Model Reduction for Discrete-Time Systems via Optimization over Grassmann Manifold
Abstract
1. Introduction
- The -optimal model reduction methods based on Grassmann manifold optimization are extended to discrete-time LTI systems.
- For one-sided projection, a gradient flow method and a sequentially quadratic approximation approach are proposed to solve the optimization problem. For two-sided projection, the optimization problem is solved by applying the strategies of alternating direction iteration and sequentially quadratic approximation.
- We present the details of implementation, such as how to efficiently solve sparse–dense discrete-time Sylvester equations.
- The effectiveness of the proposed methods in this paper is demonstrated with two numerical examples.
2. Preliminaries
2.1. -Norm of Discrete-Time LTI Systems
2.2. Stiefel and Grassmann Manifolds
2.3. The Gradient Flow Method
3. One-Sided Projection via Optimization on Grassmann Manifold
3.1. Solving the Optimization Problem via the Gradient Flow Approach
Algorithm 1 Gradient flow method |
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3.2. Solving the Optimization Problem via Sequentially Quadratic Approximation
Algorithm 2 Sequentially quadratic approximation method |
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Algorithm 3 Two-sided projection method |
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4. Two-Sided Projection via Optimization on Grassmann Manifold
Solving the Optimization Problem via an Alternating Direction Approach
- Firstly, by projecting onto the tangent space of at point , a search direction is generated by
- Secondly, we define the search matrix , which should satisfy , by
- Finally, the -th iterate is constructed by the inexact line search with the Armijo rule. That is, for some given , we find the smallest positive number l so that satisfies the adequate reduction condition
- Construct the search direction by
- Define the search matrix by
- For some given , find the smallest positive number l so that
5. Implementation Issues
5.1. Initial Projection Matrix Selection
5.2. Termination Criterion
5.3. Solving Stein Equations
- Compute the real Schur decomposition , and set .
- Finally, compute .
6. Numerical Experiments
6.1. Example 1
6.2. Example 2
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Lin, Y.; Zhou, L. Model Reduction for Discrete-Time Systems via Optimization over Grassmann Manifold. Mathematics 2025, 13, 2767. https://doi.org/10.3390/math13172767
Lin Y, Zhou L. Model Reduction for Discrete-Time Systems via Optimization over Grassmann Manifold. Mathematics. 2025; 13(17):2767. https://doi.org/10.3390/math13172767
Chicago/Turabian StyleLin, Yiqin, and Liping Zhou. 2025. "Model Reduction for Discrete-Time Systems via Optimization over Grassmann Manifold" Mathematics 13, no. 17: 2767. https://doi.org/10.3390/math13172767
APA StyleLin, Y., & Zhou, L. (2025). Model Reduction for Discrete-Time Systems via Optimization over Grassmann Manifold. Mathematics, 13(17), 2767. https://doi.org/10.3390/math13172767