
Example 8.1
The first-order system in Equation 8.1 is simulated using trapezoidal integration.
(a) Find an expression for e
l
, the fractional error in characteristic root.
(b) Find an asymptotic formula for e
l
valid for jlTj1.
(c) Over what range of values for lT is the asymptotic formula for e
l
accurate?
(a) The difference equation for trapezoidal integration is based on
x
A
(n þ 1) ¼ x
A
(n) þ
T
2
{f [x
A
(n), u(n)] þ f [x
A
(n þ 1), u(n þ 1)]} (8:25)
where f [x
A
(n), u(n)] and f [x
A
(n þ1), u(n þ1)] refer to the derivative function in Equation 8.1.
Z-transforming the difference equation and then solving for the ratio X(z)=U(z) results in the
z-domain transfer function ...