×

zbMATH — the first resource for mathematics

Stochastic algorithms for robustness of control performances. (English) Zbl 1166.93387
Summary: In recent years, there has been a growing interest in developing statistical learning methods to provide approximate solutions to “difficult” control problems. In particular, randomized algorithms have become a very popular tool used for stability and performance analysis as well as for design of control systems. However, as randomized algorithms provide an efficient solution procedure to the “intractable” problems, stochastic methods bring closer to understanding the properties of the real systems. The topic of this paper is the use of stochastic methods in order to solve the problem of control robustness: the case of parametric stochastic uncertainty is considered. Necessary concepts regarding stochastic control theory and stochastic differential equations are introduced. Then a convergence analysis is provided by means of the Chernoff bounds, which guarantees robustness in mean and in probability. As an illustration, the robustness of control performances of example control systems is computed.

MSC:
93E35 Stochastic learning and adaptive control
93E15 Stochastic stability in control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Software:
RACT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agostini, A., Balluchi, A., Bicchi, A., Piccoli, B., Sangiovanni-Vincentelli, A., & Zadarnowska, K. (2005). Randomized algorithms for platform-based design. In Proc. 44th IEEE conf. on decision and control, and the Europ. control conf. 2005(pp. 6638-6643)
[2] Bhattacharyya, S.P.; Chapellat, H.; Keel, L.H., Robust control: the parametric approach, (1995), Prentice-Hall Englewood Cliffs · Zbl 0838.93008
[3] Blondel, V.D.; Tsitsiklis, J.N., A survey of computational complexity results in systems and control, Automatica, 36, 1249-1274, (2000) · Zbl 0989.93006
[4] Brockett, R.W., Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control, IEEE transactions on automatic control, 45, 6, 1116-1130, (2000) · Zbl 0972.93076
[5] Calafiore, G., Dabbene, F., & Tempo, R. (2003). Randomized algorithms in robust control. In Proc. 42nd IEEE conf. decision contr. (pp. 1908-1912)
[6] Calafiore, G.; Dabbene, F.; Tempo, R., A survey of randomized algorithms for control synthesis and performance verification, Journal of complexity, 23, 301-316, (2007) · Zbl 1117.65180
[7] Chen, S.H.; Song, M.; Chen, Y.D., Robustness analysis of responses of vibration control structures with uncertain parameters using interval algorithm, Structural safety, 29, 94-111, (2007)
[8] Chernoff, H., A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Annals of mathematical statistics, 23, 493-507, (1952) · Zbl 0048.11804
[9] Digailova, I. A., Kurzhanski, A. B., & Varaiya, P. (2008). Stochastic reachability and measurement feedback under control-dependent noise. In Proc. 17th IFAC world congress (pp. 14336-14341)
[10] Fetiso, V.N., Some problems of robust control of a stochastic object, Automation and remote control, 65, 4, 594-602, (2004) · Zbl 1095.93008
[11] Ishii, H.; Basar, T.; Tempo, R., Randomized algorithms for synthesis of switching rules for multimodal systems, IEEE transactions on automatic control, 50, 754-767, (2005) · Zbl 1365.93400
[12] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1998), Springer-Verlag New York
[13] Knight, F.B., Essentials of Brownian motion and diffusion, (1981), American Math. Soc. Providence, Rhode Island · Zbl 0458.60002
[14] Koltchinskii, V.; Abdallah, C.T.; Ariola, M.; Dorato, P.; Panchenko, D., Improved sample complexity estimates for statistical learning control of uncertain systems, IEEE transactions on automatic control, 46, 2383-2388, (2000) · Zbl 0991.93130
[15] Milanese, M.; Tempo, R., Optimal algorithms theory for robust estimation and prediction, IEEE transactions on automatic control, AC-30, 730-738, (1985) · Zbl 0567.93024
[16] Marti, K., Numerical methods for stochastic optimization and real-time control of robots, Optimization, 47, 3-4, 251-420, (2000), (special issue) · Zbl 0952.00021
[17] Nemirovskii, A., Several NP-hard problems arising in robust stability analysis, Mathematics of control, signals and systems, 6, 99-105, (1993) · Zbl 0792.93100
[18] Oksendal, B., Stochastic differential equations. an introduction with applications, (1985), Springer Verlag New York · Zbl 0567.60055
[19] Papadimitriou, C.H.; Tsitsiklis, J.N., Intractable problems in control theory, SIAM journal of control and optimization, 24, 639-654, (1986) · Zbl 0604.90009
[20] Rico-Ramirez, V.; Diwekar, U.M., Stochastic maximum principle for optimal control under uncertainty, Computers and chemical engineering, 28, 2845-2849, (2004)
[21] Ravichandran, N., Stochastic methods in reliability theory, (1991), John Wiley & Sons · Zbl 0722.60089
[22] Ray, L.; Stengel, R., A Monte Carlo approach to the analysis of control systems robustness, Automatica, 29, 229-236, (1993) · Zbl 0800.93302
[23] Rust, J., Using randomization to break the curse of dimensionality, Econometrica, 65, 3, 487-516, (1997) · Zbl 0872.90107
[24] Tempo, R.; Calafiore, G.; Dabbene, F., Randomized algorithms for analysis and control of uncertain systems, () · Zbl 1079.93002
[25] Yong, J.; Zhou, X.Y., Stochastic controls. Hamiltonian systems and HJB equations, (1999), Springer Verlag New York · Zbl 0943.93002
[26] Vidyasagar, M., Statistical learning theory and randomized algorithms for control, IEEE control systems magazine, 18, 6, 69-85, (1998)
[27] Wang, Z.; Ho, D.W.C., Filtering on nonlinear time-delay stochastic systems, Automatica, 39, 101-109, (2003) · Zbl 1010.93099
[28] Wang, Z.; Liu, Y.; Liu, X., \(H_\infty\) filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica, 44, 1268-1277, (2008) · Zbl 1283.93284
[29] Zinober, A.S.I., Deterministic control of uncertain control system, (1990), Peter Peregrinus London · Zbl 0754.93001
[30] Zhang, W.; Chen, B.-S.; Tseng, C.-S., Robust \(H_\infty\) filtering for nonlinear stochastic systems, IEEE transactions on signal processing, 53, 589-598, (2005) · Zbl 1370.93293
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.