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REVIEW 4 major objections 8 minor 31 references

Learning-Based Tracking Perimeter Control for Two-region Macroscopic Traffic Dynamics

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that an adaptive dynamic programming controller can solve optimal tracking perimeter control for a two-region macroscopic fundamental diagram network without knowing the system dynamics, and that in simulation this…

desk verdict A plausible extension of ADP to tracking perimeter control whose headline optimality and robustness claims rest on an under-specified reference trajectory. read the letter →

arxiv 2505.21818 v1 pith:JJNQOO55 submitted 2025-05-27 eess.SY cs.SY

Can Chen , Yunping Huang , Hongwei Zhang , Shimin Wang , Martin Guay , Shu-Chien Hsu , Renxin Zhong This is my paper
classification eess.SYcs.SY MSC 93C4049L2090B20
keywords macroscopicfundamentaldiagramperimetercontroladaptivedynamicprogrammingtrajectorytrackingmodel-freeintegralreinforcementlearningtwo-regiontrafficnetworktripcompletion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that perimeter control for a two-region macroscopic traffic network should regulate accumulation along a desired time-varying trajectory rather than toward a fixed set point, and that the resulting optimal tracking problem can be solved without knowing the system dynamics. The proposed adaptive dynamic programming algorithm learns the value function and control policy from state and control data through an integral reinforcement learning Bellman equation, which the authors prove equivalent to model-based policy iteration. If the claim holds, traffic managers could adapt perimeter metering to within-day demand swings without calibrating a macroscopic fundamental diagram model. In simulation the tracking controller reduces total time spent by 20.01% and increases cumulative trip completion by 3.15% relative to set-point perimeter control, and it keeps the accumulation state on a desired reference under demand disturbance.

What carries the argument

The load-bearing object is the augmented system $N = \mathrm{col}(e_d, n_d)$ with dynamics $\dot{N} = F(N) + S(N)\mu$, together with the nonquadratic performance function whose tanh integrand enforces the actuator bound $|\mu_i| \le \lambda$. The tracking Hamilton-Jacobi-Bellman equation is solved iteratively through the integral reinforcement learning Bellman equation (21), which replaces $F(N)$ and $S(N)$ with an integral over data collected along the trajectory. Proposition III.1 proves that this data-driven update is equivalent to the model-based policy iteration, which is what lets the controller be model-free while still respecting input constraints and tracking a time-varying reference.

What would settle it

Run Example 2 again with several distinct four-dimensional reference trajectories that all reduce to the same aggregate accumulations $n_{d,1}$ and $n_{d,2}$ from (27), for example by reallocating the internal OD components. If the resulting tracking errors or cumulative trip completion differ appreciably, the claim that tracking (27) maximizes trip completion depends on reference components the paper never defines. A direct check is to compare the reported trip completion against a numerical search over admissible trajectories on the same demand pattern.

Watch

Extended reading notes

Core claim

The central claim is that the optimal tracking perimeter control problem for the two-region MFD system is solved by building an augmented system whose state combines the tracking error and the reference trajectory, then applying an integral reinforcement learning Bellman equation derived from a nonquadratic cost with tanh-saturated control. Proposition III.1 states that this IRL update is equivalent to model-based policy iteration, so the value function and control policy converge to the optimal tracking solution without knowledge of the drift and input dynamics. In the paper's simulations this yields 20.01% lower total time spent and 3.15% more cumulative trip completion than set-point perimeter control, and the controller keeps accumulation near a time-varying reference under demand noise. The paper also asserts that the reference trajectory used in Example 2 maximizes trip completion under the nominal demand pattern, so tracking it is presented as the practical traffic-management target.

Load-bearing premise

The Example 2 conclusions assume the reference trajectory (27) is fully specified and truly maximizes trip completion, but the paper defines it only for the two aggregate accumulations and does not prove the optimality claim.

