⚡ electrical engineering

Controlling the Cuk Converter using Piecewise Linear Lyapunov Functions

This paper proposes a switching control law for the Cuk converter in continuous conduction mode based on piecewise linear Lyapunov functions, demonstrating through simulations how varying the number of state variables used in their construction impacts system performance.

Original authors: Aleksandra Lekić, Nikola Petrović, Dušan Stipanović

Published 2026-04-23

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Aleksandra Lekić, Nikola Petrović, 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

Imagine you are trying to keep a very wobbly, high-speed bicycle upright while riding it down a hill. You want to reach a specific speed (the "steady state") and you want to make sure the bike doesn't wobble too much (the "ripple"). If you just use a simple rule like "pedal harder if you're slow, pedal softer if you're fast," you might overshoot, crash, or start shaking uncontrollably.

This paper is about designing a much smarter, more sophisticated "riding rule" for a specific type of electrical machine called a Ćuk converter. Think of this converter as that wobbly bicycle, but instead of wheels and pedals, it uses electricity, switches, and magnetic coils to change voltage levels.

Here is the breakdown of the paper using simple analogies:

1. The Problem: The Wobbly Bike

The authors are dealing with a device that is naturally unstable. It's like a bike that wants to fall over unless you are constantly adjusting your balance.

The Old Way: Most people use a "PID controller," which is like a rider who reacts slowly and generally. It works okay, but it can be sluggish or cause the bike to shake (oscillate) before settling down.
The New Way: The authors propose using Piecewise Linear Lyapunov Functions (PLLF). That's a mouthful, but think of it as a smart, multi-layered safety net. Instead of one simple rule, the system has a set of different "rules" (or shapes) it can switch between instantly, depending on exactly where the bike is at that moment.

2. The Solution: Building a Custom Safety Net

The core idea is to build a "polytope" (a fancy geometric shape, like a 3D box or a pyramid) around the perfect operating point.

The Goal: Keep the electrical current and voltage inside this invisible box.
The Trick: The system checks which "side" of the box the electricity is currently on. If it's getting too close to the edge, the system instantly flips a switch to push it back toward the center.
The "Lyapunov" Part: This is just a mathematical way of proving that the bike will eventually stay inside the box and won't fall out. It's like proving that no matter how hard you push the bike, the safety net will always catch it.

3. The Special Challenge: The "Unstable" Wheel

The paper highlights a specific quirk of the Ćuk converter: one part of it (the inductor current) is naturally unstable. It's like a wheel that is slightly out of round and wants to wobble more and more if you don't pay attention to it.

The authors realized that to build a good safety net, you must include this unstable wheel in your calculations. If you ignore it, the net won't work.
They found that by focusing on specific combinations of variables (like the speed of the wheel and the tension of the chain), they could create a net that is both tight (prevents shaking) and smooth (prevents jerky movements).

4. The Experiments: Testing Different Nets

The authors ran computer simulations to see what happens if they build the safety net using different numbers of variables:

The 2-Variable Net: They built a net using only two measurements (like just the wheel speed and chain tension).
- Result: It worked, but sometimes the bike jerked a little (overshoot) before settling.
The 3-Variable Net: They added a third measurement (like the handlebar angle).
- Result: The ride became much smoother. The jerky "overshoot" disappeared.
The 4-Variable Net: They used all available measurements.
- Result: The ride was the smoothest, but it took a tiny bit longer to get up to speed (the "transient" was longer).

5. The Big Takeaway

The main point of the paper is that by using these smart, switching safety nets, you can control complex electrical devices much better than with old, simple methods.

Precision: You can tell the device exactly how much it is allowed to wiggle (ripple).
Safety: You can guarantee it won't go out of control.
Flexibility: You can choose how "tight" the net is. If you want a super smooth ride, you use more variables. If you want a faster reaction, you might use fewer.

In summary: The authors figured out how to build a custom, multi-layered "guard rail" for a tricky electrical machine. By constantly checking the machine's position against this guard rail and instantly switching strategies, they ensure the machine runs smoothly, reaches its target speed quickly, and never crashes, even though it naturally wants to be unstable.

1. Problem Statement

The paper addresses the challenge of designing stabilizing controllers for DC-DC converters, specifically the Ćuk converter operating in Continuous Conduction Mode (CCM).

