Final answer:
The system's response to a step input can be found by applying the inverse Laplace transform to the product of the transfer function and the Laplace transform of the input. The settling time for this pneumatic system can be estimated using the time constant derived from the capacitance and the coefficients in its transfer function.
Step-by-step explanation:
Obtaining the Expression of System Response to a Step Input
To obtain the expression of the system response X₀(s) to a step input xᵢ, we first need to determine the system's transfer function. With capacitance C given as 1.5, the transfer function becomes:
X₀(s) / Xᵢ(s) = 5 / (2 + 4*1.5s) = 5 / (2 + 6s)
Given a step input of 3 (xᵢ = 3), the Laplace transform of a step input is 3/s. The output of the system in the Laplace domain is:
X₀(s) = (5 / (2 + 6s)) * (3/s)
Now, we apply the inverse Laplace transform to find x₀(t), the time-domain system response.
Calculating the Settling Time of the System Response to a Unit Step Input
The settling time for a first-order system is typically characterized by its time constant, which is represented by the formula τ = RC. However, in this pneumatic system, we have a constant involving capacitance (C) and the coefficient from the transfer function. Using the provided capacitance value, the equivalent formula for the time constant would be τ = 6 (as 6s represents the time-dependent part of the denominator). A commonly used rule of thumb is that the settling time, which is the time it takes for the system to settle within a certain percentage of its final value, is approximately 4 to 5 times the time constant (τ). Therefore, we would multiply the time constant by 4 or 5 to approximate the settling time. Importantly, for precise calculations, refer to the system's specific criteria for settling time, which may depend on the required percentage of the final value.