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System Identification Methods

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Applied Control Systems Design

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Abstract

In this chapter, we provide an introductory survey of the methods that have been developed for identification of continuous-time systems from samples of input–output data. The methods include

  1. 1.

    The indirect method, where first a discrete-time model is estimated from the sampled data and then an equivalent continuous-time model is calculated,

  2. 2.

    The direct method based on concepts of approximate numerical integration, where a continuous-time model is obtained directly without going through the intermediate step of first determining a discrete-time model, and

  3. 3.

    Subspace identification algorithms, where deterministic, stochastic and combined deterministic-stochastic subspace identification algorithms are treated.

These methods estimate state sequences directly from the given data, either explicitly or implicitly, through an orthogonal or oblique projection of the row spaces of certain block Hankel matrices of data into the row spaces of other block Hankel matrices, followed by a singular value decomposition (SVD) to determine the order, the observability matrix and/or the state sequence.

The purpose of this chapter is to provide an introductory survey of the methods that have been developed for identification of continuous-time systems from samples of input–output data. The two basic approaches may be described as

  1. 1.

    The indirect method, where a discrete-time model is first estimated from the sampled data and then an equivalent continuous-time model is calculated, and

  2. 2.

    The direct method based on concepts of approximate numerical integration, where a continuous-time model is obtained directly without going through the intermediate step of first determining a discrete-time model.

Next, we give a short introduction to subspace identification algorithms. Deterministic, stochastic and combined deterministic-stochastic subspace identification algorithms are treated. These methods estimate state sequences directly from the given data, either explicitly or implicitly, through an orthogonal or oblique projection of the row spaces of certain block Hankel matrices of data into the row spaces of other block Hankel matrices, followed by a singular value decomposition (SVD) to determine the order, the observability matrix and/or the state sequence. The extraction of the state space model is then achieved through the solution of a least squares problem. Each of these steps can be elegantly implemented using well-known numerical linear algebra algorithms such as the singular value decomposition and the QR decomposition.

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Notes

  1. 1.

    In this case, the predictor regressor vector is identical to the measurement vector.

  2. 2.

    Effectively, if the correction term is null, one holds the previous value of the estimated parameters.

  3. 3.

    In equations of the form of (3.24) the vector ϕ is generally called the observation vector. In this particular case, it corresponds to the measurement vector.

  4. 4.

    A positive definite matrix is characterized by: (i) each diagonal term is positive; (ii) the matrix is symmetric; (iii) the determinants of all principal matrix minors are positive. See the Appendix.

  5. 5.

    One can derives from (3.28) that an optimal value for α is α≈1/ϕ(t)t ϕ(t).

  6. 6.

    This is the real minimum with the condition that the second derivative of the criterion, with respect to \(\hat{\theta}(t)\) is positive, that is \(\frac{\partial^{2}J(t)}{\partial\hat{\theta}(t)^{2}} = 2\sum_{i=1}^{t}\phi(i-1)\phi(i-1)^{t} > 0\), as it is in general the case for \(t \geq \mathrm{dim } \theta\) (see also Sect. 3.3).

  7. 7.

    This equivalent form is particularly useful in analyzing and understanding the algorithm.

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  1. Department of Systems Engineering, King Fahad Univ. of Petroleum & Minerals, Dhahran, Saudi Arabia

    Magdi S. Mahmoud

  2. Dept. Automatic Control, Beijing Institute of Technology, Beijing, China, People’s Republic

    Prof. Yuanqing Xia

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Mahmoud, M.S., Xia, Y. (2012). System Identification Methods. In: Applied Control Systems Design. Springer, London. https://doi.org/10.1007/978-1-4471-2879-3_3

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