Under-damped second order system - CONTROL

[mem0h]

hey all,
i'm stuck with the following designing problem (Control course) :

Homework Statement

given the location of the poles , find rise time , peak time, percentage maximum overshot and settling time for each pole. pole are:
1 . pole at θ = 70 , ωn = 1

2. pole at θ = 70 , ωn = 3

3. pole at θ = 30 , ωn = 1

4. pole at θ = 30 , ωn = 3

Homework Equations

1. how to calculate tp, tr, max o.s and ts for each pole ?

2.how to find the damping ratio (zeta) ?

3. which one of the poles is the best ?

 

[rude man]

I've never see poles described this way. If θ is the angle associated with the complex s plane, then not only are all four poles in the right-hand plane but they are not complex-conjugates.

So - anybody understand this?

 

[AugustCrawl]

A step in the right direction :)

Is there no transfer function associated with the question?
I think knowing the order of the system is useful.
Maybe fourth pole means fourth order.

I referred to this http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf

If when you do the nyquist plot the transfer function does not circle -1 then I think your system is stable.
Sometimes this is called the nyquist criterion.

Im going to have a go assuming the nominator of the transfer function is 1.

I can't properly explain how to do this. But let me show a similar example:

Putting this code in Matlab
EDU>> Q2 = tf ([5],[1 204 800])
figure(99); bode(Q2)
figure(100); step(Q2)

Generates this

There are buttons for the parameters you asked for.

So from what I know the solutions to the denominator of the quadratic are the roots.
And the roots give you the location of the poles.

So if you can figure out what the transfer function is, I am happy to re-plot for you.

Unfortunately I don't have a fuller answer for you: I am studying Control Systems 1 at the moment myself. Hopefully by the time I get to Control Systems 2 I can answer this properly :)

p.s. to get the Fourier transform we substitute s=jw into the roots of the equation


[tex]\frac{1}{s{}^{\wedge}2+204 s+800}=\frac{1}{(s+200)(s+4)}[/tex]

 

[rude man]

Even after looking at your link I can't make head or tail of this way of describing pole/zero locations. Theta could be the angle from a pole on the real axis or from one of a complex-conjugate pole pair, for example. See figs. 7 and 8 in your reference.

Do you have any previous material where you were given pole/zero locations in this arcane way?

 

[rezeew]

I would approach this problem by first understanding the concept of under-damped second order systems in control theory. An under-damped system is one in which the response oscillates before reaching its steady-state value. This type of system is commonly found in engineering and can be described by a second order differential equation.

To solve this problem, we need to calculate the rise time (tr), peak time (tp), percentage maximum overshot (PMO), and settling time (ts) for each pole. These values can be calculated using the following equations:

1. tr = π / ωd, where ωd = ωn√(1-ζ²)

2. tp = π / ωd√(1-ζ²)

3. PMO = e^(-ζπ/√(1-ζ²)) * 100%

4. ts = 4 / ζωn

To find the damping ratio (ζ), we can use the following equation:

ζ = θ / 90, where θ is the angle of the pole in the complex plane.

Based on these calculations, we can determine which pole is the best. Generally, a lower PMO and shorter settling time are desirable, so the pole with the lowest values for these parameters would be considered the best. However, it is important to also consider the specific requirements and constraints of the system being controlled.