Understanding Steady-State Error Constants

For a unity feedback system, the steady-state error constants describe the system's response to different types of inputs at steady state. These constants are calculated from the open-loop transfer function $G(s)$.

The given open-loop transfer function is:

$ G(s) = \frac{1000(s+8)}{(s+7)(s+9)} $

The system is specified as type '0', meaning there are no poles at the origin ($s=0$) in the transfer function $G(s)$.

Calculating Positional Error Constant (Kp)

The positional error constant, $K_p$, is calculated as the limit of $G(s)$ as $s$ approaches 0:

$ K_p = \lim_{s \to 0} G(s) $

Substituting the given $G(s)$:

$ K_p = \lim_{s \to 0} \frac{1000(s+8)}{(s+7)(s+9)} $

$ K_p = \frac{1000(0+8)}{(0+7)(0+9)} = \frac{1000 \times 8}{7 \times 9} = \frac{8000}{63} $

$ K_p \approx 126.98 \approx 127 $

Calculating Velocity Error Constant (Kv)

The velocity error constant, $K_v$, is calculated as the limit of $s \cdot G(s)$ as $s$ approaches 0:

$ K_v = \lim_{s \to 0} s \cdot G(s) $

$ K_v = \lim_{s \to 0} s \cdot \frac{1000(s+8)}{(s+7)(s+9)} $

Since the transfer function $G(s)$ does not have a pole at $s=0$ (it's a type '0' system), the limit becomes:

$ K_v = 0 \cdot \frac{1000(0+8)}{(0+7)(0+9)} = 0 $

Calculating Acceleration Error Constant (Ka)

The acceleration error constant, $K_a$, is calculated as the limit of $s^2 \cdot G(s)$ as $s$ approaches 0:

$ K_a = \lim_{s \to 0} s^2 \cdot G(s) $

$ K_a = \lim_{s \to 0} s^2 \cdot \frac{1000(s+8)}{(s+7)(s+9)} $

Similar to $K_v$, because $G(s)$ is type '0', the limit becomes:

$ K_a = 0^2 \cdot \frac{1000(0+8)}{(0+7)(0+9)} = 0 $

Summary of Constants

The calculated values for the steady-state error constants are:

• $K_p \approx 127$
• $K_v = 0$
• $K_a = 0$

Therefore, the correct option is $K_p = 127; K_v = 0; K_a = 0$.