How Does Unity Position Feedback Affect System Dynamics in Control Theory?

[toolpusher123]

Th

 

[toolpusher123]

I've run this through numerous times & the open-loop tf I'm using has all the necessary poles & zeros:

• Poles: s = -73.51, -2, -1.25, -0.117 ± 0.0511j
• Zeros: s = -1.58, -0.452

But looking at the root locus plot, none of the branches intersect with s = -1 ± 2j. The closest branch is at least 0.3 out. I'm unsure as to where our methods diverged? I don't see a way to find K1 if there's no intersection. Any instruction on this last bit would be appreciated...

 

[milesyoung]

Finally got the root locus to come out correctly:D. I'd been using wxMaxima to multiply out my tf's so I could get them in a format that I'm familiar with inputting into MATLAB. But wxMaxima seems to be giving an incorrect result. When I did it by hand, the root locus passed through s = -1+/- 2j. There's a plot below:
Root Locus + Compensator: https://app.box.com/s/stheyn866bi43dez9rxe

I think you're making things more difficult than they need be, and in the process, you're including some errors. Let MATLAB handle some of the difficult bits, so it's easier for you to check your results.

As an example, here's how I would plot the root locus with your compensator added:

s = tf('s');

G = -0.125*(s + 0.452)/((s + 1.25)*(s^2 + 0.234*s + 0.0163));
P = -2/(s + 2)*G;

C = (s + 1)/(s + 73.51);

rlocus(C*P)

Tip: You can click the root locus to add a draggable point on it that shows numerical values.

I don't see a way to find K1 if there's no intersection.

There is an intersection, and to find $K_1$, consider that the characteristic equation for your system is true for any point that's on the root locus. If you plug your numbers into this equation, how many unknowns are there?

 

[toolpusher123]

I get the exact same plot as I did before, with no intesection at s = -1 =/- 2j ? In the example above for C, it shows the zero as s = -1 ? Is this the value I should be using, as I've been using s = 1.58 ?
If I use the s=-1, then I get a plot that quite similar but with the intersection at s= -1+/- 2j.

Edit: Yes we were given the zero at -1. Somehow I thought we'd derived a new one at s = -1.58??
Spent the whole night chasing my tail, when it was in black & white all along!

 

[toolpusher123]

When I produce the 'step' response, the overshoot is gone but it still has a settling time of 36 secs. How would I go about reducing it to the design specs?

 

[milesyoung]

When I produce the 'step' response, the overshoot is gone but it still has a settling time of 36 secs. How would I go about reducing it to the design specs?

Are there any purple/pink dots at $s = -1 \pm j2$? What does that mean?

 

[toolpusher123]

No. Are these markers of the gain (K)? I've tried to move them in order to adjust the reponse but it just becomes totally unstable?

Edit: moving the 'pink dot' to -1+/- 2j I've reduced the Ts to under 4 secs but the PO is 22.6%. Does this mean I would have to use the same process again to get closer to design specs?

 

[milesyoung]

No. Are these markers of the gain (K)? I've tried to move them in order to adjust the reponse but it just becomes totally unstable?

Edit: moving the 'pink dot' to -1+/- 2j I've reduced the Ts to under 4 secs but the PO is 22.6%. Does this mean I would have to use the same process again to get closer to design specs?

It won't be exact, since you're not working with a true second-order system. Try moving the compensator zero to -0.5 and recalculate your controller (including $K_1$).

The "fix" I mentioned, is to try and make sure there are zeros near the conjugate pair of closed-loop poles that's closer to the imaginary axis than $s = -1 \pm j2$. It's an effort to get them to cancel, so you won't have their effect in the system.

 

[toolpusher123]

One last thing I was going to ask, as I've asked you countless questions so far. Is for Q6,7 I've got to repeat the earlier Q's i.e plot root locus & do the 2nd order approx. This time I include (-K2s). I've been trying to resolve the TF I should use. Till now it's been 'open-loop',

• Would I find the closed loop for; (2/s+2) * Gs)/1+(2/s+2)*Gs*(-k2s) & then multiply by -k1?
• 40(250s+113)/(40000s^4+139360s^3+141072s^2+30039s+1630)

 

[milesyoung]

Would I find the closed loop for; (2/s+2) * Gs)/1+(2/s+2)*Gs*(-k2s) & then multiply by -k1?

No, it's good thinking, but that would only be correct if $K_2$ was a constant. You're asked to set $K_2 = K_1$, and $K_1$ is the parameter you want to plot the root locus for as it varies.

In a nutshell: You need the characteristic equation for your system to have the form:
$$
1 + K F(s) = 0 \quad (1)
$$
where $K$ is the parameter you want to plot the root locus for, and $F(s)$ is not a function of $K$.

As an example, consider the characteristic equation for your system without the velocity feedback:
$$
1 - K_1 \frac{2}{s + 2} G(s) = 0 \Leftrightarrow 1 - K_1 F(s) = 0, F(s) = \frac{2}{s + 2} G(s)
$$
All is well, since $F(s)$ isn't a function of $K_1$.

In short: You need to put the characteristic equation for your system with velocity feedback into the form $(1)$. $F(s)$ is then the function you need to use with 'rlocus' etc. in MATLAB.

 

[milesyoung]

Correction: Should be:
$$
1 - K_1 \frac{2}{s + 2} G(s) = 0 \Leftrightarrow 1 + K_1 F(s) = 0, F(s) = -\frac{2}{s + 2} G(s)
$$
I dun goofed there.

 

[toolpusher123]

Does the above represent the 'open loop' tf when the system includes K1 & K2?

 

[milesyoung]

Does the above represent the 'open loop' tf when the system includes K1 & K2?

If by above you mean post #81, then no. It was just a correction to post #80.

