{"id":"11e87979-9f6c-433f-9158-281b5149c2ba","arxiv_id":"2411.14678","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"PID controllers are reinterpreted as the combination of a state-feedback loop and a first-order lumped-disturbance observer, with gains tied to two bandwidth parameters.","lead":"This paper argues that a PID controller can be viewed as two separate parts: state feedback that stabilizes the system and an observer that cancels unknown disturbances. A reader may find it useful as a compact, ADRC-style mental model for tuning PID gains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"State-dependent lumped disturbance f breaks the transfer-function argument: Eqs. (17)-(20) treat f as exogenous, so the claimed bandwidth separation and two-bandwidth tuning rule are unsupported for the stated class f = f(x,...,u,t).","rationale":"Read in good faith, the paper's core algebraic decomposition is a valid identity for the nominal chain of integrators plus a first-order disturbance observer, and for exogenous f the transfer function (20) correctly shows that the observer attenuates the disturbance at frequencies below omega_f. The second-order PID specialization in Eq. (34) is internally consistent and supports the reinterpretation. The load-bearing weakness is the jump from this linear, exogenous-disturbance analysis to a general tuning rule for the lumped disturbances defined in Eq. (10), which explicitly includes state- and input-dependent terms. Once f depends on x or u, the Laplace-domain factorization is no longer a closed-loop model; the observer modifies the effective plant dynamics in a way that is not captured by separate bandwidths. The concrete linear test F(s) = kX(s)+D(s) shows that the closed-loop characteristic polynomial contains a destabilizing term -ks, so the proposed separation argument can fail for finite k even when both the controller and observer polynomials are Hurwitz. This is not an objection to the reinterpretation itself, which is defensible under an explicit exogenous or slowly varying disturbance assumption, but it is a real gap in the claimed tuning methodology. I agree with the reader's identification of the disturbance-estimation assumption as the weakest point, and I would keep the CONDITIONAL verdict: the central claim is plausible but needs an added assumption or proof restricting f, plus validation, before it can be accepted as stated.","tokens_in":9667,"tokens_out":15381,"duration_ms":163924,"concrete_test":"Substitute the linear state-dependent disturbance f = k x + d(t) into Eq. (10) with the proposed controller (11)-(15), and compute the roots of the closed-loop characteristic polynomial (s+omega_f)(s+omega)^n - k s = 0 for n=2, omega=1, omega_f=10, and k = 0, 1, 10, 100. If any root has positive real part for finite k, the transfer-function derivation in Eq. (20) does not cover the stated class of lumped disturbances, and the two-bandwidth tuning rule is invalid for that case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central tuning claim rests on the Laplace-domain model in Section III. The observer transfer function (17) and the closed-loop transfer function (20) are derived by treating f as an independent input. But Eq. (10) explicitly defines f = f(x,...,x^(n-1),u,t), i.e., f may depend on the state and control input. For such f, the identity F_tilde(s) = [s/(s+omega_f)] F(s) is not a valid closed-loop input-output relation, and the separation of a controller bandwidth and an observer bandwidth does not follow from the linear analysis. A minimal counterexample makes the issue precise. Let f = kx + d(t) in Eq. (10) and apply the proposed controller (11)-(15). The closed-loop characteristic equation is (s+omega_f)P(s) - k s = 0, where P(s) = s^n + sum a_i s^i. For fixed Hurwitz P and positive omega_f, choosing k sufficiently large drives roots into the right half-plane. Thus increasing omega_f does not monotonically improve disturbance rejection for state-dependent disturbances, and the claimed independence of the two bandwidths is not a theorem for the class of systems the paper emphasizes. The paper's algebraic reinterpretation of PID as homogeneous feedback plus disturbance observer survives for exogenous or slowly varying f, but the tuning prescription is not supported without an explicit assumption restricting how f depends on state and input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10040,"tokens_out":10146,"duration_ms":94571,"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":[{"comment":"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":"Sections II and III, Eqs. (2), (20)"},{"comment":"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.","section":"Section III-B, Eqs. (21)-(25)"}],"minor_comments":[{"comment":"There is a typo 'As As shown in Figure 1' in the text preceding Eq. (43).","section":"Section IV-B"},{"comment":"In step 3 of the design summary, the word 'mFrenetay' appears to be a typo for 'may', and the sentence should be rephrased.","section":"Section V"},{"comment":"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":"Section II, Eq. (2)"},{"comment":"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.","section":"Section IV-B, Eq. (51)"},{"comment":"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.","section":"Overall"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central reinterpretation is likely familiar to ADRC researchers, so the novelty is primarily pedagogical. The reliance on the authors' own prior work [12] for the VTOL error model, without derivation, is a self-citation concern that should be checked. The paper fits a control-engineering journal if the revision tightens the disturbance assumptions and makes the noise analysis rigorous; otherwise the main tuning claim is not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThere 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.