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This paper proves that a shaping filter built from known signals—without any plant model—makes data-driven feedforward tuning minimize the true model-matching cost.

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2026-08-03 06:15 UTC pith:L4T62ZXZ

load-bearing objection A clean, genuinely new filter formula for data-driven feedforward matching, but the asymptotic equivalence proof needs a compactness/coercivity patch. the 2 major comments →

arxiv 2607.29480 v1 pith:L4T62ZXZ submitted 2026-07-31 eess.SY cs.SY

Optimal shaping filter design for data-driven feedforward controller tuning

Yusuke Fujimoto This is my paper
classification eess.SY cs.SY
keywords data-driven controlfeedforward controller tuningmodel matchingshaping filterprefilter designtwo-degree-of-freedom systemsH2 normreference model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Data-driven controller tuning usually minimizes a cost that approximates the optimal controller, not the true control performance; when the controller structure is too simple to achieve perfect matching, the two objectives diverge. This paper proves that a specific shaping filter—L = (W/R)/(C_ff(ρ0)+T_d C_fb) z^{-m}—placed inside the data cost removes that gap: as data length grows, minimizing the filtered cost is equivalent to minimizing the true weighted model-matching error. The filter is constructed entirely from quantities available before the experiment: the initial feedforward controller, the feedback controller, the reference model, the weight, and the reference spectrum. If correct, this gives a parameter-free optimal prefilter for one-shot feedforward tuning, and it shows that a standard iterative method is optimal in the special case where the weight equals the reference filter.

Core claim

In a noise-free two-degree-of-freedom loop, with the reference signal generated by a known filter R (impulse or white-noise), the paper defines a filtered data cost JL(ρ) and proves (Theorem 1) that with L = (W/R)/(C_ff(ρ0)+T_d C_fb) z^{-m}, the filtered signals Lu0 and Ly0 collapse to W/(1+P C_fb) δ_{k-m} and PW/(1+P C_fb) δ_{k-m}. Consequently JL(ρ) converges, as N→∞, to the squared H2 norm of W/(1+P C_fb)(P C_ff(ρ)−T_d), which is exactly the true model-matching cost J(ρ)=∥W(T(ρ)−T_d)∥². Thus minimizing the filtered data cost yields the same parameter as minimizing the true cost, without knowing P. The optimal filter is not unique: any stable all-pass factor can be appended, and inner-oute

What carries the argument

The load-bearing identity is that in the initial experiment the input and output share the same closed-loop factor: u0 = (C_ff(ρ0)+T_d C_fb)/(1+P C_fb) Rδ and y0 = P times that same factor. The shaping filter L takes the reciprocal of the known numerator factor and divides by R, so after filtering, the unknown plant P appears only in the combination W/(1+P C_fb) multiplying P C_ff(ρ) − T_d. The filtered cost therefore becomes the squared sum of the impulse response of the true model-matching error operator, and H2 equivalence follows. The ratio-cancellation—not any identification step—is what makes the plant-free filter work.

Load-bearing premise

The entire equivalence rests on the noise-free assumption: the identities u0=(C_ff(ρ0)+T_d C_fb)/(1+P C_fb) Rδ and y0=P times that factor hold exactly only when there is no measurement or process noise; with any additive noise, the filtered data cost no longer equals the model-matching cost, and the paper's own remark notes that the optimal filter can be high-pass, which would amplify noise.

What would settle it

Simulate a known plant and a controller structure that cannot exactly match the reference model; generate noiseless data with an impulse or white-noise reference, compute the minimizers of JL and of J for increasing N, and compare them. If they do not converge to the same parameter, Theorem 1 is false.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any feedforward controller parametrization, using the proposed L makes the data-driven tuning target the true weighted H2 model-matching error rather than the error in identifying the optimal controller.
  • The optimal filter requires no plant model; it is built from the initial feedforward controller, feedback controller, reference model, weight, and the reference spectrum (A1).
  • When the weight W equals the reference filter R, the tuning procedure is exactly ERIT, so ERIT is optimal in that setting.
  • Because all-pass factors do not affect the H2 norm, the optimal filter is non-unique; a stable optimal filter can always be constructed via inner-outer decomposition even if the nominal L is unstable.
  • Under a white-noise reference, the filtered data cost converges in probability to the true cost, so the asymptotic equivalence holds for random reference sequences as well as deterministic ones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ratio-cancellation argument could generalize to feedback tuning: any data-driven cost whose signals share an unknown loop factor can be filtered by the reciprocal of that factor to recover a model-matching objective, suggesting a unified design of prefilters across existing one-shot methods.
  • Because the optimal L is high-pass when W=1, measurement noise is amplified; the paper's own experiment relies on signal projection, so a stochastic analysis of the filtered cost with noise would clarify how much data or projection is needed—this is left implicit in the paper.
  • The result implies a design principle for reference signals: since W=R makes ERIT optimal, choosing the reference filter R is equivalent to choosing the weighting W; users can shape closed-loop performance simply by choosing what signal to use in the experiment.
  • For finite N, the equivalence is only asymptotic; deriving finite-sample bounds or variance expressions for the minimizer of JL, which the paper lists as future work, would turn this into a practical uncertainty-aware tuning method.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies data-driven feedforward controller tuning in a two-degree-of-freedom control loop. The plant P is assumed unknown, but the feedback controller, the reference model, the initial feedforward controller, and the spectrum of the reference signal are known. The authors propose a shaping filter L = (1/(C_ff(ρ0)+T_d C_fb)) (W/R) z^{-m} and claim that, as N→∞ and under the reference-signal assumption (A1), minimizing the filtered data cost J_L(ρ) is equivalent to minimizing the true weighted model-matching cost J(ρ)=∥W(T(ρ)−T_d)∥². They prove algebraic identities showing pointwise convergence of J_L to J, and then claim that this implies convergence of the minimizers. They also show that ERIT corresponds to the special case W=R and hence is optimal in that setting. A numerical example and a physical experiment on a Quanser rotary servo are provided.

