RISE-based control of an uncertain nonlinear system with time-varying state delays

This paper considers a continuous control design for second-order control affine nonlinear systems with time-varying state delays. A neural network is augmented with a robust integral of the sign of the error (RISE) control structure to achieve semi-global asymptotic tracking in the presence of unknown, arbitrarily large, time-varying delays, not linear-in-the-parameters uncertainty and additive bounded disturbances. By expressing unknown functions in terms of the desired trajectories and through strategic grouping of delay-free and delay-dependent terms, Lyapunov-Krasovskii functionals are utilized to cancel the delayed terms in the analysis and obtain delay-free neural network update laws.