Tracking controller design for random nonlinear benchmark system

•Random oscillation of surrounding described by color noise instead of white noise.•Random oscillation of surrounding is transformed into the random disturbance to the controller.•The vectorial backstepping controller is designed for the quasi-lower triangle system.•Track error can be made arbitrarily small by tuning the design parameters.

This paper considers a benchmark system consisting of a rolling ball and a moving car in the oscillating surroundings. By using the Lagrange law, the dynamic model without disturbance is first constructed, then according to the relative motion principle, random oscillation of surroundings is transformed into the random noises in the constructed Lagrange equation. The special structure of the quasi-lower triangle of Lagrange equation motivates us to pay more attention to the vectorial backstepping technique. By selecting an appropriate Lyapunov-like function, a tracking controller with tunable parameters is designed such that all signals of the closed-loop system are bounded and track error can be made arbitrarily small.