# Generalization of a stability domain estimation method for nonlinear discrete systems

## 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.

- Authors:

- Tunisia Polytechnic School, University of Carthage, Laboratory of advanced systems (Tunisia)

- Publication Date:

- OSTI Identifier:
- 22769356

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; ASYMPTOTIC SOLUTIONS; MATRICES; NONLINEAR PROBLEMS; POLYNOMIALS; STABILITY

### Citation Formats

```
Zakhama, Rim, Hadj Brahim, Anis Bacha Bel, E-mail: anis.bacha@isimg.rnu.tn, and Braiek, Naceur Benhadj, E-mail: naceur.benhadj@ept.rnu.tn.
```*Generalization of a stability domain estimation method for nonlinear discrete systems*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0388-7.

```
Zakhama, Rim, Hadj Brahim, Anis Bacha Bel, E-mail: anis.bacha@isimg.rnu.tn, & Braiek, Naceur Benhadj, E-mail: naceur.benhadj@ept.rnu.tn.
```*Generalization of a stability domain estimation method for nonlinear discrete systems*. United States. doi:10.1007/S40314-016-0388-7.

```
Zakhama, Rim, Hadj Brahim, Anis Bacha Bel, E-mail: anis.bacha@isimg.rnu.tn, and Braiek, Naceur Benhadj, E-mail: naceur.benhadj@ept.rnu.tn. Tue .
"Generalization of a stability domain estimation method for nonlinear discrete systems". United States. doi:10.1007/S40314-016-0388-7.
```

```
@article{osti_22769356,
```

title = {Generalization of a stability domain estimation method for nonlinear discrete systems},

author = {Zakhama, Rim and Hadj Brahim, Anis Bacha Bel, E-mail: anis.bacha@isimg.rnu.tn and Braiek, Naceur Benhadj, E-mail: naceur.benhadj@ept.rnu.tn},

abstractNote = {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.},

doi = {10.1007/S40314-016-0388-7},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}