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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 & 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.  doi: 10.1093/imamci/dnr038.  Google Scholar

[2]

J. A. Appleby and A. Flynn, Stabilization of volterra equations by noise,, The Journal of Applied Mathematics and Stochastic Analysis, (2006).  doi: 10.1155/JAMSA/2006/89729.  Google Scholar

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L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems,, The Journal of Differential Equations, 177 (2001), 235.  doi: 10.1006/jdeq.2000.3991.  Google Scholar

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M. Bardi and D. I. Capuzzo, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Systems and Control: Foundations and Applications, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[5]

D. P. Bertsekas, Infinite time reachability of state-space regions by using feedback control,, IEEE Trans. on Automatic Control, 17 (1994), 604.   Google Scholar

[6]

A. Bensoussan, Stochastic Control of Partially Observable Systems,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511526503.  Google Scholar

[7]

F. Blanchini, Set invariance in control - a survey,, Automatica J. IFAC, 35 (1999), 1747.  doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar

[8]

F. Blanchini and S. Miani, Set Theoretic Methods in Control,, Systems & Control: Foundations & Applications. Birkhauser Boston Inc., (2008).   Google Scholar

[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.  doi: 10.1093/imamci/20.1.1.  Google Scholar

[10]

M. Davis, Linear Estimation and Stochastic Control,, Champman and Hall, (1977).   Google Scholar

[11]

T. Duncan and P. Varaiya, On the solution of a stochastic control system,, SIAM J. Control, 9 (1971), 354.  doi: 10.1137/0309026.  Google Scholar

[12]

W. Fleming and R. Rishel, Optimal Deterministic and Stochastic Control,, Springer- Verlag, (1975).   Google Scholar

[13]

U. Haussman, Some examples of optimal control, Or: Stochastic maximum principal at work,, SIAM Rev., 23 (1981), 292.  doi: 10.1137/1023062.  Google Scholar

[14]

K. Holmstrom, A. Goran and M. Edvall, User's Guide for TOMLAB/CPLEX, v12.1,, (2009)., (2009).   Google Scholar

[15]

N. V. Krylov, Controlled Diffusion Process,, Springer, (1980).   Google Scholar

[16]

A. Kurzhanskii and I. Valyi, Ellipsoidal Calculus for Estimation and Control,, Birkhauser, (1997).  doi: 10.1007/978-1-4612-0277-6.  Google Scholar

[17]

H. Kushner, Necessary condition for continuous parameter stochastic optimization problems,, SIAM J. Control, 10 (1972), 550.  doi: 10.1137/0310041.  Google Scholar

[18]

D. G. Luenberger, An introduction to observers,, IEEE Transactions on Automatic Control, 16 (1971), 596.   Google Scholar

[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.  doi: 10.1134/S0005117907030083.  Google Scholar

[20]

Y. Nesterov and A. Nemirovsky, Interior-Point Polynomial Methods in Convex Programming,, SIAM, (1994).   Google Scholar

[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., (2006), 1429.   Google Scholar

[22]

A. Polyakov and A. Poznyak, Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control,, Automatica, 47 (2011), 1450.   Google Scholar

[23]

A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques,, Vol. 1. Elsevier, 1 (2008).   Google Scholar

[24]

A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques,, Vol. 2. Elsevier, 2 (2009).   Google Scholar

[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.  doi: 10.1080/00207170210156251.  Google Scholar

[26]

F. Schweppe, Uncertain Dynamic Systems,, Prentice-Hall, (1973).   Google Scholar

[27]

V. A. Ugrinovskii, Robust H infinity control in the presence of stochastic uncertainty,, International Journal of Control, 71 (1998), 219.  doi: 10.1080/002071798221849.  Google Scholar

[28]

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999).  doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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.  doi: 10.1093/imamci/dnr038.  Google Scholar

[2]

J. A. Appleby and A. Flynn, Stabilization of volterra equations by noise,, The Journal of Applied Mathematics and Stochastic Analysis, (2006).  doi: 10.1155/JAMSA/2006/89729.  Google Scholar

[3]

L. Arnold and B. Schmalfuss, Lyapunov's second method for random dynamical systems,, The Journal of Differential Equations, 177 (2001), 235.  doi: 10.1006/jdeq.2000.3991.  Google Scholar

[4]

M. Bardi and D. I. Capuzzo, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Systems and Control: Foundations and Applications, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[5]

D. P. Bertsekas, Infinite time reachability of state-space regions by using feedback control,, IEEE Trans. on Automatic Control, 17 (1994), 604.   Google Scholar

[6]

A. Bensoussan, Stochastic Control of Partially Observable Systems,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511526503.  Google Scholar

[7]

F. Blanchini, Set invariance in control - a survey,, Automatica J. IFAC, 35 (1999), 1747.  doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar

[8]

F. Blanchini and S. Miani, Set Theoretic Methods in Control,, Systems & Control: Foundations & Applications. Birkhauser Boston Inc., (2008).   Google Scholar

[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.  doi: 10.1093/imamci/20.1.1.  Google Scholar

[10]

M. Davis, Linear Estimation and Stochastic Control,, Champman and Hall, (1977).   Google Scholar

[11]

T. Duncan and P. Varaiya, On the solution of a stochastic control system,, SIAM J. Control, 9 (1971), 354.  doi: 10.1137/0309026.  Google Scholar

[12]

W. Fleming and R. Rishel, Optimal Deterministic and Stochastic Control,, Springer- Verlag, (1975).   Google Scholar

[13]

U. Haussman, Some examples of optimal control, Or: Stochastic maximum principal at work,, SIAM Rev., 23 (1981), 292.  doi: 10.1137/1023062.  Google Scholar

[14]

K. Holmstrom, A. Goran and M. Edvall, User's Guide for TOMLAB/CPLEX, v12.1,, (2009)., (2009).   Google Scholar

[15]

N. V. Krylov, Controlled Diffusion Process,, Springer, (1980).   Google Scholar

[16]

A. Kurzhanskii and I. Valyi, Ellipsoidal Calculus for Estimation and Control,, Birkhauser, (1997).  doi: 10.1007/978-1-4612-0277-6.  Google Scholar

[17]

H. Kushner, Necessary condition for continuous parameter stochastic optimization problems,, SIAM J. Control, 10 (1972), 550.  doi: 10.1137/0310041.  Google Scholar

[18]

D. G. Luenberger, An introduction to observers,, IEEE Transactions on Automatic Control, 16 (1971), 596.   Google Scholar

[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.  doi: 10.1134/S0005117907030083.  Google Scholar

[20]

Y. Nesterov and A. Nemirovsky, Interior-Point Polynomial Methods in Convex Programming,, SIAM, (1994).   Google Scholar

[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., (2006), 1429.   Google Scholar

[22]

A. Polyakov and A. Poznyak, Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control,, Automatica, 47 (2011), 1450.   Google Scholar

[23]

A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques,, Vol. 1. Elsevier, 1 (2008).   Google Scholar

[24]

A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques,, Vol. 2. Elsevier, 2 (2009).   Google Scholar

[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.  doi: 10.1080/00207170210156251.  Google Scholar

[26]

F. Schweppe, Uncertain Dynamic Systems,, Prentice-Hall, (1973).   Google Scholar

[27]

V. A. Ugrinovskii, Robust H infinity control in the presence of stochastic uncertainty,, International Journal of Control, 71 (1998), 219.  doi: 10.1080/002071798221849.  Google Scholar

[28]

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999).  doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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