---
title: Data-driven Optimal Pole Placement via Sylvester Equations
tags: 
author: [Ikumi Banno](https://image.docswell.com/user/ibanno)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/9721KZKGER.jpg?width=480
description: 20251108 SSS25
published: November 08, 25
canonical: https://image.docswell.com/s/ibanno/KVM3J2-SSS25
---
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Data-driven Optimal Pole Placement
via Sylvester Equations
Ikumi Banno, Kaiya Irie, Shun-ichi Azuma
(Kyoto University)
The 57th ISCIE International Symposium on Stochastic Systems Theory and Its Applications (SSS25);
2025/11/8 1A1-4


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Motivation: “Robustness” for Pole Placement
Pole Placement
• Consider the system
1 −8
−5
1
0
𝑥ሶ 𝑡 = −2 −7
2 𝑢 𝑡
1 𝑥 𝑡 + −1
2 −9
0
1 −3
=𝑨
=𝑩
• Find a feedback gain 𝐹 ∈ 𝐑2×3 such that the eigenvalues of 𝑨 + 𝑩𝑭 are {−𝟏 ± 𝟐𝒊, −𝟑}
• A solution is not unique, e.g., 𝑭𝟏 = −0.417 2.966 −0.750
−0.546 2.486 −0.616
𝑭𝟐 =
−1.257 0.220
0.510
−0.935 1.280 −0.046
• Which gain should we use?
“Robustness” for Feedback Gain
• Consider the closed-loop pole sensitivity
• Eigenvalues of 𝑨 + 𝑩𝑭𝟏 + 𝜟𝒊 is sensitive wrt 𝛥𝑖
- 𝛥𝑖 ∈ 𝐑3×3 : Random perturbation matrix
• This is a motivation for robust pole placement
𝑨 + 𝑩𝑭𝟏 + 𝚫𝐢
(Sensitive)
𝑨 + 𝑩𝑭𝟐 + 𝚫𝐢
(Robust)
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Contents
1. Introduction: Pole Placement
2. Problem Setting: Data-driven Robust Pole Placement
3. Proposed Method
1. Reduction to Sylvester Equations
2. Data-driven Method for Solving Sylvester Equation
4. Numerical Example
5. Conclusion
2


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Introduction (1/2)
Pole Placement
• Computing a feedback gain that assigns the closed-loop poles to prescribed locations.
- Such feedback gain is not unique in general
How do we choose a feedback gain?
• Recently, optimizations under pole placement constraints have been investigated
- Feedback gain norm [Keel 1985], [Varga 2000]
… reducing control energy
- Closed-loop pole sensitivity [Hu 2004], [Schmid 2014] … preserving pole location from perturbation
• A Mathematical model of a system is required
When System Model Is Unavailable?
• A Data-driven approach is a possible solution
3


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Introduction (2/2)
Data-driven Optimal Pole Placement
• Compute optimal feedback gain directly from measurement data of the open-loop system
• Existing researches are summarized as follows:
Feedback gain norm
Type of
Minimization Closed-loop pole
sensitivity
Framework
Model-based
Data-based
[Keel 1985], [Varga 2000]
