Computationally Tractable Robust Nonlinear Model Predictive Control using DC Programming

We propose a computationally tractable, tube-based robust nonlinear model predictive control (MPC) framework using difference-of-convex (DC) functions and sequential convex programming. For systems with differentiable discrete time dynamics, we show how to construct systematic, data-driven DC model representations using polynomials and machine learning techniques. We develop a robust tube MPC scheme that convexifies the online optimization by linearizing the concave components of the model, and we provide guarantees of recursive feasibility and robust stability. We present three data-driven procedures for computing DC models and compare performance using a planar vertical take-off and landing (PVTOL) aircraft case study.

💡 Research Summary

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This paper introduces a novel robust nonlinear model predictive control (MPC) framework that leverages difference‑of‑convex (DC) function representations and sequential convex programming (SCP) to obtain a computationally tractable online optimization. The authors first address the difficulty of obtaining accurate nonlinear models from data. They propose three systematic, data‑driven procedures to construct DC approximations of the system dynamics: (i) a sum‑of‑squares (SOS) convex polynomial approach, (ii) an input‑convex neural network (ICNN) architecture that yields a DC neural network (DCNN) by subtracting two ICNNs, and (iii) a convex radial basis function (RBF) expansion where positive and negative coefficients are separated. Each method provides a decomposition (f = g - h) with convex components (g) and (h).

The control law is written as (u_k = v_k + K_k x_k), where (K_k) is a pre‑computed feedback gain (e.g., from an LQR on a linearized model) and (v_k) is a feed‑forward sequence to be optimized. The state tube over the prediction horizon is parameterized as a polytope (X_k) whose vertices are either element‑wise bounds (producing (2^{n_x}) vertices) or a simplex (producing (n_x+1) vertices). The original non‑convex tube propagation constraint (f_\Gamma(x_k, v_k+K_k x_k) \le q_{k+1}) is rewritten using the DC decomposition as (g(x_k, v_k+K_k x_k) - h(x_k, v_k+K_k x_k) \le q_{k+1}). Only the concave part (h) is linearized around the predicted trajectory ((x_k^\circ, u_k^\circ)), yielding a convex inequality (g - \hat h \le q_{k+1}). Because (\hat h) is an over‑approximation of (h), feasibility of the linearized constraint guarantees feasibility of the original one. Consequently, the entire MPC problem becomes a sequence of convex programs that can be solved efficiently at each sampling instant.

Recursive feasibility is proved by showing that the polytope parameters (q_k) can be updated in a way that respects the convex tube dynamics, and robust stability is established by constructing a terminal ellipsoidal set (\hat X = {x: |x|_{\hat Q} \le \hat\gamma}) whose parameters are obtained via a semidefinite program. The authors also handle additive disturbances by introducing a back‑tracking line search on the feed‑forward sequence to ensure that the tube remains invariant.

The methodology is validated on a planar vertical take‑off and landing (PVTOL) aircraft model. Three DC models are compared: the SOS‑convex polynomial model, the DCNN model, and the convex RBF model. The polynomial approach yields the fastest online solve times but may lose accuracy for highly nonlinear regions. The DCNN provides the lowest prediction error in those regions at the cost of higher offline training and slightly larger online linearization effort. The RBF method offers a middle ground in both accuracy and computational load. Across all experiments, the proposed tube‑based DC MPC achieves 5–10× faster computation than conventional robust nonlinear MPC that relies on general nonconvex solvers, while maintaining recursive feasibility and robust convergence.

In summary, the paper delivers a unified theory and practical algorithm that combine data‑driven DC modeling with a convexified tube MPC scheme. It resolves the long‑standing trade‑off between robustness, model fidelity, and real‑time tractability in nonlinear MPC, and opens avenues for extensions to stochastic uncertainties, large‑scale systems, and hardware‑in‑the‑loop demonstrations.