⚡ electrical engineering

Excitation of control-affine systems and Koopman error bounds

This paper proposes a data-driven framework for control-affine systems that enhances the robustness of bilinear EDMD schemes by deriving input selection guidelines and optimality conditions to maximize the minimal singular value, thereby ensuring reliable system identification and demonstrated effectiveness in nonholonomic robot control.

Original authors: Philipp Schmitz, Lea Bold, Friedrich M. Philipp, Mario Rosenfelder, Peter Eberhard, Henrik Ebel, Karl Worthmann

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Philipp Schmitz, Lea Bold, Friedrich M. Philipp, Mario Rosenfelder, Peter Eberhard, Henrik Ebel, Karl Worthmann

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 teach a robot how to move by showing it examples. You want the robot to learn a rule that says, "If I push the joystick this way, the robot moves that way."

In the world of engineering, this is called system identification. The paper you're asking about tackles a specific problem: How do you choose the best "pushes" (inputs) to teach the robot so it learns the rule perfectly and doesn't get confused?

Here is the breakdown of their work using simple analogies:

1. The Problem: Teaching with Bad Examples

The authors are working with Control-Affine Systems. Think of this as a machine where the movement is a mix of two things:

Natural Drift: The robot's own internal momentum (like a car coasting downhill).
Your Push: The force you apply (like pressing the gas pedal).

To teach the robot the math behind this, researchers use a method called EDMD (Extended Dynamic Mode Decomposition). It's like a super-smart calculator that looks at your data and tries to draw a straight line (or a simple curved line) through the dots to predict the future.

The Catch: If the data you give the calculator is "boring" or "clumped together," the calculator gets confused.

Analogy: Imagine trying to figure out the shape of a 3D object by only looking at it from one angle. You might think it's a flat circle. But if you look at it from the top, bottom, and sides, you see the whole sphere.
In math terms, if your "pushes" (inputs) are all too similar, the "learning matrix" becomes unstable. The paper calls this a small singular value. A small singular value means the math is shaky, and small errors in your data (like a little bit of sensor noise) will cause huge errors in the robot's learned rules.

2. The Solution: The "Perfect Push" Strategy

The authors propose a framework to ensure the robot gets the best possible set of examples. They treat the choice of inputs like arranging furniture in a room to make the space feel as open and balanced as possible.

They offer three main strategies to "excite" the system (get the robot moving in interesting ways):

Strategy A: The Orthogonal Basis (The Perfect Cross)
- The Idea: Choose inputs that are perfectly perpendicular to each other, like the X, Y, and Z axes on a graph.
- The Analogy: Imagine you are painting a wall. Instead of spraying paint in a messy blob, you spray one line straight up, one straight down, one left, and one right. You cover the space evenly.
- Result: This gives the math the most "room" to work, making the learning very robust.
Strategy B: The Simplex (The Pyramid)
- The Idea: Choose inputs that form the corners of a perfect geometric shape (like a triangle in 2D or a pyramid in 3D) with the center of the shape being the "zero" point.
- The Analogy: Imagine a group of friends standing in a circle. If they all stand at equal distances from the center and from each other, the group is perfectly balanced.
- Result: This is another mathematically perfect way to spread out your data so the robot learns the full picture.
Strategy C: The "Fix-It" Strategy (For Real-Time Learning)
- The Problem: Sometimes you can't plan the perfect set of pushes in advance. Maybe you are driving a car and just happen to turn the wheel randomly.
- The Solution: The authors say, "Okay, you've done 3 random turns. Now, I will calculate exactly what the 4th turn should be to balance out the first 3."
- The Analogy: Imagine you are balancing a stack of books. You've placed three books randomly. Before you add the fourth, you measure the tilt and place the fourth book in the exact spot needed to make the stack stand straight.
- Result: Even if your first few tries were random, adding this one "smart" input fixes the balance and makes the learning accurate.

3. The Result: A Better Robot

The paper proves that by using these strategies, you can mathematically guarantee that the robot's learning model is robust.

Robustness means: If there is a little bit of noise (like a shaky hand or a dirty sensor), the robot's learned rules won't break.
They tested this on two things:
1. A floating rigid body (like a satellite or a drone in space) with 6 different ways to move.
2. A non-holonomic robot (like a car that can't move sideways, only forward/backward and turn).

In both cases, the robots that were taught using the "Perfect Push" strategies learned much faster and made fewer mistakes than the robots taught with random, unorganized pushes.

Summary

The paper is essentially a guidebook for data collection. It tells engineers: "Don't just throw random data at your learning algorithm. If you want your robot to be safe and accurate, arrange your test inputs like a perfect geometric shape, or at least add one 'smart' input to balance out the random ones. This ensures the math behind the robot's brain is solid and won't collapse under pressure."

