Second order systems

Visual Cue Narration

Show first slide

Hello and welcome to the second tutorial on Control Systems. In this tutorial, we will show how to represent second order control systems and we'll see the responses of certain types of systems; namely, Overdamped systems, Underdamped systems, Critically damped systems and Undamped systems.

Open Scilab

Let us first define an overdamped system. To simplify, we define the denominator of the transfer function separately:

p=s^2+9*s+9
OverdampedSystem= syslin('c', 9/p)

We can verify that this is indeed a secnod order system by checking the roots of the denominator:

roots(p)

The roots of the denominator are real and distinct, as expected for a second order system.

We can also see the root locus of this system:

evans(OverdampedSystem)

We now simulate this system using a step test and plot the output:

y4 = csim('step', t, OverdampedSystem);
plot(t, y4)

This corresponds to the expected output for an overdamped system. We can also plot the Bode plot of the system. We use the same frequency bounds that we did for the first order system- 0.01 and 10:

bode(OverdampedSystem, fMin, fMax)

For a more distinctive response, we find the step response of an underdamped system.

As before, we define the denominator first.

q=s^2+2*s+9
UnderdampedSystem = syslin('c', 9/q)

Again, we check the roots of the denominator:

roots(q)

We check the root locus of this system:

evans(UnderdampedSystem)

Here, the roots of the denominator are a complex conjuate pair, which corresponds to a second order underdamped system.

Now we simulate the system and plot the response:

y5 = csim('step', t, UnderdampedSystem);
plot(t, y5)

As we can clearly see, the output oscillates before settling down to its steady state.

We also note that this response is much faster than the response for the overdamped system.

Clear screen in Scilab

We now move on to an undamped system. For an undamped system, we require the roots to be purely imaginary. This can be obtained in this way:

r = s^2+9
UndampedSystem = syslin('c', 9/r)

We verify:

roots(r)

We check the root locus of this system:

evans(UndampedSystem)

We now simulate and plot the step response of the system:

y6 = csim('step', t, UndampedSystem);
plot(t, y6)

This system has no damping, as a result of which it oscillates without settling down to a steady state.

For a critically damped system, we require the roots to be real and repeated. We choose the following denominator:

m = s^2+6*s+9
CriticallyDampedSystem = syslin('c', 9/m)
roots(m)

We check the root locus of this system:

evans(CriticallyDampedSystem)

The roots are real and repeated, as expected.

y7 = csim('step', t, CriticallyDampedSystem);
plot(t, y7)

We note that this response is the fastest response we can obtain without any oscillations. We can compare the overdamped, underdamped and critically damped responses thus:

plot(t, [y4; y5; y7])

END