A semi-algebraic approach for asymptotic stability analysis

She, Zhikun; Xia, Bican; Xiao, Rong

doi:10.1016/j.nahs.2009.04.010

Cited by 31 publications

(21 citation statements)

References 12 publications

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“…A related set of approaches directly relax the positivity of the Lyapunov form and the negativity of its derivative using Linear Matrix Inequalities (LMIs) [7-9, 27, 62]. Algebraic methods based, for example, on Gröbner basis [17], or on constructive semi-algebraic systems techniques have been explored [54,55].…”

Section: Related Workmentioning

confidence: 99%

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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Section: Related Workmentioning

confidence: 99%

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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“…To make condition 3 hold, that is, z T Q ijr z ≥ 0 ∀z ∈ R n−1 , due to the Descartes rule [47], it is equivalent to finding p i and p j such…”

Section: Computing Multiple Lyapunov Functionsmentioning

confidence: 99%

“…Note that each conjunction ∧ t h s,t (p)Δ0 can be easily transformed to a constant SAS, where p is a variable. For convenience, in our algorithm and implementation, we use SASsolver [47], which is an adaptive CAD based constant SAS solver, to compute a sample point in the corresponding constant SAS.…”

Section: Real Root Classification and Our Adaptive Cadmentioning

confidence: 99%

Discovering Multiple Lyapunov Functions for Switched Hybrid Systems

She¹,

Bai²

2014

SIAM J. Control Optim.

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In this paper we analyze local asymptotic stability of switched hybrid systems, whose subsystems have polynomial vector fields, by discovering multiple Lyapunov functions in quadratic forms. We start with an algebraizable sufficient condition for the existence of quadratic multiple Lyapunov functions. Then, since different discrete modes are considered, we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step, arriving at a set of semialgebraic sets which only involve the coefficients of the preassumed multiple Lyapunov function. Note that for each step, we additionally use the information on the different discrete modes to optimize our intermediate computation results. Afterward, we compute a sample point in the corresponding semialgebraic set for the coefficients, resulting in a multiple Lyapunov function. Finally, we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier elimination based method and the sum of squares based method. These computation and comparison results show the applicability and promise of our approach. Furthermore, our approach is extendable for piecewise affine systems and nonpolynomial switched systems by discovering multiple Lyapunov functions beyond quadratic forms.

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“…Let h(λ) = λ n + cn−1( pi, pj, Gi,j,r)λ n−1 + · · ·+c0( pi, pj, Gi,j,r) be the characteristic polynomial of Qijr. To make Vi,j( z, pi, pj, Gi,j,r) = z T Qijr z ≥ 0 for all z ∈ R n−1 , due to the Descartes rule [24], it is equivalent to finding pi and pj such that Λ3,i,j,r( pi, pj ) holds, where…”

Section: Real Root Classification Based Approach For Computing Multi-mentioning

confidence: 99%

Algebraic analysis on asymptotic stability of switched hybrid systems

She

Bai

2012

Proceedings of the 15th ACM International Conference on Hybrid Systems: Computation and Control

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In this paper we propose a mechanisable approach for discovering multiple Lyapunov functions for switched hybrid systems. We start with the classical definition on asymptotic stability, which can be assured by the existence of multiple Lyapunov functions. Then, we derive an algebraizable sufficient condition on multiple Lyapunov functions in quadratic form for asymptotic stability analysis. Since different modes are considered, in addition to real root classification, we further apply a projection operator step by step to under-approximate this sufficient condition and obtain a set of semi-algebraic sets which only involve the coefficients of the multiple Lyapunov function. Moreover, for each step, we use the information on modes to optimize our intermediate computation results. Finally, we compute a sample point in the resulting semi-algebraic sets for coefficients. We tested our approach on five examples using prototypical implementation. The computation and comparison results demonstrate the applicability and efficiency of our approach.

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A semi-algebraic approach for asymptotic stability analysis

Cited by 31 publications

References 12 publications

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Discovering Multiple Lyapunov Functions for Switched Hybrid Systems

Algebraic analysis on asymptotic stability of switched hybrid systems

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