Minimization of Pseudospectral Abscissa in Large-Scale Systems

Основные понятия

Efficiently minimize pseudospectral abscissa for large-scale systems.

Аннотация

The content discusses minimizing the pseudospectral abscissa of a matrix-valued function dependent on parameters. It introduces a subspace procedure to handle large-sized matrix functions, ensuring global convergence and superlinear convergence properties. The advantages of minimizing the pseudospectral abscissa over other stability metrics are highlighted. The article also delves into nonconvex minimax eigenvalue optimization problems and proposes a subspace framework for such scenarios. Various algorithms and methods are discussed for efficient computation of the pseudospectral abscissa, especially for large matrices. Theoretical analyses, interpolation properties, and convergence results are presented for both finite and infinite-dimensional settings.

Статистика

A classical problem is stabilization by static output feedback (SOF).
Spectral abscissa minimization is a nonconvex eigenvalue optimization problem.
Algorithms like criss-cross and fixed-point iteration are used for computing the pseudospectral abscissa efficiently.
The proposed subspace framework shows global convergence and superlinear rate of convergence.
Hermite interpolation properties exist between full and reduced problems.

Цитаты

"There are notable advantages in minimizing the pseudospectral abscissa over maximizing the distance to instability or minimizing the H∞ norm."
"The proposed subspace procedure solves a sequence of reduced problems obtained by restricting the matrix-valued function to small subspaces."
"Designing a subspace framework for a minimax problem with the eigenvalue function in the constraint requires special treatment."

Ключевые выводы из

Large-Scale Minimization of the Pseudospectral Abscissa

by Nicat Aliyev... в arxiv.org 03-15-2024

https://arxiv.org/pdf/2208.07540.pdf

Дополнительные вопросы

How does minimizing the pseudospectral abscissa relate to robust stability considerations

Minimizing the pseudospectral abscissa is closely related to robust stability considerations in control systems. The pseudospectral abscissa provides valuable information about the worst-case transient growth of a system, which is crucial for ensuring robust stability. By minimizing the pseudospectral abscissa, we are essentially minimizing a lower bound on the largest possible transient growth of the system. This optimization approach allows us to quantify and mitigate potential instabilities or excessive transients in the system's behavior.

What challenges arise when dealing with nonconvex minimax eigenvalue optimization problems

Dealing with nonconvex minimax eigenvalue optimization problems poses several challenges. One major challenge is that these problems do not have a simple convex structure, making them computationally complex and difficult to solve efficiently. Nonconvexity can lead to multiple local minima, making it challenging to find the global optimum solution. Additionally, nonsmoothness at locally optimal points can further complicate optimization algorithms' convergence and efficiency.

How can these concepts be applied in real-world engineering systems beyond theoretical analysis

The concepts of minimizing pseudospectral abscissa and solving nonconvex minimax eigenvalue optimization problems have practical applications in real-world engineering systems beyond theoretical analysis. In engineering systems such as control systems, signal processing, and structural dynamics, optimizing stability metrics like pseudospectral abscissa can help design more robust and stable systems with improved performance characteristics. These techniques are particularly useful in designing controllers for autonomous vehicles, aircraft flight control systems, power grids stabilization, robotics applications where robust stability is critical for safe operation under various conditions.