Stability and positivity analysis of singular systems with time-varying delays

Abstract

This dissertation is dedicated to the problems of stability and positivity for singular systems with time-varying delays. Standard singular system and positive singular system are both considered.

For a singular system with time-varying delays, the characterizations of robust admissibility and admissibilization are established for discrete-time cases. It is demonstrated that the time-varying delay can be estimated by an appropriate comparison model in order to tackle the uncertainty of the delay. An improved admissibility condition in terms of linear matrix inequality is proposed by means of a transformed system, which guarantees a singular system with time-varying delays to be admissible. Then, the problem of robust admissibilization is further investigated for the uncertain singular time-delay system. Moreover, an iterative linear matrix inequality algorithm is developed for obtaining a state-feedback controller.

For positive singular systems with time-varying delays, the problems of positivity, stability and performance are studied. It is proved that a singular system can be equivalently transformed to an augmented system which is more intuitive for positivity analysis by using the Drazin inverse. For a discrete-time singular system with time-varying delays, a necessary and sufficient positivity condition is proposed by the property of positive invariance. A stability condition for the constant time-delay case is further investigated by linear co-positive Lyapnunov functionals and is extended to the time-varying delay case by exploring the relation between systems with constant delay and time-varying one. For continuous-time singular systems with time-varying delays, a positivity condition is derived by considering an auxiliary system with certain initial conditions. It is shown that the solution of a stable positive singular system with constant delay is monotonically nonincreasing with some initial conditions. By comparing the trajectory of a time-varying delay system with that of a constant time-delay system, the asymptotic stability condition is established for positive singular systems with time-varying delays. Furthermore, the characterizations of $\ell_\infty$-gain and $L_\infty$-gain are presented for both discrete-time and continuous-time positive singular systems with time-varying delays after their positivity is analyzed. It is shown that they are determined by the system matrices and independent of the magnitude of delays under certain assumptions.