Abstract
In this chapter we address the issue of generating random samples of real and complex vector vectors in ℓ p norm balls, according to the uniform distribution. We present efficient algorithms based upon the theoretical developments of the Chap. 15. The presented methods are non-asymptotic, and therefore, they can be easily implemented on parallel and distributed architectures.
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Notes
- 1.
Here, p k (z) denotes a polynomial of degree k obtained at the kth step of the recursion.
References
Beadle ER, Djurić PM (1997) Uniform random parameter generation of stable minimum-phase real ARMA (p,q) processes. IEEE Signal Process Lett 4(9):259–261
Calafiore G, Dabbene F, Tempo R (1999) Radial and uniform distributions in vector and matrix spaces for probabilistic robustness. In: Miller DE, Qiu L (eds) Topics in control and its applications. Springer, New York, pp 17–31
Durbin J (1960) The fitting of time series models. Revue Inst Int Stat 28:233–243
Fam A, Meditch J (1978) A canonical parameter space for linear systems design. IEEE Trans Autom Control 23(3):454–458
Fujimoto RM (2000) Parallel and distributed simulation systems. Wiley Interscience, New York
Gantmacher FR (1959) The theory of matrices. American Mathematical Society, Providence
Hicks JS, Wheeling RF (1959) An efficient method for generating uniformly distributed points on the surface of an n-dimensional sphere. Commun ACM 2:17–19
Levinson N (1947) The Wiener RMS error criterion in filter design and prediction. J Math Phys 25:261–278
Liu W, Chen J (2010) Probabilistic estimates for mixed model validation problems with \(\mathcal{H}_{\infty}\) type uncertainties. IEEE Trans Autom Control 55(6):1488–1494
Liu W, Chen J, El-Sherief H (2007) Probabilistic bounds for uncertainty model validation. Automatica 43(6):1064–1071
Ma W, Sznaier M, Lagoa CM (2007) A risk adjusted approach to robust simultaneous fault detection and isolation. Automatica 43(3):499–504
Muller ME (1959) A note on a method for generating random points uniformly distributed on n-dimensional spheres. Commun ACM 2:19–20
Nemirovski A, Polyak BT (1994) Necessary conditions for the stability of polynomials and their use. Autom Remote Control 55(11):1644–1649
Nurges Ü (2006) Robust pole assignment via reflection coefficients of polynomials. Automatica 42:1223–1230
Nurges Ü (2009) Reflection coefficients of polynomials and stable polytopes. IEEE Trans Autom Control 54(6):1314–1318
Shcherbakov PS, Dabbene F (2011) On the generation of random stable polynomials. Eur J Control 17(2):145–161
Sznaier M, Lagoa CM, Mazzaro MC (2005) An algorithm for sampling subsets of \(\mathcal{H}_{\infty}\) with applications to risk-adjusted performance analysis and model (in)validation. IEEE Trans Autom Control 50(3):410–416
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Tempo, R., Calafiore, G., Dabbene, F. (2013). Vector Randomization Methods. In: Randomized Algorithms for Analysis and Control of Uncertain Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-4610-0_16
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DOI: https://doi.org/10.1007/978-1-4471-4610-0_16
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4609-4
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