ℒ2-gain analysis of systems with state and input delays

Sufficient conditions are presented for a system with input delays having finite ℒ₂-gain. The bounded real lemma conditions are infeasible when the actual and the delayed values of the input act on the system simultaneously. Considering the actual and delayed inputs as two independent inputs is shown to lead to very high upper-bound of the true ℒ₂-gain. Several approaches are presented involving a scaling method which is based on upper-bounds of the delayed signal norms; a method which transforms the input delays to state delays with the help of introducing additional dynamics; and a method based on integral quadratic constraints. Both time-invariant and time-varying delays are considered. The methods are evaluated on an example of interconnected vehicles.