# Active Learning for Identification of Linear Dynamical Systems

arXiv: https://arxiv.org/abs/2002.00495 · HTML: https://mlsanity.com/paper/2002.00495/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2020.

## Built on by (most cited first, 30 of 33)

- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Optimistic Active Exploration of Dynamical Systems](https://mlsanity.com/paper/2306.12371/) (2023)
- [ASID: Active Exploration for System Identification in Robotic Manipulation](https://mlsanity.com/paper/2404.12308/) (2024)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Linear Systems can be Hard to Learn](https://mlsanity.com/paper/2104.01120/) (2021)
- [Optimal Exploration for Model-Based RL in Nonlinear Systems](https://mlsanity.com/paper/2306.09210/) (2023)
- [Task-Optimal Exploration in Linear Dynamical Systems](https://mlsanity.com/paper/2102.05214/) (2021)
- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
- [Learning to Control Linear Systems can be Hard](https://mlsanity.com/paper/2205.14035/) (2022)
- [Learning Dynamical Systems by Leveraging Data from Similar Systems](https://mlsanity.com/paper/2302.04344/) (2023)
- [Active Learning for Control-Oriented Identification of Nonlinear Systems](https://mlsanity.com/paper/2404.09030/) (2024)
- [LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search](https://mlsanity.com/paper/2606.25039/) (2026)
- [Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis](https://mlsanity.com/paper/2309.14648/) (2023)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [Time varying regression with hidden linear dynamics](https://mlsanity.com/paper/2112.14862/) (2021)
- [Online Stochastic Gradient Descent Learns Linear Dynamical Systems from A Single Trajectory](https://mlsanity.com/paper/2102.11822/) (2021)
- [Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching](https://mlsanity.com/paper/2409.01416/) (2024)
- [Exact Recovery for System Identification with More Corrupt Data than Clean Data](https://mlsanity.com/paper/2305.10506/) (2023)
- [Online greedy identification of linear dynamical systems](https://mlsanity.com/paper/2204.06375/) (2022)
- [Deep active learning for nonlinear system identification](https://mlsanity.com/paper/2302.12667/) (2023)
- [Finite Sample Frequency Domain Identification](https://mlsanity.com/paper/2404.01100/) (2024)
- [Active learning of neural population dynamics using two-photon holographic optogenetics](https://mlsanity.com/paper/2412.02529/) (2024)
- [Optimal Bayesian Affine Estimator and Active Learning for the Wiener Model](https://mlsanity.com/paper/2504.05490/) (2025)
- [High Effort, Low Gain: Fundamental Limits of Active Learning for Linear Dynamical Systems](https://mlsanity.com/paper/2509.11907/) (2025)
- [Safe Linear-Quadratic Dual Control with Almost Sure Performance Guarantee](https://mlsanity.com/paper/2103.13278/) (2021)
- [System Identification via Nuclear Norm Regularization](https://mlsanity.com/paper/2203.16673/) (2022)
- [Active Learning of Dynamics Using Prior Domain Knowledge in the Sampling Process](https://mlsanity.com/paper/2403.17233/) (2024)
- [Optimal Centered Active Excitation in Linear System Identification](https://mlsanity.com/paper/2604.05518/) (2026)
- [A Memory Efficient Unified Algorithm for Online Learning of Linear Dynamical Systems](https://mlsanity.com/paper/2607.02050/) (2026)

## Abstract (arXiv)

We propose an algorithm to actively estimate the parameters of a linear dynamical system. Given complete control over the system's input, our algorithm adaptively chooses the inputs to accelerate estimation. We show a finite time bound quantifying the estimation rate our algorithm attains and prove matching upper and lower bounds which guarantee its asymptotic optimality, up to constants. In addition, we show that this optimal rate is unattainable when using Gaussian noise to excite the system, even with optimally tuned covariance, and analyze several examples where our algorithm provably improves over rates obtained by playing noise. Our analysis critically relies on a novel result quantifying the error in estimating the parameters of a dynamical system when arbitrary periodic inputs are being played. We conclude with numerical examples that illustrate the effectiveness of our algorithm in practice.