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REVIEW 5 major objections 6 minor 45 references

On Optimal Event-Triggered Distributed Control for Stochastic Multi-Agent Systems via Reinforcement Learning

T0 review · 5 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper claims a unified RL-based optimal distributed control for stochastic multi-agent systems that keeps all tracking errors bounded and avoids Zeno behavior through a hybrid event-triggered strategy.

desk verdict New combination, broken proof: the K_u metric is worth a look, but the optimality derivation and Theorem 1's SGUUB/Zeno-free argument don't survive their own equations. read the letter →

arxiv 2607.17635 v1 pith:MZSGEHXD submitted 2026-07-20 eess.SY cs.SY

Ziming Wang , Bingbing Li , Karl H. Johansson , Apostolos I. Rikos This is my paper · ORCID
classification eess.SYcs.SY MSC 93A1693D0593E2093C57
keywords event-triggeredcontrolstochasticmulti-agentsystemsreinforcementlearningactor-critic-identifierbacksteppingconsensustrackingneuralnetworksoptimal
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 sets out to prove that a distributed controller for stochastic leader-following multi-agent systems can be simultaneously optimal (in a reinforcement-learning sense), robust to unknown stochastic dynamics and non-affine faults, and event-triggered with a guaranteed minimum time between control updates. The design runs backstepping with an actor-critic-identifier neural network at every step; each virtual controller and the final controller are chosen to satisfy a Hamilton-Jacobi-Bellman optimality condition. The main theorem states that all tracking and neural-network errors are semi-globally uniformly ultimately bounded, and that the hybrid event-triggering rule (static threshold when the control magnitude is large, dynamic threshold when it is small) excludes Zeno behavior. The paper further introduces a composite index, borrowing the idea of benchmark normalization from economics, to compare triggered versus continuous control on trigger count, tracking error, and control effort. If the theorem is correct, the algorithm offers a concrete recipe for sparse, fault-tolerant distributed control.

What carries the argument

The central mechanism is the hybrid event-triggered control law with a switching threshold H_g. At triggering instants the control is held constant. Inside each inter-execution interval, the held value κ_i is related to the current optimal control u*_i by a time-varying proportional-plus-offset representation κ_i = (1 + ϖ_1 ϱ_i) u*_i + ϖ_2 λ*_i, with |ϖ_1|, |ϖ_2| ≤ 1. This representation, together with the tanh inequality 0 ≤ |x| − x tanh(x/χ) ≤ 0.2785χ, is what converts the triggering error into a residual bounded by a constant (0.557Υ_i or 0.2785Υ_i) in the Lyapunov derivative. The backstepping loops are then closed using radial-basis-function neural networks (actor, critic, identifier) wh

What would settle it

Take the saturated controller from (19), κ_i = −(1+ϱ_i)(u*_i tanh(u*_i z_i,n/Υ_i) + ϱ̄_i tanh(ϱ̄_i z_i,n/Υ_i)), and test numerically whether there exist functions ϖ_1(t), ϖ_2(t) in [−1,1] such that κ_i(t) equals (1+ϖ_1ϱ_i)u*_i + ϖ_2λ*_i along a simulated trajectory. If equality fails for any time instant, inequality (25) does not follow. Similarly, compute Δ from (30) over the same trajectory; if Δ varies with z_i,k and neural-network weights rather than staying constant, the final exponential bound (31) cannot be derived.

Watch

Extended reading notes

Core claim

At the center of the paper is a claim: with the proposed actor-critic-identifier backstepping design and the hybrid event-triggered update law (17)-(19), every follower in stochastic multi-agent system (1) tracks the leader's desired trajectory in the sense that all errors in the closed loop are SGUUB, and the sequence of controller update times has a positive uniform lower bound (no Zeno behavior). The optimality claim is that each backstepping virtual controller and the final control input solve the HJB condition ∂H/∂u* = 0 for its local performance index. The proof proceeds by a Lyapunov function that mixes quartic tracking error terms with quadratic neural-network weight errors, and conv

Load-bearing premise

The proof's error bound depends on representing the event-triggered controller as κ_i(t) = (1 + ϖ_1(t)ϱ_i)u*_i(t) + ϖ_2(t)λ*_i with bounded time-varying coefficients and on treating the residual term Δ in inequality (29) as a constant; the first representation does not match the saturated controller (19) and the displayed Δ contains state- and weight-dependent terms, so the exponential bound in (31) does not follow from the written proof.

