{R}$ is said to be $\mathbf{\text{r}}$-homogeneous approximation of $V:{\mathbb{R}}^m\mapsto \mathbb{R}$ if there exists $V_{NH}:{\mathbb{R}}^m\mapsto \mathbb{R}$ such that $V=V_H+V_{NH}$ and $\underset{ϵ\mapsto 0}{lim}\frac{V_{NH}\left({\mathrm{\Delta }}_{ϵ}^{\mathbf{\text{r}}}\mathbf{\text{x}}\right)}{{ϵ}^l}=0$ uniformly w.r.t. $\mathbf{\text{x}}\in S_c^{m-1}$ (Andrieu et al., 2008; Sepulchre & Aeyels, 1996).

2 More precisely, we invoke its extension to autonomous system $\dot{\mathbf{\text{x}}}=\mathbf{\text{f}}\left(\mathbf{\text{x}}\right)$ with continuous vector field $\mathbf{\text{f(x)}}$, see Theorem 7.2.1 in Michel et al. (2008).

3 Letters h, P and D stand for homogeneous, proportional and derivative, respectively. We also use letter s for saturated.

Additional information

Funding

This work has been partially supported by SEP-PRODEP Apoyo a la Incorporación de NPTC grant number 511-6/18-9169; Fondo de Colaboración II-FI UNAM, grant number IISGBAS-100-2015; PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica), grant number IN113617; and CONACyT basic scientific research projects CB-282807, CB-241171 and CVU 267513.