Maximal Lyapunov Functions and Domains of Attraction for Autonomous Nonlinear Systems

Vannelli, A.; Vidyasagar, M.

doi:10.1016/s1474-6670(17)63470-0

Cited by 40 publications

(61 citation statements)

References 12 publications

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“…The approach is motivated by the following result from Vannelli and Vidyasagar (Page 72, Lemma 3) [64]. A proof of this theorem is also included for the sake of completeness.…”

Section: Choose a Form V (X C)mentioning

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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“…The approach is motivated by the following result from Vannelli and Vidyasagar (Page 72, Lemma 3) [64]. A proof of this theorem is also included for the sake of completeness.…”

Section: Choose a Form V (X C)mentioning

confidence: 99%

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Sassi

Sankaranarayanan

Chen

et al. 2015

IMA J Math Control Info

View full text Add to dashboard Cite

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“…There has been significant research devoted to estimating invariant subsets of the ROA (Parrilo, 2000;Vannelli and Vidyasagar, 1985;Tibken and Fan, 2006;Tibken, 2000;Hauser and Lai, 1992;Hachicho and Tibken, 2002;Genesio et al, 1985;Davison and Kurak, 1971;Chiang and Thorp, 1989). The approach taken in this paper is to restrict the search to ellipsoidal approximations of the ROA.…”

Section: Region Of Attraction Estimationmentioning

confidence: 99%

“…Recently, significant research has been performed on the development of nonlinear analysis tools for computing regions of attraction, reachability sets, input-output gains, and robustness with respect to uncertainty for nonlinear polynomial systems (Tan, 2006;Tan et al, 2008;Topcu et al, 2007Topcu et al, , 2008Chiang and Thorp, 1989;Davison and Kurak, 1971;Genesio et al, 1985;Tibken, 2000;Tibken and Fan, 2006;Vannelli and Vidyasagar, 1985;Parrilo, 2000). These tools make use of polynomial sum-of-squares optimization (Parrilo, 2000).…”

Section: Introductionmentioning

confidence: 99%

Nonlinear region of attraction analysis for flight control verification and validation

Chakraborty

Seiler

Balas

2011

Control Engineering Practice

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Current practice for flight control validation relies heavily on linear analyses and nonlinear, high-fidelity simulations. This process would be enhanced by the addition of nonlinear analyses of the flight control system. This paper demonstrates the use of region of attraction estimation for studying nonlinear effects. A nonlinear polynomial model is constructed for the longitudinal dynamics of NASA's Generic Transport Model aircraft. A polynomial model for the short period dynamics is obtained by decoupling this mode from the nonlinear longitudinal model. Polynomial optimization techniques are applied to estimate region of attractions around trim conditions. CONTROL ENGINEERING PRACTICEPAPER CHECKLIST for the preparation of papers.Please ensure that your paper conforms to the following guidelines. PREFERRED FORMAT AND LAYOUT:

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“…(E 1 )-(E 3 ) are from (Chesi et al, 2005), (E 4 ) and (E 7 ) are from (Vannelli and Vidyasagar, 1985), and (E 5 ) and (E 6 ) are from (Hauser and Lai, 1992) and (Hachicho and Tibken, 2002), respectively. Since the dynamics in (E 1 )-(E 7 ) have no physical meaning and there is no p given, we applied SimLF G algorithm sequentially: Apply SimLF G algorithm with p(x) = x T x and N V = 1 for ∂(V ) = 2.…”

Section: Examples From the Literaturementioning

confidence: 99%

“…Therefore, researchers have focused on determining invariant subsets of the ROA. Among all other methods those based on Lyapunov functions are dominant in the literature (Davison and Kurak, 1971;Genesio et al, 1985;Vannelli and Vidyasagar, 1985;Chiang and Thorp, 1989;Chesi et al, 2005;Papachristodoulou, 2005;Tan and Packard, 2006;Hachicho and Tibken, 2002;Tibken and Fan, 2006;Tibken, 2000). These methods compute a Lyapunov function as a local stability certificate and sublevel sets of this Lyapunov function, in which the function decreases along the flow, provide invariant subsets of the ROA.…”

Section: Introductionmentioning

confidence: 99%

Local stability analysis using simulations and sum-of-squares programming

2008

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Maximal Lyapunov Functions and Domains of Attraction for Autonomous Nonlinear Systems

Cited by 40 publications

References 12 publications

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Nonlinear region of attraction analysis for flight control verification and validation

Local stability analysis using simulations and sum-of-squares programming

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