Question

The transfer function of a linear system is G(s) = Y(s) S-1 U(s) 5? + 4s +3 a. Express this system in the modal form. b. Express this system in the standard controllable form (SCF). (Parts d, e, f, and g use this system) c. In the standard controllable form, suppose the output is replaced by y=[-1 a] | [x2] Give a value for a which makes the system unobservable. d. What is y(t) if y(0-)=-3, ay = 6 and ult)= unit step starting at t=0. dt 1-0- by using any of the methods you know of. e. Compute eAt via any of the methods you know of. f. Design a state variable feedback controller using the SCF form such that the closed- loop poles are at -4+4j and -4-4j and the DC gain of the closed-loop system is 1. g. Compute the transfer function of the loop gain of the closed-loop system

Answer 1

Y(s) = (s) = 51 VS) s +45+3 x = Alt Bu y = catou [x] - state vector fal = Differential state vectus (y] = outpot vector Cu = Ye vectos A = state mator B - Yp matrix ( - olp matrix D transimssian matos state model uy Scontrollable form - 11 y=841683

y = {1, 9] (21) c = [lia] un-observable. Ler Ater])=0 CT = EA AT = 1:3) *** +603) (a) ATET = ( 3 ) let stall - 3) AN -1t49 -1 (-1+49-3a² = 0 1-49 -39" =0 3a²+40-120 a=0.215 a =-1.548

2 960) =-3 y Co-) = 6 act) = unit step two uts) - Ys [5+45 + 3]y() = (s] 015) sy®) +4546) +3463) = SUB) - USJ 59 (s) - sy (o) - Yo) + 454(3) - 49 (0) +346) = f(t) - UG) Ľt4stos[!] (5 +45 +3] 4 (6) = $1 - 35-6 - S4-3565 s[5+45+3] Y(s) = -35-554 s[s +3] [8 +13 = Yrs -(1+55 +35²) S (5+1) 8+3) = A + B to - S stl st3 = 41o =- B1+ = -[145+3) GU (2) 12 = -(1-15+2) 63) (42) = -13 6

@ Y = { 1 + 13/4) g (t) = -1/3 4 Et) - let u () - 13 ezt . ucts @ ett _ i fiss - 35 S -4 S-4 -45-3 52423 دیو S-15-3 At e