Question

using following parameters as defined

G1(s)=1/(s+50)

G2(s)=K/s

G3(s)=1/(s+10)

H(s)=1

R(s) is the unit step function

a) find the closed loop transfer function as a function of K

b) what is the maximum value of the K the system can tolerate?

c) is there an effect on the system if the pole in G1(s) is changed to :

1) G1(s)= 1/(s+500)

2) G1(s)=1/(s+11)

G1(s) G2(s) G3(s) C(s) H(s)

Answer 1

Given G_1(s) = 1/s + 50 G_2 (s) = k/s G_3(s) = 1/s + 10 H(s) = 1 solution to (a) in the given block diagram the blocks G_1(s), G_2(s), G_3(s) are in cascade connection, then their gains are multiplied then G_1(s).G_2(s).G_3(s) = 1/s + 50 k/s 1/s + 10 the G_1(s).G_2(s).G_3(s) = k/s(s + 10)(s + 50) the block diagram representation of system now the block diagram resembles, A open loop system with unity negative feedback, then over all transfer function is y(s)/R(s) = k/s(s + 10)(s + 50)/1 + k/s(s + 10)(s + 50) y(s)/R(s) = k/s(s + 10)(s + 50) + k Y(s)/R(s) = k/s(s^2 + 60s + 500) + k therefore overall transfer function (closed loop) Y(s)/R(s) = k/s^3 + 60 s^2 + 500 s + k solution to (b) the characteristic equation is s^3 + 60 s^2 + 500 s + k = 0 by Routh Hurwitz criteria for the system to be stable 60(500) - k/60 greaterthanorequalto 0 then 60(500) - k greaterthanorequalto 0 k lessthanorequalto 30000

Thankyou