Properties of delay systems

Delay systems can sometimes have non-intuitive properties, in particular when the delays appear inside of the system, i.e., not directly on the inputs or outputs.

The Nyquist plot of delay systems usually spirals towards the origin for delay systems. This is due to the phase loss at high frequencies due to the delay:

using ControlSystemsBase, Plots
w = exp10.(LinRange(-2, 2, 2000))
P = tf(1, [1, 1]) * delay(2) # Plant with delay on the input
nyquistplot(P, w)

When forming a feedback interconnection, making the delay appear in the closed loop, we may get gain ripple:

bodeplot(feedback(P), w)

If the system with delay has a direct feedthrough term, step responses may show repeated steps at integer multiples of the delay:

using ControlSystems # Load full control systems to get simulation functionality
P = tf([1, 1], [1, 0])*delay(1)
plot(step(feedback(P, 0.5), 0:0.001:20))

Indeed, if the system has a non-zero feedthrough, the output will contain a delayed step attenuated by the feedthrough term, in this case

ss(feedback(tf([1, 1], [1, 0]))).D[]

0.5

the steps will thus in this case decay exponentially with decay rate 0.5.

For a more advanced example using time delays, see the Smith predictor tutorial.

Simulation of time-delay systems

Time-delay systems are numerically challenging to simulate, if you run into problems, please open an issue with a reproducing example. The lsim, step and impulse functions accept keyword arguments that are passed along to the ODE integrator, this can be used to both select integration method and to tweak the integrator options. The documentation for solving delay-differential equations is available here and here.