Structural System Properties

If an LTI system is not completely state observable, rows of the observability matrix can be used to build a non-singular matrix that transforms the state-space representation into observability normal form, which yields the observable subspace of the state-space. The state-space representation can be transformed in a similar way into the controllability normal form which yields the controllable subspace. In case a system is neither completely state observable, nor completely state controllable, the two decompositions can be combined leading to the general Kalman decomposition of a state-space model. The transfer function matrix of the LTI system is determined by the matrices that constitute the observable and controllable submodel. These matrices build a minimal state-space representation. Structural analysis independent of the numerical values of matrix elements can be applied to check for structural observability and structural controllability for a class of LTI systems that have the same structure. The practical use is that a system that is not structurally state observable (controllable) is not numerically state observable (controllable).