\n\nIf 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.\n\nRecommendation: engage with it; ask for a revision that narrows the claim or adds the missing analysis, but don't desk-reject.","headline":"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.","tokens_in":10478,"tokens_out":6678,"would_cite":false,"duration_ms":60608,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"PID control is best read as state feedback plus a disturbance observer, with the gains fixed by two bandwidths.","keywords":["PID control","disturbance observer","lumped disturbance compensation","state feedback","bandwidth parameterization","lateral control","trajectory tracking","vehicle steering"],"falsifier":"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.","tokens_in":9468,"feed_emoji":"🎛️","tokens_out":12470,"duration_ms":102856,"temperature":0.7,"pith_summary":"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.","feed_headline":"PID is two jobs: stabilize, then cancel the disturbance","feed_subtitle":"PI/PID gains follow from two bandwidths, with aircraft and car-steering examples.","key_machinery":"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.","core_discovery":"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 $\\omega_f$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the observer-based disturbance-rejection paradigm from which the paper draws its lumped-disturbance compensation.","marker":"[10]"},{"why":"Provides the bandwidth-parameterization idea of placing all homogeneous poles at one value, giving the single controller bandwidth.","marker":"[11]"},{"why":"Supplies the VTOL trajectory-tracking error model used in the aircraft example.","marker":"[12]"},{"why":"Defines the Frenet-Serret frame used to express the vehicle's lateral error relative to the reference path.","marker":"[13]"},{"why":"Defines the companion Serret formulas used for the path-matching geometry in the lateral-control example.","marker":"[14]"},{"why":"Supplies the input-output feedback linearization tool cited in the paper's general design procedure for error models.","marker":"[16]"}],"fun_headline_variants":["PID is state feedback plus a lumped disturbance observer","PID = homogeneous control + disturbance observer","Two mechanisms, one controller: PID is feedback + observer","PID explained: stabilize the system, then cancel disturbance","From PID to state feedback and disturbance compensation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["PID is state feedback plus a lumped disturbance observer","PID = homogeneous control + disturbance observer","Two mechanisms, one controller: PID is feedback + observer","PID explained: stabilize the system, then cancel disturbance","From PID to state feedback and disturbance compensation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000804,"raw_usage":{"total_tokens":3533,"prompt_tokens":948,"completion_tokens":2585,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":2513}},"tokens_in":564,"tokens_out":2585,"duration_ms":17219,"temperature":1.0,"reasoning_tokens":2513,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:02:06.984415+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"From pid to active disturbance rejection contr ol,","cited_arxiv_id":null,"evidence_quote":"Supplies the observer-based disturbance-rejection paradigm from which the paper draws its lumped-disturbance compensation."},{"cited_title":"Scaling and bandwidth-parameterization base d controller tun- ing,","cited_arxiv_id":null,"evidence_quote":"Provides the bandwidth-parameterization idea of placing all homogeneous poles at one value, giving the single controller bandwidth."},{"cited_title":"Trajectory trackin g control based on generalized rodrigues parameter for underactuated vtol uavs,","cited_arxiv_id":null,"evidence_quote":"Supplies the VTOL trajectory-tracking error model used in the aircraft example."},{"cited_title":"Sur les courbes ` a double courbure","cited_arxiv_id":null,"evidence_quote":"Defines the Frenet-Serret frame used to express the vehicle's lateral error relative to the reference path."},{"cited_title":"Sur quelques formules relatives ` a la th ´ eorie des courbes ` a double courbure","cited_arxiv_id":null,"evidence_quote":"Defines the companion Serret formulas used for the path-matching geometry in the lateral-control example."},{"cited_title":"Khalil, Nonlinear systems (Third Edition)","cited_arxiv_id":null,"evidence_quote":"Supplies the input-output feedback linearization tool cited in the paper's general design procedure for error models."}],"review_version":1}