On bounded real matrix inequality dilation

We discuss a variation of dilated matrix inequalities for the conventional Bounded Real matrix inequality, and other similarly structured inequalities. Here, system matrices are separated from Lyapunov matrix to allow the use of different Lyapunov matrices in multiobjective and robust problems. The search involves a bounded scalar parameter that enters the problem nonlinearly, and is dealt with a line search. To demonstrate the benefits of the new dilated matrix inequalities over the conventional ones, an example of controller synthesis with L2-gain performance measure (H∞ control) for a system with polytopic uncertainty (robust problem) has been studied. It is shown that for the resulting robust problem the performance obtained via the dilated form is at least equal to those of the conventional one. Also, the connection between the proposed dilated form and the Full Block S-procedure is discussed.