Technical Summary: Amortized Nonlinear Model Predictive Control

Problem Statement

Nonlinear Model Predictive Control (NMPC) offers a powerful framework for constrained optimal control but faces a significant computational bottleneck. At every sampling instant, NMPC requires solving a constrained nonlinear program (NLP) in real-time. For nonlinear plant models, this per-step solve time can be orders of magnitude larger than for linear systems, often rendering real-time implementation infeasible on resource-constrained hardware or at high sampling rates. While Real-Time Iteration (RTI) schemes improve efficiency by performing a single Newton-type iteration, the computational cost can still be prohibitive. Furthermore, explicit nonlinear MPC is generally intractable.

Recent advances in machine learning, specifically "amortized optimization," propose replacing the online solver with a function approximator trained offline. However, a major challenge in this approach is preserving closed-loop guarantees, such as constraint satisfaction and stability, when the learned approximation deviates from the exact solution.

Methodology

The authors propose a framework for input-affine nonlinear systems (systems where the dynamics are affine in the control input, $x_{k+1} = f(x_k) + G(x_k)u_k$) that approximates the optimal control move by solving a state-dependent Quadratic Program (QP) online. The key innovation is a single-network residual-corrector architecture that separates the learning of problem parameters from the enforcement of constraints.

1. Parametric QP Formulation

For input-affine systems, the one-step state map is affine in the control input $u_0$. The authors formulate a small QP to optimize the first control move $u_0$:

• Objective: A quadratic cost function $\frac{1}{2}u_0^\top H(x_0) u_0 + c(x_0, r)^\top u_0$, where $H$ and $c$ depend on the current state and reference.
• Constraints:
  • Hard Constraints: Input bounds and affine-in-input constraints (e.g., torque limits) are enforced directly.
  • Soft Constraints: State bounds for the next step are modeled as soft constraints using slack variables to ensure the QP remains feasible.
• Size: The resulting QP has significantly fewer decision variables ($n_u + n_s$) compared to the full NLP horizon ($(N+1)n_x + N n_u$).

2. Residual-Corrector Network Architecture

Rather than learning the full optimal control sequence or the full QP parameters from scratch, the network learns only the residual corrections to an analytic baseline:

• Analytic Baseline: Derived from linearizing the system dynamics and cost function around the current state. This provides initial estimates for the Cholesky factor of the Hessian ($L_{an}$) and the linear cost vector ($c_{an}$).
• Neural Network ($\phi_\psi$): Takes the concatenated state and reference as input and predicts corrections ($\Delta L, \Delta c$).
• Correction Mechanism:
  • The full Cholesky factor is $L_{full} = L_{an} + \Delta L$.
  • The diagonal elements are processed through a softplus function to guarantee strict positivity, ensuring the resulting Hessian $H_{full} = L_{full}L_{full}^\top$ is positive definite.
  • The full cost vector is $c_{full} = c_{an} + \Delta c$.

3. Differentiable Solver and Training

• Online Inference: The corrected parameters define a small QP, which is solved online by a differentiable interior-point layer. This guarantees that the output $u_0^*$ satisfies the QP constraints by construction.
• Offline Training: The network is trained on a dataset of optimal NLP solutions (generated by a solver like IPOPT) using a hybrid loss function:
  1. Supervised Imitation ($L_{sup}$): Penalizes the deviation of the predicted first control action from the optimal NLP solution.
  2. KKT Stationarity ($L_{KKT}$): Penalizes the violation of the stationarity condition of the full NLP at the predicted solution.
  3. Fischer-Burmeister Complementarity ($L_{FB}$): Penalizes violations of complementarity conditions for inequality constraints.
  4. Regularization ($L_{reg}$): Keeps the learned corrections close to the analytic baseline.

Key Contributions

1. Residual-Corrector Architecture: A novel approach where a neural network learns only the corrections to an analytically derived QP baseline, rather than the full solution. This leverages the known structure of input-affine systems.
2. Constraint Satisfaction via Differentiable Layers: By embedding a differentiable QP solver within the network, the method guarantees that the first control action satisfies hard constraints, addressing a common failure mode in learning-based MPC.
3. Hybrid Loss Function: The training objective combines supervised imitation with physics-informed KKT and complementarity penalties, improving performance and feasibility compared to pure imitation learning.
4. Scalability: The approach reduces the online optimization problem from a complex NLP to a small QP with a state-dependent cost matrix, drastically reducing computational complexity.

Experimental Results

The method was validated on a three-link planar robotic arm with Cartesian end-effector tracking, subject to joint position, velocity, acceleration, and torque constraints.

• Performance: The amortized policy achieved a mean closed-loop cost within 4.7% of the full-horizon IPOPT optimum. In contrast, the standard RTI scheme (acados) diverged in 12% of scenarios and was 17.5% worse on average, while a direct MLP (without the QP layer) diverged in 27% of scenarios and violated hard constraints significantly.
• Constraint Satisfaction: The amortized QP satisfied hard torque and acceleration bounds to numerical precision. Soft state constraints were violated only mildly during transients (0.7% of steps).
• Computational Speed:
  • IPOPT: ~120 ms median solve time.
  • RTI (acados): ~2.4 ms median solve time.
  • Amortized QP: 0.116 ms median solve time.
  • This represents a ~21x speedup over RTI and an ~1800x speedup over the full NLP solver.

Significance

The paper claims that this approach offers a practical route to deploying NMPC on resource-constrained hardware or at high sampling rates without sacrificing constraint satisfaction. By decoupling the learning of the cost landscape (via the network) from the enforcement of constraints (via the QP solver), the method achieves orders-of-magnitude speedups over traditional NLP solvers while maintaining tracking performance comparable to the optimal solution. The use of a hybrid loss function combining supervised learning with KKT residuals is highlighted as a key factor in achieving superior feasibility and reliability compared to methods that simply fit data or learn warm-starts.