Compute the vector-relative orders of a system:

The vector-relative order of a single-output system:

The vector-relative orders of a multiple-output system:

The vector-relative orders of a StateSpaceModel:

The vector-relative order of a TransferFunctionModel:

Determine the number of states in the feedback-linearized and residual system:

The number of states in the linear subsystem:

The linearized subsystem:

The number of states in the residual subsystem:

The residual subsystem:

In AsymptoticOutputTracker, the decay rates are assigned based on relative orders:

The first three poles correspond to the first output and the last two to the second:

The first output has slower decay rates and hence takes longer to start following the reference input:

The length of the vector-relative orders is equal to the number of outputs:

Use SystemsModelDimensions to get the number of outputs:

For scalar systems with no zero dynamics, the vector-relative order equals the order:

Use SystemsModelOrder to get the order of the system:

If the vector-relative orders are finite, a linearizing feedback control can be computed:

Use FeedbackLinearize to compute the linearizing feedback:

If the sum of vector-relative orders is equal to the order, there are no zero dynamics:

The relative orders give the lengths of the integrator chains of the feedback-linearized system:

Output 1 has a single integrator and output 2 has a double integrator:

The vector-relative orders are invariant wrt a coordinate transform:

The vector-relative orders correspond to the asymptotic orders of the transfer functions:

Output 1 (first row) has asymptotic order 3, and output 2 (second row) has asymptotic order 2:

For the decoupling matrix to be invertible, the outputs must be independent:

There must be at least as many inputs as outputs:

If an operating point is specified, the decoupling matrix is evaluated at that point:

Although the system is feedback linearizable, there is a singularity at the operating point: