Preliminaries - Real-Time Rendering: Computer Graphics with Control Engineering - page 13

2.2.3 c omPuting m odel P aRameteRs
In system identification, the model parameters are estimated by minimising the func-
tion that describes errors between the derived system model output and the measured
response. Assuming a system is linear and time-invariant, the output of the linear
model y model can be expressed as

model () = () ()

y t Gsut (2.2)

where G(s) is the transfer function, y the model output and u , the input to the model.
To determine G ( s ), we can minimise the difference between the model output y model (t)
and the measured output y meas (t) . We can use the minimisation criterion which is a
weighted norm of the error v(t):

() = () − () = () − () ()

vt y t y t y t Gsut (2.3)

meas model meas

where y model ( t ) is either the model's simulated response given an input u ( t ) or its pre-
dicted response given a finite series of past output measurements, i.e., ( y meas ( t -1),
y meas ( t -1),…).
From the above, v ( t ) is otherwise known as the simulation error or prediction
error. The objective of the estimation algorithm is to generate a set of parameters in
the model structure G such that eventually this error is minimised.

2.2.4 e Valuating Q uality of d eRiVed m odel
The steps taken to evaluate the quality of a derived system model generally include
the comparison of the model response to the measured response and the analysis
of model residuals. Figure 2.6 compares the outputs of two different models with a
measured output.
Residuals are differences between a model's one-step-predicted output and the
measured data. In other words, residuals may be understood as portions of validation
data that are not well described by the model. In residual analysis, the whiteness and
independence tests are key performance indicators.
The whiteness text examines whether a model includes a residual auto-correlation
function inside the confidence interval of the estimates. If it does, the model passes
the test and the outcome indicates that the residuals are not correlated.
In addition, a model is qualified when it passes the independence test (no correla-
tion between its residuals and past inputs). If evidence indicates such a correlation,
the information revealing how the output relates to the input is incomplete. A simple
example is an output y(t) beyond the confidence interval during a lag k that originates
from the input u(t - k). A good model should perform both tests relatively well.
The system identification methodology accommodates an iterative process in the
determination of the final model structure and parameters. A real world system may
not be represented by only a single model structure. Whenever a derived model is
found inadequate, it is necessary to revisit the model selection process, reconsider