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arxiv: 2604.20242 · v1 · submitted 2026-04-22 · 📡 eess.SY · cs.SY

Controlling the \'{C}uk Converter using Piecewise Linear Lyapunov Functions

Aleksandra Leki\'c , Nikola Petrovi\'c , Du\v{s}an Stipanovi\'c This is my paper

Pith reviewed 2026-05-10 00:06 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords Ćuk converterpiecewise linear Lyapunov functionsswitching control lawcontinuous conduction modestabilizationpower converter control
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The pith

Switching control laws for the Ćuk converter can be designed using piecewise linear Lyapunov functions built from varying numbers of state variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to create a switching control law for the Ćuk converter while it operates in continuous conduction mode. The law relies on piecewise linear Lyapunov functions, which can be made from different combinations of the converter's state variables. The choice of how many state variables to include changes the control performance. Simulations in the paper demonstrate several such constructions and their results.

Core claim

The authors establish that piecewise linear Lyapunov functions, constructed using different numbers of state variables, can be used to design a switching control law that stabilizes the Ćuk converter dynamics in continuous conduction mode, with the construction process illustrated through representative simulations.

What carries the argument

Piecewise linear Lyapunov functions built from subsets of state variables, which serve as the basis for deriving the switching control law.

If this is right

  • The designed control law stabilizes the converter in continuous conduction mode.
  • Using fewer state variables in the Lyapunov function affects the system's performance.
  • Multiple constructions of the piecewise linear Lyapunov functions are feasible.
  • Simulations validate the approach for various choices of state variables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar piecewise linear approaches might apply to controlling other switched power converters.
  • Simplifying the Lyapunov function by using fewer variables could reduce computational requirements in implementation.
  • The method leaves open whether it extends to discontinuous conduction mode or other operating conditions.

Load-bearing premise

That the piecewise linear Lyapunov functions derived from state variable subsets will produce a switching law capable of stabilizing the converter's dynamics.

What would settle it

If the closed-loop system trajectories fail to converge to the equilibrium point or exhibit instability under the switching law derived from the piecewise linear Lyapunov function, the claim would be falsified.

Figures

Figures reproduced from arXiv: 2604.20242 by Aleksandra Leki\'c, Du\v{s}an Stipanovi\'c, Nikola Petrovi\'c.

Figure 1
Figure 1. Figure 1: Ćuk converter. Since we want not only to stabilize the switched system but also to control the ripple, a control law based on the switching among multiple piecewise linear Lyapunov functions [1], [8] having polytopic level sets, is designed in this paper. The earlier results reported in [1], [8] provide different algorithms for the fast construction and computation of the level set polytopes thus proving f… view at source ↗
Figure 2
Figure 2. Figure 2: Ćuk converters' time diagrams, with applied control using PLLF with controlling variables: 𝑖𝐿 and 𝑖𝐿 : left - during first 5ms after circuit's startup and right - during the time interval 4.95ms - 5ms [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ćuk converters' time diagrams, with applied control using PLLF with controlling variables: 𝑖𝐿 and 𝑖𝐿 : left - during first 5ms after circuit's startup and right - during the time interval 4.95ms - 5ms [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ćuk converters' time diagrams, with applied control using PLLF with controlling variables 𝑖𝐿 , 𝑖𝐿 and 𝑣𝐶 : left - during first 5ms after circuit's startup and right - during the time interval 4.95ms - 5ms [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ćuk converters' time diagrams, with applied control using PLLF with controlling variables 𝑖𝐿 , 𝑖𝐿 , 𝑣𝐶 and 𝑣𝐶 : left - during first5ms after circuit's startup and right - during the time interval 4.95ms - 5ms [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

In this paper we design a switching control law for the \'Cuk converter in the continuous conduction mode using piecewise linear Lyapunov functions. These Lyapunov functions can be constructed using different number of state variables affecting the system's performance. In the paper, some representative simulations covering construction of different piecewise Lyapunov functions, are provided.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper designs a switching control law for the Ćuk converter operating in continuous conduction mode (CCM) by means of piecewise linear (PWL) Lyapunov functions. These functions are constructed from different subsets of the state variables, with the number of states affecting closed-loop performance; representative simulations are provided to illustrate the resulting trajectories.

