Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System

Ji, Zhenyi; Wu, Wenyuan; Feng, Yong

doi:10.1155/2013/859578

Cited by 10 publications

(5 citation statements)

References 19 publications

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“…An overview about computational methods for Lyapunov functions is given in [5]. Several methods are limited to a special case of systems, such as positive dimensional polynomial systems [6]. A well-established method to synthesize Lyapunov functions is Sum of Squares Decomposition (SOS) [7].…”

Section: Related Workmentioning

confidence: 99%

Synthesis of Lyapunov Functions using Formal Verification

Lukas¹,

Devadze²,

Streif³

2021

Preprint

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Recent employments of SMT solvers within the Lyapunov function synthesis provided effective tools for automated construction of Lyapunov functions alongside with sound computer-assisted certificates. The main benefit of the suggested approach is the formal correctness and elimination of the numerical uncertainty. In the present work we extend the SMTbased synthesis approach for wider classes of continuous and discrete-time systems. Additionally, we address constructions of Lyapunov functions for state-dependent switching systems. We illustrate our approach by means of various examples from the control systems literature.

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Section: Related Workmentioning

confidence: 99%

Synthesis of Lyapunov Functions using Formal Verification

Lukas¹,

Devadze²,

Streif³

2021

Preprint

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“…The explicit solution and stability properties of a linear dynamical system can be readily obtained from the eigenvalue decomposition of the dynamic matrix. However, the results can hardly be extended to homogeneous polynomial dynamical systems due to its nonlinear nature [1,2,19,34,37]. In terms of stability, many methods such that generalized characteristic value problems [34] and optimization-based Lyapunov functions [1] have been proposed to establish stability of some special homogeneous polynomial dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

Chen¹

2021

Preprint

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In this paper, we investigate the explicit solutions and stability properties of certain continuous-time homogeneous polynomial dynamical systems via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonal decomposable tensor, we can write down its explicit solution in a simple fashion by exploiting tensor Z-eigenvalues and Z-eigenvectors. In addition, according to the form of the explicit solution, we explore the stability properties of the homogeneous polynomial dynamical system. We discover that the Z-eigenvalues from the orthogonal decomposition of the corresponding dynamic tensor can offer necessary and sufficient stability conditions, similar to these from linear systems theory. Finally, we demonstrate our results with numerical examples.

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“…The mentioned method also provides a lower bound for the absorption region. Establishing a Lyapunov function in the square form for polynomial systems of positive dimensions has taken in [11]. The square Lyapunov function defined in this method is such that some coefficients are unknown, which was calculated using the Homotopy continuation algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…A method to determine the Lyapunov function for the desired switching dynamical systems is given in [12]. In [11], by using the theory of normal forms, a method for determining the Lyapunov function for nonlinear systems is presented in the form of normal coordinates.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of the homogeneous nonlinear dynamical systems using Lyapunov function generation based on the basic functions

2020

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In this paper, based on Lyapunov functions candidates, a new approach in the stability analysis of homogeneous nonlinear systems is proposed in which instead of concentrating on the positive definiteness of the Lyapunov candidate functions, we stress on the negative definiteness of its derivative. Having ensured of negative definiteness of the derivative function, based on sign assignment of the primitive function, the stability of the equilibrium is analyzed wherein the necessary and sufficient conditions are declared simultaneously. Selecting the trend of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions whose unknown coefficients in the candidate function structure are computed based on negative definiteness of the derivative function. Afterward, using these determined coefficients in the Lyapunov function, the sign of the primitive function in the state space is argued. Therefore, the triple sign attitudes of the candidate function can be used to deduce the stability/instability of the equilibrium point. Moreover, in the process of the negative definiteness of the derivative function, the coefficients are obtained using two independent methods. Numerical simulations support the proposed theoretical results and show their effectiveness.

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Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System

Cited by 10 publications

References 19 publications

Synthesis of Lyapunov Functions using Formal Verification

Synthesis of Lyapunov Functions using Formal Verification

Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

Stability analysis of the homogeneous nonlinear dynamical systems using Lyapunov function generation based on the basic functions

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