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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