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Exact and approximate approaches to the identification of stochastic max-plus-linear systems

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Abstract

Stochastic max-plus linear systems, i.e., perturbed systems that are linear in the max-plus algebra, belong to a special class of discrete-event systems that consists of systems with synchronization but no choice. In this paper, we study the identification problem for such systems, considering two different approaches. One approach is based on exact computation of the expected values and consists in recasting the identification problem as an optimization problem that can be solved using gradient-based algorithms. However, due to the structure of stochastic max-plus linear systems, this method results in a complex optimization problem. The alternative approach discussed in this paper, is an approximation method based on the higher-order moments of a random variable. This approach decreases the required computation time significantly while still guaranteeing a performance that is comparable to the one of the exact solution.

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Notes

  1. Intuitively, this can be characterized as follows. Note that Eqs. 3 and 4 imply that each component of x(k + 1) can be written as a max expression of terms in which the unknown parameters θ and λ appear. An input signal is then said to be sufficiently rich if it is such that each of these terms is the maximal one sufficiently often (this is also related to the idea of persistent excitation in conventional system identification (Ljung 1999)).

  2. If there are two identical affine arguments in the max expression in Eq. 10, then the corresponding sets Ω ij coincide. So in general the Ω ij sets either coincide or they only overlap at the boundaries (see also Remark 1). For the sake of simplicity of the exposition, we assume here that any identical arguments in the max expression in Eq. 10 have already been eliminated.

  3. For the case that f(e) is a continuous function, \(\hat{\eta}_i(k+1,\hat{\theta},\hat{\lambda})\) is continuously differentiable and since it is a convex function, the subgradients are unique and they are equal to the gradients.

  4. In the rest of this section, the index k will be dropped (except for η i and \(\hat \eta_i\)) for the sake of simplicity of notation.

  5. This theorem is in fact a special case of the results appeared in Pečarić and Beesack (1987), as explained in Ivelić and Pečarić (2011)

  6. These times were obtained running Matlab 7.5.0 (R2007b) on a 2.33 GHz Intel Core Duo E655 processor.

  7. The relative error is defined as \(\frac{|x_0 - x|}{|x|}\) where x is the true value and x 0 is the estimated value.

  8. Note that here due to the 3σ-rule, we choose λ one third of the one in Example 7.1.

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Acknowledgements

The authors would like to thank Dr. Ioan Landau for his useful comments and suggestions and Dr. Hans van der Weide for his help in the derivation of the approximation method presented in Farahani et al. (2010). This research is partially funded by the Dutch Technology Foundation STW project “Model-predictive railway traffic management” (11025), and by the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement no. 257462 HYCON2 Network of Excellence.

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  1. Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands

    Samira S. Farahani, Ton van den Boom & Bart De Schutter

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  1. Samira S. Farahani
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  2. Ton van den Boom
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  3. Bart De Schutter
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Correspondence to Samira S. Farahani.

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Farahani, S.S., van den Boom, T. & De Schutter, B. Exact and approximate approaches to the identification of stochastic max-plus-linear systems. Discrete Event Dyn Syst 24, 447–471 (2014). https://doi.org/10.1007/s10626-013-0164-4

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