Editorial extensions

If this is right

  • If Theorem 1 is correct, a group of agents with stochastic disturbances and unknown non-affine faults can achieve practical leader-following consensus with no centralized coordinator and with control inputs updated only at discrete events.
  • The positive minimum inter-execution interval excludes Zeno behavior, so the event-triggered implementation is physically realizable without accumulating updates in finite time.
  • Optimality at each backstepping step means the learned virtual and actual control policies are local minimizers of their performance indices, a property the paper argues is absent in non-optimized backstepping consensus.
  • The benchmark-normalized index K_u gives a single scalar for trading off communication savings against tracking accuracy and actuator wear, enabling designers to choose thresholds by weighting α, β, γ.
  • The simulation results suggest that the hybrid ETC can reduce controller activation frequency by roughly 80% or more compared with continuous updating while keeping the combined index near 1.

Reading between the lines

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

  • The per-step HJB optimality is local optimality of each backstepping virtual controller, not a global optimality certificate for the full multi-agent consensus problem; treating 'optimal distributed control' as global would overread the claim.
  • The saturation in (19) is a tanh-based approximation of the sign/linear law; if the proof's linear-representation shortcut fails, the stability conclusion may still be recoverable through a more direct Lyapunov analysis of the saturated error, but that is not what the paper shows.
  • The K_u index is a portable measurement tool: it could be applied to any event-triggered consensus controller to compare designs on the same three-axis trade-off, independent of the neural-network specifics.
  • A natural testable extension is to vary H_g and the weighting (α, β, γ) in the simulation to map the Pareto frontier of communication savings versus tracking accuracy; the paper gives the tool but does not explore the trade-off surface.
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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

5 major / 6 minor

Summary. The paper proposes a distributed leader-following consensus controller for stochastic nonlinear multi-agent systems subject to fault-induced uncertainties. The design combines backstepping with an RL-based actor-critic-identifier architecture and a hybrid event-triggering strategy. The authors claim that, at each backstepping step, the virtual and actual control laws are optimal solutions of the associated HJB equations, that all closed-loop errors are SGUUB, and that the event-triggered scheme is Zeno-free. A simulation study on a single-axis robotic manipulator compares the proposed controller with a non-optimal baseline and introduces a weighted economic index K_u for evaluating event-triggered performance.

Significance. If the theoretical claims were correct, the paper would address an active and difficult problem: simultaneously handling stochastic disturbances and non-affine faults in distributed multi-agent systems while preserving some notion of optimality and event-triggered resource savings. The simulation section is clearly presented, includes a practical robotic example, and proposes an interesting quantitative ETC evaluation index K_u. However, the core theoretical contributions are not established. The optimality derivation is internally inconsistent, and the stability proof relies on unsupported algebraic representations and treats a state-dependent quantity as a constant. These are load-bearing flaws affecting the central claims of Theorem 1, so the paper cannot be accepted in its current form.

