Reinterpreting PID Controller From the Perspective of State Feedback and Lumped Disturbance Compensation

Xinyu Shi

REVIEW 2 major objections 5 minor 17 references

Reinterpreting PID Controller From the Perspective of State Feedback and Lumped Disturbance Compensation

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read PID control is best read as state feedback plus a disturbance observer, with the gains fixed by two bandwidths.

desk verdict A pedagogically nice but largely known PID-as-ADRC derivation whose main tuning claim overreaches because the Laplace analysis treats a state-dependent disturbance as exogenous. read the letter →

arxiv 2411.14678 v1 pith:VX3CYIPC submitted 2024-11-22 eess.SY cs.SYmath.OC

Xinyu Shi This is my paper · ORCID

classification eess.SYcs.SYmath.OC

keywords PIDcontroldisturbanceobserverlumpedcompensationstatefeedbackbandwidthparameterizationlateraltrajectorytrackingvehiclesteering

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 a PID controller is not simply three gains to be tuned by trial and error; it is a mechanism with two jobs. For a system written as $x^{(n)}=f+bu$, the control splits into a state-feedback term that stabilizes the disturbance-free homogeneous system and an observer term that estimates and cancels the lumped disturbance $f$. From that split, the PI and PID gain formulas follow algebraically: the gains are combinations of a controller bandwidth $\omega$ and an observer bandwidth $\omega_f$, so tuning reduces to choosing two meaningful numbers. The paper also traces how measurement noise enters the loop and recommends keeping the controller bandwidth below the observer bandwidth. Two worked examples, a VTOL trajectory tracker and a distance-domain vehicle lateral controller, show how the split is applied.

What carries the argument

The load-bearing object is the decomposition of the solution of a non-homogeneous linear differential equation into a zero-input response and a zero-state response. The zero-input response converges when the characteristic polynomial $s^n+\sum a_i s^i$ is Hurwitz, meaning all its roots lie in the left half-plane, and the zero-state response is ultimately bounded by $\limsup_{t\to\infty}|f(t)|/\omega^n$, so the paper reads the state as an $n$-th-order low-pass filtered version of $f$. The observer is a first-order low-pass filter with transfer function $\hat{F}/F=\omega_f/(s+\omega_f)$, built from the measurable state $x^{(n-1)}$ and the integral of the homogeneous control, and the whole closed loop is the series connection of the homogeneous-system transfer function $1/(s^n+\sum a_i s^i)$ and the observer error filter $s/(s+\omega_f)$. This mechanism turns disturbance rejection into a bandwidth choice and turns PID tuning into pole placement plus observer bandwidth selection.

What would settle it

Drive a second-order plant with a sinusoidal disturbance whose frequency is well above the observer bandwidth $\omega_f$; the predicted disturbance-to-output transfer $s/((s^2+2\omega s+\omega^2)(s+\omega_f))$ says the disturbance will be weakly attenuated, so the tracking error should remain large. If a PID tuned by this paper's formulas eliminates that error anyway, the observer mechanism is not what is doing the work.

Watch

Extended reading notes

Core claim

The central claim is that a PI/PID controller is equivalent to a homogeneous state-feedback controller in series with a first-order lumped-disturbance observer. For $x^{(n)}=f+bu$, the paper writes $u=(u_x-\hat{f})/b$, with $u_x=-\sum_{i=0}^{n-1}a_i x^{(i)}$ and observer $\hat{f}=\omega_f(x^{(n-1)}-\int_0^t u_x\,dt)$. Substituting the model into the observer shows that $\hat{f}$ is a low-pass filtered version of $f$ with transfer function $\omega_f/(s+\omega_f)$, so the closed-loop map from $f$ to $x$ becomes $s/((s^n+\sum_{i=0}^{n-1}a_i s^i)(s+\omega_f))$. For a first-order plant the formulas reduce to a PI controller with $k_p=a_0$ and $k_i=\omega_f a_0$; for a second-order plant they reduce to a PID controller with $k_d=a_1+\omega_f$, $k_p=a_0+\omega_f a_1$, and $k_i=\omega_f a_0$. When the homogeneous poles are all placed at $-\omega$, the whole controller has just the two bandwidth parameters $\omega$ and $\omega_f$.

