Topic: Control Systems
Find the values of $K_{P}$, $K_{I}$, and $K_{D}$ for PID controller whose characteristics equation has real roots at 10, $\xi=0.8$ and $\omega_{n}=2~\text{rad/s}$
- A. $K_{D}=1.8, K_{P}=8, K_{I}=10$
- B. $K_{D}=1.8, K_{P}=4, K_{I}=5$
- C. $K_{D}=3.6, K_{P}=8, K_{I}=20$
- D. $K_{D}=1.8, K_{P}=8, K_{I}=20$
Answer: A. $K_{D}=1.8, K_{P}=8, K_{I}=10$
Explanation: For PID controller, $G_c(s) = K_p + K_D s + \frac{K_I}{s}$.
The characteristic equation is $1 + G_c(s)G(s) = 0$.
$1 + \left(K_P + K_D s + \frac{K_I}{s}\right) \left(\frac{4}{s²+6s+1}\right) = 0$.
$s³ + 6s^2 + s + 4K_P s + 4K_D s² + 4K_I = 0$.
$s³ + (6+4K_D)s² + (1+4K_P)s + 4K_I = 0$ ... (i)
Given roots are $s = -10$ and complex roots corresponding to $\xi=0.8, \omega_n=2$.
The desired characteristic equation is $(s² + 2(0.8)(2)s + 2^2)(s+10) = 0$.
$(s² + 3.2s + 4)(s+10) = s³ + 13.2s^2 + 36s + 40 = 0$ ... (ii)
Comparing (i) and (ii):
$6 + 4K_D = 13.2 \Rightarrow K_D = 1.8$.
$1 + 4K_P = 36 \Rightarrow K_P \approx 8$.
$4K_I = 40 \Rightarrow K_I = 10$.