Question

Consider a unity-gain negative feedback system consisting of the plant \(G(s)\) and a proportional-integral (PI) controller. \[ G(s) = \frac{1}{s-1} \] PI controller: \(C(s) = K_p + \frac{K_i}{s} = \frac{K_p s + K_i}{s}\). Given \(K_p = 3, K_i = 1\). For a unit step reference input, the final values of the controller output and the plant output are asked.

• A. \(\infty, \infty\)
• B. \(1, 0\)
• C. \(1, -1\)
• D. \(-1, 1\)

Hint:

Always check pole locations: if the plant is unstable, controller output may diverge to infinity despite zero error condition.

Answer 1

The Correct Option is A

Step 1: Open-loop transfer function.
\[ L(s) = C(s)G(s) = \frac{3s+1}{s} \cdot \frac{1}{s-1} = \frac{3s+1}{s(s-1)}. \]

Step 2: Closed-loop transfer function.
\[ T(s) = \frac{L(s)}{1+L(s)}. \] Since system has a pole at origin (due to PI controller), system type = 1.

Step 3: Steady-state error for step input.
For type-1 system, steady-state error for unit step is zero. So plant output \(\to 1\).

Step 4: Controller output.
But plant \(G(s) = \frac{1}{s-1}\) is unstable (pole at \(+1\)). So PI control drives input unbounded \(\to\) controller output tends to \(\infty\).

Final Answer: \[ \boxed{\infty, \infty} \]