An achievability condition for n-dimensional behaviors with a finite frequency specification: Dissipation inequalities approach

Many of practical design specifications are provided by finite frequency properties described by inequalities over restricted finite frequency intervals. The properties are known as the important property in integrated design of plant and controller. Recently, it was proved that the property is equivalent to the dissipation inequality in time and space domain, where the dissipation inequality is an inequality which describes the energy and power interaction of a dynamical system with its outside environment. In this paper, we consider a synthesis of a controller for finite frequency properties for n-dimensional systems described by partial differential equations. As a main result, we derive a necessary and sufficient condition for the achievability of a controller satisfying a finite frequency specification in terms of dissipation inequalities. This result clarifies a physical interpretation of the controller from the dissipativity viewpoint.