Complexity: The Ćuk converter is a fourth-order system with two switches (transistor and diode), resulting in four operating subsystems. In CCM, it switches between two marginally stable subsystems.
Limitations of Existing Methods:
- PID Control: While popular and often implemented digitally, it relies on averaging models and may not optimally handle the inherent nonlinearity of switching systems, potentially leading to larger dynamic losses.
- Current Mode Control: Approaches like peak limiting can lead to chaotic behavior if not carefully designed.
Specific Challenge: The Ćuk converter in CCM contains an unstable state variable (inductor current $i_{L2}$ ). The control strategy must not only stabilize the system to a desired equilibrium but also strictly control the ripple of state variables and minimize overshoot during transients.

2. Methodology

The authors propose a control law based on Piecewise Linear Lyapunov Functions (PLLF) with polytopic level sets. This approach treats the converter as a switched nonlinear system rather than an averaged linear one.

A. System Modeling

The Ćuk converter is modeled using two state-space equations corresponding to the two CCM modes (switch ON and switch OFF).
The system is defined by four state variables:
1. Inductor current $i_{L1}$
2. Capacitor voltage $v_{C1}$
3. Inductor current $i_{L2}$ (identified as the unstable variable)
4. Capacitor voltage $v_{C2}$
The equilibrium point is determined using Volt-second and Ampere-second balance principles.

B. Control Design

Lyapunov Function: A piecewise linear function is defined as $V(x) = \max_i |c_i^T (x - \bar{x})|$ , where $x$ is the state vector and $\bar{x}$ is the equilibrium.
Polytope Construction: The control aims to keep the system state within a specific polytope (attractive set) defined by the level sets of the Lyapunov function.
Switching Law: A switching rule is derived to ensure the Lyapunov function decreases (or remains bounded) on the facets of the polytope. The rule switches between subsystems based on the sign of the projection of the state error onto specific vectors.
Coefficient Calculation: The coefficients for the Lyapunov function are calculated to satisfy specific equilibrium and ripple constraints. The authors prove that for the system to be stable, the unstable variable $i_{L2}$ must be included in the polytope construction.

C. Simulation Strategy

The authors tested four different control configurations by varying the number of state variables included in the polytope construction:

2 Variables: Only inductor currents ( $i_{L1}, i_{L2}$ ).
3 Variables: $i_{L1}, i_{L2}, v_{C1}$ .
4 Variables: All state variables ( $i_{L1}, i_{L2}, v_{C1}, v_{C2}$ ).

3. Key Contributions

Handling Marginal Stability: The paper provides a specific methodology for controlling systems that switch between marginally stable subsystems, a common issue in CCM Ćuk converters.
Unstable Variable Management: It explicitly demonstrates that the unstable inductor current ( $i_{L2}$ ) is critical for stability and must be part of the Lyapunov function construction, regardless of the other variables chosen.
Ripple and Overshoot Control: The method allows for precise estimation and control of steady-state ripple and initial transient overshoot by adjusting the number of state variables in the polytope.
Analytical Coefficient Derivation: The paper offers a direct calculation method for the Lyapunov coefficients (Eq. 9) based on desired equilibrium and ripple, avoiding complex iterative algorithms.

4. Simulation Results

The simulations compared the transient and steady-state behaviors of the different PLLF configurations:

2-Variable Control ( $i_{L1}, i_{L2}$ ):
- Achieved stability and met ripple specifications.
- Observation: When the coefficient for $i_{L2}$ was negative, no overshoot occurred. However, a small overshoot in the ripple of capacitor voltage $v_{C1}$ was observed during the initial transient.
3-Variable Control ( $i_{L1}, i_{L2}, v_{C1}$ ):
- Improvement: Eliminated the overshoot in $v_{C1}$ seen in the 2-variable case.
- Trade-off: The transient response was slightly different, but the steady-state performance remained consistent.
4-Variable Control (All states):
- Result: Provided the most robust transient response with no overshoot on any state variable.
- Trade-off: The initial transient duration was slightly longer due to the inclusion of the additional state variable ( $v_{C2}$ ) in the control logic.
General Findings: In all cases, the value of the switched Lyapunov function remained below 1, satisfying the design specification. The control successfully stabilized the system without the chaotic behavior associated with poorly tuned peak-limiting controls.

5. Significance

This paper demonstrates that Piecewise Linear Lyapunov Functions offer a superior alternative to classical PID control for complex, high-order DC-DC converters like the Ćuk converter.

Nonlinear Handling: It effectively manages the nonlinear switching nature of the converter without relying on averaging approximations that can obscure dynamic losses.
Design Flexibility: Engineers can trade off transient speed against overshoot by selecting the number of state variables in the polytope.
Robustness: The method guarantees stability even when the underlying subsystems are only marginally stable, provided the unstable modes are correctly accounted for in the Lyapunov construction.
Practical Application: The approach provides a systematic way to design controllers that meet strict ripple and overshoot specifications, which is crucial for sensitive power electronics applications.