Estabilidad de Sistemas No Lineales Basada en la Teoría de Liapunov

Francisco Gordillo

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 artículo 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, C. y F. Gordillo (2007). Estimation of the region of attraction for a boost DC-AC converter control law. En: Proceedings of the 7th IFAC Simposium. Nonlinear Control System (NOLCOS). pp. 874–879.

Aracil, J., F. Gordillo y E. Ponce (2005). Stabilization of oscillations through backstepping in high–dimensional systems. IEEE Tr. on Automat. Control 50(5), 705–710.

Aracil, J. y F. Gordillo (2005). El péndulo invertido: un desafío para el control no lineal. Revista Iberoamericana de Automática e Informática Industrial 2(2), 8 – 19.

Aracil, J. y Gordillo, F., Eds. (2000). Stability Issues in Fuzzy Control. Physica-Verlag.

Aström, K. J. y K. Furuta (2000). Swinging up a pendulum by energy control. Automatica 36, 287–295.

Axelby, G. S. y P. C. Parks (1992). Lyapunov centenary. Automatica 28(5), 863 – 864.

Barbashin, E. A. y N.N. Krasovskii (1952). On the stability of motion in the large. Dokl. Akad. Nauk. 86, 453–456.

Cao, J. y J. Wang (2005). Global asymptotic and robust stability of recurrent neural networks with time delays. IEEE Transactions on Circuits and Systems I: Regular Papers 52(2), 417–426.

Choi, M. D., T. Y. Lam y B. Reznick (1995). Sums of squares of real polynomials. En: K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (B. Jacob, A. Rosenberg, Eds.), Proc. Symp. Pure Math. Vol. 58. pp. 103–126.

Cuesta, F., F. Gordillo, J. Aracil y A. Ollero (1999). Global stability analysis of a class of multivariable Takagi-Sugeno fuzzy control systems. IEEE Trans. Fuzzy Systems 7(5), 508– 520.

Davison, E. J. y E. M. Kurak (1971). A computational method for determining quadratic Lyapunov functions for non-linear systems. Automatica 7, 627–636.

Davrazos, G. y NT Koussoulas (2001). A review of stability results for switched and hybrid systems. En: Mediterranean Conference on Control and Automation.

Espada, A. y A. Barreiro (1999). Robust stability of fuzzy control systems based on conicity conditions. Automatica 35(4), 643–654.

Feng, G. (2006). A survey on analysis and design of modelbased fuzzy control systems. IEEE Transactions on Fuzzy Systems 14(5), 676–697.

Forti, M. y A. Tesi (1995). New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 42(7), 354–366.

Fuller, A. T. (1992). Lyapunov Centenary Issue. International Journal of Control 55(3), 521–527.

Furuta, K. (2003). Control of pendulum: From super mechanosystem to human adaptive mechatronics. En: Proceedings of the 42nd IEEE CDC. pp. 1498–1507.

Genesio, R., M. Tartaglia y A. Vicino (1985). On the estimation of asymptotic stability regions: State of the art and new proposals. IEEE Transactions on Automatic Control 30(8), 747– 755.

Haddad, W. M. y V. Chellaboina (2008). Nonlinear dynamical systems and control. A Lyapunov-based approach. Princeton University Press.

Hahn, W. (1967). Stability of motion. Springer-Verlag.

Isidori, A. (1999). Nonlinear control systems II. Communications and control engineering series. Springer-Verlag.

Jarvis-Wloszek, Z., R. Feeley, W. Tan, K. Sun y A. Packard (2003). Some controls applications of sum of squares programming. En: 42nd IEEE Conference on Decision and Control, 2003. Proceedings. Vol. 5.

Jarvis-Wloszek, Z. W. (2003). Lyapunov based analysis and controller synthesis for polynomial systems using sum-ofsquares optimization. PhD thesis. University of California.

Johansson, M., A. Rantzer y K. E. Arzen (1999). Piecewise quadratic stability of fuzzy systems. IEEE Transactions on Fuzzy Systems 7(6), 713–722.

Johansson, M. y A. Rantzer (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE transactions on automatic control 43(4), 555–559.

Kaliora, Georgia (2002). Control of nonlinear systems with bounded signals. PhD thesis. Imperial College of Sicnece, Technology and Medicine. London.

Kalman, R. E. y J. E. Bertram (1960). Control system analysis and design via the second method of Lyapunov. Journal of Basic Engineering 82(2), 371–393.

Khalil, H. K. (1996). Nonlinear Systems. 2a ed. Prentice Hall.

Khalil, H. K. (2002). Nonlinear Systems. 3a ed. Prentice Hall.

Krasovskii, N.N. (1959). Some problems of the motion stability theory. Moscow: Fizmatgiz. Traducido al ingl´es por Stanford University Press, 1963.

LaSalle, J.P. (1960). Some extensions of Liapunov’s second method. Circuit Theory, IRE Transactions on 7(4), 520–527.

Liberzon, D. (2003). Switching in systems and control. Birkhauser.

Liberzon, D. y A. S. Morse (1999). Basic problems in stability and design of switched systems. IEEE control systems magazine 19(5), 59–70.

Loparo, K. y G. Blankenship (1978). Estimating the domain of attraction of nonlinear feedback systems. IEEE Transactions on Automatic Control 23(4), 602–608.

