⚡ electrical engineering

LMI Approach for Sliding Mode Control and Analysis of DC-DC Converters

This paper analyzes the steady-state behavior of DC-DC converters, specifically the Cuk converter, using an equivalent control modeling approach within a sliding mode regime, where stability is assessed via linear matrix inequalities to determine the limits of linear ripple approximation and validated through simulations of practical switching surfaces.

Original authors: Aleksandra Lekić, Dušan Stipanović

Published 2026-04-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Aleksandra Lekić, Dušan Stipanović

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Taming a Chaotic Power Switch

Imagine you have a very complex machine (a Ćuk converter) that takes electricity from a battery and changes its voltage to power your phone or laptop. This machine works by flipping a switch on and off thousands of times a second.

Because the switch is flipping so fast, the electricity inside the machine doesn't flow smoothly; it jitters, ripples, and behaves unpredictably. This is like trying to drive a car where the steering wheel vibrates violently.

The authors of this paper, Aleksandra Lekić and Dušan Stipanović, wanted to answer a simple question: "How can we predict exactly how this machine will behave, even when it's acting crazy?"

They used a mathematical tool called LMI (Linear Matrix Inequalities) to create a "safety net" that tells engineers exactly how much the machine can jitter before it becomes unstable.

The Core Concepts (Translated)

1. The "Sliding Mode" (The Magic Track)

Imagine the machine is a skier going down a mountain. The skier wants to stay on a specific, narrow path (the sliding surface).

The Problem: The wind (electrical noise) keeps trying to blow the skier off the path.
The Solution: The skier has a super-fast reflex. If they drift left, they instantly steer right. If they drift right, they steer left. This creates a "sliding" motion where they stay on the path, even though they are technically zig-zagging.
In the paper: The "zig-zag" is the switch turning on and off. The "path" is the math equation the engineers want the voltage to follow.

2. The "Linear vs. Nonlinear" Problem (The Smooth Road vs. The Bumpy Road)

To analyze the machine, engineers usually try to pretend the road is perfectly smooth (Linear). This makes the math easy.

The Reality: The road is actually full of potholes and bumps (Nonlinear).
The Paper's Trick: The authors say, "Let's pretend the road is smooth, but we will add a 'Bump Factor' to our math."
They calculate exactly how big those bumps can get before the car crashes. If the bumps are small, the "smooth road" math works. If the bumps are huge, the math breaks, and the machine might fail.

3. The "LMI" (The Safety Net Calculator)

This is the fancy math part. Think of LMI as a super-smart calculator that draws a safety net around the "Bump Factor."

The calculator tries to make the net as big as possible.
If the machine's actual behavior stays inside the net, the system is stable (safe).
If the behavior pokes outside the net, the system is unstable (dangerous).

4. The "Ripple" (The Wobbly Water)

When the switch flips, the electricity "ripples" like water in a bucket being shaken.

Engineers often use a "Linear Ripple Approximation," which is like assuming the water waves are perfect, straight lines.
The Paper's Discovery: They found a limit. If you shake the bucket too hard (a large "hysteresis" or switching gap), the waves become chaotic and curved. The "straight line" math no longer works.
Using their LMI safety net, they calculated the exact limit of how hard you can shake the bucket before the math stops working.

The Experiment: Testing Two Different "Tracks"

The authors tested their theory on the Ćuk converter using two different "tracks" (switching surfaces) to see how the machine behaved.

Scenario A: The Simple Track

They tried to control just one thing (the input current).
Result: When the "shaking" (hysteresis) was small, the machine behaved perfectly like the smooth math predicted. But when they increased the shaking, the machine started to wobble wildly, and the "straight line" math failed. The LMI safety net caught this failure and told them, "Stop! You've crossed the limit."

Scenario B: The Complex Track

They tried to control a mix of currents and voltages.
Result: Similar to Scenario A, they found a specific limit. If they stayed within the limit, the machine was stable. If they went over, the machine entered a "discontinuous" mode (like the car losing traction), and the simple math couldn't predict it anymore.

The "Aha!" Moment

The most important takeaway is this: You don't have to guess how big your safety margins should be.

Before this paper, engineers might have guessed, "Let's keep the switching gap small just to be safe."
With this paper, they can now calculate the exact maximum size of that gap.

Analogy: It's like knowing the exact speed limit on a winding road. You don't have to drive at 10 mph just in case; you can drive at 55 mph safely because you know exactly where the curve ends.

Summary for the General Audience

This paper is about taming the chaos of electricity.

The Problem: Power converters switch on and off so fast that they create messy, unpredictable ripples.
The Tool: The authors used a mathematical "safety net" (LMI) to measure exactly how messy the ripples can get.
The Result: They found the precise "tipping point" where simple math stops working and complex chaos begins.
The Benefit: Engineers can now design these power converters to be more efficient and smaller, because they know exactly how much "wiggle room" they have before the system breaks.

In short, they turned a chaotic, jittery electrical system into a predictable, safe, and optimizable machine.

1. Problem Statement

The paper addresses the challenge of analyzing and controlling the steady-state behavior of DC-DC converters, specifically the complex fourth-order Ćuk converter, under Sliding Mode Control (SMC).

The Core Issue: Sliding mode control introduces nonlinear switching behavior. While the "equivalent control" method is used to model the system as a continuous average, the actual system contains nonlinearities and perturbations due to switching.
The Gap: Traditional analysis often relies on the linear ripple approximation, which assumes that state variable ripples (currents and voltages) are linear and that system time constants are negligible compared to the switching period. However, there is no rigorous mathematical method to determine when this approximation holds true or to quantify the limits of its validity for complex, high-order systems like the Ćuk converter.
Objective: To develop a rigorous stability analysis framework that quantifies the bounds of nonlinearity and determines the maximum allowable hysteresis (switching frequency/ripple) for which linear analysis remains valid.

