MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems--Fall 2003 PROBLEM SET 10 SOLUTIONS Home study exercise(E1)O&W 11.32 (a) H(s XO 1+KG(SH(s) (via Black's equation NISD D1 sD2(s)+K (multiply through by D(s)D, s)) The zeros of this closed-loop system are the roots of Ni(the zeros of H) and the roots of D2 (the poles of G) (b) If K=0, then the system is operating without feedback of any sort. Naturally the poles and zeros of the system without feedback are the poles and zeros of H(s) More formally, we can take limits of the above equation N1(s)D2(s)N1(s) 小(5)D2(8)+kM1()M2()=D2(s)D2()=D1()=H() (e)Check by simple substitution Q(s) (8)1+kG(s)f(s 1+K H(s 1+KG(SH(s (d)We are given the +2 (s+4)(s+2) In this simple example, we can find p and q by inspection p(s)=8+1,q(s)=s+2,B()-s+4,()=1 Root locus equation: H(sG(s) 8+4 K →s=-(K+4)� MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 10 Solutions Home study exercise (E1) O&W 11.32 (a) Y (s) H(s) = (via Black’s equation) X(s) 1 + KG(s)H(s) N1(s)D2(s) = (multiply through by D1(s)D2(s)) D1(s)D2(s) + KN1(s)N2(s) The zeros of this closed-loop system are the roots of N1 (the zeros of H) and the roots of D2 (the poles of G). (b) If K = 0, then the system is operating without feedback of any sort. Naturally the poles and zeros of the system without feedback are the poles and zeros of H(s). More formally, we can take limits of the above equation: N1(s)D2(s) N1(s)D2(s) N1(s) lim = = = H(s) K�0 D1(s)D2(s) + KN1(s)N2(s) D1(s)D2(s) D1(s) (c) Check by simple substitution: ⎣ ⎤ ˆ p(s) H(s) Q(s) = q(s) · 1 + KGˆ(s)H(s) ˆ � N1(s)/p(s) p(s) D1(s)/q(s) = q(s) · � 1 + K N2(s)/q(s) N1(s)/p(s) ⎡ D2(s)/p(s) D1(s)/q(s) · · H(s) = 1 + KG(s)H(s) (d) We are given the following: s + 1 s + 2 H(s) = , G(s) = (s + 4)(s + 2) s + 1 In this simple example, we can find p and q by inspection: 1 ˆ ˆ p(s) = s + 1, q(s) = s + 2, H(s) = , G(s) = 1 s + 4 Root locus equation: Hˆ (s)Gˆ(s) = 1 = 1 =� s = −(K + 4) s + 4 −K 1