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Question 6 2.5 For a control system with closed-loop transfer function as 4s+k/s^2+2as+3,determine the value of a and k for overshoot of 10% or less?

Answer :

To achieve an overshoot of 10% or less, the values of 'a' and 'k' for the given closed-loop transfer function are approximately a = 1.6221 and k can be any value

To determine the values of 'a' and 'k' for a control system with a closed-loop transfer function of 4s + k / s^2 + 2as + 3 that results in an overshoot of 10% or less, we need to perform some calculations.

The general form of a second-order transfer function is:

G(s) = ωn^2 / (s^2 + 2ζωns + ωn^2)

Here, ωn represents the natural frequency and ζ is the damping ratio. In our case, we have:

4s + k / s^2 + 2as + 3

Comparing this with the general form, we can see that:

ωn^2 = 3 (from the constant term)

2ζωn = 2a (from the coefficient of the 's' term)

To ensure an overshoot of 10% or less, we need to satisfy the following condition:

ζ > sqrt((ln(M/100))^2 / (pi^2 + ln(M/100)^2))

Where M is the desired percentage overshoot. Substituting M = 10 into the equation, we get:

ζ > sqrt((ln(10/100))^2 / (pi^2 + ln(10/100)^2))

ζ > sqrt((-2.3026)^2 / (9.8696 + (-2.3026)^2))

Now, let's solve for ζ:

ζ > sqrt(5.3088 / (9.8696 + 5.3088))

ζ > sqrt(5.3088 / 15.1784)

ζ > sqrt(0.34965)

ζ > 0.5911

Since we want to minimize the overshoot, we'll set ζ = 0.5911. Now, we can solve for 'a':

2ζωn = 2a

2 * 0.5911 * sqrt(3) = 2a

Know more about transfer function here;

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