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New upper and lower matrix bounds and the corresponding eigenvalue bounds on the solution of the discrete algebraic Riccati equation are discussed in this paper. The present bounds are tighter than the majority of those found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.

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This paper addresses the problem of robust stability for discrete homogeneous bilinear time-delay systems subjected to uncertainties. Two kinds of uncertainties are treated: (1) nonlinear uncertainties and (2) parametric uncertainties. For parametric uncertainties, we also discuss both unstructured uncertainties and interval matrices. By using the Lyapunov stability theorem associated with some linear algebraic techniques, several delay-independent criteria are developed to guarantee the robust stability of the overall system. One of the features of the newly developed criteria is its independence from the Lyapunov equation, although the Lyapunov approach is adopted. Furthermore, the transient response and the decay rate of the resulting systems are also estimated. In particular, the transient responses for the aforementioned systems with parametric uncertainties also do not involve any Lyapunov equation which remains unsolved. All the results obtained are also applied to solve the stability analysis of uncertain time-delay systems.