The task of inducing, via continuous static state-feedback, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system, that is, to a smooth curve in its state--control space that is consistent with the system dynamics and which connects two stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver's orbit as to combine two linearization techniques for this purpose: the Jacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. This allows for the computation of stabilizing control gains offline by solving a semidefinite programming problem. The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and the final equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching and has a familiar feedforward plus feedback--like structure. The method is also complemented by synchronization function--based arguments for planning such maneuvers for mechanical systems with one degree of underactuation. Numeric simulations of the non-prehensile manipulation task of a ball rolling between two points upon the "butterfly" robot demonstrates the efficacy of the full synthesis.
翻译:在非线性控制系统中,通过连续静止的静态状态回溯,引导非线性控制系统中无症状稳定的异地临床轨道的任务在本文中得到考虑。主要动力来自确保与所谓的点对点操作趋同的问题,即确保在一个未完全起动的机械系统中,即与其与系统动态一致并连接两个可稳定平衡点的状态控制空间的平稳曲线趋同,由此产生的非线性控制空间与系统动态一致,同时稳定轨道和最后平衡点。拟议方法使用特定的参数化,同时将状态投射到演习轨道上,为此目的将两种线性技术结合起来:在边界上的叶柯比亚线性线性,以及轨道上的反向线性线性线性线化。这通过解决半不完全起动的机械系统,从而可以计算离线上的稳定控制增益。由此产生的非线性控制器,同时稳定轨道和最后平衡点,是时间变异的,本地的利普西茨基茨连续,不需要转换,并且有一个熟悉的反馈和反馈式结构。该方法还得到了“在一种水平上同步的滚式机性操作系统下,在一种机尾部的滚动的滚动操作中,在两个方向上进行一个方向的滚动的滚动的滚动操作中,在两个系统之间,这个方法得到了补充。