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Controllability Analysis and Optimal Control of Mixed Traffic Flow with Human-driven and Autonomous Vehicles
Jiawei Wang, Yang Zheng, Qing Xu, Jianqiang Wang, Keqiang Li
TL;DR
Mixed traffic systems with heterogeneous HDVs raise unresolved questions about controllability and stabilizability, especially when only one CAV is available. This paper uses PBH analysis and structured optimal control with explicit communication constraints, showing stabilizability, deriving reachability conditions, and supporting CAV-based traffic-flow smoothing through numerical experiments.
Problem
The controllability and stabilizability of ring-road mixed traffic with one CAV and multiple heterogeneous HDVs have not been well understood.
Method
The paper combines PBH-based controllability analysis with structured optimal controller synthesis that explicitly incorporates the CAV’s communication topology.
Results
The system is not completely controllable but is stabilizable under a very mild condition, while reachability analysis yields conditions for desired traffic velocity.
Takeaways & Limitations
The results support using a single CAV as a mobile actuator to stabilize mixed traffic and dampen traffic waves without changing HDV behavior.
Abstract
from arXiv · showhide
Connected and automated vehicles (CAVs) have a great potential to improve traffic efficiency in mixed traffic systems, which has been demonstrated by multiple numerical simulations and field experiments. However, some fundamental properties of mixed traffic flow, including controllability and stabilizability, have not been well understood. This paper analyzes the controllability of mixed traffic systems and designs a system-level optimal control strategy. Using the Popov-Belevitch-Hautus (PBH) criterion, we prove for the first time that a ring-road mixed traffic system with one CAV and multiple heterogeneous human-driven vehicles is not completely controllable, but is stabilizable under a very mild condition. Then, we formulate the design of a system-level control strategy for the CAV as a structured optimal control problem, where the CAV's communication ability is explicitly considered. Finally, we derive an upper bound for reachable traffic velocity via controlling the CAV. Extensive numerical experiments verify the effectiveness of our analytical results and the proposed control strategy. Our results validate the possibility of utilizing CAVs as mobile actuators to smooth traffic flow actively.
I. INTRODUCTION
Mixed traffic research examines how a small number of CAVs can influence heterogeneous HDVs, while this paper develops controllability analysis and system-level control for one-CAV ring-road traffic. The model combines heterogeneous car-following dynamics with CAV communication constraints.
- Motivation: CAVs can act as mobile actuators that influence surrounding HDVs and potentially control global traffic flow in mixed traffic systems.This motivates Lagrangian traffic-flow control during the transition to mixed CAV–HDV traffic.
- Contributions: The proposed control strategy targets entire traffic-flow performance rather than only the CAV’s driving behavior.The paper describes this as a shift from local-level to system-level CAV control.
- Contributions: The paper studies one CAV and multiple heterogeneous HDVs on a ring road, using a linearized car-following model and PBH-based controllability analysis.It also formulates CAV design as a structured optimal control problem that accounts for communication abilities.
- Contributions: A single CAV with heterogeneous HDVs is not completely controllable but is stabilizable under a very mild condition.The result extends prior theoretical work beyond homogeneous HDV assumptions.
- System setup: The ring-road setup represents a closed traffic system with controlled average density and can reproduce traffic waves without infrastructure bottlenecks or lane changes.The setup also has theoretical advantages for analyzing mixed traffic flow.
- Modeling: The HDV model represents acceleration as depending on relative distance, relative velocity, and the vehicle’s own velocity, with vehicle-specific equilibrium spacings and dynamics.Different HDVs may have different spacing relationships and model blocks at the same equilibrium velocity.
B. Problem Statement
The problem statement asks whether the mixed traffic system is controllable or stabilizable and how to design a CAV controller under limited communication. It models the CAV as a single driving node with feedback gains constrained by the available communication topology.
- Controllability and stabilizability: The first problem is to determine whether the mixed traffic system is controllable or stabilizable.These properties govern whether feedback can move system modes and stabilize the dynamics.
