Voltage Control of the Boost Converter: PI vs. Nonlinear Passivity-based Control

Leyan Fang; Robert Gri\~n\'o; Romeo Ortega

REVIEW 2 major objections 5 minor 19 references

Voltage Control of the Boost Converter: PI vs. Nonlinear Passivity-based Control

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Direct voltage PI control of a boost converter cannot stabilize the useful low-current equilibrium: the paper proves it unstable for all PI gains and presents passivity-based substitutes.

desk verdict The instability of the useful equilibrium under direct voltage PI control is solid and worth publishing; the companion stabilizability claim is unproven in this version because the Maple appendix is missing. read the letter →

arxiv 2505.23112 v1 pith:HDZKBY5N submitted 2025-05-29 eess.SY cs.SY

Leyan Fang , Romeo Ortega , Robert Griñó This is my paper

classification eess.SYcs.SY MSC 93C1093D0593D15

keywords boostconvertervoltagecontrolPIpassivity-basednon-minimumphasezerodynamicspowerconvertersnonlinear

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 proves a negative result about the most common control strategy for DC-DC boost converters: closing a PI loop directly on the output voltage cannot stabilize the operating point engineers actually want. With no inductor parasitic resistance the closed-loop system has a single equilibrium and it is unstable for every choice of PI gains; with parasitic resistance there are two equilibria, the low-current/useful one is always unstable, and the only stabilizable one forces impractically large inductor current and integrator state. The paper then recalls three passivity-based voltage-feedback controllers that do stabilize the desired equilibrium with simple tuning, two static nonlinear laws and a PID-PBC with a finite-time state observer. The practical upshot is that direct voltage PI control should be avoided, while the passivity-based alternatives provide a workable substitute.

What carries the argument

The argument runs on the scaled port-Hamiltonian model (2) with damping d1 and d2, the assignable equilibrium set (6) derived from the PID-PBC theory, and the zero dynamics equation (8), whose only nonzero equilibrium is unstable. For the PI analysis, the decisive object is the Jacobian of the closed-loop three-state system and the sign of the constant term a0 of its characteristic polynomial: a0 = -2KI r at the minimal-current equilibrium, which is negative for all gains, while at the maximal-current equilibrium a0 = +2KI r and the remaining Routh-Hurwitz terms are shown positive under the gain bounds (12). The power-balance separatrix surfaces S0 and Su, where the stored energy derivative vanishes, separate trajectories that collapse to zero from those that diverge.

What would settle it

Simulate or build the averaged closed-loop system (2)-(9) with d1 = 0 and any positive PI gains; if the unique equilibrium (d2 y*^2, y*) is locally asymptotically stable for some gains, Proposition 1 is false. Alternatively, for d1 > 0 with condition (7) satisfied, check the zero dynamics (8): if its nonzero equilibrium is stable, Lemma 1 is false.

Watch

Extended reading notes

Core claim

For the averaged, time-scaled boost converter model (2) in closed loop with the voltage PI controller (9), the paper establishes the following dichotomy. If the inductor has no parasitic resistance (d1 = 0), Proposition 1 shows there is a unique equilibrium and its Jacobian always has a root with positive real part, so it is unstable for all KP ≥ 0, KI > 0, and u0. If d1 > 0 and the equilibrium-existence condition d1d2 < 1/(4 y*^2) holds, Proposition 2 shows two equilibria: the minimal-current one is unstable for all gains—its characteristic polynomial has a0 = -2KI r < 0—while the maximal-current one is asymptotically stable provided KP ≥ 1/(2 $d1^{2}$) and KI ≥ (5/16)(d1/y*^2), with a Lyapunov-defined domain of attraction. The paper also proves the zero dynamics of the voltage output is non-minimum phase with an unstable equilibrium (Lemma 1), and, in contrast, presents three PBC schemes that render the desired equilibrium asymptotically stable, one of them using a finite-convergence-time observer to avoid current measurement.

Load-bearing premise

The instability results are proven for the averaged, time-scaled continuous-conduction-mode model (1)-(2), which ignores switching ripple, discrete switching events, parasitics other than the inductor resistance, and load dynamics; if the physical converter departs from this model, the instability conclusions need not carry over.

