# 2nd order approximation

## Homework Statement

With unity position feedbck, i.e. make K2=0, plot root locus as a function of pitch gain (K1). By imposing 2nd order system approximation, estimate settling time, rise time, peak time of the closed-loop system with 20% overshoot.

## The Attempt at a Solution

I have derived the open & closed loop transfer functions for the system, when K2=0. From this I have determined the poles & zeros:
• 1 pole at, s=-2
• 1 pole at, s=-1.25
• 1 zero at, s= -0.452
• A complex conjugate pair at, -0.177+/- 0.051j
• Plotted the root locus: https://app.box.com/s/6o5y65btkp3zbvc31cqg
• I now have no idea how to impose 2nd order approx & estimate rise time etc. Please help...

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A complex conjugate pair at, -0.177+/- 0.051j
It's probably just a typo, but I think that should be -0.117 instead.

You have a conjugate pair of poles that are approximately dominant. As I understand it, you're asked to only consider that conjugate pair (approximate with a second-order system) and place them in the complex plane along the root locus where it intersects with an isoline of constant damping ratio corresponding to 20% overshoot, i.e. what value of $K_1$ puts the conjugate pair at a spot with the damping factor you want?

From the resulting system you can determine the rest of the parameters. This assumes you're not using a more sophisticated method of reducing the order of your system.

Does this mean I should take the quadratic out of the transfer function & just adjust K1 until I get a step response= the original tf's step response?

Is there a specific equation or set of equations that I can use to determine the appropriate value for K1?

It's probably just a typo, but I think that should be -0.117 instead.

You have a conjugate pair of poles that are approximately dominant. As I understand it, you're asked to only consider that conjugate pair (approximate with a second-order system) and place them in the complex plane along the root locus where it intersects with an isoline of constant damping ratio corresponding to 20% overshoot, i.e. what value of $K_1$ puts the conjugate pair at a spot with the damping factor you want?

From the resulting system you can determine the rest of the parameters. This assumes you're not using a more sophisticated method of reducing the order of your system.
Is there a formula to find K1?

Does this mean I should take the quadratic out of the transfer function & just adjust K1 until I get a step response= the original tf's step response?
No, I just meant that you should focus solely on the conjugate pair of poles. The root locus shows you that those poles will probably always remain dominant.

Is there a specific equation or set of equations that I can use to determine the appropriate value for K1?
You don't necessarily have to determine $K_1$. The dynamics of a second-order system is fully determined by its undamped natural frequency $\omega_n$ and damping ratio $\zeta$. Have you seen a plot of how these two quantities determine pole locations and vice versa in the complex plane? Hint: Think radius and angle.

The 20% overshoot gives you $\zeta$ and a specific location on the root locus for that conjugate pair of poles. From that location you can eyeball $\omega_n$, which lets you estimate the parameters you need.

I'm pretty confused as fom a suggestion from another site, I was pointed to 'Routh-Hurwitz Stability Criteria'. I was then told that its too complex to explain in a post? I'm really unsure what to do next. By the way your help is very much appreciated

I'm pretty confused as fom a suggestion from another site, I was pointed to 'Routh-Hurwitz Stability Criteria'.
The Routh-Hurwitz criterion can be used to find the value of $K_1$ at the intersection of the root locus with the imaginary axis, i.e. where the system becomes marginally stable, but I'm not sure what it'll solve in your case.

I think the easiest solution to your problem is the one I've already hinted at.

I'm still uncertain of the actual procedure.
• If I follow your method, I just plot the root locus using the closed loop tf.
• I know how to determine ζ from % overshoot, but how do I find ωnwithout first having ts, tr, tp etc?
• In what way does ζ give me the location of the 'complex conjugate pair'? Does it just show where they already are, or is it a new position I must move them to?
• Sorry if I come across as 'slow' but the longer I spend at this, the more confused I get....Thanks for your patience

I'm still uncertain of the actual procedure.
• If I follow your method, I just plot the root locus using the closed loop tf.
• I know how to determine ζ from % overshoot, but how do I find ωnwithout first having ts, tr, tp etc?
• In what way does ζ give me the location of the 'complex conjugate pair'? Does it just show where they already are, or is it a new position I must move them to?
• Sorry if I come across as 'slow' but the longer I spend at this, the more confused I get....Thanks for your patience
You've found that your system has 4 poles and a single zero. Two of the poles are complex conjugates and are somewhat closer to the imaginary axis (slower) than the other poles and zero. This conjugate pair will thus tend to dominate the response, i.e. your system will behave approximately as if only the conjugate pair of poles was present. The poles are dominant.

