Estimating the domain of attraction based on the invariance principle

Han, Dong; El-Guindy, Ahmed; Althoff, Matthias

doi:10.1109/cdc.2016.7799125

Cited by 15 publications

(11 citation statements)

References 32 publications

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“…s are radially unbounded and continuously differentiable functions in R n . Suppose that s 2,2,i,m,0 (x), s 2,4,i,m,0 (x), s 9,i, j,0 (x), s 10,i, ,0 (x) ∈ ∑ and s 7,i, j,0 (x), s 8,i, ,0 (x) ∈ R(x) satisfy the constraints (13) and (14). Moreover, If there exist radially unbounded and continuously differentiable functions V k + 1,i (x)s (i ∈ Γ) satisfying the constraints (15)- (19), then V k + 1,i (x) satisfies constraints (8)- (12) and…”

Section: Computing Larger Inner Estimates Of Domains Of Attractionmentioning

confidence: 99%

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Zheng

She

et al. 2018

Intl J Robust & Nonlinear

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Summary Domain of attraction plays an important role in stability analysis and safety verification of nonlinear control systems. In this paper, based on the concept of multiple Lyapunov‐like functions, we propose iteration algorithms for computing inner estimates of domains of attraction for a class of switched hybrid systems, where the state space is composed of several regions and each region is described by polyhedral sets. Starting with an initial inner estimate of domain of attraction, we firstly present a theoretical framework for obtaining a larger inner estimate by iteratively computing multiple Lyapunov‐like functions. Successively, the theoretical framework is underapproximatively realized by using S‐procedure and sums of squares programming, associated with the coordinatewise iteration method. Afterwards, for obtaining a required initial inner estimate of domain of attraction, we propose an alternative higher‐order truncation and linear semidefinite programming based method for computing a common Lyapunov function. Especially, a bisection method based improvement is proposed for obtaining better estimates in each iteration step. Finally, we implement proposed algorithms and test them on numerical examples with comparisons. These computation and comparison results show that the advantages of our multiple Lyapunov‐like functions based algorithm. Especially, we provide alternative underapproximations for avoiding the possible numerical problem.

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Section: Computing Larger Inner Estimates Of Domains Of Attractionmentioning

confidence: 99%

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Zheng

She

et al. 2018

Intl J Robust & Nonlinear

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“…Among the various DoA approximation methods proposed in the literature, methods using the subset of Lyapunov-like functions, such as quadratic Lyapunov functions [20] and rational polynomial Lyapunov functions [3], are proved to be effective [12]. Further improvements on the Lyapunov sublevel set based methods are developed in [6], [21], [5] to reduce the conservativeness with invariant sets. In this paper, the set invariance property is established with barrier certificates, which are allowed to take arbitrary shapes rather than the sublevel set of the Lypapunov function.…”

Section: Introductionmentioning

confidence: 99%

Permissive Barrier Certificates for Safe Stabilization Using Sum-of-squares

Wang

Han

Egerstedt

2018

2018 Annual American Control Conference (ACC)

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Motivated by the need to simultaneously guarantee safety and stability of safety-critical dynamical systems, we construct permissive barrier certificates in this paper that explicitly maximize the region where the system can be stabilized without violating safety constraints. An optimization strategy is developed to search for the maximum volume barrier certified region of safe stabilization. The barrier certified region, which is allowed to take any arbitrary shape, is proved to be strictly larger than safe regions generated with Lyapunov sublevel set based methods. The proposed approach effectively unites a Lyapunov function with multiple barrier functions that might not be compatible with each other. Iterative search algorithms are developed using sum-of-squares to compute the most permissive, that is, the maximum volume, barrier certificates. Simulation results of the iterative search algorithm demonstrate the effectiveness of the proposed method.

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“…In Han et al [3], a new method for estimation the domain of attraction was shown by using non-strict Lyapunov functions. In Bretas and Alberto [4], an extension of invariance principle with non-strict Lyapunov functions was provided for power systems with transmission losses.…”

Section: Introductionmentioning

confidence: 99%

Time-Varying Lyapunov Function for Mechanical Systems

Zhang

Jia

2019

JRNAL

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In this paper, a general method for constructing time-varying Lyapunov functions is provided for mechanical systems. A new class of time-varying Lyapunov functions is provided and sufficient conditions of uniform asymptotical stability are established. Different from the existing results on this subject, we remove the periodic restrictions and persistency of excitation restrictions in our work. Moreover, the results are extended into the robustness cases with unmodeled dynamics. Based on the Lyapunov methods, it is shown that uniform asymptotical stability can be obtained by using the control input provided in our work.

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Estimating the domain of attraction based on the invariance principle

Cited by 15 publications

References 32 publications

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Permissive Barrier Certificates for Safe Stabilization Using Sum-of-squares

Time-Varying Lyapunov Function for Mechanical Systems

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