Editorial extensions

If this is right

Direct voltage PI control of a boost converter is unusable at the operationally meaningful equilibrium: practitioners must keep using indirect current-mode schemes or switch to a passivity-based voltage loop.
When inductor resistance is ignored, no PI gain setting can stabilize the system; the model itself forces a redesign of the controller.
Even with parasitic resistance present, the only stabilizable PI equilibrium has extremely large inductor current and integrator state, which is impractical for real converters.
The two static nonlinear PBC laws (20) and (21) stabilize the desired voltage without knowledge of circuit parameters and with simple tuning gains.
The PID-PBC with observer (28) stabilizes the desired equilibrium using only voltage measurement, with the observer converging in finite time under an interval-excitation condition.

Reading between the lines

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

The instability of the minimal-current equilibrium is a structural consequence of the non-minimum phase zero dynamics, so any direct voltage feedback that does not actively reshape the energy may encounter the same barrier; the paper's PBC laws work precisely by reshaping the interconnection and damping.
The separatrix surface Su that the authors observe, coinciding with the zero-energy-derivative set, suggests a geometric design rule: a controller that places the desired equilibrium on the stable side of this surface could make the low-current point asymptotically stable.
The same PBC design principles and the finite-time observer could plausibly extend to Buck-Boost and Čuk converters, which share the non-minimum phase voltage behavior; the authors list this as future work.

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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 core negative result is real and useful. Direct voltage PI control of the boost converter cannot stabilize the low-current (efficient) equilibrium, and the proof is short and checkable. The companion claim that the high-current equilibrium can be stabilized is currently unverified because the Appendix A Maple computation that carries the proof is missing from this version. That needs to be fixed before the paper is publishable as is.

What's new: Lemma 1 gives the zero dynamics for d1>0 without linearization, which I don't recall seeing elsewhere. Propositions 1 and 2 show that a0 is strictly negative at the useful equilibrium, so local instability holds for all PI gains. This cleanly refutes the stability claim in [1]. The three PBC schemes in Section 4 are recalled from prior work, mostly by the same authors, so they are background rather than new. The simulations illustrate the algebra but don't add much beyond what the equations already say.

Soft spots: the proof of C3, that the maximal-current equilibrium can be stabilized, rests entirely on a Maple computation that is not included. The text says f1 exists and p1,p2,p3 dominate the negative terms, but f1 is never shown and the domination argument is only sketched. Since C4 (domain of attraction) also relies on the same Hurwitz conclusion, the whole positive half of the PI story is unverifiable in this version. That is a completeness problem, not a cosmetic one. Also, inequality (12a) is garbled in the typeset manuscript, leaving the reader to guess the actual condition. The abstract overstates the observer-based PID-PBC as "we present a modified version" when it is actually taken from [2]; the paper only assembles the certainty-equivalent version. All of these are fixable.

The averaged-model assumption is standard, and the instability result is a property of that model. I don't consider that a flaw; it is the right starting point. The paper is honest about its scope.

Who this is for: people working on boost converter control, especially those who have wrestled with why direct voltage PI is so fragile in practice. The negative result will be a useful reference for them. The PBC sections will be familiar to the Ortega school, but they serve as a fair comparison.

Recommendation: send it to peer review. The main instability result deserves to be published, but the missing appendix must be supplied — either the Maple worksheet or a hand-checkable proof of (18) — and the typesetting cleaned up.

Referee Report

2 major / 5 minor

Summary. This paper studies direct voltage PI control of the averaged, scaled boost-converter model (2) and contrasts it with three passivity-based controllers. The main analytical results are: for d1=0, the unique equilibrium of the PI-controlled system is unstable for all nonnegative PI gains (Proposition 1); for d1>0, there are two equilibria and the minimal-current (practical) equilibrium is unstable for all gains, while the high-current equilibrium is claimed to be asymptotically stable under the gain inequalities (12) (Proposition 2). The paper also derives a zero-dynamics equation (Lemma 1) and recalls from prior work two static nonlinear voltage controllers and an observer-based PID-PBC.

Significance. If the claimed stability of the high-current equilibrium is properly verified, the paper contributes a rigorous, parameter-free negative result: the practically relevant low-current equilibrium of a PI voltage-controlled boost converter is locally unstable for every PI gain in the averaged continuous-conduction-mode model. This is a useful and nontrivial caution, and it is derived from the model without fitted parameters. The instability proofs for d1=0 and for the minimal-current equilibrium are concise and checkable. The treatment of the PBC alternatives is largely a review of published results and does not add new design tools; the novelty rests on the PI analysis and the zero-dynamics discussion. Since the positive stability claim C3 currently depends on an absent Maple computation, the significance of the paper cannot be fully assessed until that computation is supplied.

major comments (2)

