Providing a Basin of Attraction to a Target Region of Polynomial Systems by Computation of Lyapunov-Like Functions

Ratschan, Stefan; She, Zhikun

doi:10.1137/090749955

Cited by 101 publications

(67 citation statements)

References 34 publications

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“…In [7], Eghbal, Pariz, and Karimpour formulate the computation of piecewise quadratic Lyapunov functions for planar piecewise affine systems as linear matrix inequalities. In [32], Ratschan and She give an interval based branch-and-relax algorithm to compute polynomial Lyapunov-like functions for polynomial ODE. Another approach to numerically investigate the stability of nonlinear systems is, for example, given by Oishi in [27], where he considers the probabilistic computation of a stable control for systems that are parameter dependent, linear, and discrete.…”

Section: Introductionmentioning

confidence: 99%

Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium's basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C 2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

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Section: Introductionmentioning

confidence: 99%

Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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“…(3.10) to (3.12) with other existing linear relaxations including Handelman and interval LP relaxations. More precisely, we will show the benefit of using Bernstein relaxations instead of the LP relaxations given by Ratschan et al [49] and our previous work [50].…”

Section: Comparison Of Bernstein Relaxations With Other Linear Represmentioning

confidence: 67%

“…Interval relaxations are presented by Ratschan et al [49] and in our previous work [50]. Notably, let K be a hyper-rectangular domain.…”

Section: Comparison With Interval Representationsmentioning

confidence: 87%

“…Our future work will consider the application of the LP relaxations to those considered in Kamyar et al, facilitating an experimental comparison. Ratschan and She use interval arithmetic relaxations with branch-and-bound to discover Lyapunov like functions to prove a notion of region stability of polynomial systems [49]. This is extended in our previous work to find LP relaxations using the notion of Handelman representations [50].…”

Section: Related Workmentioning

confidence: 99%

“…The interval relaxation replaces each monomial x I with an interval over K. The interval for a polynomial p is obtained by summing up the interval for each term. While there exist many approaches to evaluate a polynomial over an interval, we will consider the scheme (implicitly) adopted by Ratschan et al [49] and in our previous work [50] that uses an interval arithmetic based LP relaxation for (parametric) polynomial optimization problems. Here we will fix K : [0, 1] n , mapping arbitrary, bounded hyper-rectangles to this domain through a linear change of variables (see Section 5.3).…”

Section: Comparison With Interval Representationsmentioning

confidence: 99%

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Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Sassi

Sankaranarayanan

Chen

et al. 2015

IMA J Math Control Info

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We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two classes of relaxations for encoding polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations based on Handelman representations and Bernstein polynomials, that produce linear programs. Next, we present a series of increasingly powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. Subsequently, we show how these LP relaxations can be used to search for Lyapunov functions for polynomial ODEs by formulating LP instances. We compare our techniques with approaches based on SOS on a suite of automatically synthesized benchmarks. Posi-

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Practical stability analysis of sliding‐mode control with explicit computation of sampling time

Osinenko

Devadze

Streif

2019

Asian Journal of Control

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Sliding‐mode control is a wide‐spread approach to a number of practical problems. The classical stability analyses of sliding‐mode control had to face the difficulty of defining a trajectory of a system whose dynamics are discontinuous in the state variable. Different generalizations of the system trajectory were suggested, such as Filippov solutions. Another generalization is based on the sample‐and‐hold framework, where the system dynamics are in continuous time and the control is changed only at certain discrete times. The current work addresses the sample‐and‐hold stability analysis of sliding‐mode control with special attention to an explicit computation of the required controller sampling time. A computational example is provided.

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Providing a Basin of Attraction to a Target Region of Polynomial Systems by Computation of Lyapunov-Like Functions

Cited by 101 publications

References 34 publications

Revised CPA method to compute Lyapunov functions for nonlinear systems

Revised CPA method to compute Lyapunov functions for nonlinear systems

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Practical stability analysis of sliding‐mode control with explicit computation of sampling time

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