Arbitrary-order sliding modes with finite-time convergence

Sliding mode is used in order to retain a dynamic system accurately at a given constraint and is the main operation mode in variable structure systems. Such mode is a motion on a discontinuity set of a dynamic system and features theoretically-infinite-frequency switching. The standard sliding modes are known to feature finite time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. In realization their sliding precision is proportional to the time interval between measurements. Having generalized the notion of sliding mode, higher order sliding modes preserve or generalize its main properties and remove the chattering effect. With discrete measurements rth-order sliding mode realization may provide up to the rth order of sliding precision with respect to the measurement interval. Such finite-time convergence modes were known only for r = 1, 2, 3. A sample of an arbitraryorder sliding mode attracting trajectories in finite time and featuring the above-mentioned utmost accuracy is presented for the first time in the present paper. The proposed controllers provide for full real-time control of the output variable if the relative degree r of the uncertain dynamic