Sums-of-squares (SOS) optimization is a promising tool to synthesize certifiable controllers for nonlinear dynamical systems. Building upon prior works, we demonstrate that SOS can synthesize dynamic controllers with bounded suboptimal performance for various underactuated robotic systems by finding good approximations of the value function. We summarize a unified SOS framework to synthesize both under- and over- approximations of the value function for continuous-time, control-affine systems, use these approximations to generate approximate optimal controllers, and perform regional analysis on the closed-loop system driven by these controllers. We then extend the formulation to handle hybrid systems with contacts. We demonstrate that our method can generate tight under- and over- approximations of the value function with low-degree polynomials, which are used to provide stabilizing controllers for continuous-time systems including the inverted pendulum, the cart-pole, and the quadrotor as well as a hybrid system, the planar pusher. To the best of our knowledge, this is the first time that a SOS-based time-invariant controller can swing up and stabilize a cart-pole, and push the planar slider to the desired pose.

翻译：平方和（Sums-of-Squares，SOS）优化是综合非线性动力系统可认证控制器的一种有前景的工具。基于先前研究，我们证明SOS能够通过寻找价值函数的良好近似，为多种欠驱动机器人系统综合具有有界次优性能的动态控制器。我们总结了一个统一的SOS框架，用于综合连续时间控制仿射系统价值函数的下近似和上近似，利用这些近似生成近似最优控制器，并对这些控制器驱动的闭环系统进行区域分析。随后，我们将该公式扩展至处理带接触的混合系统。我们证明，该方法可以用低次多项式生成价值函数的紧致上下近似，从而为连续时间系统（包括倒立摆、推车杆和四旋翼飞行器）以及混合系统（平面推杆）提供镇定控制器。据我们所知，这是首次基于SOS的非时变控制器能够实现推车杆的摆起与镇定，以及将平面滑块推至期望位姿。