Load-bearing premise

The derivation assumes the plant is exactly a linear nth-order system $x^{(n)}=f+bu$ with known order $n$ and a known nonzero input coefficient $b$, and that the lumped disturbance $f$ varies slowly enough for the first-order observer to track it.

Editorial extensions

If this is right

Tuning a PID reduces to choosing a controller bandwidth $\omega$ and an observer bandwidth $\omega_f$; the three gains are then fixed by the algebra of the split.
The integral term in a PID is the disturbance observer's accumulated estimate, not merely a steady-state-error patch.
Measurement noise favors keeping the controller bandwidth below the observer bandwidth, because high-frequency noise enters through the feedback path and is partly filtered by the observer's integrator.
The same two-part construction extends to higher-order generalized PID controllers and to observers other than the first-order one.
In the vehicle example, working in the distance domain gives a lateral-steering law whose observer integrates over arc length, so the same tuning logic applies to path following without time-domain speed planning.

Reading between the lines

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

One step beyond the paper: replacing the first-order observer with a higher-order observer should improve high-frequency disturbance rejection while leaving the homogeneous controller untouched, and that upgrade is directly testable.
The paper's bound $\limsup_{t\to\infty}|f(t)|/\omega^n$ suggests a quantitative design rule: choose $\omega_f$ from the disturbance's high-frequency content and $\omega$ from actuator limits, a spectrum-based calculation the paper does not carry out.
The distance-domain reparameterization used for vehicle steering could transfer to other geometric path-following problems, such as marine or mobile-robot guidance, where time-domain dynamics obscure the error cascade.

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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

You can read this in an afternoon. It re-derives PID as two parts: a state-feedback term that stabilizes the homogeneous system and a first-order disturbance observer that estimates and cancels the lumped disturbance f. The gain formulas come out as kd = a1 + omega_f, kp = a0 + omega_f*a1, ki = omega_f*a0, which is exactly Gao's bandwidth parameterization. The paper is upfront about this: it cites Han and Gao and says the reinterpretation is 'inspired by' ADRC. The derivation path through the solution of a non-homogeneous ODE is different from the usual ESO presentation, and the bound on the zero-state response is clean. The distance-domain lateral controller for vehicles is a nice touch.

The soft spots are real but not fatal. The main one: Section III derives the observer and closed-loop transfer functions by treating f as an exogenous signal, but Eq. (10) explicitly allows f to depend on the state and input. For state-dependent f, the identity F_tilde(s) = [s/(s+omega_f)]F(s) is not a valid input-output relation. A simple counterexample, f = kx + d(t), gives a closed-loop characteristic equation (s+omega_f)(s+a0) - ks = 0; a large positive k destabilizes the loop no matter how large omega_f is. So the claimed separation of controller and observer bandwidths is not a theorem for the stated class. The paper would need a nonlinear stability analysis or an explicit slow-variation assumption to support the tuning rule. As a heuristic, the rule is fine; as a formal claim, it overshoots.

The noise analysis in Eqs. (21)-(25) is also rougher than the rest: notation slips and an approximation that is hard to follow. The conclusion (set omega < omega_f) is sensible, but the derivation is not the paper's strong point. The two design examples illustrate the workflow but include no simulation or experimental data, so they remain templates.

There is no data-fitting or hidden fitted parameters; the gains come from two user-chosen bandwidths. The self-citation to the VTOL error model is peripheral and not a problem.

If I were editing, I would send it to review. A referee can push for the disturbance-dependence assumption to be stated precisely and for a simulation to validate the examples. The paper is honest, mathematically clean where it actually makes assumptions, and useful as a pedagogical bridge between PID and ADRC. It just claims a bit more than the linear analysis proves.

Recommendation: engage with it; ask for a revision that narrows the claim or adds the missing analysis, but don't desk-reject.

Referee Report

2 major / 5 minor

Summary. The paper analyzes the motion of solutions to non-homogeneous linear differential equations and uses this to reinterpret PID control as a combination of homogeneous state feedback and a first-order lumped-disturbance observer. The authors derive an observer transfer function, propose a two-bandwidth tuning rule (controller bandwidth omega and observer bandwidth omega_f), extend the idea to PI and PID forms in Eqs. (30) and (34), and illustrate the approach on a VTOL trajectory tracking problem and a vehicle lateral control problem. The central claim is that a PID controller essentially comprises a homogeneous controller and a disturbance observer, so that its gains can be computed from two bandwidths rather than from heuristic loop-shaping.

