Abstract
In Chap. 9 we provided an overview of some key results of statistical learning theory. In this chapter we discuss their specific application to the design of systems affected by uncertainty formulating a learning-based approach. In particular, we develop a randomized technique for solving a nonconvex semi-infinite optimization problem and compute the related sample complexity. A sequential algorithm which provides a solution to the same problem is also presented.
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Notes
- 1.
We recall that bounds on the VC dimension have been computed for various control problems. For example, in [408], the VC dimension for static output feedback is given by \(2 n_{i} n_{o} \log_{2} [2 \mathrm{e}n_{s}^{2} (n_{s}+1)]\).
References
Alamo T, Tempo R, Camacho EF (2009) A randomized strategy for probabilistic solutions of uncertain feasibility and optimization problems. IEEE Trans Autom Control 54:2545–2559
Alamo T, Tempo R, Ramirez DR, Camacho EF (2008) A new vertex result for robustness problems with interval matrix uncertainty. Syst Control Lett 57:474–481
Anderson BDO, Moore JB (1990) Optimal control: linear quadratic methods. Prentice-Hall, Englewood Cliffs
Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. SIAM, Philadelphia
Calafiore G (2010) Random convex programs. SIAM J Optim 20(6):3427–3464
Calafiore G, Dabbene F (2008) A reduced vertex set result for interval semidefinite optimization problems. J Optim Theory Appl 139:17–33
El Ghaoui L, Oustry F, AitRami M (1997) A cone complementary linearization algorithm for static output-feedback and related problems. IEEE Trans Autom Control 42:1171–1176
Fujisaki Y, Kozawa Y (2006) Probabilistic robust controller design: probable near minmax value and randomized algorithms. In: Calafiore G, Dabbene F (eds) Probabilistic and randomized methods for design under uncertainty. Springer, London, pp 317–329
Koltchinskii V, Abdallah CT, Ariola M, Dorato P (2001) Statistical learning control of uncertain systems: theory and algorithms. Appl Comput Math 120:31–43
Koltchinskii V, Abdallah CT, Ariola M, Dorato P, Panchenko D (2000) Improved sample complexity estimates for statistical learning control of uncertain systems. IEEE Trans Autom Control 46:2383–2388
Vapnik VN (1998) Statistical learning theory. Wiley, New York
Vidyasagar M (1998) Statistical learning theory and randomized algorithms for control. IEEE Control Syst Mag 18:69–85
Vidyasagar M (2001) Randomized algorithms for robust controller synthesis using statistical learning theory. Automatica 37:1515–1528
Vidyasagar M (2002) Learning and generalization: with applications to neural networks, 2nd edn. Springer, New York
Vidyasagar M, Blondel V (2001) Probabilistic solutions to some NP-hard matrix problems. Automatica 37:1397–1405
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Tempo, R., Calafiore, G., Dabbene, F. (2013). Learning-Based Probabilistic Design. 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_13
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DOI: https://doi.org/10.1007/978-1-4471-4610-0_13
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