WOLFRAM

represents the model of the transfer-function matrix g[s] with complex variable s.

TransferFunctionModel[{n[s],d[s]},s]

specifies the numerator n[s] and denominator d[s] of a transfer-function model.

specifies the zeros z, poles p, and gain g of a transfer-function model.

gives the transfer-function model of the systems model sys.

Details and Options

Examples

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Basic Examples  (5)Summary of the most common use cases

A single-input, single-output system:

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A system with two inputs and one output:

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Obtain the transfer-function representation of a state-space model:

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A discrete-time transfer function with a sampling period of 1:

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Evaluate a transfer function over a range of frequencies:

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Plot the magnitudes:

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Scope  (19)Survey of the scope of standard use cases

A first-order continuous-time system:

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A second-order system:

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A fifth-order system:

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A system with three zeros and six poles:

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A first-order discrete-time system:

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A two-input, one-output system:

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A one-input, two-output system:

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A two-input, two-output system:

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Specify a transfer function using its numerator and denominator:

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A MIMO transfer function specified in terms of its numerators and denominators:

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A denominator polynomial that is the least common multiple:

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Specify the transfer function, using its algebraic poles, zeros, and gains:

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A multivariable system:

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A constant gain of 10:

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A discrete-time gain:

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A symbolic gain:

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The transfer-function representation of a state-space model:

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Taylor linearize an AffineStateSpaceModel and obtain its transfer function representation:

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The linearization of an AffineStateSpaceModel with nonzero equilibrium values:

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Taylor linearize a NonlinearStateSpaceModel:

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The list of available properties:

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Generalizations & Extensions  (2)Generalized and extended use cases

SISO systems can also be specified as a single-element list:

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Or just as a rational function:

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A single-output system can be given as a list:

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Options  (5)Common values & functionality for each option

SamplingPeriod  (3)

Specify a continuous-time system:

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A discrete-time system with sampling period 1:

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A system with a symbolic sampling period:

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Set the sampling period to a numeric value:

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SystemsModelLabels  (1)

Label the input and output variables:

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By default, the appearance is selected to fit the display in the notebook:

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Applications  (18)Sample problems that can be solved with this function

A proportional-integral (PI) controller:

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A proportional-derivative (PD) controller:

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A function to construct a proportional-integral-derivative (PID) controller:

A PID with specific gain values:

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A function to construct a discrete-time PID controller:

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A function for a continuous-time lead compensator:

A lead compensator for specific values of gain and pole-zero locations:

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A function for a continuous-time lag compensator:

A specific lag compensator:

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A digital lag compensator defined in terms of its zero and pole locations:

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A general formula for analog lowpass Butterworth filters:

Filters of specific orders:

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A third-order Bessel filter:

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The general second-order transfer function:

Variations in damping ratio lead to qualitatively different responses:

A linearized inverted pendulum model:

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A spring-mass-damper system:

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Transfer function between the input voltage and the shaft angular position of a DC motor:

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The aileron-to-roll-rate transfer function of an aircraft:

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A temperature-controlled chemical reactor:

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An RLC circuit:

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A MIMO transfer function describing an aircraft's longitudinal dynamics:

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A ball mill grinding system with delay due to material transport:

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Properties & Relations  (8)Properties of the function, and connections to other functions

TransferFunctionModel behaves as a pure function of one argument:

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The value of the transfer-function matrix at a specific frequency:

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The values at several frequencies:

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Use TransferFunctionFactor to obtain the factored form:

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Obtain the expanded form:

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Use TransferFunctionCancel to cancel any common poles and zeros:

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Collect terms with similar powers:

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Collect terms in any variable:

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Find the element zeros and poles of a transfer-function matrix:

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Obtain a state-space form of a transfer-function model:

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Possible Issues  (3)Common pitfalls and unexpected behavior

In TransferFunctionModel[m,var], pole-zero pairs may cancel before being processed:

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Use Unevaluated to prevent cancellations:

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Or use TransferFunctionModel[{num,den},var]:

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Or TransferFunctionModel[{z,p,g},var]:

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TransferFunctionModel[m,var] might result in a system with higher order:

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Simplify the system:

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Or simplify m before passing it to TransferFunctionModel:

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If the complex variable var is not specified, it is assumed to be s for continuous-time systems:

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Specify the transfer function using s:

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For discrete-time systems, use z:

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Wolfram Research (2010), TransferFunctionModel, Wolfram Language function, https://reference.wolfram.com/language/ref/TransferFunctionModel.html (updated 2014).
Wolfram Research (2010), TransferFunctionModel, Wolfram Language function, https://reference.wolfram.com/language/ref/TransferFunctionModel.html (updated 2014).

Text

Wolfram Research (2010), TransferFunctionModel, Wolfram Language function, https://reference.wolfram.com/language/ref/TransferFunctionModel.html (updated 2014).

Wolfram Research (2010), TransferFunctionModel, Wolfram Language function, https://reference.wolfram.com/language/ref/TransferFunctionModel.html (updated 2014).

CMS

Wolfram Language. 2010. "TransferFunctionModel." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/TransferFunctionModel.html.

Wolfram Language. 2010. "TransferFunctionModel." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/TransferFunctionModel.html.

APA

Wolfram Language. (2010). TransferFunctionModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TransferFunctionModel.html

Wolfram Language. (2010). TransferFunctionModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TransferFunctionModel.html

BibTeX

@misc{reference.wolfram_2025_transferfunctionmodel, author="Wolfram Research", title="{TransferFunctionModel}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/TransferFunctionModel.html}", note=[Accessed: 03-June-2025 ]}

@misc{reference.wolfram_2025_transferfunctionmodel, author="Wolfram Research", title="{TransferFunctionModel}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/TransferFunctionModel.html}", note=[Accessed: 03-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_transferfunctionmodel, organization={Wolfram Research}, title={TransferFunctionModel}, year={2014}, url={https://reference.wolfram.com/language/ref/TransferFunctionModel.html}, note=[Accessed: 03-June-2025 ]}

@online{reference.wolfram_2025_transferfunctionmodel, organization={Wolfram Research}, title={TransferFunctionModel}, year={2014}, url={https://reference.wolfram.com/language/ref/TransferFunctionModel.html}, note=[Accessed: 03-June-2025 ]}