Minimizing the Homogeneous ℒ_p-Gain of the Continuous Super-Twisting-Like Algorithm Subject to Noise

We consider homogeneous systems with inputs and outputs, i.e. homogeneous input-output mappings, and observe that the classical ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> -gain is not suitable. Hence, the recently introduced dilation-invariant homogeneous ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> -gain (ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ph</inf> -gain) is regarded. We focus on the continuous super-twisting-like algorithm (CSTLA) acting as a differentiator and propose a Lyapunov function to prove stability of the free system. This is used as a candidate storage function to estimate the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ph</inf> -gain for p ≥ 2 with the homogeneous dissipation inequality. In a high-gain setup, scaled gains are designed that minimize the effect of noise and disturbance on the second state in terms of the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ph</inf> -gain estimate. A larger family of Lyapunov functions leads to less conservatism for the stability and ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ph</inf> -gain analysis. Further, given the frequency response measurements for a periodic input with constant amplitude, the response for any other amplitude is determined using homogeneity. In contrast, the maximum of the scaling-invariant homogeneous frequency response yields a lower bound on the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ph</inf>-gain.