Question

F(S) + X(S) + + Suppose that the system shown has a plant for which Gr=-- and a PI con- S+2 troller, so that Ge=K,+". (40) (a) Using the expressions given for G, and Gc, solve the block diagram shown to find an expression for the output X(s) in terms of F(s), Kp, K, and s. (20) (b) For the system's natural frequency to be 0, = 3 rad/s, what would be the integral gain, K,? (20) (c) For W, = 5 rad/s and G = 0.15, what would be the proportional gain, K,? (20) (d) For the system to be critically damped, what would be the relation- ship between the gains K, and K,?

Answer 1

Given block diagram Given block diagram XC) 1-Gp (4) XC) Gp GC 1-4Gp Gp Gc fcs) 1-4 Go +GpGc [xes) = X(S) = Geoke fcs) 1-4 Gp + Spac substuting Gp = 5th Ас - Кет к Scanned with CamScanner

Xcs)= stz * (kp + Bate 1-4 (58) + (Stax (ket han foy (Kf8+ĶI) X (s)- str 5+2 -14 + (ket K2) ste ket K2 XCS)= - fo s- 2tke + kr Kostka Xcs) - fcs) s²_ast kest ki. kostki st kp stki XCS) = 4 Fco I sht (kp-2) 5+ ko KI: 9 ③ comparing denome nator with shtz&wnston we get whl= kz., given wh=3&ad/sec, so ki= a 0 2&wn = (ke-2) => kp=240.15x5+2 co kane 35 & CamScanner

for system to be critically damped §=1 but whark? =) 2x &x Wn = kp-2. 2x1x Ski= kp-2 25 ki=kp-2 or kp= zu Ki +2. CS scanduk pz[rka+] K + Scanned with CamScaner