Significance. If the central claim held as stated, the paper would offer a simple and pedagogically attractive bridge between classical PID and ADRC. The algebraic decomposition into PI/PID forms in Eqs. (30) and (34) is explicit and easy to verify, and the bound in Section II is a clean derivation of the low-pass filtering interpretation of linear systems. The distance-domain modeling in the vehicle example is a creative idea. However, the main theoretical result is not established for the class of state-dependent lumped disturbances that the paper emphasizes, and the noise analysis is too informal to support the tuning recommendations. The paper would be a useful conceptual note after substantial revision, but in its current form the load-bearing transfer-function arguments require an explicit restriction of the disturbance class or a robust stability analysis.

major comments (2)

[Sections II and III, Eqs. (2), (20)] The transfer-function analysis treats f as an exogenous input, but Eq. (10) explicitly allows f = f(x,...,x^(n-1),u,t). For state-dependent f, the closed-loop relation G(s) = s / ((s^n + sum a_i s^i)(s + omega_f)) in Eq. (20) is not a valid input-output transfer function, and the separation principle invoked in Section V does not follow. A minimal counterexample is f = kx + d(t): the closed-loop characteristic equation becomes (s + omega_f)P(s) - k s = 0, where P(s) is the homogeneous polynomial. For any fixed Hurwitz P and positive omega_f, sufficiently large k drives roots into the right half-plane, so increasing omega_f cannot compensate for state-dependent feedback. The paper must either restrict the lumped disturbance to be exogenous or slowly varying, or provide a stability analysis (e.g., small-gain or passivity conditions) that covers state-dependent f.
[Section III-B, Eqs. (21)-(25)] The noise analysis is not self-consistent. Eq. (23) defines uw = -sum_{i=0}^{n-1} a_i w_{n-i}, but w_i is only defined for i = 0,...,n-1, so w_n is undefined; the intended expression is likely uw = -sum_{i=0}^{n-1} a_i w_i. In addition, Eq. (25) is asserted after an approximation w_hat ≈ omega_f w_{n-1} that neglects the integral of uw; the conditions under which this approximation is valid are not given, and the algebra leading to Eq. (25) is not shown in detail. The qualitative claim that high-frequency noise is filtered through the observer is plausible, but it is not a rigorous result in the current form.

minor comments (5)

[Section IV-B] There is a typo 'As As shown in Figure 1' in the text preceding Eq. (43).
[Section V] In step 3 of the design summary, the word 'mFrenetay' appears to be a typo for 'may', and the sentence should be rephrased.
[Section II, Eq. (2)] The statement 'Clearly, when the unknown dynamics f are bounded, the state x is also bounded' is only true if f is an exogenous signal; for f = f(x,...,t) this claim is circular. The paper should distinguish exogenous disturbances from state-dependent f and clarify that the Laplace-domain results apply to the former.
[Section IV-B, Eq. (51)] In the observer definition, the integral over distance is rewritten as ∫_0^t v ux dt; this relies on the change of variables s = ∫ v dt and should be stated explicitly to avoid confusion.
[Overall] The paper presents no simulation or experimental validation of the proposed tuning rule; the examples end at controller design. A small simulation or numerical demonstration would substantially strengthen the practical claims.

Circularity Check

0 steps flagged · score 2.0 of 10

The central PID-as-state-feedback-plus-lumped-disturbance-observer derivation is an algebraic reformulation with no fitted prediction; the only self-citation is peripheral to the main claim.

