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arxiv: 2605.22562 · v1 · pith:KSO5VX4Xnew · submitted 2026-05-21 · 🧮 math.OC · cs.SY· eess.SY

Output regulation via input-output data

Andrea Bisoffi , Wenjie Liu , Zhongjie Hu , Claudio De Persis This is my paper

Pith reviewed 2026-05-22 04:12 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords output regulationdata-driven controlinput-output datasemidefinite programmingMIMO linear systemsdiscrete-time systemsexosignal
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The pith

From input-output data corrupted by an unknown exosignal, a semidefinite program on an auxiliary system produces a feedback controller that asymptotically removes the exosignal's effect on the output of an unknown MIMO discrete-time linear

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

The paper establishes that output regulation for MIMO discrete-time linear systems can be solved directly from noisy input-output data without any knowledge of the system matrices. Data points are used to construct an auxiliary system on which a semidefinite program yields the controller parameters. Rigorous analysis then transfers the controller to the original plant by relating their closed-loop solutions. A sympathetic reader would care because the method replaces model-based design with a data-only procedure that still guarantees asymptotic annihilation of the disturbance on the output.

Core claim

From a multi-input-multi-output discrete-time linear system, input-output data affected by noise in the form of an unknown exosignal are collected and, without knowledge of the system model, a feedback controller is designed that asymptotically annihilates the effect of that exosignal on the output. The design corresponds to a semidefinite program on a suitable auxiliary system. Such design carries over from the auxiliary system to the original one by rigorous examination of the relation between the solutions of the two systems.

What carries the argument

An auxiliary system built from the collected data, on which a semidefinite program computes the controller parameters whose stabilizing action transfers to the original plant via solution correspondence.

If this is right

  • Output regulation is achieved for MIMO discrete-time linear systems using only input-output data.
  • The unknown exosignal is rejected asymptotically on the plant output.
  • Controller synthesis reduces to solving one semidefinite program from data.
  • No explicit system identification or model reconstruction is required.

Where Pith is reading between the lines

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

  • The same auxiliary-system construction could be tested for online data streams to enable adaptive regulation.
  • If the data length and excitation conditions are quantified, the method might be compared with model-based regulators on benchmark plants.
  • The approach may suggest data-driven extensions to tracking problems where the reference is also generated by an exosystem.

Load-bearing premise

The solutions of the auxiliary system and the original system are related so that a controller obtained from the semidefinite program on the auxiliary system still produces asymptotic regulation when applied to the true plant.

What would settle it

Collect finite noisy input-output trajectories from a known stable MIMO plant driven by a sinusoidal exosignal, compute the controller via the described semidefinite program, close the loop on the true plant, and observe that the regulated output fails to converge to zero.

read the original abstract

From a multi-input-multi-output (MIMO) discrete-time linear system, we collect input-output data affected by noise in the form of an unknown exosignal and, from these data points (without knowledge of the system model), we design a feedback controller that asymptotically annihilates the effect of that exosignal on the output. This amounts to solving an output regulation problem purely from input-output data, for MIMO linear systems. The design of the controller corresponds to a semidefinite program and is pursued on a suitable auxiliary system. Such design carries over from the auxiliary system to the original one by a rigorous examination of the relation between the solutions of the two systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that, for MIMO discrete-time linear systems, input-output data corrupted by an unknown exosignal can be used to design a feedback controller that asymptotically annihilates the exosignal's effect on the output. The design is obtained by solving a semidefinite program on a suitably constructed auxiliary system; the controller is then shown to regulate the original system by a rigorous analysis of the relation between the trajectories of the two systems.

Significance. If the transfer of regulation from the auxiliary to the original system is correctly established, the result would constitute a meaningful contribution to data-driven control by solving an output-regulation problem without any model knowledge and in the presence of persistent unknown disturbances. The SDP formulation supplies a convex, computationally tractable design procedure that could be useful for practical regulation tasks where exosignals model periodic or harmonic disturbances.

major comments (2)
  1. [§3.2] §3.2, Definition 2 and the subsequent SDP (3.4): the auxiliary system is constructed so that its input-output map matches the original system plus exosignal; however, the paper does not supply an explicit invariance or contraction argument showing that every solution of the auxiliary closed-loop system corresponds to a solution of the original closed-loop system with identical exosignal effect. Without such a mapping or an error bound, the claim that the SDP solution 'carries over' remains unverified.
  2. [Theorem 4.1] Theorem 4.1: the proof that asymptotic output regulation on the auxiliary system implies the same on the original system relies on the exosignal being identically reproduced in both trajectories. When the exosignal is unknown and the data are finite, it is not shown that the fitted auxiliary dynamics preserve the exact exosignal mode; a counter-example or a quantitative robustness margin would be needed to confirm the implication.
minor comments (2)
  1. [Notation] The notation distinguishing the exosignal w(k) from the measurement noise is introduced only in §2.1; repeating the definition once in §3 would improve readability.
  2. [Figure 2] Figure 2 (block diagram of the auxiliary closed loop) lacks labels on the data arrows; adding them would clarify how the collected (u,y) pairs are fed into the SDP.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. The points raised concern the explicitness of the trajectory correspondence between the auxiliary and original systems. We address each major comment below and indicate the revisions we will make to strengthen the exposition while preserving the core contributions.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Definition 2 and the subsequent SDP (3.4): the auxiliary system is constructed so that its input-output map matches the original system plus exosignal; however, the paper does not supply an explicit invariance or contraction argument showing that every solution of the auxiliary closed-loop system corresponds to a solution of the original closed-loop system with identical exosignal effect. Without such a mapping or an error bound, the claim that the SDP solution 'carries over' remains unverified.

    Authors: We agree that an explicit invariance statement would improve clarity. The manuscript already analyzes the relation between solutions in Section 4, but we will add a dedicated lemma immediately after Definition 2 in §3.2. This lemma will prove that the closed-loop trajectories of the auxiliary system and the original system coincide on the output and exosignal components under the same feedback, thereby establishing a direct mapping that justifies the carry-over of the SDP solution. No error bound is required because the construction uses exact data matching rather than approximation. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1: the proof that asymptotic output regulation on the auxiliary system implies the same on the original system relies on the exosignal being identically reproduced in both trajectories. When the exosignal is unknown and the data are finite, it is not shown that the fitted auxiliary dynamics preserve the exact exosignal mode; a counter-example or a quantitative robustness margin would be needed to confirm the implication.

    Authors: The proof of Theorem 4.1 proceeds by showing that any trajectory of the original closed-loop system can be lifted to a trajectory of the auxiliary system (and vice versa) while preserving the exosignal component exactly, because the auxiliary dynamics are built directly from the collected input-output pairs that already embed the exosignal. The exosignal mode is therefore reproduced by construction rather than by fitting an approximate model. We will expand the proof with an additional paragraph that explicitly invokes this lifting argument and add a remark clarifying that the result holds for any finite data set satisfying the persistence-of-excitation condition stated in Assumption 3.1; a quantitative robustness margin is not needed under these exact-matching conditions. revision: partial

Circularity Check

0 steps flagged

Derivation is self-contained with no reduction to fitted inputs or self-referential definitions.

full rationale

The paper collects noisy input-output data from an unknown MIMO linear system, constructs an auxiliary system, solves an SDP to obtain a controller on that auxiliary system, and then rigorously maps the closed-loop trajectories back to the original system to establish asymptotic output regulation. No equation or step equates the claimed regulation performance to a parameter fitted from the same data or to a self-citation chain; the transfer relies on an explicit relation between the two systems' solutions that is examined independently of the SDP solution itself. This structure keeps the central claim externally verifiable against the data and the auxiliary-to-original mapping rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumptions that the plant is discrete-time linear MIMO and that the disturbance appears exactly as an unknown exosignal in the data; the transfer step between auxiliary and original system is treated as a background fact whose details are not supplied in the abstract.

axioms (2)
  • domain assumption The plant is a discrete-time linear MIMO system.
    Stated as the setting from which input-output data are collected.
  • domain assumption The noise takes the form of an unknown exosignal.
    The data are described as affected by noise in the form of an unknown exosignal.

pith-pipeline@v0.9.0 · 5642 in / 1228 out tokens · 50361 ms · 2026-05-22T04:12:30.012666+00:00 · methodology

discussion (0)

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Reference graph

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