First Order System

First order system transfer function equals to:

H(s) = K/(s-p1)

Low pass RC filter is an example of a first order system with one pole: p = -1/(R*C)

Impulse response is determined via Inverse Laplace transform:

Gain and Phase Margins

Example

% Matlab code

% Gain and Phase Margins

clear all;close all

N=1000;f=logspace(0,2,N);w=2*pi*f;s=j*w;w0=2*pi*10;

H=3./((1+s/w0).*(1+s/w0).*(1+s/w0));

f0dB=max(find(20*log10(abs(H))>0));

f180=min(find((angle(H)/pi*180)<-179));

gainmargin=abs(20*log10(abs(H(f180))));

phasemargin=(angle(H(f0dB))-angle(H(f180)))/pi*180;

disp(['Gain Margin = ' num2str(gainmargin) ' dB'])

disp(['Phase Margin = ' num2str(phasemargin) ' degrees'])

subplot(2,1,1);semilogx(f,20*log10(abs(H)));grid on;axis tight

hold on;line([f(f180(1)) f(f180(1))],[min(20*log10(abs(H))) max(20*log10(abs(H)))])

hold on;line([f(f0dB(1)) f(f0dB(1))],[min(20*log10(abs(H))) max(20*log10(abs(H)))])

hold on;line([f(1) f(N)],[0 0])

hold on;line([f(1) f(N)],[-gainmargin -gainmargin])

text(f(round(N/4)),round(-gainmargin/2),'Gain Margin','fontsize',20)

title('\bf Open Loop Gain','fontsize',20)

xlabel('Frequency (Hz)','fontsize',20);ylabel('|A_0|_d_B','fontsize',20)

subplot(2,1,2);semilogx(f,180/pi*angle(H));grid on;axis tight

hold on;line([f(f180(1)) f(f180(1))],[min(180/pi*angle(H)) max(180/pi*angle(H))])

hold on;line([f(f0dB(1)) f(f0dB(1))],[min(180/pi*angle(H)) max(180/pi*angle(H))])

hold on;line([f(1) f(N)],[angle(H(f0dB))/pi*180 angle(H(f0dB))/pi*180])

text(f(round(N/4)),-180+round(phasemargin/2),'Phase Margin','fontsize',20)

title('\bf Open Loop Phase','fontsize',20)

xlabel('Frequency (Hz)','fontsize',20);ylabel('Phase[A_0]','fontsize',20)

Root Locus

Root locus is a plot of system poles in terms of damping ratio zeta.

Example

Define second order system transfer function:

H(s) = K/((s/w0)^2+2*zeta*s/w0+1)

H(s) = K/((s-p1)*(s-p2))

mathworks.com/access/helpdesk/help/toolbox/simulink/slref/zeropole.html

Gain is infinite for s = p1,p2.

Second order system poles equal to:

p1,2 = -zeta*w0 + sqrt(zeta^2 - 1)*w0, = -zeta*w0 - sqrt(zeta^2 - 1)*w0

where: zeta = damping ratio

thus p2 = conj(p1)

Assuming w0 = 1 rad/s we get root locus plot:

Underdamped system transient impulse response:

h(t) = 2*Re[A*exp(p1*t)] = 2*|A|*exp(-zeta*w0*t)*cos(w0*sqrt(1-zeta^2)*t+phase(A))

where: A = |A|*exp(j*phase(A))

% Matlab code

% Root Locus

clear all;close all

figure;A=axes;set(A,'FontSize',20,'LineWidth',2);hold on

zeta=linspace(0,1,100);w0=1;

plot(-zeta.*w0,w0.*sqrt(1-zeta.^2));hold on;plot(-zeta.*w0,-w0.*sqrt(1-zeta.^2))

zeta=linspace(-1,0,100);p=-zeta.*w0+sqrt((zeta.^2)-1).*w0;

plot(real(p),imag(p));hold on;plot(real(conj(p)),imag(conj(p)))

text(0.05,0.1,'p_1=-\zeta\omega_0+sqrt(\zeta^2-1)\omega_0','fontsize',20)

text(-0.5,0.1,'p_1=-\zeta\omega_0+jsqrt(1-\zeta^2)\omega_0','fontsize',20)

line([-1 1],[0 0])

text(0.05,-0.1,'p_2=-\zeta\omega_0-sqrt(\zeta^2-1)\omega_0','fontsize',20)

text(-0.5,-0.1,'p_2=-\zeta\omega_0-jsqrt(1-\zeta^2)\omega_0','fontsize',20)

text(-0.2,0.5,'0<\zeta<1','fontsize',20)

line([0 0],[-1 1])

text(0.05,0.5,'-1<\zeta<0','fontsize',20)

title('\bf Second Order System Root Locus, p(\zeta,\omega_0=1)','fontsize',20)

xlabel('\sigma=-\zeta\omega_0','fontsize',20);ylabel('j\omega=sqrt(\zeta^2-1)\omega_0,-sqrt(\zeta^2-1)\omega_0','fontsize',20)