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.

中文翻译：

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使用虚拟轨迹和拉萨尔不变性原理对非完整系统进行实际调节

本文研究了一类非完整系统的调节问题，该系统包括作为特殊情况的幂形式系统和滚球近似系统。其基本思想是首先引入一个虚拟周期运动轨迹，该轨迹满足一定的持续激励条件（PE）并且在某些时刻具有零值。基于拉萨尔不变性原理，相关的跟踪问题在稳定的必要条件下得到解决，特别是对于动力形式系统和滚动球。借助虚拟轨迹，将获得的跟踪结果应用于调节问题并用于保证实际稳定性。所提出的控制器具有简单而明确的形式，因此很容易实现。同时保证快速收敛，感谢 $\boldsymbol {\mathcal {K}}$ - 指数收敛。更有趣的是，所使用的方法是通过考虑虚拟跟踪信号向系统添加足够令人兴奋的信号，以便基于拉萨尔不变性原理保证原点的吸引力。因此，可以将建议的结果扩展到更一般的系统。为了验证所提出方案的有效性，展示了有趣的仿真结果。