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A unity feedback system is characterized by an open loop transfer function: K G(s) = s(s + 10). Determine the gain K so that the system will have a damping ratio of 0.5. For this value of K, determine settling time, peak percent overshoot, and time to peak overshoot for a unit step input.

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Transcript

00:01 Hello students, in this question it is given that the open loop transfer function is equal to k into g of x is equal to s into s plus 10.

00:06 The damping ratio is given as zeta is equal to 0 .5.

00:09 We are asked to find out these credentials.

00:10 So, let us start by writing out the characteristics equation that is for unity feedback system, we have 1 plus g of s into h of s.

00:21 As the system is unity feedback, h of s is equal to 1.

00:25 Therefore, 1 plus g of s is equal to 0.

00:29 Substituting the function open loop transfer function, we have 1 plus s into s plus 10 divided by k is equal to 0.

00:42 We have k plus s square plus 10 s is equal to 0.

00:47 That is rearranging s square plus 10 s plus k is equal to 0.

00:52 Comparing this equation by the standard equation of a unity feedback system that is s square plus 2 zeta omega n is the natural frequency into s plus omega n square is equal to 0.

01:08 This is the standard equation.

01:11 Comparing the equation, we have omega n square is equal to root k.

01:18 So, each k implies omega n the natural frequency is equal to root k.

01:27 Also while comparing we have 2 zeta omega n is equal to 10.

01:34 From this we can find the value of omega n and k that is 2 zeta root k is equal to 10...