So, let us pretend that we only see the conjugate pair of poles and their part of the root locus. This is the second-order system approximation (which only has a pair of complex conjugate poles, assuming it's oscillatory).

Recall that the root locus is a curve in the complex plane that shows you all the possible locations for the closed-loop poles as you vary a single parameter, which in your case is chosen to be $K_1$. Given a value of $K_1$, the closed-loop poles will be at some specific location on the root locus.

Now, you're asked to find some parameters subject to the condition that the closed-loop system has 20 % overshoot. This is a constraint on the location of the closed-loop poles in the complex plane! You know that the closed-loop poles must lie on the root locus, but they can't just be anywhere on it. They have to be at the location on the root locus that gives the conjugate pair of poles a damping factor corresponding to 20 % overshoot. If you could find this location, you also know what $\omega_n$ is for the system. From $\omega_n$ and $\zeta$, you can find $t_s, t_r, t_p$ and whatnot.

Have you maybe seen an expression such as:
$$\zeta = \cos(\alpha)$$
where $\alpha$ is the angle a line from the origin into the left-half plane makes with the real axis?

Any location on one such line will have the same damping factor. One of these lines corresponds to the damping factor you want, and its intersection with the root locus shows you the location where the second-order system has 20 % overshoot. What is $\omega_n$ at this location? $\omega_n$ also has a simple geometric property in the complex plane.

Edit:
A picture might help:

http://jpkc.zju.edu.cn/k/506/econtext/image/02/032.jpg

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I follow you so far, but if I'm now looking at the 'root locus' with only the dominant pair. Any point on these lines representing a different value for K1 & the 'closed-loop' poles.
• How do I use the trig relationship above with the 'curved' lines on the root locus?
• Pic: https://app.box.com/s/31jwzwc3apchfaiayks3
• What values do I use for the 'opposite' & 'adjacent' in order to find length of the 'hypotenuse=ωn'?

Obviousy I need the length of one of the sides. What value would I use for the 'opposite' or 'adjacent' to find 'hypotenuse' i.e. ωn?
Yes, that's where I wanted you to eyeball either the real or imaginary part of the intersection point on your plot. An analytical solution is going to be very tedious work.

If you're familiar with the angle criterion for the root locus, you could let $s = \omega_n e^{j (\pi - \theta)}$ and build a small table of values of $\omega_n$ that shows it converge to something that approximately satisfies the angle criterion. If your instructor really wants an analytical solution, I'd be curious to see what he/she had in mind.

I managed by plotting root locus in Matlab & using 'sgrid' i.e specifying (ζ) & estimating (ωn) to find a very close value for natural frequency. By using an iterative process I eventually got all lines to intersect at poles on root locus. See below:
• https://app.box.com/s/qp1gh26st9viazefeya6
• ts=33.649 sec (±5%)
• tp=13.539 sec
• tr=6.904 sec (10-90%)
• I now have to find K1 at 20% overshoot. It says to generate 'step response' of the closed-loop system & estimate steady state error. Measure actual PO, ts, tp, tr. See if their values are close to the ones found in previous step.

Could you perhaps include a picture of the assignment?

Here it is:
• Questions: https://app.box.com/s/lnh0jr9yl291ie8dhi8x [Broken]

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By using an iterative process I eventually got all lines to intersect at poles on root locus. See below:
So first thing, it looks like you're plotting the root locus of the closed-loop system, which isn't right. That shows you the pole locations of a system that has an outer feedback loop around the feedback system that's shown in your assignment. You need to plot the root locus of the open-loop system, i.e. 'rlocus(-2/(s+2)*G)' in MATLAB.

