Hi,

I've done a search & found a few references for '2nd order system approximation'. The issue I'm having is that none are for 4th order to 2nd, & the one that's closest shows how to transform a 5th to 2nd but the transfer function is much simpler.

• Should it be the 'G(s) open-loop' that gets transformed from 4th to 2nd?
• Or should it be 'G(s) closed-loop' that gets transformed from 4th to 2nd?
• Link to G(s) open-loop: https://app.box.com/s/kh9dw3cpszaubioa44ia
  
• Link to G(s) closed-loop: https://app.box.com/s/28lg4l9ojm6xzau7r5ud
• Any instruction/help that could be offered would be really appreciated as I'm 100% stuck at this bit...

 

Limit the gain K1 to the range [0,K1max] and ignore the quadratic polynomial. You are left with a 2nd order approximation. This is the practical pilot speaking.

 

Sorry for being so slow but does this mean;

1. Take the quadratic out the denominator, or just ignore?
2. How do I represent in the transfer function 'limit the K1 to range'? Should I take it out of the tf?

 

Sorry for being so slow but does this mean;

1. Take the quadratic out the denominator, or just ignore?
2. How do I represent in the transfer function 'limit the K1 to range'? Should I take it out of the tf?

It means take the quadratic out of the picture altogether. Limiting the range of K1 is just a mathematical statement about where the 2nd order approximation is valid. You should set the limit on K1 to keep the roots of the quadratic out of the right half plane.

 

To do this in Matlab/sisotool, could I just enter the new transfer function (minus quadratic) & use the compensator to adjust K?

 

Yes, just enter the new 2nd order transfer function without the quadratic. You need to keep the zero however. Not that familiar with Matlab. What is the "compensator"?

 

If I do this, should the 'step response' of the new root locus/system= the original, or be pretty close?

 

I thought I had a tool in Matlab that would find K1., I don't. Is there a specific equation or set of equations that I can use to determine the new K1?

 

Yes, there is a technique to find the root locus intercept on the jω axis. From the value of the intercept at 0 ± jb you can compute the value of K at that point. the method is based on the Routh-Hurwitz Stability criteria. You use the original closed loop transfer function and make the Routh-Hurwitz table. From this table you look for any row that can contain all zeros. This results in an equation for K which can be solved. Using that value of K you form the polynomial indicated by the row above the one will all zeros and solve for the values of the axis crossings.

You need to look up the method of completing the Roth-Hurwitz table because the method requires more explanation than can conveniently fit into a post.

 

If I remove quadratic from tf I get: https://app.box.com/s/jbaf2vogusbwmfjbdih8

So if I use this & then derive a new value for K1 that results in 20% overshoot?

Is there a formula or group of formula's that I can use to find K1?

 

Sorry, I posted before your response came up. I thought I was getting close but now the finnish seems further than ever! I'll go & look this up but it doesn't sound comforting that it's too big to post....Your patience is appreciated

 

Sorry, I posted before your response came up. I thought I was getting close but now the finnish seems further than ever! I'll go & look this up but it doesn't sound comforting that it's too big to post....Your patience is appreciated

Once you understand the technique, it is relatively easy to apply especially for higher order systems where the roots may be difficult to obtain.

http://en.wikipedia.org/wiki/Routh–Hurwitz_stability_criterion

You can also find a value of K that will result in a critically damped system. In practical terms you want to be careful with the critically damped choice, as the pilot would say that the system was sluggish.

 

just finished doing a few examples, will check my work in the morning & post results.....

 

Hi,

I've done a search & found a few references for '2nd order system approximation'. The issue I'm having is that none are for 4th order to 2nd, & the one that's closest shows how to transform a 5th to 2nd but the transfer function is much simpler.

• Should it be the 'G(s) open-loop' that gets transformed from 4th to 2nd?
• Or should it be 'G(s) closed-loop' that gets transformed from 4th to 2nd?
• Link to G(s) open-loop: https://app.box.com/s/kh9dw3cpszaubioa44ia
  
• Link to G(s) closed-loop: https://app.box.com/s/28lg4l9ojm6xzau7r5ud
• Any instruction/help that could be offered would be really appreciated as I'm 100% stuck at this bit...

I'm puzzled by your Link to G(s) closed-loop: https://app.box.com/s/28lg4l9ojm6xzau7r5ud

Is this indicating that K1=1 for 20% overshoot with step input?

I obtain K1=1.67359 for the 20% overshoot case...

