⚡ electrical engineering

$H_\infty$ Performance Analysis for Almost Periodic Piecewise Linear Systems with Application to Roll-to-Roll Manufacturing Control

This paper proposes an $H_\infty$ performance analysis method and a corresponding controller synthesis algorithm for almost periodic piecewise linear systems with uncertain dwell-times, demonstrating through roll-to-roll manufacturing experiments that the approach yields significantly less conservative and higher-performing controllers than baseline methods that ignore the system's switching structure.

Original authors: Christopher Martin, Edward Kim, Enrique Velasquez, Wei Li, Dongmei Chen

Published 2026-02-13

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CC BY 4.0

Original authors: Christopher Martin, Edward Kim, Enrique Velasquez, Wei Li, Dongmei Chen

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 driving a car on a very specific, repeating route. This route has three distinct zones: a smooth highway, a bumpy construction zone, and a slippery icy patch.

In a perfect world (what engineers call a Periodic system), you would know exactly when you hit the construction zone and when you leave it. You could set your cruise control to be perfect for that exact timing.

But in the real world (what this paper calls an Almost Periodic system), things are messy. Maybe the construction zone starts 5 seconds early or 5 seconds late. Maybe the icy patch is a little longer or shorter than expected. You know the order of the zones (Highway → Construction → Ice), but you don't know the exact timing of the switches.

This paper solves a major problem: How do you drive safely and smoothly when the road keeps changing its length unpredictably, and you have to fight against wind and bumps (disturbances)?

Here is the breakdown of the paper's solution using simple analogies:

1. The Problem: The "Guessing Game" Controller

The researchers looked at a high-tech manufacturing machine called Roll-to-Roll (R2R). Imagine a giant conveyor belt peeling a sticker off one roll and sticking it onto another.

The Challenge: The sticker isn't uniform. It has patterns (like stripes of different materials). As the machine peels the sticker, the physics change depending on which "stripe" is currently being peeled.
The Mess: Because the machine vibrates and the material stretches, the machine doesn't switch from "Stripe A" to "Stripe B" at the exact same millisecond every time. It's an Almost Periodic system.
The Old Way: Previous controllers were like a driver who ignores the timing uncertainty. They assume the road is always the same length. To be safe, they drive very cautiously (using high "gain"), which makes the car jerk around and waste energy. It's like slamming the brakes every time you think you might see a pothole, even if you aren't sure.

2. The Solution: The "Smart Navigator" (The New Algorithm)

The authors developed a new mathematical tool (an $\mathcal{H}_\infty$ Performance Analysis) that acts like a super-smart navigator.

It knows the map: It knows the sequence of zones (Stripe A, then Stripe B).
It accepts the uncertainty: It knows the switch might happen between 10 seconds and 15 seconds, not just exactly at 12.
It calculates the "Worst Case": Instead of guessing, it calculates the absolute worst possible scenario (e.g., "What if the icy patch is longest and the wind is strongest?") and designs a controller that handles that specific worst case without panicking.

The Analogy:
Imagine you are walking through a dark forest with a flashlight.

The Old Controller: You shine the light wildly in every direction, terrified of a bear appearing anywhere. You are exhausted and jittery.
The New Controller: You know the bear only appears in specific clearings, but you don't know exactly when you will reach the clearing. Your new strategy is to walk confidently, knowing exactly how to react if the bear jumps out at any point within that clearing. You aren't jittery; you are prepared for the uncertainty.

3. The Result: Smoother, Faster, Less Wear

The researchers tested this new "Smart Navigator" on a real machine peeling 2D materials (like graphene or MoS2, which are super-thin, advanced materials used in electronics).

They compared their new method against the old "cautious" method:

Accuracy: The new method reduced the "peeling angle" errors by nearly 20%. Imagine trying to peel a sticker perfectly straight; the old method made it wobble, while the new one kept it razor-sharp.
Smoothness: The machine's motors and rollers moved much more smoothly. The "jitter" (variations in speed and force) dropped by about 63-70%.
Longevity: Because the machine isn't jerking around as much, the parts won't wear out as fast. It's the difference between a car engine that revs wildly and one that idles smoothly.

Why This Matters

This isn't just about peeling stickers. This math applies to any system that switches modes but has timing errors:

Power grids switching between solar and wind power (where the sun doesn't shine on a perfect schedule).
Robots that switch between walking, running, and jumping.
Manufacturing lines that handle different products in a batch.

The Takeaway

The paper proves that you don't have to be conservative and sluggish just because the world is uncertain. By mathematically understanding how the uncertainty works (the "Almost Periodic" structure), you can build a controller that is less conservative, more precise, and much more efficient.

In short: They taught the machine to dance gracefully even when the music tempo keeps changing slightly, rather than just standing still and hoping for the best.

1. Problem Statement

The paper addresses the control of Almost Periodic Piecewise Linear Systems (APPLSs). These are systems that cyclically switch between different linear dynamic modes, but unlike standard Periodic Piecewise Linear Systems (PPLSs), the dwell-times (the duration the system stays in each mode) are uncertain but bounded.

