Patent for Application Method and System for Adaptive Control of Real-Time Computer Graphics Rendering - Real-Time Rendering: Computer Graphics with Control Engineering - page 157

time-invariant dynamic models may be derived using the aforementioned system
identification technique.
To obtain the individual line segments for curves, we can describe this non-linear
relationship represented by a polynomial model:

+
∑

n 1

ni +− 1

y = p x

i

, uxu N

0 ≤≤

(1)

i = 1

where n is the degree of the polynomial and ( n + 1) is the degree that gives the
highest power of the predictor variable. Since straight line segments are used to
fit the curve, the order of the polynomial is chosen as 1. The objective is to derive
a series of line segments which fulfills the approximation of this relationship by
the following:

abx
abx

+
+

uxu
uxu

≤≤
≤≤

1 1 0 1

2 2 1 2

y = (2)

... ...

u ≤≤ x u

abx +

N − 1 N

NN

where the variables a and b are to be found that minimise the following equation (a
constrained optimisation problem):

( )

Fa aabb , ..., ,, ,..., buuu , ...,

12 N 12 N , 12 N − 1

2

N

u

∑ =

j

∫

( )

() −−

= fx abxdx

j

(3)

j

u

j − 1

j 1

and the right hand side of the equation represents the least square error of the
approximation.
Given the solution to the optimisation problem in Equation (3), the input-output
data pairs in each line segment shall be used for the derivation of the corresponding
system model which may be expressed as in the following state space representation
or its ARX model representation as shown in Equations (4), (5), and (6), respectively:

( ) = () + ()

xk + 1 Ax k uk (4)

() = () + ()

yk Cx kDuk (5)

Here x is the state variable of the system, u is the input to the system, y is the output
of the system, and k is the time step. The ARX model representation is given by

() + ( ) +… ( ) = ( ) +…+ ( 1

yt ay t − 1 a yt n − ut n − b ut n − − n b + (6)

1 n a 1 k n k

a b