Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

Meng, Fanwei; Wang, Dini; Yang, Penghui; Xie, Guanzhou; Guo, Feng

doi:10.1109/access.2020.2966566

Cited by 9 publications

(3 citation statements)

References 34 publications

(63 reference statements)

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“…Considering that the loss of control of aircraft is becoming a major cause of catastrophic accidents in aviation, scholars have carried out a lot of research work on this problem, which involves the accurate determination of dynamic boundaries in complex situations (upset conditions). By analyzing the current state of research at home and abroad, the following methods are commonly used to determine the dynamical boundaries: Bifurcation mutation theory [21], Phase plane method [22], Reachable Set estimation method [23][24][25], Stable region estimation method based on differential manifold [26], Region of Attraction (ROA) estimation method based on the sum of squares (SOS) [27], Neural Network (NN) method [28], etc. Among them, Reachable Set is favored by many scholars for its wide application and outstanding performance, and is applied in the determination of flight safety boundaries.…”

Section: Methods Of Determining Flight Safety Boundariesmentioning

confidence: 99%

Research on the Determination Method of Aircraft Flight Safety Boundaries Based on Adaptive Control

2022

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Icing is one of the main external environmental factors causing loss of control (LOC) in aircraft. To ensure safe flying in icy conditions, modern large aircraft are all fitted with anti-icing systems. Although aircraft anti-icing technology is becoming more sophisticated as research continues to expand and deepen, the scope of protection provided by anti-icing systems based on existing anti-icing technology is still relatively limited, and in practice, it is difficult to avoid flying with ice even when the anti-icing system is switched on. Therefore, it is necessary to consider providing additional safety strategies in addition to the anti-icing system, i.e., to consider icing safety from the aerodynamic, stability, and control points of view during the aircraft design phase, and to build a complete ice-tolerant protection system combining aerodynamic design methods, flight control strategies and implementation equipment. Based on the modern control theory of adaptive control, this paper presents a new method of envelope protection in icing situations based on a case study of icing, which has the advantages of strong real-time performance and good robustness, and has high engineering application value.

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Section: Methods Of Determining Flight Safety Boundariesmentioning

confidence: 99%

Research on the Determination Method of Aircraft Flight Safety Boundaries Based on Adaptive Control

2022

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“…Intuitively, this means that if the dynamical system starts inside G, it will converge to the goal state, x g , as time t goes to infinity. Sum-of-squares techniques are the basis for a large number of algorithms estimating regions of attractions for non-linear systems [30].…”

Section: Region Of Attractionmentioning

confidence: 99%

Robust Entry Vehicle Guidance with Sampling-Based Invariant Funnels

Derollez¹,

Cleac’h²,

Manchester³

2020

Preprint

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Managing uncertainty is a fundamental and critical issue in spacecraft entry guidance. This paper presents a novel approach for uncertainty propagation during entry, descent and landing that relies on a new sum-of-squares robust verification technique. Unlike risk-based and probabilistic approaches, our technique does not rely on any probabilistic assumptions. It uses a set-based description to bound uncertainties and disturbances like vehicle and atmospheric parameters and winds. The approach leverages a recently developed sampling-based version of sum-of-squares programming to compute regions of finite time invariance, commonly referred to as "invariant funnels". We apply this approach to a three-degree-of-freedom entry vehicle model and test it using a Mars Science Laboratory reference trajectory. We compute tight approximations of robust invariant funnels that are guaranteed to reach a goal region with increased landing accuracy while respecting realistic thermal constraints.

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“…Importantly, by finding the positively invariant set where the limit cycle lies, and by requiring conditions to hold only in this set thereby easing the computation, we are able to prove the stability of limit cycles for systems of higher dimension. Indeed, SOS programmes have recently been used to establish region of attractions but have been demonstrated using a system of dimension two only [20]. On the other hand, the method presented recently in [29] does offer means to obtain excellent approximation for attractors of higher dimensions using semidefinite programming; hence, we think of the approach presented in this paper as an alternative.…”

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confidence: 99%

Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions

August¹,

Barahona²

2023

JCD

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<p style='text-indent:20px;'>The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by using semidefinite programming to solve <i>sum-of-squares (SOS)</i> programmes. With the emergence of SOS programmes, it is possible to efficiently search for Lyapunov functions that guarantee stability of polynomial systems. Yet, SOS computations often fail to find functions, such that the conditions hold in the entire state space. We show here that restricting the SOS optimisation to specific domains enables us to obtain positively invariant sets, thus facilitating the analysis of the dynamics by considering separately each positively invariant set. In addition, we go beyond classical Lyapunov stability analysis and use SOS decompositions to computationally implement sufficient positivity conditions that guarantee existence, uniqueness, and exponential stability of a limit cycle. Importantly, this approach is applicable to systems of any dimension and, thus, goes beyond classical methods that are restricted to two-dimensional phase space. We illustrate our different results with applications to classical systems, such as the van der Pol oscillator, the Fitzhugh-Nagumo neuronal equation, and the Lorenz system.</p>

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Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

Cited by 9 publications

References 34 publications

Research on the Determination Method of Aircraft Flight Safety Boundaries Based on Adaptive Control

Research on the Determination Method of Aircraft Flight Safety Boundaries Based on Adaptive Control

Robust Entry Vehicle Guidance with Sampling-Based Invariant Funnels

Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions

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