Patent for Application Method and System for Adaptive Control of Real-Time Computer Graphics Rendering - Real-Time Rendering: Computer Graphics with Control Engineering - page 158

where:

a 1 … a n a and b 1 … b n b are parameters to be estimated.
y(t) is the output of the system at time t.
yt −

( 1 … yt n a ( ) are the previous outputs on which the current output depends. −

( ) … ut nn ( 1 are the previous inputs on which the current output

ut n k − −−+

k b

depends.
n a is the number of poles or the order of the system.
n b is the number of zeroes plus one.
n k is the delay in the system.

The proportional, integral, and derivative (PID) controller is well reputed for its adop-
tion in over 90% of the world's real control systems. There are several advantages
in using the PID controller, namely its efficiency attributed by the relatively simpler
computation and the ease of implementation compared to other more elaborate con-
trol schemes. In brief, the PID control action in a closed-loop feedback system takes
the form (parallel mode):

t

d
dt et

∫

() = () + () + () ut Ket K ed K ττ (7)

p i d

0

K
T

p

where K

d p = with T i and T d as the time constants of the integral
and derivative controls. At the implementation level, the PID controller's discrete
time form may be expressed as:

= and KKT

i

i

n

KT
T

KT
T

∑ 0

( ) () = () +

p

() +

pd

() −− ( )

un Ken ek en en 1 (8)

p

i

k =

where T is the sampling period and

KT
T

KT
T

p pd

K = , K =

i d

i

where u(n) is the control action. The PID controller's gain values may be derived
either empirically via trial and adjustments or by using the model derived in the
previous section in a closed-loop feedback system as shown in Figure B.2 with an
auto-tuning algorithm.
The derivation of both the system model and the PID controller is exercised
for each linear operation range corresponding to the line segments derived from
the solution to Equation (3). By cascading the series of PID controllers, an overall
control system may be derived as shown in Figure B.6. The object 601 represents
the cascaded PID controller array in which only one PID controller is active at
any time. The object 602 represents a switch agent that channels and activates the