major comments (5)
  1. [Section III-A, Eqs. (8)-(13)] The expressions for the HJB derivative dJ*/dz are mutually inconsistent. Substituting (9) into (8) gives u* = -(1/(2 eta_i))(zeta z + J0 + 2h*). With the NN approximations (11), this becomes u* approximately -(1/(2 eta_i))(zeta z + W_J^T Q_J + 2W_h^T Q_h). The implemented controller (12) is u_hat* = -(1/eta_i)(zeta z + W_hat_h^T Q_h + 0.5 W_hat_u^T Q_J), which would correspond to dJ*/dz = (1/eta_i^2)(2 zeta z + 2 W_hat_h^T Q_h + W_hat_u^T Q_J). This differs by a factor of 2 on the zeta z term and replaces the J0 approximation by W_hat_u. Equation (13) gives yet a third expression, dJ*/dz = (1/eta_i^2)(2 zeta z + 2 W_hat_h^T Q_h + W_hat_c^T Q_J). Thus the 'optimal' controller (12) does not follow from the HJB solution (8)-(9), and the critic expression (13) contradicts (9). The optimality claim is therefore unsupported.
  2. [Section III-C, Eqs. (29)-(31)] The application of Lemma 3 is invalid because Delta in Eq. (30) is not a constant. Lemma 3 requires V_dot <= -Psi1 V + Psi2 with Psi1, Psi2 positive constants, and Eq. (31) uses Delta/Weierstrass-p as a uniform ultimate bound. However, Delta in (30) explicitly contains state- and weight-dependent terms: sum 0.5 z_{i,k}^6, sum 0.25 (W_hat_{u_i,k}^T Q_{J_i,k})^2, and sum z_{i,k+1}^4. These positive nonlinearities are not canceled by the negative terms in (28), which are of order z^4 and quadratic in the weight errors. For sufficiently large |z| or ||W_hat||, V_dot can become positive, so no SGUUB bound follows from (29). Bounding Delta by a constant would require the very a priori bounds the theorem is supposed to establish, which is circular. Consequently, the exponential bound (31) and the SGUUB conclusion are not proven.
  3. [Section III-C, before Eq. (25); Eq. (19)] The event-triggering error analysis rests on a representation that contradicts the actual controller. The proof assumes that when |u_i| >= H_g, kappa_i(t) = (1 + varpi_1(t) rho_i) u*_i(t) + varpi_2(t) lambda*_i with |varpi_1|, |varpi_2| <= 1. But the controller actually defined in (19) is kappa_i = -(1+rho_i)(u*_i tanh(u*_i z_{i,n}/Upsilon_i) + bar_rho_i tanh(bar_rho_i z_{i,n}/Upsilon_i)). This is a saturated nonlinear function of u*_i, not a linear function. For u*_i = 0, (19) gives kappa_i = -(1+rho_i)bar_rho_i tanh(bar_rho_i z/Upsilon), whereas the assumed representation gives varpi_2 lambda*, which may be nonzero. For large |u*|, the representation grows linearly while (19) saturates. Hence the bound (25) does not follow from Lemma 1, and the ETC-related terms entering (28) are unsupported. The subsequent Zeno-free conclusion, which depends on (25) and on a bound for pi_dot, is also
  4. [Section III-C, Lyapunov proof, Eq. (28)] The stability proof ignores the stochastic nature of the system. The plant in (1) is driven by a Wiener process, yet the proof computes only the deterministic derivative V_dot in (28). For stochastic differential equations, the infinitesimal generator must include the second-order Ito correction term (1/2)Tr(sigma^T V_xx sigma). Such a term appears in the HJB derivation (7) but is absent from the Lyapunov analysis. Lemma 3 is stated for the infinitesimal generator, but the proof does not compute it. Therefore the claim that all errors are SGUUB in mean square is not justified. This is a separate, load-bearing gap from the Delta issue.
  5. [Section III-A, Eqs. (4)-(14)] The paper's central claim of 'optimality' is constructed rather than derived. The decomposition (9) is tautological: J0_{i,1} is defined as the leftover of eta_i^2 dJ*/dz - 2 zeta z - 2h*, so (9) holds for any choice of zeta and h*. The NN approximation of J0 then merely replaces an unknown quantity with a basis-function expansion, and no HJB residual is minimized. The critic and actor update laws (12)-(14) are chosen for Lyapunov stability, not to drive the Hamiltonian or any optimality error to zero. Thus the controller (12)/(16) is an adaptive backstepping law with adjustable gains, and the label 'optimal' is not supported by any optimization step. This concern affects the main contribution claimed in the abstract and introduction.
minor comments (6)
  1. [Sections II-III] The abstract and introduction mention a 'low-pass filter' that 'effectively suppresses problems stemming from non-affine nonlinear faults', but no low-pass filter appears in the design equations (12)-(19) or in the stability proof. The claim is never substantiated.
  2. [Section IV, Parameter Setting] The simulation parameters phi_{u_i,k}=13 and phi_{c_i,k}=15 violate the design condition stated in Sections III-A and III-B, which requires phi_u > phi_c > phi_u/2. With 13 < 15, the parameter condition used in the Lyapunov analysis is not satisfied in the experiment.
  3. [Eqs. (4)-(6)] The notation for the running cost is inconsistent: Eq. (4) uses o_{i,1}, while Eqs. (5)-(6) use h_{i,1}. The relationship between o, h, and the earlier value function is not clarified.
  4. [Eq. (8)] In Eq. (8), the left-hand side is written as u*_{i,k}, but the right-hand side depends on J*_{i,1}; this appears to be an indexing error. It should be u*_{i,1} in Step 1, and analogous expressions for later steps should be stated consistently.
  5. [Section III-C, Zeno-free proof] The final paragraph of the proof of Theorem 1 asserts that there exists a constant Lambda_i such that t* >= max{lambda_i, rho_i |u_i| + lambda*_i}/Lambda_i, but no argument is given for the existence or boundedness of Lambda_i. A rigorous Zeno-free proof requires a bound on the growth rate of pi_i, which is not established.
  6. [Section III-C, Eq. (25)] The inequality (25) uses the numerical constant 0.557 Upsilon_i, apparently from 2 times 0.2785 Upsilon_i, but Lemma 1 is stated for |chi| - chi tanh(chi/chi_2) <= epsilon* chi_2 with epsilon* approximately 0.2785. The manuscript does not show how the two tanh terms in (19) lead to the stated factor of 0.557 in the presence of the factor (1+rho_i).

