Dictionary-free Koopman Predictive Control for Autonomous Vehicles in Mixed Traffic

TL;DR

The paper targets autonomous vehicle control in mixed HDV/CAV traffic, where nonlinear dynamics complicate linear data-driven approaches. It introduces dictionary-free KMPC (DF-KMPC), which learns a data-driven Koopman representation directly from trajectories without selecting lifting functions or updating for changing equilibria, using an iterative Hankel-based procedure inspired by Willems' fundamental lemma. The method handles both exact and inexact Koopman embeddings, projecting initial conditions to the data-driven trajectory space to guarantee feasibility and avoiding slack. Numerical results on CF-LCC traffic demonstrate that DF-KMPC yields superior traffic-wave mitigation and trajectory tracking compared with EDMD-K and DNN-K, highlighting practical advantages for robust, data-driven CAV control in mixed traffic.

Abstract

Koopman Model Predictive Control (KMPC) and Data-EnablEd Predictive Control (DeePC) use linear models to approximate nonlinear systems and integrate them with predictive control. Both approaches have recently demonstrated promising performance in controlling Connected and Autonomous Vehicles (CAVs) in mixed traffic. However, selecting appropriate lifting functions for the Koopman operator in KMPC is challenging, while the data-driven representation from Willems' fundamental lemma in DeePC must be updated to approximate the local linearization when the equilibrium traffic state changes. In this paper, we propose a dictionary-free Koopman model predictive control (DF-KMPC) for CAV control. In particular, we first introduce a behavioral perspective to identify the optimal dictionary-free Koopman linear model. We then utilize an iterative algorithm to compute a data-driven approximation of the dictionary-free Koopman representation. Integrating this data-driven linear representation with predictive control leads to our DF-KMPC, which eliminates the need to select lifting functions and update the traffic equilibrium state. Nonlinear traffic simulations show that DF-KMPC effectively mitigates traffic waves and improves tracking performance.

Figures (5)

• Figure 1: Schematic of a CF-LCC system. The construction of the Koopman linear model requires selection of suitable lifting functions (i.e., $z = \Phi(x)$), while the proposed dictionary-free representation bypasses this process.
• Figure 2: Simulation scenario. The CF-LCC system consists of $5$ vehicles, where the yellow node, blue node, and grey nodes represent the head vehicle, the CAV, and other HDVs, respectively.
• Figure 3: Schematic of trajectories and velocities of vehicles in Experiment A. The black line and blue line represent the position of the head vehicle and the CAV, respectively. The darker the color, the higher the velocity. The disturbance applied on the head vehicle starts at $t = 10 \, \mathrm{s}$ and ends at $t = 20 \, \mathrm{s}$. (a) All vehicles are HDVs. (b)-(d) The CAV utilizes predictive controllers with different linear representations.
• Figure 4: Velocity profiles of head vehicle and realization cost of predictive controller with different linear models. (a) Real traffic velocity profiles of vehicles. (b) Realized control cost.
• Figure 5: Velocity profiles in Experiment B. The black profile denotes the head vehicle, while the blue profile and gray profiles represent the CAV and HDVs respectively. (a) The CAV utilizes the proposed DF-KMPC with DF-K. (b) The CAV utilizes a predictive controller with EDMD-K.

Theorems & Definitions (7)

• Definition 1: Persistently exciting
• Lemma 1: Willems' fundamental lemma willems2005note
• Remark 1: Local linearization and the model update
• Remark 2: Deep Hankel matrix and no explicit lifting
• Definition 2: Lifted excitation
• Theorem 1
• Remark 3