Abstract
The aim of the study is to present an estimation of the domain of attraction (DA) for nonlinear delay systems. The proposed approach is based on the assumption that on a subset of instant states the system is represented by a continuous-time Takagi–Sugeno (TS) system with delay. Because of the constraints defining the subset under consideration, an important issue is to estimate the set of initial points such that the trajectories starting at those points remain to satisfy the constraints. Using Lyapunov functions with the Razumikhin technique, we describe the problem of constructing an inner estimation of the DA in terms of invariant ellipsoids. As the result, the problem reduces to linear matrix inequalities and semidefinite programs which can be solved numerically. The method is also applied to constructing some outer estimate for the attractor of a nonlinear delay system subject to asymptotic stability for an approximating TS system. The resulting ellipsoidal estimate for the attractor depends on the estimate of the approximation error.
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References
La Salle, J., Lefschetz, S.: Stability by Liapunov’s Direct Method with Applications. Academic Press, New York (1961)
Kamenetskii, V.A.: Parametric stabilization of nonlinear control systems under state constraints. Autom. Remote Control 57(10), 1427–1435 (1996)
Polyak, B., Shcherbakov, P.: Ellipsoidal approximations to attraction domains of linear systems with bounded control. In: American Control Conference, pp. 5363–5367 (2009)
Prasolov, A.V.: On an estimate of the domain of attraction for systems with after effect. Ukr. Math. J. 50(5), 761–769 (1998)
González, T., Bernal, M.: Progressively better estimates of the domain of attraction for nonlinear systems via piecewise Takagi–Sugeno models: stability and stabilization issues. Fuzzy Sets Syst. 297, 73–95 (2016)
Ting, C.-S.: A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs. Int. J. Fuzzy Syst. 10(1), 50–60 (2008)
Yang, R.-M., Wang, Y.-Z.: Stability analysis and estimate of domain of attraction for a class of nonlinear systems with time-varying delay. Acta Autom. Sin. 38(5), 716–723 (2012)
Andreev, A.S.: The Lyapunov functionals method in stability problems for functional differential equations. Autom. Remote Control 70(9), 1438–1486 (2009)
Hale, J.: Theory of Functional Differential Equations. Springer, Berlin (1977)
Leng, S., Lin W., Kurths, J.: Basin stability in delayed dynamics. Nature Scientific Reports, vol. 6 (2016). https://doi.org/10.1038/srep21449
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM, Philadelphia (1994)
Nesterov, Y., Nemirovski, A.: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (1994)
Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, New York (2001)
Khlebnikov, M.V., Polyak, B.T., Kuntsevich, V.M.: Optimization of linear systems subject to bounded exogenous disturbances: the invariant ellipsoid technique. Autom. Remote Control 72(11), 2227–2275 (2011)
Fridman, E., Shaked, U.: An ellipsoid bounding of reachable systems with delay and bounded peak inputs. IFAC Proc. Vol. 36(19), 269–274 (2003)
Sontag, E. D.: Stability and stabilization: discontinuities and the effect of disturbances. In: Nonlinear analysis, differential equations and control (Eds. Clarke, F.H. and Stern, R.J.) Springer, Dordrecht, pp. 551–598 (1999)
Sedova, N.O.: On the principle of reduction for the nonlinear delay systems. Autom. Remote Control 72(9), 1864–1875 (2011)
Fantuzzi, C., Rovatti, R.: On the approximation capabilities of the homogeneous Takagi–Sugeno model, fuzzy systems, 1996. In: Proceedings of the Fifth IEEE International Conference on, vol. 2 (1996)
Buckley, J.: Universal fuzzy controllers. Automatica (J. IFAC) 28(6), 1245–1248 (1992)
Wang, L.-X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)
Egrashkina, J.E., Sedova, N.O.: On approximate Takagi–Sugeno linear representations of nonlinear functions. Matem. Mod. 29(1), 20–32 (2017). [in Russian]
Razumikhin, B.S.: On stability on systems with delay. Prikl. Mat. Mekh. 20, 500–512 (1956) (in Russian)
Druzhinina, O.V., Sedova, N.O.: Analysis of stability and stabilization of cascade systems with time delay in terms of linear matrix inequalities. J. Comput. Syst. Sci. Int. 56(1), 19–32 (2017)
Sedova, N.O.: The design of digital stabilizing regulators for continuous systems based on the Lyapunov function approach. Autom. Remote Control 73(10), 1734–1743 (2012)
Xie, X.-P., Liu, Z.-W., Zhu, X.-L.: An efficient approach for reducing the conservatism of LMI-based stability conditions for continuous-time TS fuzzy systems. Fuzzy Sets Syst. 263, 71–81 (2015)
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Sedova, N., Pertseva, I. LMI and SDP technique for stability analysis of nonlinear delay systems subject to constraints. Optim Lett 13, 1937–1952 (2019). https://doi.org/10.1007/s11590-018-1352-9
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DOI: https://doi.org/10.1007/s11590-018-1352-9