H ∞ model reduction for discrete time-delay systems: delay-independent and dependent approaches

This paper investigates the problem of H ∞ model reduction for linear discrete-time state-delay systems. For a given stable system, our attention is focused on the construction of reduced-order models, which guarantee the corresponding error system to be asymptotically stable and have a prescribed H ∞ error performance. Both delay-independent and dependent approaches are developed, with sufficient conditions obtained for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to LMI constraints, which can be readily solved in standard numerical software. In addition, the development of reduced-order models with special structures, such as delay-free models and zeroth-order models, is also addressed. The approximation methods presented in this paper can be further extended to cope with systems with uncertain parameters. Two numerical examples have been provided to show the effectiveness of the proposed theories.