Multi-Timescale Model Predictive Control for Slow-Fast Systems

TL;DR

This work tackles the computational burden of real-time MPC for slow-fast systems by introducing a multi-timescale MPC (MTS-MPC) that combines an initial full, stiff-model phase with a subsequent reduced slow-model phase. The horizon is discretized with an exponential step-size schedule $\Delta t_k = \Delta t_0\,α^{k}$ ($α>1$) and a state-projection mapping $\phi$ to transition between models at a switching stage $\bar{k}$ chosen to keep the closed-loop cost below a tolerance $\epsilon$. The key contribution is showing that this two-phase scheme preserves control performance while delivering up to an order of magnitude speed-up in simulation across differential-drive, drone, and trunk-like robotic tasks, aided by an open-source interface on top of acados. The findings suggest that EDS-informed, non-hierarchical slow-fast MPC can enable real-time deployment on limited hardware without sacrificing safety or accuracy. The approach balances accuracy and efficiency by exploiting the exponential decay of sensitivities along the horizon and selectively simplifying dynamics where fast effects are less influential for the initial control input.

Abstract

Model Predictive Control (MPC) has established itself as the primary methodology for constrained control, enabling autonomy across diverse applications. While model fidelity is crucial in MPC, solving the corresponding optimization problem in real time remains challenging when combining long horizons with high-fidelity models that capture both short-term dynamics and long-term behavior. Motivated by results on the Exponential Decay of Sensitivities (EDS), which imply that, under certain conditions, the influence of modeling inaccuracies decreases exponentially along the prediction horizon, this paper proposes a multi-timescale MPC scheme for fast-sampled control. Tailored to systems with both fast and slow dynamics, the proposed approach improves computational efficiency by i) switching to a reduced model that captures only the slow, dominant dynamics and ii) exponentially increasing integration step sizes to progressively reduce model detail along the horizon. We evaluate the method on three practically motivated robotic control problems in simulation and observe speed-ups of up to an order of magnitude.

Figures (6)

• Figure 1: Multi-Timescale MPC for slow-fast systems. (a) The prediction horizon is split into two phases—first using the full model, then switching to a reduced model of the slow dynamics, enabling efficient discretization with exponentially increasing step sizes and explicit integration schemes. (b) Tuning of the switching stage: for a given step size profile, sweep over candidate switching stages and select the first stage for which the closed-loop cost increase relative to the full-resolution baseline is below a tolerance $\epsilon$.
• Figure 2: Mean closed-loop cost increase (relative to the full-resolution baseline) versus switching stage $\bar{k}$. The orange pentagon indicates the selected switching stage.
• Figure 3: System schematics and Pareto frontiers. MTS-MPC (orange) consistently outperforms all baselines in the trade-off between mean closed-loop stage costs and computational efficiency across three slow-fast robotic systems.
• Figure 4: Closed-loop trajectories: MTS-MPC achieves the same performance as the full-resolution baseline while being 16×, 3.3×, and 5.4× more computationally efficient for the differential-drive, drone, and trunk-like systems, respectively.
• Figure 5: Frobenius norm of the sensitivity matrix $\frac{du_0^\star}{dp_k}$ plotted against prediction horizon stages for randomly generated instances of \ref{['eq:QP_EDS_example']}. Lines of different colors correspond to different problem instances. $S$ is chosen to be the cost-to-go of the corresponding discrete-time .

Theorems & Definitions (3)

• Proposition 1: Sensitivity of the Optimal Control Input
• proof
• Theorem 1: EDS (Theorem 4.5 in shin2022exponential)