Solution 10.8.9.7c The MATLAB program w=logspace(-2,3,200);; s=j*w;; K =138 z1 = 3.26 p1 = 0 p2 = 1 p3 = 30 mag = 20*log10( abs((K.*( +z1))./((s+p1).*(s+p2).*(s+p3))));; phase=(angle(s+z1)-angle(s+p1)-angle(s+p2)-angle(s+p3))*180/pi;; % % 0.5 dB % M=10^(0.5/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[gamma'];; R=[rho'];; % % 1dB % M=10^(1/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; 1 G=[G gamma'];; R=[R rho'];; M=1.0593;; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; % % 2dB % M=10^(2/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; % % 3dB % M=10^(3/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; 2 c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; % % 4dB % M=10^(4/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; % % 6dB % M=10^(6/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; % % 9dB % M=10^(9/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; 3 theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; % % 12 dB % M=10^(12/20);; c=-M^2/(M^2 - 1);; r =abs(M/(M^2-1));; theta = linspace(0.00001,2*pi,360);; x=c+r*cos(theta);; y=r*sin(theta);; s=x+j*y;; gamma = angle(s)*180/pi;; b=zeros(1,360);; c=gamma > b;; gamma = gamma -360*c;; rho = 20*log10(abs(s));; G=[G gamma'];; R=[R rho'];; plot(phase,mag,'k-',G,R,'k--') grid on axis([-360 0 -10 30]) print -deps 10897c.eps gcgp = zpk([-z1],[-p1 -p2 -p3],K) tc = feedback(gcgp,1) t=linspace(0,4,800);; [yhat,t] = step(tc,t);; plot(t,yhat) grid on axis([0,4,0,1.4]) print -deps 10897csr.eps Draws the plot shown in Figure 1. As can be seen the log magnitude plot passes just outside the 2 dB circle. 4 -350 -300 -250 -200 -150 -100 -50 0 -10 -5 0 5 10 15 20 25 30 0.5 dB 1 dB 2 dB 3 dB 4 dB 6 dB 9 dB 12 dB Figure 1: Nichols plot If weestimate 1.8 dB, then Then 10 1:8=20 =1:2303: Based on this calculation wewould expect the maximum overshoot to be about 23%. The step response is shown in Figure 2. The actual overshoot is about 18% so the Nichols chart gives a conservative estimate in this case. 5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 Figure 2: Step response 6