We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in Reproducing Kernel Hilbert Spaces, we formulate stabilizable dynamics learning as a functional optimization with convex objective and bi-convex functional constraints. Under a mild structural assumption and relaxation of the functional constraints to sampling-based constraints, we derive the optimal solution with a modified Representer theorem. Finally, we utilize random matrix feature approximations to reduce the dimensionality of the search parameters and formulate an iterative convex optimization algorithm that jointly fits the dynamics functions and searches for a certificate of stabilizability. We validate the proposed algorithm in simulation for a planar quadrotor, and on a quadrotor hardware testbed emulating planar dynamics. We verify, both in simulation and on hardware, significantly improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when learning from small supervised datasets. The results support the conjecture that the use of stabilizability constraints as a form of regularization can help prune the hypothesis space in a manner that is tailored to the downstream task of trajectory generation and feedback control, resulting in models that are not only dramatically better conditioned, but also data efficient.

中文翻译：

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使用基于收缩的正则化学习可稳定的非线性动力学

我们提出了一种新的框架，用于学习机器人技术中连续控制任务的可稳定非线性动力系统。关键贡献是基于稳定性概念的动力学拟合的控制理论正则化器，这是一种约束，可保证对学习系统生成的任意开环轨迹存在稳健的跟踪控制器。利用 Reproduced Kernel Hilbert Spaces 中收缩理论和统计学习的工具，我们将可稳定动态学习公式化为具有凸目标和双凸函数约束的函数优化。在温和的结构假设和基于采样的约束的功能约束的放松下，我们使用修改后的 Representer 定理推导出最优解。最后，我们利用随机矩阵特征近似来降低搜索参数的维数，并制定一个迭代凸优化算法，该算法联合拟合动力学函数并搜索稳定性证明。我们在模拟平面四旋翼飞行器和模拟平面动力学的四旋翼硬件测试台上验证了所提出的算法。我们在模拟和硬件上都验证了使用控制理论正则化模型显着改善了轨迹生成和跟踪性能，而不是使用传统回归技术学习的模型，尤其是在从小型监督数据集学习时。