Significance. If the minimizer-equivalence claim can be rigorously established, the paper gives an attractive closed-form prefilter for data-driven feedforward tuning: it is constructed from known quantities only, does not depend on the unknown plant, and connects a standard method (ERIT) to a model-matching objective. The algebraic core of the paper is transparent and the derivation of the filter is elegant. The ERIT optimality result for W=R is a useful interpretation. The paper also includes both simulation and experimental demonstrations. The main weakness is that the proof of Theorem 1 only establishes pointwise convergence of cost functions and does not establish the asserted convergence of minimizers.

major comments (2)
  1. [Sec. III, proof of Theorem 1, after Eq. (11)] The proof shows that, for each fixed ρ, J_L(ρ) converges to J(ρ) as N→∞ (or in probability in the random case). The proof then concludes that the minimizer of J_L converges to the minimizer of J. This inference is not valid without additional assumptions: pointwise convergence of objective functions does not imply convergence of argmins unless the parameter set is compact and the convergence is uniform (or epi-convergence holds), or unless coercivity and a uniform law of large numbers are established. No such conditions are stated. The numerical section optimizes ρ with unconstrained fminsearch (Sec. V), so the gap is not merely formal. This issue is load-bearing for Theorem 1 and propagates to Theorem 2.
  2. [Sec. III, Eq. (13), stochastic case] For the random-reference case, the paper only establishes pointwise convergence in probability of the normalized cost ar J_L(ρ) to J(ρ) for each fixed ρ. The statement that 'the optimal solution of ar J_L converges to the one of J' requires uniform convergence over the parameter set, stochastic equicontinuity, or an equivalent consistency argument. This is not provided. The asymptotic equivalence of minimizers is therefore not proven as stated.
minor comments (5)
  1. [Sec. IV-B] In the reduction of ERIT to J_L, the paper writes L=1/(C_ff(ρ0)+T_d C_fb) 'or possibly L=(1/(C_ff(ρ0)+T_d C_fb)) z^{-m}'. For a rigorous properness statement, z^{-m} should be included in the theorem statement, as it is in Theorem 1.
  2. [Sec. V-A] Typo: 'C_ff(ρ0)) = 0' has an extra closing parenthesis. Also, the initial condition for fminsearch is not stated, so the numerical results may depend on the initialization.
  3. [Sec. III, Eq. (14)] The phrase 'the optimal shaping filter satisfies |L|²=...' should clarify that the phase is arbitrary up to an all-pass factor. The paper mentions this in the following paragraph, but the statement itself would benefit from that qualification.
  4. [General] The title and abstract use 'optimal shaping filter' without a formal optimality criterion. The theorem proves that a particular filter achieves asymptotic cost equivalence, but it does not prove that no other filter achieves the same. If 'optimal' is intended to mean 'cost-matching prefilter', this should be stated explicitly.
  5. [Sec. VI] The practical experiment uses signal projection to reduce noise, while the theory is developed under a noise-free assumption. This is a reasonable limitation, but the paper should state that the experiment validates a heuristic extension rather than the theorem itself.

Circularity Check

0 steps flagged

No significant circularity: the optimal shaping filter is constructed from known quantities and its equivalence to the H2 model-matching objective is shown by direct algebra, not by assuming the conclusion.

