Stability of MIMO PID With Backward Differences Under Fast Sampling: An Exact Spectral Criterion

Peng Zhang; Yanyi Wu; Zhaobo Liu

Stability of MIMO PID With Backward Differences Under Fast Sampling: An Exact Spectral Criterion

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that the standard backward-difference PID derivative can destabilize a stable multivariable loop at arbitrarily fast sampling, and gives an exact spectral test for when it does.

desk verdict The exact fast-mode criterion rho(CBK_D)<1 for MIMO backward-difference PID is correct, clearly proved, and worth publishing, with only its explicitly stated boundary exclusions limiting the otherwise clean iff. read the letter →

arxiv 2608.08318 v1 pith:ZTK5NVLQ submitted 2026-08-08 math.OC

Zhaobo Liu , Yanyi Wu , Peng Zhang This is my paper · ORCID

classification math.OC MSC 93C0593C3593C5793D20

keywords derivativefeedbackfastmodesfinitedifferencesmultivariablePIDcontrolsampled-datasingularperturbationbackwarddifferencespectralcriterion

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

The paper asks whether the most common digital way to implement derivative action—replacing $\dot y(t)$ by the backward difference $(y_k-y_{k-1})/h$—keeps a well-designed continuous-time MIMO PID loop stable as the sampling period $h$ shrinks. The answer it proves is 'not necessarily': the previous output sample becomes a controller state, and in the exact lifted discrete model it creates fast eigenvalues that converge to $-\operatorname{spec}(CBK_D)$ rather than to the continuous-time spectrum. Consequently, if any eigenvalue of $CBK_D$ lies outside the unit disk, the sampled loop is unstable for every sufficiently small $h$, no matter how stable the ideal loop is. Away from boundary spectra the criterion is exact: stability for all small sampling periods holds if and only if the ideal closed-loop matrix $A_{\rm id}$ is Hurwitz and $CBK_D$ is Schur. This matters because it turns a practical rule of thumb—'sample faster to get closer to continuous time'—into a checkable spectral condition on the implementation itself.

What carries the argument

The engine is the exact lifted discrete-time matrix $A_h$ from (8), whose state includes the stored previous output sample $r_k=y_{k-1}$. As $h\to0^+$, $A_h$ tends to $A_0$ with spectrum $\{1\}^{n+p}\cup(-\operatorname{spec}(M_D))$, where $M_D=CBK_D$; the factor $h^{-1}\Gamma_h\to I_n$ keeps the derivative-memory block alive. A first-order perturbation reduction, applied to the semisimple cluster at 1, shows the slow eigenvalues move as $1+h\mu+o(h)$ with $\mu\in\operatorname{spec}(A_{\rm id})$, while the simple-eigenvalue perturbation formula gives the boundary drift of a fast eigenvalue at $-\nu$, $|\nu|=1$, in Proposition 2. This split turns a sampled-data problem into two finite-dimensional spectral checks: Hurwitz stability of $A_{\rm id}$ and Schur stability of $M_D$.

What would settle it

If the claimed instability criterion were false, the scalar integrator $\dot x=u$, $y=x$ with $k_p,k_i>0$ and $k_d>1$ would be a counterexample: the ideal matrix $A_{\rm id}$ is Hurwitz while $|k_d|>1$, and direct computation of the eigenvalues of the lifted matrix $A_h$ for $h=0.1,0.01,0.001$ gives $\rho(A_h)>1$ in every case, confirming the prediction that no sufficiently small sampling period restores stability.

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Extended reading notes

Core claim

For the plant $\dot x=Ax+Bu$, $y=Cx$ with the output PID law $u=-K_P y-K_I z-K_D \dot y$, $\dot z=y$, the paper's central discovery is a spectral split of the sampled closed-loop dynamics. With the exact lifted transition matrix $A_h$ over one sampling interval, $n+p$ eigenvalues track the ideal loop, $\lambda_j(h)=1+h\mu_j+o(h)$ with $\mu_j\in\operatorname{spec}(A_{\rm id})$, while the remaining $p$ eigenvalues converge to $-\operatorname{spec}(CBK_D)$. Under $E=I_n+BK_DC$ nonsingular, no imaginary-axis eigenvalue of $A_{\rm id}$, and no unit-circle eigenvalue of $CBK_D$, Theorem 2 states that $A_h$ is Schur for all sufficiently small $h$ if and only if $A_{\rm id}$ is Hurwitz and $\rho(CBK_D)<1$. If $A_{\rm id}$ has a positive-real-part eigenvalue or $CBK_D$ has a modulus-greater-than-one eigenvalue, the sampled loop is unstable for all small $h$; the relative-degree case $CB=0$ makes the extra condition automatic.

