Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach

This letter presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems

Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach

This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems.Comment: IEEE Control Systems Letters (L-CSS). Preprint version, accepted June 202

Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimization

This paper presents ConVex optimization-based Stochastic steady-state Tracking Error Minimization (CV-STEM), a new state feedback control framework for a class of Ito stochastic nonlinear systems and Lagrangian systems. Its strength lies in computing the control input by an optimal contraction metric, which greedily minimizes an upper bound of the steady-state mean squared tracking error of the system trajectories. Although the problem of minimizing the bound is nonlinear, its equivalent convex formulation is proposed utilizing state-dependent coefficient parameterizations of the nonlinear system equation. It is shown using stochastic incremental contraction analysis that the CV-STEM provides a sufficient guarantee for exponential boundedness of the error for all time with L₂-robustness properties. For the sake of its sampling-based implementation, we present discrete-time stochastic contraction analysis with respect to a state- and time-dependent metric along with its explicit connection to continuous-time cases. We validate the superiority of the CV-STEM to PID, H∞, and given nonlinear control for spacecraft attitude control and synchronization problems

Contraction Theory for Robust Learning-Based Control: Toward Aerospace and Robotic Autonomy

Machine learning and AI have been used for achieving autonomy in various aerospace and robotic systems. In next-generation research tasks, which could involve highly nonlinear, complicated, and large-scale decision-making problems in safety-critical situations, however, the existing performance guarantees of black-box AI approaches may not be sufficiently powerful. This thesis gives a mathematical overview of contraction theory, with some practical examples drawn from joint projects with NASA JPL, for enjoying formal guarantees of nonlinear control theory even with the use of machine learning-based and data-driven methods. This is not to argue that these methods are always better than conventional approaches, but to provide formal tools to investigate their performance for further discussion, so we can design and operate truly autonomous aerospace and robotic systems safely, robustly, adaptively, and intelligently in real-time. Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. Its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, resulting in many parallels drawn between linear systems theory and contraction theory for nonlinear systems. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit the systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The first two parts of this thesis are about a theoretical overview of contraction theory and its advantages, with an emphasis on deriving formal robustness and stability guarantees for deep learning-based 1) feedback control, 2) state estimation, 3) motion planning, 4) multi-agent collision avoidance and robust tracking augmentation, 5) adaptive control, 6) neural net-based system identification and control, for nonlinear systems perturbed externally by deterministic and stochastic disturbances. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks. In the third part of the thesis, we present several numerical simulations and empirical validation of our proposed approaches to assess the impact of our findings on realizing aerospace and robotic autonomy. We mainly focus on the two joint projects with NASA JPL: 1) Science-Infused Spacecraft Autonomy for Interstellar Object Exploration and 2) Constellation Autonomous Space Technology Demonstration of Orbital Reconfiguration (CASTOR), where we also perform hardware demonstrations of our methods using our thruster-based spacecraft simulators (M-STAR) and in high-conflict, distributed, intelligent UAV swarm reconfiguration with up to 20 UAVs (crazyflies).</p

Neural Stochastic Contraction Metrics for Learning-based Control and Estimation

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable learning-based control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric and its differential Lyapunov function, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The trained NSCM model allows autonomous systems to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic NCM, as shown in simulation results