Observability Gramians

Determine the so-called controllability and observability gramians, the matrices P and Q. These are solutions to the equations ... 

If an adaptronic system is modelled in a state-space description (5.3), (5.4), its observability and controllability can be determined numerically by various methods. A common way is to compute the eigenvalues of the controllability and observability 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. 

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