Immersion and invariance adaptive velocity observer for a class of Euler–Lagrange mechanical systems

Abstract

This paper addresses the problem of velocity estimation for a class of uncertain mechanical systems. Using advantages of immersion and invariance technique with input–output filtered transformation, a proper immersion and dynamical auxiliary filter have been constructed in the designed estimator. Uniform global asymptotic convergence of the velocity estimator has been proved for the system with parametric uncertainties. In the presence of perturbations on the input and output, the performance analysis of the estimator has been theoretically investigated and illustrated by simulation results.

Appendix

Proof of Proposition 1: Differentiating $\mathcal {L}(q,\dot{q} )$ in (2) yields

$$\begin{aligned} {\frac{d{\mathcal {L}}}{\mathrm{d}t} }= & {} {\ddot{q}}^T \frac{\partial {}\mathcal {L}}{\partial {}\dot{q}} + {\dot{q}}^T \frac{\partial {}\mathcal {L}}{\partial {}q} \nonumber \\= & {} {\dot{q}}^T \left( {\frac{d}{\mathrm{d}t}\frac{\partial {}\mathcal {L}}{\partial {}\dot{q}}-\frac{\partial {}\mathcal {L}}{\partial {}q}}\right) ={\frac{d{\mathcal {K}}}{\mathrm{d}t} }-{\dot{q}}^T \frac{\partial {}\mathcal {P}}{\partial {}q}. \end{aligned}$$

(51)

Replacing (1) into (51) yields

$$\begin{aligned} \dot{q} ^T \tau = {\frac{d{\mathcal {K}}}{\mathrm{d}t} }-{\dot{q}}^T \frac{\partial {}\mathcal {P}}{\partial {}q}. \end{aligned}$$

(52)

From the definition of $\mathcal {K} \left( q,\dot{q} \right)$ in (2) and the definition of $\upsilon$ in Proposition 1, we get

$$\begin{aligned} {\frac{d{\mathcal {K}}}{\mathrm{d}t} }= & {} \frac{d{}}{\mathrm{d}t} \left( \frac{1}{2}{\dot{q}}^T{H(q)} \dot{q} \right) \nonumber \\= & {} \frac{d{}}{\mathrm{d}t} \left( \frac{1}{2}{\upsilon ^T \upsilon } \right) = \upsilon ^T \dot{\upsilon }= \dot{q} ^T {T(q)}^T \dot{\upsilon }. \end{aligned}$$

(53)

Replacing (53) in (52) and eliminating $\dot{q} ^T$ from two sides, we conclude the proof.