# SystemsModelLinearity

gives the linearity of the systems model sys.

SystemsModelLinearity[{sys,{in1,},{out1,},{s1,}}]

only considers the subsystem associated with inputs ini, outputs outj, and states sk.

# Details

• SystemsModelLinearity is typically used to determine whether a NonlinearStateSpaceModel or AffineStateSpaceModel satisfies additional linearity conditions, which would allow it to be exactly converted to a more specialized form and thus making a wider range of design and analysis techniques applicable.
• Possible systems models sys include TransferFunctionModel, StateSpaceModel, AffineStateSpaceModel, and NonlinearStateSpaceModel.
• A state space model with state , input , state equations and output equations can be classified based on what variables in and occur linearly.
• Possible values and the structural form required for both and are given below:
•  "Linear" linear in states and inputs, "Bilinear" linear in states and inputs separately, "StateLinear" linear only in states, "InputLinear" linear only in inputs, "Nonlinear" not linear in either states or inputs

# Examples

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## Basic Examples(2)

A linear mass-spring-damper (MSD) model:

A MSD model with the spring having cubic nonlinearity:

## Scope(7)

A linear model:

A NonlinearStateSpaceModel that is linear:

A bilinear model:

An input-linear model:

A model that is not linear:

Frequency domain models are linear models:

A model with linear state equations:

## Applications(2)

Verify the linearization obtained using StateTransformationLinearize:

Check if it is input-output linear:

Check if it is state-output linear:

Check if it is input-state linear:

Compute the linearization after systems connections:

Connect the two systems in a series:

The linearity of the original and connected systems:

## Properties & Relations(4)

A linear system obeys the principles of superposition and homogeneity:

Its response to the input signal :

Its response to :

The response to is the same as :

Typically StateSpaceModel is used to model linear state-space models:

The model can be exactly converted to any other systems model:

Typically AffineStateSpaceModel is used for models that are bilinear or input-linear:

It can be exactly converted to NonlinearStateSpaceModel:

Conversion to other systems models is approximate:

Use NonlinearStateSpaceModel for models that are not linear:

Conversion to any other systems model is approximate:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_systemsmodellinearity, author="Wolfram Research", title="{SystemsModelLinearity}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/SystemsModelLinearity.html}", note=[Accessed: 30-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_systemsmodellinearity, organization={Wolfram Research}, title={SystemsModelLinearity}, year={2014}, url={https://reference.wolfram.com/language/ref/SystemsModelLinearity.html}, note=[Accessed: 30-May-2024 ]}