Circularity Check

2 steps flagged · score 6.0 of 10

Optimality is constructed via the tautological Eq. (9), and the SGUUB proof treats the state-dependent residual Δ as a constant, making the two load-bearing claims circular.

  1. self definitional [Section III-A, Step 1, Eqs. (8)-(13)]
    "Then, the optimal virtual controller u*_{i,1} can be derived by solving equation ∂H_{i,1}/∂u*_{i,1}=0 as u*_{i,k}=−η_i/2 dJ*_{i,1}/dz_{i,1}. (8) ... we introduce the consensus error term (dJ*_{i,1}/dz_{i,1}) with a designed positive parameter ζ_{i,1} as dJ*_{i,1}/dz_{i,1}=1/η_i^2(ζ_{i,1}z_{i,1}+J'_{i,1}+2h*_{i,1}), (9) where J'_{i,1}=η_i^2 dJ*_{i,1}/dz_{i,1}−2ζ_{i,1}z_{i,1}−2h*_{i,1}. (10)"

    Substituting (10) into (9) makes (9) an identity, so dJ*/dz is not being determined by the HJB stationarity condition (8); any desired backstepping controller can be represented this way by choosing ζ and the NN terms. The subsequent 'optimal' virtual controller (12) is therefore the chosen stabilizing law relabeled as optimal, and the HJB equation (6) is never verified after the NN substitutions. The critic update in (13) is only ˙W_c = −φ_c Q^T Q W_c, with no cost or value term, so the 'critic' does not evaluate J*. The claimed derivation of optimal control reduces by construction to the designer's choice of ζ and NN functions.

  2. other [Section III-C, Theorem 1 proof, Eqs. (28)-(31)]
    "Then we have \V\dot≤−℘V+Δ (29) where ℘=min{4ζ_{i,k}, φ_{u_i,k}Θ_min(J), φ_{c_i,k}Θ_min(J), σ_{i,k}/Θ_max(δ)}, Δ=Σ[(ρ^2_{h_i,k}+z^6_{i,k})/2 + (\W\hat^T_{u_i,k}Q_{J_i,k})^2/4 + η_i z^4_{i,2}/4 + z^4_{i,k+1} + ...] + Σ 0.8355Υ_i. (30) And (30) satisfies 0≤V(t)≤e^{−℘t}V(0)+Δ/℘. (31)"

    Lemma 3 requires V\dot≤−Ψ_1 V+Ψ_2 with Ψ_1,Ψ_2 constants to conclude the exponential bound (23). Here Δ in (30) explicitly contains z^6_{i,k}, z^4_{i,k+1}, and (\W\hat^T_{u_i,k}Q_{J_i,k})^2, which are functions of the tracking errors and NN weight estimates, not constants. Passing from (29) to (31) requires bounding Δ by a constant independent of the state and weights, and that bound is exactly the a priori SGUUB boundedness of z_{i,k} and \W\hat_{u_i,k} that Theorem 1 is meant to prove. Thus the SGUUB conclusion is used to justify the very bound needed for the proof.

