【CDC2025】Data Informativity for Output Controllability Gramians and Its Duality

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

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 1

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 2

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 3

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 4

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) 5

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 6

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 7

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 8

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) 9

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

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 11

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! 12