On the connection between input-output resonances and internal modes of linear time-invariant systems

This paper demonstrates that in general Linear Time-Invariant systems, particularly those with high or infinite dimensions, there is no necessary correspondence between internal modes and input-output resonance peaks, a phenomenon driven by system zeros in SISO cases and matrix non-normality in specific MIMO cases.

The Gist

Imagine you have a complex machine, like a giant, intricate musical instrument or a massive suspension bridge. In the world of engineering, we usually look at this machine in two ways:

1. The Inside: We look at the "internal modes," which are like the machine's natural, hidden vibrations. If you tap it, these are the specific notes it wants to hum.
2. The Outside: We look at the "input-output" response. This is what happens when you push or shake the machine. We measure how much the output amplifies at different frequencies. These amplification points are called resonances (the loudest, most dramatic reactions).

The Old Belief
For a long time, engineers believed there was a direct, one-to-one map between the inside and the outside. The rule of thumb was: "If the machine has a hidden, wobbly vibration (a lightly damped internal mode) at a specific pitch, then pushing the machine at that pitch will make it scream loudly (a resonance peak)."

Think of it like a swing. If the swing has a natural rhythm (internal mode), pushing it at that rhythm makes it go high (resonance).

The Paper's Big Surprise
This paper by Bassam Bamieh shatters that simple rule. It argues that in very large, complex systems, the inside and the outside can be completely disconnected.

Here is the twist:

• You can have a machine with a very wobbly, sensitive internal vibration, but when you push it, it doesn't react much at all.
• Conversely, you can have a machine that screams loudly (huge resonance) when you push it, even though it doesn't have any wobbly internal vibrations nearby to explain why.

The "Why" (Using Analogies)

The paper explains that this disconnect happens for two main reasons, depending on the type of system:

1. The "Silent Switch" (For Simple Systems): In simpler systems, imagine the internal vibration exists, but the "switch" to turn it on is broken or hidden. The paper calls these "zeros." It's like having a piano key that is stuck; you can press it (input), but the string (internal mode) never vibrates, so no sound comes out.
2. The "Tangled Web" (For Complex Systems): In massive, complex systems (like the ones with many inputs and outputs), the internal parts are so tangled that they cancel each other out. The paper calls this "non-normality." Imagine a room full of people shouting. Even if one person has a very loud voice (a sensitive mode), if everyone else is shouting in a way that perfectly cancels that voice out, the room sounds quiet. The internal "loudness" is there, but the outside world hears nothing.

The "High-Dimensional" Catch
The paper is careful to note that this weird disconnect is rare in small, simple machines. It mostly happens in huge, high-dimensional systems—think of systems with thousands or even infinite moving parts, like fluids flowing through a pipe or air moving over a wing.

To prove this, the author shows a mathematical trick: You can take any set of internal vibrations you want (even if they are all very stable and quiet) and build a massive, complex machine that mimics any loud, wild external behavior you desire. It's like being able to build a giant, chaotic orchestra that plays a specific song perfectly, even if every single musician inside is trying to play a completely different, quiet tune.

Real-World Example Mentioned
The paper points to one specific real-world example where this matters: Airflow over a wall (like in a jet engine or wind tunnel).

• The Inside: There is a very specific, wobbly wave in the air (called a Tollmien-Schlichting wave) that should theoretically cause turbulence.
• The Outside: However, the biggest, most dangerous turbulence actually happens at a different frequency entirely, where the air seems perfectly stable internally.
• The Result: For decades, scientists were confused because they were looking at the "inside" vibrations to predict the "outside" chaos, but the two didn't match. This paper explains that the "inside" and "outside" are simply speaking different languages in these massive systems.

In Summary
This paper tells us: Don't assume that just because a system has a sensitive internal vibration, it will react strongly to the outside world. And don't assume that a huge reaction means there is a sensitive vibration inside. In the world of giant, complex systems, the internal "personality" of the machine and its external "behavior" can be completely unrelated.