2. Methodology

The authors propose a methodology combining Equivalent Control, Linearization, and Linear Matrix Inequalities (LMI).

A. System Modeling

Equivalent Control: The switching system is modeled using the concept of equivalent control ( $u_{eq}$ ), which represents the average control signal required to keep the system on the sliding surface ( $S(x) = 0$ ).
Linearization & Decomposition: The nonlinear dynamic model is linearized around an equilibrium point ( $x^*$ $x^{*}$ ). The system is then decomposed into:
- A linear part (the Jacobian matrix).
- A nonlinear remainder ( $h(z)$ ), treated as an additive perturbation.
Sector Bounding: The nonlinear remainder is treated as a sector-bounded perturbation. The goal is to find the largest "sector" (bound) within which the nonlinearity resides.

B. Stability Analysis via LMI

Lyapunov Function: A quadratic Lyapunov function candidate $V(z) = z^T P z$ is used to prove stability.
S-Procedure: The authors employ the S-procedure to formulate the stability conditions as an LMI optimization problem.
Optimization Goal: The LMI problem is set up to maximize the sector bound ( $\alpha$ ).
- If the sector bound is large, the system is robustly stable, and the linear approximation is valid over a wider range.
- If the bound is exceeded, the linear ripple approximation fails, and the system may exhibit complex nonlinear dynamics or instability.
Transformation: A nonsingular transformation matrix $T$ (based on eigenvectors) is applied to the state space to maximize the computed sector bound $\alpha$ , providing a tighter and more accurate limit.

C. Application to the Ćuk Converter

The methodology is applied to the Ćuk converter in two configurations:

Unidirectional Switch + Diode: Operates in Continuous Conduction Mode (CCM), Discontinuous Inductor Current Mode (DICM), and Discontinuous Capacitor Voltage Mode (DCVM).
Bidirectional Switches (Synchronous): Operates only in CCM.

Two specific switching surfaces are analyzed:

$S_1(x) = m_1 x_1 - m_5$ (Input current control).
$S_2(x) = m_1 x_1 + m_2 x_2 - m_5$ (Weighted sum of inductor currents).

3. Key Contributions

Rigorous Limit for Linear Ripple Approximation: The paper provides a mathematical derivation for the maximum hysteresis bound ( $\Delta$ ) and the corresponding ripple limit for which the linear ripple approximation is valid. This is derived directly from the maximized sector bound $\alpha$ .
LMI-Based Stability for Switching Systems: It demonstrates how LMI techniques can be effectively used to analyze the stability of high-order, nonlinear switching systems by treating switching effects as sector-bounded perturbations.
Validation of Simulation vs. Theory: The authors validate the theoretical LMI bounds against PLECS simulations, showing that the theoretical sector bound accurately predicts when the system behavior deviates from linearity.
Analysis of Discontinuous Modes: The paper analyzes how the system behaves when entering discontinuous modes (DICM/DCVM) and how the switching surface trajectory changes in these regimes compared to bidirectional implementations.

4. Results

The authors conducted simulations with specific parameters ( $V_{in}=10V$ , $L_1=L_2=1mH$ , $C_1=1\mu F$ , $C_2=20\mu F$ , $R=5\Omega$ ) and varying hysteresis bounds ( $\Delta$ ).

Case 1 (Small $\Delta$ ): When the hysteresis is small (e.g., $\Delta = 1m$ or $10m$ ), the system waveforms are highly linear. The nonlinear error term ( $h^*$ ) stays well within the calculated sector bound. The linear ripple approximation is valid.
Case 2 (Large $\Delta$ ): When the hysteresis is large (e.g., $\Delta = 100m$ $Δ = 100 m$ ), the system exhibits significant nonlinearity.
- The error term exceeds the sector bound.
- The time constants of the system (determined by eigenvalues) become comparable to the switching period, invalidating the linear ripple assumption.
- Waveforms show distortion, particularly in the inductor current $i_{L1}$ .
Quantitative Limits:
- For the first switching surface, the limit for linear approximation was calculated as a voltage ripple $\Delta v_{C1} \approx 4.3V$ , corresponding to a hysteresis limit of $\Delta \approx 17m$ .
- For the second switching surface, the limit was found to be $\Delta \approx 90m$ .
Phase Diagrams: The phase plots confirm that when the sector bound is respected, the system converges smoothly to the equilibrium. When exceeded, trajectories show overshoots and complex paths, especially when entering DCVM.

5. Significance

This paper bridges the gap between theoretical control design and practical implementation for power electronics:

Design Guidance: It offers engineers a concrete method to calculate the maximum allowable switching ripple (hysteresis) before the system's behavior becomes too nonlinear for standard linear analysis tools.
Safety and Stability: By defining the "safe zone" via LMI, designers can ensure that their converters operate in a regime where stability is mathematically guaranteed, preventing unexpected oscillations or instability in high-power applications.
Generalizability: While focused on the Ćuk converter, the LMI-based approach for sector-bounded perturbations is applicable to other complex DC-DC topologies and switching systems, providing a robust alternative to heuristic tuning.

In conclusion, the paper successfully demonstrates that LMI-based stability analysis is a powerful tool for determining the operational limits of sliding mode controlled DC-DC converters, specifically quantifying the validity of the linear ripple approximation.