REVIEW 4 major objections 4 minor 1 cited by
On Sufficient Richness for Linear Time-Invariant Systems
T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read The paper proves that for asymptotically stable reachable linear time-invariant systems, persistent excitation of the state or state-input regressor is governed exactly by the partial-PE degree of a windowed stack of input time shifts…
desk verdict The new multi-input necessity proofs collapse on a rank-nullity step—full row rank is asserted for a matrix with more rows than columns—but the sufficient conditions and the single-input characterization are worth keeping. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the stacked history operator: $Q_k(u)$ collects the last $k$ time-shifted copies of the input (discrete time), and $D_k(u)$ collects the derivatives of the input up to order $k-1$ (continuous time). The new notion of partial persistent excitation (PPE) of degree $d'$ says that some surjective linear projection of a signal is persistently exciting, so the signal persistently spans only a $d'$-dimensional subspace. Lemmas 2 and 4 are the workhorse: once the PPE degree of the $n$-stack falls below the critical value $d(n-1)$, increasing the stack length cannot raise the PPE degree. That 'stacking cannot create new persistently spanned directions' property converts a lack of PPE in the input into a lack of PE in the state in the necessity directions; in the sufficiency directions, a non-PE state is shown to force the input to be nearly a static state feedback plus a low-dimensional residual, which prevents the stacked input from being PE.
What would settle it
Set $d' < d(n-1)$ and choose $N$ so large that the stacked matrix $[M - \widetilde{E}]$ has $ndN$ rows and $Nd + d'(N+1)$ columns; since $ndN > Nd + d'(N+1)$ when $d' < d(n-1)$, full row rank is impossible, so the kernel-dimension formula $d'' = N(d+d'-nd)+d'$ cannot hold. Checking this rank for one such system settles whether Theorem 3's necessity direction is backed by a valid argument.
Extended reading notes
Core claim
The paper's central claim is that, for asymptotically stable reachable LTI systems, PE of the output regressor is exactly a property of the stacked input history $Q_n(u)$ (discrete time) or $D_n(u)$ (continuous time). The necessity theorems (Theorem 1 discrete, Theorem 3 continuous) state that if $x$ is PE then $Q_n(u)$ (or $D_n(u)$) must be partially persistently exciting of degree $n$, and if $(x,u)$ is PE then $Q_{n+1}(u)$ (or $D_{n+1}(u)$) must be partially PE of degree $n+m$. The sufficiency theorems (Theorems 2 and 4) run in the opposite direction: full PE of the stacked input implies PE of the state, and full PE of the $n+1$ stack implies PE of the state-input pair. In the single-input case the necessary and sufficient conditions coincide, giving the explicit description of sufficiently rich signals in Lemma 6; in the multi-input case the set of sufficiently rich signals is bracketed between full PE and partial PE of degree $n$ in Lemma 7. Numerical examples show that using a weaker sufficient condition or a stronger necessary condition fails.
Load-bearing premise
The continuous-time necessity proof assumes that a certain stacked matrix built from sampled derivatives has full row rank, but in the parameter range $d' < d(n-1)$ that matrix can have more rows than columns, so full row rank is impossible; if that rank argument cannot be repaired, the new continuous-time necessary condition loses its proof.
Editorial extensions
If this is right
- For any stable reachable single-input system, an input is sufficiently rich if and only if its recent $n$-shift window (or $n$ derivatives in continuous time) is persistently exciting, so richness can be certified without knowing the system matrices.
- The same condition with $n+1$ shifts or derivatives certifies persistent excitation of the full state-input regressor $(x,u)$, the regressor used in many data-driven and adaptive schemes.
- In multi-input systems, full PE of $Q_n(u)$ is sufficient and PE of $x$ forces at least partial PE of degree $n$; these two conditions bracket the set of sufficiently rich inputs.
- Because the same proof architecture covers discrete and continuous time, conditions previously stated separately for time shifts and for derivatives are shown to be one unified result.
- Numerical examples show the conditions cannot be improved without adding knowledge of the specific system: weakening the sufficient stack condition or strengthening the necessary PPE condition produces inputs that fail to excite.
