🔢 mathematics

Bilinear Systems with Quadratic Outputs: $\mathcal{H}_2$ Analysis, Optimality Conditions for Model Reduction, and Algorithmic Solutions

This paper establishes a comprehensive $\mathcal{H}_2$ framework for bilinear systems with quadratic outputs by defining their norm and deriving first-order optimality conditions, which are then utilized to develop an algorithm for optimal model reduction that generalizes existing methods for linear and bilinear systems.

Original authors: Heike Faßbender (Institute for Numerical Analysis, TU Braunschweig), Serkan Gugercin (Department of Mathematics and Division of Computational Modeling and Data Analytics, Academy of Data Science, Virg

Published 2026-07-23

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Heike Faßbender (Institute for Numerical Analysis, TU Braunschweig), Serkan Gugercin (Department of Mathematics and Division of Computational Modeling and Data Analytics, Academy of Data Science, Virginia Tech), Till Peters (Institute for Numerical Analysis, TU Braunschweig)

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 understand a massive, chaotic orchestra playing a symphony. The music is beautiful, but the orchestra has 40,000 musicians, and you only have a tiny speaker system that can handle a few dozen. You want to create a smaller, simpler version of the orchestra that still sounds exactly like the original when it plays a song. This is the heart of "model order reduction," a field in engineering and mathematics dedicated to shrinking complex systems without losing their essential behavior.

Usually, these systems are like simple machines: you push a button, and a gear turns. But in the real world, things are messier. Sometimes, the way a machine moves depends on how hard you push it and how fast it's already moving. This is called a "bilinear" system. Even trickier, sometimes what you are measuring (the "output") isn't just a simple line, but a curve that depends on the square of the machine's state—like measuring the total energy or the "loudness" of a sound, which grows much faster than the volume knob itself. This paper focuses on these specific, tricky systems: ones with complex, squiggly movements and outputs that curve like a parabola. The goal is to figure out how to shrink these complicated mathematical models down to a manageable size while keeping them accurate enough to be useful.

The Paper's Mission: Taming the Squiggly Systems

The authors, Heike Faßbender, Serkan Gugercin, and Till Peters, tackle a problem that had been missing a crucial piece of the puzzle. While scientists knew how to shrink simple linear systems and even some of the squiggly "bilinear" ones, they didn't have a complete, systematic way to handle systems that combined both complex movements and curved, squared outputs. They call these "Bilinear Systems with Quadratic Outputs" (BQO systems).

Think of a BQO system like a very sensitive, high-tech drone. Its flight path (dynamics) changes based on wind speed and throttle (bilinear), but the sensor you are using to track it doesn't just report its position; it reports the energy of its movement, which involves squaring the numbers (quadratic output). If you try to build a tiny, simplified version of this drone's math, you can't just use the old rules for simple drones or even the rules for squiggly ones. You need a new set of instructions.

The New Toolkit: A "H2" Ruler

To solve this, the team built a new mathematical "ruler" called the H2 norm. In the world of control theory, the H2 norm is like a measure of how much "noise" or "error" a system produces when you run it through a standard test. For simple systems, we already had this ruler. For bilinear systems, we had a slightly different one. But for BQO systems, the ruler was missing.

The authors did three main things to fix this:

They defined the ruler: They created a precise mathematical formula to measure the "size" or "energy" of a BQO system. This involved looking at how the system reacts to inputs over time, breaking it down into layers of complexity (like peeling an onion), and adding up the results.
They proved the ruler works: They showed that if you know the H2 norm of a system, you can predict how big the output error will be. It's like saying, "If the blueprint says the error is small, the actual building won't wobble much." This gives engineers confidence that if they shrink the model, the results won't go haywire.
They found the "Perfect Fit" conditions: This is the big discovery. They figured out exactly what conditions a tiny, reduced model must meet to be the best possible version of the original. Imagine trying to fit a key into a lock. The authors wrote down the exact shape the key (the reduced model) needs to have to fit the lock (the original system) perfectly. These are called "optimality conditions."

The Algorithm: The "BQO-TSIA" Dance

Knowing the rules for the perfect key is great, but how do you actually carve it? The authors proposed an algorithm they call BQO-TSIA (Bilinear Quadratic Output Two-Sided Iteration Algorithm).

