Signals and Systems - Electrical Engineering

376 C H A P T E R 6: Application to Control and Communications

We want to create a MATLAB function that has as inputs the coefficients of the numeratorN(s)

and of the denominatorD(s)of the system’s transfer functionH(s)=N(s)/D(s)(the coefficients

are ordered from the highest order to the lowest order or constant term). The other input of the

function is the type of responsetwheret=1 corresponds to the impulse response,t=2 to the

unit-step response, andt=3 to the response to a ramp. The output of the function is the desired

response. The function should show the transfer function, the poles, and zeros, and plot the cor-

responding response. We need to figure out how to compute the ramp response using thestep

function.

Consider the following transfer functions:

(a)H 1 (s)=

s+ 1

s^2 +s+ 1

(b)H 2 (s)=

s

s^3 +s^2 +s+ 1

Determine the stability of these systems.

Solution

The following script is used to look at the desired responses of the two systems and the location

of their poles and zeros. We consider the second system; you can run the script for the first system

by putting % at the numerator and the denominator afterH 2 (s)and getting rid of % afterH 1 (s)

in the script. The functionresponsecomputes the desired responses (in this case the impulse, step,

and ramp responses).

%%%%%%%%%%%%%%%%%%%

% Example 6.4 -- Control toolbox

%%%%%%%%%%%%%%%%%%%

clear all; clf

% % H1(s)

% nu = [1 1]; de = [1 1 1];

%% H2(s)

nu = [1 0]; de = [1 1 1 1]; % unstable

h = response(nu, de, 1);

s = response(nu, de, 2);

r = response(nu, de, 3);

function y = response(N, D, t)

sys = tf(N, D)

poles = roots(D)

zeros = roots(N)

figure(1)

pzmap(sys);grid

if t == 3,

D1 = [D 0]; % for ramp response

end