Significance. If the PWL Lyapunov functions are shown to be valid certificates (i.e., positive definite with strictly negative Lie derivatives along the closed-loop affine vector fields in each polyhedral region), the approach would supply a tunable, low-complexity method for stabilizing switched power converters and could extend standard Lyapunov-based switching design to systems where full-state information is costly. The simulations indicate practical feasibility, but the absence of explicit stability verification limits the immediate contribution.

major comments (1)
  1. [Lyapunov function construction and simulation results sections] The central claim requires that each constructed PWL Lyapunov function certifies asymptotic stability of the switched CCM dynamics. The manuscript reports only simulation trajectories and does not provide the necessary verification that the Lie derivative of V is negative in every region defined by the switching law (or that the decrease condition holds across switches). Without such checks—e.g., analytic sign analysis, per-region LMI feasibility, or common Lyapunov decrease—the simulations cannot rule out hidden instability or chattering outside the tested initial conditions.
minor comments (1)
  1. The abstract states that performance is affected by the number of state variables but does not quantify the observed trade-offs (e.g., settling time or ripple versus number of states); adding a brief table or sentence would clarify the contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting the need for explicit stability verification. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Lyapunov function construction and simulation results sections] The central claim requires that each constructed PWL Lyapunov function certifies asymptotic stability of the switched CCM dynamics. The manuscript reports only simulation trajectories and does not provide the necessary verification that the Lie derivative of V is negative in every region defined by the switching law (or that the decrease condition holds across switches). Without such checks—e.g., analytic sign analysis, per-region LMI feasibility, or common Lyapunov decrease—the simulations cannot rule out hidden instability or chattering outside the tested initial conditions.

    Authors: We agree that the manuscript currently presents only representative simulation trajectories without explicit verification that each PWL Lyapunov function satisfies the required conditions for asymptotic stability. The switching laws are constructed from the PWL functions with the intent that the Lie derivative is negative in each polyhedral region, but this is not demonstrated analytically or numerically in the current version. In the revised manuscript we will add a dedicated subsection providing per-region verification of the decrease condition (via sign analysis of the affine vector fields or feasibility of the associated LMIs for the chosen gains). We will also explicitly note that the PWL functions are continuous by construction, so V does not increase at switching instants; this, combined with the strict decrease inside regions, precludes chattering. These additions will directly address the concern and strengthen the theoretical claims. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Lyapunov construction for switched systems

full rationale

The paper constructs piecewise-linear Lyapunov functions from subsets of the Ćuk converter states to synthesize a switching law in CCM, then illustrates behavior via simulation. This follows the conventional template of positing a candidate V (here piecewise linear), invoking the standard decrease condition along affine vector fields in each region, and verifying via trajectories. No equation reduces to a prior fit or self-citation by construction; the functions are chosen rather than derived from the target stability result itself. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. Standard Lyapunov stability theory is implicitly used.

pith-pipeline@v0.9.0 · 5349 in / 954 out tokens · 23530 ms · 2026-05-10T00:06:01.902714+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    In general, there are many approaches designed to guarantee converters' outputs satisfying the prescribed specifications

    INTRODUCTION Designing stabilizing controllers for DC -DC converters has been a very popular and demanding research topic [3] over a number of years. In general, there are many approaches designed to guarantee converters' outputs satisfying the prescribed specifications. Most popular approaches are based on first performing the averaging of the converters...

    work page

  2. [2]

    Operation of the Ćuk converter Ćuk con verter is one of the most complex fourth order DC -DC converters

    CONTROL OF THE ĆUK CONVERTER 2.1. Operation of the Ćuk converter Ćuk con verter is one of the most complex fourth order DC -DC converters. It is constructed using two switches being transistor S and a diode D. The switchings produce four operating subsystems and two of them are operating in the CCM. First operating subsystem occ urs when the PWM signal on...

    work page

  3. [3]

    1” when the switch conducts and “0

    SIMULATION RESULTS In order to control the converter, we apply the piecewise linear Lyapunov functions control design approach with the following parameters: , , , and . In this section we will focus on constructing polytope with different number of state variables, so that the set * +. This control produces Lyapunov -like functions and it may be linked t...

    work page

  4. [4]

    The procedure for constructing these functions with respect to the desired equilibrium and ripple values is g iven in detail