Think of this algorithm as a dance between two partners.

Partner A is the giant, original system (the 40,000-musician orchestra).
Partner B is the tiny, reduced model (the 12-musician band).

The dance works like this:

They start with a guess for the tiny band.
They ask the giant orchestra, "How do you react to this tiny band?" and the tiny band, "How do you react to the giant orchestra?"
Based on this conversation, they adjust the tiny band's instruments and sheet music.
They repeat this process over and over. With each step, the tiny band gets better at mimicking the giant orchestra.
Eventually, the dance stops when the tiny band is so good that it satisfies the "perfect fit" conditions the authors discovered.

Does it Work? The Test Drive

The authors didn't just stop at the math; they tested their new method on two real-world scenarios to see if it actually worked better than the old ways.

Test 1: The Nonlinear RC Circuit
They used a model of an electrical circuit with 40,200 components. This is a huge system. They compared their new "BQO-TSIA" method against an older method called "Balanced Truncation" (which is like trying to shrink the orchestra by just cutting out the quietest musicians without listening to the music).

The Result: The new method produced a smaller error in the output. In other words, the tiny band sounded more like the giant orchestra.
Speed: The new method was also much faster. While the old method took about 150 seconds to run, the new method (specifically the version using a solver called GLGMRES) took less than 5 seconds for most sizes. It was a clear win for both accuracy and speed.

Test 2: The Heat Equation
Next, they simulated heat moving through a square plate (like a hot pan). This was a "MIMO" system, meaning it had multiple inputs and outputs. The original system had 2,500 variables.

The Result: Again, the new method beat the old one. The error in the new method went down steadily as they made the model slightly larger, whereas the old method's error jumped up and down unpredictably.
Speed: Once again, the new method was significantly faster, taking less than a second to a few seconds, compared to the 16 seconds the old method took.

The Takeaway

This paper doesn't just suggest a new idea; it provides a complete, proven framework. The authors have shown that by building a new mathematical ruler (the H2 norm) and a new dance (the BQO-TSIA algorithm), we can shrink these incredibly complex, squiggly systems into manageable sizes without losing the plot.

The results from their simulations suggest that this approach is not only more accurate than the current standard (Balanced Truncation) but also much more efficient. For engineers and scientists dealing with complex systems—whether they are designing better batteries, simulating heat flow, or controlling drones—this new toolkit offers a way to get high-quality answers without needing a supercomputer to do the math. The "perfect fit" conditions they found serve as a guidepost, ensuring that when we simplify the world, we don't lose the magic of how it actually works.

Technical Summary: H2 Analysis, Optimality Conditions, and Algorithmic Solutions for Bilinear Systems with Quadratic Outputs

Problem Statement
Bilinear systems with quadratic outputs (BQO systems) represent a class of dynamical systems where both the state dynamics and the measured response depend nonlinearly on the state. These systems arise naturally in applications involving state-dependent nonlinearities, such as bilinear port-Hamiltonian systems, approximations of nonlinear systems via Carleman bilinearization, and the modeling of variances in stochastic systems or energy-related quantities. The general state-space form is given by:
$\dot{x}(t) = Ax(t) + \sum_{k=1}^m N_k x(t) u_k(t) + Bu(t), \quad x(0)=0$
$y(t) = Cx(t) + \sum_{j=1}^p x(t)^T M_j x(t)$
While BQO systems unify bilinear systems (where $M_j=0$ ) and linear systems with quadratic outputs (LQO, where $N_k=0$ ), a systematic framework for $H_2$ optimal model order reduction (MOR) for this specific class has been lacking. Existing methods for bilinear or LQO systems do not directly apply to the combined BQO structure. The objective is to construct a reduced-order BQO system of dimension $r \ll n$ that accurately approximates the input-output behavior of the original large-scale system.