full rationale

The paper's central derivation is self-contained. Theorem 1 proposes L = 1/(Cff(ρ0)+Td Cfb) · W/R · z^{-m}, and every ingredient is an available input to Problem 1: the initial controller parameter ρ0, the reference model Td, the feedback controller Cfb, the known reference-shaping filter R, and the weight W. In contrast, the unknown plant P, the noise, and the optimized parameter ρ do not enter the filter definition. The proof then uses the measured-signal identities u0_k = (Cff(ρ0)+Td Cfb)/(1+P Cfb) Rδ_k and y0_k = P(Cff(ρ0)+Td Cfb)/(1+P Cfb) Rδ_k (Eq. 9) to obtain exactly JL(ρ) = || W/(1+P Cfb) (Td − Cff(ρ)P) ||², which is J(ρ) rewritten via Eq. (8). The cancellation of Cff(ρ0)+Td Cfb makes the equality a direct algebraic identity, not a fitted equivalence; no parameter is estimated from the target cost and then renamed a prediction. Theorem 2 similarly follows by substituting W=R into the same algebraic reduction of the ERIT cost, and it does not import an unverified uniqueness or optimality claim. The self-citations in the paper, e.g., [18] for signal projection in the practical experiment and [20] for classification context, are not load-bearing for the main theorem; the theorem is proved from displayed equations rather than cited. The acknowledged limitations (noise-free setting in Sec. II-A, high-pass filter noise sensitivity in Remark 1, and the deferral of finite-sample statistical analysis in Sec. VII) concern assumptions and proof completeness, not circularity. In particular, the skeptic's point about pointwise convergence not automatically implying argmin convergence is an internal proof-technique gap, not a reduction of the result to its own inputs. Therefore the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No fitted free parameters appear in the derivation: W, T_d, C_fb, C_ff(ρ0), and R are problem data, and ρ is the decision variable. The result leans on the noise-free and known-reference-spectrum assumptions, a stability/invertibility condition on L, and an unproved minimizer-convergence step. No new entities are postulated.

axioms (5)
  • domain assumption Noise-free input-output data (Sec. II-A).
    Theorem 1's proof uses exact relations y0=P(C_ff0+T_d C_fb)/(1+P C_fb) Rδ and u0=(C_ff0+T_d C_fb)/(1+P C_fb) Rδ (Eq. 9); additive noise would break the equivalence, and the high-pass L amplifies noise (Remark 1).
  • domain assumption (A1) reference signal has known spectrum: r=Rδ or r=Rε with ε white noise.
    The filter contains 1/R, and the equivalence proof needs the deterministic impulse or white-noise character of r; unknown or colored reference spectra invalidate the result.
  • domain assumption The feedback loop is internally stable with C_fb, and L in (7) is stable; unstable L is only treated by a remark claiming spectral factorization.
    The theorem states 'If this L is stable'; the stable case keeps H2 norms finite. The unstable-L extension via all-pass/spectral factorization is stated without proof.
  • ad hoc to paper Pointwise convergence of J_L to J implies convergence of the minimizers as N→∞.
    The proof shows convergence of the cost functions (Eqs. 11, 13) but not uniform convergence or compactness of the feasible set, so the asserted equivalence of minimizers is an extra unproven premise.
  • domain assumption The plant P is LTI and the closed loop is stable; perfect model matching may be infeasible (C* not in C).
    Problem setup; if C* is in C the shaping filter is unnecessary.

pith-pipeline@v1.3.0-daily-deepseek · 8222 in / 14692 out tokens · 135359 ms · 2026-08-03T06:15:36.388321+00:00 · methodology

0 comments
read the original abstract

This paper discusses the data-driven model matching problem. In particular, this paper focuses on two-degree-of-freedom control systems, and consider to design feedforward controller from input-output data. An intuitive solution to this problem would be identifying the optimal controller using data, but this does not give the exact solution to the original problem. A shaping filter is required to compensate for this gap, and the main contribution of this paper is to give the optimal shaping filter. The proposed shaping filter is constructed from available information under reasonable assumptions, and its effectiveness is shown through a numerical example and a practical experiment. The relation between the proposed shaping filter and Estimated Response Iterative Tuning (ERIT) is also discussed, and it is shown that ERIT is optimal for a special case.

Figures

Figures reproduced from arXiv: 2607.29480 by Yusuke Fujimoto.

Figure 1
Figure 1. Figure 1: Two-degrees of freedom control system [Notation] The imaginary unit is denoted by j throughout the paper. The complex frequency in the z-transform is denoted by z. In this paper, we use ∥G∥ to show the H2 norm of a single-input-single-output transfer function G, i.e., if the impulse response of G is denoted by gk, ∥G∥ 2 = P∞ k=0 g 2 k . Throughout the paper, we slightly abuse the notation and regard the tr… view at source ↗
Figure 2
Figure 2. Figure 2: Bode diagram of optimal shaping filter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bode diagram of T(ρ) B. Case 1: W = 1 We first consider the case with W = 1 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bode diagram of feedforward controllers the reference model. Recall that the shaping filter plays a crucial role in such a case. If the oracle perfectly matches the reference model, it means C ∗ ∈ C and the shaping filter is not required. In the low frequency range (lower than 0.2 [rad/sample]), the case without a shaping filter shows closer behavior to the reference model than the others. In the high￾freq… view at source ↗
Figure 6
Figure 6. Figure 6: shows its specific interval (300 ≤ k ≤ 600). The horizontal axes show the time step k, and the vertical axes show yk. The dotted and the broke lines show Tdrk and T(ρ ∗ )rk where ρ ∗ denotes the oracle parameter, respectively. The thin and thick solid lines show the result without and with the shaping filter, respectively. The output without the shaping filter shows oscillating behavior, and the one with t… view at source ↗
Figure 7
Figure 7. Figure 7: Outputs with initial/updated controllers [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

discussion (0)

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