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Data-driven Internal Model Control for Output Regulation

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Output regulation of unknown systems becomes stabilization of an augmented plant: one data-based LMI yields zero tracking error from noisy data, plus kth-order nonlinear and multi-agent extensions.

desk verdict A nice internal-model trick for data-driven output regulation, but the main guarantee rests on a noise bound that cannot be verified from data and the MAS section has a complex-eigenvalue bug. read the letter →

arxiv 2505.09255 v1 pith:4U4NOP3O submitted 2025-05-14 eess.SY cs.SY

Wenjie Liu , Yifei Li , Jian Sun , Gang Wang , Keyou You , Lihua Xie , Jie Chen This is my paper · ORCID
classification eess.SYcs.SY
keywords data-drivencontroloutputregulationinternalmodelprinciplelinearmatrixinequalitynoisydatamulti-agentsystemsnonlinearkth-order
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

This paper claims that an unknown linear system can be made to track a reference and reject a disturbance with zero steady-state error using only noisy input-state data and a known description of the reference/disturbance generator. The proposed route is to embed an internal model of that generator into the controller, which turns output regulation into a stabilization problem, and then to compute the stabilizing gain from a single data-based linear matrix inequality (LMI). No output regulation equations are solved and no plant model is identified. The same idea is extended to give kth-order approximate tracking for a class of nonlinear systems and to solve linear and nonlinear cooperative output regulation for multi-agent systems.

What carries the argument

The load-bearing object is the augmented system $\dot\xi=A_\xi\xi+B_\xi u+E_\xi v$, $e=C_\xi\xi+Fv$, formed by appending the internal model $\dot z=G_1 z+G_2 e$ to the unknown plant, with $\xi=\mathrm{col}(x,z)$. The paper's key move is the internal-model-principle reduction: finding a controller that solves output regulation is replaced by finding $K_\xi$ that makes $A_\xi+B_\xi K_\xi$ Hurwitz, so no output regulation equations are needed. On the data side, the workhorse is a robust data-driven stabilization result: from noisy trajectories of $\xi$ one builds matrices $\Psi$, $\Upsilon$, $\Sigma$ and a quadratic inequality describing the set of all augmented systems consistent with the data; feasibility of LMI (23) then yields a gain that stabilizes every system in that set, including the true plant.

What would settle it

Simulate the robot example with known matrices, record the true combined disturbance-exosignal matrix $E_\xi V+D$, and take $\Delta$ from its largest singular value; then rerun the LMI with a deliberately smaller $\Delta$ while holding the data fixed. If the LMI remains feasible and the closed-loop tracking error fails to converge to zero, Theorem 2's claim is contradicted, because the true augmented system no longer belongs to the data-consistent set on which the proof relies.

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Extended reading notes

Core claim

At the core is a reduction: for an unknown LTI plant whose reference and disturbance signals are generated by a known matrix $S$, the paper designs a controller $u=K_x x+K_z z$ with $\dot z=G_1 z+G_2 e$, where $(G_1,G_2)$ is an $n_y$-copy internal model of $S$. Embedding the generator dynamics into the controller is the key step: by the internal model principle, whenever $A_\xi+B_\xi K_\xi$ is Hurwitz, the output regulation equations are automatically solvable and $e(t)\to 0$. The main theorem states that, under a persistency-of-excitation condition and a known energy bound on the combined noise and exosignal contribution, feasibility of the data-based LMI (23) produces $K_\xi=YP^{-1}$ that stabilizes the true augmented system and solves the regulation problem. The same construction, using a $k$-fold internal model of the exosystem, solves the $k$th-order nonlinear output regulation problem, and distributed versions solve linear and nonlinear cooperative output regulation for heterogeneous multi-agent systems.

Load-bearing premise

The result depends on the user first knowing a bound on the combined effect of the measurement noise and the reference/disturbance signal during data collection; because that effect involves the unknown plant, the bound cannot be verified from data alone, and if it is chosen too small the zero-error conclusion collapses.

Editorial extensions

If this is right

  • For unknown LTI systems, zero steady-state tracking error is achievable from noisy data by solving one convex LMI; the controller neither solves output regulation equations nor requires online measurements of the reference or disturbance signal.
  • For smooth nonlinear systems, a $k$-fold internal model makes the tracking error converge to a term of order $O(v^{k+1})$, so higher-order internal models give higher-order asymptotic tracking without identifying the nonlinear functions.
  • For heterogeneous linear multi-agent systems under a directed spanning-tree graph, the distributed protocol makes every agent's output track the exosystem output exactly; for nonlinear agents the same protocol gives $k$th-order cooperative output regulation.
  • Because the design reduces output regulation to robust stabilization, the data-based stabilization LMI is the only plant-dependent computation, and the resulting controller is obtained directly from offline noisy data rather than from a system identification step.