Editorial extensions

If this is right

  • Calibrated MFD dynamics become unnecessary: the IRL Bellman equation collects data along the actual trajectory, so the same algorithm can be applied when the drift and input dynamics are unknown or time-varying.
  • Trajectory tracking changes the control objective from stabilizing one equilibrium to following a demand-fitted path, which the simulations show avoids unnecessary queueing when congestion is offset.
  • The nonquadratic tanh cost keeps the feedback correction inside the actuator limits, so the learned policy respects $0 \le u_{ij} \le 1$ without post-hoc clipping.
  • Because the policy is updated online from measured state and control data, the controller can compensate for demand disturbances that differ from the pattern used to design the reference.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves the design of the reference trajectory largely open; defining it from economic or OD-specific objectives would be a natural next step and would also make the trip-completion optimality claim checkable.
  • The same augmented-system and IRL derivation does not use the two-region structure, so it should transfer to multi-region or trip-based MFD networks, though the convergence guarantee would need to be re-established.
  • The chattering in the metering inputs under demand noise suggests the learned policy could benefit from a control-rate penalty or a filter on the demand estimate; the paper does not test these modifications.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 8 minor

Summary. The paper extends set-point perimeter control for a two-region macroscopic fundamental diagram (MFD) traffic system to an optimal tracking perimeter control problem (OTPCP). The authors formulate an augmented system composed of the tracking error and the reference trajectory, introduce a nonquadratic cost to handle actuator constraints, and propose an adaptive dynamic programming (ADP) algorithm based on integral reinforcement learning (IRL) that is claimed to solve the OTPCP without knowledge of the system dynamics. Numerical experiments compare the proposed tracking perimeter control (TPC) with set-point perimeter control (SPC) in a time-varying demand scenario, reporting a 20.01% reduction in total time spent and a 3.15% improvement in cumulative trip completion, and a second example is intended to demonstrate robust tracking of a time-varying reference that is asserted to maximize trip completion.

Significance. If the claims are validated, the paper makes a useful contribution by generalizing MFD perimeter control from equilibrium stabilization to trajectory tracking, which is better suited to time-varying demand. The IRL derivation and the equivalence proof in Proposition III.1 follow standard policy-iteration arguments and are internally consistent apart from notation slips. The use of a nonquadratic performance function for constrained inputs is appropriate, and the numerical comparison in Example 1 gives a concrete performance improvement over SPC. The main advertised strengths are the model-free learning formulation and the ability to track a time-varying reference under demand disturbances. However, the strongest version of these claims, presented in Example 2, is not supported as written because the reference trajectory is incompletely specified and its optimality is asserted rather than proved; the 'model-free' label also requires qualification.

