Generalization of a stability domain estimation method for nonlinear discrete systems
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Generalization of a stability domain estimation method for nonlinear discrete systems Rim Zakhama1,2,3 · Anis Bacha Bel Hadj Brahim1,2,3 · Naceur Benhadj Braiek1
Received: 18 March 2016 / Revised: 23 June 2016 / Accepted: 3 October 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In this paper, we present a generalization approach of an algebraic existing method to estimate stability regions for discrete nonlinear polynomial systems of degree 3. The existing method is based on the enlargement of a guaranteed stability region by applying various steps of a proposed algorithm. Its main limitation is that the initial result has only been subsequently developed in a particular case of a single iteration. The stability domain obtained is consequently not the widest one. Our main contribution in this paper is to develop generalized functions that allow the enlargement of the guaranteed stability region after k iterations, for any value of k. A required fundamental tool is developed and consists in a general formula allowing to give the result of the Kronecker power calculation of two matrices sum. The advantages of this generalization are to reach a larger region of asymptotic stability and to improve the existing methods results. Two application examples illustrate the proposed method. Keywords Asymptotic stability region · Discrete time systems · Guaranteed stability region · Nonlinear systems Mathematics Subject Classification 15A72 · 34D20 · 93D20
Communicated by Luz de Teresa.
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Rim Zakhama [email protected] Anis Bacha Bel Hadj Brahim [email protected] Naceur Benhadj Braiek [email protected]
1
Laboratory of advanced systems, Tunisia Polytechnic School, University of Carthage, La Marsa 2078, Tunisia
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ENISO, 4023 Sousse, Tunisia
3
University of Sousse, 4002 Sousse, Tunisia
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R. Zakhama et al.
1 Introduction Several research studies have been directed to estimate stability domains of nonlinear continuous systems. These methods fall into two different categories. The first category considers methods using the Lyapunov candidate function for the enlargement of an invariant domain around the stable equilibrium point. These methods are generally known as Lyapunov methods (Giesl and Hafstein 2015; Khodadadi et al. 2014; Matallana et al. 2011; Papachristodoulou et al. 2013; Vanelli and Vidyasagar 1985; Wu et al. 2014). The improvement of these techniques consists in the search for new parametric Lyapunov functions to obtain larger stability domain of nonlinear continuous systems by the means of optimization techniques. The second category considers geometric concepts for determination and estimation of the stability domain. It studies the behavior of trajectories around the stable equilibrium point by handling some topological considerations. These methods are known as “nonLyapunov” methods (Loccufier and Noldus 2000; Genesio et al. 1985; Noldus and Loccufier 1995). Found results are widely applied in the field of engineering, particularly to the ele
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