
k
3
¼ fx(n) þ
2T
3
k
2
, unþ
2
3
(8:211)
x(n þ 1) ¼ x(n) þ
T
4
(k
1
þ 3k
3
)(8:212)
Using this RK-3 integrator with a sampling interval
^
T ¼ T=3 to simulate the first-order continuous-
time system dx=dt ¼lx þu results in the third-order difference equation (see Exercise 8.27)
x(n þ 3) ¼ 1 þ lT þ
(lT)
2
2
þ
(lT)
3
6
x(n) þ
T
4
þ
l
2
T
3
6
u(n)
þ
lT
2
2
u(n þ 1) þ
3T
4
u(n þ 2), n ¼ 0, 1, 2, 3, ... (8:213)
where x( n ), n ¼0, 3, 6, 9, . . . are the RK-3 states updated once every T(s).
z-Transforming Equation 8.213 leads to the z-domain transfer function
H(z) ¼
(3T=4)z
2
þ ( lT
2
=2)z þ (T=4) þ (l
2
T
3
=6)
z
3
[ 1 þ lT þ (lT)
2
=2 þ (lT)
3
=6
(8:214)
Consider the RK-4 integrator presented in Section