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January  2016, 12(1): 169-186. doi: 10.3934/jimo.2016.12.169

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, Mexico
2. 

Centro de Investigacin y de Estudios Avanzados del I.P.N. (Cinvestav-IPN), Mexico

Received  December 2013 Revised  November 2014 Published  April 2015

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: attractive ellipsoid method., Stochastic differential equations, matrix inequalities.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Hussain Alazki, Alexander Poznyak. Robust output stabilization for a class of nonlinear uncertain stochastic systems under multiplicative and additive noises: The attractive ellipsoid method. Journal of Industrial and Management Optimization, 2016, 12 (1) : 169-186. doi: 10.3934/jimo.2016.12.169
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.

show all references

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.

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