An idea for Lyapunov function design for arbitrary order continuous twisting algorithms

Continuous Twisting Algorithm (CTA) for systems of order n allows to compensate theoretically exactly Lipschitz time perturbations, generating a continuous control signal and ensuring finite-time convergence to a sliding mode of order (n + 1). In this paper an idea of a recursive procedure for Lypaunov Function (LF) and gains design for n — th order CTA is proposed. The procedure consists in constructing a homogeneous LF for the n — th order CTA as a sum of three generalized homogeneous polynomials of the same degree: (i) a LF for the continuous state feedback controlled system of order n; (ii) a LF for CTA of order (n — 1) and (iii) some extra cross terms. The two first terms can in turn be constructed in a recursive form. The proposed idea is illustrated with the design of a LF and gains for the second order CTA. Furthermore, the proposed procedure allows to select the control gains of the CTA for third order systems and to prove global finite-time stability.