This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# TransferFunctionModel

 TransferFunctionModelrepresents the model of the transfer-function matrix m with complex variable var. TransferFunctionModelspecifies the numerator num and denominator den of a transfer-function model. TransferFunctionModel[ss]gives the transfer-function model of the StateSpaceModel object ss.
• TransferFunctionModel can represent either a multiple-input, multiple-output (MIMO) or a single-input, single-output (SISO) system.
• The system can be in either continuous-time or discrete-time domain.
• The default value of var is for continuous-time systems and for discrete-time systems.
• In TransferFunctionModel num must be a polynomial matrix and den can be specified as a polynomial matrix or just the common denominator polynomial.
• The following options can be given:
 Method Automatic the method to obtain the transfer function of a state-space model SamplingPeriod None the sampling period of the system SystemsModelLabels None labels for the input and output variables
• Settings for the Method option include , , , and . With a setting Method, the transfer-function model is computed using determinant expansion.
A single-input, single-output system:
A system with two inputs and one output:
Obtain the transfer-function representation of a state-space model:
A discrete-time transfer function with a sampling period of 1:
Evaluate a transfer function over a range of frequencies:
Plot the magnitudes:
A single-input, single-output system:
 Out[1]=

A system with two inputs and one output:
 Out[1]=

Obtain the transfer-function representation of a state-space model:
 Out[1]=

A discrete-time transfer function with a sampling period of 1:
 Out[1]=

Evaluate a transfer function over a range of frequencies:
 Out[1]=
Plot the magnitudes:
 Out[2]=
 Scope   (18)
A first-order continuous-time system:
A second-order system:
A fifth-order system:
A system with three zeros and six poles:
A first-order discrete-time system:
A second-order discrete-time system with symbolic sampling period:
A two-input, one-output system:
A one-input, two-output system:
A two-input, two-output system:
Specify a transfer function using its numerator and denominator:
A MIMO transfer function specified in terms of its numerators and denominators:
A denominator polynomial that is the least common multiple:
Specifying the numerators and denominators avoids pole-zero cancellations:
Alternatively, use Unevaluated:
A constant gain of 10:
A discrete-time gain:
A symbolic gain:
The transfer-function representation of a state-space model:
The default complex variable for continuous-time systems is :
For discrete-time systems it is :
SISO systems can also be specified as a single-element list:
Or just as a rational function:
A single-output system can be given as a list:
 Options   (4)
Specify a continuous-time system:
A discrete-time system with sampling period 1:
A system with a symbolic sampling period:
Set the sampling period to a numeric value:
Label the input and output variables:
 Applications   (17)
A proportional-integral (PI) controller:
A proportional-derivative (PD) controller:
A function to construct a proportional-integral-derivative (PID) controller:
A PID with specific gain values:
A function to construct a discrete-time PID controller:
A function for a continuous-time lead compensator:
A lead compensator for specific values of gain and pole-zero locations:
A function for a continuous-time lag compensator:
A specific lag compensator:
A digital lag compensator defined in terms of its zero and pole locations:
A general formula for analog low-pass Butterworth filters:
Filters of specific orders:
A third-order Bessel filter:
The general second-order transfer function:
Variations in damping ratio lead to qualitatively different responses:
A linearized inverted pendulum model:
A spring-mass-damper system:
Transfer function between the input voltage and the shaft angular position of a DC motor:
The aileron to roll-rate transfer function of an aircraft:
A temperature-controlled chemical reactor:
An RLC circuit:
A MIMO transfer function describing an aircraft's longitudinal dynamics:
TransferFunctionModel behaves as a pure function of one argument:
The value of the transfer-function matrix at a specific frequency:
The values at several frequencies:
Use TransferFunctionFactor to obtain the factored form:
Obtain the expanded form:
Use TransferFunctionCancel to cancel any common poles and zeros:
Collect terms with similar powers:
Collect terms in any variable:
Find the element zeros and poles of a transfer-function matrix:
Obtain a state-space form of a transfer-function model:
In TransferFunctionModel pole-zero pairs may cancel before being processed by TransferFunctionModel:
Use Unevaluated to prevent cancellations:
TransferFunctionModel[Unevaluated[m], var] and TransferFunctionModel only prevent pole-zero cancellations:
TransferFunctionModel might result in a system with higher order:
Simplify the system:
Or simplify m before passing it to TransferFunctionModel:
If the complex variable var is not specified it is assumed to be for continuous-time systems:
Specify the transfer function using :
For discrete-time systems use :
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