Reading between the lines
- Editorial inference: if the rank issue in the continuous-time necessity lemma is repairable, the same sampling construction could yield finite-time, quantitative certificates for continuous-time systems, connecting classical PE certificates to finite-horizon data-driven ones.
- Editorial inference: the gap between the two inclusions in Lemma 7 likely depends on how the input channels enter through the matrix $B$; a system-dependent invariant, probably the controllability index, may determine the minimal stack length needed for multi-input sufficient richness.
- Editorial inference: the shift-derivative analogy suggests that nonlinear versions of the result, if they exist, would need a time-varying or Lipschitz replacement for Lemmas 2 and 4, where the critical stack length would depend on the system's rate of variation rather than on the state dimension alone.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies sufficient richness for linear time-invariant systems. For discrete- and continuous-time stable, reachable systems with output x or (x,u), it claims necessary and sufficient conditions on the input for the output to be persistently exciting: PE of the state forces the n-window stack (or derivative stack) of the input to be partially PE of degree n (or n+m for state-input), and full PE of that stack is sufficient. The paper also characterizes the single-input sufficiently rich set exactly and gives inclusions for multi-input systems. The main novelty claimed is the multi-input necessary direction, which relies on two lemmas (Lemma 2 and Lemma 4) asserting that the partial-PE degree of a stacked signal does not increase when the window length grows.
Significance. If the results were correct, the paper would unify and extend existing sufficient-richness conditions, in particular giving the first necessary conditions for multi-input systems and a clean single-input characterization. The paper is clearly written, uses a unifying notation, and includes numerical examples demonstrating tightness. However, the central new necessary conditions for multi-input systems rest on rank-nullity arguments in Lemmas 2 and 4 that are invalid: the matrices involved do not have full row rank in exactly the regime where the lemmas are applied, and the displayed kernel dimensions can be negative. The proofs of several theorem statements are also explicitly omitted or only sketched (Theorem 3(ii) is not proved at all). Consequently, the main claimed contribution is not established as written. The sufficient-condition directions and the single-input characterization appear more robust, but those are largely known or are not the paper's principal novelty.
major comments (4)
- [Appendix VI-B, Eq. (43)] Lemma 2 is not proved correctly. The matrix [M − \tilde E] has dimensions nd(k−n+1) × (kd + d′(k−n+1)). The proof asserts that it has full row rank and computes dim ker = kd + d′(k−n+1) − nd(k−n+1) = (k−n)(d+d′−nd) + d′. Full row rank is impossible whenever the number of rows exceeds the number of columns, which is exactly the regime d′ ≤ d(n−1) for the multi-input applications. For example, with d=2, n=3, d′=1, k=4, the matrix is 12×10 and the displayed formula gives a kernel dimension of −2. Therefore the conclusion d′′ ≤ d′ is unsupported, and the necessity direction of Theorem 1, which invokes Lemma 2 with d=m and d′=n′≤n−1, is not established for multi-input systems.
- [Appendix VI-D, Eq. (66)] Lemma 4 has the same defect as Lemma 2. The matrix [M − \tilde E] has ndN rows and Nd + d′(N+1) columns, and the proof asserts full row rank, leading to d′′ = N(d+d′−nd) + d′. For d=2, n=3, d′=1, N=2, the matrix is 12×7 and the claimed kernel dimension is negative. Since Theorem 3 (continuous-time necessity) invokes Lemma 4 with d=m and d′=n′≤n−1, the new continuous-time necessary conditions for multi-input systems are unsupported.
- [Section III-B and Appendix VI-H] Theorem 3(ii) is stated as a theorem but its proof is not given: Appendix VI-H ends with 'We omit the second statement'. Similarly, Theorem 1(ii) is only sketched (Appendix VI-F) with the final step omitted, and Theorem 4(ii) is sketched with explicit omissions (Appendix VI-I). A published theorem requires a complete or at least a rigorous proof of all its claims; an explicit omission is not acceptable in a journal submission. This is particularly serious because those second statements are part of the claimed characterization for the (x,u) output.
