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Annealed Ensemble Kalman Inversion for Constrained Nonlinear Model Predictive Control: An ADMM Approach

Ahmed Khalil, Mohamed Safwat, Efstathios Bakolas

TL;DR

This work introduces ADMM-EKI, a derivative-free, parallelizable approach for constrained nonlinear model predictive control. By embedding nonlinear stagewise constraints with nonnegative slack variables and solving the resulting unconstrained augmented Lagrangian via Ensemble Kalman Inversion inside an ADMM framework, the method handles nonconvex dynamics and inequality constraints with a principled primal-dual scheme. The authors provide a Bayesian interpretation that links the primal update to MAP estimation and employ annealing to balance exploration and convergence. Empirical results on a 2D nonconvex problem and an autonomous racing task show that ADMM-EKI can outperform a DIAL-MPC variant in speed and lap-time metrics while maintaining feasible constraint satisfaction, highlighting its potential for real-time, derivative-free constrained NMPC.

Abstract

This work proposes a novel Alternating Direction Method of Multipliers (ADMM)-based Ensemble Kalman Inversion (EKI) algorithm for solving constrained nonlinear model predictive control (NMPC) problems. First, the stage-wise nonlinear inequality constraints in the NMPC problem are embedded via an augmented Lagrangian with nonnegative slack variables. We then show that the unconstrained augmented Lagrangian formulation of the NMPC admits a Bayesian interpretation: under a Gaussian observation model, its minimizers coincide with MAP estimators, enabling solution via EKI. However, since the nonnegativity constraint on the slacks cannot be enforced via Gaussian noise, our proposed algorithm results in a two-block ADMM that alternates between (i) a primal step that minimizes the unconstrained augmented Lagrangian, (ii) a nonnegativity projection for the slacks, and (iii) a dual ascent step. To balance exploration and convergence, an annealing schedule tempers covariances and penalty weights, thereby encouraging global search early and precise constraint satisfaction later. To demonstrate the performance of the proposed method, we compare it with another iterative sampling-based approach based on Model Predictive Path Integral (MPPI) control, called DIAL-MPC.

Annealed Ensemble Kalman Inversion for Constrained Nonlinear Model Predictive Control: An ADMM Approach

TL;DR

This work introduces ADMM-EKI, a derivative-free, parallelizable approach for constrained nonlinear model predictive control. By embedding nonlinear stagewise constraints with nonnegative slack variables and solving the resulting unconstrained augmented Lagrangian via Ensemble Kalman Inversion inside an ADMM framework, the method handles nonconvex dynamics and inequality constraints with a principled primal-dual scheme. The authors provide a Bayesian interpretation that links the primal update to MAP estimation and employ annealing to balance exploration and convergence. Empirical results on a 2D nonconvex problem and an autonomous racing task show that ADMM-EKI can outperform a DIAL-MPC variant in speed and lap-time metrics while maintaining feasible constraint satisfaction, highlighting its potential for real-time, derivative-free constrained NMPC.

Abstract

This work proposes a novel Alternating Direction Method of Multipliers (ADMM)-based Ensemble Kalman Inversion (EKI) algorithm for solving constrained nonlinear model predictive control (NMPC) problems. First, the stage-wise nonlinear inequality constraints in the NMPC problem are embedded via an augmented Lagrangian with nonnegative slack variables. We then show that the unconstrained augmented Lagrangian formulation of the NMPC admits a Bayesian interpretation: under a Gaussian observation model, its minimizers coincide with MAP estimators, enabling solution via EKI. However, since the nonnegativity constraint on the slacks cannot be enforced via Gaussian noise, our proposed algorithm results in a two-block ADMM that alternates between (i) a primal step that minimizes the unconstrained augmented Lagrangian, (ii) a nonnegativity projection for the slacks, and (iii) a dual ascent step. To balance exploration and convergence, an annealing schedule tempers covariances and penalty weights, thereby encouraging global search early and precise constraint satisfaction later. To demonstrate the performance of the proposed method, we compare it with another iterative sampling-based approach based on Model Predictive Path Integral (MPPI) control, called DIAL-MPC.

Paper Structure

This paper contains 12 sections, 1 theorem, 29 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Ensemble Kalman Inversion using ADMM
  4. Augmented Lagrangian Formulation
  5. ADMM Outer Loop
  6. Bayesian Estimation for Nonlinear MPC
  7. Primal Update via Ensemble Kalman Inversion
  8. Receding-Horizon Implementation of ADMM-EKI
  9. Numerical Simulations
  10. Illustrative 2D Example
  11. Autonomous Racing with Obstacles Example
  12. Conclusion

Key Result

Proposition 1

Let $\Phi^{\ell} (U)$ denote the objective function of the primal problem eq_primal_step, which is defined as in eq:PhiU, and let us assume that the set of minimizers of $\Phi^{\ell} (U)$ is nonempty. Then, the set of MAP (Maximum A Posteriori) estimators for the posterior $p(U \mid y^{\ell})$ under

Figures (1)

  • Figure 1: Numerical simulation results.

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2: Efficient inversion via Woodbury matrix identity
  • Remark 3