- SystemsModelDelay[δ] makes it possible to represent systems involving time delays and efficiently manipulate, approximate, and simulate these systems.
- SystemsModelDelay[δ] is typeset in StandardForm as δ and can be entered using delay.
- For a StateSpaceModel, SystemsModelDelay[δ] can occur linearly in any of the system matrices. For a signal , SystemsModelDelay[δ]w[t] is taken to be .
- For a TransferFunctionModel, SystemsModelDelay[δ] can occur linearly in the coefficients of the polynomials. It is taken to represent a transformed time delay:
|-δ s||in a continuous-time system|
|z-δ||in a discrete-time system|
- Here, s is the Laplace-transform variable, and z is the z-transform variable.
- For discrete-time systems, the delay δ is taken to be a multiple of the SamplingPeriod.
Examplesopen allclose all
Basic Examples (3)
A state-space model with an input delay:
A transfer-function model with delay:
A time-delay state-space model created from delay differential equations:
A continuous-time state-space model with a state delay:
A discrete-time state-space model with an output delay:
A continuous-time transfer-function model with delay:
Or represent the delay as an exponential:
A discrete-time transfer-function model with delay:
A state-space model with a delay in the descriptor matrix:
A continuous-time state-space model created from delay differential equations:
A discrete-time system from a difference equation including SystemsModelDelay:
The delay differential equation below describes the motion of the cutting tool in a lathe where the force on the tool depends on the position of the tool from the previous rotation:
The delay creates peaks in the frequency response:
Properties & Relations (5)
A zero delay reduces to 1:
Delays in continuous-time transfer functions are equivalent to exponentials:
Delays in discrete-time transfer-function models are equivalent to additional poles:
State-space systems with neutral delays have delays in the descriptor matrix:
When converting to a transfer function, state delays appear in the denominator:
Possible Issues (1)
In discrete-time systems, delays should have integer values:
When necessary, zero-order approximations are used:
Approximating delays before simulation can give a more accurate result: