Differentiable Robust Model Predictive Control

TL;DR

This work addresses the challenge of robust real-time control under disturbances by unifying differentiable optimization with tube-based model predictive control. By leveraging the implicit function theorem, it develops a Differentiable Optimal Control (DOC) framework that backpropagates through a two-layer MPC (nominal and ancillary) and couples it with embedded barrier states (DBaS) to enforce safety. The resulting Differentiable Tube-based MPC (DT-MPC) enables online adaptation of both nominal and ancillary controller parameters, achieving robust performance with linear-time complexity in the horizon and constant memory for gradients. The approach is validated across five nonlinear robotic systems in simulation and on hardware (Robotarium), showing markedly improved safety and task success compared to non-adaptive tube MPC, and demonstrating practical impact for real-time, safe autonomous control. Theoretical guarantees on numerical precision of the implicit differentiation are complemented by empirical results, and the framework generalizes to learning-based adaptations of control policies while maintaining real-time feasibility.

Abstract

Deterministic model predictive control (MPC), while powerful, is often insufficient for effectively controlling autonomous systems in the real-world. Factors such as environmental noise and model error can cause deviations from the expected nominal performance. Robust MPC algorithms aim to bridge this gap between deterministic and uncertain control. However, these methods are often excessively difficult to tune for robustness due to the nonlinear and non-intuitive effects that controller parameters have on performance. To address this challenge, we first present a unifying perspective on differentiable optimization for control using the implicit function theorem (IFT), from which existing state-of-the art methods can be derived. Drawing parallels with differential dynamic programming, the IFT enables the derivation of an efficient differentiable optimal control framework. The derived scheme is subsequently paired with a tube-based MPC architecture to facilitate the automatic and real-time tuning of robust controllers in the presence of large uncertainties and disturbances. The proposed algorithm is benchmarked on multiple nonlinear robotic systems, including two systems in the MuJoCo simulator environment and one hardware experiment on the Robotarium testbed, to demonstrate its efficacy.

Figures (12)

• Figure 1: Proposed differentiable robust MPC architecture. Orange dashed arrows show how gradients are passed in our architecture.
• Figure 2: Controlled quadrotor trajectories subject to large disturbances. 50 trajectories are plotted for each algorithm. 'Nominal' corresponds to the reference trajectory being tracked by the two algorithms. Our proposed differentiable tube-based MPC (DT-MPC) is safer and more robust than the baseline nonlinear tube-based MPC (NT-MPC).
• Figure 3: Jacobian estimate errors on the quadrotor system as a function of DDP iterate error.
• Figure 4: Controlled Dubins vehicle trajectories subject to large noise. NT-MPC trajectories diverge from the nominal trajectory and the uncertainty increases over time. Meanwhile, DT-MPC adapts to the environment, maintaining safety and robust task performance.
• Figure 5: Environment for the robot arm task.

Theorems & Definitions (30)

• Theorem 1: Implicit function theorem krantz2002implicit
• proof
• Remark 1
• Proposition 2: Optimality conditions of \ref{['prob:parameterized_oc']}
• proof
• Proposition 3: Implicit derivative of \ref{['prob:parameterized_oc']}
• proof
• Remark 2
• Corollary 4: Pontryagin differentiable programming jin2020pontryagin
• proof