[Paper Review] Memory-Efficient Learning of Stable Linear Dynamical Systems for Prediction and Control

Giorgos Mamakoukas, Orest Xherija+1 more|arXiv (Cornell University)|Jun 6, 2020

Model Reduction and Neural Networks36 references10 citations

TL;DR

This paper proposes a memory-efficient, gradient-based algorithm for learning stable linear dynamical systems (LDS) that enforces stability at every optimization step using a recent characterization of stable matrices. By jointly updating both state and control matrices, the method achieves orders-of-magnitude lower reconstruction error and superior control performance compared to state-of-the-art approaches, with O(n²) space complexity—scaling effectively to high-dimensional systems like robotic manipulators and video sequences.

ABSTRACT

Learning a stable Linear Dynamical System (LDS) from data involves creating models that both minimize reconstruction error and enforce stability of the learned representation. We propose a novel algorithm for learning stable LDSs. Using a recent characterization of stable matrices, we present an optimization method that ensures stability at every step and iteratively improves the reconstruction error using gradient directions derived in this paper. When applied to LDSs with inputs, our approach---in contrast to current methods for learning stable LDSs---updates both the state and control matrices, expanding the solution space and allowing for models with lower reconstruction error. We apply our algorithm in simulations and experiments to a variety of problems, including learning dynamic textures from image sequences and controlling a robotic manipulator. Compared to existing approaches, our proposed method achieves an orders-of-magnitude improvement in reconstruction error and superior results in terms of control performance. In addition, it is provably more memory-efficient, with an O(n^2) space complexity compared to O(n^4) of competing alternatives, thus scaling to higher-dimensional systems when the other methods fail.

Motivation & Objective

• Address the critical need for stable, data-driven linear dynamical systems in control and prediction tasks, especially in safety-critical robotics and vision applications.
• Overcome the limitations of existing methods that either fail to scale due to high memory complexity (O(n⁴)) or do not jointly optimize state and control matrices.
• Enable online and real-time learning of stable LDS models by ensuring stability from the first optimization step, crucial for time-sensitive and safety-critical systems.
• Improve model fidelity and control performance by expanding the solution space through joint optimization of state and control matrices.
• Demonstrate the method’s effectiveness on high-dimensional systems, including dynamic texture modeling and robotic control, where prior methods fail due to memory or stability constraints.

Proposed method

• Leverages a recent characterization of stable matrices (based on Lyapunov equations) to define a feasible set of stable system matrices, enabling projection-based optimization.
• Derives closed-form gradient directions for both the state matrix A and control matrix B using a projected gradient descent framework that maintains stability at every iteration.
• Applies a projected optimization scheme where each update step projects the current estimate onto the set of stable matrices, ensuring stability is preserved throughout training.
• Uses a least-squares objective for reconstruction error minimization, with gradients computed via matrix calculus under the stability constraint.
• Employs a fast gradient method with efficient Hessian approximation to reduce computational overhead, enabling scalability to high-dimensional systems.
• Extends the method to input-affine LDSs by deriving joint gradients for A and B, allowing for better representation of control dynamics than methods that fix B or only optimize A.

Experimental results

Research questions

• RQ1 Can a stable linear dynamical system be learned from data with provably lower memory complexity than existing methods?
• RQ2 Does jointly optimizing both state and control matrices lead to significantly lower reconstruction error and better control performance compared to methods that only optimize the state matrix?
• RQ3 Can a gradient-based method enforce stability at every optimization step, enabling faster convergence and earlier stability in online learning settings?
• RQ4 How does the proposed method perform in real-world control tasks, such as robotic manipulation, compared to state-of-the-art stable LDS learning algorithms?
• RQ5 Can the method scale to high-dimensional systems (e.g., video sequences, robotic arms) where existing O(n⁴) methods fail due to memory constraints?

Key findings

• The proposed method achieves an orders-of-magnitude lower reconstruction error than the current top-performing methods (CG and WLS) on benchmark dynamic texture datasets.
• In robotic control tasks, the method achieves a mean tracking error of 0.0799 over 50 trials, outperforming WLS (38.73) and LS (7556), with CG scoring 0.0810, showing near-identical performance.
• The method demonstrates robustness in low-resource settings, maintaining high performance even with limited training data (e.g., 600 measurements for the Franka Panda robot).
• The algorithm is provably more memory efficient, with O(n²) space complexity, compared to O(n⁴) of competing methods, enabling scalability to high-dimensional systems.
• The simulation results are validated experimentally on the Franka Emika Panda robot, showing that the control performance in simulation accurately predicts real-world behavior.
• The method is the first to successfully evaluate control performance on stable LDS models, demonstrating that stability and fidelity together are essential for reliable control.