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Contents
1 Introduction to System Identification^{[1]}
In order to control a system a model of that system must be created. There are 2 options for creating a model
 Physics, chemistry, biology, etc. which explains the dynamics (usually in time) of the system to be controlled
 Experimental data
The process of constructing models from experimental data is called system identification. These experimentally derived models are not intended to explain the physical system in totality or in any meaningful way. In order to control a system we must have a model of its behavior, understanding the details of that behavior is useful but unnecessary. We need a model adequate to develop a controller for the real system which provides stability and the desired performance.
Typically we want linear, timeinvariant (LTI) models for controls modeling. There are 2 categories of linear model
 Parametric
 Nonparametric
For design via root locus or pole placement a transfer function (parametric description) or a statevariable description that cna provide poles and zeros of the plant. Parametric models are completely defined by their coefficients or parameters. Parameter identification methods are used to determine these parameters.
Measured transfer functions are considered nonparametric models because no finite set of parameters defines them. A controller can be designed directly from these transfer functions.
2 Models and Criteria for Parameteric Identification^{[2]}
To formulate a problem for parametric identification, parameters to be estimated must be selected, and a criteria determined by which the quality of the estimation can be evaluated.
2.1 Parameter Selection
Parameter selection is a matter of estimating the order of the transfer function and solving for those coefficients. See Introduction to Parameter Identification for more information.
2.2 Error Definition
Some measure of the goodness of a fit must be decided upon. Since, by definition, we do not know the true system parameters the error cannot be the difference between the estimated and true parameters. The error definition must be computed from the measured inputs and the outputs. There are 3 well understood definitions
 Equation Error
 Output Error
 Prediction Error
2.2.1 Equation Error
This definition requires knowledge of the input, output, and system states. Using this information, a nonnegative error function is defined
Equation Error 
A search over can be performed and a chosen at the minimum .
Remember this requires measurements of the states and state derivatives, which frequently is not measured or impossible to measure.
2.2.2 Output Error
The definition does not require any knowledge of the states. Instead the estimated parameter is used in the model so that the model becomes a function of just as the actual output is. The error function definition is
Output Error 
A search over can be performed and a chosen at the minimum .
2.2.3 Prediction Error
The output error definition has the weakness of an unbounded result when an input outside the data set used to create the model is used. The prediction error is used to create a model fit that best estimate the system output.
3 System Identification Deterministic Estimation^{[3]}
Now that we've defined an error that depends on performance criteria can be formulated. Within the defined performance a best estimate, , can be found using the error functions described above.
Note that the error definitions above are a sum of squares. These sum of squares of the error leads directly to the motivation for least squares solutions.
 Least Squares (Deterministic)
 Recursive Least Squares (Deterministic)
 Stochastic Least Squares (Stochastic)
 Maximum Likelihood (Stochastic)
4 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