Reading between the lines

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

  • The practical bottleneck implied by this approach is the choice of the energy bound in Assumption 5; a data-only procedure that validates or adaptively tightens $\Delta$ would make the zero-error guarantee usable in applications, but no such procedure is given.
  • Because the reduction is modular, other data-driven stabilization designs with different noise models or Lyapunov functions could be substituted for the robust stabilization step used here, potentially reducing conservativeness; the paper points toward this in Remark 6 but does not develop it.
  • A natural stress test not reported in the paper is to fix a known plant, collect data with a large exosignal, choose $\Delta$ from a smaller exosignal, and check whether the tracking error still converges; the theory predicts it will not, because the true augmented system then falls outside the data-consistent set.
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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

2 major / 5 minor

Summary. The paper addresses the output regulation problem for unknown linear time-invariant systems, kth-order nonlinear systems, and multi-agent systems using noisy input-state data. The main idea is to augment the plant with an internal model of the exosystem, transforming output regulation into a stabilization problem, and then to design the stabilizing gain via the robust data-driven LMI based on Petersen's lemma. The paper claims exact zero tracking error without solving output regulation equations. The claims are supported by four numerical examples and comparisons with a polytopic data-driven synchronization method.

Significance. The internal-model reformulation is an interesting and potentially impactful way to avoid the generally infeasible data-based output regulation equations, and the reduction to a convex LMI is elegant. If the core guarantees were fully data-driven, the paper would be a significant contribution to direct data-driven control. However, as discussed in the major comments, the main theorem is conditional on an unverifiable matrix bound that involves the unknown plant matrices, and a proof step in the multi-agent extension assumes diagonalizability of the graph matrix H. With these gaps repaired or the claims appropriately qualified, the paper would be a useful contribution for the data-driven control community.

major comments (2)
  1. [III-A3, Eq. (20); Theorems 2–5] Assumption 5 requires a known matrix Δ such that (EξV+D)(EξV+D)^T ⪯ ΔΔ^T, but Eξ = col(E, G2F) is defined in terms of the unknown plant matrices E and F. The paper does not provide a data-only procedure to construct or certify Δ; it only states existence. For finite data such a Δ always exists, so the substantive content is that a certified bound is available, and the examples do not establish this because E and F are known to the authors when the data are generated. If Δ is chosen too small, the true pair [Aξ,Bξ] may lie outside Cξ in (21), so LMI (23) gives no guarantee for the actual closed loop; if Δ is chosen too large, the LMI may be infeasible. The same issue propagates through Assumption 9 and Theorems 3, 4, and 5. The central claim of exact model-free output regulation is therefore not established as stated.
  2. [IV-A, proof of Theorem 4] The proof assumes a unitary T1 with T1 H T1^{-1} = diag{λ1,...,λN}. This is not guaranteed for the matrix H = L+Λ of a directed graph satisfying Assumption 7; H can be non-diagonalizable (for example, a directed path with the leader as root gives a Jordan block). The stability conclusion can still be obtained by using a Schur triangular form of H and the fact that a block-triangular matrix is Hurwitz if all diagonal blocks are Hurwitz, but the proof as written is not valid for general directed graphs. This also affects Theorem 5, which relies on Theorem 4.
minor comments (5)
  1. [III-A3, around Eq. (19)] Please clarify how the augmented state ξ = col(x,z) data are collected; in particular, specify that v is measured during the offline experiment and that z is obtained by simulating the internal model (15b) using the measured error e, since z is not a physical state.
  2. [Theorem 2 proof, Eq. (24)] The term "-P" on the left-hand side is not a consequence of Theorem 1's LMI (7b)/(23); either remove it or provide a separate argument for the stronger inequality.
  3. [Theorem 4 proof, Eq. (56)] The stabilizing gain obtained from LMI (55) is Y_i P_i^{-1} = λ_i Kξ_i, so the displayed inequality should involve λ_i B̄ξi Kξ_i (or a tilde gain) to match the definition Kξ_i := Y_i P_i^{-1}/λ_i.
  4. [Assumption 5] The wording "there exists some matrix Δ" should be changed to "a known matrix Δ is available such that ..." to avoid a vacuous-existence interpretation that does not yield a constructor.
  5. [General] There are several minor typos: "diagraph" in Problems 3 and 4 should be "digraph"; in Eq. (42b), the term "h0x1x2/4" appears to have missing or ambiguous parentheses; and in Property 2 of Problem 2, the expression `(e(t)− O(v(k+1)(t))` has an unbalanced parenthesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a chain of external robustness and internal-model theorems, and the earlier self-citation is only a comparison.

full rationale

The paper's derivation chain is self-contained and externally supported. The data equation Ξ+ = AξΞ− + BξU− + EξV + D, together with Assumption 5 bounding EξV+D, places the true augmented pair [Aξ,Bξ] in the data-consistent set Cξ defined in (21)-(22). Feasibility of LMI (23) is then exactly the hypothesis of the external Petersen-lemma result Theorem 1 from [16], applied to the augmented system (18). That yields a Hurwitz Aξ+BξKξ. The final zero-error conclusion is the standard internal-model implication [1, Lemma 1.27], imported as an external theorem. Nothing in the paper fits a parameter to the target tracking error and then renames the fit as a prediction; the controller gain is synthesized from the LMI, not fitted to the regulated output. The only prior-work overlap, [36], appears as a comparison baseline rather than a load-bearing premise. Assumption 5 does require a bound on EξV+D, which depends on unknown plant matrices, but this is a verifiability/correctness concern about whether the user can certify Δ, not circularity: the theorem's implication from data plus assumption to output regulation is not equivalent to its input by construction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard internal-model theorems, the Petersen-lemma LMI, and persistency-of-excitation assumptions. The only user-chosen quantity is the energy bound Δ, which is not fitted to data but is assumed valid; it is the primary unverifiable input.

