From ControlTheoryPro.com

Contents
1 Introduction to Parameter Identification
In order to control or properly analyze a complex system a mathematical model is required. There are 2 ways to create a model for any given system. The first method is to examine each component directly and analytically describe each. The second method is to use input and output data from the actual system and construct a mathematical approximation.^{[1]}
When constructing an analytical model each component must be described, mathematically, in detail. Once each component is described they can be integrated into a complete endtoend system model. Constructing each component is nontrivial and integrating the components can be significantly more difficult than constructing the individual components. Additionally, a system could be made up of components whose internal workings are unknown or whose internal workings change dramatically based on external conditions. Parameter identification techniques determine an approximate model from measured input and output data. There are 2 basic methods of parameter identification. The fastest is generally OffLine identification. Typically this involves a leastsquares technique for determining a relationship between the input and output data. The slower, more versatile, method is online, or recursive, identification. Recursive identification involves sequential input and output data to construct a model which adjusts over time to the changing conditions. Both methods are linear but the recursive techniques can often handle some nonlinearities. Handling of these nonlinear portions of a model would be akin to making a small angle assumption around a desired operating point but the recursive algorithms allow that operating point to change over time.^{[2]}
2 Basic Mathematics of Parameter Identification^{[3]}
In this section a basic introduction to the mathematics of parameter identification. If it is assumed that 2^{nd} order difference equation then
where
 is an unknown system input,
 is a boundary condition,
 is a boundary condition, and
 are unknown equation coefficients.
Since there are 3 unknowns, a minimum of 3 equations is needed
Define
Note that
if
If the data, and , contain measurement errors then more data points than unknowns should be used to find the optimal parameters.
where
 is abbreviated as and
Then
but
To find the optimal parameter set,
where
pseudoinverse 
is the pseudoinverse.
It can be seen, from what follows, that is the optimal solution. The error vector is defined as
error 
and the squared error is defined as
where
 is a scalar constant.
The necessary condition for optimality is
When then
The sufficient condition for optimality is
for minimum
3 References
 Franklin, G. F., Powell, J. D., and Workman, M. 1998 Digital Control of Dynamic Systems. 3^{rd}. AddisonWesley Longman Publishing Co., Inc. ISBN 0201331535
 Spradlin, Gabriel T. '"AN EXPLORATION OF PARAMETER IDENTIFICATION TECHNIQUES: CMG TEMPERATURE PREDICTION THEORY AND RESULTS." Master's Thesis, University of Houston, Houston, TX December 2005.