For a feedback control system of type 2, the steady state error for a ramp input is: 

Explanation:

Feedback Control System of Type 2

Definition: A feedback control system of type 2 is a system where the open-loop transfer function has two integrators. This means that the system has two poles at the origin in the s-plane. The presence of these two integrators allows the system to handle certain types of input signals with zero steady-state error.

Steady-State Error: The steady-state error of a control system is the difference between the desired output and the actual output as time approaches infinity. This error is a critical parameter in evaluating the performance of control systems.

Ramp Input: A ramp input is a type of input signal that increases linearly with time. Mathematically, it can be represented as R(t) = At, where A is a constant. When analyzing control systems, it is crucial to understand how the system responds to different types of inputs, including step, ramp, and parabolic inputs.

Type 2 System and Ramp Input: For a feedback control system of type 2, the steady-state error for a ramp input can be determined using the final value theorem and the concept of system type. The system type is determined by the number of integrators in the open-loop transfer function.

Let's denote the open-loop transfer function as G(s)H(s). For a type 2 system:

G(s)H(s) = K / s2 * G'(s)

where K is a gain constant and G'(s) is a polynomial that does not have poles at the origin.

When a ramp input R(t) = At is applied, the Laplace transform of the ramp input is R(s) = A / s².

The steady-state error for a ramp input can be found using the final value theorem:

e_ss = lim (s → 0) s * E(s)

where E(s) is the Laplace transform of the error signal. The error signal E(s) is given by:

E(s) = R(s) / (1 + G(s)H(s))

Substitute R(s) and G(s)H(s):

E(s) = (A / s2) / (1 + K / s2 * G'(s))

For a type 2 system, the dominant term in the denominator for small values of s is K / s². Therefore:

E(s) ≈ (A / s2) / (K / s2) = A / K

Using the final value theorem:

e_ss = lim (s → 0) s * (A / K) = 0

Conclusion: The steady-state error for a ramp input in a type 2 feedback control system is zero. Hence, the correct answer is option 2: Zero.

Additional Information

To further understand the analysis, let's evaluate the other options:

Option 1: Infinite

This option suggests that the steady-state error for a ramp input is infinite. This would be true for a type 0 system (a system with no integrators in the open-loop transfer function). However, for a type 2 system, the steady-state error is zero, so this option is incorrect.

Option 3: Unity

This option suggests that the steady-state error for a ramp input is unity. This would be true for a type 1 system (a system with one integrator in the open-loop transfer function). However, for a type 2 system, the steady-state error is zero, so this option is incorrect.

Option 4: More than unity but not infinite

This option suggests that the steady-state error for a ramp input is more than unity but not infinite. This is not accurate for a type 2 system, where the steady-state error is zero for a ramp input. Therefore, this option is incorrect.

Conclusion:

Understanding the relationship between the type of a feedback control system and its steady-state error for different types of inputs is essential for evaluating system performance. For a type 2 system, the steady-state error for a ramp input is zero, making option 2 the correct choice. This analysis helps in designing and tuning control systems to achieve desired performance criteria based on the type of input signals they are expected to handle.