ToContinuous - Maple Help
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DynamicSystems

  

ToContinuous

  

convert a discrete-time system to continuous-time system

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

ToContinuous(sys, opts)

Parameters

sys

-

System; system object

opts

-

(optional) equation(s) of the form option = value; specify options for the ToContinuous command

Options

• 

method = forward, backward, bilinear, prewarp, matched, zoh, or foh

  

Specifies the conversion method.

• 

frequency = realcons

  

Specifies the warping or critical frequency in rad/s. This is used only if method = prewarp, otherwise it is ignored.

Description

• 

The ToContinuous command converts a discrete-time system, sys, to a continuous-time system.

• 

The method option specifies the discrete-time to continuous-time conversion method. The following methods are supported: forward rectangle rule (forward), backward rectangle rule (backward), bilinear rule (bilinear), bilinear with prewarping (prewarp), matched poles and zeros (matched), zero-order hold (zoh), and first-order hold or non-causal triangle-hold (foh).

• 

The bilinear rule is also known as the Tustin or Trapezoid rule.

• 

Not all inverse-discretization methods can be applied to all system types. The following table describes the usage. An x indicates that the method can be used by the system type. A D indicates that the method is the default for the system type.

System Type

forward

backward

bilinear

prewarp

matched

zoh

foh

 

 

 

 

 

 

 

 

State-Space

   x

   x

   D

 

 

 x

 x

Transfer-Function

   x

   x

   D

   x

 

 x

 x

Coefficients

   x

   x

   D

   x

 

 x

 x

Zero-Pole-Gain

 

 

 

 

   D

 

 

Transformations

Frequency Domain Transformations

The frequency-domain transformations convert rational functions in z to rational functions in s, where z and s are the discrete and continuous complex frequency variables, respectively. The following functions describe the mappings from the z plane to the s plane for each of the defined transformations where T is the sampling period of the discrete-time system.

• 

forward  : z -> 1 + s*T

• 

backward : z -> 1/(1-s*T)

• 

bilinear : z -> (2+s*T)/(2-s*T)

• 

prewarp  : z -> (|| + s*tan(||*T/2))/(|| - s*tan(||*T/2)) where  is the critical frequency in rad/s.

• 

matched  : poles and zeros in the z-plane map to the s-plane as  z = exp(s*T).

The Laplace transform is used in the following hold equivalents:

• 

zoh : H(s) = s*laplace(invztrans(H(z)/(1-1/z)))

• 

foh : H(s) = s^2*laplace(invztrans(T*z*H(z)/(z-1)^2))

Time Domain Transformations

The time-domain transformations are done on the discrete-time state-space matrices Ad, Bd, Cd, Dd, and T is the sampling period of the discrete-time system. The matrices Ac, Bc, Cc, Dc represent the continuous-time matrices. I is the identity matrix.

• 

forward

Ac = (Ad - I)/T

Bc = Bd/T

Cc = Cd

Dc = Dd

• 

backward

Ac = (I - Ad^(-1))/T

Bc = Ad(-1).Bd/T

Cc = Cd.Ad(-1)

Dc = Dd - Cd.Bc*T

• 

bilinear

M = (Ad + I)*T/2

Ac = M^(-1).(Ad - I)

Bc = Bd/sqrt(T) - sqrt(T)/2*Ac.Bd

Cc = Cd/sqrt(T) + sqrt(T)/2*Cd.Ac

Dc = Dd - sqrt(T)/2*Cd.Bc

• 

zoh

Ac = ln(Ad)/T

Bc = (exp(Ac*T) - I)^(-1).Ac.Bd

Cc = Cd

Dc = Dd

• 

foh

Ac = ln(Ad)/T

Bc = (exp(Ac*T) - I)^(-2).Ac^2.Bd*T

Cc = Cd

Dc = Dd - Cc.(Ac^(-1).(exp(Ac*T)/T - I) - I)Ac^(-1).Bc

Examples

Convert discrete-time transfer function to continuous-time using different conversion methods.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Compare the frequency responses of all systems.

Compatibility

• 

The DynamicSystems[ToContinuous] command was introduced in Maple 18.

• 

For more information on Maple 18 changes, see Updates in Maple 18.

See Also

DynamicSystems

DynamicSystems[Resample]

DynamicSystems[SystemOptions]

DynamicSystems[ToDiscrete]

invztrans

laplace

 


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