Time delays are common in a variety of systems, often caused by communication lags, material transport, delayed sensing, etc. Time delays can cause instabilities and are generally harder to control unless compensated for.  The Wolfram Language makes it easy to model and simulate delay systems by using a delay operator for either transfer function or state-space models. By either approximating delays or compensating for them, the full suite of design tools can be used for delay control systems.

Modeling and Simulation

SystemsModelDelay — represents a time-delay operator _q

StateSpaceModel — state-space model with delays

TransferFunctionModel — transfer-function model with delays

SystemsModelSeriesConnect  ▪  SystemsModelFeedbackConnect  ▪  ...

OutputResponse  ▪  StateResponse  ▪  ...

BodePlot  ▪  NyquistPlot  ▪  ...

Controller and Estimator Design

SystemsModelDelayApproximate — higher-order, delay-free system approximation

SmithDelayCompensator — compensates for internal delays in the system

StateFeedbackGains  ▪  LQRegulatorGains  ▪  KalmanEstimator  ▪  ...