EXTENSION OF MODAL ANALYSIS TO LINEAR TIME-VARYING SYSTEMS

Abstract

Linear time-varying (LTV) systems have been often dealt with on a case-by-case basis. Many well-developed concepts and analytic methods of linear time-invariant (LTI) systems cannot be applied to LTV systems. For example, the conventional definition of modal parameters is invalid for LTV systems. The first part of this paper explores the possibility of extending the modal concept of LTV systems. The discrete-time state-space model is used to represent LTV systems. By analogy to LTI systems, the pseudo-modal parameters are defined using the eigenvalues of the discrete-time state transition matrix. The paper shows that the pseudo-modal parameters preserve certain properties of the conventional modal parameters defined for LTI systems. The second part of the paper extends a previously developed algorithm to identify the pseudo-modal parameters using forced responses and forcing inputs. For a general LTV system, the input and output Hankel matrices formed by an ensemble of data satisfy a matrix facotrization relation. The key step of the method is to modify the output matrix in such a way that the range space of the observability matrix can be extracted. A robotic manipulator with varying inertia links is used as an example. The first part of the numerical simulation illustrates the applications of the pseudo-modal parameters. The second part of the simulation tests the identification algorithm under different conditions. Several practical issues are addressed.