The decoupling of second-order linear systems with a singular mass matrix

Abstract

TitleThe decoupling of second-order linear systems with a singular mass matrix a b s t r a c tIt was demonstrated in earlier work that a nondefective, linear dynamical system with an invertible mass matrix in free or forced motion may be decoupled in the configuration space by a real and isospectral transformation. We extend this work by developing a procedure for decoupling a linear dynamical system with a singular mass matrix in the configuration space, transforming the original differential-algebraic system into decoupled sets of real, independent, first-and second-order differential equations. Numerical examples are provided to illustrate the application of the decoupling procedure.

Linear Stability Analysis of Transient Electrodeposition in Charged Porous Media: Suppression of Dendritic Growth by Surface Conduction

“…The infinite eigenvalues are physically irrelevant to the linear stability analysis [191,192], therefore we would want to focus on solving for the finite eigenvalues. This can be achieved by mapping the infinite eigenvalues to other arbitrarily chosen points in the complex plane via simple matrix transformations [193].…”

Robust pole assignment using velocity–acceleration feedback for second-order dynamical systems with singular mass matrix

“…However, in [1,2] the mass matrix M is regular while in the present study the mass matrix M is singular, rankðMÞ ¼ q o n; q 4 0. Then, the system is called singular, generalized, or descriptor second-order system; see [3][4][5][6][7][8][9][10][11][12]. One inherent characteristic of descriptor second-order linear systems is their impulsive natural response which is generated by infinite eigenvalues.…”

“…Second-order systems with singular mass matrix arise naturally in mechanical multi-body systems and a variety of other practical applications; see [3][4][5][6][7][8][9][10][11][12]. In 1982, the second order generalized systems are applied to power systems by Campbell and Rose [3].…”

“…Especially in modeling complex multibody systems, it is useful to use redundant coordinates, and in such situations singular mass matrices can arise when describing the unconstrained system [7]. In addition, when a mass matrix M is invertible but contains very small terms such that M is ill-conditioned, it is common practice to set these terms to zero, rendering M singular [11]. The controllability and observability conditions for descriptor second-order linear systems have been analyzed in [8].…”

“…The output regulation for matrix secondorder descriptor systems using output feedback measurement is presented [10]. Moreover, a procedure for decoupling a secondorder linear system with a singular mass matrix is developed [11]. Recently, some new results on the eigenstructure assignment for the second-order system with a singular mass matrix are developed in [12].…”