Modern Control Engineering

Example Problems and Solutions 59

A–2–13. Linearize the nonlinear equation

in the region defined by 8x10, 2y4.

Solution.Define

Then

where we choose

Since the higher-order terms in the expanded equation are small, neglecting these higher-

order terms, we obtain

where

z–=x–^2 + 4 x–y–+ 6 y–^2 = 92 + 4 * 9 * 3 + 6 * 9 = 243

K 2 =

0 f

0 y

2

x=x–,y=y–

= 4 x–+ 12 y–= 4 * 9 + 12 * 3 = 72

K 1 =

0 f

0 x

2

x=x–,y=y–

= 2 x–+ 4 y–= 2 * 9 + 4 * 3 = 30

z-z–=K 1 (x-x–)+K 2 (y-y–)

x–=9, y–=3.

z=f(x,y)=f(x–,y–)+ c

0 f

0 x

(x-x–)+

0 f

0 y

(y-y–)d

x=x–,y=y–

+p

f(x, y)=z=x^2 +4xy+6y^2

z=x^2 +4xy+6y^2

MATLAB Program 2–5

A = [0 1;-25 -4];

B = [1 1;0 1];

C = [1 0;0 1];

D = [0 0;0 0];

[NUM,den] = ss2tf(A,B,C,D,1)

NUM =

01 4

0 0 –25

den =

1425

[NUM,den] = ss2tf(A,B,C,D,2)

NUM =

0 1.0000 5.0000

0 1.0000 -25.0000

den =

1425