Higher-order sliding modes, differentiation and output-feedback control

TLDR

Sliding mode control maintains constraints with finite‑time convergence and robustness, but requires relative degree one and can cause chattering. The authors use arbitrary‑order robust exact differentiators that provide finite‑time convergence and noise‑optimal asymptotics to estimate the higher‑order output derivatives needed for sliding‑mode control. Higher‑order sliding modes eliminate the relative‑degree restriction, achieve r‑th order precision, enable full output‑feedback control of uncertain systems, and are validated by simulation.

Abstract

Being a motion on a discontinuity set of a dynamic system, sliding mode is used to keep accurately a given constraint and features theoretically-infinite-frequency switching. Standard sliding modes provide for finite-time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. Yet the relative degree of the constraint has to be 1 and a dangerous chattering effect is possible. Higher-order sliding modes preserve or generalize the main properties of the standard sliding mode and remove the above restrictions. r-Sliding mode realization provides for up to the rth order of sliding precision with respect to the sampling interval compared with the first order of the standard sliding mode. Such controllers require higher-order real-time derivatives of the outputs to be available. The lacking information is achieved by means of proposed arbitrary-order robust exact differentiators with finite-time convergence. These differentiators feature optimal asymptotics with respect to input noises and can be used for numerical differentiation as well. The resulting controllers provide for the full output-feedback real-time control of any output variable of an uncertain dynamic system, if its relative degree is known and constant. The theoretical results are confirmed by computer simulation.