# Guaranteed Trajectory Tracking under Learned Dynamics with Contraction Metrics and Disturbance Estimation

arXiv: https://arxiv.org/abs/2112.08222 · HTML: https://mlsanity.com/paper/2112.08222/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 15 Dec 2021.

## Builds on (most cited first, 14 of 14)

- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach](https://mlsanity.com/paper/2006.04361/) (2020)
- [Deep Model Reference Adaptive Control](https://mlsanity.com/paper/1909.08602/) (2019)
- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [Safe Feedback Motion Planning: A Contraction Theory and L1-Adaptive Control Based Approach](https://mlsanity.com/paper/2004.01142/) (2020)
- [Robust Controller Design for Stochastic Nonlinear Systems via Convex Optimization](https://mlsanity.com/paper/2006.04359/) (2020)
- [Probabilistic Safety Constraints for Learned High Relative Degree System Dynamics](https://mlsanity.com/paper/1912.10116/) (2019)
- [Nonlinear Stabilization via Control Contraction Metrics: a Pseudospectral Approach for Computing Geodesics](https://mlsanity.com/paper/1607.04340/) (2016)
- [Tube-Certified Trajectory Tracking for Nonlinear Systems With Robust Control Contraction Metrics](https://mlsanity.com/paper/2109.04453/) (2021)

## Abstract (arXiv)

This paper presents an approach to trajectory-centric learning control based on contraction metrics and disturbance estimation for nonlinear systems subject to matched uncertainties. The approach uses deep neural networks to learn uncertain dynamics while still providing guarantees of transient tracking performance throughout the learning phase. Within the proposed approach, a disturbance estimation law is adopted to estimate the pointwise value of the uncertainty, with pre-computable estimation error bounds (EEBs). The learned dynamics, the estimated disturbances, and the EEBs are then incorporated in a robust Riemann energy condition to compute the control law that guarantees exponential convergence of actual trajectories to desired ones throughout the learning phase, even when the learned model is poor. On the other hand, with improved accuracy, the learned model can help improve the robustness of the tracking controller, e.g., against input delays, and can be incorporated to plan better trajectories with improved performance, e.g., lower energy consumption and shorter travel time.The proposed framework is validated on a planar quadrotor example.