Subspace identification method: Difference between revisions

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{{short description|Mathematical concept}}

In mathematics, specifically in [[control theory]], '''subspace identification''' ('''SID)''') aims at identifying [[linear time-invariant theory|linear time invariant]] (LTI) state space models from input-output data. SID does not require that the user parametrizes the system matrices before solving a parametric optimization problem and, as a consequence, SID methods do not suffer from problems related to local minima that often lead to unsatisfactory identification results.

== History ==

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SID methods are rooted in the work by the German mathematician [[Leopold Kronecker]] (1823–1891). Kronecker<ref>L. Kronecker, "Algebraische reduktion der schaaren bilinearer formen", S. B. Akad. Berlin, pp. 663–776, 1890.</ref> showed that a power series can be written as a rational function when the rank of the Hankel operator that has the power series as its symbol is finite. The rank determines the order of the polynomials of the rational function.

In the 1960s the work of Kronecker inspired a number of researchers in the area of Systems and Control, like Ho and Kalman, Silverman and Youla and Tissi, to store the [[Markov parameter]]s of an LTI system into a finite dimensional [[Hankel matrix]] and derive from this matrix an (A,B,C) realization of the LTI system. The key observation was that when the Hankel matrix is properly dimensioned versus the order of the LTI system, the rank of the Hankel matrix is the order of the LTI system and the SVD of the Hankel matrix provides a basis of the column space observability matrix and row space of the controllability matrix of the LTI system. Knowledge of this key spaces allows to estimate the system matrices via linear least squares.<ref>M.{{Citation |last=Verhaegen, "|first=Michel |title=Subspace Techniques in System Identification", in|date=2013 |url=https://doi.org/10.1007/978-1-4471-5102-9_107-1 |work=Encyclopedia of Systems and Control, https://link.springer.com/referenceworkentry/|pages=1–13 |editor-last=Baillieul |editor-first=John |access-date=2023-10-12 |place=London |publisher=Springer |language=en |doi=10.1007/978-1-4471-5102-9_107-1 |isbn=978-1-4471-5102-9 |editor2-last=Samad |editor2-first=Tariq|url-access=subscription }}</ref>

An extension to the stochastic realization problem where we have knowledge only of the Auto-correlation (covariance) function of the output of an LTI system driven by white noise, was derived by researchers like Akaike.<ref>H. Akaike, "A new look at the statistical model identification", IEEE Transactions on Automatic Control, vol. 19, pp. 716–723, 1974.</ref>