\(D\)-Stability Analysis of Structured Matrices Appearing in First and Second Order Economy Models

Abstract

The $D$-stability of structured matrices has significant implications in system theory and decision-making systems. The notation of $D$-stability refers to the characterization of a dynamical model where structured stability is maintained when each eigenvalue of the system matrix remains within a designated region, we consider it in half of right complex plane $\mathbb{C}$ subject to various perturbations. The structured $D$-stability of time varying dynamical systems implies that the system will remain stable under certain diagonal transformations. This concept is fundamental in system theory and ensures the robust stability and  performance even if system experiences structural changes or parameter uncertainties. In this paper, novel results are obtained on the computation of structured stability and structured $D$-stability of first order and second dynamical models with mathematical forms
$$\frac{d}{dt}(x(t)) = Ax,\,\,\,\,\frac{d^2}{dt^2}(x(t)) = A\frac{d}{dt}(x(t)) + Bx,\,\,x\in \mathbb{R}^{n,1},$$
with matrices $A,B \in \mathbb{R}^{n,n}$.
New results are developed with the necessary conditions for interconnection among stable, structured $D$-stable matrices and structured singular values. The numerical experimentation show how structured singular values ($\mu$-values) behaves. The EigTool is used to sketch the behavior of pseudo-spectrum.