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Envelope Characterizations for Trajectories of the \'Cuk Converter

Aleksandra Lekić, Dušan Stipanović

arXiv:2610.04451v1eess.SYcs.SY

TL;DR

The paper addresses characterization of stability and trajectory-envelope boundaries for the Cuk converter under time- and state-dependent switching. It discretizes the converter’s subsystem equations and uses Lyapunov functions to analyze stability and periodic behavior, deriving envelope bounds and DCM conditions for both switching types.

  • Problem

    The Cuk converter’s complex transient behavior and multiple operating modes motivate a complete analytical account of trajectory envelopes and stability across switching types.

    Source: §I · p. 1

  • Method

    The paper discretizes the converter’s subsystem differential equations into difference equations and constructs discrete Lyapunov functions for time- and state-dependent switching.

    Source: §I · p. 1, §IV · p. 5

  • Results

    The analysis provides exact transient envelopes and stability and periodic-behavior characterizations for both switching types, including conditions for discontinuous conduction modes.

    Source: §IV · p. 5

  • Takeaways & Limitations

    Envelope bounds yield worst-case inductor-current and output-voltage ripple estimates as functions of switching period and duty ratio, enabling analytical design without time-domain simulation.

    Source: §IV · p. 5

Abstract

from arXiv · show

In this paper, a complete characterization of the stability and the boundaries of the envelopes for trajectories of the \'Cuk converter is provided in the cases of time and state-dependent switching. In particular, difference equations are developed and used to compute boundaries for the system's response envelopes as well as Lyapunov functions to show the system's stability in both continuous and discontinuous regimes of operation. Each case is illustrated by multiple simulations.

I. INTRODUCTION

The paper addresses stability and periodic behavior of the Cuk converter across time- and state-dependent switching, where complex transients and multiple operating modes motivate analysis beyond control design. It discretizes the converter’s dynamics to characterize trajectory envelopes and stability across operating modes and switching types.

I. INTRODUCTION

  • The converter’s complex transient behavior, two marginally stable subsystems, and four possible operating modes have sustained research interest.Prior work includes single-variable, sliding-mode, hysteresis, and polytopic control approaches.Source: §I · p. 1
  • The work extends prior hysteresis-control studies from time-domain simulations and experiments to analytical characterization of trajectory envelopes and stability conditions.The analysis uses discretized continuous-time models and Lyapunov functions for discrete-time systems.Source: §I · p. 1
  • The proposed envelope characterization covers all four operating modes and both switching types, with closed-form DCM boundary conditions and a unified Lyapunov framework.The paper’s analysis proceeds through circuit models, switching behavior, and stability conditions.Source: §I · p. 1

II. CONVERTER’S OPERATING MODES AND STATE-SPACE

The converter has four operating modes represented by linear time-invariant state-space models, arising from two continuous-conduction switch configurations and two discontinuous modes. Its Qn-PWM operation permits switching sequences that combine these modes, and the mode matrices differ in whether they are singular.

II. CONVERTER’S OPERATING MODES AND STATE-SPACE

  • All four converter operating modes are represented in linear time-invariant form, with the state vector defined for the mode models.The modes are indexed i ∈ {1, 2, 3, 4}.Source: §II · p. 1
  • In the two CCM modes, either switch S conducts while diode D is off, or S is off while D conducts.These configurations use matrices A1, B1 and A2, B2, respectively.Source: §II · pp. 1–2
  • DCVM begins when capacitor C1 voltage reaches zero while S conducts, forcing D on and defining matrices A3 and B3.Source: §II · p. 2
  • DICM begins when the sum of inductor currents reaches zero, turning D off while S is already off; both switches are then off.This mode uses matrices A4 and B4.Source: §II · p. 2
  • Matrices A1, A3, and A4 are singular, whereas A2 is nonsingular; the exact ODE solution is simplified when Ai is nonsingular.The paper gives the exact solution under an initial condition x(0).Source: §II · p. 2
  • Qn-PWM operation permits four within-period switch sequences, including sequences that combine DCVM and DICM.The listed sequences also include the two-switch transition for CCM and the three-state sequence for each individual discontinuous mode.Source: §II · p. 2

III. ENVELOPE SWITCHING CHARACTERIZATION

The section characterizes trajectory-envelope boundaries under switching and uses discrete-time Lyapunov analysis to assess stability. It reports stability across the considered CCM intervals and boundary stability for the simulated DCVM case.