[3.2.1] The verification of C3 in Proposition 2 is incomplete. The proof of the Hurwitz condition a1a2 - a0 > 0 (Eq. (18)) is deferred to "Appendix A: Maple computation," but Appendix A is empty in this version of the manuscript. The in-text argument refers to an unspecified expression f1 and to terms "which appear in the last right hand side product of f1," so the claimed domination and the derivation of inequalities (12a) and (12b) cannot be checked. Because C3 is the only support for the abstract's statement that the high-current equilibrium "may be rendered stable," and because C4 relies on the same Hurwitz conclusion, this is a load-bearing gap. The appendix, or an analytic proof of (18), must be supplied before the claim can be evaluated.
[2.4] The zero-dynamics statement in Lemma 1, items F2 and F3, is mathematically inaccurate as written. Under the strict inequality (7), the polynomial w(u)=u^2 - u/y_star + d1d2 has two positive roots, not one; at the smaller root the linearization of (8) is unstable, while at the larger root it is stable. The proof's observation that w(u_min)<0 only establishes the existence of two roots and does not justify the singular claim "this equilibrium is unstable." This issue does not affect Propositions 1-2, but it is presented as a new result and should be corrected.

minor comments (5)

[3.2.1] In claim C4, the proof of the domain-of-attraction statement is only sketched: after Eq. (19) it is asserted that V(delta chi)=delta chi^T P delta chi is a Lyapunov function, but no argument shows that the ellipsoid {chi: chi^T P chi <= rho_D} is contained in the region where the derivative is negative for the nonlinear system. Also, Q is said to be in R^{2x2} while A and P are 3x3, and A is written as nabla f(chi3) instead of nabla f(chi_s).
[3.2.1] The typesetting of inequality (12a) is ambiguous; please state clearly whether the intended condition is KP >= d1^2/2 or KP >= 1/(2 d1^2), and check that it matches the domination argument in the proof, which requires KP >= d1^2/2.
[Abstract] The phrase "We carry-out" should read "We carry out."
[Remark 4] The assertion that Proposition 1 disproves Proposition 3 and Theorem 2 of reference [1] would be easier to verify if the authors identified the precise controller and statement in [1] being refuted, since [1] appears to concern current-mode control rather than direct voltage PI control.
[4] Propositions 5 and 6 are quoted from reference [2]; the paper should clarify what, if anything, is new in the "modified version" of the PID-PBC beyond the result already reported in [2].

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the PI instability analysis is self-contained; the PBC sections rely on prior self-citations only as background alternatives.

full rationale

The central PI analysis is a direct mathematical derivation from the averaged boost-converter model (2) and the PI controller (9). Proposition 1 computes the closed-loop Jacobian and its characteristic polynomial, obtaining a0 = -KI < 0 at the unique equilibrium for d1 = 0, which violates the Routh-Hurwitz condition and proves instability for all PI gains. Proposition 2 repeats the computation for d1 > 0 and obtains a0 = -2KIr < 0 at the minimal-current equilibrium, again proving unconditional local instability. These are algebraic consequences of the model and controller, with no fitted parameters and no input quantity renamed as a prediction. The companion stability claim C3 for the maximal-current equilibrium is deferred to an absent Maple computation in Appendix A ('In Appendix A we present the Maple calculations that compute (18) multiplied by a positive factor'); this is an omitted-proof completeness problem that affects verifiability, but it is not circularity, because inequality (18) is an independent algebraic condition rather than an assumed input. The PBC stability propositions in Section 4 are recalled from prior work by overlapping authors ([14], [19], [2]) and are stated without proof, but they are presented as background alternatives and do not support the new PI instability results; hence the self-citations are not load-bearing for the paper's central derivation. Fact 2 cites [13, Proposition B.1] for the assignable equilibrium set, a standard algebraic fact that is also directly verifiable, so this citation does not smuggle in the target conclusion. Overall, the derivation chain of the main instability theorem is self-contained, and the self-citations are peripheral. The missing Appendix A is the main weakness, but it is a correctness/completeness concern, not a circular-reasoning concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central instability result depends only on standard Lyapunov analysis of the averaged model; no free parameters are fit to data, and no new physical entities are introduced. The main modeling burden is the validity of the averaged CCM model (1), plus the local nature of linearization-based stability tests.

