Optimization of the Control System Parameters with Use of the New Simple Method of the Largest Lyapunov Exponent Estimation

• Autorzy: Balcerzak M. , Dąbrowski A. , Kapitaniak T. , Jach A.
• Języki publikacji: EN

Abstrakty

EN

This text covers application of Largest Lapunov Exponent (LLE) as a criterion for control performance assessment (CPA) in a simulated control system. The main task is to find a simple and effective method to search for the best configuration of a controller in a control system. In this context, CPA criterion based on calculation of LLE by means of a new method [3] is compared to classical CPA criteria used in control engineering [1]. Introduction contains references to previous publications on Lyapunov stability. Later on, description of classical criteria for CPA along with formulae is presented. Significance of LLE in control systems is explained. Moreover, new efficient formula for calculation of LLE [3] is shown. In the second part simulation of the control system used for experiment is described. The next part contains results of the simulation in which typical criteria for CPA are compared with criterion based on value of LLE. In the last part results of the experiment are summed up and conclusions are drawn.

Słowa kluczowe

EN

largest Lyapunov exponent; LLE; control system; parameters; optimization; controller; PID; inverted pendulum; IAE; ISE; ITAE

PL

największy wykładnik Lapunowa; system sterowania; parametry; optymalizacja; kontroler; PID; wahadło odwrócone

• Wydawca: Wydawnictwo Politechniki Łódzkiej
• Czasopismo: Mechanics and Mechanical Engineering
• Rocznik: 2013
• Tom: Vol. 17, nr 4
• Strony: 225--239
• Opis fizyczny: Bibliogr. 40 poz.

Twórcy

autor

Balcerzak M.

• Department of Dynamics Łódź University of Technology

autor

Dąbrowski A.

• Department of Dynamics Łódź University of Technology

autor

Kapitaniak T.

• Department of Dynamics Łódź University of Technology

autor

Jach A.

