Question

Q-2 Suppose a system, as shown in the block diagram of a unity feedback loop, below, has a transfer function, P(S) = . We want to stabilize this system with a proportional integral (PI) controller, such that the poles of the characteristic equation are to be: s, = s2 = S3 = -1. If we let the controller, C(s) = , determine all parameters (a, a, b , K, and K) in the following transfer function in order to satisfy both given and required conditions. [28 pts.] bo (Kps + KD Try(8) = 33 + ,82 + (Qo+Kpbo)s + Kibo

Answer 1

given that, PCS) - 5+ 35-1 CCS) - KpS+KI is 5,= 'ses :- we know formula for closed loop transfer füriction Y(5) - P(s) c(s) RCS) itpcs)(5), ..:' kps + KI s($+35-1) .. kps+KI s($+36-1) It Y(S) kes+ Кт RES) . 3+38+5+kph+ kg ys) - KpS+ ks RCS) 5343$+s(kp-1)4 kq o characterstic equation of the system after stabilizing with PI Corrlroller = (Śh) 3. (5+1)3 = ($+25 +i)(+) CE = '$+5+28+ $+1+25: : ce (required)'s 3+.38+.35+17e

comparing coefficients of eqnco and . . kp-1=3 and kg = 1 Kp = 3+1 and ķy = 1 kp=u and kj=1 Substuding thease values an eqno given resi= bo (kps+1) staš+(qat kepbuds+ kybo comparing coefficients of eqn3) and eqn@ and ka bo=i : [a,=3] qtk bo=3 9 +461)= 3 kgbo=1 KI XI = 1 [kg = 1 ... kp= 4, 1, 9-3, q= -1, k1=1 There fare all Parameters are a=-1, a=3, bol, kp 4 and k = 1