# SystemsModelVectorRelativeOrders

gives the vector-relative orders of the systems model sys.

# Details and Options • SystemsModelVectorRelativeOrders is also known as vector-relative degrees.
• The relative order of an output of sys is essentially the number of times it must be differentiated for the input of sys to appear explicitly.
• Output has relative order if after differentiations of the output, .
• The vector-relative orders are the list of the relative orders for each output.
• The matrix is called the decoupling matrix and needs to have rank .
• The system sys can be a TransferFunctionModel, StateSpaceModel, or AffineStateSpaceModel.
• The following options can be given:
•  MaxIterations Automatic maximum number of differentiations ZeroTest Automatic zero test

# Examples

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## Basic Examples(1)

Compute the vector-relative orders of a system:

## Scope(4)

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:

## Applications(2)

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:

## Properties & Relations(7)

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:

## Possible Issues(3)

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: