Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals

Daniele Astolfi; Giordano Scarciotti; Zirui Niu

REVIEW 2 major objections 4 minor 1 cited by

Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that for a stabilizable SISO LTI plant, the full-information output regulation problem with non-smooth, non-periodic exogenous signals is solvable if and only if the regulator equations (11) admit bounded…

desk verdict The paper tackles a real gap and introduces a genuinely new integral-based immersion idea, but the proof of Lemma 1 bounds the wrong ratio, so the central theorems are unsupported until Assumption 1 is strengthened. read the letter →

arxiv 2505.21209 v1 pith:DLO3OEFN submitted 2025-05-27 eess.SY cs.SYmath.OC

Zirui Niu , Daniele Astolfi , Giordano Scarciotti This is my paper

classification eess.SYcs.SYmath.OC

keywords outputregulationinternalmodelprinciplenon-smoothexogenoussignalsnon-periodicexplicitexosystemregulatorequationsrobustcontroltime-varyingsystems

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

Classical output regulation theory handles exogenous signals generated by smooth differential equations or by periodic jumps; this paper removes both restrictions. It considers signals of the form $\omega(t)=\Lambda(t,t_0)\omega_0$, where $\Lambda$ is piecewise continuous, nonsingular, finite-time bounded, and has bounded inverse, so square waves, PWM waveforms, and diverging triangular waves are allowed. The central claim is that the full-information regulation problem for a stabilizable SISO LTI plant is solvable if and only if the associated regulator equations admit bounded piecewise-continuous solutions, and that solvability for every disturbance and reference pair is characterized by a new non-resonance condition. The paper then obtains error-feedback regulators under a minimum-phase, relative-degree-one assumption, and two robust internal-model constructions (an augmentation-based one and an integral-based immersion) for uncertain parameters. A reader should care because non-smooth non-periodic signals are common in power electronics and robotics, and this is a systematic regulator design for them.

What carries the argument

The argument rests on three objects. First, the explicit exosystem (1), with $\Lambda(t,t_0)$ a piecewise continuous, nonsingular, finite-time bounded matrix whose inverse is bounded; this replaces the implicit generator $\dot\omega=S\omega$ and can express non-smooth, non-periodic, even diverging signals. Second, Lemma 1, which asserts that any exponentially stable linear time-varying system driven by such a $\Lambda$ admits a bounded steady-state response matrix $\Pi_g$; this lemma converts the classical regulator equations into the integral form (11), and via Corollary 1 into the differential-algebraic form (16) that is used computationally. Third, the non-smooth non-resonance condition (17), a boundedness requirement on the integral $\Omega(t)=\int_{t_0}^{t} (\Lambda(\tau,t_0)\Lambda(t,t_0)^{-1})^\top \otimes e^{A_z(t-\tau)}d\tau$, where $A_z$ has eigenvalues equal to the transmission zeros of the plant; this replaces the classical eigenvalue non-resonance condition. For the robust part, the integral-based immersion identity (43) plays the central role, requiring that the internal-model output $R^*(t,\mu)$ be annihilated by a linear combination with its iterated integrals.

What would settle it

Construct a piecewise-constant $\Lambda$ satisfying Assumption 1 but breaching the decaying-envelope requirement: let $\Lambda(t,t_0)=1$ except on narrow intervals around times $t_n=n$, where it jumps to $e^{e^n}$; then $\Lambda^{-1}$ is bounded and $\Lambda$ is finite-time bounded. For a scalar system with $A_g=-1$ and $B_g=1$, the integral in (7) at a time just after the spike at $t_n$ contains a term of order $e^{e^n}$, so $\Pi_g(t)$ is unbounded as $n\to\infty$, which would contradict Lemma 1 and the regulator-equation theorems that rely on it.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the output regulation problem—full-information, error-feedback, and robust—can be posed and solved for linear systems whose exogenous signals are generated by an explicit exosystem $\omega(t)=\Lambda(t,t_0)\omega_0$ with $\Lambda$ piecewise continuous, nonsingular, finite-time bounded, and $\Lambda^{-1}$ bounded. Theorem 1 establishes an if-and-only-if: a stabilizing state-feedback regulator exists exactly when bounded piecewise-continuous matrix functions $\Pi_x$ and $\Delta$ solve the regulator equations (11). Theorem 3 gives the solvability condition for all $P$ and $Q$: the non-smooth non-resonance condition (17), plus either $D\neq 0$ or $D=0$ with Assumptions 3 and 4. For error feedback, Theorem 4 shows that any exponentially stabilizing controller solves the problem if and only if it has the explicit internal model property defined by (20), a condition that can be met by a canonical realization; and for uncertainties, the paper constructs two robust internal models, one by augmentation along the lines of hybrid regulation and one by a new integral-based immersion (43). The key structural message is that non-smoothness forces the regulator equations into integral form and makes relative degree and feedthrough play a necessary role that they do not play in the classical smooth theory.

Load-bearing premise

The paper's key lemma assumes that an exponentially stable system driven by an exosystem that is finite-time bounded with bounded inverse always has a bounded steady-state response; this needs the extra, unstated condition that past values of the exosystem decay exponentially relative to the present.

Editorial extensions

If this is right

If the regulator equations (11) are solvable, the full-information control law $u=Kx+(\Delta-K\Pi_x)\omega$ solves Problem 1 with any stabilizing $K$; regulator design reduces to proving boundedness of an integral equation rather than solving a matrix Sylvester equation.
When $D=0$, Theorem 2 implies that no finite-time-bounded, piecewise-continuous regulator can track a genuinely discontinuous signal; a feedthrough term ($D\neq 0$) or a higher-relative-degree construction is necessary.
Under minimum-phase and relative-degree-one assumptions, the error-feedback regulator (32) with a canonical realization of the internal model and a sufficiently large high-gain $k>\kappa$ achieves exponential stability and asymptotic regulation, and the stabilization is robust to parameter uncertainty as long as the sign of the high-frequency gain is known.
The two robust internal models in Section V—augmentation-based and immersion-based—both solve Problem 3, and in cases where the uncertainty satisfies the factorization structure (39), the immersion-based regulator can have strictly smaller dimension (dimension 4 versus 15 in the worked circuit example).
The new non-resonance condition (17) reduces to the classical condition $\sigma(A_z)\cap\sigma(S)=\emptyset$ when $\Lambda(t,t_0)=e^{S(t-t_0)}$, so the paper's results contain the classical linear output regulation theory as a special case while allowing unbounded and non-smooth signals.

Reading between the lines

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

A natural technical strengthening would be to add to Assumption 1 an explicit uniform exponential-decay condition on $\Lambda(\tau,t_0)\Lambda(t,t_0)^{-1}$; with that amendment, Lemma 1's proof goes through as written and all theorems built on it would hold.
The integral-based immersion (43) suggests a data-driven route: the coefficients $a_i(t)$ could be estimated online from measurements of $R^*(t,\mu)$ by least-squares fitting on iterated integrals, avoiding the need to know $\Lambda$ and $\Delta^*$ in closed form.
Because Theorem 2 shows that discontinuous outputs require $D\neq 0$ or impulsive inputs, the practical design message for PWM-driven and switching power electronics is that a small feedthrough or an approximate high-bandwidth actuator is needed, not just a clever regulator.
The non-resonance condition (17) can be checked numerically from samples of $\Lambda$, potentially giving a model-free pre-test for whether a given non-smooth exosystem is regulable by a given plant.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is my read of arXiv:2505.21209. The problem is real: output regulation for non-smooth, non-periodic exogenous signals (PWM, sawtooth, time-varying square waves) has not been treated in a unified way. The explicit-generator formulation and the integral-based regulator equations are a sensible way in, and the integral-based immersion idea is genuinely new. The circuit example is worked in enough detail to show the machinery moves.

The trouble is Lemma 1. The proof in Appendix A defines W_Lambda(t,t0) = lim_{tf->inf} Lambda(t,t0) Lambda(tf,t0)^{-1} and then bounds the integrand in (51) by ||W_Lambda(tau,t0)||. But the expression for Pi_g(t) contains Lambda(tau,t0) Lambda(t,t0)^{-1}, not W_Lambda. Assumption 1 gives finite-time boundedness of Lambda and boundedness of Lambda^{-1}; it does not give a uniform bound on that ratio. A scalar piecewise-constant Lambda with spikes of increasing height followed by drops back to 1 satisfies Assumption 1, yet Pi_g grows without bound. So the statement of Lemma 1 is false as written, and since Theorems 1, 4, 5, and 6 lean on it, the central claims are not established. This is not a cosmetic gap; it is the load-bearing issue.

The good news: the fix is likely local. Add a condition like ||Lambda(tau,t0) Lambda(t,t0)^{-1}|| <= c e^{gamma(t-tau)} for some gamma < beta (or a polynomial bound), and the proof of Lemma 1 goes through. The examples in the paper—a linearly growing triangular wave, bounded square waves—satisfy such a condition, so the intended applications survive. But the current Assumption 1 is too weak.

Two smaller complaints: the non-resonance condition in Definition 1 is essentially defined as boundedness of the integral that appears in the solution, so Theorem 3's 'if and only if' is partly definitional. And the immersion condition (43) is an existence assumption; the paper gives a computational way to find the a_i in examples but no general constructive test. These are worth mentioning but they are not fatal; the Lemma 1 issue is.

Who is this for? People working on internal-model-based output regulation, especially for hybrid or non-smooth signals. It deserves a serious referee: the topic matters and the ideas are worth developing. I recommend sending it out, with a clear request to fix Assumption 1 and Lemma 1 before acceptance. If the authors do that, the paper could be a solid contribution.

Referee Report

2 major / 4 minor

Summary. The paper studies output regulation of SISO LTI systems in the presence of non-periodic, non-smooth exogenous signals generated by an explicit-form exosystem ω(t)=Λ(t,t0)ω0. Under Assumption 1 (Λ piecewise continuous, nonsingular, finite-time bounded, with Λ^{-1} bounded), it claims a full-information solution characterized by the regulator equations (Theorem 1), a solvability analysis based on a new non-resonance condition (Theorem 3), an error-feedback internal-model design with high-gain stabilization (Theorem 4 and Proposition 3), and two robust internal-model designs, one by augmentation and one by integral-based immersion (Propositions 4 and Lemma 2). The results are illustrated on an RLC circuit example with diverging triangular and rectangular waveforms.

Significance. The problem class is timely: non-periodic non-smooth reference/disturbance signals are relevant to power electronics, robotics, and hybrid systems, and the explicit-generator formulation is a genuine attempt to go beyond periodic hybrid exosystems. The paper is also transparent about not assuming periodicity or the semigroup property, and the integral-based immersion idea in Section V-C is interesting. The circuit example is detailed and appears to support the constructive parts of the paper. However, the central technical lemma, Lemma 1, is false as stated, and because Theorems 1, 2, 4, 5, and 6 all invoke it, the main theoretical claims are not established. The paper's advertised relaxation to finite-time bounded Λ, with Λ diverging faster than exponentially, is precisely what the faulty proof tries to accommodate, and the counterexample in my major comment shows that this relaxation is untenable without additional structure.

major comments (2)

[Appendix A, Eq. (51) and Lemma 1] The proof of Lemma 1 bounds the wrong object. The integral term in (7) contains Λ(τ,t0)Λ(t,t0)^{-1}, but the proof bounds the integrand by ||W_Λ(τ,t0)||, where W_Λ(τ,t0)=lim_{tf→∞} Λ(τ,t0)Λ(tf,t0)^{-1}. Assumption 1 neither guarantees that this limit exists nor implies a uniform bound on Λ(τ,t0)Λ(t,t0)^{-1} for τ≤t. A concrete scalar counterexample is Λ(s)=1+M_n on [n,n+h] and Λ(s)=1 on (n+h,n+h+ε], with M_n→∞. This Λ is piecewise continuous, nonsingular, finite-time bounded, and Λ^{-1} is bounded, so Assumption 1 holds. Taking A_g=-β and B_g=1 in (6), the contribution to Π_g(t) at t=n+h+ε from the n-th pulse is (1+M_n)β^{-1}(1-e^{-βh})e^{-β(h+ε)}, which is unbounded. Hence Lemma 1 is false as stated, and the boundedness of Π_x, Π_ξ, and related matrices used in Theorems 1, 2, 4, 5, and 6 is unsupported. Since the 'if and only if' in Theorem 1 depends on this boundedness, the central claim of the paper is not established.
[Definition 1 and Theorem 5 (Appendix C)] The new non-resonance condition is essentially definitional. Definition 1 defines non-resonance as the boundedness of the integral Ω in (17), and the proof of Theorem 5 in Appendix C then shows that solvability of the regulator equations is equivalent to boundedness of that same integral. This makes the 'if and only if' in Theorem 3 close to a restatement, with the only nontrivial work being the reduction of (16) to (57a). In addition, the proof of Theorem 5 claims that a bounded solution to (57a) with arbitrary initial condition Π_z(ˆt) exists iff the non-resonance condition holds, but the homogeneous term e^{A11(t-ˆt)}Π_z(ˆt)Λ(t,t0)^{-1} is not controlled by boundedness of Ω under Assumption 1; extra decay or dichotomy assumptions on Λ relative to A11 are needed and are not stated or proved.

minor comments (4)

[Appendix C, Eq. (56)] Equation (56) has a missing integration variable: the term CA^{j*+1}Ψ_x(t) inside the integral should be CA^{j*+1}Ψ_x(τ), and the displayed expression is missing a closing bracket.
[Section V-B] The notation 'vect Iν' should be 'vec Iν' to match the notation introduced in the beginning of the paper; the current text is inconsistent.
[Appendix A, proof of Lemma 1] The sentence beginning 'for any pair of ((xg(t0), Πg(t0)ω(t0)),' is malformed and appears to be missing a phrase; this should be corrected.
[Example 1 and Section VI] The existence of the immersion coefficients a_i is justified only by a numerical rank check in (46) and by Fig. 1; the text should state the numerical tolerance and the verification method used to conclude that (43) holds for all t≥ˆt.

Circularity Check

1 steps flagged · score 3.0 of 10

Non-resonance condition is partly self-definitional, but the main output-regulation theorems are not circular.

self definitional [Definition 1 (Section III-B) and Theorem 5 proof (Appendix C, after eq. (57a))]
"Systems (2) and (1) are non-resonant if the matrix-valued function Ω(t) = ∫_{t0}^{t}(Λ(τ,t0)Λ(t,t0)^{-1})^⊤ ⊗ e^{Az(t−τ)}dτ, is bounded for all t ≥ t0. ... When Assumptions 1 and 3 hold, the vectorization of (57a) yields that a bounded solution arΠ_z exists for any arΠ(ˆt) and any P and Q if and only if the non-smooth non-resonance condition is satisfied."

Definition 1 defines 'non-resonant' as the boundedness of Ω, which is exactly the integral kernel appearing in the explicit solution (57a) after vectorization: vec(arΠ_z(t)) = (∫ (Λ(τ,t0)Λ(t,t0)^{-1})^⊤ ⊗ e^{A11(t−τ)}dτ)vec(G1) + ... with A11 playing the role of Az. Theorem 5 then proves that a bounded arΠ_z exists if and only if the non-resonance condition holds simply by vectorizing (57a). The boundedness of the realized solution and the boundedness of Ω are the same assertion, so the 'if and only if' in Theorem 3 is true by construction: the new condition names the solution kernel rather than providing an independent criterion. This does not make the regulator-equation derivation in Theorem 1 circular, but it weakens the claimed novelty of the non-resonance characterization.

full rationale

The core derivation chain for Theorem 1 is self-contained: Π_x and Δ are defined by the explicit solution formula and the limit condition, and the sufficiency/necessity proof uses Lemma 1 directly. The robust internal-model and immersion constructions are design procedures, and the numerically computed a_i in Example 1 are controller-design parameters, not predictions. Self-citations to [30], [31], [32], and [43] are background or side remarks and are not load-bearing; Lemma 1 has its own proof (though with a separate correctness concern in eq. (51) that is a mathematical error, not circularity, and is not scored here). The only definitional shortcut is the non-resonance condition, which absorbs the boundedness of the solution kernel into Definition 1 and then restates it in Theorem 5/3. Overall, the paper is mostly self-contained, with one partly definitional characterization that does not undermine the main regulator-equation and internal-model results.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; the explicit generator and integral-based immersion are mathematical constructions without independent empirical handles. The central claim rests on Assumptions 1-5 and on the existence of immersion coefficients, plus the flawed Lemma 1.

free parameters (4)

a_i(t) (immersion coefficients) = computed numerically via pseudoinverse; see Fig 1 and Fig 5
Functions satisfying the integral relation (43); existence is assumed and values are fitted to the signal in the examples, not derived analytically.
d (immersion order) = 4 in the examples
Chosen so that the rank condition (46) holds; affects the dimension of the internal model.
high-gain k = 100
Stabilization gain in Proposition 3, chosen as a large constant in the simulation.
Fim eigenvalues = randomly selected conjugate pairs listed in Section VI
Design degrees of freedom for the canonical realization; values influence Him(t) but are not derived from the problem data.

