The block schematic uses PID controller to design a system

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The block schematic uses PID controller to design a system whose characteristics equation has real root at – 10 and ξ = 0.8 and ω_n = 2 rad/sec. Find the value of K_P, K_I and K_d –

1. K_d = 1.8, K_P = 8, K_I = 10
2. K_d = 1.8, K_P = 4, K_I = 5
3. K_d = 3.6, K_P = 8, K_I = 20
4. K_d = 1.8, K_P = 8, K_I = 20

Correct Option: A

Given G(s) =	4
	s² + 6s + 4

PID → KP +	KI	+ Kd.s
	s

From figure

C(s)	=	K_P +(K_I/s)+ K_d.s
R(s)	=	s²+ 6s + 4(K_P +(K_I/s)+ K_d.s) + 4

C(s)	=	(KP +(K_I/s)+ K_d.s)s
R(s)	=	s³+ 6s² + 4K_Ps + 4K_I + 4K_d.s² + 4s

C(s)	=	(KP +(K_I/s)+ K_d.s)s
R(s)	=	s³+ (6s + 4K_P)s² + 4(1 + K_d)s + 4K_I

C.E. s³ + (6 + 4K_d)s² + 4(1 + K_p)s + 4K_I = 0 …....(i)
As given that C.E. has real root at – 10,
i.e., (s + 10) (s² + 2ξω_n s + ω²_n) = 0
or
s³ + (2ξω_n + 10)s² + (ω²_n + 20ξω_n)s + 10ω²_n = 0 . . .…(ii)
on comparing equation (i) with the standard equation of 3rd order, we get
6 + 4K_d = 2ξω_n + 10
on putting
ξ = 0·8
and
ω_n = 2 rad/sec

K_d =	2 × 0·8 × 2 + 10 – 6	= 1·8
	4

and 4(1 + K_P) = ω²_n + 20ξω_n

K_P =	2² + 20 × 0·8 × 2	– 1 = 8
	4

and
4K_I = 10ω²_{n<.sub>
or
K_I = 10 × 2² = 10
4

Hence alternative (A) is the correct choice.}