44 Chapter 2 / Mathematical Modeling of Control Systemswhere the derivatives are evaluated at If the variation
is small, we may neglect the higher-order terms in Then Equation (2–43) may be
written as
(2–44)
where
Equation (2–44) may be rewritten as
(2–45)
which indicates that is proportional to Equation (2–45) gives a linear math-
ematical model for the nonlinear system given by Equation (2–42) near the operating
point
Next, consider a nonlinear system whose output yis a function of two inputs x 1 and
x 2 ,so that
(2–46)
To obtain a linear approximation to this nonlinear system, we may expand Equation (2–46)
into a Taylor series about the normal operating point Then Equation (2–46)
becomes
where the partial derivatives are evaluated at Near the normal oper-
ating point, the higher-order terms may be neglected. The linear mathematical model of
this nonlinear system in the neighborhood of the normal operating condition is then
given by
y-y–=K 1 Ax 1 - x– 1 B+K 2 Ax 2 - x– 2 B
x 1 =x– 1 ,x 2 =x– 2.
+
02 f
0 x^22
Ax 2 - x– 2 B^2 d +p
+
1
2!
c
02 f
0 x^21
Ax 1 - x– 1 B^2 + 2
02 f
0 x 1 0 x 2
Ax 1 - x– 1 BAx 2 - x– 2 B
y=fAx– 1 ,x– 2 B+ c
0 f
0 x 1
Ax 1 - x– 1 B+
0 f
0 x 2
Ax 2 - x– 2 Bd
x– 1 ,x– 2.
y=fAx 1 ,x 2 B
x=x–,y=y–.
y-y– x-x–.
y-y–=K(x-x–)
K=
df
dx
2
x=x–y–=f(x–)
y=y–+K(x-x–)
x-x–.
dfdx,d^2 fdx^2 ,p x=x–. x-x–
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