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Inicio Revista Iberoamericana de Automática e Informática Industrial RIAI Estabilidad de Sistemas No Lineales Basada en la Teoría de Liapunov
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Vol. 6. Núm. 2.
Páginas 5-16 (abril 2009)
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Vol. 6. Núm. 2.
Páginas 5-16 (abril 2009)
Open Access
Estabilidad de Sistemas No Lineales Basada en la Teoría de Liapunov
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13235
Francisco Gordillo
Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092, Sevilla, España
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Resumen

El comportamiento dinámico de los sistemas no lineales es mucho más rico que el de los lineales y su análisis mucho más complicado. Para el análisis de estabilidad, las técnicas basadas en la teoría de Liapunov tienen un lugar destacado. En este articulo se revisa parte de esta teoría incluyendo las técnicas de estimación de la cuenca de atracción. También se repasan los resultados que han aparecido en los últimos años sobre la aplicación a este campo de los métodos numéricos de optimización de suma de cuadrados.

Palabras clave:
Estabilidad de Liapunov
Análisis de estabilidad
Cuenca de atracción
Análisis numérico
Problemas de optimización
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Referencias
[Albea and Gordillo, 2007]
C. Albea, F. Gordillo.
Estimation of the región of attraction for a boost DC-AC converter control law.
Proceedings of the 7th IFAC Simposium. Nonlinear Control System (NOLCOS), pp. 874-879
[Aracil et al., 2005]
J. Aracil, F. Gordillo, E. Ponce.
Stabilization of oscillations through backstepping in high-dimensional systems.
IEEETr. onAutomat. Control, 50 (2005), pp. 705-710
[Aracil and Gordillo, 2005]
J. Aracil, F. Gordillo.
El péndulo invertido: un desafío para el control no lineal.
Revista Iberoamericana de Automática e Informática Industrial, 2 (2005), pp. 8-19
[Aracil and Gordillo, 2000]
Stability Issues in Fuzzy Control,
[Ástróm and Furuta, 2000]
K.J. Ástróm, K. Furuta.
Swinging up a pendulum by energy control.
Automática, 36 (2000), pp. 287-295
[Axelby and Parks, 1992]
G.S. Axelby, P.C. Parks.
Lyapunov centenary.
Automática, 28 (1992), pp. 863-864
[Barbashin and Krasovskii, 1952]
E.A. Barbashin, N.N. Krasovskii.
On the stability of motion in the large.
Dokl. Akad. Nauk, 86 (1952), pp. 453-456
[Cao and Wang, 2005]
J. Cao, J. Wang.
Global asymptotic and robust stability of recurrent neural networks with time delays.
IEEE Transactions on Circuits and Systems I: Regular Papers, 52 (2005), pp. 417-426
[Choi et al., 1995]
M.D. Choi, T.Y. Lamy, B. Reznick.
Sums of squares of real polynomials.
K-Theory and Algébrale Geometry: Connections with Quadratic Forms and División Algebras. Proc. Symp. Pure Math. Vol. 58, pp. 103-126
[Cuesta et al., 1999]
F. Cuesta, F. Gordillo, J. Aracil, A. Ollero.
Global stability analysis of a class of multivariable Takagi-Sugeno fuzzy control systems.
IEEE Trans. Fuzzy Systems, 7 (1999), pp. 508-520
[Davison and Kurak, 1971]
E.J. Davison, E.M. Kurak.
A computational method for determining quadratic Lyapunov functions for non-linear systems.
Automática, 7 (1971), pp. 627-636
[Davrazos and Koussoulas, 2001]
G. Davrazos, N.T. Koussoulas.
A review of stability results for switched and hybrid systems.
Mediterranean Conference on Control and Automation, (2001),
[Espada and Barreiro, 1999]
A. Espada, A. Barreiro.
Robust stability of fuzzy control systems based on conicity conditions.
Automática, 35 (1999), pp. 643-654
[Feng, 2006]
G. Feng.
A survey on analysis and design of model-based fuzzy control systems.
IEEE Transactions on Fuzzy Systems, 14 (2006), pp. 676-697
[Forti and Tesi, 1995]
M. Forti, A. Tesi.
New conditions for global stability of neural networks with application to linear and quadratic programming problems.
IEEE Transactions on Cir-cuits and Systems I: Fundamental Theory and Applications, 42 (1995), pp. 354-366
[Fuller, 1992]
A.T. Fuller.
Lyapunov Centenary Issue.
International Journal of Control, 55 (1992), pp. 521-527
[Furuta, 2003]
K. Furuta.
Control of pendulum: From super mechano-system to human adaptive mechatronics.
Proceedings of the 42ndIEEE CDC, pp. 1498-1507
[Genesio et al., 1985]
R. Genesio, M. Tartagliay, A. Vicino.
On the estimation of asymptotic stability regions: State of the art and new proposals.
