Eigenvalues and Eigenvectors in Controllability Analysis

This chapter demonstrates the effectiveness of spectral theory in analyzing controllability property of linear and non-linear systems. System design, which is at the core of control system theory, relies heavily on spectral properties of the system matrices. With the aid of eigenvalues and eigenvectors, the state, input, and output matrices can be chosen so that the system behaves in a desired manner. We start by introducing the notion of controllability for linear time-variant (LTI) systems in terms of the spectral properties of the controllability Gramian derived from the state transition matrix of the system. We also define steering control using eigenvalues and eigenvectors of the controllability Gramian, which steers the system from an arbitrary initial state to a desired final state. For LTI systems, the eigenvalues and eigenvectors of the state matrix are used to characterize controllable and observable systems. Key concepts such as the Popov-Belevitch-Hautus(PBH) eigenvector test and Kalman’s rank condition are introduced to illustrate how these spectral tools guide the analysis and design of control systems. For nonlinear systems, the spectral properties of the controllability Gramian help in developing a computational algorithm for steering control. This chapter explores how spectral methods help in developing suitable control strategies and give a better understanding of system dynamics and system design.