gives the linearity of the systems model sys.


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


  • 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


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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 TemplateBox[{t}, UnitStepSeq]:

The response to 3 sin(t)+5 TemplateBox[{t}, UnitStepSeq] 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,


Wolfram Research (2014), SystemsModelLinearity, Wolfram Language function,


Wolfram Language. 2014. "SystemsModelLinearity." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2014). SystemsModelLinearity. Wolfram Language & System Documentation Center. Retrieved from


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


@online{reference.wolfram_2024_systemsmodellinearity, organization={Wolfram Research}, title={SystemsModelLinearity}, year={2014}, url={}, note=[Accessed: 30-May-2024 ]}