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Time-Delay Systems

Time-delay systems can be specified in multiple ways and conveniently converted between state-space and transfer-function representations.

A time-delay system specified as an ODE:
Wolfram Language code: StateSpaceModel[x'[t] + x[t - 1] == 2 u[t] + u[t - 3], x[t], u[t], x[t], t]
A system specified using state-space matrices and the SystemsModelDelay operator:
Wolfram Language code: StateSpaceModel[{{{-SystemsModelDelay[2]}}, {{3 + SystemsModelDelay[1]}}, {{1}}, {{0}}}]
A system specified as a transfer function:
Wolfram Language code: TransferFunctionModel[(3 + Exp[-s]/s + Exp[-2 s]), s]
A transfer function with delays specified using SystemsModelDelay:
Wolfram Language code: TransferFunctionModel[(3 + SystemsModelDelay[1]/s + SystemsModelDelay[2]), s]
A discrete-time system is determined with the SamplingPeriod option:
Wolfram Language code: StateSpaceModel[{{{-SystemsModelDelay[2]}}, {{3 + SystemsModelDelay[1]}}, {{1}}, {{0}}}, SamplingPeriod -> 2]
Wolfram Language code: TransferFunctionModel[(3 + SystemsModelDelay[1]/z + SystemsModelDelay[2]), z, SamplingPeriod -> 2]
Convert between state-space and transfer-function representations:
Wolfram Language code: StateSpaceModel[TransferFunctionModel[{{{3 + SystemsModelDelay[1]}}, s + SystemsModelDelay[2]}, s]]
Wolfram Language code: TransferFunctionModel[StateSpaceModel[{{{-Subscript[, 2]}}, {{3 + Subscript[, 1]}}, {{1}}, {{0}}}, SamplingPeriod -> 2, SystemsModelLabels -> None], z]
Obtain the discrete-time approximations of continuous-time systems and vice versa:
Wolfram Language code: ToContinuousTimeModel[StateSpaceModel[{{{-Subscript[, 2]}}, {{1}}, {{3 + Subscript[, 1]}}, {{0}}}, SamplingPeriod -> 2, SystemsModelLabels -> None], Method -> "BilinearTransform"]
Wolfram Language code: ToDiscreteTimeModel[StateSpaceModel[{{{-Subscript[, 1]}}, {{2 + Subscript[, 3]}}, {{1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], 2]