ℒ₂-gain tuning of variable structure SISO systems of relative degree n

The present paper extends the L₂-gain analysis towards variable structure systems (VSS) with sliding-mode (SM) dynamics. The developed analysis is applied to feedback linearisable VSS, driven by standard SM algorithms of the first order. Constructive parameter-tuning of the resulting closed-loop system to possess the L₂-gain less than a desired attenuation level is additionally deduced. The proposed tuning is shown to be capable not only of rejecting matched uniformly bounded disturbances but also of attenuating unbounded ones, including mismatched disturbances. Numerical simulations of an inverted pendulum, driven by a first-order SM controller, are involved to support the theory in the presence of the position measurement inaccuracy, being a source of a mismatched disturbance.