High-Order Sliding Mode Control of DFIG-Based Wind Turbines - Wind Turbine Control and Monitoring - page 31

or in matrix form

x ¼ Ax þ u ð x ; u Þþ e ð t Þ
y ¼ Cx

where

8
<

01
00

A ¼

C ¼ 10 ½

Kx 1 u
J
0

:

u ð x ; u Þ¼

:

0
f ð t Þ

e ð t Þ¼

A candidate observer could be [ 18 ]

x ¼ Ax þ u ð x ; u Þ hD h S 1 C T C ð x x Þ

ð 2 : 20 Þ

; S ¼

10
0

1 1
12

where D h ¼

1
h

Let S be the unique solution of the algebraic Lyapunov equation

S þ A T S þ SA C T C ¼ 0

ð 2 : 21 Þ

Let us define x ¼ D h ð x x Þ then

x ¼ h ð A S 1 C T C Þ x þ D h u ð x Þ u ð x Þ

_

ð Þ D h e ð t Þ ð 2 : 22 Þ

Consider the quadratic function

V ¼ x T Sx ð 2 : 23 Þ

then

8
<

V ¼ 2x T S _

x
V ¼ hV x T C T Cx þ 2x T SD h u ð x Þ u ð x Þ

ð Þ

:

2x T SD h e ð t Þ

Therefore,

k D h u ð x Þ u ð x Þ ð Þ k

V hV þ 2 kk k max ð S Þ

ð 2 : 24 Þ

þ D h e ð t Þ k k