# ToContinuousTimeModel

ToContinuousTimeModel[lsys]

gives the continuous-time approximation of the discrete-time systems models lsys.

ToContinuousTimeModel[tfm,s]

specifies the transform variable s.

# Details and Options  • ToContinuousTimeModel is also known as inverse sampling.
• The systems model lsys can be a TransferFunctionModel or StateSpaceModel.
• ToContinuousTimeModel accepts a Method option that can be used to specify the approximation method.
• Possible settings for the Method option include:
•  "ForwardRectangularRule" Euler forward method "BackwardRectangularRule" Euler backward method "BilinearTransform" Tustin bilinear approximation "ZeroPoleMapping" exact matching of zeros and poles "ZeroOrderHold" piecewise constant approximation "FirstOrderHold" piecewise linear (triangular) approximation
• The default setting implements the bilinear transformation.
• The setting Method->"ZeroPoleMapping" does not support time delays.
• A bilinear transform with critical frequency ω radians per time unit can be specified by setting Method->{"BilinearTransform","CriticalFrequency"->ω}.
• The "FirstOrderHold" method gives the non-causal first-order hold equivalent.
• When approximating TransferFunctionModel objects, "StateSpaceConversion" can be specified as a suboption.
• The setting Method->{m,"StateSpaceConversion"->True} computes the approximation using the state-space representation and converts the result to the transfer-function representation.
• The setting "StateSpaceConversion"->False computes using the transfer-function object.
• The setting "StateSpaceConversion"->Automatic computes the approximation using the transfer-function representation, except for the "ZeroOrderHold" method.

# Examples

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## Basic Examples(1)

A continuous-time approximation of a discrete-time system:

## Scope(6)

Convert a discrete-time transfer-function model to a continuous-time model:

Convert a discrete-time state-space model to a continuous-time model:

Convert a multiple-input, multiple-output system to a continuous-time system:

Convert a symbolic system:

Convert a time-delay StateSpaceModel:

Convert a singular descriptor StateSpaceModel:

## Options(6)

### Method(6)

By default, the approximation is based on the bilinear transformation:

Specify the approximation method:

Compare various approximation methods:

An approximation that preserves the transmission at a specified frequency:

The backward Euler method adds states if the state and descriptor matrices are both singular:

The bilinear method adds states if the sum of the state and descriptor matrices is singular:

## Applications(1)

Various continuous-time approximations to a fourth-order Chebyshev II bandstop filter:

Bode plots:

## Properties & Relations(2)

ToDiscreteTimeModel is essentially the inverse of ToContinuousTimeModel:

ToContinuousTimeModel may add states to systems with neutral time delays: