Question

The steady-state error of Type-2 system for a standard unit velocity input is

• A. \(\frac{1}{1+K_v}\)
• B. \(\infty\)
• C. \(\frac{1}{K_v}\)
• D. \(0\)

Hint:

Keep this reference table in mind for steady-state errors ($e_{ss}$): {|c|c|c|c|} System Type & Step Input (Position) & Ramp Input (Velocity) & Parabolic Input (Accel.)
Type 0 & $\frac{1}{1+K_p}$ & $\infty$ & $\infty$
Type 1 & 0 & $\frac{1}{K_v}$ & $\infty$
Type 2 & 0 & 0 & $\frac{1}{K_a}$

Answer 1

The Correct Option is D

Concept: The steady-state error ($e_{ss}$) evaluates a control system's tracking accuracy as time approaches infinity ($t \rightarrow \infty$). It depends on the input signal type (position, velocity, acceleration) and the System Type, which is defined by the number of open-loop poles located exactly at the origin ($s = 0$) of the complex s-plane.

Step 1: Identify the input signal and its Laplace domain form.
The input provided is a standard unit velocity input (also known as a unit ramp function, $r(t) = t$). Its Laplace transform is: \[ R(s) = \frac{1}{s^2} \]

Step 2: Set up the steady-state error formula.
Using the Final Value Theorem for a unity feedback control system: \[ e_{ss} = \lim_{s \to 0} \frac{s \cdot R(s)}{1 + G(s)H(s)} = \lim_{s \to 0} \frac{s \cdot \left(\frac{1}{s^2}\right)}{1 + G(s)H(s)} = \lim_{s \to 0} \frac{1}{s + s \cdot G(s)H(s)} \]

Step 3: Account for a Type-2 system configuration.
A Type-2 system contains exactly two pure integrators at the origin. Its open-loop transfer function can be written as: \[ G(s)H(s) = \frac{K \prod (s + z_i)}{s^2 \prod (s + p_i)} \] Substituting this expression into the error equation yields: \[ e_{ss} = \lim_{s \to 0} \frac{1}{s + s \cdot \left[ \frac{K \prod (s + z_i)}{s^2 \prod (s + p_i)} \right]} = \lim_{s \to 0} \frac{1}{s + \frac{K \prod (s + z_i)}{s \prod (s + p_i)}} \] Evaluating the limit as $s \to 0$: \[ e_{ss} = \frac{1}{0 + \frac{K \cdot \text{constant}}{0}} = \frac{1}{\infty} = 0 \] Because a Type-2 system has high open-loop loop gain at low frequencies, it can track a ramp velocity input with zero steady-state tracking error. This corresponds to Option (D).