Secondly, the plot you've included here does not show the intersection point of the isoline of constant damping ratio and the root locus. It just shows what the natural frequency is ($\omega_n = 0.261$) of the closed-loop poles with $K_1 = 1$.

After you sort this out, you can find what $K_1$ is at the intersection point, which is some value of $s$, by inserting this value into the characteristic equation for your system and solve for $K$. It's not going to be exact, since you don't know the exact intersection point, but you can just discard any imaginary part in the result as an approximation.

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Thanks, I'll get on that now 7 post asap.....

I reworked the problem with the 'open loop' tf. My values are:
For the poles:
• s = -2
• s = -1.25
• s = -0.117 ± 0.0511j
For the zeros:
• s = -0.452
If I now plot the root locus to include a damping ratio ζ = 0.456 & natural frequency ωn = 0.44:
Everything intersects on the root locus, but the 'step response' has an incorrect overshoot. See below:
I have to make K1=1.67 in order for it to have the specified 20% overshoot. But if I want the pole to sit at the intersection of ζ & ωn isolines, I have to make K1=2.79. Again this changes my 'step response' overshoot to 28.1%.

Should I use the above values for ζ & ωn to derive tp, ts, tr? I'm unsure because at these values the % overshoot wasn't 20%.

If I now plot the root locus to include a damping ratio ζ = 0.456 & natural frequency ωn = 0.44:
Yes! :)

Everything intersects on the root locus, but the 'step response' has an incorrect overshoot. See below:
I think you attached the wrong plot. This one shows the step response with $K_1 = 1$.

I have to make K1=1.67 in order for it to have the specified 20% overshoot. But if I want the pole to sit at the intersection of ζ & ωn isolines, I have to make K1=2.79. Again this changes my 'step response' overshoot to 28.1%.

Should I use the above values for ζ & ωn to derive tp, ts, tr? I'm unsure because at these values the % overshoot wasn't 20%.
You should use the values at the intersection point. Remember, the whole point of this exercise is to show the difference between a true second-order system and the approximation you're using. You'd only have 20 % overshoot if that conjugate pair of poles were the only ones present (with no zeros).

Have a read-through again of the problem statement for (3). See how you're now asked to compare the values of those parameters for the approximation and the full-order system?

At last! I thought I'd spent two days working on this, only to arive at the answers I had when I originally posted:(. I think I'm getting my head around it a bit more now.
• Does this mean I've already found K1 asked for in question 3?
• Is the steady state error just the % difference between steady state value & 1.0?

Does this mean I've already found K1 asked for in question 3?
Yes, I thought that was the part of the assignment you were doing now:
I now have to find K1 at 20% overshoot. It says to generate 'step response' of the closed-loop system & estimate steady state error. Measure actual PO, ts, tp, tr. See if their values are close to the ones found in previous step.
I guess your instructor wants to know that you're able to?
Is the steady state error just the % difference between steady state value & 1.0?
The steady-state error is usually defined as:
$$e_\mathrm{ss} = \lim_{t\rightarrow \infty}\left[r(t) - c(t)\right]$$where $r(t)$ and $c(t)$ is the input and output signal, respectively.

So if, for instance, the input step was equal to $2$ for $t \geq 0$ and the output tends to $4$ as $t \rightarrow \infty$, you'd have $e_\mathrm{ss} = 2 - 4 = -2$.
No need to be sorry. I wouldn't be posting if I didn't want to help.

For question 3 it asks me to design a 'phase-lead compensator'. I have to get the 'step response' to give, PO<16%, ts<4 secs. I've got to then put a zero at (-1) & find the remaining unknown parameter. I also have to select a pair of 'dominant closed loop poles'.

• Is this asking me to find a region to insert a pole, that will result in the correct PO & ts?
• If not, how do I go about designing a 'phase-lead compensator' for the specs above?

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