Please see attached slide for a Routh-Array. I'm struggling to see why the encircled element is 4. Rather, according to my calculations it is 0 as the determinant will give zero. Unless I'm doing something fundamentally wrong...
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2 Answers
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I think in your question there are two special cases that you need to take care while you are approaching to solve this problem.
First thing is that in case if your entire row is zero than you need to take the coefficient of previous row and differentiate it in order to obtain the coefficients of the new row.
Second thing is that if you are geeting a zero in the first column of your routh array then you need to approximate it by a small no e (eplison) and then after calculating all the limits you need to find its limit as epsilon tends to zero..
Using these two properties i hope it will be a lot easier to arrive at your answer .
Hope this helps!!
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I don't know which book you are using to study, but on Ogata's Modern Control Theory, 5th edition page 216 there is a list of the special cases that may happen while applying the Routh-Hurwitz stability criteria to a system. Your example has a need to apply both special cases explained in the book. The first case is when you get the first column of a row equal to zero (the other values on this row are different than zero), the second case is when all of the elements on a row are zero. Each case has a different approach on how to continue the Routh-Hurwitz criterion.
So, if you understood how to proceed when all the elements of a row are zeros (using the auxiliary polynomial) you get the 4 on row s^3(represented as 0->4). When you evaluate the next row(s^2) you will find (s^2| 0 4), the 4 you circled came from the evaluation of (4*4-1*0)/4.
