Qualitative differences of two classes of multivariable super-twisting algorithms

In this paper we compare two families of multivariable super-twisting algorithms. The first family is an implementation of independent generalised super-twisting algorithms, whereas the second one presents nonlinearities that couple all the states. For the latter, the Lyapuov stability proof boils down to the positive-definitiveness test of a 2 × 2 matrix, in spite of the number of states. This second family is a generalisation of the multivariable super-twisting algorithm available in the literature. Furthermore, we highlight differences between these families regarding i) the discontinuity of the state space; ii) the number of design parameters; and iii) their convergence properties. We show their applicability and differences by designing an observer for a continuous-time, linear, time-invariant system. To conclude, we present a motivating example that suggests the robust stability of the feedback interconnection of generalised super-twisting algorithms.