# 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

open allclose all

## 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:

Wolfram Research (2014), SystemsModelVectorRelativeOrders, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.

#### Text

Wolfram Research (2014), SystemsModelVectorRelativeOrders, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.

#### CMS

Wolfram Language. 2014. "SystemsModelVectorRelativeOrders." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.

#### APA

Wolfram Language. (2014). SystemsModelVectorRelativeOrders. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html

#### BibTeX

@misc{reference.wolfram_2024_systemsmodelvectorrelativeorders, author="Wolfram Research", title="{SystemsModelVectorRelativeOrders}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html}", note=[Accessed: 12-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_systemsmodelvectorrelativeorders, organization={Wolfram Research}, title={SystemsModelVectorRelativeOrders}, year={2014}, url={https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html}, note=[Accessed: 12-August-2024 ]}