The Wolfram Language provides a complete set of functions needed to compute and verify the controllability and observability properties of linear systems, as well as advanced functions that yield decompositions with desired controllability and observability characteristics.

Controllability and Observability Properties

ControllableModelQ — test whether the states are controllable from the inputs

ObservableModelQ — test whether the states are observable from outputs

ControllabilityMatrix  ▪  ObservabilityMatrix  ▪  OutputControllableModelQ  ▪  OutputControllabilityMatrix  ▪  ControllabilityGramian  ▪  ObservabilityGramian

Controllability and Observability Transformations

MinimalStateSpaceModel — give the subsystem that is controllable and observable

InternallyBalancedDecomposition — balance state controllability and observability

DualSystemsModel  ▪  ControllableDecomposition  ▪  ObservableDecomposition

General Transformations

StateSpaceTransform — coordinate transformation of the states of a model

JordanModelDecomposition — transform state matrix to Jordan form

KroneckerModelDecomposition — decouple fast and slow subsystems