Context: The problem is motivated by advanced Roll-to-Roll (R2R) manufacturing processes, specifically the dry transfer of 2D materials (e.g., graphene, MoS2) and printed electronics.
Challenge: In R2R processes, the material often has a patterned structure (different adhesion or stiffness in different sections), creating a switched system. While the sequence of modes is known, the exact timing of the switch (dwell-time) is uncertain due to position errors (register errors) inherent in high-speed manufacturing.
Gap: Existing control methods for switched systems either assume deterministic switching times (PPLS) or do not provide rigorous $\mathcal{H}_\infty$ performance analysis (guaranteed worst-case disturbance rejection) for systems with both switching uncertainty and modeling uncertainty. Current controllers for R2R are often overly conservative because they ignore the specific switching structure.

2. Methodology

The authors propose a novel framework for analyzing and synthesizing controllers for uncertain APPLSs.

A. System Modeling

Uncertain APPLS Structure: The system is modeled as a set of linear differential inclusions with additive, norm-bounded modeling uncertainty ( $\Delta(t)$ ) in each mode.
Switching Signal: The switching sequence is fixed and cyclic, but the switching times $t_{l,i}$ are unknown variables bounded within intervals $[t_i, \bar{t}_i]$ .
Control Structure: A switched full-state feedback controller is used, where the gain matrix changes based on the active mode and the specific time segment (dwell phase vs. switching transition phase).

B. Theoretical Framework: $\mathcal{H}_\infty$ Performance Analysis

Weighted $\mathcal{H}_\infty$ Definition: The authors define a weighted $\mathcal{H}_\infty$ performance metric using a decaying exponential weight. This allows for the use of discontinuous Lyapunov functions, which are necessary for switched systems.
Theorem 1 (LMI Conditions): The core theoretical contribution is a set of Linear Matrix Inequalities (LMIs) that guarantee:
1. Exponential stability when disturbances are zero.
2. A bounded weighted $\mathcal{H}_\infty$ norm (disturbance rejection capability) for non-zero disturbances.
Key Mechanism: The proof utilizes a Mode-Dependent Lyapunov Function (MDLF) that is discontinuous at switching instants. The LMIs account for:
- System dynamics during the "must-be-active" dwell times.
- System dynamics during the "uncertain" switching transition intervals.
- The growth/decay of the Lyapunov function across mode switches (using scalar parameters $\mu$ ).

C. Controller Synthesis Algorithm

The paper presents Algorithm 1, a two-phase iterative procedure to synthesize the controller gains:

Phase 1 (Stability): Iteratively finds controller gains and decay rates ( $\lambda$ ) to ensure the system is exponentially stable, satisfying the LMI conditions without the $\mathcal{H}_\infty$ objective.
Phase 2 (Optimization): With stability guaranteed, the algorithm minimizes the $\mathcal{H}_\infty$ performance bound ( $\gamma$ ) to maximize disturbance rejection.

Practicality: The algorithm includes constraints on the norm of the controller gains to prevent excessive control effort and is designed for offline computation, making it suitable for industrial implementation.

3. Key Contributions

First Rigorous $\mathcal{H}_\infty$ Analysis for APPLSs: The paper provides the first theoretical framework to analyze and guarantee disturbance rejection for cyclically switched linear systems with bounded dwell-time uncertainty.
Handling Dual Uncertainty: The method simultaneously handles dwell-time uncertainty (switching time jitter) and modeling uncertainty (parameter variations within modes).
Switched Controller Synthesis: Development of a practical algorithm to synthesize switched controllers that exploit the known switching sequence to achieve less conservative performance than time-invariant controllers.
Experimental Validation: The first experimental demonstration of a switching control method applied to R2R dry transfer of patterned 2D materials.

4. Results

The proposed method was validated on an experimental R2R testbed transferring patterned tape (simulating 2D materials).

Comparison: The proposed switched $\mathcal{H}_\infty$ controller was compared against a baseline time-invariant (LTI) $\mathcal{H}_\infty$ controller that ignores the switching pattern and assumes worst-case uncertainty across all modes.
Performance Metrics:
- Peeling Angle Error: The proposed controller reduced the mean RMS error of the peeling angles by 17.5% to 19.6% compared to the baseline.
- System Variation: The proposed controller significantly reduced variations in web velocity and motor torque (standard deviations reduced by 62.7% to 69.9%).
Observations:
- The baseline controller was overly conservative. To guarantee stability under the assumed worst-case uncertainty (ignoring the pattern), it required high gains, which excited unmodeled dynamics and caused severe oscillations.
- The proposed controller utilized the a-priori knowledge of the switching pattern and bounded uncertainty intervals to find an optimal trade-off, resulting in smoother operation and better disturbance rejection.

5. Significance

Manufacturing Impact: The results demonstrate that accounting for the specific switching structure of patterned materials in R2R manufacturing leads to significantly higher transfer quality (precision) and reduced mechanical wear on actuators.
Theoretical Advancement: The work bridges the gap between switched system theory and practical industrial control, proving that incorporating dwell-time uncertainty into the control design is feasible and highly beneficial.
Future Applications: The framework is generalizable to other advanced manufacturing processes involving patterned materials, flexible electronics, and any system where the operating mode changes cyclically with timing jitter. The authors suggest future work could integrate data-driven learning to further refine the mode dynamics.