Gain Scheduled H Control of Wind Turbines for the Entire Operating Range - Wind Turbine Control and Monitoring - page 83

X ( s ) − I

u d

u

y d −

G ( s ) X ( s )

y lin

w e u

K ( s ) G ( s )

−

Fig. 4.13 Equivalent representation of the anti-windup compensation scheme in Fig. 4.12

be proved, after some system manipulations, that the block diagram in Fig. 4.13 is
equivalent to the scheme in Fig. 4.12 by defining

¼ T aw ð s Þ u ð s Þ¼ X ð s Þ I u ð s Þ;

u d ð s Þ
y d ð s Þ

ð 4 : 13 Þ

Y ð s Þ

where X and Y are the coprime factors of G, i.e., G ¼ X 1 Y. Therefore, the anti-
windup compensator can be expressed as

ð 4 : 14 Þ

where F is chosen for A þ B u F to be Hurwitz.
In this way X must be designed to ensure the closed-loop stability of X I and
the deadzone nonlinear operator. At the same time, X must be designed to mini-
mize the effect of y d on the controlled variable. It can be proved that using the
Lyapunov function V ð x aw Þ¼ x aw Px aw [ 0 and forcing

V ð x aw Þþ y d y d mu T u\0 ;

ð 4 : 15 Þ

with x aw the state of the system T aw , the previously mentioned objectives are
satisfied. Because of the sector boundedness of the dead-zone nonlinearity, the
following condition is satisfied

2 ^ T U 1 ð u Fx aw ^ Þ 0 ;

ð 4 : 16 Þ

with U a diagonal matrix.