III. ENVELOPE SWITCHING CHARACTERIZATION

  • With time-dependent switching of period T and duty ratio d, ∆1 = dT and ∆2 = (1−d)T; x1 and x2 mark the ends of the two CCM intervals on the trajectory envelope.The states correspond respectively to switch-on/diode-off and switch-off/diode-on intervals.Source: §III · p. 2
  • The Fig. 2 transient plots the difference-equation envelope border alongside a PLECS simulation for the stated converter parameters, fS = 100 kHz, and d = 0.8.Source: §III · p. 2, Figure 2 · p. 2
  • The CCM system is reported stable for every pair of switching intervals (∆1, ∆2), as indicated by determinant and maximal-eigenvalue diagrams.The diagrams show the dependence of det A and |λ|max A on (∆1, ∆2).Source: §III · p. 2, Figure 3 · p. 2, Figure 4 · p. 3
  • The matrix function B identifies x1 and x3 as critical: large ∆1 makes their values large, forcing operation into discontinuous modes.Source: §III · p. 2
  • For DCVM, simulations give max(|λ|max(A(∆1, ∆2, ∆3))) < 1 and max(det(A(∆1, ∆2, ∆3))) = 0.9970, implying at least boundary stability.The same analysis can be performed for DICM and DQRM.Source: §III · p. 3
  • The same Lyapunov function applies to x1 and x2; its matrix P also establishes discrete-time stability in CCM when inequalities (13) hold.For the Fig. 2 operating point, Q is positive definite, confirming CCM operation.Source: §III · p. 3
  • The paper reports all Q eigenvalues inside the unit circle in CCM, whereas a determined Lyapunov function in DCM yields at least one eigenvalue outside it.Source: §III · p. 3

A. Time-dependent switching

For time-dependent switching, a fixed period T induces periodic behavior, and Lyapunov conditions are tested across duty ratios and switching periods. DCM boundaries and negative-eigenvalue regions provide operating-mode checks.

A. Time-dependent switching

  • A constant switching period T induces periodic behavior, while Lyapunov conditions are tested by fixing duty ratios at d1 = 0.01, d2 = 0.99, and d3 = 0.5.The analysis varies d at fixed T using the same Lyapunov matrix.Source: A. Time-dependent switching · p. 3
  • Fig. 6 maps DCVM and DICM borders in (d, T); negative-eigenvalue dots mostly avoid the CCM region, aside from cases near the border attributed to numerical errors.The red border marks DCVM and the green border DICM.Source: A. Time-dependent switching · p. 3, Figure 6 · p. 4
  • For a desired T and duty ratio d, CCM operation is checked by locating the operating point below both boundary curves; approaching the red or green border warns of DCVM or DICM entry, respectively.Near the red border, reduce T or adjust d.Source: Figure 7 · p. 4, A. Time-dependent switching · p. 4
  • Figure 7 depicts the DCM border in (∆1, ∆2) coordinates, characterized by λ(Q(∆1, ∆2)) < 0, and identifies it as a red border line.Source: A. Time-dependent switching · p. 3, Figure 7 · p. 4

B. State-dependent switching

State-dependent switching uses state-threshold crossings and can produce nonhomogeneous interval durations. Its envelope and Lyapunov analyses characterize ripple bounds and operating conditions across conduction modes.

B. State-dependent switching

  • Hysteresis switching sets the switching instant when the state crosses a threshold, and the derived envelope bounds give guaranteed ripple limits without time-domain simulation.Source: B. State-dependent switching · p. 4
  • State-dependent switching yields a global result because interval durations can be nonhomogeneous, with results similar to time-dependent switching.Source: B. State-dependent switching · p. 4
  • For fixed ∆2, formal stability is tested with LMIs using a matrix P computed for three values of ∆1.Source: B. State-dependent switching · p. 4
  • The CCM-calculated Lyapunov function indicates stability in CCM and non-stability in DCM, although the converter itself remains stable in DCM.This distinction provides information about the converter’s operating conditions.Source: Figure 8 · p. 5, B. State-dependent switching · p. 5
  • Fig. 8 uses blue circles for cases with a negative eigenvalue of Ax and shows the same property as Fig. 7.Source: Figure 8 · p. 5, B. State-dependent switching · p. 5
  • The matrix Q can act as an online operating-mode indicator: negative eigenvalues signal transition toward DCM before it occurs, supporting preemptive adjustment of d or T.The stated computation requires matrix-vector multiplications at each switching instant.Source: B. State-dependent switching · p. 5

IV. CONCLUSION

The conclusion presents discretized subsystem equations and discrete Lyapunov functions for analyzing the Cuk converter under time- and state-dependent switching. The envelope bounds support ripple estimates, while Lyapunov-based DCM detection is suited to lightweight supervisory control.

IV. CONCLUSION

  • Discretized subsystem equations provide an exact transient envelope, and discrete Lyapunov functions are constructed for both switching types to determine DCM operating conditions.Source: §IV · p. 5
  • The envelope bounds yield worst-case estimates of inductor-current and output-voltage ripple as functions of switching period and duty ratio, supporting analytical design without time-domain simulation.Source: §IV · p. 5
  • Lyapunov-based DCM detection provides a computationally lightweight criterion for real-time supervisory control.Source: §IV · p. 5
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