Amplitude Dependent Bode Diagrams via Scaled Relative Graphs

This paper introduces a method for computing less conservative $L_2$-gain bounds for nonlinear Lur'e systems over restricted frequency and amplitude ranges by combining Scaled Relative Graphs with Sobolev theory, resulting in a three-dimensional nonlinear generalization of the Bode diagram that recovers both the standard LTI Bode plot and the global $L_2$-gain as limiting cases.

The Gist

Imagine you are trying to understand how a car behaves.

If the car were a simple, predictable machine (like a toy car on a track), you could use a standard map to predict exactly how fast it goes at every speed. In engineering, this is called a Linear Time-Invariant (LTI) system, and we have a famous map for it called the Bode Diagram. It tells you: "If you push the gas at this frequency, the car will respond with this amount of speed."

But real cars (and real-world systems like power grids, robots, or communication loops) are Nonlinear. They have quirks. If you push the gas too hard, the engine might sputter. If you turn the wheel too sharply, the tires might skid. The old maps (Bode diagrams) fail here because they assume the car behaves the same way no matter how hard you push.

This paper introduces a new, 3D map called the Amplitude-Dependent Bode Diagram. Here is how it works, explained simply:

1. The Problem: The "Worst-Case" Trap

Previously, engineers tried to analyze these tricky nonlinear systems by looking at the worst-case scenario.

• Analogy: Imagine a bouncer at a club. To be safe, the bouncer assumes every person trying to enter is a giant, 7-foot-tall bodybuilder, even if most are average-sized. So, the bouncer sets the door height to accommodate the biggest possible person.
• The Result: The door is huge. It's safe, but it's also overly conservative. It doesn't tell you what happens to the average person. In engineering, this meant the safety margins were so wide they weren't very useful for designing high-performance systems.

2. The Solution: The "Scaled Relative Graph" (SRG)

The authors use a tool called a Scaled Relative Graph (SRG).

• Analogy: Think of the SRG as a magical, flexible net. Instead of just drawing a flat line on a piece of paper (like the old Bode diagram), this net can stretch and shrink in 3D space. It captures how the system reacts not just to how fast you push (frequency), but also how hard you push (amplitude/energy).

3. The Secret Ingredient: Sobolev Theory (The "Smoothness" Check)

To make this 3D map accurate, the authors used a branch of math called Sobolev theory.

• Analogy: Imagine you are driving a car. You care about two things:

  1. How far you go (Energy/Amplitude).
  2. How jerky the ride is (Rate of change/Derivative).

  If you only look at how far you go, you might miss the fact that the car is shaking apart. Sobolev theory is like a "smoothness check." It looks at the input signal and says, "Okay, this signal has a certain amount of energy, but it also changes very quickly. Because it changes quickly, the output can't get too wild."

  By combining the Energy (how hard you push) with the Smoothness (how fast the push changes), the authors can draw a much tighter, more accurate boundary around what the system can do.

4. The Result: A 3D Bode Diagram

The paper creates a new kind of graph that has three dimensions:

1. Frequency: How fast the input is oscillating (like the pitch of a sound).
2. Energy/Amplitude: How strong the input is (like the volume of the sound).
3. Gain: How much the system amplifies that input.

• At the bottom (Low Energy): The graph looks like the old, simple 2D Bode diagram. This is the "Linear" world where things are predictable.
• At the top (High Energy): The graph shows the "Worst-Case" nonlinear behavior.
• In the middle: The graph shows exactly what happens for specific combinations of speed and strength.

5. Why Does This Matter? (The PLL Example)

The authors tested this on a Phase-Locked Loop (PLL).

• Analogy: A PLL is like a DJ trying to match the beat of two different songs. If the songs are slightly off, the DJ adjusts the speed. But if the DJ tries to adjust too quickly or too violently, the system might get confused and lose the beat entirely.
• Using their new 3D map, engineers can see exactly how much they can "push" the DJ (the system) before it loses the beat, without having to assume the DJ is going to go crazy. This allows for safer, faster, and more efficient designs.

Summary

In short, this paper takes a flat, 2D map of a system's behavior and upgrades it to a 3D hologram.

• Old Way: "If you push hard, the system might break. So, let's assume it breaks and design for that." (Too cautious).
• New Way: "If you push this hard at this speed, the system will do exactly this." (Precise and useful).

It bridges the gap between simple, predictable systems and complex, real-world chaos, giving engineers a better tool to design things that are both safe and high-performing.

Technical Summary

Here is a detailed technical summary of the paper "Amplitude Dependent Bode Diagrams via Scaled Relative Graphs" by Krebbekx et al.

1. Problem Statement

Linear Time-Invariant (LTI) systems rely heavily on frequency-domain tools like the Bode and Nyquist diagrams for analysis and controller design. However, these tools do not directly apply to Nonlinear (NL) systems. While methods like the Describing Function exist, they are approximate. Modern exact methods, such as those based on Scaled Relative Graphs (SRG), provide rigorous stability and performance bounds but often rely on the $L_2$-gain.

The primary limitation of the standard $L_2$-gain is that it is too conservative for nonlinear systems because it considers the worst-case gain over all possible input signals, regardless of their amplitude or energy content. Existing attempts to generalize the Bode diagram to nonlinear cases (e.g., restricting inputs to periodic signals) often fail to account for amplitude dependence, computing gains as the worst-case over all amplitudes for a given frequency.

