---
title: 【CDC2025】Data Informativity for Output Controllability Gramians and Its Duality
tags: 
author: [Ikumi Banno](https://image.docswell.com/user/ibanno)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/9J2RNRGM7R.jpg?width=480
description: 20251211 cdc2025
published: November 07, 26
canonical: https://image.docswell.com/s/ibanno/K9NQP7-cdc2025
---
# Page. 1

![Page Image](https://bcdn.docswell.com/page/9J2RNRGM7R.jpg)

Data Informativity for Output
Controllability Gramians and Its Duality
Ikumi Banno (Kyoto University)
IEEE CDC2025 ThA08.7 (Dec 11, 2025)


# Page. 2

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Main Topic: Data-driven Controllability Analysis
• Consider the network system
𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡
𝑦 𝑡 = 𝐶𝑥 𝑡
- Dynamics is unknown
- Impulse responses for each
input nodes are available
as a dataset
: input nodes
: sensor nodes
• Can we compute the controllability Gramian of the system 𝐴, 𝔹, 𝐶
from the dataset?
- 𝔹: given input matrix
• The solvability of the problem depends on the choice of 𝔹
- For example, solvable if 𝔹 = [𝒆𝟏 − 𝒆𝟑 𝒆𝟕 ] but not if 𝔹 = 𝑒15
- What condition is needed for 𝔹? − The main topic of this talk
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# Page. 3

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Existing Results
• This corresponds to “data informativity” for controllability analysis
Data Informativity [van Waarde, 2020]
• This represents whether a dataset contains enough information
to solve a given problem
• It is characterized for controllability problems, e.g.,
- Determining controllability [van Waarde, 2020], [Eising, 2022], [Li, 2025]
- Computing controllability Gramians [Banno (presenter), 2023]
Limitation of Existing Results

Full-state measurement is required for Gramian computation

Output controllability has never been investigated
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# Page. 4

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Outline of This Study
Purpose of Study
Characterize the informativity of input-to-output data
for computing an output controllability Gramian
- In other words: What data condition is needed for determining
the Gramian?
Main Contributions
• An informativity condition on impulse response data
• Data-driven computation for the output controllability Gramian
• Based on the “duality”, informativity results for generalized
observability Gramians are derived
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# Page. 5

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Problem Setting (1/3)
System
Y𝑖𝑗 (𝑡)
𝓓
𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡
𝑦 𝑡 = 𝐶𝑥 𝑡
𝑡
Dataset 𝓓
- (𝐴, 𝐶) is unknown but 𝐵 is known
- The dataset is given by the impulse response data
𝒀(𝒕) ≔ 𝑪𝒆𝑨𝒕 𝑩
𝑡 ∈ 0, 𝑡𝑓
- Just for simplicity, we assume 𝒕𝒇 = ∞ in this talk
Output Controllability Gramian
• We estimate the output controllability for
𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝔹𝕦 𝑡
𝑦 𝑡 = 𝐶𝑥 𝑡
∞
• Output controllability Gramian: 𝑮𝐨𝐜 (𝔹) ≔ 𝐶
⊤
𝑒 𝐴𝑡 𝔹𝔹⊤ 𝑒 𝐴 𝑡 𝑑𝑡 𝐶 ⊤
0
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# Page. 6

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Problem Setting (2/3)
Problem: Data-driven Computation of 𝑮𝒐𝒄 (𝔹)
• Impulse response of the system (𝐴, 𝐵, 𝐶)
Given
• Input matrix 𝐵 (used in an experiment)
• Input matrix 𝔹 (used for controllability analysis)
Output controllability Gramian of (𝐴, 𝔹, 𝐶):
Find
∞
⊤
𝑒 𝐴𝑡 𝔹𝔹⊤ 𝑒 𝐴 𝑡 𝑑𝑡 𝐶 ⊤
𝐺(𝔹) ≔ 𝐶
0
Approach for This Problem
• We focus on a necessary and sufficient condition for this problem
to be solvable (⇔ the data is informative for this problem)
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# Page. 7

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Problem Setting (3/3)
What Is Solvable Problem? (Data Informativity)
• The situations of the problem are classified as two cases:
(𝐴1 , 𝐶1 )
𝒟
sysid
(𝐴2 , 𝐶2 )
𝐺1
𝐺2
𝒟
sysid
⋮
Data
Set Σ(𝒟)
 NOT solvable
= 𝐺oc (𝔹)
(𝐴1 , 𝐶1 )
𝐺∗
(𝐴2 , 𝐶2 )
⋮
𝐺oc (𝔹)
Data
Set Σ(𝒟)
𝐺oc (𝔹)
✓ Solvable
• If the right situation is occurred, we say the dataset 𝓓 is
informative for computing 𝑮𝒐𝒄 𝔹
Main Problem:
• Derive a necessary and sufficient condition for the informativity
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# Page. 8

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Key Idea for Data Informativity
• Consider the special case ∃𝐾 s.t. 𝐵 = 𝔹𝐾
• Then,
∞
𝐺oc 𝔹 =
𝐶𝑒 𝐴𝑡 𝔹𝔹⊤ 𝑒 𝐴𝑡 𝐶 𝑑𝑡
… definition of 𝐺oc (𝔹)
𝐶𝑒 𝐴𝑡 𝐵 𝐾𝐾 ⊤ (𝐶𝑒 𝐴𝑡 𝐵) 𝑑𝑡
… 𝐵 = 𝔹𝐾
𝑌 𝑡 𝐾𝐾 ⊤ 𝑌 𝑡 ⊤ 𝑑𝑡
… definition of 𝑌(𝑡)
0
∞
=
0
∞
=
0
=: 𝐹(𝒟, 𝐾)
• Thus, the dataset 𝒟 is informative
for computing 𝑮𝒐𝒄 (𝔹)
• 𝐺oc (𝔹) is computed by
∞
𝒀 𝒕 𝑲𝑲⊤ 𝒀 𝒕 ⊤ 𝒅𝒕
𝑮𝐨𝐜 𝔹 =
𝟎
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# Page. 9

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Main Theorems
• When ∃𝐾 s.t. 𝐵 = 𝔹𝐾
does not hold
• We can find that the situation in the figure
- Thus, the dataset is NOT Informative
Theorem [Data Informativity]
The dataset 𝒟 is informative for 𝐺𝑜𝑐 (𝔹) ⇔ ∃𝑲 s.t. 𝑩 = 𝔹𝑲
• 𝐈𝐦𝐚𝐠𝐞 𝔹 ⊆ 𝐈𝐦𝐚𝐠𝐞(𝑩) (= directly-excited state subspace)
• Useful for an experiment design to acquire informative datasets
Theorem [Data-driven Computation]
∞
If 𝒟 is informative for 𝐺oc 𝔹 ,
𝒀 𝒕 𝑲𝑲⊤ 𝒀 𝒕 ⊤ 𝒅𝒕
𝑮𝐨𝐜 𝔹 =
𝟎
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# Page. 10

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Numerical Example (1/2)
• Consider the network system
𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡
𝑦 𝑡 = 𝐶𝑥 𝑡
- (𝐴, 𝐶) is unknown
- 𝑩 = 𝒆𝟏 𝒆𝟑 𝒆𝟒 𝒆𝟕 𝒆𝟏𝟖 is known
: input nodes
: sensor nodes
- Impulse response 𝑌 𝑡 is available
• Compute 𝐺oc 𝔹 with 𝔹 = [𝒆𝟏 − 𝒆𝟑 𝒆𝟕 ] from the data
(Step 1) Informativity Analysis
•
𝑩 = 𝔹𝑲
holds for
𝑲=
𝟏
−𝟏
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝟏
𝟎
• From the theorem, the dataset is informative for computing 𝑮𝒐𝒄 (𝔹)
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# Page. 11

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Numerical Example (2/2)
• Consider the network system
𝑥 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢 𝑡
𝑦 𝑡 = 𝐶𝑥 𝑡
- (𝐴, 𝐶) is unknown
: input nodes
: sensor nodes
- 𝑩 = 𝒆𝟏 𝒆𝟑 𝒆𝟒 𝒆𝟕 𝒆𝟏𝟖 is known
- Impulse response 𝑌 𝑡 is available
• Compute 𝐺oc 𝔹 with 𝔹 = [𝒆𝟏 − 𝒆𝟑 𝒆𝟕 ] from the data
(Step 2) Data-driven Computation
• From
𝑲=
𝟏
−𝟏
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝟏
𝟎
∞
, we obtain
• Successfully computed!
𝐺oc 𝔹 =
=
𝑌 𝑡 𝐾𝐾 ⊤ 𝑌 𝑡 ⊤ 𝑑𝑡
0
𝟑. 𝟓𝟗𝟒 𝟎. 𝟑𝟎𝟑 𝟏. 𝟕𝟒𝟓
𝟎. 𝟑𝟎𝟑 𝟎. 𝟎𝟔𝟗 𝟎. 𝟐𝟐𝟑
𝟏. 𝟕𝟒𝟓 𝟎. 𝟐𝟐𝟑 𝟐. 𝟒𝟓𝟐
10


# Page. 12

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Duality Results
• There is the duality between controllability and observability
• Based on this duality, the data informativity for generalized
observability is also characterized
Output Controllability
Gramian
Definition
Generalized Observability
∞
∞
𝐴𝑡
𝐺𝑜𝑐 𝔹 ≔ 𝐶
⊤ 𝐴𝑡
𝑒 𝔹𝔹 𝑒 𝑑𝑡 𝐶
⊤
𝐺go ℂ ≔ 𝐵
⊤
⊤
𝑒 𝐴 𝑡 ℂ⊤ ℂ𝑒 𝐴𝑡 𝑑𝑡 𝐵
0
0
Assumption
𝐵: known;
𝐴, 𝐶 : unknown
𝐶: known;
𝐴, 𝐵 : unknown
[Theorem]
Informativity
Condition
∃𝑲 𝐬. 𝐭. 𝑩 = 𝔹𝑲
∃𝑮 𝐬. 𝐭. ℂ = 𝑮𝑪
[Theorem]
Data-driven
Computation
∞
∞
⊤
𝐺oc 𝔹 =
𝒀 𝒕 𝑲𝑲 𝒀 𝒕
𝟎
⊤
𝒅𝒕
𝒀 𝒕 ⊤ 𝑮⊤ 𝑮𝒀 𝒕 𝒅𝒕
𝐺go ℂ =
𝟎
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# Page. 13

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Conclusion
Summary
• Data informativity for computing the output controllability Gramian
𝐺oc (𝔹) is characterized: ∃𝐾 s.t. 𝐵 = 𝔹𝐾
• Computation of 𝐺𝑜𝑐 (𝔹) and dual results are also derived
• Generalized theorems found in manuscript
- The case for finite-time-length dataset
- Weighted output controllability Gramians
Future Work
• Extension to arbitrary input-to-output data
• Noisy dataset
Thank You for Attention!
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