Practical Regulation of Nonholonomic Systems Using Virtual Trajectories and LaSalle Invariance Principle

This paper investigates the regulation problem for a class of nonholonomic systems that includes power form systems and an approximated system of the rolling sphere as special cases. The basic idea is first to introduce a virtual periodic moving trajectory that satisfies certain persistent excitation condition (PE) and has the zero value at some time instants. Based on LaSalle invariance principle, the associated tracking problem is then solved under a necessary condition for stabilization and particularly true for the power form systems and the rolling sphere. With the help of virtual trajectory, the achieved tracking result is applied to the regulation problem and used to guarantee practical stability. The proposed controllers have a simple and explicit form, and hence are easily implemented. Simultaneously, fast convergence is guaranteed, thanks to the $\boldsymbol {\mathcal {K}}$ -exponential convergence. More interestingly, the used approach is adding sufficiently exciting signals to the systems by considering virtual tracking signals so that the attractivity of the origin can be guaranteed based on LaSalle invariance principle. Thus, it is possible to extend the proposed results to more general systems. To verify the effectiveness of the proposed scheme, interesting simulation results are presented.