- [Section III-C, Lemmas 6 and 7] The characterization of the sufficiently rich sets depends directly on the necessity and sufficiency theorems. Since Theorems 1 and 3 are not established for multi-input systems, the inclusions in Lemma 7 for the multi-input classes rest on unproved necessary directions. The single-input characterization in Lemma 6 may survive, but the paper's broader claim to characterize sufficiently rich inputs for multi-input systems is not supported by the current proofs.
minor comments (4)
- [Throughout] There are several typographical and terminology issues, such as 'Shur' in place of 'Schur' in Remarks 5 and 7, and the inconsistent use of Q_n(u) in place of D_n(u) in Remark 8.
- [Section IV, Figures 1 and 2] Two different plots are both labelled 'Figure 1' in the text; the second should be Figure 2. The captions are also very brief and do not fully explain the plotted quantities.
- [Section II, Definition 5] The discrete-time PE definition uses a sum over τ = t, ..., t+T, but the window length should be specified consistently with the continuous-time definition; a short clarification would avoid ambiguity.
- [Appendix VI-F, Eq. (82)] In the proof of Theorem 1(ii), the matrix dimensions of the stacked vector (u_{τ+1}, U_τ, ..., U_{τ-(K-1)n}) are not explicitly given; adding the dimensions would make the argument easier to follow.
Circularity Check
No significant circularity: the paper's PE/SR results are derived from explicit system equations and standard external lemmas, not from fitted data or self-referential definitions.
full rationale
This paper derives its PE conditions from first principles: the state-space equations (13), reachability/stability assumptions, and explicit proofs in the appendix. The central theorems are proven directly: Theorem 1 proceeds by contraposition through Lemma 2, Theorem 2 through a feedback decomposition, Theorem 3 through the continuous-time analogue Lemma 4, and Theorem 4 through derivative-based arguments. The definitions of PE, PPE, and SR are standard and not mutually defining: SR is defined as the property that an input makes a given output PE, and the later characterizations of SR sets (Lemma 6 and Lemma 7) are obtained by applying the theorems, not by renaming or assuming the conclusion. No parameter is fitted to any dataset, and no prediction is made from tuned values; the numerical examples are explicit counterexamples illustrating tightness, not fitted predictions. The external references used, e.g., [6, Lemma 4.8.3] and [54, Lemma 2.2], are standard, parameter-free, and not authored by the present authors, so they constitute independent support rather than circular input. The only self-citations are contextual references to prior work on reinforcement learning and are not load-bearing for the main derivation. The reader's rank-nullity objection to Lemma 2 and Lemma 4, if valid, describes a potential correctness flaw in the proof (an invalid dimension count), not circularity: the claimed theorem would be unsupported or false, not equivalent to its assumptions by construction.
Assumptions & free parameters
assumptions (4)
- domain assumption A is Schur (discrete) or Hurwitz (continuous), and (A,B) is reachable, for every system in the classes L_x and L_xu.
- domain assumption All signals are bounded, and continuous-time signals are smooth with bounded derivatives (C_infinity_b), with PE defined by a uniform lower bound over all windows.
- standard math A surjective linear image of a PE signal is PE, and PE of (x,u) implies PE of any surjective constant linear map of (x,u).
- standard math For a reachable pair (A,B), there exists a feedback gain F such that (A+BF,b) is reachable with b in the image of B.
Cite this review
Pith. "Pith review of On Sufficient Richness for Linear Time-Invariant Systems." pith.science (2026). https://pith.science/paper/LDN6IWXH
@misc{pith2026250204062,
author = {Pith},
title = {Pith review of: On Sufficient Richness for Linear Time-Invariant Systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/LDN6IWXH}},
note = {Machine review of arXiv:2502.04062}
}
read the original abstract
Persistent excitation (PE) is a necessary and sufficient condition for uniform exponential parameter convergence in several adaptive, identification, and learning schemes. In this article, we consider, in the context of multi-input linear time-invariant (LTI) systems, the problem of guaranteeing PE of commonly-used regressors by applying a sufficiently rich (SR) input signal. Exploiting the analogies between time shifts and time derivatives, we state simple necessary and sufficient PE conditions for the discrete- and continuous-time frameworks. Moreover, we characterize the shape of the set of SR input signals for both single-input and multi-input systems. Finally, we show with a numerical example that the derived conditions are tight and cannot be improved without including additional knowledge of the considered LTI system.
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