On the Fragility of Multivariable Super-Twisting Algorithm for non-Symmetric Uncertain Input Matrix

A non-decoupled MIMO Super-Twisting Algorithm (STA) extension, not requiring the exact knowledge of the system input matrix was recently developed. However, a symmetry assumption had to be made on the uncertain gain matrix. In this paper, we exploit how far from the symmetric case the stability of the multivariable algorithm can be preserved. Some asymmetry bounds are theoretically established showing that for small enough asymmetry, stability can be guaranteed. Moreover, the Lyapunov function used in the analysis, leads to a rather sharp detection of instability, when asymmetry increases. Surprisingly, it also reveals that the actual asymmetry allowance can be quite small suggesting some fragility of the algorithm for the class of uncertainty considered. This motivates the need of new developments for more robust multivariable STA as an interesting open problem.