Approximate Optimal Controller Synthesis for Cart-Poles and Quadrotors via Sums-of-Squares

TL;DR

This paper presents a sum-of-squares optimization framework to synthesize certifiable, near-optimal controllers for nonlinear and hybrid robotic systems, including cart-poles and quadrotors, with demonstrated stability and performance improvements.

Contribution

It introduces a unified SOS-based method for approximating value functions and synthesizing controllers for both continuous and hybrid systems, enabling new control capabilities.

Findings

Successfully stabilizes cart-pole and quadrotor systems.

Provides tight under- and over-approximations of the value function.

First SOS-based controller to swing up and stabilize a cart-pole.

Abstract

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.