Realization (systems)
Encyclopedia
Realization, in the system theory context refers to a state space
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...

 model implementing a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of (time-varying
Time-variant system
A time-variant system is a system that is not time invariant . Roughly speaking, characteristics of its output depend explicitly upon time.- Overview :...

) matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

  such that

with describing the input and output of the system at time .

LTI System

For a linear time-invariant system specified by a transfer matrix
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

, , a realization is any quadruple of matrices such that .

Canonical realizations

Any given transfer function which is strictly proper
Strictly proper
In control theory, a strictly proper transfer function is a transfer function where the degree of the numerator is less than the degree of the denominator.-Example:...

 can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)):

Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:.

The coefficients can now be inserted directly into the state-space model by the following approach:
.

This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable
Controllability
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control....

 (i.e., because the control enters a chain of integrators, it has the ability to move every state).

The transfer function coefficients can also be used to construct another type of canonical form
.

This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable
Observability
Observability, in control theory, is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals. The concept of observability was introduced by American-Hungarian scientist Rudolf E...

 (i.e., because the output exits from a chain of integrators, every state has an effect on the output).

If we have an input , an output , and a weighting pattern
Weighting pattern
A weighting pattern for a linear dynamical system describes the relationship between an input u and output y. Given the time-variant system described by...

  then a realization is any triple of matrices such that where is the state-transition matrix
State-transition matrix
In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems...

 associated with the realization.

System identification

System identification techniques take the experimental data from a system and output a realization. Such techniques can utilize both input and output data (e.g. eigensystem realization algorithm
Eigensystem realization algorithm
The Eigensystem realization algorithm is a system identification technique popular in civil engineering, in particular in structural health monitoring...

) or can only include the output data (e.g. frequency domain decomposition). Typically an input-output technique would be more accurate, but the input data is not always available.
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