• Department of Dynamics Łódź University of Technology

Bibliografia

• [1] Dębowski, A.: Automatyka podstawy teorii, Warszawa: WNT, 2008.
• [2] Kapitaniak, T.: Wstęp do teorii drgań, Łódź: Wydawnictwo Politechniki Lódzkiej, 2005.
• [3] Dąbrowski, A. and Stefański, A.: Estimation of the largest Lyapunov exponent
• from the perturbation vector and its derivative dot product, Nonlinear Dynamics, 67, 283–291, 2012.
• [4] Bennettin, G. and Froeschle, C. and Scheidecker, J.P.: Kolmogorov entropy of a dynamical system with increasing number of degrees of freedom, Phys. Rev. A, 19, 2454–60, 1979.
• [5] Grassberger, P. and Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett., 50:346–9, 1983.
• [6] Alligood, K.T., Sauer, T.D. and Yorke, J.A.: Chaos an introduction to dynamical systems, New York, Springer-Verlag, 2000.
• [7] Eckmann, J-P. and Ruelle, D.: Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617–56, 1985.
• [8] Oseledec, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans Moscow Math. Soc., 19, 197–231, 1968.
• [9] Henon, M. and Heiles, C.: The applicability of the third integral of the motion: some numerical results, Astron. J., 69, 77, 1964.
• [10] Benettin, G., Galgani, L. and Strelcyn, J.M.: Kolmogorov entropy and numerical experiment, Phys. Rev. A, 14, 2338–45, 1976.
• [11] Shimada, I. and Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems, Prog. Theor. Phys., 61(6), 1605–16, 1979.
• [12] Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J.M.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them, Part I: theory, Meccanica, 15, 9–20, 1980.
• [13] Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J.M.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems; a method for computing all of them, Part II: numerical application, Meccanica, 15, 21–30, 1980.
• [14] Wolf, A.: Quantifying chaos with Lyapunov exponents. In: Holden, V., editor Chaos, Manchester: Manchester University Press, 273–90, 1986.
• [15] Takens, F.: Detecting strange attractors in turbulence, Lect. Notes Math., 898, 366, 1981.
• [16] Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A.: Determining Lyapunov exponents from a time series, Physica D, 16, 285–317, 1985.
• [17] Sano, M. and Sawada, Y.: Measurement of the Lyapunov spectrum from a chaotic time series, Phys. Rev. Lett., 55, 1082–5, 1985.
• [18] Eckmann, J.P., Kamphorst, S.O., Ruelle, D. and Ciliberto, S.: Lyapunov exponents from a time series, Phys. Rev. Lett., 34(9), 4971–9, 1986.
• [19] Rosenstein, M.T., Collins, J.J. and De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65(1, 2), 117–34, 1993.
• [20] Parlitz, U.: Identification of true and spurious Lyapunov exponents from time series, J. Bifurcat. Chaos, 2(1), 155–65, 1992.
• [21] Young, L.: Entropy, Lyapunov exponents, and Hausdorff dimension in differentiable dynamical systems. IEEE Trans Circuits Syst., CAS–30, 599–607, 1983.
• [22] Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series, Phys. Lett. A, 185, 77–87, 1994.
• [23] Kim, B.J. and Choe, G.H.: High precision numerical estimation of the largest Lyapunov exponent, Commun Nonlinear Sci. Numer. Simulat., 15, 1378–1384, 2010.
• [24] Stefański, A.: Estimation of the largest Lyapunov exponent in systems with impacts, Chaos, Solitons and Fractals, 11 (15), 2443–2451, 2000.
• [25] Stefański, A. and Kapitaniak, T.: Estimation of the dominant Lyapunov exponent of non–smooth systems on the basis of maps synchronization, Chaos, Solitons and Fractals, 15, 233–244, 2003.
• [26] Stefański, A., Dąbrowski, A. and Kapitaniak, T.: Evaluation of the largest Lyapunov exponent in dynamical systems with time delay, Chaos, Solitons and Fractals, 23, 1651–1659, 2005.
• [27] Stefański, A.: Lyapunov exponents of the systems with noise and fluctuating parameters, Journal of Theoretical and Applied Mechanics, 46(3), 665–678, 2008.
• [28] Yilmaz, D., Gülerb, N.F.: Analysis of the Doppler signals using largest Lyapunov exponentand correlation dimension in healthy and stenosed internal carotid artery patients, Digital Signal Processing, 20, 401–409, 2010.
• [29] Gharavia, R. and Anantharamb, V.: An upper bound for the largest Lyapunov exponent of a Markovian product of nonnegative matrices. Theoretical Computer Science, 332, 543–557, 2005.
• [30] Ronga, H.W., Mengb, G.,Wanga, X.D., Xuc, W. and Fangc, T.: Largest Lyapunov exponent for second–order linear systems under combined harmonic and random parametric excitations, Journal of Sound and Vibration, 283, 1250–1256, 2005.
• [31] Vallejos, R.O. and Anteneodo, C.: Largest Lyapunov exponent of long–range XY systems, Physica A, 340, 178–186, 2004.
• [32] Petry, A, and Barone, D.A.C.: Preliminary experiments in speaker verification usingtime–dependent largest Lyapunov exponents, Computer Speech and Language, 17, 403–413, 2003.
• [33] Yilmaz, D. and Gülerb, N.F.: Analysis of the Doppler signals using largest Lyapunov exponent and correlation dimension in healthy and stenosed internal carotid artery patients. Digital Signal Processing, 20, 401–409, 2010.
• [34] Chlouverakis, K.E. and Adams, M.J.: Stability maps of injection–locked laser diodes usingthe largest Lyapunov exponent, Optics Communications, 216, 405–412, 2003.
• [35] Pavlov, A.N., Janson,N.B., Anishchenko,V.S., Gridnev,V.I. and Dovgalevsky, P.Y.: Diagnostic of cardio–vascular disease with help of largest Lyapunov exponent of RR–sequences, Chaos, Solitons and Fractals, 11, 807–814, 2000.
• [36] Giovanni, A., Ouaknine, M. and Triglia, J.M.: Determination of Largest Lyapunov Exponents of Vocal Signal: Application to Unilateral Laryngeal Paralysis, Journal of Voice, 13(3), 341–354, 1999.
• [37] Feng, Z.H. and Hu, H.Y.: Largest Lyapunov exponent and almost certain stability analysis of slender beams under a large linear motion of basement subject to narrowband parametric excitation, Journal of Sound and Vibration, 257(4), 733–752, 2002.
• [38] Stefański, A., Perlikowski, P., Kapitaniak, T.: Ragged synchronizability of coupled oscillators, Physical Review E, 75, 016210:1–7, 2007.
• [39] Stefański, A., Perlikowski, P. and Kapitaniak, T.: Discontinuous synchrony in an array of Van der Pol oscillators, International Journal of Non-Linear Mechanics, 45, 895–901, 2010.
• [40] Dąbrowski, A.: The largest Transversal Lyapunov Exponent and Master Stability Function from the perturbation Vector and its derivative Dot Product (TLEVDP), Nonlinear Dyn., 69, pp. 1225–1235, 2012.