Balanced reduction relies on the internal balancing of the controllability and observability grammians to eliminate weakly observable and controllable modes. Weighting functions are often used to improve this procedure. Several benefits exist in implementing low-order rather than high-order controllers. It is, therefore, imperative to reduce any errors that result from using the infinite grammians in the controller reduction process. By fixing the structure of the high-order controller and choosing an appropriate frequency interval for the balanced reduction a system that assumes a second-order controller is obtained, forming the basis for an efficient approximation/optimization towards obtaining a robust PID controller. This paper establishes a procedure for choosing the weighting functions.
