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


open allclose all

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,


@misc{reference.wolfram_2021_systemsmodellinearity, author="Wolfram Research", title="{SystemsModelLinearity}", year="2014", howpublished="\url{}", note=[Accessed: 01-December-2021 ]}


@online{reference.wolfram_2021_systemsmodellinearity, organization={Wolfram Research}, title={SystemsModelLinearity}, year={2014}, url={}, note=[Accessed: 01-December-2021 ]}


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


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