assumptions (7)

domain assumption Assumption 1: Lambda is piecewise continuous, non-singular, finite-time bounded, with Lambda^{-1} bounded.
Used throughout; the proof of Lemma 1 additionally requires a uniform bound on Lambda(tau,t0)Lambda(t,t0)^{-1}, which is not stated and is false for some Lambda satisfying the assumption.
domain assumption Assumption 2: (A,B) stabilizable.
Standard for state feedback stabilization.
domain assumption Assumption 3: QLambda piecewise differentiable and QLambda_dot Lambda^{-1} bounded.
Needed for solvability of regulator equations in the D=0 case.
domain assumption Assumption 4: unitary relative degree.
Used in Theorem 3 and in the high-gain stabilization proof.
domain assumption Assumption 5: minimum phase.
Sufficient for the non-resonance condition and the high-gain stabilization.
ad hoc to paper Lemma 1: for an exponentially stable LTV system with Lambda satisfying Assumption 1, the steady-state matrix Pi_g in (7) is bounded and x_g converges to Pi_g omega.
The proof is flawed; boundedness does not follow from Assumption 1. This lemma is load-bearing for Theorems 1, 4, 5, and 6.
ad hoc to paper Existence of integral-immersion coefficients a_i in (43) for the uncertain signal.
The immersion-based robust internal model relies on this existence; in the example the coefficients are found numerically via a rank condition, not guaranteed by a general theorem.

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Cite this review

Pith. "Pith review of Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals." pith.science (2026). https://pith.science/paper/DLO3OEFN

@misc{pith2026250521209,
  author       = {Pith},
  title        = {Pith review of: Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DLO3OEFN}},
  note         = {Machine review of arXiv:2505.21209}
}

read the original abstract

We address the output regulation problem of linear systems with non-smooth and non-periodic exogenous signals. Specifically, we first formulate and solve the full-information problem by designing a state-feedback controller. We study the solvability of the regulator equations, providing a new non-resonance condition. We then focus on the error-feedback problem, for which we design a (non-robust) internal model leveraging the concept of canonical realisation and applying a high-gain method for the stabilisation of the closed-loop system under the minimum-phase assumption. Finally, we study the regulation problem involving model parameter uncertainties. Drawing ideas from both hybrid and time-varying (smooth) output regulation, we propose two methods to establish an internal model that is robust to uncertainties. The first method is an extension of the hybrid internal model, while the second relies on a new concept of immersion. In this non-smooth case, the immersion is established based on integrals rather than derivatives. The effectiveness of the proposed solutions is illustrated by a circuit regulation example.

Figures

Figures reproduced from arXiv: 2505.21209 by the authors.

**Figure 1.** Time histories of the functions a1(t), a2(t), a3(t), and a4(t). A valid explicit immersion can be established if (43), with R∗ (t, µ) replaced by F(t), is satisfied by a group of bounded piecewise continuous functions a1, a2, . . . , ad. These functions, grouped as v(t) = col(ad(t), ad−1(t), · · · , a1(t)), can be found if there exist a positive integer d and a positive constant tˆ such that for all t ≥ tˆ, rank … view at source ↗

**Figure 2.** Schematics of the RLC circuit. -0.2 -0.1 0 0.1 0 2 4 6 8 10 12 t 0 0.5 1 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Top: time history of solution Πx(t) of the regulator equations. Bottom: time history of solution ∆(t) of the regulator equations. [dth(t), rf (t), 0]⊤ = Λ(t, 0)ω0 where Λ = blkdiag(Λd, Λˆ r) with Λˆ r(t, 0) = Λr(t, 0) −Λ˜ r(t, 0) Λ˜ r(t, 0) Λr(t, 0) , (49) where Λ˜ r(t, 0) = (t + 1)∇( 2π 3 t) is selected to guarantee the invertibility of Λˆ r (and therefore Λ). Then we are ready to derive the input-output dynami… view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Top: Time histories of functions a1(t), a2(t), a3(t), a4(t). Middle: Time history of Him(t) for the canonical realization. Bottom: Time history of the input u(t) generated by the robust immersion-based regulator [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Top: Time history of system state x(t). Bottom: Time history of the regulation error e(t). the problem cannot be solved by the normal error-feedback controller as U ′ (µ) and Λ are not commutative. However, this can be solved by either an augmentation-based robust regu…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Robust Output Regulation of Uncertain Linear Time-Varying Systems
eess.SY 2026-01 conditional novelty 7.0 of 10

Exact robust output regulation for uncertain linear time-varying systems generally requires an infinite-dimensional internal model, but a finite truncation can achieve arbitrarily small tracking error.

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