Controllability and observability grammians for balancing linear singularly perturbed systems

In this article we present a method to obtain an improved balancing transformation for linear singularly perturbed (SP) systems. First, an algorithm which utilizes the Chang transformation is introduced. The algorithm is used to obtain the exact controllability and observability Grammians (identical under balanced coordinates) of a linear system. The exact Grammians obtained in terms of reduced order Lyapunov/Sylvester equations corresponding to slow and fast subsystems are then used to construct the improved balanced realization of the SP system. A case study is provided at the end of the article to demonstrate the difference between the algorithm present in current literature and the method we develop.