- Network representation: The CAV is modeled as the only driving node in a network system, while HDV interactions remain part of the traffic dynamics.This representation captures control through one vehicle within the ring-road network.
- Communication constraints: Limited CAV communication is represented by a communication network specifying which vehicle states the CAV can receive for feedback.The controller uses static state feedback u(t) = −Kx(t).
- Structured controller: The communication topology imposes a block-sparsity pattern on the feedback gain, setting gains to zero for vehicles whose information is unavailable.This structural constraint is illustrated for neighboring vehicles accessible to the CAV.
- Optimal controller synthesis: The second problem is to compute a structured optimal controller that uses the available communication topology to dampen undesired traffic perturbations.Perturbations are modeled as disturbance signals in vehicle accelerations and may generate traffic waves and congestion.
III. CONTROLLABILITY AND STABILIZABILITY
The analysis transforms the CAV-controlled system into an equivalent system with an externally controlled HDV, preserving controllability properties. It then identifies a single stable uncontrollable mode associated with the ring-road spacing constraint.
- Controllability: The PBH-based analysis reduces controllability questions to the existence of left eigenvectors satisfying the uncontrollability condition.This criterion identifies modes that cannot be influenced by the control input.
- System transformation: The system transformation preserves controllability, stabilizability, and uncontrollable modes between the original and transformed systems.The transformed system replaces the CAV input with an external input applied to an HDV.
- Controllability: The mixed traffic system is not completely controllable because it has one uncontrollable mode associated with the zero eigenvalue.The zero eigenvalue has algebraic multiplicity one and corresponds to the sole uncontrollable mode.
- Controllability: The uncontrollable mode remains constant and is stable in the Lyapunov sense.Physically, this mode reflects the ring-road constraint that the sum of vehicle spacings remains constant.
B. Stabilizability Analysis
Under a sufficient mild condition on the heterogeneous vehicle parameters, every nonzero mode is controllable while the remaining zero mode is stable. Consequently, the mixed traffic system is stabilizable, and the condition holds with probability one for randomly chosen parameters.
- B. Stabilizability Analysis: Under condition (13), every nonzero eigenvalue mode is controllable.The proof rules out nonzero eigenvalues having uncontrollable left eigenvectors under the stated parameter condition.
- B. Stabilizability Analysis: The mixed traffic system is stabilizable when condition (13) holds.The only uncontrollable mode is the stable zero-eigenvalue mode established in the controllability analysis.
- B. Stabilizability Analysis: Condition (13) is satisfied with probability one when the heterogeneous vehicle parameters are chosen randomly.The condition is sufficient and restricts the locations of the closed-loop poles.
- B. Stabilizability Analysis: The result extends prior homogeneous-vehicle analysis to heterogeneous mixed traffic systems using PBH eigenvalue-eigenvector analysis.For homogeneous traffic, the stated theorems are consistent with earlier results.
- B. Stabilizability Analysis: The stabilizability result imposes no requirement on system size or on the stability of the uncontrolled traffic system.This contrasts with prior approaches that imposed a lower bound on CAV penetration under specific controllers.
IV. OPTIMAL CONTROLLER SYNTHESIS
The paper designs a CAV control input to dampen traffic-flow perturbations. It formulates the task as a structured optimal control problem and uses convex relaxation for numerical solution, with desired-state selection informed by reachability analysis.
- IV. OPTIMAL CONTROLLER SYNTHESIS: The controller synthesis aims to design a CAV input that dampens undesired perturbations in traffic flow.The task is formulated at the system level rather than only for the CAV itself.
- IV. OPTIMAL CONTROLLER SYNTHESIS: The design is formulated as a structured optimal control problem and addressed with a numerical approach based on convex relaxation.The section also discusses designing the desired system state using reachability analysis.
A. System-level Performance and Structured Optimal Control
The proposed controller optimizes a system-level performance index that penalizes traffic deviations and CAV control effort while respecting communication sparsity. This structured constraint makes the problem generally non-convex, although convex reformulations are available when the sparsity constraint is removed.
