systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...

, a realization of a

state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the t ...

model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of ( time-varying)

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

(t),B(t),C(t),D(t) /math> such that
: \dot(t) = A(t) \mathbf(t) + B(t) \mathbf(t) : \mathbf(t) = C(t) \mathbf(t) + D(t) \mathbf(t) with (u(t),y(t)) describing the input and output of the system at time t .

LTI System

For a

linear time-invariant system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly define ...

specified by a transfer matrix,

H(s)

, a realization is any quadruple of matrices

(A,B,C,D)

such that

H(s) = C(sI-A)^B+D

Canonical realizations

Any given transfer function which is

strictly proper In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusiv ...

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: :

H(s) = \frac

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

\dot(t) = \begin
                               -d_& -d_& -d_& -d_\\
                                1&      0&      0&      0\\
                                0&      1&      0&      0\\
                                0&      0&      1&      0
                             \end\textbf(t) + 
                             \begin 1\\ 0\\ 0\\ 0\\ \end\textbf(t)

\textbf(t) = \begin n_& n_& n_& n_ \end\textbf(t)

. This state-space realization is called controllable canonical form (also known as phase variable canonical form) because the resulting model is guaranteed to be controllable (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 :

\dot(t) = \begin
                               -d_&   1&  0&  0\\
                               -d_&   0&  1&  0\\
                               -d_&   0&  0&  1\\
                               -d_&   0&  0&  0
                             \end\textbf(t) + 
                             \begin n_\\ n_\\ n_\\ n_ \end\textbf(t)

\textbf(t) = \begin 1& 0& 0& 0 \end\textbf(t)

. This state-space realization is called observable canonical form because the resulting model is guaranteed to be

observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...

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

General System

''D'' = 0

If we have an input

u(t)

, an output

y(t)

, and a

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 : \dot(t) = A(t)x(t) + B(t)u(t) : y(t) = C(t)x(t), then the output can be written as : y(t) = ...

T(t,\sigma)

then a realization is any triple of matrices

(t),B(t),C(t) /math> such that T(t,\sigma) = C(t) \phi(t,\sigma) B(\sigma) where \phi is the state-transition matrix 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) or can only include the output data (e.g.

frequency domain decomposition The frequency domain decomposition (FDD) is an output-only system identification technique popular in civil engineering, in particular in structural health monitoring. As an output-only algorithm, it is useful when the input data is unknown. FDD ...

). Typically an input-output technique would be more accurate, but the input data is not always available.

References