major comments (4)
  1. [Section IV-B, Eq. (27)] The command generator (27) defines differential equations only for nd,1 and nd,2, the regional accumulations, whereas the tracking error ed = n - nd in Eq. (5) is four-dimensional and nd(t) was defined in Section II-B as col(nd_11, nd_12, nd_21, nd_22). The OD-specific reference components nd_12 and nd_21 enter (27) as ratios, but their dynamics and initial conditions are never specified, so (27) is not a well-posed four-dimensional reference. Consequently, Example 2 cannot be implemented as stated and the claim that the controller regulates the four-dimensional accumulation state to the desired reference trajectory is not testable. The authors should either specify a complete four-dimensional command generator or reformulate the tracking problem in terms of the two regional accumulations n1 and n2 and correspondingly reduce the dimension of ed.
  2. [Section IV-B] It is asserted without proof or citation that tracking (27) 'maximizes trip completion.' The right-hand side of (27) is the free-running accumulation dynamics under a nominal demand pattern; no argument shows that this trajectory maximizes cumulative trip completion or the integral of the trip completion rates. In fact, a free-running trajectory generally settles at a demand-determined equilibrium rather than at the trip-completion-maximizing accumulation, which is typically near the critical accumulation of the MFD. Since the optimality of the tracked reference is load-bearing for the robust-optimality conclusion of Example 2, an explicit optimality argument or a precise definition of the trip-completion-maximizing reference is required.
  3. [Section II-B and Section III] The label 'model-free' overstates what Algorithm 1 actually achieves. The total perimeter control is u = us + µ, and the feedforward part us is computed from (6) or (7) using f(nd), s(nd), and θ(nd). If the system dynamics are unknown, us cannot be computed from these formulas. Furthermore, the policy improvement relation (17) still contains S(N); the text states that V(k+1) and D(k+1) are solved simultaneously from (21), but does not explain how S(N) is eliminated or approximated in the actor update. Please clarify the exact meaning of 'model-free' in this context and state which quantities are assumed known and which are learned from data.
  4. [Section III, Proposition III.1 and Algorithm 1] The paper states that the convergence of the policy iteration sequence (V(k), µ(k)) to (V*, µ*) 'has been checked by [21]', but no convergence theorem or sufficient conditions (initial admissibility, excitation condition, approximation error bounds) are provided in this manuscript. Because Algorithm 1's optimality claim is central, the authors should either state the relevant convergence result with assumptions or give a precise reference to a theorem in [21] that covers the constrained, augmented-system setting used here.
minor comments (8)
  1. [Eq. (20)] In the second-to-last line of (20), the term '−(∇V(k))^T S(µ(k)−µ)' should be '−(∇V(k+1))^T S(µ(k)−µ)' to be consistent with the preceding line; also, the notation tanh^{-T}(υ/λ) is unusual and should be defined.
  2. [Proposition III.1 proof] The proof divides (21) by Δt and takes the limit Δt → 0; this interchange requires differentiability of V(k+1) and continuity of the integrand. These regularity conditions are not stated.
  3. [Section II-A] Equation (7) requires s(nd)^T s(nd) to be invertible, i.e., s(nd) must have full column rank; this assumption should be stated explicitly.
  4. [Figures 5 and 6] The legend labels in Figures 5 and 6 use abbreviated subscripts 'ntpc', 'nref', and 'nspc' that are not defined in the captions; please spell out the quantities being plotted.
  5. [Reference [31]] Reference [31] is a preprint under review, and it is used to motivate MFD uncertainty; please provide a published version or additional details so that the reader can verify the claim.
  6. [Eq. (27)] The notation nd,1 and nd,2 in (27) is inconsistent with the four-dimensional nd(t) defined earlier; clarify that nd,i = nd,i1 + nd,i2 and specify the relation between the regional reference accumulations and the OD-specific reference components.
  7. [Algorithm 1] The step 'According to the control policy D(k), D(k+1) and V(k+1) can be solved simultaneously' is presented as an equation, not as an implementable algorithm; please describe the function approximators and the least-squares or recursive least-squares procedure used to solve (21).
  8. [Abstract and Table II] The abstract reports a 20.01% TTS reduction and a 3.15% CTC improvement without stating that these numbers are for a single scenario (Example 1); please qualify them accordingly and, ideally, provide multiple simulation runs or error bars.

Circularity Check

2 steps flagged · score 3.0 of 10

The ADP derivation (Proposition III.1) is self-contained, but the convergence guarantee is imported from the authors' own [21] and Example 2's claim that tracking (27) maximizes trip completion is an unproven, under-specified assertion rather than a circular reduction.

  1. self citation load bearing [Section III, paragraph immediately after Eq. (17)]
    "The convergence of sequence {(V (k), µ(k))} (i.e., {(V (k), D(k))}) to the optimal value and policy function (V ∗, µ∗) (i.e., (V ∗, D∗)) applying (16)-(17) has been checked by [21]."

    [21] is the authors' own prior paper (Chen et al., Transportation Research Part C 2022). Algorithm 1's optimality claim depends on this convergence assertion, and the manuscript provides no independent proof of convergence here, nor does it reproduce or machine-check the cited result. The underlying policy-iteration convergence is also standard in the external ADP literature, so this is a load-bearing self-citation rather than a full circular collapse, but it is the kind of support that should ideally cite a non-overlapping source or give a self-contained theorem.

  2. other [Section IV-B, Eq. (27) and the paragraph following Fig. 10]
    "The desired trajectory reference is given by ˙nd,1(t) = qˆ1(t) − G1(nd,1(t)) + nd,21(t)/nd,2(t) G2(nd,2(t)), ˙nd,2(t) = qˆ2(t) − G2(nd,2(t)) + nd,12(t)/nd,1(t) G1(nd,1(t)) ... and the trip completion of the system is thus maximized."

