⚡ electrical engineering

Experiment Design for Set-membership Identification: From Prior Knowledge to Universal Inputs

This paper proposes new methods for designing universal input signals that guarantee accurate identification of unknown linear time-invariant systems within a finite horizon by leveraging general prior knowledge, thereby extending beyond traditional persistently exciting inputs and enabling exact identification even in noisy environments.

Original authors: Amir Shakouri, Henk J. van Waarde, M. Kanat Camlibel

Published 2026-07-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amir Shakouri, Henk J. van Waarde, M. Kanat Camlibel

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 a detective trying to figure out how a mysterious machine works. You can't see inside it, but you can push buttons (inputs) and watch what happens (outputs). Your goal is to build a perfect mathematical model of this machine so you can predict its future behavior.

This paper is about how to push those buttons in the smartest way possible to solve the mystery quickly and accurately, even when you don't know exactly what kind of machine you are dealing with.

Here is the breakdown of their ideas using simple analogies:

1. The Problem: Guessing the Machine's Rules

Usually, to learn how a machine works, you have to push buttons randomly and hope you get enough data. This is like trying to learn a new language by shouting random words at a native speaker and hoping they eventually understand you.

The authors ask: "Can we design a specific sequence of button pushes that guarantees we will learn the machine's rules, no matter which specific machine it is, as long as it fits a few basic clues we already have?"

These "clues" are called Prior Knowledge.

Example: "We know the machine is stable," or "We know the mass of the object is between 10 and 20 kg."

2. The Magic Solution: "Universal Inputs"

The paper introduces a concept called a Universal Input. Think of this as a "Master Key."

The Old Way (Persistently Exciting Inputs): In the past, scientists said, "To learn the machine, you must push the buttons in a very complex, chaotic, and constant pattern that never repeats." This is like shaking a box of Legos violently to see how they fit. It works, but it takes a long time and uses a lot of energy.
The New Way (Universal Inputs): The authors found that if you know some specific clues about the machine, you don't need to shake the box violently. You can use a much simpler, shorter, or even "lazy" sequence of button pushes that still reveals the secrets.

The Analogy:
Imagine you are trying to identify a specific type of fruit in a dark room.

The Old Way: You grab every fruit in the basket, squeeze them all, smell them, and taste them (a lot of work).
The New Way: You know beforehand that the fruit is either a red apple or a green apple. You only need to check the color once. You don't need to squeeze every single fruit. The "Universal Input" is that single, smart color check that works for any apple in that specific basket.

3. The "Lazy" Button Push (Hands-Off Inputs)

One of the coolest findings is that sometimes, the best way to learn is to do nothing for a while.

The paper shows that for certain types of machines, you can push a button, wait for a long time (doing nothing), and then push it again. This "hands-off" approach is often safer and cheaper than constantly pushing buttons.

Real-world example mentioned: Spacecraft. You don't want to fire thrusters constantly because it burns fuel. The authors show that a "fire, wait, fire" strategy can still perfectly identify the spacecraft's physics if you know the basic laws of orbital motion.

4. Dealing with Noise (The Static on the Radio)

In the real world, your measurements are never perfect. There is always "noise" (static, sensor errors, wind).

The Challenge: If there is noise, you can't always find the exact answer. You can only find an answer that is "close enough."
The Solution: The paper provides a recipe to design button pushes that guarantee your answer is within a specific "error margin" (e.g., "We know the weight is within 0.1 kg of the truth").
The Catch: If you want the exact answer in a noisy world, the "clues" you start with must be very specific (like knowing the parameters are whole numbers). If the clues are too vague, noise will always leave some uncertainty.

5. Why This Matters

The authors prove that by using what you already know (prior knowledge), you can:

Save Time: You need fewer data points (fewer button pushes) to learn the system.
Save Energy: You can use simpler, "lazy" inputs instead of complex, chaotic ones.
Be Safer: In fields like aerospace or biology, you can't always blast a system with maximum force. This method allows you to learn the system gently.

Summary

This paper is a guidebook for smart experimentation. It tells engineers: "Don't just throw data at the wall. If you know a little bit about the system beforehand, you can design a specific, efficient test that guarantees you'll learn exactly what you need to know, even if the data is a bit messy."

They provide the mathematical formulas (the "recipes") to calculate exactly what that perfect test looks like for different types of machines and different types of prior knowledge.

Technical Summary: Experiment Design for Set-membership Identification

Problem Statement

This paper addresses the offline experiment design problem for set-membership identification of unknown linear time-invariant (LTI) systems. The objective is to design an input signal $u[0, T-1]$ such that the resulting input-state (or input-output) data, collected over a finite horizon, is sufficiently informative to identify the system parameters with a desired accuracy.

The study operates under two distinct noise regimes:

Noise-free ( $\varepsilon = 0$ ): The goal is exact identification, where the data must uniquely determine the true system parameters.
Noisy ( $\varepsilon > 0$ ): The goal is $\rho$ -accuracy identification, where the data must constrain the set of possible systems to a bounded set with a Chebyshev radius (worst-case error) less than or equal to a prescribed tolerance $\rho$ .

The central challenge is that the true system is unknown. Therefore, the design must rely on prior knowledge ( $\Sigma_{pk}$ ), a set of systems consistent with available information (e.g., bounds on parameters, structural constraints, or controllability properties). The paper introduces the concept of universal inputs: an input signal is universal if, when applied to any system within the prior knowledge set $\Sigma_{pk}$ , it generates data suitable for the desired identification task.

