On estimating the Robust Domain of Attraction for uncertain non-polynomial systems: An LMI approach

Han, Dong; Althoff, Matthias

doi:10.1109/cdc.2016.7798586

Cited by 13 publications

(4 citation statements)

References 27 publications

(34 reference statements)

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“…To handle with the above issue, effective methods are proposed to compute the ROA for partially unknown systems. Lyapunov-based methods are developed and extended to uncertain systems, where a Lyapunov certified ROA (LCROA) estimation is conducted for uncertain, polynomial systems [14], [15]. Furthermore, learning-based methods have also been studied in the literature.…”

Section: Introductionmentioning

confidence: 99%

On Estimating the Probabilistic Region of Attraction for Partially Unknown Nonlinear Systems: An Sum-of-Squares Approach

Huang¹,

Han²

2021

Preprint

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Estimating the region of attraction for partially unknown nonlinear systems is a challenging issue. In this paper, we propose a tractable method to generate an estimated region of attraction with probability bounds, by searching an optimal polynomial barrier function. Chebyshev interpolants, Gaussian processes and sum-of-squares programmings are used in this paper. To approximate the unknown non-polynomial dynamics, a polynomial mean function of Gaussian processes model is computed to represent the exact dynamics based on the Chebyshev interpolants. Furthermore, probabilistic conditions are given such that all the estimates are located in certain probability bounds. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

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Section: Introductionmentioning

confidence: 99%

On Estimating the Probabilistic Region of Attraction for Partially Unknown Nonlinear Systems: An Sum-of-Squares Approach

Huang¹,

Han²

2021

Preprint

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“…For convergence analysis of DT dynamical systems, [7,8] used Banach fixed-point principle together with a contraction mapping theorem. 15 At the same time, there exist alternative numerical Lyapunov-based approaches to determine forward invariant subsets of the state-space, for example, [9,10] proposed a simulation-guided LF computation method for nonlinear switched and DT systems, respectively, by applying some linear constraints obtained from the execution traces of the dynamics in discrete sample points of a 20 bounded subset of the state-space. Based on multi-resolution state-space sampling approach, [11] considered an initial quadratic finite-step Lyapunov function to systematically find a LF in a general quadratic form of nonlinear terms alongside with a (possibly non-convex) bounded invariant region.…”

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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“…In [21], a method is proposed by using the truncated Taylor expansion, and the largest estimate of the DA is obtained by using polynomial Lyapunov functions. Based on this idea, the DA of rational non-polynomial systems is computed, also using the truncated Taylor expansion [22].…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the work in [18], in contrast to our previous work [22], [23] using Taylor approximation, this paper deploys higher order derivatives of Lyapunov functions via Chebyshev approximation. The main contributions are displayed as follows:…”

Section: Introductionmentioning

confidence: 99%

Chebyshev approximation and higher order derivatives of Lyapunov functions for estimating the domain of attraction

Han

Panagou

2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

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Estimating the Domain of Attraction (DA) of nonpolynomial systems is a challenging problem. Taylor expansion is widely adopted for transforming a nonlinear analytic function into a polynomial function, but the performance of Taylor expansion is not always satisfactory. This paper provides solvable ways for estimating the DA via Chebyshev approximation. Firstly, for Chebyshev approximation without the remainder, higher order derivatives of Lyapunov functions are used for estimating the DA, and the largest estimate is obtained by solving a generalized eigenvalue problem. Moreover, for Chebyshev approximation with the remainder, an uncertain polynomial system is reformulated, and a condition is proposed for ensuring the convergence to the largest estimate with a selected Lyapunov function. Numerical examples demonstrate that both accuracy and efficiency are improved compared to Taylor approximation.

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On estimating the Robust Domain of Attraction for uncertain non-polynomial systems: An LMI approach

Cited by 13 publications

References 27 publications

On Estimating the Probabilistic Region of Attraction for Partially Unknown Nonlinear Systems: An Sum-of-Squares Approach

On Estimating the Probabilistic Region of Attraction for Partially Unknown Nonlinear Systems: An Sum-of-Squares Approach

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

Chebyshev approximation and higher order derivatives of Lyapunov functions for estimating the domain of attraction

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