    This is not an equation-level circular reduction, but it is a load-bearing asserted gap. The tracking error ed = n − nd in Eq. (5) is four-dimensional, while (27) defines only the two aggregate reference accumulations nd,1 and nd,2; the OD-specific components nd,12, nd,21, and nd,22 are never specified, so the controller is not shown to track a complete 4D reference trajectory. Moreover, no derivation or optimization shows that the uncontrolled aggregate dynamics (27) is the trajectory that maximizes trip completion; the statement that tracking it 'maximizes trip completion' is asserted without proof. Thus the Section IV-B conclusions on robust optimality are unsupported even if the ADP feedback loop itself is sound.

full rationale

The core of the paper, Proposition III.1, is a self-contained equivalence proof between the model-based policy iteration (16)-(17) and the model-free IRL Bellman equation (21); it follows from the Bellman optimality principle and standard ADP arguments, and it does not reduce to the paper's own inputs. The main circularity risk is the citation of the authors' own [21] for the convergence of the policy-iteration sequence underlying Algorithm 1. That citation is load-bearing because the optimality of the proposed controller requires convergence, but it is not the only possible support and the underlying result is standard in the external literature, so it counts as a minor self-citation concern rather than a demonstration that the algorithm is circular. A separate, non-circular but serious gap is Example 2: the reference trajectory (27) is under-specified (only two aggregate states are defined, while the tracking error is four-dimensional) and its claimed trip-completion-maximizing property is asserted, not derived. This makes the paper's headline robustness/optimality conclusion for Example 2 unsupported, but it is a correctness gap, not a self-referential reduction. Overall, no equation in the derivation is equivalent to its own output by construction, so the score is 3 rather than higher.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The central claim rests on the affine two-region MFD model taken from prior literature (eqs. (3), (26)), the command-generator reference model from [27], a convergence citation to the authors' own [21], an unproven and under-specified reference trajectory in Example 2, and several unreported tuning parameters (Q, R, lambda, critic architecture) that affect the reported 20.01% TTS improvement.

free parameters (5)
  • Q (tracking-error weight matrix) = not reported
    Symmetric positive definite weight in (8)/(10); the reported TTS/CTC improvements depend on this choice. No value or tuning procedure is given.
  • R = diag(gamma1, gamma2) (control-effort weights) = not reported
    Weights in the nonquadratic cost (11); no values are reported.
  • lambda (actuator saturation bound) = not reported
    Saturating bound for the feedback control mu in (11); must be consistent with the umin/umax margins after adding the steady-state term us, but no margin analysis is given.
  • Function-approximation architecture for V and D = not reported
    Algorithm 1 requires solving for V(k+1) and D(k+1) simultaneously from (21); no basis functions, network structure, learning rates, or convergence criteria are described.
  • Initial admissible policy mu(0) = not reported
    Input to Algorithm 1; admissibility is required for policy-iteration convergence, but no construction or verification procedure is given.
assumptions (6)
  • domain assumption The two-region MFD dynamics are exactly affine (3) with the given f(n) and s(n), and the MFD (26) is well-calibrated and identical for both regions.
    Eqs. (3) and (26), Section IV. The model-free claim is tested only against this fixed model with demand noise, not against MFD parametric uncertainty, which is the motivating uncertainty in Section III.
  • domain assumption The reference trajectory nd is bounded and generated by a Lipschitz continuous command generator (4) with theta(0)=0.
    Eq. (4), from [27]. Example 1 uses piecewise-constant set-point jumps, which are not representable by a Lipschitz generator as written; the framework's applicability to discontinuous references is not discussed.
  • domain assumption s(nd)^T s(nd) is invertible, equivalently n12 and n21 are positive along the reference, so that us in (6)/(7) is well-defined.
    Eqs. (6)-(7). The feedforward control requires the pseudo-inverse; if an OD accumulation vanishes under strong gating, the solution degenerates. This is not analyzed.
  • domain assumption The policy iteration (16)-(17) converges to the optimal solution of the tracking HJB equation (15).
    Section III states convergence 'has been checked by [21]', the authors' own prior work. No convergence conditions or admissibility requirements are stated for this constrained tracking setting.
  • standard math The tracking HJB (15) has a smooth solution V* and the value function is continuously differentiable along closed-loop trajectories.
    Standard ADP assumption inherited from [27]-[29]; needed for the stationary condition (13) and for differentiating V along trajectories.
  • ad hoc to paper The reference trajectory (27), defined as the free-running dynamics under a nominal demand pattern, maximizes trip completion.
    Section IV-B. Asserted without derivation and without comparison against the analytical optimality conditions for perimeter control already cited as [5]. If false, the 'trip completion is maximized' claim fails.