Load-bearing premise

The exact if-and-only-if criterion holds only when the ideal derivative feedback is well posed (a certain matrix is nonsingular) and when neither the ideal loop nor the derivative-memory matrix has a borderline eigenvalue; at the boundary, higher-order terms decide stability, so the simple criterion is silent.

Editorial extensions

If this is right

Any PID design for a plant with $CB\neq0$ must verify $\rho(CBK_D)<1$; exponential stability of the ideal continuous-time law alone is not enough to certify the backward-difference implementation.
Decreasing $h$ improves the slow approximation—$\rho(A_h)=1+h\alpha_{\rm id}+o(h)$ when $M_D$ is Schur—but cannot move the fast eigenvalues, so smaller sampling periods cannot rescue an unstable derivative channel.
For plants with relative degree at least two ($CB=0$), the obstruction vanishes: $M_D=0$ and every Hurwitz ideal loop yields a stable fast-sampled backward-difference PID.
In the SISO case the fast-sampling check reduces to $|CB\,k_D|<1$, whereas ideal well-posedness only requires $1+CB\,k_D\neq0$, so the sampled constraint is genuinely stronger.
When a fast limit sits on the unit circle ($|\nu|=1$), Proposition 2 gives the first-order drift that decides inward or outward crossing, and both outcomes can occur.

Reading between the lines

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

Extension: For alternative discrete differentiators—forward differences, filtered derivatives, or predictor-based estimators—the same lifted-state construction would likely replace $M_D=CBK_D$ by a realization-dependent fast matrix, so an analogous spectral condition should be re-derived for each implementation rather than inherited.
Extension: The criterion opens a design route the paper does not pursue: when $CB\neq0$, choose $K_D$ so that $CBK_D$ is Schur, turning the derivative gain into a tuning knob for the sampling-induced fast mode.
Extension: For nonlinear or time-varying plants there is no fixed matrix $CBK_D$, so the fast-sampling limit may show a similar but state-dependent obstruction; a testable prediction is that very small-$h$ simulations of a plant with direct feedthrough in $\dot y$ would exhibit persistent high-frequency oscillations even as $h\to0$.

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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

Bottom line: the paper is correct and worth publishing. It proves that the standard MIMO backward-difference PID realization has fast sampled eigenvalues converging to -spec(CBK_D), so ideal-loop stability plus Schur stability of CBK_D is necessary and sufficient for small-h stability, away from boundary spectra. The mechanism is real: the previous output sample is controller memory, and it does not disappear as h -> 0. This also explains why relative-degree-two treatments with CB=0 miss the obstruction, since then the fast spectrum collapses to zero. That is a genuine new result relative to the cited artificial-delay/LMI literature.

Strengths. I checked the lifted matrix (8), the spectral factorization in Proposition 1, the semisimple reduction W A1 V = Aid, and the boundary drift formula (12). They are consistent. The MIMO counterexample with D0 is well chosen because it shows the condition is spectral, not entrywise. Proposition 2 is a nice extra: it gives first-order drift for a simple fast limit on the unit circle and honestly notes that second-order terms decide when the drift vanishes. The paper does not oversell the criterion; Remark 5 correctly says it is realization-specific.

Soft spots are mostly at the boundaries, and the paper labels them. The iff in Theorem 2(3) excludes imaginary-axis Aid eigenvalues and unit-circle MD eigenvalues. Proposition 2 handles only simple unit-circle fast limits; repeated eigenvalues or vanishing first-order drift are left open. That is a limitation, but it is explicit and does not weaken the interior result. E nonsingular is a standard well-posedness condition for derivative feedback. The only thing I would push on is the novelty claim: 'first such characterization' is credible for MIMO, but the authors should do a more careful SISO prior-art search, since scalar backward-difference integrator examples may be folklore. That is a revision request, not a substantive flaw. Citations are appropriate; no self-citation is load-bearing.

Who benefits: sampled-data and PID control researchers, especially people who teach backward-difference derivative action as a harmless implementation. It deserves a serious referee and likely acceptance after a modest literature pass. I would cite it and would bring it to reading group.

