Robust Model Predictive Control for nonlinear discrete-time systems using iterative time-varying constraint tightening

TL;DR

This paper introduces a shrinking-horizon robust NMPC framework for nonlinear discrete-time systems that accounts for disturbances and linearization errors via iterative time-varying constraint tightening. By deriving error sets around a reference trajectory and applying LQR-based constraint tightening, it guarantees robust constraint satisfaction while enabling real-time operation. The method iteratively optimizes a reference trajectory using Problem 2 and employs a fallback control when online NLP solutions fail, thereby maintaining safety and feasibility. Numerical experiments on a nonlinear FTMS aircraft model show comparable performance across NLP solvers, with notable gains in computation speed and scalability relative to existing robust NMPC techniques. The work advances practical robust NMPC by balancing performance, conservatism, and computational tractability, with promising potential for long-horizon and mission-based applications.

Abstract

Robust Model Predictive Control (MPC) for nonlinear systems is a problem that poses significant challenges as highlighted by the diversity of approaches proposed in the last decades. Often compromises with respect to computational load, conservatism, generality, or implementation complexity have to be made, and finding an approach that provides the right balance is still a challenge to the research community. This work provides a contribution by proposing a novel shrinking-horizon robust MPC formulation for nonlinear discrete-time systems. By explicitly accounting for how disturbances and linearization errors are propagated through the nonlinear dynamics, a constraint tightening-based formulation is obtained, with guarantees of robust constraint satisfaction. The proposed controller relies on iteratively solving a Nonlinear Program (NLP) to simultaneously optimize system operation and the required constraint tightening. Numerical experiments show the effectiveness of the proposed controller with three different choices of NLP solvers as well as significantly improved computational speed, better scalability, and generally reduced conservatism when compared to an existing technique from the literature.

Figures (12)

• Figure 1: Dual-tank FTMS architecture modified with Air Cycle Machine.
• Figure 2: Disturbance profiles used in the simulations. Top: $\Delta d_k$ is a square wave. Bottom: $\Delta d_k$ is a random signal.
• Figure 3: State and input trajectories obtained when a nominal NMPC is applied to Test Case 1, with multiple temperature constraint violations. Original constraints are shown in blue.
• Figure 4: Error sets $\mathcal{E}_{k+i|k}$ computed for $k=0$ (red) and the actual error $e_k$ (black) obtained for 100 random realizations of $\Delta d_k$ and with the FTMS system controlled using only the fallback control input \ref{['eq:MPCplusLQRControl']} based on the same reference trajectory used to compute these error sets.
• Figure 5: State and input trajectories obtained when applying IPOPT-MPC1 (black), IPOPT-MPC1 with the fallback control applied alternately for periods of 500 seconds (orange), and fallback control only (dark green) to Test Case 1. Original constraints are shown in blue.

Theorems & Definitions (10)

• definition 1: Reference Trajectory
• proposition 1: Morphological Opening Property castroAnalyticalSolutionsMinkowski2013
• definition 2: Valid Reference Trajectory
• theorem 1
• proof
• remark 1
• theorem 2
• proof
• remark 2
• remark 3