Graphics Reference
In-Depth Information
problem with a minimal number of line segments. First, we present this non-linear
relationship represented by a polynomial model:
+
n
1
ni
+−
1
y
=
p x
i
, uxu N
0 ≤≤
(5.3)
i
=
1
where ( n + 1) is the order of the polynomial and n is the degree of the polynomial.
The order denotes the number of coefficients to be fit, and the degree represents the
highest power of the predictor variable. Since straight line segments are used to fit
the curve, the degree of the polynomial is chosen as 1. The objective is to derive a
series of line segments that fulfill the approximation of this relationship by:
abx
abx
+
+
,
,
uxu
ux
≤≤
1
1
1
0
1
u
2
2
2
1
2
y
=
(5.4)
...
...
abx
+
,
u
≤≤
x
u
NNN
N
1
N
where the variables a and b minimise the following equation:
(
)
Fa aabb
, ...,
,, ,...,
buuu
, ...,
12
N
12
N
,
12
N
1
2
N
u
=
j
(
)
() −−
=
fx
abxdx
j
(5.5)
j
u
j
1
j
1
The right side of the equation represents the least square error of the approximation.
The different approaches to solving this problem are offered in previous research by
Stone [58], Bellman [59], and Chan and Chin [60].
From the linear ranges derived, the corresponding input-output data set is used
for model identification. The model structure is represented by the state-space [57]
Equations [(5.6) and (5.7)]. The parameters of this system model structure may be
obtained using the subspace algorithm (N4SID) [1].
(
) =
() +
()
xk
+
1
Ax k
uk
(5.6)
() =
() +
()
yk
Cx kDuk
(5.7)
Based on the non-linear operation characteristics of the rendering process, a single
PID controller would be inadequate to provide reasonable control performance over
the entire operating range. Therefore we approach the problem by scheduling dif-
ferent gain values for the PID controller according to the respective linear operating
ranges (Figure  5.8). In this design, the configuration of the PID controller can be
stored in a look-up table so that the relevant values may be set into the PID controller
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