Design of global smooth implicit control Lyapunov function for multiple-integrator system with input constraint
In order to deal with multiple-integrator systems with input constraints and with a variety of restrictions on the virtual input for the backstepping method, the authors modify the approach of [\textit{A. Polyakov} et al., Automatica 51, 332--340 (2015; Zbl 1309.93135)] to obtain a smooth Lyapunov functional and a smooth controller, in exchange for homogeneity and finite-time stability requirements being dropped. The newly introduced controller is almost linear with respect to the state near the origin, while approximating the homogeneous controller obtained via the implicit Lyapunov function method of [loc. cit.] far from the origin. A bounded virtual control law is then obtained using a transformation on the restricted virtual input, no boundedness condition on the actual input being needed. It is also observed that the proposed method can be combined with bounded-input backstepping methods to obtain a bounded feedback law provided that certain additional assumptions are satisfied. The proposed implicit control Lyapunov function is then applied to the stabilizing control of a magnetic levitation system, the advantages of the approach being highlighted.
scientific article; zbMATH DE number 7290544