Finite time control of multi-input nonlinear uncertain systems

The sliding mode control of nonlinear multi-input systems with uncertain control direction has been recently addressed by the authors. The case in which the matrix through which the action of the control is exerted, the so-called High Frequency Gain Matrix (HFGM) is unknown yet constant, has been dealt with exploiting the existence of the so-called finite unmixing spectrum. In the more general case of a time and state dependent HFGM a solution can be found if the time and state dependent eigenvalues belong to the positive complex half plane. In such case two type of integrators are added to the original system: integrators in the output channel required to avoid the so-called reaching condition by an integral sliding mode control and integrators in the output channel which are devoted to reduce the chattering phenomenon and to transform a non-affine system in affine with respect to a new control. In this last case, in order to maintain the original relative degree, the use of differentiators turns out to be crucial. The problem cannot be dealt with by the well known super-twisting algorithm and its more recent modifications. The paper is mainly devoted to provide a novel “super-twisting like” algorithm guaranteeing finite time convergence of the differentiation error despite the severe uncertainty condition presented by the considered problem.