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

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


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  • [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.


    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.


    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.


    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.


    D. P. Bertsekas, Infinite time reachability of state-space regions by using feedback control, IEEE Trans. on Automatic Control, 17 (1994), 604-613.


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


    F. Blanchini, Set invariance in control - a survey, Automatica J. IFAC, 35 (1999), 1747-1767.doi: 10.1016/S0005-1098(99)00113-2.


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


    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.


    M. Davis, Linear Estimation and Stochastic Control, Champman and Hall, New York, 1977.


    T. Duncan and P. Varaiya, On the solution of a stochastic control system, SIAM J. Control, 9 (1971), 354-371.doi: 10.1137/0309026.


    W. Fleming and R. Rishel, Optimal Deterministic and Stochastic Control, Springer- Verlag, Berlin, 1975.


    U. Haussman, Some examples of optimal control, Or: Stochastic maximum principal at work, SIAM Rev., 23 (1981), 292-307.doi: 10.1137/1023062.


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


    N. V. Krylov, Controlled Diffusion Process, Springer, New York. 1980.


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


    H. Kushner, Necessary condition for continuous parameter stochastic optimization problems, SIAM J. Control, 10 (1972), 550-565.doi: 10.1137/0310041.


    D. G. Luenberger, An introduction to observers, IEEE Transactions on Automatic Control, 16 (1971), 596-602.


    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.


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


    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.


    A. Polyakov and A. Poznyak, Invariant ellipsoid method for minimization of unmatched disturbances effects in sliding mode control, Automatica, 47 (2011), 1450-1454.


    A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Vol. 1. Elsevier, London - New York, 2008.


    A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques, Vol. 2. Elsevier, London - New York, 2009.


    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.


    F. Schweppe, Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs, N.J., 1973.


    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.


    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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