Attainability analysis in the problem of stochastic equilibria synthesis for nonlinear discrete systems

Bashkirtseva, I. ; Ryashko, L.

International Journal of Applied Mathematics and Computer Science, Tome 23 (2013) no. 1, pp. 5-16.

Résumé

A nonlinear discrete-time control system forced by stochastic disturbances is considered. We study the problem of synthesis of the regulator which stabilizes an equilibrium of the deterministic system and provides required scattering of random states near this equilibrium for the corresponding stochastic system. Our approach is based on the stochastic sensitivity functions technique. The necessary and important part of the examined control problem is an analysis of attainability. For 2D systems, a detailed investigation of attainability domains is given. A parametrical description of the attainability domains for various types of control inputs in a stochastic Henon model is presented. Application of this technique for suppression of noise-induced chaos is demonstrated.

Détail

Keywords: stochastic sensitivity function, attainability, chaos suppression

Bibliographie

[1] Albert, A. (1972). Regression and theMoore–Penrose Pseudoinverse, Academic Press, New York, NY.

[2] Alexandrov, D. and Malygin, A. (2011). Nonlinear dynamics of phase transitions during seawater freezing with false bottom formation, Oceanology 51(6): 940–948.

[3] Åström, K. (1970). Introduction to the Stochastic Control Theory, Academic Press, New York, NY.

[4] Bashkirtseva, I. and Ryashko, L. (2000). Sensitivity analysis of the stochastically and periodically forced Brusselator, Physica A 278(1–2): 126–139.

[5] Bashkirtseva, I. and Ryashko, L. (2005). Sensitivity and chaos control for the forced nonlinear oscillations, Chaos Solitons and Fractals 26(5): 1437–1451.

[6] Bashkirtseva, I. and Ryashko, L. (2009). Constructive analysis of noise-induced transitions for coexisting periodic attractors of Lorenz model, Physical Review E 79(4): 041106–041114.

[7] Bashkirtseva, I., Ryashko, L. and Stikhin, P. (2010). Noise-induced backward bifurcations of stochastic 3D-cycles, Fluctuation and Noise Letters 9(1): 89–106.

[8] Bashkirtseva, I., Ryashko, L. and Tsvetkov, I. (2009). Sensitivity analysis of stochastic equilibria and cycles for the discrete dynamic systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 17(4): 501–515.

[9] Chen, G. and Yu, X.E. (2003). Chaos Control: Theory and Applications, Springer-Verlag, New York, NY.

[10] Digailova, I. and Kurzhanskii, A. (2004). Attainability problems under stochastic perturbations, Differential Equations 40(11): 1573–1578.

[11] Elaydi, S. (1999). An Introduction to Difference Equations, Springer-Verlag, New York, NY.

[12] Fedotov, S., Bashkirtseva, I. and Ryashko, L. (2004). Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo, Physica A 342(3–4): 491–506.

[13] Fradkov, A. and Pogromsky, A. (1998). Introduction to Control of Oscillations and Chaos, World Scientific Series of Nonlinear Science, Singapore.

[14] Gao, J., Hwang, S. and Liu, J. (1999). When can noise induce chaos?, Physical Review Letters 82(6): 1132–1135.

[15] Gassmann, F. (1997). Noise-induced chaos-order transitions, Physical Review E 55(3): 2215–2221.

[16] Goswami, B. and Basu, S. (2002). Transforming complex multistability to controlled monostability, Physical Review E 66(2): 026214–026223.

[17] Henon, M. (1976). A two-dimensional mapping with a strange attractor, Communications in Mathematical Physics 50(1): 69–77.

[18] Karthikeyan, S. and Balachandran, K. (2011). Constrained controllability of nonlinear stochastic impulsive systems, International Journal of Applied Mathematics and Computer Science 21(2): 307–316, DOI: 10.2478/v10006-011-0023-0.

[19] Kučera, V. (1973). Algebraic theory of discrete optimal control for single-variable systems, III: Closed-loop control, Kybernetika 9(4): 291–312.

[20] Martínez-Zéregaa, B. and Pisarchik, A. (2002). Stochastic control of attractor preference in a multistable system, Communications in Nonlinear Science and Numerical Simulation 17(11): 4023–4028.

[21] Matsumoto, K. and Tsuda, I. (1983). Noise-induced order, Journal of Statistical Physics 31(1): 87–106.

[22] McDonnell, M., Stocks, N., Pearce, C. and Abbott, D. (2008). Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization, Cambridge University Press, Cambridge.

[23] Mil’shtein, G. and Ryashko, L. (1995). A first approximation of the quasipotential in problems of the stability of systems with random non-degenerate perturbations, Journal of Applied Mathematics and Mechanics 59(1): 47–56.

[24] Nayfeh, A. and Balachandran, B. (2006). Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, Wiley, New York, NY.

[25] Ott, E., Grebogi, C. and Yorke, J. (1990). Controlling chaos, Physical Review Letters 64(11): 1196–1199.

[26] Ryagin, M. and Ryashko, L. (2004). The analysis of the stochastically forced periodic attractors for Chua’s circuit, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 14(11): 3981–3987.

[27] Ryashko, L. (1996). The stability of stochastically perturbed orbital motions, Journal of Applied Mathematics and Mechanics 60(4): 579–590.

[28] Ryashko, L. and Bashkirtseva, I. (2011a). Analysis of excitability for the FitzHugh–Nagumo model via a stochastic sensitivity function technique, Physical Review E 83(6): 061109–061116.

[29] Ryashko, L. and Bashkirtseva, I. (2011b). Control of equilibria for nonlinear stochastic discrete-time systems, IEEE Transactions on Automatic Control 56(9): 2162–2166.

[30] Sanjuan, M. and Grebogi, C. (2010). Recent Progress in Controlling Chaos, World Scientific, Singapore.

[31] Wonham, W. (1979). Linear Multivariable Control: A Geometric Approach, Springer-Verlag, Berlin.

[32] Zhirabok, A. and Shumsky, A. (2012). An approach to the analysis of observability and controllability in nonlinear systems via linear methods, International Journal of Applied Mathematics and Computer Science 22(3): 507–522, DOI: 10.2478/v10006-012-0038-1.