Technical Summary

Technical Summary: On the Connection Between Input-Output Resonances and Internal Modes of Linear Time-Invariant Systems

Problem Statement
In the analysis of Linear Time-Invariant (LTI) systems, a prevailing intuition suggests a direct correspondence between the system's internal modes (eigenvalues of the state matrix $A$) and its input-output frequency response. Specifically, it is commonly assumed that under-damped internal modes (eigenvalues with small negative real parts) manifest as resonance peaks in the frequency response near the imaginary parts of those eigenvalues. While this relationship holds for Single Input Single Output (SISO) systems with specific pole-zero configurations and for Multi Input Multi Output (MIMO) systems where the state matrix is normal, the paper investigates whether this correspondence is a universal property. The central problem addressed is the potential disconnect between the location of internal modes in the Left Half Plane (LHP) and the shape of the system's external frequency response, particularly in high-dimensional systems.

Methodology
The paper employs a mathematical approximation approach within the framework of Hardy spaces. The authors focus on the space $H^2$, defined as the subspace of $L^2$ functions on the imaginary axis that are analytic in the right half of the complex plane. This space encompasses stable rational transfer functions.

The core methodology involves proving a density result (Lemma 1). The authors demonstrate that for any specified set of pole locations $Z = \{z_1, \dots, z_n\}$ in the LHP, and for any target frequency response function $G(j\omega)$ belonging to $H^2_{p \times q}$, one can construct a transfer function with poles strictly restricted to the set $Z$ that approximates $G$ arbitrarily closely in the $H^2$ norm.

The proof utilizes a generalization of a known technique involving elementary matrices and Cauchy's integral formula. It establishes that the set of functions formed by terms of the form $\frac{1}{(s-z_\ell)^k}E_{ij}$ (where $E_{ij}$ are elementary matrices) spans a dense subspace in $H^2_{p \times q}$. Consequently, any function in this space can be approximated by a linear combination of these basis functions, which share the same pole locations.

Key Contributions and Results

1. Decoupling of Internal Modes and Resonances: The primary result is the demonstration that, in general, there is no necessary connection between the location of internal modes and the peaks of the input-output frequency response. The authors show that one can approximate any frequency response shape using a system whose poles are fixed at pre-specified locations in the LHP.
2. Role of System Dimension and Structure:
   • SISO Systems: The disconnect can be attributed to the presence of system zeros which may cancel or modify the effect of nearby poles.
   • MIMO Systems: The phenomenon can occur even in systems without zeros. In these cases, it is attributed to the non-normality of the matrix $A$ generating the internal dynamics. When $A$ is non-normal, the singular value plot of the frequency response is not simply inversely proportional to the distance to the nearest eigenvalue.
3. High-Dimensional Nature: The paper establishes that this lack of correspondence is fundamentally a high-dimensional (or infinite-dimensional) phenomenon. The construction required to achieve arbitrary approximation with fixed pole locations typically necessitates high-order systems, often involving high pole multiplicities or clusters of poles.
4. Specific Counter-Intuitive Behaviors: The results indicate two surprising possibilities:
   • A lightly damped internal mode may not produce a resonance peak in the frequency response.
   • A resonance peak may appear in the frequency response without the presence of a nearby lightly damped internal mode.

Significance and Claims
The paper claims that the lack of correspondence between under-damped internal modes and input-output resonances, while atypical in low-dimensional systems, is a critical factor in understanding certain complex physical systems.

The authors highlight a specific application in the input-output analysis of wall-bounded shear flow transition to turbulence. In the linearized Navier-Stokes equations for channel flows, there exists a well-known "anomaly" in hydrodynamic stability analysis:

• The system possesses a very lightly damped pole (Tollmien-Schlichting waves) with a non-zero imaginary part.
• However, the most significant input-output (spatio-temporal) resonances occur near zero frequency ($\omega = 0$), far from the lightly damped pole.
• The physical structures associated with these external resonances (observed in bypass transition regimes) are distinct from those predicted by the internal modes.

The paper posits that the phenomenon described—where internal modes do not dictate external resonance peaks—provides a mathematical explanation for this long-standing paradox in hydrodynamic stability. The work underscores that relying solely on internal mode analysis can be misleading for high-dimensional systems where non-normality and high-order dynamics dominate the input-output behavior.