full rationale

The paper is not built on a self-citation chain: Lemma 3 is a standard stochastic Lyapunov result, and the simulation provides an external benchmark. However, the two load-bearing theoretical reductions are circular. First, the claimed HJB-derived optimal controller is not actually derived from the HJB equation: Eq. (9) introduces dJ*/dz through an identity, because J' is defined in Eq. (10) to make Eq. (9) tautologically true, and the subsequent NN and actor/critic updates are selected for Lyapunov stability rather than to satisfy ∂H/∂u*=0 or to minimize J*. The critic update (13) contains no value or cost term, so the 'optimal' label is attached to the chosen stabilizing backstepping controller by construction. Second, Theorem 1's SGUUB proof applies Lemma 3 to inequality (29), but the residual Δ in (30) contains state- and weight-dependent terms (z^6, z^4, (\W\hat^T Q)^2); converting (29) into the exponential bound (31) requires Δ to be a constant, which in turn requires the a priori boundedness of the states and weights—exactly the conclusion being proven. There are additional non-circular mathematical gaps, such as the linear representation of κ_i used at (25) being inconsistent with the saturated controller (19), but those are correctness concerns rather than circularity. Accordingly, the central optimality claim and the stability proof each reduce by construction to their own assumptions, yielding a partial circularity score of 6.

Assumptions & free parameters 6 free parameters · 8 assumptions · 1 invented entities

The paper's claimed contribution (a provably optimal, event-triggered, RL-based scheme) rests on a large stack of hand-set gains (ζ, φ, δ, σ, all ETC thresholds, αβγ) and unspecified NN bases, plus two ad hoc mathematical assertions in the proof of Theorem 1 (the ϖ-representation of the saturated controller and the constancy of Δ). Assumptions 1-2 and Lemmas 1-3 are standard background. No new physical entities are posited; the only novel object is the K_u index, which is defined on the validation data itself.

free parameters (6)
  • ζ_{i,k} (tracking/consensus gain in the 'optimal' controller) = 1.2
    Introduced in Eq. (9) as the 'designed positive parameter' that prescribes the shape of dJ*/dz and hence defines the 'optimal' controller (12); hand-chosen in Section IV. The factor-of-2 discrepancy between (9) and (13) shows the claim is a gain choice, not a derivation.
  • φ_{ui,k}, φ_{ci,k} (actor/critic adaptation gains) = 13, 15
    Hand-set in Section IV; constrained only by φ_u > 1/2 and φ_u > φ_c > φ_u/2.
  • δ_{i,k}, σ_{i,k} (identifier adaptation/leakage gains) = 1.5, 1.5
    Hand-set in Section IV; no learning-rate tuning rule given.
  • Hybrid ETC thresholds H_g, ϱ_i, ϱ̄_i, Υ_i, λ_i, λ̄_i, λ*_i = 6, 0.28, 3, 0.3, 5.2, 3, 3.2
    Hand-set in Section IV; these directly determine the trigger counts in Table I (86.53%, 81.20%, 79.80%) and the Zeno-free bound t* ≥ max{...}/Λ.
  • K_u weights α, β, γ = 0.3, 0.6, 0.1
    Hand-set in Section IV; the reported K_u convergence to the [0.9, 1.1] band in Fig. 6 is a consequence of this choice.
  • RBF basis parameters (number of neurons, centers, widths) = unspecified
    Eq. (11) invokes RBF approximation J⁰ = W*^T Q + ρ and h* = W*^T Q + ρ, but the networks are never specified; the identifier's approximation quality and the residual bounds ρ depend on this missing choice.
assumptions (8)
  • domain assumption Assumption 1: y_r(t) and ẏ_r(t) are bounded and known
    Section II-B; standard for tracking; bounds the reference and its derivative.
  • domain assumption Assumption 2: there exists unknown nonnegative Φ_i with |h_{i,n} + c_iξ_i| ≤ Φ_i
    Section II-B; confines the fault/uncertainty magnitude; the text states 'without these assumptions the algorithm cannot be realized.'
  • standard math Lemma 1 (tanh inequality 0 ≤ |χ₁| − χ₁ tanh(χ₁/χ₂) ≤ ε*χ₂)
    Used to bound ETC terms (25)-(26).
  • standard math Lemma 2 (Lyapunov equation existence for Hurwitz A)
    Used in the weight-error matrix elimination in (28).
  • standard math Lemma 3 (comparison lemma giving SGUUB from V̇ ≤ −Ψ₁V + Ψ₂)
    Basis of the boundedness conclusion (31).
  • domain assumption RBF universal approximation holds on a compact set containing the closed-loop trajectory
    Eq. (11); the compact set where W*, ρ bounds hold is assumed while boundedness of the trajectory is exactly what Theorem 1 must prove.
  • ad hoc to paper κ_i(t) = (1+ϖ₁ϱ_i)u*_i(t) + ϖ₂λ*_i with |ϖ₁|,|ϖ₂| ≤ 1 under the hybrid ETC
    Section III-C before (25); this linear-in-u* representation is not derivable from the saturated controller (19) κ = −(1+ϱ)(u* tanh(u*z/Υ) + ϱ̄ tanh(ϱ̄z/Υ)); the stability/ETC bound (25) rests on it.
  • ad hoc to paper Δ in (29)-(30) is a constant independent of z and Ŵ
    Needed for V ≤ e^{−℘t}V(0) + Δ/℘; but (30) contains z⁶, (Ŵ^TQ)², ηz⁴, z⁴_{k+1} terms so Δ is state-dependent as written.
invented entities (1)
  • K_u composite performance index
    purpose: A single metric combining triggering cost, tracking degradation, and actuation wear, normalized against the continuous-control run (αK_trigger + βK_track + γK_control).
    Defined in Section IV; the weights (0.3, 0.6, 0.1) are hand-set and the index is evaluated on the same single simulation used to validate the controller, so it cannot provide independent evidence for the control claims; it is a presentation layer, not a predictive quantity.

