Absolute stability analysis for negative-imaginary systems

This paper deals with absolute stability of a Lur'e system with positive feedback where the linear subsystem exhibits negative-imaginary frequency response and the nonlinearity connected in feedback is time-invariant, memoryless and slope-restricted. The proposed absolute stability criterion requires the linear subsystem to belong to the strongly strict negative-imaginary class. Along with that, positive definiteness of a symmetric matrix needs to be ensured, where the symmetric matrix is obtained by subtracting the dc-gain matrix of the linear subsystem from a strictly positive diagonal matrix with elements indicating the reciprocal of the maximum slope bounds of the nonlinearities. The stability criterion is proved using a Lur'e-Postnikov-type Lyapunov function. Numerical examples are presented to demonstrate the proposed results.