Methodology
The paper develops a comprehensive $H_2$ framework for BQO systems, extending existing theories for bilinear and LQO systems. The methodology proceeds through the following steps:

Theoretical Foundations: The authors establish the existence and uniqueness of reachability and observability Gramians for BQO systems, defined via generalized Lyapunov equations. They also introduce generalized Sylvester equations to characterize the connection between full and reduced systems.
$H_2$ Norm Definition: A new $H_2$ $H_{2}$ inner product and norm are defined for BQO systems using Volterra kernels. Crucially, the paper derives explicit formulas for the $H_2$ $H_{2}$ inner product and norm in terms of the solutions to the generalized Lyapunov and Sylvester equations, avoiding the need for direct integration of kernels.
- The $H_2$ norm squared is given by $\|S\|_{H_2}^2 = \text{tr}(CP C^T) + \sum_{j=1}^p \text{tr}(P M_j P M_j)$ , where $P$ is the reachability Gramian.
Output Error Bounds: The authors derive bounds on the system output and the output error between a full and reduced system in terms of the $H_2$ norm. Specifically, the output error is bounded by the $H_2$ distance between the systems multiplied by a term dependent on the input norms. This motivates minimizing the $H_2$ system error to ensure small output errors.
Optimality Conditions: By formulating the model reduction problem as a constrained optimization of the $H_2$ error, the paper derives first-order necessary optimality conditions. These conditions characterize locally $H_2$ optimal reduced systems.
Algorithmic Solution (BQO-TSIA): Based on the optimality conditions, the authors propose an iterative algorithm called BQO-TSIA (Bilinear Quadratic Output Two-Sided Iteration Algorithm). This algorithm generalizes IRKA, BIRKA, and LQO-TSIA. It iteratively solves generalized Sylvester equations to compute projection subspaces ( $V$ and $W$ ) that enforce the optimality conditions upon convergence. Two variants are implemented: one using fixed-point iterations for the Sylvester equations and another using a global GMRES solver.

Key Contributions
The paper claims the following primary contributions:

$H_2$ Theory: Introduction of an $H_2$ inner product and norm specifically for BQO systems, which recovers existing definitions for bilinear and LQO systems as special cases.
Error Bounds: Derivation of output bounds for BQO systems in terms of the $H_2$ norm and corresponding error bounds between two BQO systems.
Optimality Conditions: Derivation of first-order necessary optimality conditions for $H_2$ optimal model reduction of BQO systems.
Algorithm: Proposal of the BQO-TSIA algorithm, which computes a reduced BQO system satisfying these optimality conditions, thereby generalizing existing $H_2$ optimal methods.

Results
The effectiveness of the proposed framework is demonstrated through two numerical examples:

Nonlinear RC Circuit: A single-input single-output (SISO) system with dimension $n=40,200$ derived from Carleman bilinearization.
Heat Equation: A multi-input multi-output (MIMO) system with dimension $n=2,500$ representing a bilinear controlled heat transfer system.

In both examples, the proposed $H_2$ -based methods (BQO-TSIA and BQO-TSIA GLGMRES) were compared against Balanced Truncation (BT) variants (using full and truncated Gramians).

Accuracy: The $H_2$ -inspired methods generally achieved lower relative $H_2$ norm errors and, for higher reduced orders, smaller maximal output errors compared to the balanced truncation methods. The BT methods showed oscillatory behavior in output error for the heat equation, whereas the $H_2$ methods exhibited monotonic decay.
Efficiency: The BQO-TSIA GLGMRES variant was consistently the fastest method, particularly for larger reduced orders, outperforming the fixed-point iteration version and the balanced truncation approaches. The computational cost of balanced truncation was noted to be prohibitive for large-scale systems due to the requirement of solving full generalized Lyapunov equations.

Significance
The paper asserts that it provides the first complete $H_2$ theory for BQO systems. By establishing the $H_2$ norm, deriving optimality conditions, and proposing a convergent algorithm, the work fills a theoretical gap that previously prevented the systematic application of $H_2$ optimal model reduction to this important system class. The framework naturally unifies and extends existing $H_2$ theories for bilinear and LQO systems. The numerical results suggest that the proposed approach offers a competitive alternative to balanced truncation, often yielding higher fidelity reduced models with lower computational costs, particularly when efficient solvers for the underlying generalized Sylvester equations are employed.

The Paper's Mission: Taming the Squiggly Systems

The New Toolkit: A "H2" Ruler

The Algorithm: The "BQO-TSIA" Dance

Does it Work? The Test Drive

The Takeaway

More like this