Linearization of Nonlinear Functions

53.7 Linearization of Nonlinear Functions

Frequently, the equations expressing the geometric and physical conditions of a problem are nonlinear, which makes their direct solution difficult and uneconomical. We linearize these equations using series expansion, usually Taylor’s series, which in general is given by the following for y = f (x):

y = fx 0 ( ) df + ------ Dx + ---- 1 d -------- y

2 ( Dx ) + L + ----- 1 --------

This gives the value of y at (x 0 + Dx), given the value of the function f (x 0 ) at x 0 . Equation (53.121) includes still higher-order terms, and therefore we usually drop the second- and higher-order terms and use the approximation

0 y 0 ª fx ( ) dy + ------ Dx y ª + j Dx

dx

with obvious correspondence in terms. The technique of linearization is demonstrated in Fig. 53.4 . The curve represents the original nonlinear function f(x), whereas the straight line represents the linearized form, Eq. (53.122). That line is tangent to the curve at the given point a, (x 0 ,y 0 ). When Dx is given (or evaluated), the value of the function would be approximated by point b, whose ordinate is (y 0 +j Dx), and the exact value from the nonlinear function is point c, with ordinate f (x 0 + Dx). The error arising from using the linear form is the line segment bc.

One Function of Two Variables

y = fx ( 1 , x 2 ) = fx 0 ( 1 , x 0 2 ) + -------- ∂y

Dx 1 + -------- ∂ y

2 + ---- 1 -------- y

2 + ---- 1 -------- y ∂ ( Dx 1 ) ∂

+ -------- ∂y

-------- ∂y

( Dx 1 ) Dx ( 2 )L +

∂x 1 0 0 ∂x 2 0 0

x1 , x2

x1 , x2

For the linearized form, Eq. (53.123) is truncated to y = 0 y + j 1 Dx 1 + j 2 Dx 2 (53.124)

where

0 0 y 0 = fx (

j 1 = -------- ∂y

j 2 = -------- ∂y

Equation (53.124) can be rewritten in matrix form as

J = ------ ∂y = -------- ∂y

-------- yx y ∂

∂x

∂x 1 ∂x 2

is the Jacobian of y with respect to x.

Two Functions of One Variable

[ j 1 2 ] j = ------- ------- dx

Two Functions of Two Variables Each

1 ª y 1 + j 11 12 j Dx 1 (53.127b)

y 2 y 0 2 j 21 22 j Dx 2

or

y = y 0 + J yx Dx

(53.127c) (53.127c)

and

∂y -------- 1 -------- ∂y 1 J = ∂y ------ = ∂x 1 xy ∂x 2

∂x

∂y -------- 2 ∂y -------- 2 ∂x 1 ∂x 2

0 evaluated at 0 x

General Case of m Functions of n Variables

With the auxiliaries,

-------- ∂y 1 ∂y -------- L 1 -------- ∂y 1 ∂x 1 ∂x 2 ∂x n

J yx = ∂y ------ =

M 0 O M evaluated at x

∂x --------- ∂y m ∂y --------- L m --------- ∂y m ∂x 1 ∂x 2 ∂x n

Dx 1 Dx = Dx 2

M Dx n

the linearized form of Eq. (53.128a) becomes

y y 0 ª + J yx Dx

(53.128b) which represents the general form, with y, y 0 being m ¥ 1 vectors, J an m ¥ n Jacobian matrix, and Dx

an n ¥ 1 vector. Equations (53.122), (53.125), and (53.127c) are special cases of Eq. (53.128b).

Differentiation of a Determinant

The partial derivative of a determinant with respect to a scalar is composed of the sum of p p ¥ p determinants, each having the elements of only one row or one column replaced by their derivatives.

Thus, given the determinant in which represents its p columns, then d = D 1 D 2 LD p D i , i = , 1 2Lp

D ----------L ∂D 2 2 ∂D p + 1 D P + L + D 1 D 2 L --------- p ∂x (53.129) ∂x ∂x ∂x

------ ∂d = ----------D ∂D 1 LD

An expression similar to Eq. (53.129) can be written in which rows instead of the columns of d are partially differentiated.

Differentiation of a Quotient

The partial derivative of g = U/W with respect to a variable x is given by

------ ∂g = ----- ∂ 1 ------- U – ----- U -------- W ∂

∂x (53.130)

W ∂x W ∂x

Both U and W can be general functions, including determinants, of several variables.