full rationale

The central derivation chain is self-contained and algebraic. Section III-A proposes u=(1/b)(u_x-\hat f), Eq. (11), with u_x given by Eq. (12) and \hat f by Eq. (15). Substituting into Eq. (10) gives Eq. (14), and the Laplace transforms produce the observer transfer function Eq. (17), the error transfer function Eq. (19), and the closed-loop relation Eq. (20) through direct manipulation; no empirical fitting occurs. Section III-B instantiated for n=1 and n=2 yields Eqs. (30) and (34), which are exact identities connecting PI/PID gains to the design parameters a0, a1, and \omega_f. Thus the paper's 'prediction' is an equivalence between a constructed two-part controller and the classical PID structure; it does not define PID in terms of the conclusion. The only self-citation, 'By simplifying our previous work [12]' in Section IV-A, supplies the VTOL error model; it is an example-specific and peripheral reliance, not the load-bearing step for the PID interpretation, so it does not make the central claim circular. The skeptical issue that f may depend on x and u, as in Eq. (10), is a validity gap in treating F(s) as exogenous in Eqs. (17)-(20), but that is a correctness concern, not a circularity. The score of 2 reflects only the non-load-bearing self-citation; there are no circular steps.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation relies on standard linear systems theory (Laplace transforms, Hurwitz polynomials) and on the ADRC-style plant assumption. The paper introduces two user-selected bandwidths as design parameters, assumes a separation principle between controller and observer, and imports the VTOL error model from the author's prior work. No invented entities are introduced.

free parameters (2)

Controller bandwidth ω
User-selected pole-placement parameter; the tuning scheme sets all homogeneous poles to -ω (Section III-A).
Observer bandwidth ω_f
User-selected bandwidth of the first-order disturbance observer (Eq. 15).

assumptions (6)

domain assumption The plant can be represented as x^(n) = f + b u with known relative degree n and nonzero constant input coefficient b.
Used throughout Section III, especially Eq. (10). This is the standard ADRC plant assumption; the analysis fails if b is unknown or if the system has zeros or unmodeled dynamics.
domain assumption The lumped disturbance f is bounded and slowly varying relative to observer bandwidth ω_f, so that a first-order low-pass observer with transfer function ω_f/(s+ω_f) can estimate it.
Eqs. (15)-(19). The observer error transfer function s/(s+ω_f) has large gain for high-frequency f, so the disturbance compensation degrades if f varies fast.
ad hoc to paper Measurement noise can be treated as high-frequency and the approximation in Eq. (25) (neglecting ω_f/s terms and filtering of \hat w) is valid.
Section III-A, Eqs. (21)-(25). The noise analysis relies on this approximation to conclude that controller bandwidth should be smaller than observer bandwidth; no rigorous bound is given.
ad hoc to paper Separation principle: the homogeneous controller and the disturbance observer can be designed independently.
Section III-A: 'we can simplistically assume that the reference of the homogeneous controller only determines the system's poles, while the observer parameters only determine the observer's bandwidth.' This is stated heuristically, not proven.
ad hoc to paper The VTOL error model from Eq. (37) is obtained by simplifying the authors' prior work [12] without derivation in this paper.
Section IV-A: 'By simplifying our previous work [12], we obtain the error model'. This is a self-citation that supplies a load-bearing model.
domain assumption For the vehicle example, the heading error satisfies eθ ∈ (-π/2, π/2) and the arc-length ratio rs≈1.
Section IV-B, around Eqs. (44)-(48). These assumptions simplify the error model but exclude large tracking errors and sharp paths.

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Cite this review

Pith. "Pith review of Reinterpreting PID Controller From the Perspective of State Feedback and Lumped Disturbance Compensation." pith.science (2026). https://pith.science/paper/VX3CYIPC

@misc{pith2026241114678,
  author       = {Pith},
  title        = {Pith review of: Reinterpreting PID Controller From the Perspective of State Feedback and Lumped Disturbance Compensation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VX3CYIPC}},
  note         = {Machine review of arXiv:2411.14678}
}

read the original abstract

This paper analyzes the motion of solutions to non-homogeneous linear differential equations. It further clarifies that a proportional-integral-derivative (PID) controller essentially comprises two parts: a homogeneous controller and a disturbance observer, which are responsible for stabilizing the homogeneous system and compensating for the lumped disturbances (non-homogeneous components) of the system respectively. Based on this framework, the impact of measurement noise on control performance is examined, and a parameter tuning scheme for the traditional PID controller is provided. Finally, as examples, controllers are designed for two representative control problems: a trajectory tracking controller for an underactuated vertical takeoff and landing (VTOL) aircraft in the time domain, and a lateral controller for a vehicle in the distance domain.

Figures

Figures reproduced from arXiv: 2411.14678 by the authors.

**Figure 2.** Schematic diagram of vehicle lateral path tracking i [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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

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