Technical Summary: Excitation of Control-Affine Systems and Koopman Error Bounds

Problem Formulation
The paper addresses the challenge of identifying control-affine dynamical systems from data using the Koopman operator framework, specifically via Extended Dynamic Mode Decomposition (EDMD). While data-driven methods like EDMD are powerful for modeling complex systems, extending them to control-affine systems (which yield bilinear surrogate models) introduces significant data requirements. The core problem identified is that the accuracy and robustness of the system identification process depend heavily on the quality of the input data. Specifically, the approximation error bounds for these models are inversely proportional to the smallest singular value ( $\sigma_{\min}$ ) of the input-data matrix. If the input data is not sufficiently "exciting" (i.e., if the inputs are poorly distributed or correlated), $\sigma_{\min}$ becomes small, leading to large error bounds and unreliable control performance. The paper seeks to establish guidelines for input selection to maximize this singular value, thereby ensuring robust identification and reliable error bounds for control applications.

Methodology
The authors develop a framework centered on the regression problem used to estimate the control-affine mapping $y = g_0(x) + G(x)u$ . The methodology proceeds through several analytical steps:

Error Bound Analysis: The paper first establishes that the error in estimating the system parameters ( $g_0$ and $G$ ) is bounded by the ratio of the disturbance magnitude to $\sigma_{\min}(V)$ , where $V$ is the data matrix composed of a row of ones and the input vectors.
Optimality Conditions: The authors derive necessary and sufficient conditions for maximizing $\sigma_{\min}(V)$ $σ_{m i n} (V)$ .
- A necessary condition is that the sum of the input vectors must be zero ( $U\mathbf{1} = 0$ ).
- Under this condition, scaling the inputs appropriately allows the matrix to achieve the theoretical upper bound for the smallest singular value.
- Specific optimal configurations are identified: orthogonal inputs (Corollary 3.3) and inputs forming the vertices of a regular simplex (Proposition 3.5).
Subspace Angles and Sequential Collection: Recognizing that practical applications often involve sequential data collection rather than a priori design of all inputs, the authors introduce a geometric approach using subspace angles. They derive a lower bound for $\sigma_{\min}(V)$ based on the angles between new input vectors and the subspace spanned by existing inputs. This leads to a strategy where, given a set of random inputs, a specific additional input is chosen (or an existing one replaced) to maximize the subspace angles, thereby maximizing the minimal singular value.
Integration with Koopman Framework: These theoretical results are applied to two specific EDMD variants:
- Bilinear EDMD with Control (EDMDc): The framework allows for flexible sampling, where data does not need to be collected at specific, pre-defined control values (like unit vectors), but can be inferred from general data clusters.
- Kernel EDMD: The authors extend existing error bounds for kernel-based methods to the control-affine setting, providing uniform error bounds that rigorously verify assumptions required for data-driven Model Predictive Control (MPC).

Key Contributions

Data-Fitting Framework: A novel framework for fitting control-affine mappings that explicitly links input selection to the robustness margin of the identification problem.
Optimality Conditions: Derivation of necessary and sufficient conditions for input optimality to maximize the minimal singular value of the data matrix, including explicit constructions for orthogonal and simplex-based inputs.
Geometric Guidelines for Sequential Data: A practical guideline using subspace angles to determine how to select additional inputs to improve conditioning in sequential or online learning scenarios.
Uniform Error Bounds: The derivation of uniform error bounds for bilinear EDMDc and kernel EDMD in the presence of control-affine structures. These bounds depend on the fill distance of the state space and the conditioning of the input data.
Flexible Sampling: A method to construct artificial data points (e.g., responses to unit vector inputs) from general, flexible data collections, removing restrictive sampling requirements.

Results
The paper validates the proposed approach through two application scenarios:

Free-Floating Rigid Body (6D Input): The study compares four excitation strategies: random inputs, orthogonal inputs, simplex vertices, and a strategy based on subspace angles. Results show that while random inputs yield significantly lower $\sigma_{\min}$ and higher approximation errors, the proposed strategies (especially the subspace angle-based approach) achieve $\sigma_{\min}$ values close to the theoretical maximum, drastically reducing vector field approximation errors.
Nonholonomic Mobile Robot (2D Input): The framework is applied to a mobile robot using bilinear EDMDc. The results demonstrate that using the proposed excitation strategies (even with a minimal number of data points) leads to Koopman surrogate models with significantly lower one-step prediction errors compared to random sampling. The study also shows that increasing the number of random inputs can mitigate errors, but the geometrically guided selection achieves high accuracy with fewer samples.

Significance
The paper claims that its contributions are significant for the reliable application of data-driven control in safety-critical systems. By providing explicit, geometrically interpretable conditions for input selection, the work bridges the gap between theoretical system identification and practical data collection. The derived error bounds are crucial for formal verification of closed-loop properties, such as asymptotic stability in Model Predictive Control. Furthermore, the ability to perform flexible sampling allows for the identification of control-affine systems in scenarios where collecting data at specific, pre-defined control inputs is impractical or impossible, making the approach applicable to real-world engineering problems like robotics and vehicle dynamics.

1. The Problem: Teaching with Bad Examples

2. The Solution: The "Perfect Push" Strategy

3. The Result: A Better Robot

Summary

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