# zbMATH — the first resource for mathematics

Lyapunov stability for continuous-time multidimensional nonlinear systems. (English) Zbl 1283.93215
Summary: This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

##### MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D30 Lyapunov and storage functions
Matlab
Full Text:
##### References:
 [1] Shanks, J.; Treitel, S.; Justice, J., Stability and synthesis of two-dimensional recursive filters, IEEE Trans. Audio Electroacoust., AU-20, 115-128, (1972) [2] Huang, T., Stability of two-dimensional recursive filters, IEEE Trans. Audio Electroacoust., AU-20, 158-163, (1972) [3] Anderson, B.; Jury, E., Stability test for two-dimensional recursive filters, IEEE Trans. Audio Electroacoust., AU-21, 366-372, (1973) [4] Pandolfi, L., Exponential stability of 2-D systems, Syst. Control Lett., 4, 381-385, (1984) · Zbl 0543.93051 [5] Goodman, D., Some stability properties of two dimensional linear shift invariant digital filters, IEEE Trans. Circuits Syst., CAS-24, 201-208, (1977) · Zbl 0382.93063 [6] Roesser, R., Discrete state-space model for linear image processing, IEEE Trans. Autom. Control, AC-20, 1-10, (1975) · Zbl 0304.68099 [7] Fornasini, E.; Marchesini, G., State-space realization theory of two-dimensional filters, IEEE Trans. Autom. Control, AC-21, 484-492, (1976) · Zbl 0332.93072 [8] Fornasini, E.; Marchesini, G., Doubly-indexed dynamical systems: state-space models and structural properties, Math. Syst. Theory, 12, 59-72, (1978) · Zbl 0392.93034 [9] Hinamoto, T., The Fornasini-Marchesini model with no over flow oscillations and its application to 2D digital filter design, Portland, USA [10] Du, C.; Xie, L., Stability analysis and stabilization of uncertain two-dimensional discrete systems: an LMI approach, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 46, 1371-1374, (1999) · Zbl 0970.93037 [11] Bliman, P.-A., Lyapunov equation for the stability of 2-D systems, Multidimens. Syst. Signal Process., 13, 201-222, (2002) · Zbl 1021.93021 [12] Galkowski, K.; Lam, J.; Xu, S.; Lin, Z., LMI approach to state feedback stabilization of multidimensional systems, Int. J. Control, 76, 1428-1436, (2003) · Zbl 1048.93084 [13] Paszke, W.; Lam, J.; Galkowski, K.; Xu, S.; Lin, Z., Robust stability and stabilization of 2D discrete state-delayed systems, Syst. Control Lett., 51, 277-291, (2004) · Zbl 1157.93472 [14] Shaker, H.R.; Tahavori, M., Stability analysis for a class of discrete-time two-dimensional nonlinear systems, Multidimens. Syst. Signal Process., 21, 293-299, (2010) · Zbl 1202.93144 [15] Kaczorek, T., LMI approach to stability of 2D positive systems, Multidimens. Syst. Signal Process., 20, 39-54, (2009) · Zbl 1169.93022 [16] Avelli, D.N.; Rapisarda, P.; Rocha, P., Lyapunov stability of 2D finite-dimensional behaviours, Int. J. Control, 84, 737-745, (2011) · Zbl 1245.93109 [17] Yeganefar, N.; Yeganefar, N.; Ghamgui, M.; Moulay, E., Lyapunov theory for 2D nonlinear Roesser models: application to asymptotic and exponential stability, IEEE Trans. Autom. Control, 58, 1299-1304, (2013) · Zbl 1369.93544 [18] Lewis, F.L., A review of 2-D implicit systems, Automatica, 28, 345-354, (1992) · Zbl 0766.93035 [19] Marszalek, W., Two-dimensional state-space discrete models for hyperbolic partial differential equations, Appl. Math. Model., 8, 11-14, (1984) · Zbl 0529.65039 [20] Marszalek, W., On modelling of distributed processes with two-dimensional discrete linear equations, Rozpr. Elektrotech., 33, 627-640, (1987) [21] Lewis, F.L., Walsh function analysis of 2-D generalized continuous systems, IEEE Trans. Autom. Control, 35, 1140-1144, (1990) · Zbl 0724.93045 [22] Khalil, H.: Nonlinear Systems, 3rd edn. Prentice Hall, Englewood Cliffs (2002) · Zbl 1003.34002 [23] Leine, R.I., The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability, Nonlinear Dyn., 59, 173-182, (2010) · Zbl 1183.70002 [24] Aniszewska, D.; Rybaczuk, M., Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems, Nonlinear Dyn., 54, 345-354, (2008) · Zbl 1170.70010 [25] Ling, Q.; Jin, X.L.; Wang, Y.; Li, H.F.; Huang, Z.L., Lyapunov function construction for nonlinear stochastic dynamical systems, Nonlinear Dyn., (2013) · Zbl 1284.93207 [26] Kidouche, M.; Habbi, H., On Lyapunov stability of interconnected nonlinear systems: recursive integration methodology, Nonlinear Dyn., 60, 183-191, (2010) · Zbl 1189.70082 [27] Vandenberghe, L.; Boyd, S., Semidefinite programming, SIAM Rev., 38, 49-95, (1996) · Zbl 0845.65023 [28] Papachristodoulou, A.; Prajna, S., A tutorial on sum of squares techniques for systems analysis, No. 4, 2686-2700, (2005) [29] Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P.A.: Sum of squares optimization toolbox for MATLAB—user’s guide (2004). Available at: http://www.cds.caltech.edu/sostools/sostools.pdf · Zbl 0724.93045 [30] Papachristodoulou, A.; Prajna, S.; Henrion, D. (ed.); Garulli, A. (ed.), Analysis of non-polynomial systems using the sum of squares decomposition, 23-44, (2005), Berlin [31] Papachristodoulou, A.; Prajna, S., On the construction of Lyapunov functions using the sum of squares decomposition, No. 3, 3482-3487, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.