Methodology

The authors develop a framework based on data informativity and input universality, leveraging set-membership identification principles.

1. Data Informativity and Universal Inputs

Data Informativity: A dataset $D$ is $\Sigma_{pk}$ -informative for $\rho$ -accuracy if the set of data-consistent systems $\Sigma_D$ intersected with the prior knowledge $\Sigma_{pk}$ has a Chebyshev radius $\le \rho$ .
Universal Inputs: An input $u[0, T-1]$ is $\Sigma_{pk}$ -universal if, for every system $(A, B) \in \Sigma_{pk}$ , all possible datasets generated by that system under the input are $\Sigma_{pk}$ -informative.

2. Noise-Free Analysis

In the absence of noise, the authors analyze conditions under which universal inputs exist and their relationship to persistency of excitation (PE).

Open Prior Knowledge: For open sets of prior knowledge contained within the set of controllable systems ( $\Sigma_{cont}$ ), the paper establishes that universal inputs are equivalent to inputs that are persistently exciting of order $n+1$ (where $n$ is the state dimension).
General Prior Knowledge: The authors derive sufficient conditions (Theorems 9 and 14) for universality that depend on the specific structure of $\Sigma_{pk}$ . These conditions utilize the minimal polynomial of the system matrix $A$ and specific matrix constructions ( $M_{A,B}$ , $D_A$ ).
Key Insight: For certain structured prior knowledge (e.g., specific norm bounds or known minimal polynomial degrees), universal inputs can be designed with a length $T$ significantly shorter than the $nm+n+m$ samples required by standard PE. In some cases, inputs that are not persistently exciting (e.g., "hands-off" inputs with zero values over large intervals) are proven to be universal.

3. Noisy Analysis

In the presence of $\ell_\infty$ -bounded noise, the analysis shifts to bounding the identification error.

$\rho$ -Accuracy: The paper provides conditions (Theorem 21) ensuring that the worst-case error is bounded by $\rho$ . This involves ensuring the smallest singular value of a specific data-dependent matrix is sufficiently large relative to the noise level $\varepsilon$ .
Feasibility Conditions: The authors show that universal experiment design for $\rho$ -accuracy is feasible if the prior knowledge set is bounded and satisfies a controllability measure $\gamma_{pk} > 0$ . They also propose a partitioning strategy (Theorem 24) to handle cases where the condition cannot be satisfied for the entire set $\Sigma_{pk}$ but holds for a subset, provided the remaining subset has a small intrinsic radius.
Exact Identification with Noise: The paper proves that exact identification ( $\rho=0$ ) in the presence of noise is only feasible if the prior knowledge set is uniformly discrete (Theorem 29, 30). This means distinct systems in the prior set must be separated by a minimum distance, a condition often met in systems with integer parameters.

4. Extension to Input-Output Data

The framework is extended to ARX systems (input-output models) by reformulating them into state-space representations. This allows the application of the derived input-state results to input-output data scenarios.

Key Contributions

Framework for Universal Inputs: A novel framework is developed to study offline experiment design by explicitly incorporating general prior knowledge, moving beyond the limitations of controllability-only assumptions.
Characterization of Universality vs. PE:
- For single-input systems with open prior knowledge, universality is shown to be equivalent to persistency of excitation of order $n+1$ .
- For multi-input systems and specific prior knowledge structures, the paper demonstrates that universal inputs exist that are not persistently exciting of order $n+1$ , thereby reducing the required sample size (e.g., from $nm+n+m$ to $2n+m$ or even $k+1$ ).
Noise-Robust Design: Sufficient conditions are provided for designing inputs that guarantee identification within a prescribed error bound $\rho$ in noisy settings.
Exact Identification in Noise: The paper establishes that exact identification in the presence of noise is feasible if and only if the prior knowledge set is uniformly discrete, and provides a design method for such cases.
Hands-off Inputs: The methodology enables the design of "hands-off" inputs (signals with significant zero intervals), which are practically valuable in safety-critical or resource-constrained applications where persistently exciting inputs are infeasible.

Results and Examples

Theoretical Bounds: The paper derives lower bounds on the number of samples $T$ required for universal identification. For instance, under specific prior knowledge, $T \ge 2n+m$ suffices, improving upon the standard $T \ge nm+n+m$ .
Example 1 (Orbital Motion): A spacecraft relative motion model is used to demonstrate that a two-impulse input (zero for most of the horizon) is $\Sigma_{pk}$ -universal for identifying orbital rate and mass, whereas standard PE would require a much longer, continuous excitation.
Example 2 (Network of Systems): A network of 1000 identical systems is considered. The results show how to design a universal input for $\rho$ -accuracy identification in a high-dimensional setting with noise, utilizing the structure of the network to reduce complexity.

Significance

The paper claims that its results generalize the experiment design approach based on Willems et al.'s fundamental lemma. While the fundamental lemma relies on persistency of excitation and is limited to prior knowledge regarding controllability, this work accommodates general prior knowledge.

The significance lies in the finding that for many practical scenarios where prior knowledge is available (e.g., bounds on parameters, structural constraints, or discrete parameter sets), persistently exciting inputs are not necessary. The proposed methods can yield sample-efficient experiment designs that outperform existing approaches. This is particularly relevant for applications where applying persistently exciting inputs is impractical due to:

Safety considerations.
Physical constraints.
Limited resources (e.g., in chemical/biological systems, clinical settings, large-scale networks, and aerospace systems).

By providing conditions under which shorter, structured, or "hands-off" inputs are sufficient, the paper offers a more flexible and efficient approach to system identification in constrained environments.