Referee Report

0 major / 5 minor

Summary. This note studies the fast-sampling limit of a sampled-data MIMO PID controller in which the derivative term is realized by a backward difference. The authors derive the exact lifted discrete-time model (8) and show that its state matrix has n+p eigenvalues near 1 with first-order drift given by the ideal continuous-time PID matrix A_id, plus p fast eigenvalues converging to the negated spectrum of M_D = C B K_D (Proposition 1, Theorem 1). The main result, Theorem 2, states that away from boundary spectra the sampled loop is Schur stable for all sufficiently small h if and only if A_id is Hurwitz and M_D is Schur. Minimal counterexamples illustrate the obstruction, a boundary-drift formula is provided for simple fast eigenvalues on the unit circle, and the relative-degree case C B = 0 is shown to be obstruction-free.

Significance. The result is a precise and useful caution for sampled PID implementation: a backward-difference realization can be unstable for arbitrarily small h even when the ideal continuous-time loop is exponentially stable, unless the simple matrix condition rho(C B K_D) < 1 holds. The paper's method is exact lifting followed by standard perturbation theory, and the appendix proofs are self-contained and checkable. The main theorem is honestly stated with explicit spectral exclusions, and Proposition 2 partially resolves the simple unit-circle boundary case. The scalar and MIMO examples are minimal and clearly isolate the mechanism. The contribution is appropriately scoped: Remark 5 correctly notes that the condition is realization-specific, not a universal property of all digital PID implementations.

minor comments (5)

[Authors and affiliations] The author block contains 'Y . Wu' with a space before the period; this should read 'Y. Wu'.
[Section II-B, Eq. (8)] Several expressions involving h are typeset ambiguously, for example 'h−1Γh' should be read and typeset as h^{-1} Γ_h; please ensure all such expressions use clear superscript notation.
[Section IV-C, Eq. (13)] In Eq. (13) and the surrounding text, the terms written as 'h2kiλ' should be typeset as h^2 k_i λ; the missing superscripts appear repeatedly and should be corrected.
[Proposition 2] The notation 'q, ℓ∈ Cp' should read 'q, ℓ \in C^p', and similarly 'ℓ∗' should be typeset as ℓ^* for clarity.
[Appendix A] In the proof of Proposition 1, the Schur-complement calculation is performed for λ \neq 0 and then extended by polynomial continuation; this is correct, but a sentence explicitly noting that the extension is justified by continuity of the determinant would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spectral criterion is derived from the exact lifted matrix and standard perturbation theory; no fitted inputs, no load-bearing self-citations, and no definitional reduction.

full rationale

The paper's central claim, Theorem 2(3), is derived self-containedly from the exact lifted discrete-time model (equation (8)), which is obtained directly from the plant discretization (7) and the controller update law (6). No parameter is fitted to data, and no empirical quantity is renamed as a prediction; the matrix MD = CBK_D is forced by the lifted structure rather than chosen to reproduce the instability. Proposition 1 computes the limit spectrum via block-triangularization and Sylvester's determinant identity, giving spec(A0) = {1}^{n+p} union (-spec(MD)); this is a mathematical consequence of the block structure of A0, not an imposed ansatz. Theorem 1's slow eigenvalue estimate uses the standard semisimple perturbation formula (Lemma 1, citing Kato's textbook) with explicit dual bases V and W, and the direct calculation W A1 V = A_id identifies the first-order reduction; this is a coordinate calculation, not a definitional equivalence. The sufficiency and necessity directions of Theorem 2 follow from root continuity and the stated spectral exclusions, which are transparently disclosed: if A_id has no imaginary-axis eigenvalues, then non-Hurwitz implies a positive real-part eigenvalue, and if MD has no unit-circle eigenvalues, then non-Schur implies a spectral radius greater than one. The boundary cases with eigenvalues on the imaginary axis or unit circle are explicitly excluded from the iff statement and treated separately in Proposition 2, so the theorem does not overclaim. References [11] and [17] are standard external texts on perturbation theory and are used only for the textbook eigenvalue perturbation formula, not as a self-citation chain. No load-bearing self-citation, uniqueness import, or ansatz smuggling is present. The finding is therefore no significant circularity.