IEEE Transactions on Automatic Control, 30 (1985), pp. 747-755
[Haddad and Chellaboina, 2008]
W.M. Haddad, V. Chellaboina.
Nonlinear dynamical systems and control. A Lyapunov-based approach.
Princeton University Press, (2008),
[Hahn, 1967]
W. Hahn.
Stability ofmotion.
Springer-Verlag, (1967),
[Isidori, 1999]
A. Isidori.
Nonlinear control systems II. Communications and control engineering series.
Springer-Verlag, (1999),
[Jarvis-Wloszek et al., 2003]
Z. Jarvis-Wloszek, R. Feeley, W. Tan, K. Sun, A. Packard.
Some controls applications of sum of squares programming.
42nd IEEE Conference on Decisión and Control, 2003. Proceedings, pp. 5
[Jarvis-Wloszek, 2003]
Jarvis-Wloszek, Z. W. (2003). Lyapunov based analysis and controller synthesis for polynomial systems using sum-of-squares optimization. PhD thesis. University of California.
[Johansson et al., 1999]
M. Johansson, A. Rantzer, K.E. Arzen.
Piecewise quadratic stability of fuzzy systems.
IEEE Transactions on Fuzzy Systems, 7 (1999), pp. 713-722
[Johansson and Rantzer, 1998]
M. Johansson, A. Rantzer.
Computation of piecewise quadratic Lyapunov functions for hybrid systems.
IEEE transactions on automatic control, 43 (1998), pp. 555-559
[Kaliora, 2002]
Kaliora, Georgia (2002). Control of nonlinear systems with bounded signáis. PhD thesis. Imperial College of Sicnece, Technology and Medicine. London.
[Kalman and Bertram, 1960]
R.E. Kalman, J.E. Bertram.
Control system analysis and design via the second method of Lyapunov.
Journal of Basic Engineering, 82 (1960), pp. 371-393
[Khalil, 1996]
H.K. Khalil.
Nonlinear Systems.
2a, Prentice Hall, (1996),
[Khalil, 2002]
H.K. Khalil.
Nonlinear Systems.
3 a, Prentice Hall, (2002),
[Krasovskii, 1959]
N.N. Krasovskii.
Some problems of the motion stability theory.
Traducido al inglés por Stanford University Press, (1959),
[LaSalle, 1960]
J.P. LaSalle.
Some extensions of Liapunov's second method.
Circuit Theory, IRÉ Transactions on, 7 (1960), pp. 520-527
[Liberzon, 2003]
Liberzon, D. (2003). Switching in systems and control. Birkhauser.
[Liberzon and Morse, 1999]
D. Liberzon, A.S. Morse.
Basic problems in stability and design of switched systems.
IEEE control systems ma-gazine, 19 (1999), pp. 59-70
[Loparo and Blankenship, 1978]
K. Loparo, G. Blankenship.
Estimating the domain of attraction of nonlinear feedback systems.
IEEE Transactions on Automatic Control, 23 (1978), pp. 602-608
[Lyapunov, 1892]
Lyapunov, A. M. (1892). El problema general de la estabilidad del movimiento. PhD thesis. Kharkov Mathematical Society. En ruso.
[Lyapunov, 1992]
A.M. Lyapunov.
The General Problem of the Stability of Motion, 1892.
International Journal of Control: Lyapunov Centenary, (1992),
[Margolis and Vogt, 1963]
S. Margolis, W. Vogt.
Control engineering applications of V. I. Zubov's construction procedure for Lyapunov functions.
IEEE Transactions on Automatic Control, 8 (1963), pp. 104-113
[Martynyuk, 2000]
A.A. Martynyuk.
A survey of some classical and modern developments of stability theory.
Nonlinear Analysis, 40 (2000), pp. 483-496
[Martynyuk, 2007]
A.A. Martynyuk.
Stability of Motion. The role ofMulti-component Liapunov's Functions. Stability, Oscillations and Optimization of Systems.
Cambridge Scienfic Publishers, (2007),
[Michel, 1996]
A.N. Michel.
Stability: the common thread in the evolution of feedback control.
IEEE Control Systems Magazine, 16 (1996), pp. 50-60
[Narendra and Balakrishnan, 1994]
K.S. Narendra, J. Balakrishnan.
A common Lyapunov function for stable LTI systems with commuting A-matrices.
IEEE Transactions on automatic control, 39 (1994), pp. 2469-2471
[Papachristodoulou and Prajna, 2002]
A. Papachristodoulou, S. Prajna.
On the construction of Lyapunov functions using the sum of squares decomposition.
Decisión and Control, 2002, Proceedings of the 41st IEEE Conference on,
[Papachristodoulou and Prajna, 2005]
A. Papachristodoulou, S. Prajna.
A tutorial on sumof squares techniques for systems analysis.
2005 American Control Conference, pp. 2686-2700
[Parrilo, 2000]
Parrilo, R A. (2000). Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis. California Institute of Technology, Pasadena, California.
