A Lyapunov approach for attraction domain estimation of polynomial discrete nonlinear systems with quadratic and cubic t
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A Lyapunov approach for attraction domain estimation of polynomial discrete nonlinear systems with quadratic and cubic terms Rim Zakhama1,2
· Anis Belhadj Brahim Bacha1,2 · Naceur Benhadj Braiek1
Received: 18 August 2017 / Revised: 22 June 2019 / Accepted: 6 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we propose a Lyapunov method for estimating asymptotic stability domain of polynomial discrete nonlinear systems, drawn from an existing approach for attraction domain estimation of continuous nonlinear systems. Proceeding from two different forms of a parametric equation, we present two development methods with the aim of identifying a sufficient condition to obtain a part of the stability domain for quadratic and cubic n-dimensional polynomial discrete systems. The results consist in solving a polynomial equation and unconstrained nonlinear optimization problems, where in case of quadratic or cubic systems, significant simplifications are obtained. The main notable advantages of this approach are its applicability on high-dimensional systems and obtaining an ellipsoidal stability domain which approaches the exact attraction domain limits. Two application examples are illustrated to validate each development. Keywords Asymptotic stability domain · Discrete nonlinear system · Lyapunov function · Quadratic and cubic systems · Vector function · Kronecker product Mathematics Subject Classification 34D20 · 93D20
Communicated by Antonio José Silva Neto.
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Rim Zakhama [email protected] Anis Belhadj Brahim Bacha [email protected] Naceur Benhadj Braiek [email protected]
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Laboratory of Advanced Systems, Tunisia Polytechnic School, University of Carthage, 2078 La Marsa, Tunisia
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ENISO, University of Sousse, 4023 Sousse, Tunisia 0123456789().: V,-vol
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1 Introduction The problem of asymptotic stability domain estimation for nonlinear systems is widely known in the field of research, especially for nonlinear systems analysis and control. During the past several years, a significant interest has been dedicated to the study of asymptotic stability domain estimation for nonlinear continuous systems. For example, see Hachicho (2007), Matallana et al. (2010), Wu et al. (2014), Chesi (2012), Purschely et al. (2016), and Chesi (2007). However, fewer research works have been studied with the intention of estimating asymptotic stability domain of nonlinear discrete systems. Among these works, we can cite, Coutinho and De Souza (2013), Bobiti and Lazar (2014), and Zakhama et al. (2016). For both continuous and discrete nonlinear systems, research works fall into two categories. The first category concerns methods based on topological considerations, named “non Lyapunov methods”, studied in Genesio et al. (1985), Benhadj Braiek et al. (2008), and Bacha et al. (2008). The second category involves methods using the “Lyapunov function” notion to estimate asymptotic stability domain, named “Lyapunov methods”, st
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