If you can obtain or borrow a copy of Schaum's Outline of Theory and Problems of Feedback and Control Systems this makes mention of dominant pole-zero approximation in the chapter on Root Locus and gives examples. However, I'm not sure whether your system fits the approximation criteria. The approximation in the aforementioned examples is applied to the closed loop TF. There is a caveat on the use of such approximations as they may satisfy the dynamic response for the conditions imposed but they may not indicate potential instabilities inherent in the real system.

 

It means take the quadratic out of the picture altogether. Limiting the range of K1 is just a mathematical statement about where the 2nd order approximation is valid. You should set the limit on K1 to keep the roots of the quadratic out of the right half plane.

Suppose, as suggested, we take the quadratic out of the open loop TF. This leaves a zero at -0.452 and poles at -1.25 and -2.0. The root locus for this arrangement gives only real closed loop poles for any K1 value. How would this produce a transient response with overshoot?

 

Hello,

I am not sure about the methods given here, but seeing an example helps to explain what our main goal is when converting from a higher order system to an approximate lower order system, so i am including a picture of a fourth order system reduced to a second order system using one method where we solve for the new system by taking derivatives and solving for various things. I've shown pole cancellation and partial pole cancellation as well as the actual computed new 2nd order response for comparison.

What this might remind us of could be a few different things, like linearization, filtering, or even just local averaging. I like the local averaging view because if you notice any of the smoother lines look like the local averaging of the true response. In fact, if we just draw a line through the original response where we see roughly the average of two adjacent peaks, we get an approximation to the original response that is second order. What we end up doing is using the dominent pole pair, which can be found by finding the slowest varying pair. The higher frequency sets are left out. We could then even go as far as to solve for the new transfer function using a few well chosen points along that average line, which would again give us a second order system. So we could call this a 'graphical' method of sorts, but it is interesting because if we do it right we can get the response to look just like the original except without the fast bumps and since it came from a direct inspection of the old response it must be a reasonable approximation.

The technique for converting from say fourth or fifth order to second order mathematically that i wanted to show is a little complicated so im not sure if you want to see this or not. It's not like it is difficult, but there are a lot of 'little' calculations that have to be done and results kept in perfect order.

 

Sorry I've not posted in response quicker. I've been jumping between multiple sites trying to figure things out. I've managed to determine the '2nd order approx'. I'd assumed that I had to somehow change the tf into 2nd order from 4th, but this wasn't the case. The 2nd order approx was just the 'complex conjugate' root locus branches. I then used Matlab's 'sisotool' to plot Zeta, this gave me an approx intersection for isolines for Wn. With Wn I could find Ts, Tp, Tr etc.

I now have to design a 'phase-lead compensator' to fit specified design requirements but I'll open that in a seperate post. Thanks for everyones input, I'd be completely lost without the help...

 

Suppose, as suggested, we take the quadratic out of the open loop TF. This leaves a zero at -0.452 and poles at -1.25 and -2.0. The root locus for this arrangement gives only real closed loop poles for any K1 value. How would this produce a transient response with overshoot?

After reading further I realized that ignoring the quadratic factor was only one possible approximation. It was based on the very small values of both the real and imaginary parts of the complex conjugate pair and some practical experience using pitch control in a GA aircraft. It occurred to me that you could replace the pair with a single pole on the real axis at say s = -0.1 + j0; now we have one zero and three poles which changes the flavor of the system. Now there is a breakaway between the poles at -1.25 + j0 and -2 + j0 and we have our 2nd order effects back.

I didn't imagine for a second that a "2nd order approximation" was unique, or even a precise and unambiguous term. I would discuss this with the professor to see what he had in mind. You wouldn't want to tell him that you just listened to some guy on the internet -- would you?

 

Sorry I've not posted in response quicker. I've been jumping between multiple sites trying to figure things out. I've managed to determine the '2nd order approx'. I'd assumed that I had to somehow change the tf into 2nd order from 4th, but this wasn't the case. The 2nd order approx was just the 'complex conjugate' root locus branches. I then used Matlab's 'sisotool' to plot Zeta, this gave me an approx intersection for isolines for Wn. With Wn I could find Ts, Tp, Tr etc.

I now have to design a 'phase-lead compensator' to fit specified design requirements but I'll open that in a seperate post. Thanks for everyones input, I'd be completely lost without the help...

Hello,

I would be interested in seeing the time response of your original function and approximated function. This tells us right away if it is a reasonable approximation. If you dont have the time response then just post the original function and the new function. This will be interesting to compare.