Solution 6.8.3.9 The root locus is shown in Figure 1. The real part of the complex poles is Im(s) Re(s) -4 s + 4 s + 2 -2 Figure 1: Root locus -3 the imaginary part is ! d = ! n q 1; 2 = 3 p 1; 2  = 3: ! n =3 q (2) = 4:2426 The gain that places the poles at s = ;3j3is K = js +2jjs+4jj s=;3+j3 =10: The MATLAB dialogue p1 =2 p2 = 4 zeta = 1/sqrt(2) sigma = 3 omegad = (sigma/zeta)*sqrt(1-zeta^2) K=1+omegad^2 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 2: Step response omegan = sigma/zeta s=-zeta*omegan + j* omegan *sqrt(1-zeta^2);; sc = conj(s) tn2 = zpk([],[s sc],omegan^2) g=zpk([],[-p1 -p2],K) tc = feedback(g,1) T=linspace(0,5,200);; [Yn2,T] = step(tn2,T);; [Ytc,T] = step(tc,T);; plot(T,Yn2,'k-',T,Ytc,'k--') print -deps sr6839.eps t=0:0.01:1;; u=t;; [Yrtc,t] = lsim(tc,u,t);; [Yrn2,t] = lsim(tn2,u,t);; plot(t,Yrn2,'k-',t,Yrtc,'k--') print -deps rr6839.eps plots and saves the step and ramp responses are shown in Figure 2 and 3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 3: Ramp response The partial fraction expansion for the step response is C(s)= 0:5556 s + 0:3928 6 2:3562 s +3; j3 + 0:3928 6 2:3562 s +3+j3 Then the time response is c(t)=[0:55+ 0:7857e ;3t cos(3t +2:3562)]1(t): The partial fraction expansion for the ramp response is C(s)= ;0:1852 s + 0:5556 s + 0:09259 6 0 s +3; j3 + 0:09259 6 0 s +3+j3 Then the time response is c(t)=[;0:1852+ 0:5555t+0:1852e ;3t cos(3t+2:3562)]1(t): The complete MATLAB porgram to nd the step and ramp responses and the partial fraction expansions is p1 =2 p2 = 4 3 zeta = 1/sqrt(2) sigma = 3 omegad = (sigma/zeta)*sqrt(1-zeta^2) K=1+omegad^2 omegan = sigma/zeta s=-zeta*omegan + j* omegan *sqrt(1-zeta^2);; sc = conj(s) tn2 = zpk([],[s sc],omegan^2) g=zpk([],[-p1 -p2],K) tc = feedback(g,1) T=linspace(0,5,200);; [Yn2,T] = step(tn2,T);; [Ytc,T] = step(tc,T);; plot(T,Yn2,'k-',T,Ytc,'k--') print -deps sr6839.eps t=0:0.01:1;; u=t;; [Yrtc,t] = lsim(tc,u,t);; [Yrn2,t] = lsim(tn2,u,t);; plot(t,Yrn2,'k-',t,Yrtc,'k--') print -deps rr6839.eps v1 = [1 0] v2 = [1 -sc] v3 = [1 conj(-sc)] B= K A=conv(v1,v2) A=conv(A,v3) roots(A) [R,P,K1] = residue(B,A) M=R(2) absm =abs(M) abs2m = 2*abs(M) angm =angle(M) t=0 dt = 0.025 kount = 1 test = R(3) +abs2m*cos(angm) while t<5 c(kount) = R(3) +abs2m*exp(-sigma*t) *cos(sigma*t + angm);; time(kount) = t;; 4 t=t+dt;; kount = kount + 1;; end plot(time,c) pause v1 = [1 0] v2 = [1 -sc] v3 = [1 conj(-sc)] B= K A=conv(v1,v1) A=conv(A,v2) A=conv(A,v3) roots(A) [R,P,K1] = residue(B,A) M=R(2) absm =abs(M) abs2m = 2*abs(M) angm =angle(M) t=0 dt = 0.025 kount = 1 test = R(3) +abs2m*cos(angm) while t<5 c(kount) = R(3) + R(4)* t +abs2m*exp(-sigma*t) *cos(0.9338*t + angm);; time(kount) = t;; t=t+dt;; kount = kount + 1;; end plot(time,c) 5