Balancing of Gramians

High-order models are often a result of models consisting of many differential equations or partial differential equations that have been converted into ordinary differential equations. These types of model are adequate for simulations studies but are not suitable for online use. A popular technique of model reduction that does not make use of error minimization is the model balancing method. The procedure is to find observability and controllability Gramians so as to determine which states have the largest overall contribution to the model. In systems theory and linear algebra, a Gramian matrix is a real-values symmetric matrix that can be used for a test for linear independence of functions. A system is called controllable if all states X can be influenced by the control input vector w, a system is observable if all states can be determined from the measurement vector jp. 

Balancing of Gramians can effectively be seen as a method of model reduction while maintaining observability and controllability. 

Assume a state space model has the following structure  

The gramians can only be computed for stable systems. A system is called frequency weighted balanced if the matrices Wc and Wo are both equal and diagonal  

The transformed systems equations, also called the balanced realization, consequently become  

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Hahn, J., Edgar, T. F., An improved method for nonlinear model reduction using balancing of empirical gramians, Computers Chem. Eng. 2002, 26 1379— 1397. 

This is a transformation of the linearized system, and can be derived by substituting xb = Txi in the linearized system. The gramians Pb and Qb of the balanced system satisfy ... 

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