[Peleties and DeCarlo, 1992]
R. Peleties, R. DeCarlo.
Asymptotic stability of m-switched systems using Lyapunov functions.
Decisión and Control, 1992., Proceedings ofthe 31st IEEE Conference on, pp. 3438-3439
[Powers and Wormann, 1998]
V. Powers, T. Wormann.
An algorithm for sums of squares of real polynomials.
Journal of puré and applied algebra, 127 (1998), pp. 99
[Prajna et al., 2005]
S. Prajna, A. Papachristodoulou, R. Seiler, R.A. Parrilo.
SOSTOOLS and its control applications.
Positive Polynomials in Control, (2005), pp. 273-292
[Prajna et al., 2002]
S. Prajna, A. Papachristodoulou, P.A. Parrilo.
Introducing SOSTOOLS: a general purpose sum of squares programming solver.
Proceedings of the 41 st IEEE Conference on Decisión and Control,
[Rodden, 1964]
J.J. Rodden.
Numerical Applications of Lyapunov Stability Theory.
En: Joint Automatic Control Conference, (1964), pp. 261-268
[Routh, 1882]
E.J. Routh.
The Advanced Part of a Treatise on the Dynamics ofa System ofRigid Bodies.
Macmillan, (1882),
[Salam, 1988]
F.M.A. Salam.
A formulation for the design of neural processors.
IEEE International Conference on Neural Networks, pp. 173-180
[Sastry, 1999]
S. Sastry.
Nonlinear systems: analysis, stability, and control.
Springer, (1999),
[Shcherbakov, 1992]
R.S. Shcherbakov.
Alexander MikhailovitchLyapunov: On the centenary of his doctoral dissertation on stability of motion.
Automática, 28 (1992), pp. 865-871
[Slotine, 1991]
Slotine, J. J. E. y W. Li (1991). Applied nonlinear control. Prentice Hall Englewood Cliffs, NJ.
[Sontag, 1995]
E.D. Sontag.
On the input-to-state stability property.
European J. Control, 1 (1995), pp. 24-36
[Sontag and ED, 1989]
Sontag, ED.
Smooth stabilization implies coprime factorization.
IEEE Transactions on Automatic Control, 34 (1989), pp. 435-443
[Stein, 2003]
G. Stein.
Respect the unstable.
IEEE Control Systems Magazine, 23 (2003), pp. 12-25
[Sturm, 1999]
J.F. Sturm.
Using SeDuMi 1. 02, a MATLAB toolbox for optimization over symmetric cones.
Optimization Methods and Software, 11–12 (1999), pp. 625-653
[Tanaka and Sugeno, 1992]
K. Tanaka, M. Sugeno.
Stability analysis and design of fuzzy control systems.
Fuzzy sets andsystems, 45 (1992), pp. 135-156
[Tibken, 2000]
B. Tibken.
Estimation of the domain of attraction for polynomial systems via LMIs.
Decisión and Control, 2000, Proceedings of the 39th IEEE Conference on, Vol 4, pp. 3860-3864
[Tibken and Dilaver, 2002]
B. Tibken, K.F. Dilaver.
Computation of subsets ofthe domain of attraction for polynomial systems.
Decisión and Control, 2002, Proceedings ofthe 41 st IEEE Conference on, 3 (2002), pp. 2651-2656
[Toh et al., 1999]
K.C. Toh, M.J. Todd, R.H. Tutuncu.
SDPT3-a Matlab software package for semidefinite programming.
Optimization Methods and Software, (1999), pp. 545-581
[Topcu et al., 2008]
U. Topcu, A. Packardy, P. Seiler.
Local stability analysis using simulations and sum-of-squares programming.
Automática, 44 (2008), pp. 2669-2675
[Vidyasagar, 1993]
M. Vidyasagar.
Nonlinear sytems analysis.
Prentice-Hall, (1993),
[Wan et al., 1993]
C.J. Wan, V.T. Coppola, D.S. Bernstein.
A Lyapunov function for the energy-Casimir method.
Decision and Control, 1993., Proceedings of the 32nd IEEE Conference on, pp. 3122-3123
[Wong et al., 2000]
L.K. Wong, F.H.F. Leung, P.K.S. Tam.
Stability analysis of fuzzy control systems.
Stability Issues in Fuzzy Control, pp. 255-284
[Zubov, 1955]
V.I. Zubov.
Problems in the theory of the second method of Lyapunov, construction of the general solution in the domain of asymptotic stability.
Prikladnaya Matematika iMekhanika, 19 (1955), pp. 179-210
[Zubov, 1962]
V.I. Zubov.
Mathematical methods for the study of automatic control systems.
Pergamon, (1962),
[Zubov, 1964]
Zubov, V. I. (1964). Methods of A. M. Lyapunov and their Application. P. Noordhoff Groningen.
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