    CONCLUSION In this paper we provide a construction of the piecewise linear Lyapunov functions to be used for control of the Ćuk converter. The procedure for constructing these functions with respect to the desired equilibrium and ripple values is g iven in detail. It is shown how the initial circuit transient behavior changes when the Lyapunov functions a...

    work page

  5. [5]

    ACKNOWLEDGEMENT This work is supported by project TR33020 of the Ministry of Education, Science and Technological Development of the Republic of Serbia

    work page

  6. [6]

    Nonquadratic Lyapunov functions for robust control,

    F. Blanchini, “Nonquadratic Lyapunov functions for robust control,” Automatica, vol. 31, no. 3, pp. 45 1– 461, 1995

    work page 1995

  7. [7]

    PI and sliding mode control of a Ćuk converter,

    Z. Chen, “ PI and sliding mode control of a Ćuk converter,” IEEE Transactions on Power Electronics, vol. 27, no. 8, pp. 3695–3703, 2012

    work page 2012

  8. [8]

    A review on stability analysis methods for switching mode power converters ,

    A. El Aroudi, D. Giaouris, H. H. -C. Iu, and I. A. Hiskens, “A review on stability analysis methods for switching mode power converters ,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 5, no. 3, pp. 302–315, 2015

    work page 2015

  9. [9]

    R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics . Springer Science & Business Media, 2001

    work page 2001

  10. [10]

    Hysteresis switching control of the Ćuk converter ,

    A. Leki ć and D. M. Stipanovi ć, “ Hysteresis switching control of the Ćuk converter ,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 11, pp. 2048–2061, 2016. 5

    work page 2048

  11. [11]

    Controlling the Ćuk converter using polytopic lyapunov functions,

    A. Leki ć and D. M. Stipanovi ć, “ Controlling the Ćuk converter using polytopic lyapunov functions,”IEEE Transactions on Circuits and Systems II: Express Briefs – submitted for review, 2017

    work page 2017

  12. [12]

    Switching stabilizability for continuous -time uncertain switched linear systems,

    H. Lin and P. J. Antsaklis, “ Switching stabilizability for continuous -time uncertain switched linear systems,” IEEE Transactions on Automatic Control, vol. 52, no. 4, pp. 633–646, 2007

    work page 2007

  13. [13]

    On the construction of piecewise linear Lyapunov functions ,

    Y. Ohta, “ On the construction of piecewise linear Lyapunov functions ,” in Proceedings of the 40th IEEE Conference on Decision and Control, 2001, vol. 3. IEEE, 2001, pp. 2173–2178

    work page 2001

  14. [14]

    Design of a digital PID regulator based on look -up tables for control of high-frequency DC -DC converters ,

    A. Prodic and D. Maksimovic, “ Design of a digital PID regulator based on look -up tables for control of high-frequency DC -DC converters ,” in Computers in Power Electronics, 2002. Proceedings. 2002 IEEE Workshop on. IEEE, 2002, pp. 18–22

    work page 2002

  15. [15]

    Chaos from a current - programmed Ćuk converter ,

    C. Tse and W . Chan, “ Chaos from a current - programmed Ćuk converter ,” International Journal of Circuit Theory and Applications, vol. 23, no. 3, pp. 217–225, 1995

    work page 1995

  16. [16]

    V. I. Vorotnikov, Partial stability and control . Springer Science & Business Media, 2012

    work page 2012

  17. [17]

    Constrained switching stabilization of a dc -dc boost converter using piecewise -linear lyapunov functions ,

    C. Yfoulis, D. Giaouris, S. Voutetakis, and S. Papadopoulou, “ Constrained switching stabilization of a dc -dc boost converter using piecewise -linear lyapunov functions ,” in 2013 21st Mediterranean Conference on Control & Automation (MED). IEEE, 2013, pp. 814–823

    work page 2013

  18. [18]

    A numerical technique for the stability analysis of linear switched systems,

    C. A. Yf oulis and R. Shorten*, “ A numerical technique for the stability analysis of linear switched systems,” International Journal of Control, vol. 77, no. 11, pp. 1019–1039, 2004. Fig. 4. Ćuk converters' time diagrams, with applied control using PLLF with controlling variables 𝑖𝐿 , 𝑖𝐿 and 𝑣𝐶 : left - during first 5ms after circuit's startup and right ...

    work page 2004