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Differentiable-by-design Nonlinear Optimization for Model Predictive Control

Riccardo Zuliani, Efe C. Balta, John Lygeros

TL;DR

The paper addresses the difficulty of differentiating solution maps of nonlinear NLP-based controllers in MPC when standard differentiability conditions fail. It introduces a differentiable-by-design regularized NLP (P3) that preserves the original solution locally and provides a surrogate derivative computed via a structured linear system arising from the KKT conditions. The authors establish theoretical guarantees on differentiability, convergence of the surrogate derivative to the true derivative as the regularization parameter rho vanishes, and practical computational procedures. Numerical experiments in QP, nonlinear trajectory optimization, and MPC demonstrate accurate sensitivities, improved reliability under non-differentiability, and favorable computational efficiency for gradient-based policy optimization."

Abstract

Nonlinear optimization-based control policies, such as those those arising in nonlinear Model Predictive Control, have seen remarkable success in recent years. These policies require solving computationally demanding nonlinear optimization programs online at each time-step. The resulting solution map, viewed as a function of the measured state of the system and design parameters, may not be differentiable, which poses significant challenges if the control policy is embedded in a gradient-based policy optimization scheme. We propose a principled way to regularize the nonlinear optimization problem, obtaining a surrogate derivative even if when the original problem is not differentiable. The surrogate problem is differentiable by design and its solution map coincides with the solution of the unregularized problem. We demonstrate the effectiveness of our approach in a free-final-time optimal control problem and a receding-horizon nonlinear MPC example.

Differentiable-by-design Nonlinear Optimization for Model Predictive Control

TL;DR

The paper addresses the difficulty of differentiating solution maps of nonlinear NLP-based controllers in MPC when standard differentiability conditions fail. It introduces a differentiable-by-design regularized NLP (P3) that preserves the original solution locally and provides a surrogate derivative computed via a structured linear system arising from the KKT conditions. The authors establish theoretical guarantees on differentiability, convergence of the surrogate derivative to the true derivative as the regularization parameter rho vanishes, and practical computational procedures. Numerical experiments in QP, nonlinear trajectory optimization, and MPC demonstrate accurate sensitivities, improved reliability under non-differentiability, and favorable computational efficiency for gradient-based policy optimization."

Abstract

Nonlinear optimization-based control policies, such as those those arising in nonlinear Model Predictive Control, have seen remarkable success in recent years. These policies require solving computationally demanding nonlinear optimization programs online at each time-step. The resulting solution map, viewed as a function of the measured state of the system and design parameters, may not be differentiable, which poses significant challenges if the control policy is embedded in a gradient-based policy optimization scheme. We propose a principled way to regularize the nonlinear optimization problem, obtaining a surrogate derivative even if when the original problem is not differentiable. The surrogate problem is differentiable by design and its solution map coincides with the solution of the unregularized problem. We demonstrate the effectiveness of our approach in a free-final-time optimal control problem and a receding-horizon nonlinear MPC example.

Paper Structure

This paper contains 7 sections, 5 theorems, 49 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Preliminaries on derivatives of NLPs
  4. Regularization of the NLP
  5. Differentiating the regularized NLP
  6. Simulations
  7. Conclusion

Key Result

Lemma 1

If $\phi\in \Phi(\theta)$ satisfies the SSOSC, LICQ, and SCS, then there exists a neighborhood $N$ of $\theta$ and a continuously differentiable function $\phi:N \to \mathbb{R}^{n_\phi}$, such that $\phi(\tilde{\theta}) \in \Phi(\tilde{\theta})$ satisfies the SSOSC, LICQ, and SCS for all $\tilde{\th with

Figures (4)

  • Figure 1: Optimal position and input for $\theta=\bar{\theta}$ (orange), $\theta=\bar{\theta}+\delta \theta$ (blue), and first order approximation (red).
  • Figure 2: Derivative of the closed-loop state trajectory computed with finite differences (orange) and our method (dashed blue).
  • Figure 3: Relative error and cosine similarity between the sensitivity of $x(\theta)$ obtained with finite difference and with our method.
  • Figure 4: Relative error and cosine similarity between the sensitivity of $u_0(\theta)$ obtained with finite difference and with our method.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Example 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 6 more