- A. System-level Performance and Structured Optimal Control: The optimization models disturbances with finite-energy acceleration signals affecting vehicles throughout the traffic flow.The performance output accounts for deviations of all vehicles.
- A. System-level Performance and Structured Optimal Control: The performance output penalizes spacing error, velocity error, and CAV control input across the entire traffic flow.The disturbance-to-performance transfer is evaluated using the H2 norm.
- A. System-level Performance and Structured Optimal Control: The controller explicitly uses a prescribed communication topology and can incorporate information from vehicles ahead and behind the CAV.This system-level design differs from local controllers that primarily use information from vehicles ahead.
- A. System-level Performance and Structured Optimal Control: The structured optimal control problem is generally non-convex and computationally hard because the feedback gain must satisfy the communication sparsity constraint.Without the structured constraint, the relaxed formulation is convex and can be solved with conic solvers.
- A. System-level Performance and Structured Optimal Control: Increasing state-error weights generally favors faster traffic-flow stabilization, whereas increasing the control-input weight generally keeps the CAV input lower.These weights provide a direct tuning mechanism for balancing performance and feasible control actions.
B. Numerical Solution Approach
The paper replaces a non-convex structured-controller constraint with sparsity-invariance conditions that yield a convex optimization problem. The resulting controller respects the prescribed communication structure but is generally suboptimal, while numerical experiments indicate satisfactory performance.
- Convex relaxation: Sparsity invariance replaces the non-convex constraint ZX^-1 ∈K with separate constraints on Z and X, enabling a convex relaxation.The relaxation is formulated through binary sparsity patterns T and S satisfying the sparsity-invariance property.
- Convex relaxation: The relaxed problem minimizes Tr(QX) + Tr(RY) subject to linear matrix inequalities involving A, B, H, X, Y, and Z.
- Sparsity-pattern design: The method can use a diagonal Sparse(S) assumption or a more general sparsity pattern, with the latter avoiding overly restrictive block-diagonal Lyapunov requirements.The block-diagonal assumption may fail to return feasible solutions in some instances.
- Sparsity-pattern design: T is chosen to induce the controller sparsity pattern K, while S is selected to satisfy sparsity invariance and maximize non-zero entries for the given T.The resulting S* is described as an optimal choice for maximizing non-zero entries while preserving sparsity invariance.
- Controller recovery: After S and T are obtained, conic solvers solve the convex problem, and the controller K = ZX^-1 ∈K is recovered while satisfying the structural constraint.This controller is generally suboptimal for the original problem, although numerical experiments report satisfactory performance; removing the structural constraint yields a globally optimal convex problem.
C. Reachability Analysis: Desired Traffic Velocity
The analysis determines when a desired traffic velocity is reachable under the proposed controller and derives a maximum reachable velocity. It shows that the CAV’s desired spacing must satisfy a condition tied to the desired velocity and HDV equilibrium dynamics.
- Reachability condition: The traffic flow can maintain stability at velocity v* if and only if the CAV’s desired spacing satisfies the reachability condition.The result assumes a static feedback gain from the structured control problem and a non-singular coefficient matrix.
- Reachability condition: If the CAV’s desired spacing is not selected to satisfy the condition, the system stabilizes at another equilibrium velocity vf instead of v*.
- Maximum reachable velocity: The desired traffic velocity is constrained to a reachable range with a maximum value v*max determined by vehicle number, HDV dynamics, and ring-road circumference.The upper bound follows from the reachability analysis and the positivity of the CAV’s equilibrium spacing.
- Maximum reachable velocity: As vehicle density increases, the maximum reachable traffic velocity decreases under the monotonicity assumption on the HDV spacing functions.
- Practical design: A moderate desired velocity is required because higher values reduce CAV spacing and increase rear-end collision risk, whereas lower values leave more space for cut-ins.The recommended range is 0 ≤ v* < v*max.
