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Computation of Lyapunov functions for smooth nonlinear systems using convex optimization. (English) Zbl 0971.93069
This work presents a computational approach for searching a Lyapunov function for the equilibrium point $x= 0$ for a class of nonautonomous nonlinear systems $\dot x= f(x,\theta)$, where $x\in\bbfR^n$ is the state vector, $\theta\in \bbfR^\alpha$ is a possibly time-varying parameter vector, $f(0,\theta)= 0$, and for all $\theta$, $f(x,\theta)$ is smooth. The Lyapunov function is considered in the form $V(x)= x^T P(x)x$, where $P(x)= \sum^N_{i=1} P_i\rho_i(x)$, $\rho_i(x)$ are smooth basis-functions and $P_i$ are parameter matrices. The parameter matrices are sought from the condition, $\dot V\le -\gamma V(x)$, where $\gamma> 0$. The last condition is the condition of exponential stability of the equilibrium point. The main contribution of this paper is the utilization of a flexible and general smooth parameterization of the Lyapunov function candidates that does not introduce significant conservativeness and the problem is reduced to a convex optimization problem involving linear inequality constraints at each point in the state space.

93D30Scalar and vector Lyapunov functions
93C10Nonlinear control systems
93B40Computational methods in systems theory
15A39Linear inequalities of matrices
90C25Convex programming
Full Text: DOI
[1] Abrahamson, R.; Marsden, J. E.; Ratiu, T.: Manifolds, tensor analysis and applications. (1988) · Zbl 0875.58002
[2] Blanchini, F.: Nonquadratic Lyapunov functions for robust control. Automatica 31, 451-461 (1995) · Zbl 0825.93653
[3] Blanchini, F., & Miani, S. (1996). A new class of universal Lyapunov functions for the control of uncertain linear systems. Proceedings of the 35th conference on decision and control, Kobe, Japan (pp. 1027-1032). · Zbl 0858.93063
[4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balahrishnan, V.: Linear matrix inequalities in system and control theory. (1994) · Zbl 0816.93004
[5] Brayton, R. K.; Tong, C. H.: Constructive stability and asymptotic stability of dynamical systems. IEEE transactions on circuits and systems 27, 1121-1130 (1980) · Zbl 0458.93047
[6] Gahinet, P.; Apkarian, P.; Chilali, M.: Affine parameter-dependent Lyapunov function and real parametric uncertainty. IEEE transactions on automatic control 41, 436-442 (1996) · Zbl 0854.93113
[7] Johansson, M.; Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE transactions on automatic control 43, 555-559 (1998) · Zbl 0905.93039
[8] Julian, P., Guivant, J., & Desages, A. (1999). A parameterization of piecewise linear Lyapunov functions via linear programming, International Journal of Control, 72, 702-715. · Zbl 0963.93036
[9] Kamenetskii, V. A.; Pyatnitskii, E. S.: Gradient method of constructing Lyapunov functions in problems of absolute stability. Automation and remote control 48, 1-9 (1987)
[10] Khalil, H. K.: Nonlinear systems. (1992) · Zbl 0969.34001
[11] Luenberger, D. G.: Introduction to linear and nonlinear programming. (1989)
[12] Molchanov, A. P.: Lyapunov functions for nonlinear discrete-time control systems. Automation and remote control 48, 728-736 (1987) · Zbl 0667.93075
[13] Molchanov, A. P., & Pyatnitskii, E. S. (1986). Lyapunov functions that specify necessary and sufficient conditions of absolute stability of nonlinear nonstationary control systems. Parts I-III. Automation and Remote Control, 47, 344-354, 443-451, 620-630. · Zbl 0607.93039
[14] Murray-Smith, R.; Johansen, T. A.: Multiple model approaches to modelling and control. (1997)
[15] Ohta, Y.; Imanishi, H.; Gong, L.; Haneda, H.: Computer generated Lyapunov functions for a class of nonlinear systems. IEEE transactions on circuits and systems 40, 343-354 (1993) · Zbl 0790.93120
[16] Petterson, S., & Lennartson, B. (1997). Exponential stability analysis of nonlinear systems using LMIs. In Proceedings of the IEEE conference on decision and control, San Diego. · Zbl 0981.93033
[17] Polak, E.: Optimization: algorithms and consistent approximation. (1997) · Zbl 0899.90148
[18] Pyatnitskii, E. S.; Skordodinskii, V. I.: A criterion of absolute stability of nonlinear sampled-data control systems in the form of numerical procedures. Automation and remote control 48, 1190-1198 (1987)
[19] Rovatti, R. (1996). Takagi-Sugeno models as approximators in Sobolev norms -- the SISO Case, Proceedings of the IEEE Conference Fuzzy Systems, New Orleans (pp. 1060-1066).
[20] Tanaka, Y.; Fukushima, M.; Ibaraki, T.: A comparative study of several semi-infinite nonlinear programming algorithms. European journal of operational research 36, 92-100 (1988) · Zbl 0643.90079
[21] Watanabe, R., Uchida, K., & Fujita, M. (1996). A new LMI approach to analysis of linear systems with scheduling parameter -- Reduction to finite number of LMI conditions. In Proceedings of the 35th conference on decision and control. Kobe (pp. 1663-1665).
[22] Wu, F.; Yang, X. H.; Packard, A.; Becker, G.: Induced L2-norm control for LPV systems with bounded parameter variation rates. International journal on nonlinear and robust control 6, 983-998 (1996) · Zbl 0863.93074
[23] Zelentsovsky, A. L.: Nonquadratic Lyapunov function for robust stability analysis of linear uncertain systems. IEEE transactions on automatic control 39, 135-138 (1994) · Zbl 0796.93101