
Example 4.30
A discrete-time system is described by the difference equation
y
k
þ a
1
y
k1
þ a
2
y
k2
¼ b
0
u
k
, k ¼ 0, 1, 2, 3, ... (4:594)
Initial conditions y
1
¼y
2
¼0. The input sequence is given by
u
k
¼ 1 þ (0:1)
k
, k ¼ 0, 1, 2, 3, ... (4:595)
Find the z-domain transfer function H(z) and its poles, the impulse response h
k
, k ¼0,1,2,3,...,
the total response y
k
, k ¼0, 1, 2, 3, . . . , and the natural and forced components of the total
response, and comment on stability for the following cases:
(a) a
1
¼ 0, a
2
¼0:25, b
0
¼ 1
(b) a
1
¼0:5, a
2
¼0:5, b
0
¼ 1
(c) a
1
¼1:5, a
2
¼1, b
0
¼ 1
(a) z-transforming the difference equation y
k
0.25 y
k2
¼u
k
, k ¼0, 1, 2, 3, . . . yields
H