Finite-time stabilisation of simple mechanical systems using continuous feedback

Abstract

Stabilisation of simple mechanical systems in finite time with continuous state feedback is considered here. The dynamics are represented by generalised (local) coordinates. A general methodology to construct control Lyapunov functions that are Hölder continuous and that can be used to show finite-time stability of the feedback controlled system, is presented. This construction also gives the feedback control law, and results in the feedback system being Hölder continuous as well. Unlike Lipschitz continuous feedback control systems, the feedback control scheme given here converges to the desired equilibrium in finite time. Moreover, unlike discontinuous and hybrid control schemes, the feedback control law does not lead to chattering in the presence of measurement noise, does not excite unmodelled high-frequency dynamics, and can be implemented with actuators that can only deliver continuous control inputs. The advantages of continuous finite-time stabilisation over continuous asymptotic stabilisation of mechanical systems, has been described in some prior research on finite-time stabilisation of the double integrator. The finite-time stabilisation scheme given here generalises this prior research to multiple degree-of-freedom mechanical systems. A numerical comparison is carried out through numerical simulations on two example systems that are representative of a broad class of simple mechanical systems.