Analyzing the system

Visual Cue Narration

In this tutorial, we will see how to analyze a system that we have modeled in Scilab.

First, we see what the step response of the system is. In the physical example that we've chosen, this corresponds to applying a constant amount of force to the system. The response of the system corresponds to the displacement that the system will undergo.

t = 0:0.01:3;
y = csim('step', t, System);
plot(t, y)

From this plot, we see that this is the expected response since we had earlier noticed that the system was overdamped. This corresponds to the mass block moving by an amount 1 by k and coming to rest at a steady state.

Let us see what happens when we add unity feedback to this system

t = 0:0.01:3;
yFeedback = csim('step', t, System/. 1);
plot(t, yFeedback)

Note the space between the point and the one.

We notice that the transient performance improves- that is to say that the system reaches its steady state faster. However, we notice that the steady state error degrades.

Let us investigate the steady state error of the system further.

We check the steady state response of the original system:

Show slide about final value theorem

We know from the final value theorem that <read final value theorem expression>

Rs = 1/s
Gs = System
css = horner(s*System*Rs, 0)

We see that the steady state value is 1, as expected.

Let us investigate the degradation in steady state value after adding unity feedback.

t = 0:0.01:3;
yFeedback = csim('step', t, System/. 1);
plot(t, yFeedback)

We notice that the steady state value decreases to half, in this case.

We compare the open loop transfer function response and the closed loop transfer function response:

plot(t, [y; yFeedback])

We note that it would be useful to improve the steady state response of this system using a controller and, if needed, improve the transient response as well.

Let us now take a look at the root locus of the system

evans(System)

We note that we can find the gain at a particular point along the root locus using the datatip manager by clicking on "Edit" and "Start Datatip Manager"

Click on "Edit" and "Start Datatip Manager"

Then click carefully on a point on the root locus to find the gain, K and location of that point.

END

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