Question

\(\displaystyle \lim_{s\rightarrow0}s^{2}G(s)\) defines

• A. Positional error constant
• B. Velocity error coefficient
• C. Acceleration error constant
• D. Mean square error coefficient

Hint:

Remember: \[ K_p=\lim_{s\to0}G(s) \] \[ K_v=\lim_{s\to0}sG(s) \] \[ K_a=\lim_{s\to0}s^2G(s) \] These are position, velocity and acceleration error constants respectively.

Answer 1

The Correct Option is C

Concept: In control systems, static error constants are used to determine the steady-state error for different standard inputs. The three important error constants are: \[ K_p=\lim_{s\to0}G(s) \] \[ K_v=\lim_{s\to0}sG(s) \] \[ K_a=\lim_{s\to0}s^2G(s) \] where \(G(s)\) is the open-loop transfer function.

Step 1: Recall the standard error constants.
For a unity feedback system: \[ K_p=\lim_{s\to0}G(s) \] represents the position error constant. \[ K_v=\lim_{s\to0}sG(s) \] represents the velocity error constant. \[ K_a=\lim_{s\to0}s^2G(s) \] represents the acceleration error constant.

Step 2: Compare with the given expression.
The question provides \[ \lim_{s\to0}s^2G(s) \] which exactly matches the definition of \[ K_a. \]

Step 3: Write the conclusion.
Hence, \[ \boxed{K_a=\lim_{s\to0}s^2G(s)} \] which is called the acceleration error constant. \[ \boxed{\text{Correct Option (C)}} \]