Spectral Decompositions of Controllability Gramian and Its Inverse based on System Eigenvalues in Companion Form

TL;DR

This work develops spectral decompositions (SDSE) of the controllability Gramian and its inverse for continuous LTI systems in the controllability (companion) form, expressing them as sums of Hermitian components associated with system eigenvalues and resonances. It provides explicit, unique SDSEs for both algebraic and differential Lyapunov/Riccati equations, including finite-time extensions with arbitrary initial conditions and extensions to multiple eigenvalues. The spectral components have a measurable interpretation in minimum-energy control and enable closed-form insights into eigenmode influence, resonance effects, and practical design tasks such as actuator placement and model reduction. By unifying modal analysis with Lyapunov-based stability and control theory, the results enhance accuracy and tractability in stability analysis, minimum-energy control, and network design applications.

Abstract

Controllability and observability Gramians, along with their inverses, are widely used to solve various problems in control theory. This paper proposes spectral decompositions of the controllability Gramian and its inverse based on system eigenvalues for a continuous LTI dynamical system in the controllability canonical (companion) form. The Gramian and its inverse are represented as sums of Hermitian matrices, each corresponding to individual system eigenvalues or their pairwise combinations. These decompositions are obtained for the solutions of both algebraic and differential Lyapunov and Riccati equations with arbitrary initial conditions, allowing for the estimation of system spectral properties over an arbitrary time interval and their prediction at future moments. The derived decompositions are also generalized to the case of multiple eigenvalues in the dynamics matrix spectrum, enabling a closed-form estimation of the effects of resonant interactions with the system's eigenmodes. The spectral components are interpreted as measurable quantities in the minimum energy control problem. Therefore, they are unambiguously defined and can quantitatively characterize the influence of individual eigenmodes and associated system devices on controllability, observability, and the asymptotic dynamics of perturbation energy. The additional information obtained from these decompositions can improve the accuracy of algorithms in solving various practical problems, such as stability analysis, minimum energy control, structural design, tuning regulators, optimal placement of actuators and sensors, network analysis, and model order reduction.