[Celi 2023], [Irie, 2025]
[Hu 2004], [Schmid 2014]
(No literature)
• No research for a data-driven method for minimizing closed-loop pole sensitivity
Objective of Our Research
Derive a data-driven method for minimizing closed-loop pole sensitivity
subject to a pole placement constraint
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Problem Setting (1/2)
System and Optimal Pole Placement
• Consider the system
𝑥ሶ 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢(𝑡)
- 𝑥 𝑡 ∈ 𝐑𝑛 : state; 𝑢 𝑡 ∈ 𝐑𝑚 : input
Weighted sum of
𝐢 closed-loop pole sensitivity
𝐢𝐢 feedback gain norm
• Optimal pole placement [Varga 2000] is described as:
Find 𝑭, 𝑽 ∈ 𝐑𝑚×𝑛 × 𝐑𝑛×𝑛 that minimizes
𝐽 𝐹, 𝑉 ≔ 𝛼
𝑽
𝟐
𝐅 +
−𝟏 𝟐
𝑽
𝐅
+ 1−𝛼
subject to the pole placement constraint
𝑭 𝟐𝐅
𝑽−𝟏 𝐴 + 𝐵𝑭 𝑽 = diag 𝜆1 , … , 𝜆𝑛
- 𝐢 corresponds to an upper bound of the closed-loop pole sensitivity, i.e.,
sup 𝜆𝑖 𝐴 + 𝐵𝐹 + Δ − 𝜆𝑖 𝐴 + 𝐵𝐹
𝑖,Δ
Δ 2
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Problem Setting (2/2)
• Our interest is: when the system dynamics is unknown
System Assumptions
• For the system
𝑥ሶ 𝑡 = 𝐴𝑥 𝑡 + 𝐵𝑢(𝑡),
we assume that:
- 𝐴 is unknown;
𝑥(𝑡)
- The dataset 𝓓 is available for pole placement
𝑡
𝒟: 𝒙 𝒕, 𝒙𝟎 𝒕 ∈ 𝟎, 𝒕𝐟
- 𝑥 𝑡, 𝑥0 : the state trajectory of the system with 𝑥 0 = 𝑥0 and 𝑢 𝑡 ≡ 0
Data-driven Problem for Optimal Pole Placement
Given
the dataset 𝓓, closed-loop pole location Λ ≔ diag 𝜆1 , … , 𝜆𝑛
Find
𝑭, 𝑽 that minimizes
s.t.
the pole placement constraint
𝐽 𝐹, 𝑉 ≔ 𝛼
𝟐
𝑽 𝟐𝐅 + 𝑽−𝟏 𝐅 + 1 − 𝛼
𝑽−𝟏 𝐴 + 𝐵𝑭 𝑽 = Λ
𝑭 𝟐𝐅
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Convert to Unconstrained Optimization [Varga 2000]
Original Problem (constrained optimization)
Find
s.t.
𝑭, 𝑽 that minimizes
𝐽 𝐹, 𝑉 ≔ 𝛼
𝟐
𝑽 𝟐𝐅 + 𝑽−𝟏 𝐅 + 1 − 𝛼
𝑭 𝟐𝐅
𝑽−𝟏 𝐴 + 𝐵𝑭 𝑽 = Λ
Equivalent (under a reasonable assumption)
Converted Problem (unconstrained optimization)
Find
The matrix 𝑮 that minimizes
𝑱෠ 𝑮 ≔ 𝐽 𝐹 𝐺 , 𝑉 𝐺
• Gradient of 𝑱෠ is characterized by the solutions 𝑿, 𝒀 to the Sylvester equations
Λ𝒀 − 𝒀𝐴 = −𝑄
𝐴𝑿 − 𝑿Λ = −𝑄
• The optimization is reduced to data-driven methods for solving these equations
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Data-driven Solution to Sylvester Equation (1/2)
• We aim to find a solution 𝒀 to the Sylvester equation Λ𝒀 − 𝒀𝐴 = −𝑄,
1
where
- 𝐴 ∈ 𝐑𝑛×𝑛 is unknown
- the state trajectory 𝑥 𝑡 of 𝑥ሶ 𝑡 = 𝐴𝑥(𝑡)
2
is available
• Multiplying both sides of Sylvester equation (1) on the right by 𝑥 𝑡 :
𝛬𝒀 − 𝒀𝐴 𝑥 𝑡 = −𝑄𝑥(𝑡)
2
⇔
𝛬𝒀𝑥 𝑡 − 𝒀𝑥ሶ 𝑡 = −𝑄𝑥(𝑡)
• By integrating the both sides on [𝑡1 , 𝑡2 ], we have
𝑡2
𝛬𝒀 න 𝑥 𝑡 𝑑𝑡 − 𝒀 𝑥 𝑡2 − 𝑥 𝑡1
𝑡1
𝑡2
= −𝑄 න 𝑥 𝑡 𝑑𝑡
𝑡1
- A linear equation with unknown matrix 𝒀
