I have been going through Ogata Modern Control Engineering book and working through several exercises to improve my understanding of basic control principles. I came across the following example which I am struggling to solve.
I am stuck with reducing order of my transfer function while maintaining the same (or very similar) response.
I am modelling a system which includes a mechanical and an electrical part. The overall transfer function for my system is 5th order.
The time constant for the electrical part is very small when compared to the mechanical time constants and due to this we can replace the electrical system with a gain and get an equivalent 4th order system.
As an example, my system is similar to this (taken from Ogata's control book):
This is my transfer function, and it includes both the mechanical and electrical parts of my system:
I want to reduce the 5th order system, to a 4th order system, while maintaining the same response.
What I have tried:
In order to know whether I have reduced the system correctly, I designed the block diagram for my model on Simulink.
This is the response to a chirp signal:
My idea was to use the above response, to see whether I have reduced the transfer function correctly.
Next, I proceeded to find the poles, zeros and gain of my system. To do so, I used this piece of code available on MathWorks, with the 5th order TF above:
b = [0.0001 10]; a = [0.005 5.00010060 0.661600001 61.01102010 2.1101 10]; fvtool(b,a,'polezero') [b,a] = eqtflength(b,a); [z,p,k] = tf2zp(b,a)
The output was as follows, which is exactly what I had expected:
and the equivalent PZ-map:
The above results show the pole associated with the electrical circuit, which is far to the left. This can be removed, thus reducing the order of the transfer function from 5th order to 4th order.
Next, I proceeded by using zp2tf to eliminate the pole on the far left as follows, however the output does not seem to make sense.
z = [-100000]'; p = [ -0.04 + 0.0347i % *100 -0.04 - 0.0347i -0.02 + 0.0041i -0.02 - 0.0041i ]; k = 0.0200; [b,a] = zp2tf(z,p,k); [bnew,anew] = zp2tf(z,p,k); bnew/200 anew/200 ans = 0 0 0 0 0.0001 10.0000 ans = 0.0050 0.0500 0.0001 0.0001 0.0000 0.0000
I was expecting the above to result in a 4th order system, but clearly I am doing something wrong with my approach.
I am stuck with replacing the electrical part of the block diagram with a simple gain block, while maintaining the same (or very similar) output response.
How can I get the equivalent block diagram model for the fourth order system?
Any pointers, tips and/or advice on what I should do would be appreciated.