Construct the Kalman controllable decomposition:

ControllableDecomposition picks out the controllable subsystem only:

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

Compute the dimension of the controllable subspace:

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

Find the controllable subspace for the system below and explain the possible state trajectories that can occur in the system:

The system is uncontrollable, so only a subspace is controllable:

The range of the transformation matrix p gives the controllable subspace:

Trajectories can be controlled in this subspace:

But you cannot control the drift between parallel copies of the controllable subspace:

Design a controller for a system that is not completely controllable, by building a controller for the controllable subsystem:

Since a force applied only on the first mass , the second mass is not controllable:

Design a controller using the controllable subsystem:

Use the transformation to obtain the controller for the original system:

The simulation shows controlled as well as oscillatory modes in the closed-loop system:

Compute the modes of the closed-loop system:

The oscillatory modes are the uncontrollable ones:

Construct the triangular controllability decomposition:

ControllableDecomposition picks out the controllable subsystem only:

Triangular controllability decomposition puts the controllable subsystem first and keeps the rest:

Compute the dimension of the controllable subspace:

The dimension can be obtained from the inverse transformation :

The controllable decomposition gives the reachable subspace and subsystem. This can be used to visualize the motion of the system from one subspace to another:

The generic reachable subspace:

A specific reachable subspace:

Select several initial points on this surface:

All these initial points end up at exactly the same final surface because that motion is not controllable:

A 3D plot shows the motion from the initial surface to the final one: