Step and Ramp Response Sampled 1st Order Type 0

Cuthbert Nyack

Applets show the step and ramp response of a first order, type 0 system. The continuous system has a K dependent offset but is always stable.

The sampled system is described by the closed loop transfer function

with difference equation

The pole of the sampled system is shown by the eqn below and the system may become unstable if the closed loop pole moves out of the unit circle.

The output is given by the following eqn.

The system has an offset identical to that of the continuous system.
eg parameters (3.0, 1.0, 0.04, 50.0) shows the continuous and sampled step response are practically identical. The closed loop pole is on the real axis and close to z = 1( in the range 0 < z < 1.0).

eg parameters (3.12, 1.0, 0.28, 50.0) show the closed loop pole at the origin and the response reaches its final value after 1 sample time.

eg parameters (3.0, 1.0, 0.6, 50.0) show a case where the closed loop pole has moved to the region -1 < z < 0 and the response has an initial damped oscillation.

eg parameters (3.0, 1.0, 0.69, 50.0) show the closed loop close to the -1 point and the response is close to oscillating between 0 and twice its steady value.

eg parameters (3.0, 1.0, 0.70, 50.0) show the closed loop pole has moved outside the circle and the system has become unstable.

Besides Ts, the location of the pole is also affected by K and T1 as shown in the equation above.

Ramp response is shown below. The output is given by the following eqn.

The gradient of the output is less than that of the input by K/(K + 1).
eg parameters (3.0, 1.0, 0.05, 70.0) show the closed loop pole in the range 0 < z < 1.0. The sampled and continuous response are practically equal.

eg parameters (3.12, 1.0, 0.28, 70.0) show the closed loop pole is at the origin and the response is a ramp, from the origin, with smaller slope than the input.

eg parameters (3.0, 1.0, 0.69, 100.0) show the closed loop pole is close to the -1 point ant the response is begining to show some variation from the ramp.

eg parameters (3.0, 1.0, 0.72, 100.0) show the closed loop pole is outside the unit circle and the system is unstable.

Gif images below show how the applets should appear.

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