Introduction to System Identification


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Introduction to System Identification
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System Identification Parameter Identification
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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

  1. Physics, chemistry, biology, etc. which explains the dynamics (usually in time) of the system to be controlled
  2. 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, time-invariant (LTI) models for controls modeling. There are 2 categories of linear model

  1. Parametric
  2. Nonparametric

For design via root locus or pole placement a transfer function (parametric description) or a state-variable 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

  1. Equation Error
  2. Output Error
  3. Prediction Error

2.2.1 Equation Error

This definition requires knowledge of the input, output, and system states. Using this information, a non-negative error function is defined

LaTeX: J\left(\mathbf{\theta}\right)=\int_{T}^{0} \mathbf{e}^T\left(t, \mathbf{\theta} \right) \mathbf{e}\left(t, \mathbf{\theta} \right) dt Equation Error

A search over LaTeX: \mathbf{\theta} can be performed and a LaTeX: \hat{\mathbf{\theta}} chosen at the minimum LaTeX: J\left(\mathbf{\theta}\right).

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 LaTeX: \mathbf{\theta} is used in the model so that the model becomes a function of LaTeX: \mathbf{\theta} just as the actual output is. The error function definition is

LaTeX: J\left(\mathbf{\theta}\right)=\sum_{k=0}^{N}\mathbf{e_0}^2\left(k; \mathbf{\theta}\right) Output Error

A search over LaTeX: \mathbf{\theta} can be performed and a LaTeX: \hat{\mathbf{\theta}} chosen at the minimum LaTeX: J\left(\mathbf{\theta}\right).

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 LaTeX: \mathbf{\theta} performance criteria can be formulated. Within the defined performance a best estimate, LaTeX: \hat{\mathbf{\theta}}, can be found using the error functions described above.

Note that the error definitions LaTeX: J\left(\mathbf{\theta}\right) above are a sum of squares. These sum of squares of the error leads directly to the motivation for least squares solutions.

4 References

  • Franklin, G. F., Powell, J. D., and Workman, M. 1998 Digital Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201331535

4.1 Notes

  1. Franklin et all, pp. 479-480
  2. Franklin et all, pg. 495-502
  3. Franklin et all, pg. 502-535