Modelling Non-Linear Rendering Processes - Real-Time Rendering: Computer Graphics with Control Engineering - page 65

The linearised approximation of the model at an operating (trim) point is:

( ) = () +

Δ xk + 1 AxkBuk Δ Δ ( ) (4.21)

() = () +

Δ yk CxkDuk Δ Δ () (4.22)

where Δ x 1 , Δ x 2 , Δ u , and Δ y are small deviations:

() = () −

Δ xk xk x trim (4.23)

1 1 1

() = () −

Δ xk xk x trim (4.24)

2 2 2

() = () −

Δ uk uk u trim (4.25)

() = () −

Δ yk yk y trim (4.26)

with

x = u (4.27)

1 trim trim

( )

1 1 1

x = F Wu + B (4.28)

2 trim trim

( )

2 2 2

y = FWx + B (4.29)

trim 2 trim

∂
∂

( )

2

C =

x Fxu xy

, | (4.30)

trim , trim

∂
∂

( )

2

D =

u Fxu xu

, | (4.31)

trim , trim

From the above equations, we have:

d
dx

{ }

() = () −= () +

2 2 2

Δy kyky FWxk B ( ) (4.32)

trim 2

2

and

0 0

=

1
0

A = d

dx

, B (4.33)

{ }

1 1 1

FWx ( + B ) 0

1 trim

1