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Cite this review

Pith. "Pith review of On Optimal Event-Triggered Distributed Control for Stochastic Multi-Agent Systems via Reinforcement Learning." pith.science (2026). https://pith.science/paper/MZSGEHXD

@misc{pith2026260717635,
  author       = {Pith},
  title        = {Pith review of: On Optimal Event-Triggered Distributed Control for Stochastic Multi-Agent Systems via Reinforcement Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZSGEHXD}},
  note         = {Machine review of arXiv:2607.17635}
}
read the original abstract

We propose a reinforcement learning (RL) based optimal distributed control algorithm for the multi-agent systems (MASs) with stochastic uncertainties. Unlike existing methods, during the optimized backstepping design process, we use the actor-critic-identifier structure. The actor neural network is used to reflect control behavior, the critic neural network works to evaluate control performance and the unknown stochastic uncertainties are handled by identifier neural network. Furthermore, a low-pass filter effectively suppresses problems stemming from non-affine nonlinear faults and a hybrid event-triggered control (ETC) strategy is proposed to reduce control frequency. We analyze our algorithm's operation, and we provide a Lyapunov-based stability proof that guarantees all errors are bounded, ensuring precise tracking between the leader and followers. We validate its correctness in a single-axis robotic manipulator simulation and finally, we compare against the non-optimal control algorithm highlighting our optimal control algorithm's operational advantages.

Figures

Figures reproduced from arXiv: 2607.17635 by the authors.

Figure 1
Figure 1. Communication topology [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 4
Figure 4. The control signal of followers under the hybrid ETC [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 2
Figure 2. The tracking performance under the RL-based optimal [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: The intervals of the hybrid ETC strategy for followers. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 3
Figure 3. Figure 3: The weight norms of the actor, critic, identifier NNs [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]
Figure 6
Figure 6. Figure 6: The value of Ku for each follower in the control process. while preserving tracking accuracy and control effort at levels comparable to the continuous-control benchmark. In this paper, the weight parameters for the proposed analytical framework are set as α = 0.3, β = …
Figure 7
Figure 7. Figure 7: Comparison of tracking performance between the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.