assumptions (5)

domain assumption The averaged boost converter model (1)-(2) is a valid representation of the physical switched converter in continuous conduction mode.
Section 2.1 introduces equations (1) as the average model; all subsequent equilibrium and stability results are derived for this model.
standard math Lyapunov's indirect method and the Routh-Hurwitz criterion provide valid local stability conclusions.
Used in Propositions 1 and 2, citing Khalil [6, Theorem 4.7].
domain assumption The load is a purely resistive conductance G>0 and the source is a constant voltage with possible series resistance R.
Model (1) has no load dynamics; the practical judgment about which equilibrium is useful depends on this resistive load assumption.
domain assumption The equilibrium existence condition d1 d2 < 1/(4 y*^2) holds for the d1>0 analysis.
Proposition 2 explicitly assumes condition (7); the two-equilibrium structure exists only under this condition.
domain assumption Assumption 1 (interval excitation) holds for the FCT observer of Proposition 5.
Recalled from [2]; the PID-PBC result requires (22), which is stated but not established for the boost converter in this paper.

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Cite this review

Pith. "Pith review of Voltage Control of the Boost Converter: PI vs. Nonlinear Passivity-based Control." pith.science (2026). https://pith.science/paper/HDZKBY5N

@misc{pith2026250523112,
  author       = {Pith},
  title        = {Pith review of: Voltage Control of the Boost Converter: PI vs. Nonlinear Passivity-based Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDZKBY5N}},
  note         = {Machine review of arXiv:2505.23112}
}

read the original abstract

We carry-out a detailed analysis of direct voltage control of a Boost converter feeding a simple resistive load. First, we prove that using a classical PI control to stabilize a desired equilibrium leads to a very complicated dynamic behavior consisting of two equilibrium points, one of them always unstable for all PI gains and circuit parameter values. Interestingly, the second equilibrium point may be rendered stable -- but for all tuning gains leading to an extremely large value of the circuit current and the controller integrator state. Moreover, if we neglect the resistive effect of the inductor, there is only one equilibrium and it is always unstable. From a practical point of view, it is important to note that the only useful equilibrium point is that of minimum current and that, in addition, there is always a resistive component in the inductor either by its parasitic resistance or by the resistive component of the output impedance of the previous stage. In opposition to this troublesome scenario we recall three nonlinear voltage-feedback controllers, that ensure asymptotic stability of the desired equilibrium with simple gain tuning rules, an easily defined domain of attraction and smooth transient behavior. Two of them are very simple, nonlinear, static voltage feedback rules, while the third one is a variation of the PID scheme called PID-Passivity-based Control (PBC). In its original formulation PID-PBC requires full state measurement, but we present a modified version that incorporates a current observer. All three nonlinear controllers are designed following the principles of PBC, which has had enormous success in many engineering applications.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Alvarez and G

J. Alvarez and G. Espinosa, Stability of current-mode control for DC-DC power converters, Systems and Control Letters, vol. 45, pp. 113-119, 2002

work page 2002

[2] [2]

Bobtsov, R

A. Bobtsov, R. Ortega, N. Nikolayev and W. He, A globally stable practically implementable PI passivity-based controller for switched power converters, Int. J. on Adaptive Control and Signal Processing, vol. 35, no. 11, pp. 2155-2174, 2021

work page 2021

[3] [3]

Fliess, Generalized controller canonical forms for linear and nonlinear dynamics, IEEE Trans

M. Fliess, Generalized controller canonical forms for linear and nonlinear dynamics, IEEE Trans. Automatic Control, vol. 35, pp. 994-1001, 1990

work page 1990

[4] [4]

Hernandez-Gomez, R

M. Hernandez-Gomez, R. Ortega, F. Lamnabhi-Lagarrigue and G. Escobar, Adaptive PI stabilization of switched power converters, IEEE Trans. Control Systems Technology, vol. 18, no. 3, pp. 688-698, 2008

work page 2008

[5] [5]

Isidori, Nonlinear Control Systems, Springer-Verlag, 1995

A. Isidori, Nonlinear Control Systems, Springer-Verlag, 1995

work page 1995

[6] [6]

Khalil, Nonlinear Systems, Third Ed., Pearson, 2005

H. Khalil, Nonlinear Systems, Third Ed., Pearson, 2005

work page 2005

[7] [7]

Kreisselmeier and G

G. Kreisselmeier and G. Rietze-Augst, Richness and excitation on an interval–with application to continuous-time adaptive control, IEEE Transactions on Automatic Control, vol. 35, no. 2, pp. 165-171, 1990

work page 1990

[8] [8]

R. D. Middlebrook and S. ˇCuk, A general unified approach to modelling switching-sonverter power stages, PESC, pp. 18-34, 1976

work page 1976