V. NUMERICAL EXPERIMENTS
The numerical experiments evaluate the theoretical results using a realistic nonlinear human-driven-vehicle model. Three simulation types are conducted in MATLAB to assess the proposed analysis and controller.
- Numerical experiments: The experiments test the analytical results and controller effectiveness under nonlinearities from realistic car-following dynamics.The theoretical results use a linearized mixed-traffic model, while the simulations use a nonlinear HDV model.
- Numerical experiments: All numerical experiments are conducted in MATLAB using a realistic nonlinear model of human-driven-vehicle car-following behavior.
A. Experimental Setup
The experimental setup uses a 400 m single-lane ring road with one CAV and 19 heterogeneous HDVs, with feedback from five vehicles ahead and behind. A nonlinear optimal velocity model and randomized driver parameters represent heterogeneous stop-and-go behavior.
- Experimental Setup: The setup contains one CAV and 19 HDVs on a 400 m ring road, giving the CAV a 5% penetration rate.
- Experimental Setup: The CAV receives information from five vehicles ahead and five vehicles behind for feedback control.
- Experimental Setup: HDV behavior is modeled with a nonlinear optimal velocity model whose desired velocity depends on inter-vehicle spacing.
- Experimental Setup: Randomized sensitivity, acceleration, and spacing parameters produce heterogeneous HDVs exhibiting stop-and-go car-following behavior.The parameters are sampled from uniform distributions, with vi,max = 30 and si,st = 5.
- Experimental Setup: A standard automatic emergency braking system is assumed for all vehicles, with maximum acceleration 2 m/s2 and deceleration −5 m/s2.
B. Stabilizing Mixed Traffic Flow
Experiments show that one CAV can stabilize heterogeneous mixed traffic, dissipate stop-and-go waves, and dampen perturbations. Careful desired-spacing design and system-level optimal control improve reachability and transient spacing compared with heuristic strategies.
- Experiment A: One CAV stabilizes heterogeneous traffic and can precisely drive it to a pre-specified velocity when its desired spacing satisfies Theorem 3.Without the required spacing condition, the system remains stabilized but its final velocity differs unpredictably from the desired velocity.
- Experiment B: Activating the proposed controller dissipates a growing stop-and-go wave and returns traffic to equilibrium within 100 seconds.The wave develops while the controller is inactive and all vehicles follow human-driven behavior.
- Experiment C: The proposed CAV controller prevents upstream wave propagation and dampens a braking perturbation within a short time.This contrasts with all-human control, where the perturbation generates a persistent upstream traffic wave.
- Experiment C: All three controllers dampen traffic waves, but the optimal strategy keeps CAV spacing moderate while FollowerStopper and PI with Saturation exceed 50 m.The heuristic strategies’ large gaps may induce adjacent-lane cut-ins; Fig. 11 further compares maximum spacing and linear-quadratic cost across perturbation positions.
- Analytical results: The analysis proves that a ring-road system with one CAV and heterogeneous HDVs is stabilizable under a very mild condition.The paper frames this result as evidence for using CAVs as mobile actuators for traffic control.
- Scope and limitations: Model mismatch and drivers’ reaction time remain future directions, while the controllability results are specifically established for a ring-road setting.The structured optimal-control formulation can be adapted to a straight road, but the controllability conclusions may differ.
APPENDIX A PROOF OF (9)
The appendix analyzes the null-space structure of the squared transformed system matrix and then rules out nonzero eigenvalues satisfying the vehicle-specific characteristic equation. These steps establish the required eigenvector and kernel-dimension properties for the proof of (9).
- Kernel characterization: The equation ˆA2p = 0 is reduced to coupled vehicle-indexed relations involving coefficients Fi1, Fi2, and Fi3.The matrix ˆA2 is first written in compact block form before the componentwise expansion.
- Eigenvalue exclusion: The solution space of ˆAp = 0 has dimension one, giving dim ker(ˆA − 0·I)^2 = 1 and completing the proof of (9).The appendix then examines left eigenvectors for nonzero λ and derives a contradiction if λ2 + αi2λ + αi1 = 0 for any vehicle.