An alternative approach for stability analysis of discrete time nonlinear dynamical systems

Bouyekhf, Rachid; Gruyitch, Lyubomir

doi:10.1080/10236198.2017.1391239

Cited by 2 publications

(1 citation statement)

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“…[1,2,3,4,5], where the authors used analytical techniques to iteratively obtain a LF for continuous-time (CT) and/or hybrid switched systems. In [6], an analytical Lyapunov-like solution 10 is proposed for discrete-time (DT) nonlinear dynamical systems by introducing the so-called G-functions. Unlike Lyapunov functions, G-functions do not need to be positive or negative definite.…”

Section: Introductionmentioning

confidence: 99%

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

Polcz

Péni

Szederkényi

2019

European Journal of Control

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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 (Trofino and Dezuo, 2013), 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 (D'Andrea and Khatri, 1997). Finally, additional tuning knobs are proposed to obtain more degrees of freedom in the LMI conditions. The method is illustrated on two benchmark examples. 45 to check the feasibility of the obtained LMIs only in the corner points of the polytope. Based on this work, [19] analysed the synthesis of sufficient conditions for finite-time stability of nonlinear quadratic systems using polynomial LFs, furthermore, [20] used truncated Taylor expansion to estimate the robust DOA for non-polynomial nonlinear systems. 50The results of [18] were developed further in [21,22], which proposed an LFT-based systematic procedure to construct the required differential-algebraic system representation needed for stability analysis. The authors proposed an efficient method to generate so-called maximal annihilators needed for LMI computations and introduced a model simplification technique, which resulted in a 55 dimensionally reduced optimization problem compared to other known LMIbased solutions in the literature [23,24,25, 26, 18].In this paper, we extend the ideas presented in [18,21,22] to discretetime uncertain rational systems. The parameter dependent LF is searched in 3 a general quadratic form of rational terms obtained from the linear fractional 60 representation of the dynamic equation. An estimate for the robust DOA is computed within a predefined bounded polytopic subset of the state-space and it is computed as the intersection of the largest level set of the Lyapunov function (inside the polytope) for the different values of the uncertain parameters. To give sufficient LMI conditions for the required properties of the LF we used Finsler's 65 lemma with maximal annihilators proposed by [21]. The LMI condition for the decreasing property of the LF along the system trajectories is composed in two different ways. Both methods are based on the expansion of the parameter space of the LMI problem by introducing new rational terms into the model.The paper is organized as follows: Section 2 presents the LMI approach 70 for the c...

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