Upper and Lower Bounds of Frequency Interval Gramians for a Class of Perturbed Linear Systems

The notions of controllability and observability play an important role in different problems within feedback control analysis and design. To verify the controllability and observability of a system, several techniques have been introduced. However, often it is not only important to verify if the system is controllable or observable, but also it is required to know the degree of controllability or observability of the system. Gramian matrices were introduced to address this issue by providing a quantitative measure for controllability and observability. In many applications, the information on the controllability and observability properties of a system is needed within a specific frequency interval rather than the whole frequency-domain. The frequency interval gramians provide such information. While this concept were originally introduced for fixed known systems, it needs to be investigated for the case of uncertain systems. In this paper, we derive upper and lower bounds of frequency interval gramians under perturbations of an A-matrix in the state-space form. These bounds are obtained by solving algebraic Riccati equations. The results are further used to obtain upper and lower bounds of the frequency interval Hankel singular values for perturbed systems.