free parameters (1)
  • noise-energy bound Δ = not given; user-supplied in Assumptions 2, 5, and 9
    The LMI and the stabilizing gains depend on the bound Δ on (EξV+D)(EξV+D)^T. For unknown Eξ, a valid Δ is not constructible from data; examples choose it with knowledge of the simulator. The closed-loop stability guarantee holds only if the chosen Δ is a true bound.
assumptions (6)
  • domain assumption Internal model principle for linear systems ([1, Lemma 1.27]): if the augmented system (Aξ,Bξ) is stabilized by Kξ and (G1,G2) is an ny-copy internal model of S, then the OREs (14) have a solution and the controller solves Problem 1.
    Invoked in the Theorem 2 proof and in the construction before Eq. (18).
  • domain assumption Internal model principle for kth-order nonlinear servomechanism ([8, Theorem 3.12]): stabilizing the linearized augmented system with a k-fold internal model solves Problem 2.
    Invoked in Theorem 3 and Remark 6.
  • standard math Petersen's lemma robust stabilization result of Bisoffi, De Persis, and Tesi (Theorem 1): feasibility of LMI (7) yields a gain stabilizing all systems in the data-consistent set C.
    Used to prove Theorems 2, 3, 4, and 5.
  • standard math Persistency of excitation, i.e., full row rank of the data matrix [U; X] (Assumptions 1, 4, and 8).
    Standard data-driven assumption; ensures the data-consistent set is well-defined.
  • domain assumption Existence of a known energy bound Δ for the combined exosignal and disturbance term (Assumptions 2, 5, and 9).
    The LMI certificate and stabilizing guarantee depend on this bound; it is not verifiable when the plant matrices are unknown.
  • domain assumption Exosystem conditions: S is known with no eigenvalues in the open left half-plane for the linear case (Assumption 3), all eigenvalues on the imaginary axis for the nonlinear case (Assumption 6), and the graph contains a directed spanning tree rooted at the exosystem (Assumption 7).
    Standard output regulation and cooperative control conditions.

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Pith. "Pith review of Data-driven Internal Model Control for Output Regulation." pith.science (2026). https://pith.science/paper/4U4NOP3O

@misc{pith2026250509255,
  author       = {Pith},
  title        = {Pith review of: Data-driven Internal Model Control for Output Regulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4U4NOP3O}},
  note         = {Machine review of arXiv:2505.09255}
}
abstract

Output regulation is a fundamental problem in control theory, extensively studied since the 1970s. Traditionally, research has primarily addressed scenarios where the system model is explicitly known, leaving the problem in the absence of a system model less explored. Leveraging the recent advancements in Willems et al.'s fundamental lemma, data-driven control has emerged as a powerful tool for stabilizing unknown systems. This paper tackles the output regulation problem for unknown single and multi-agent systems (MASs) using noisy data. Previous approaches have attempted to solve data-based output regulation equations (OREs), which are inadequate for achieving zero tracking error with noisy data. To circumvent the need for solving data-based OREs, we propose an internal model-based data-driven controller that reformulates the output regulation problem into a stabilization problem. This method is first applied to linear time-invariant (LTI) systems, demonstrating exact solution capabilities, i.e., zero tracking error, through solving a straightforward data-based linear matrix inequality (LMI). Furthermore, we extend our approach to solve the $k$th-order output regulation problem for nonlinear systems. Extensions to both linear and nonlinear MASs are discussed. Finally, numerical tests validate the effectiveness and correctness of the proposed controllers.

Figures

Figures reproduced from arXiv: 2505.09255 by the authors.

Figure 1
Figure 1. Tracking performance under the controller (15). [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Data-driven output regulation of nonlinear systems. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Data-driven local output regulation of nonlinear systems. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: , where the node 0 represents the exosystem v(t). It is evident that Assumption 7 holds. We collected each agent’s trajectories of length T = 20 from the perturbed system (50) under the same conditions as in Example 1, assuming that the noise di is bounded by [−0.01, 0…
Figure 5
Figure 5. Figure 5: Tracking performance under the proposed data-driven control approach. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Tracking error ei(t) under the proposed control approach and polytopic control approach [36]. Let us rewrite the nonlinear MAS (58) and the exosystem (44) as follows x˙ i = Aixi + Biui + Eiv + αi(xi , ui , v) (60a) yi = Cixi + µi(xi) (60b) v = Sv (60c) y0 = −F v (60d) …
Figure 9
Figure 9. Figure 9: Tracking error of the nonlinear MAS with [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 7
Figure 7. Figure 7: Tracking performance of the nonlinear MAS with [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Tracking performance of the nonlinear MAS with [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.