Robust output stabilizationfor a class of nonlinear uncertain stochastic systems under multiplicativeand additive noises: The attractive ellipsoid method

Hussain Alazki; Alexander Poznyak

doi:10.3934/jimo.2016.12.169

Article Contents

2016, Volume 12, Issue 1: 169-186. Doi: 10.3934/jimo.2016.12.169

This issue Previous Article Component allocation cost minimization for a multistate computer network subject to a reliability threshold using tabu search Next Article Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate

Robust output stabilization for a class of nonlinear uncertain stochastic systems under multiplicative and additive noises: The attractive ellipsoid method

Hussain Alazki^1, and
Alexander Poznyak^2,

1.
Departamento de Ingenieria Mecatronica, Universidad Autonoma del Carmen, Capmeche
2.
Centro de Investigacin y de Estudios Avanzados del I.P.N. (Cinvestav-IPN)

Received: December 2013

Revised: November 2014

Published: January 2016

Abstract / Introduction Related Papers Cited by

Abstract

This work concerns the robust stabilization of a class of ``Quasi-Lipschitz" nonlinear uncertain systems governed by stochastic Differential Equations (SDE) subject to both multiplicative and additive stochastic noises modeled by a vector Brownian motion. The state-vector is admitted to be non-completely available, and be estimated by a Luenberger-type filter. The stabilization around the origin is realized by a linear feedback proportional to the current state-estimates. First, the class of feedback matrices and filter matrix-gains, providing the boundedness of the stochastic trajectories with probability one in a vicinity of the origin, is specified. Then a corresponding ellipsoid, containing these trajectories, is found. Its ``size" (the trace of the ellipsoid matrix) is derived as a function of the applied gain matrices. To make this ellipsoid ``as small as possible" the corresponding constrained optimization problem is suggested to be solved. These constraints are given by a system of Matrix Inequalities (MI's) which under a specific change of variables may be converted into a conventional system of Bilinear Matrix Inequalities (BMI's). The last may be resolved by the standard MATLAB toolboxes such as ``penbmiTL, Tomlab toolbox". Finally, a numerical example, containing the arctangent-type nonlinearities, is presented to illustrate the effectiveness of the suggested methodology.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[1]	H. Alazki and A. Poznyak, Inventory constraint control with uncertain stochastic demands: Attractive ellipsoid technique application, IMA Journal of Mathematical Control and Information, 29 (2012), 399-425.doi: 10.1093/imamci/dnr038.
[2]	J. A. Appleby and A. Flynn, Stabilization of volterra equations by noise, The Journal of Applied Mathematics and Stochastic Analysis, (2006), Art. ID 89729, 29 pp.doi: 10.1155/JAMSA/2006/89729.
[3]	L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems, The Journal of Differential Equations, 177 (2001), 235-265.doi: 10.1006/jdeq.2000.3991.
[4]	M. Bardi and D. I. Capuzzo, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhauser, Boston, 1997.doi: 10.1007/978-0-8176-4755-1.
[5]	D. P. Bertsekas, Infinite time reachability of state-space regions by using feedback control, IEEE Trans. on Automatic Control, 17 (1994), 604-613.
[6]	A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.doi: 10.1017/CBO9780511526503.
[7]	F. Blanchini, Set invariance in control - a survey, Automatica J. IFAC, 35 (1999), 1747-1767.doi: 10.1016/S0005-1098(99)00113-2.
[8]	F. Blanchini and S. Miani, Set Theoretic Methods in Control, Systems & Control: Foundations & Applications. Birkhauser Boston Inc., Boston, MA, 2008.
[9]	A. El Bouhtouri and K. El Hadri, Robust stabilization of jump linear systems with multiplicative noise, IMA Journal of Mathematical Control and Information, 20 (2003), 1-19.doi: 10.1093/imamci/20.1.1.
[10]	M. Davis, Linear Estimation and Stochastic Control, Champman and Hall, New York, 1977.
[11]	T. Duncan and P. Varaiya, On the solution of a stochastic control system, SIAM J. Control, 9 (1971), 354-371.doi: 10.1137/0309026.
[12]	W. Fleming and R. Rishel, Optimal Deterministic and Stochastic Control, Springer- Verlag, Berlin, 1975.
[13]	U. Haussman, Some examples of optimal control, Or: Stochastic maximum principal at work, SIAM Rev., 23 (1981), 292-307.doi: 10.1137/1023062.
[14]	K. Holmstrom, A. Goran and M. Edvall, User's Guide for TOMLAB/CPLEX, v12.1, (2009).
[15]	N. V. Krylov, Controlled Diffusion Process, Springer, New York. 1980.
[16]	A. Kurzhanskii and I. Valyi, Ellipsoidal Calculus for Estimation and Control, Birkhauser, Boston, MA, 1997.doi: 10.1007/978-1-4612-0277-6.
[17]	H. Kushner, Necessary condition for continuous parameter stochastic optimization problems, SIAM J. Control, 10 (1972), 550-565.doi: 10.1137/0310041.
[18]	D. G. Luenberger, An introduction to observers, IEEE Transactions on Automatic Control, 16 (1971), 596-602.
[19]	S. A. Nazin, B. T. Polyak and M. V. Tpopunov, Rejection of bounded exogenous disturbances by the method of invariant ellipsoids, Autom. Remote Control, 68 (2007), 467-486.doi: 10.1134/S0005117907030083.
[20]	Y. Nesterov and A. Nemirovsky, Interior-Point Polynomial Methods in Convex Programming, SIAM, 1994.
[21]	B. Polyak, A. V. Nazin, M. V. Topunov and S. A. Nazin, Rejection of bounded disturbances via invariant ellipsoids technique, In Proc. 45th IEEE Conf. Decision Contr., San Diego. USA, (2006), 1429-1434.
[22]	A. Polyakov and A. Poznyak, Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control, Automatica, 47 (2011), 1450-1454.
[23]	A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Vol. 1. Elsevier, London - New York, 2008.
[24]	A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques, Vol. 2. Elsevier, London - New York, 2009.
[25]	A. S. Poznyak, T. E. Duncan, B. Pasik-Duncan and V. G. Boltyanskii, Robust stochastic maximum principle for multi-model worst case optimization, International Journal of Control, 75 (2002), 1032-1048.doi: 10.1080/00207170210156251.
[26]	F. Schweppe, Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs, N.J., 1973.
[27]	V. A. Ugrinovskii, Robust H infinity control in the presence of stochastic uncertainty, International Journal of Control, 71 (1998), 219-237.doi: 10.1080/002071798221849.
[28]	J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, 1999.doi: 10.1007/978-1-4612-1466-3.