$\mathcal{H}_{\infty}$ Optimal Parameters for the Super-Twisting Algorithm with Intermediate Disturbance Bound Mismatch

The super-twisting algorithm is studied for the case when a disturbance whose derivative is $\mathcal{L}_{2}$-norm bounded and violates its presumed Lipschitz bound from time to time. In such interim time span the state may be driven away from the origin and converge again once the derivative of the disturbance reenters the presumed Lipschitz bound. We present an $\mathcal{H}_{\infty}$-norm optimal parameter range for choosing the super-twisting gains to minimize the $\mathcal{L}_{2}$-gain of a matched disturbance in this worst case scenario. Simulations show that such parameter choice may provide better results, compared to other choices.