Problem 4 Consider the following unity feedback...

Problem 4 Consider the following unity feedback system. G(s) = 1 / (4s^2(s^2 + 2)) a) Using the Routh–Hurwitz criterion, find the region of the s-plane where the poles of the closed-loop system are located. b) Use MATLAB to verify your answer and find the poles of T(s). Indicate whether this system is stable, unstable, or marginally stable. Show all your workings, MATLAB codes and results.

Transcript

00:01 Hi, so we are given that for the following unit feedback system, the g of s is equals to 1 over 4 s square into the bracket s square plus 2.

00:18 In a part, we are using ruther routh hurwitz criteria, we have to find the region of s plane.

00:27 So, let's see the solution.

00:28 So, first of all, this can be simplified as the characteristics equation 1 plus g of s into h of s, this should be equals to 0.

00:43 Now, h of s is unity given to us, so we get 1 plus 1 over 4 s square s square plus 2 is equals to 0.

00:54 We get 4 s to the power 4 plus 8 s square plus 1 is equals to 0.

00:59 So, according to routh array, we say s cube s square s1 and s0 terms.

01:10 So, as you can see, okay, we just left s to the power 4 term.

01:14 So, s to the power 4 will have 4, 8 and 1, then s cube is corresponding to 0, 0, 0 and s square s1 is 0 is missing here.

01:27 So, by means of ruther's array, we get the p of s is equals to 4 s to the power 4 plus 8 s square plus 1 as our auxiliary polynomial.

01:39 So, let's say s square is equals to s to the power 1 means s dash.

01:45 So, here it will become 4 s dash whole square plus 8 s dash plus 1 is equals to 0.

01:53 By using b square minus 4 is formula, s dash should be equals to minus of b means minus 8 under root over b square 8 square will be 64 minus 4 ac.

02:03 So, 4 into 4 divided by twice of a is what 4.

02:08 So, on solving, we get the roots of s dash is equals to minus 1 plus minus 0 .866.

02:16 So, clearly once we can subtract once we can add it up.

02:21 So, s dash will have two roots negative of 0 .134 and the second one will be negative of 1 .866...

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