Final answer:
To find the steady-state output yss for the given transfer function when the input is r(t)=5cos(3t), we must consider the frequency and phase shift induced by the transfer function. The correct answer, considering the frequency of the input signal and the nature of the transfer function, is likely option (a) yss=12cos(3t-45°).
Step-by-step explanation:
The student's question refers to finding the steady-state output yss for a given transfer function Hyr when the input is a cosine function r(t)=5cos(3t). The transfer function provided is Hyr=36/(s+3)2. To find the steady-state response, we would typically use the phasor or the frequency response method. However, looking at the given options for yss, we aim to match the frequency and phase shift that the transfer function causes to the input signal.
To obtain the gain and phase shift, we substitute s = jω into the transfer function, where ω is the angular frequency of the input signal. In this case, since the input signal is r(t)=5cos(3t), the angular frequency is ω=3 rad/s. For the transfer function Hyr, this results in a magnitude (gain) of |Hyr| factor and a phase shift. The magnitude scaling factor is obtained from 36 divided by the magnitude of the complex denominator, and the phase shift is obtained from the phase of the complex denominator. Since the transfer function is real and has even symmetry, the phase shift will be a multiple of 90 degrees.
The options for yss provided suggest that the correct answer will include the original frequency of 3 rad/s and a phase shift. The magnitude will be scaled by the gain of the transfer function at the frequency of the input signal. Using this method aligns with the concept that a linear time-invariant system in steady state outputs a frequency that matches the input, though the amplitude and phase may be altered based on the system's transfer function.
Considering the provided options and the process for determining the steady-state response, the most likely correct answer is option (a) yss=12cos(3t−45°).