# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

##### MSC:
 93D30 Scalar and vector Lyapunov functions 93C10 Nonlinear control systems 93B40 Computational methods in systems theory 15A39 Linear inequalities of matrices 90C25 Convex programming
Full Text:
##### References:
 [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