Learning Stabilizable Dynamical Systems via Control Contraction Metrics

Abstract

We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key idea is to develop a new control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, which guarantees that the learned system can be accompanied by a robust controller capable of stabilizing any open-loop trajectory that the system may generate. By leveraging tools from contraction theory, statistical learning, and convex optimization, we provide a general and tractable semi-supervised algorithm to learn stabilizable dynamics, which can be applied to complex underactuated systems. We validated the proposed algorithm on a simulated planar quadrotor system and observed notably improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when using a small number of demonstration examples. The results presented illustrate the need to infuse standard model-based reinforcement learning algorithms with concepts drawn from nonlinear control theory for improved reliability.

Learning-based Robust Motion Planning With Guaranteed Stability: A Contraction Theory Approach

“…5) Computational Complexity: Since (a) and LAG-ROS (c) are implementable with one neural net evaluation at each t, their performance is also compared with (b) which requires solving motion planning problems to get its control input. Its time horizon is selected to make the trajectory optimization [42] solvable online considering the current computational power, for the sake of a fair comparison. We denote the computational time as ∆t in this section, and it should be less than the maximum control time interval ∆t max , i.e., ∆t ≤ ∆t max = 0.1(s).…”

Linearly Constrained Linear Quadratic Regulator from the Viewpoint of Kernel Methods

“…The kernel Hilbert spaces have also been used to define suitable domains for operators (Rosenfeld et al, 2019;Giannakis et al, 2019). In most cases, the kernel is taken off-the-shelf, as with Gaussian kernels in connection with Bayesian inference (Singh et al, 2018;Bertalan et al, 2019).…”

Learning stabilizable nonlinear dynamics with contraction-based regularization

“…A preliminary version of this article was presented at WAFR 2018 (Singh et al, 2018). In this revised and extended version, we include the following additional contributions: (i) rigorous derivation of the stabilizability-regularized finite-dimensional optimization problem using RKHS theory and random matrix features; (ii) extensive additional numerical studies into the convergence behavior of the iterative algorithm and comparison with traditional ridge-regression techniques; and (iii) validation of the algorithm on a quadrotor testbed with partially closed control loops to emulate a planar quadrotor.…”