The Core Problem: How to develop a rigorous, non-conservative, graphical method to analyze nonlinear feedback systems that captures performance as a function of both input frequency and input amplitude (energy).

2. Methodology

The authors propose a three-dimensional generalization of the Bode diagram where the $L_2$-gain is plotted as a function of input frequency ($\omega$) and a harmonic energy constraint ($U$). The methodology combines Sobolev theory with Scaled Relative Graph (SRG) analysis.

A. Signal Space and Energy Constraints

• Input Restriction: Instead of considering the entire $L_2$ space, the input space is restricted to signals with bounded frequency and bounded "harmonic energy."
• Harmonic Energy ($U$): Defined as $U = \|u\|_T \|u'\|_T$, where $\|u\|_T$ is the $L_2$ norm over a period and $\|u'\|_T$ is the norm of the derivative. This quantity captures both the signal's energy and its rate of variation (harmonic content).
• Sobolev Space ($H^1$): The analysis restricts inputs to the Sobolev space $H^1$, where both the signal and its derivative have finite energy. This allows the use of embedding theorems to bound the signal's peak amplitude ($\|u\|_\infty$).

B. Amplitude Bounding via Sobolev Theory

Using the Gagliardo-Nirenberg interpolation inequality, the authors derive a bound on the output amplitude ($\|y\|_\infty$) based on the input's harmonic energy $U$:
$$\|y\|_\infty \leq \sqrt{2} \|y\|_T \|y'\|_T \leq \sqrt{2} \lambda_\omega \lambda^\partial_\omega U$$
Here, $\lambda_\omega$ and $\lambda^\partial_\omega$ are gain bounds from input to output and from input derivative to output derivative, respectively. This creates a feedback loop: a bound on amplitude allows for a tighter bound on the nonlinearity, which in turn yields a tighter gain estimate.

C. SRG for Gain Computation

The authors utilize SRGs to compute the required gain bounds ($\lambda_\omega$ and $\lambda^\partial_\omega$) for Lur'e systems (an LTI system $G$ in feedback with a static nonlinearity $\phi$).

1. LTI Component: The SRG of the LTI system $G$ is the hyperbolic convex hull of its Nyquist diagram.
2. Nonlinear Component: The nonlinearity $\phi$ is assumed to be slope-restricted and sector-bounded. The SRG of $\phi$ is bounded by a disk $D[a, b]$.
3. Derivative Maps: A key theoretical contribution is proving that the SRG of the derivative map ($\partial G$ and $\partial \phi$) is identical to the SRG of the original maps under specific conditions. This allows the stability analysis to be applied simultaneously to the signal and its derivative.
4. Iterative Solution: The method solves an optimization problem to find the smallest amplitude bound $A$ that satisfies the gain constraints for a given frequency $\omega$ and energy $U$.

3. Key Contributions

1. 3D Amplitude-Dependent Bode Diagram: The paper introduces a method to plot $L_2$-gain as a surface over the axes of frequency and input energy (amplitude).
2. Integration of Sobolev Theory and SRG: It bridges the gap between functional analysis (Sobolev spaces for amplitude bounding) and graphical control theory (SRGs for stability/gain analysis).
3. Derivative Map SRG Equivalence: It establishes that for stable LTI systems and slope-restricted nonlinearities, the SRG of the derivative map is equal to the SRG of the original map. This is crucial for bounding the derivative of the output.
4. Recovery of Limits: The method rigorously recovers known limits:
   • As $U \to 0$ (zero energy), the diagram converges to the standard LTI Bode diagram (linearization at the origin).
   • As $U \to \infty$ (infinite energy) and $\omega \to 0$, it converges to the standard worst-case $L_2$-gain.
   • For intermediate values, it provides a less conservative, amplitude-specific performance metric.

4. Results

The method was demonstrated on a Phase-Locked Loop (PLL) model, consisting of a first-order LTI filter and a sinusoidal nonlinearity ($\phi(x) = \sin(x)$).

• Gain Surface (Fig 2a): The resulting plot shows the gain $\gamma_{\omega, U}$ varying with frequency and energy.
  • At low energy ($U \to 0$), the gain matches the linearized system $|G(s)/(1+G(s))|$.
  • At high energy, the gain approaches the worst-case nonlinear gain.
  • The surface smoothly interpolates between these two extremes, providing a more accurate performance prediction than worst-case bounds.
• Amplitude Bounds (Fig 2b): The method successfully computes the maximum output amplitude ($A_{\omega, U}$) for specific input constraints. This allows engineers to determine the maximum allowable input energy to guarantee a specific output amplitude limit.

5. Significance

• Reduced Conservatism: By restricting the input space to specific energy and frequency ranges, the method avoids the excessive conservatism of global $L_2$-gain analysis, which is often too pessimistic for practical nonlinear control design.
• Intuitive Design Tool: It extends the familiar Bode diagram intuition to nonlinear systems, allowing engineers to visualize how system performance degrades or changes as signal amplitude increases.
• Rigorous Foundation: Unlike the Describing Function (which is approximate), this method provides exact bounds based on the SRG framework, ensuring mathematical rigor for stability and performance certification.
• Practical Application: The ability to bound output amplitude directly from input energy constraints is highly valuable for safety-critical systems (like PLLs) where signal saturation or large-signal behavior must be guaranteed.

In summary, this paper provides a novel, mathematically rigorous framework for analyzing nonlinear feedback systems that unifies frequency-domain intuition with amplitude-dependent performance constraints, offering a powerful tool for the design and analysis of modern nonlinear control systems.