Final answer:
To find the values of K and a in the transfer function, we can equate the steady state responses to different inputs and solve for the coefficients. By comparing the Laplace transforms of the input and output, we can set up equations to find the values of K and a. We then solve the equations simultaneously to obtain the values of K and a.
Step-by-step explanation:
To find the values of K and a, we can use the steady state responses to different inputs and compare them.
Given that the steady state response to x(t) = 4sin(t) is yss(t) = 20sin(t + p1) and the steady state response to x(t) = 5sin(4t) is yss(t) = 10sin(4t + p2), where p1 and p2 are the phase shifts, we can use the relationship between the input and output in the frequency domain.
H(s) represents the transfer function, which relates the Laplace transform of the output to the Laplace transform of the input.
For the input x(t) = 4sin(t) and the corresponding output yss(t) = 20sin(t + p1), we can write the Laplace transforms as Yss(s) = H(s)X(s), where X(s) and Yss(s) are the Laplace transforms of x(t) and yss(t) respectively. Substituting the given values, we have 20sin(t + p1) = K/s + a * 4sin(t).
Taking the Laplace transform of both sides, we get Yss(s) = 20/(s^2+1) + aK/(s^2+1).
Similarly, for the input x(t) = 5sin(4t) and the corresponding output yss(t) = 10sin(4t + p2), we can write the Laplace transforms as Yss(s) = H(s)X(s), where X(s) and Yss(s) are the Laplace transforms of x(t) and yss(t) respectively. Substituting the given values, we have 10sin(4t + p2) = K/s + a * 5sin(4t).
Taking the Laplace transform of both sides, we get Yss(s) = 10/(s^2+16) + 5aK/(s^2+16).
Now we can equate the two expressions for Yss(s) to find the values of K and a. Setting 20/(s^2+1) + aK/(s^2+1) = 10/(s^2+16) + 5aK/(s^2+16), we can find a common denominator and equate the numerators.
20(s^2+16) + aK(s^2+16) = 10(s^2+1) + 5aK(s^2+1)
Expanding and rearranging the terms, we get:
20s^2 + 320 + aKs^2 + 16aK = 10s^2 + 10 + 5aKs^2 + 5aK
Combining like terms, we have:
(20 + aK)s^2 + (320 + 16aK - 10)s + (16aK - 10) = 0
This is a quadratic equation in s^2. To find the values of K and a, we need to equate the coefficients of s^2, s, and the constant term to 0.
Eq1: 20 + aK = 0
Eq2: 320 + 16aK - 10 = 0
Eq3: 16aK - 10 = 0
Solving these equations simultaneously, we can find the values of K and a.