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

All ingredients of the criterion come from the lifted model: no constants are fitted to data, and no new entities are postulated. The gains and plant matrices are problem inputs, not fitted parameters. The paper's own assumptions, including E nonsingular, A_id Hurwitz, spectral exclusions, and the ZOH/backward-difference realization, are listed as domain assumptions; the only mathematical imports are standard perturbation and determinant identities.

assumptions (6)

domain assumption E = I_n + B K_D C is nonsingular, equivalently I_p + C B K_D is nonsingular.
Invoked in Section II-A to make the ideal derivative-feedback law an ODE and in Appendix A to keep the eigenvalue 1 separated from -spec(M_D). Without it A_id is not defined as in Eq. (4).
domain assumption The ideal closed-loop matrix A_id is exponentially stable.
The problem starts from a stable continuous-time PID design and asks whether fast sampling preserves stability; A_id Hurwitz is one half of the criterion, assumed in the problem statement and Corollary 2.
domain assumption Spectral exclusions: A_id has no eigenvalues on the imaginary axis and M_D has no eigenvalues on the unit circle.
Theorem 2(3) and Corollary 2 require these exclusions for the if-and-only-if statement. Proposition 2 partially handles simple unit-circle fast limits, but second-order terms would be needed for full boundary classification.
domain assumption The digital realization uses zero-order hold (7), rectangular integration (5), and backward difference (6).
The lifted matrix (8) is exact only for this realization. Remark 5 states that filtered derivatives or other differentiators change the fast spectrum, so the criterion is implementation-specific rather than universal.
standard math Standard matrix perturbation theory and determinant identities are used.
Appendix A uses Kato's reduction [11] for the semisimple eigenvalue cluster at 1 and Sylvester's determinant identity for the spectral factorization. These are background theorems, not assumptions tuned to the result.
domain assumption The previous output sample r_k is an independent controller state with arbitrary initial value r_0.
Stability is defined over all initial triples (x_0, z_0, r_0) in Section II-B, which is the standard internal stability requirement for sampled controllers with memory. This is what makes the extra p eigenvalues genuine modes.

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Cite this review

Pith. "Pith review of Stability of MIMO PID With Backward Differences Under Fast Sampling: An Exact Spectral Criterion." pith.science (2026). https://pith.science/paper/ZTK5NVLQ

@misc{pith2026260808318,
  author       = {Pith},
  title        = {Pith review of: Stability of MIMO PID With Backward Differences Under Fast Sampling: An Exact Spectral Criterion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTK5NVLQ}},
  note         = {Machine review of arXiv:2608.08318}
}

read the original abstract

Backward differences are a standard digital realization of derivative action in proportional-integral-derivative control. This note proves that, for multivariable state-space plants, this implementation can be unstable no matter how small the sampling period is. The exact lifted model reveals fast eigenvalues created by the stored previous output sample. Away from boundary spectra, stability is equivalent to ideal loop stability plus Schur stability of the product of the output, input, and derivative gain matrices.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

K. J. ˚Astr¨om and T. H ¨agglund, PID Controllers: Theory, Design, and Tuning, 2nd ed. Research Triangle Park, NC, USA: ISA, 1995

work page 1995

[2] [2]

The future of PID control,

K. J. ˚Astr¨om and T. H ¨agglund, “The future of PID control,” Control Eng. Pract., vol. 9, no. 11, pp. 1163–1175, 2001

work page 2001

[3] [3]

K. J. ˚Astr¨om and T. H ¨agglund, Advanced PID Control . Research Triangle Park, NC, USA: ISA, 2006

work page 2006

[4] [4]

K. J. ˚Astr¨om and B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd ed. Upper Saddle River, NJ, USA: Prentice-Hall, 1997

work page 1997

[5] [5]

Quasi feedback forms for differential-algebraic systems,

T. Berger, A. Ilchmann, and S. Trenn, “Quasi feedback forms for differential-algebraic systems,” IMA J. Math. Control Inf., vol. 39, no. 2, pp. 533–563, 2022

work page 2022

[6] [6]

Chen and B

T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems . London, U.K.: Springer, 1995

work page 1995

[7] [7]

Stabilization by using artificial delays: An LMI approach,

E. Fridman and L. Shaikhet, “Stabilization by using artificial delays: An LMI approach,” Automatica, vol. 81, pp. 429–437, 2017

work page 2017

[8] [8]

Robust sampled-data stabiliza- tion of linear systems: An input delay approach,

E. Fridman, A. Seuret, and J.-P. Richard, “Robust sampled-data stabiliza- tion of linear systems: An input delay approach,” Automatica, vol. 40, no. 8, pp. 1441–1446, 2004

work page 2004