# Computational method for estimating the domain of attraction of discrete-time uncertain rational systems

Péter Polcz, Tamás Péni, G. Szederkényi

Research output: Contribution to journalArticle

### Abstract

Using linear matrix inequality (LMI) conditions, we propose a computational method to generate Lyapunov functions and to estimate the domain of attraction (DOA) of uncertain nonlinear (rational) discrete-time systems. The presented method is a discrete-time extension of the approach first presented in , where the authors used Finsler's lemma and affine annihilators to give sufficient LMI conditions for stability. The system representation required for DOA computation is generated systematically by using the linear fractional transformation (LFT). Then a model simplification step not affecting the computed Lyapunov function (LF) is executed on the obtained linear fractional representation (LFR). The LF is computed in a general quadratic form of a state and parameter dependent vector of rational functions, which are generated from the obtained LFR model. The proposed method is compared to the numeric n-dimensional order reduction technique proposed in . Finally, additional tuning knobs are proposed to obtain more degrees of freedom in the LMI conditions. The method is illustrated on two benchmark examples.

Original language English European Journal of Control https://doi.org/10.1016/j.ejcon.2018.12.004 Accepted/In press - Jan 1 2018

### Fingerprint

Lyapunov functions
Computational methods
Linear matrix inequalities
Knobs
Rational functions
Tuning

### Keywords

• Domain of attraction
• Linear matrix inequality
• Lyapunov function
• Model simplification
• Nonlinear system

### ASJC Scopus subject areas

• Engineering(all)

### Cite this

In: European Journal of Control, 01.01.2018.

Research output: Contribution to journalArticle

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abstract = "Using linear matrix inequality (LMI) conditions, we propose a computational method to generate Lyapunov functions and to estimate the domain of attraction (DOA) of uncertain nonlinear (rational) discrete-time systems. The presented method is a discrete-time extension of the approach first presented in , where the authors used Finsler's lemma and affine annihilators to give sufficient LMI conditions for stability. The system representation required for DOA computation is generated systematically by using the linear fractional transformation (LFT). Then a model simplification step not affecting the computed Lyapunov function (LF) is executed on the obtained linear fractional representation (LFR). The LF is computed in a general quadratic form of a state and parameter dependent vector of rational functions, which are generated from the obtained LFR model. The proposed method is compared to the numeric n-dimensional order reduction technique proposed in . Finally, additional tuning knobs are proposed to obtain more degrees of freedom in the LMI conditions. The method is illustrated on two benchmark examples.",
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