- Constructed by the dataset (without the matrix 𝐴)
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Data-driven Solution to Sylvester Equation (2/2)
• We can construct a linear equation from (a fragment of) 𝑥 𝑡
𝑡2
𝑡2
𝛬𝒀 න 𝑥 𝑡 𝑑𝑡 − 𝒀 𝑥 𝑡2 − 𝑥 𝑡1
= −𝑄 න 𝑥 𝑡 𝑑𝑡
𝑡1
3
𝑡1
• We obtain 𝑵 linear equations by dividing a dataset 𝒟
𝑥(𝑡)
𝑥(𝑡)
𝑡
The dataset 𝒟
#1
#2
⋯
#𝑵
𝑡
Fragments of Trajectory
Equation 3 for data #1
Equation 3 for data #2
⋮
Equation 3 for data #𝑁
4
𝑁 linear equations
Theorem: Data-driven Solution to Sylvester Equation
If the dataset 𝒟 has sufficient information, 1) a solution 𝒀 to (4) is unique and
2) 𝒀 is equal to the unique solution to the Sylvester equation Λ𝒀 − 𝒀𝐴 = −𝑄
• Moreover, similar theorem holds for the other Sylvester equation 𝐴𝑿 − 𝑿Λ = −𝑄
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Proposed Algorithm
Algorithm: Data-driven Optimal Pole Placement
Input: a dataset 𝒟, an initial matrix 𝐺0
Output: Feedback gain 𝐹∞
1 For 𝒌 = 𝟎, 𝟏, 𝟐, …
2
𝑋෠𝑘 ← data-driven solution to Sylvester equation
𝐴𝑿 − 𝑿Λ = −𝐵𝐺𝑘
3
𝑌෠𝑘 ← data-driven solution to Sylvester equation
Data-based Gradient Computation
Λ𝒀 − 𝒀𝐴 = −𝑆(𝑋෠𝑘 )
4
∇𝐽መ𝑘 ← 𝑔(𝑋෠𝑘 , 𝑌෠𝑘 )
5
𝐺𝑘+1 ← 𝐺𝑘 − 𝛾𝑘 ∇𝐽መ𝑘
6
𝐹𝑘+1 ← 𝐹 𝐺𝑘+1
7 end
// gradient computation
// update of parameter 𝐺𝑘
Update of Feedback Gain 𝑭𝒌
10


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Numerical Example (1/2)
System
• Consider the system
1 −8
−5
1
0
𝑥ሶ 𝑡 = −2 −7
2 𝑢 𝑡
1 𝑥 𝑡 + −1
2 −9
0
1 −3
=𝑩
=𝑨
- Assume that 𝐴 is unknown, but the following dataset is available:
Optimal Pole Placement Problem
Find
s.t.
𝑭, 𝑽 that minimizes
1
𝐽 𝐹, 𝑉 = 2
the pole placement constraint
𝟐
1
𝑽 𝟐𝐅 + 𝑽−𝟏 𝐅 + 2 𝑭 𝟐𝐅
−3
𝑽−𝟏 𝐴 + 𝐵𝑭 𝑽 = 0
0
0
−1
−2
0
2
−1
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Numerical Example (2/2)
Application of Proposed Method
• Objective function value is monotonically decreasing
• After the iteration, we finally obtain the feedback gain
𝐹53 =
−1.257 0.220
0.510
−0.935 1.280 −0.046
- 𝐹53 satisfies pole placement constraint
Verification for Closed-pole Sensitivity Reduction
• We plot the eigenvalue locations of
𝐴 + 𝐵𝑭𝟎 + Δ𝑖
𝐴 + 𝐵𝑭𝟓𝟑 + Δ𝑖
optim.
- Δ𝑖 𝑖 = 1,2, … , 1000 : random perturbation matrix
• “Robust” feedback gain is obtained
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Conclusion
Summary
• A data-driven robust pole placement is presented
- Based on Data-driven solution to the Sylvester equation
• The results can be extended to input-to-state data (in prep.)
Future Work
• Algorithm for noisy data
• Robustness for structured perturbation
Thank You for Your Attention!
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