Lyapunov, A. M. (1892). El problema general de la estabilidad del movimiento. PhD thesis. Kharkov Mathematical Society. En ruso.

Lyapunov, A. M. (1992). The General Problem of the Stability of Motion, 1892. International Journal of Control: Lyapunov Centenary.

Margolis, S. y W. Vogt (1963). Control engineering applications of V. I. Zubov’s construction procedure for Lyapunov functions. IEEE Transactions on Automatic Control 8(2), 104– 113.

Martynyuk, A. A. (2000). A survey of some classical and modern developments of stability theory. Nonlinear Analysis 40(1-8), 483 – 496.

Martynyuk, A. A. (2007). Stability of Motion. The role of Multicomponent Liapunov’s Functions. Stability, Oscillations and Optimization of Systems. Cambridge Scienſc Publishers.

Michel, A.N. (1996). Stability: the common thread in the evolution of feedback control. IEEE Control Systems Magazine 16(3), 50–60.

Narendra, K. S. y J. Balakrishnan (1994). A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Transactions on automatic control 39(12), 2469–2471.

Papachristodoulou, A. y S. Prajna (2002). On the construction of Lyapunov functions using the sum of squares decomposition. Decision and Control, 2002, Proceedings of the 41st IEEE Conference on.

Papachristodoulou, A. y S. Prajna (2005). A tutorial on sum of squares techniques for systems analysis. En: 2005 American Control Conference. pp. 2686–2700.

Parrilo, P. A. (2000). Structured Semideſnite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis. California Institute of Technology, Pasadena, California.

Peleties, P. y R. DeCarlo (1992). Asymptotic stability of mswitched systems using Lyapunov functions. En: Decision and Control, 1992., Proceedings of the 31st IEEE Conference on. pp. 3438–3439.

Powers, V. y T. Wormann (1998). An algorithm for sums of squares of real polynomials. Journal of pure and applied algebra 127(1), 99.

Prajna, S., A. Papachristodoulou, P. Seiler y P. A. Parrilo (2005). SOSTOOLS and its control applications. Positive Polynomials in Control pp. 273–292.

Prajna, S., A. Papachristodoulou y P. A. Parrilo (2002). Introducing SOSTOOLS: a general purpose sum of squares programming solver. Proceedings of the 41st IEEE Conference on Decision and Control.

Rodden, J. J. (1964). Numerical Applications of Lyapunov Stability Theory. En: Joint Automatic Control Conference. pp. 261–268.

Routh, E. J. (1882). The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies. Macmillan.

Salam, F. M. A. (1988). A formulation for the design of neural processors. En: IEEE International Conference on Neural Networks, 1988. pp. 173–180.

Sastry, S. (1999). Nonlinear systems: analysis, stability, and control. Springer.

Shcherbakov, P. S. (1992). Alexander Mikhailovitch Lyapunov: On the centenary of his doctoral dissertation on stability of motion. Automatica 28(5), 865 – 871.

Slotine, J. J. E. y W. Li (1991). Applied nonlinear control. Prentice Hall Englewood Cliffs, NJ.

Sontag, E. D. (1995). On the input-to-state stability property. European J. Control 1, 24–36.

Sontag, ED (1989). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control 34(4), 435–443.

Stein, G. (2003). Respect the unstable. IEEE Control Systems Magazine 23(4), 12–25.

Sturm, J. F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11–12, 625–653.

Tanaka, K. y M. Sugeno (1992). Stability analysis and design of fuzzy control systems. Fuzzy sets and systems 45(2), 135– 156.

Tibken, B. (2000). Estimation of the domain of attraction for polynomial systems via LMIs. En: Decision and Control, 2000, Proceedings of the 39th IEEE Conference on. Vol. 4. pp. 3860–3864.

Tibken, B. y K. F. Dilaver (2002). Computation of subsets of the domain of attraction for polynomial systems. En: Decision and Control, 2002, Proceedings of the 41st IEEE Conference on. Vol. 3. pp. 2651–2656.

Toh, K. C., M. J. Todd y R. H. Tutuncu (1999). SDPT3– a Matlab software package for semideſnite programming. Optimization Methods and Software pp. 545–581.

Topcu, U., A. Packard y P. Seiler (2008). Local stability analysis using simulations and sum-of-squares programming. Automatica 44(10), 2669–2675.

Vidyasagar, M. (1993). Nonlinear sytems analysis. PrenticeHall.

Wan, C. J., V. T. Coppola y D. S. Bernstein (1993). A Lyapunov function for the energy-Casimir method. En: Decision and Control, 1993., Proceedings of the 32nd IEEE Conference on. pp. 3122–3123.

Wong, L. K., F. H. F. Leung y P. K. S. Tam (2000). Stability analysis of fuzzy control systems. En: Stability Issues in Fuzzy Control (J. Aracil y F. Gordillo, Eds.). pp. 255–284. Physica-Verlag.

Zubov, V. I. (1955). Problems in the theory of the second method of Lyapunov, construction of the general solution in the domain of asymptotic stability. Prikladnaya Matematika i Mekhanika 19, 179–210.

Zubov, V. I. (1962). Mathematical methods for the study of automatic control systems. Pergamon.

Zubov, V. I. (1964). Methods of A. M. Lyapunov and their Application. P. Noordhoff Groningen.

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