Again:

In short: You need to put the characteristic equation for your system with velocity feedback into the form $(1)$. $F(s)$ is then the function you need to use with 'rlocus' etc. in MATLAB.

 

[toolpusher123]

Therefore F(s) = 2/(s+2)*G(s)

 

[milesyoung]

Therefore F(s) = 2/(s+2)*G(s)

No, $F(s) = -\frac{2}{s + 2} G(s)$ for the system without velocity feedback.

You need to figure out what the characteristic equation is for your system when you add the inner feedback loop.

 

[toolpusher123]

But that's exactly where I'm stuck, I've done everything but I'm not sure how to get the 'open loop' with velocity feedback i.e. K1=K2. For the 1st system you just multiply :

• -1*2/(s+2)*G(s)
• But how do you work out the open loop tf with velocity feedback? Not sure how to deal with the closed loop part with K2
• Would it be: 1+Ks(s)G(s)(2/(s+2))?

 

[milesyoung]

I'm all for helping, but you have to start showing more of your thought process behind your work. Right now, it just looks like you're guessing.

If you've studied feedback systems in some form, lectures/textbook etc., you should know what the characteristic equation is for such a system, e.g.:

What is the characteristic equation for this system?

 

[toolpusher123]

-K1*H(s)*(G(s)/1-H(s)*G(s)*K2s ?

• H(s) = 2/(s+2)
• G(s) = -0.125(s+0.452)/(s+1.25)(s^2+0.234s+0.0163)
• Therefore as I originally said , find closed loop of H(s), G(s) & -k2s
• the open loop is then: the closed loop above * -k1
• Root Locus Plot:
• Open Loop Bode Plot: https://app.box.com/s/iaqcgq70apqf6v2staw5

 

[milesyoung]

The picture has nothing to do with the system in your assignment.

In terms of the functions $G(s)$ and $H(s)$, what is the characteristic equation for the system in the picture?

 

[toolpusher123]

G(s): https://app.box.com/s/s7szsp7rztjndh58oolc
H(s): https://app.box.com/s/8hn2fryguhnis8mr0469
F(s) open loop tf when K1=K2: https://app.box.com/s/arwnrxwopi6y2cjls42q
Root Locus of F(s): https://app.box.com/s/wzaupxumo3j96lvjwobo
2nd Order Approx K1=K2: https://app.box.com/s/p2vyncwxhil3252g07d8

I'm hoping that F(s) above is correct as this is what I've based everything afterwards on...

 

[donpacino]

G(s): https://app.box.com/s/s7szsp7rztjndh58oolc
H(s): https://app.box.com/s/8hn2fryguhnis8mr0469
F(s) open loop tf when K1=K2: https://app.box.com/s/arwnrxwopi6y2cjls42q
Root Locus of F(s): https://app.box.com/s/wzaupxumo3j96lvjwobo
2nd Order Approx K1=K2: https://app.box.com/s/p2vyncwxhil3252g07d8

I'm hoping that F(s) above is correct as this is what I've based everything afterwards on...

While you did not answer Milesyoung's question, your F(s) open loop is correct.

 

[toolpusher123]

In response to 'milesyoung' post #87;

• Closed Loop Tf: https://app.box.com/s/twk1h23blxebgawzaw7d

• Characteristic Equation for post #87: https://app.box.com/s/qivkcx0b0csz6xx8oe8e

Miles I have submitted my report. I must thank you for all your help & suggestions. Now that the project is complete, I have a much better understanding of the processes/concepts involved. At the beginning I really did not have a clue. So thanks again for your patience & commitment...

Edit: Thanks to anyone else on the site who added any input.

 

[milesyoung]

While you did not answer Milesyoung's question, your F(s) open loop is correct.

That's the equivalent system for the forward path of the outer feedback loop, but that's not the function you should use in place of $F(s)$, since it depends on $K_1,K_2$.

Miles I have submitted my report. I must thank you for all your help & suggestions. Now that the project is complete, I have a much better understanding of the processes/concepts involved. At the beginning I really did not have a clue. So thanks again for your patience & commitment...

You're very welcome. Since you've submitted it, I can show you what I was getting at:

The characteristic equation for your system with velocity feedback is (using your definitions):
$$
1 - \frac{K_1 H(s) G(s)}{1 - K_1 s H(s) G(s)} = 0 \Leftrightarrow 1 - K_1 s H(s) G(s) - K_1 H(s) G(s) = 0\\
\Leftrightarrow 1 + K_1\left[-s H(s) G(s) - H(s) G(s)\right] = 0 \Leftrightarrow 1 + K_1 F(s) = 0, F(s) = -s H(s) G(s) - H(s) G(s)
$$
If you plot the root locus for $F(s)$, you should notice some significant differences from the system without velocity feedback, e.g. it's stable for any $K_1$, and the dominant branch lies much further into the LHP.

 

[toolpusher123]

No my post was in response to your post (post # 87).
My solution regarding my system i.e. K1 = K2 was to find the open-loop tf. The equation I used to do this was: https://app.box.com/s/arwnrxwopi6y2cjls42q

• Did I enter an incorrect tf in order to plot 'root locus' of system when K1 = K2 ?
• Is this the correct root locus i.e. with 'velocity feedback': https://app.box.com/s/wzaupxumo3j96lvjwobo

Edit: From post #87 ( "your F(s) open loop is correct").

 

[milesyoung]

Did I enter an incorrect tf in order to plot 'root locus' of system when K1 = K2 ?

Yes, it should be $F(s) = -s H(s) G(s) - H(s) G(s)$, as per my post #93.

Is this the correct root locus i.e. with 'velocity feedback': https://app.box.com/s/wzaupxumo3j96lvjwobo

No, it should look like this: