Final answer:
To determine the closed-loop transfer function Φ(s) from the pulse response, perform a Laplace transform on the pulse response k(t). The transfer function, represented as K(s), is found by integrating k(t)e⁻ˣt from 0 to infinity. This process will reveal the system's transfer function.
Step-by-step explanation:
The question asks us to determine the closed-loop transfer function Φ(s) for a system given its pulse response k(t)=0.0125e⁻¹²⁵ᵗ. To find the closed-loop transfer function from a pulse response, we need to perform a Laplace transform of the pulse response k(t).
The Laplace transform of k(t) is K(s), which can be found by integrating k(t)e⁻ˣt with respect to t from 0 to ∞. Assuming the system is linear and time-invariant, K(s) should provide us with the system's transfer function.
For k(t)=0.0125e⁻¹²⁵ᵗ, the Laplace transform is:
K(s) = ∫0∞ 0.0125e⁻(t¹²⁵ + s)t dt
By solving this integral, we get the transfer function Φ(s).