The response of a system to a sudden excitation is often modeled as a step response. The following is an example of how to obtain the step response of a simple system. It illustrated the difference between a system with and without so-called *numerator dynamics*. Such dynamics represent an interaction due to velocity induced forces, such as those in viscous fluid dynamics or caused by a dashpot (also fluid-driven).

Consider the transfer function of a system given by

is

while for a second system given by

is

Find the response to an input of \(5u(t)\).

Given that the amplitude of the step input is 5, it's easiest to simply
multiply the transfer function by five and use a unit step function,
allowing us to use the Matlab `step` function.

sys1=tf([1 1]*5,[4 1]) sys2 = tf([1]*5,[4 1]) step(sys1); hold on; step(sys2)

sys1 = 5 s + 5 ------- 4 s + 1 Continuous-time transfer function. sys2 = 5 ------- 4 s + 1 Continuous-time transfer function.

Alternatively, using `lsim`.

sys1=tf([1 1]*5,[4 1]) sys2 = tf([1]*5,[4 1]) t=0:.01:40; u=t*0+1; lsim(sys1,u, t); hold on; lsim(sys2,u,t)

sys1 = 5 s + 5 ------- 4 s + 1 Continuous-time transfer function. sys2 = 5 ------- 4 s + 1 Continuous-time transfer function.

In both cases, the blue line represents the *sys1* response, and the
orange line the *sys2* response. This can be demonstrated by plotting
them individually.

The effect of the \(\dot{g}(t)\) term is to effectively jump start
the response at a higher level, equivalent to 5/4, which are two numbers
you should see in the *sys1* transfer function.

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