Example H3.3

Determine the stepresponse in the time domain from a pz-map

• Date : 11/08/2021
• Revision : 1.0

The given situation

Finding the stepresponse y(t)

a) Symbolic through Inverse laplace transform

% clear all variables from Workspace and close all figures

clear variables;

close all;

% Define 's' variable

s=tf('s'); % Make s of type tf

 

H=10/(2*s+8);

X=5/s;

Y=X*H;

zpk(Y)

ans = 25 ------- s (s+4) Continuous-time zero/pole/gain model.

figure(101);

pzmap(Y);

grid on;

title('pz map of Y(s)');

ylim([-1 1]);

xlim([-5 1]);

% Replace s of type tf with s of type sym

syms s; % Make s of type syms

% Create symbolic time functions

syms t x(t) y(t);

% Create symbolic laplace functions

syms X(s) Y(s) H(s);

 

 

% Create symbolic transferfunction

H(s)=10/(2*s+8);

 

% x input is step of 5

x(t)=5*heaviside(t);

figure(102);

fplot(x(t));

grid on;

title('inputsignal x(t)');

xlabel('t [sec]');

ylabel('x(t)');

ylim([-1 6]);

xlim([-1 4]);

% Transform to laplace

X(s)=laplace(x,t,s)

X(s) = 

Y(s)=X(s)*H(s);

y(t)=ilaplace(Y)

y(t) = 

figure(103);

fplot(y(t));

grid on;

title('stepresponse y(t)');

xlabel('t [sec]');

ylabel('y(t)');

ylim([0 7]);

xlim([0 4]);

b) Stepresponse using the pz-map

where and

belongs to the pole in zero and the multiplication of all the distances from that pole to the other poles equals 4. As there are no zeros the denumerator equals 1.

% Create symbolic time functions

syms t x(t) y(t);

% State the values of the pz-map

p1=0;

p2=-4;

Kpn=25;

% calculate the Aj coefficients

A1=1/(p1-p2);

A2=1/(p2-p1);

% define the stepresponse signal

y(t)=Kpn*(A1*exp(p1*t)+A2*exp(p2*t))

y(t) = 

 

% Plot it

figure(105);

fplot(y(t));

grid on;

title('stepresponse y(t)');

xlabel('t [sec]');

ylabel('y(t)');

ylim([0 7]);

xlim([0 4]);