Construct the Kalman observable decomposition:

ObservableDecomposition picks out the observable subsystem only:

Kalman observable decomposition puts the observable subsystem first and keeps the rest:

Compute the dimension of the observable subspace:

The observable subspace is the range of p, i.e. the column dimension:

Find the observable subspace for the system below and show what state trajectories you can tell apart from observing the output only:

The system is unobservable, so only a subspace is observable from output:

The range of the transformation p gives the observable subspace:

Simulate trajectories whose initial value projects to a single point on the observable subspace:

From observing the output, all these trajectories look identical:

Determine states that can be estimated using available measurements and design an estimator:

Only the position of mass is measured, and so the system is not completely observable:

The states associated with zero rows in the transformation matrix cannot be observed:

An estimator can be designed to estimate any combination of the first four states:

An estimator that estimates :

Compute the response of for a set of input signals and initial conditions:

The estimated state trajectories:

Construct the triangular observability decomposition:

ObservableDecomposition picks out the observable subsystem only:

Triangular observability decomposition puts the observable subsystem first and keeps the rest:

Compute the dimension of the observable subspace:

The dimension can be obtained from the inverse transformation :

Find the subspaces whose outputs are indistinguishable:

The system is unobservable, so only a subspace is observable from output:

The indistinguishable subspace:

Two points on the subspace:

The trajectories of the outputs from the two points are the same: