Perfecting Periodic Trajectory Tracking: Model Predictive Control with a Periodic Observer ($Π$-MPC)

TL;DR

This paper addresses the problem of achieving perfect tracking in MPC for periodic references despite large model mismatch by introducing a lifted periodic disturbance observer. The proposed Pi-MPC augments the nominal model with period-wide disturbances and uses an observer to generate targets that guide a tracking MPC, with convergence guarantees under feasibility and periodicity assumptions. The key contributions are (i) a linear periodic disturbance observer with observability analysis, (ii) a tracking MPC that attains zero asymptotic tracking error for periodic references, and (iii) extensions to nonlinear models with practical observer designs and MPC formulations. The method is validated on a high-dimensional soft robot and a small-scale race car, showing substantial reductions in tracking error and real-time feasibility, highlighting its practical impact for repetitive robotic tasks with limited model accuracy.

Abstract

In Model Predictive Control (MPC), discrepancies between the actual system and the predictive model can lead to substantial tracking errors and significantly degrade performance and reliability. While such discrepancies can be alleviated with more complex models, this often complicates controller design and implementation. By leveraging the fact that many trajectories of interest are periodic, we show that perfect tracking is possible when incorporating a simple observer that estimates and compensates for periodic disturbances. We present the design of the observer and the accompanying tracking MPC scheme, proving that their combination achieves zero tracking error asymptotically, regardless of the complexity of the unmodelled dynamics. We validate the effectiveness of our method, demonstrating asymptotically perfect tracking on a high-dimensional soft robot with nearly 10,000 states and a fivefold reduction in tracking errors compared to a baseline MPC on small-scale autonomous race car experiments.

Figures (2)

• Figure 1: Left: Picture of the Diamond robot mesh. The reference figure-eight trajectory for the tip is shown with a dashed black line. The red arrows indicate the actuator inputs, i.e., applied forces at the elbows. Right: Simulation results illustrate tracking performance over ten periods for a high-frequency, 2D figure-eight trajectory. We compare a standard MPC scheme (left), offset-free MPC (center), and our proposed MPC with a periodic disturbance observer ($\Pi$-MPC) (right), all shown in blue. The bottom plots show the tracking error over time. The shading indicates time progression, with lighter shades representing earlier portions of the trajectory. The dashed black line represents the reference trajectory.
• Figure 2: Experimental results illustrate tracking with a race car. We compare a standard MPC scheme and the same MPC with the proposed periodic disturbance observer ($\Pi$-MPC), all shown in blue. The shading indicates time progression, with lighter shades representing earlier portions of the trajectory. The dashed black line represents the reference trajectory.

Theorems & Definitions (10)

• Proposition 1
• proof
• Remark 1
• Proposition 2
• proof
• Theorem 1
• proof
• proof : Proof of Prop. \ref{['prop:augObs']}
• Proposition 3
• proof