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Pith. "Pith review of Learning-Based Tracking Perimeter Control for Two-region Macroscopic Traffic Dynamics." pith.science (2026). https://pith.science/paper/JJNQOO55

@misc{pith2026250521818,
  author       = {Pith},
  title        = {Pith review of: Learning-Based Tracking Perimeter Control for Two-region Macroscopic Traffic Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJNQOO55}},
  note         = {Machine review of arXiv:2505.21818}
}
read the original abstract

Leveraging the concept of the macroscopic fundamental diagram (MFD), perimeter control can alleviate network-level congestion by identifying critical intersections and regulating them effectively. Considering the time-varying nature of travel demand and the equilibrium of accumulation state, we extend the conventional set-point perimeter control (SPC) problem for the two-region MFD system as an optimal tracking perimeter control problem (OTPCP). Unlike the SPC schemes that stabilize the traffic dynamics to the desired equilibrium point, the proposed tracking perimeter control (TPC) scheme regulates the traffic dynamics to a desired trajectory in a differential framework. Due to the inherent network uncertainties, such as heterogeneity of traffic dynamics and demand disturbance, the system dynamics could be uncertain or even unknown. To address these issues, we propose an adaptive dynamic programming (ADP) approach to solving the OTPCP without utilizing the well-calibrated system dynamics. Numerical experiments demonstrate the effectiveness of the proposed ADP-based TPC. Compared with the SPC scheme, the proposed TPC scheme achieves a 20.01% reduction in total travel time and a 3.15% improvement in cumulative trip completion. Moreover, the proposed adaptive TPC approach can regulate the accumulation state under network uncertainties and demand disturbances to the desired time-varying equilibrium trajectory that aims to maximize the trip completion under a nominal demand pattern. These results validate the robustness of the adaptive TPC approach.

Figures

Figures reproduced from arXiv: 2505.21818 by the authors.

Figure 1
Figure 1. The two-region MFD system of two homogeneous regions that both admit well-defined MFDs. A two-region MFD system can be used to model the macroscopic traffic dynamics. The MFD is a function that depicts a nonlinear relationship between the regional accu￾mulation ni(t) (veh) and the trip completion rate Gi(ni(t)) (veh/s) at time t, i = 1, 2. The regional accumulation ni(t) represents the number of vehicles in region i… view at source ↗
Figure 2
Figure 2. Demand pattern and desired state trajectory [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The diverse time-varying demand patterns [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: MFDs under demand variations the dynamics in the iteration procedure. Denote µ (k) as the policy to be updated and µ as the behavior policy that is actually implemented to generate the data for learning. Then we can rewrite the augmented system as: N˙ = F(N) + S(N) · µ…
Figure 5
Figure 5. Figure 5: Accumulation state evolutions of Example 1 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: Performances in TTS and CTC of Example 1 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 6
Figure 6. Figure 6: OD-specific state evolutions of Example 1 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: The nominal demand pattern 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 11
Figure 11. Figure 11: Perimeter control inputs of Example 2 ADP-based TPC scheme can well adapt to the changes in the traffic condition (e.g., time-varying travel demand) and regulate the accumulation state to a desired reference trajectory that better fits the dynamics of the demand. More…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.