Haar Wavelets and \(D\)-Stability of Lumped-Parameter Dynamical Systems

Abstract

$D$-stability is a well-known mathematical tool used to analyze and characterize dynamical systems. It plays an important role in the stability analysis of dynamical systems, particularly in cases where stability is preserved under various types of perturbation, especially those involving positive diagonal scaling. The analysis of $D$-stability ensures the stability of dynamical systems. In this paper, we present new results on the characterization of $D$-stability and strong $D$-stability for structured matrices of the form $(I_n - A \otimes P^t)$, where $I_n$ is an $n \times n$ identity matrix and the matrices $A$ and $P$ associated with a lumped-parameter dynamical system
\[\begin{cases}
  x(t) = A~x(t) + B ~u(t),\,\,\,\, x(0)=x_0\\
  y(t) = C~x(t) + D~u(t). 
\end{cases}\]
The results on $D$-stability and strong $D$-stability are obtained using mathematical tools from linear algebra, matrix analysis, system theory and their interactions with the computation of structured singular values. Furthermore, we present the numerical approximations to singular values and pseudo-spectrum of Haar wavelet matrices associated with a lumped-parameter dynamical system.