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A branch of science concerned with the construction of mathematical models of dynamical systems from measured input/output data. The constructed models are mostly of finite-dimensional difference or differential equation form. The area has close connections with statistics and time-series analysis, and also offers a very wide spectrum of applications.

From a formal point of view, a system identification method is a mapping from sets of data to sets of models. An example of a simple model is the discrete-time ARX-model

\begin{equation} \tag{a1} y ( t ) + a _ { 1 } y ( t - 1 ) + \ldots + a _ { n } y ( t - n ) = \end{equation}

\begin{equation*} = b _ { 1 } u ( t - 1 ) + \ldots + b _ { m } u ( t - m ) + e ( t ), \end{equation*}

where $y$ and $u$ are the outputs and inputs, respectively, of the system and $e$ is a realization of a [[Stochastic process|stochastic process]] (often assumed to be a sequence of independent random variables, cf. also [[Random variable|Random variable]]). Another example is the continuous-time state-space model, described by the linear [[Stochastic differential equation|stochastic differential equation]]

\begin{equation} \tag{a2} \left\{ \begin{array} { l } { d x ( t ) = A x ( t ) d t + B u ( t ) d t + d w ( t ), } \\ { d y ( t ) = C x ( t ) d t + D u ( t ) d t + d v ( t ), } \end{array} \right. \end{equation}

where $x$ is the vector of (internal) state variables and $w$ and $v$ are Wiener processes (cf. also [[Wiener process|Wiener process]]). Artificial neural networks form an example of common non-linear black-box models for dynamical systems.

In any case, the model can be associated with a predictor function $f$ that predicts $y ( t )$ from past (discrete-time) observations

\begin{equation*} Z ^ { t - 1 } = \{ y ( t - 1 ) , u ( t - 1 ) , \dots , y ( 0 ) , u ( 0 ) \}: \end{equation*}

\begin{equation} \tag{a3} \hat{y} ( t | t - 1 ) = f ( Z ^ { t - 1 } , t ). \end{equation}

A set of smoothly parametrized such predictor functions, $f ( Z ^ { t - 1 } , t , \theta )$, forms a model structure $\mathcal{M}$ as $\theta$ ranges over a subset $D _ {\cal{ M} }$ of $\mathbf{R} ^ { d }$. The mapping (estimator or identification method) from observed data $Z ^ { N }$ to $D _ {\cal{ M} }$, yielding the estimate $\hat { \theta } _ { N }$, can be chosen based on a least-squares fit or as a maximum-likelihood estimator (cf. also [[Least squares, method of|Least squares, method of]]; [[Maximum-likelihood method|Maximum-likelihood method]]). This leads to a mapping of the kind

\begin{equation} \tag{a4} \hat { \theta } _ { N } = \operatorname { arg } \operatorname { min } _ { \theta \in D _ { \mathcal{M} } } \sum _ { \mathcal{M} } ^ { N _ { t } = 1 } \text{l} \left( y ( t ) - f ( Z ^ { t - 1 } , t , \theta ) \right), \end{equation}

with a positive scalar-valued function $\operatorname{l}$.

When the data $Z ^ { N }$ are described as random variables, the [[Law of large numbers|law of large numbers]] and the [[Central limit theorem|central limit theorem]] can be applied under weak assumptions to infer the asymptotic (as $N \rightarrow \infty$) properties of the random variable $\hat { \theta } _ { N }$. The covariance matrix of the asymptotic (normal) distribution of the estimate takes the typical form

\begin{equation} \tag{a5} P = \operatorname { lim } _ { N \rightarrow \infty } N . \operatorname{Cov} ( \hat{\theta}_ N ) = \end{equation}

\begin{equation*} = \lambda \operatorname { lim } _ { N \rightarrow \infty } \sum _ { t = 1 } ^ { N } \mathsf{E} \frac { \partial } { \partial \theta } f ( Z ^ { t - 1 } , t , \theta ) \left( \frac { \partial } { \partial \theta } f ( Z ^ { t - 1 } , t , \theta ) \right) ^ { T }, \end{equation*}

where $\lambda$ is the variance of the resulting model's prediction errors, and $\mathsf{E}$ denotes [[Mathematical expectation|mathematical expectation]]. Explicit expressions for $P$ form the basis for experiment design and other user-oriented issues. For general treatments of system identification, see, e.g., [[#References|[a5]]], [[#References|[a7]]], and [[#References|[a3]]].

By adaptive system identification (also called recursive identification or sequential identification) one means that the mapping from $Z ^ { N }$ to $\hat { \theta } _ { N }$ is constrained to be of the form

\begin{equation} \tag{a6} \left\{ \begin{array} { r l r l } { X _ { N } = H ( N , X _ { N - 1 } , y ( N ) , u ( N ) ), }  \\ { \hat{\theta}_{N} = h ( X _ { N } ), } \end{array} \right. \end{equation}

where $X ( t )$ is a vector of fixed dimensions. This structure allows the computation of the estimate at step (time) $N$ with a fixed amount of calculations. This is instrumental in an application where the model is required  "on-line"  as the data is measured. Such applications include adaptive control, adaptive filtering, supervision, etc. The structure (a6) often takes the more specific form

\begin{equation} \tag{a7} \left\{ \begin{array} { c c c c }{   \hat{ \theta }_{N} =\hat{\theta }_{N-1}+ \gamma (N) Q_1(X(N),y(N),u(N)),    }\\{X_{N}= X _ { N - 1 } + \mu _ { N } Q _ { 2 } ( X _ { N-1} ,y(N), u(N)), }\end{array} \right.  \end{equation}

to reflect that the estimate is adjusted from the previous one, usually by a small amount. The convergence analysis of algorithms like (a7) is treated in e.g. [[#References|[a6]]], [[#References|[a1]]], [[#References|[a4]]], and [[#References|[a8]]]. The underlying theory is typically based on averaging, relating (a7) to an associated differential equation, and the subsequent stability analysis of this equation, or on stochastic Lyapunov functions (cf. also [[Lyapunov stochastic function|Lyapunov stochastic function]]). It is also of interest to determine the asymptotic distribution of the estimate as $\gamma$ and $\mu$ become small, see, e.g., [[#References|[a2]]] and [[#References|[a8]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Benveniste,   M. Métivier,   P. Priouret,   "Adaptive algorithms and stochastic approximations" , Springer  (1990) {{ZBL|0752.93073}}</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  L. Guo,   L. Ljung,   "Performance analysis of general tracking algorithms."  ''IEEE Trans. Automat. Control'' , '''40'''  (1995)  pp. 1388–1402</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E.J. Hannan,   M. Deistler,   "The statistical theory of linear systems" , Wiley  (1988)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H.J. Kushner,   D.S. Clark,   "Stochastic approximation methods for constrained and unconstrained systems" , Springer  (1978)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  L. Ljung,   "System identification: Theory for the user" , Prentice-Hall  (1999)  (Edition: Second)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  L. Ljung,   T. Söderström,   "Theory and practice of recursive identification" , MIT  (1983)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  T. Söderström,   P. Stoica,   "System identification" , Prentice-Hall  (1989)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  V. Solo,   X. Kong,   "Adaptive signal processing algorithms" , Prentice-Hall  (1995)</td></tr></table>