Question 11.15.7: Consider the discrete-time system described by the transfer ...
Consider the discrete-time system described by the transfer function
H(z)=\frac{8z^2\ +\ 10z\ -\ 6}{z^3\ +\ 2z^2\ -\ z\ -\ 2}.
a. Find out if the system is stable.
b. Express the transfer function in zero/pole/gain form.
c. Express the transfer function in partial fraction form.
The command zplane plots one extra zero at the axis origin when the order of the numerator polynomial is lower than the order of the denominator polynomial.
a.
num=[8 10 -6];
den=[1 2 -1 -2];
zplane(num,den)
One pole (plotted with the asterisk) is not inside the unit circle, hence the system is not stable.
b.
Ts=0.1;
H=tf(num,den,Ts);
H=zpk(H)
Zero/pole/gain:
\frac{8(z\ +\ 1.693)(z\ -\ 0.443)}{(z\ -\ 1)(z\ +\ 1)(z\ +\ 2)}
Sampling time: 0.1
c.
[R,P,K]=residue(num,den)
R=2.0000 2.0000 4.0000
P=-2.0000 1.0000
-1.0000
K=[]
The transfer function in partial fraction form is given by
H(z)=\frac{2}{z\ +\ 2}+\frac{2}{z\ -\ 1}+\frac{4}{z\ +\ 1}
