A new convergence analysis for the Volterra series representation of nonlinear systems

Zhu, Yunpeng; Lang, Zi–Qiang

doi:10.1016/j.automatica.2019.108599

Cited by 16 publications

(10 citation statements)

References 31 publications

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“…Four amplitudes are tested, chosen such that ∥x 1 ∥ X /ρ ∈ {0.8; 1; 1.2; 2}. The truncated Volterra expansion (8-9) that solves the nonlinear beam model (30)(31) provides a finite cascade of linear interconnected systems, with a controlled approximation error. However, these linear systems are infinite dimensional PDEs, so that their real-time simulation may still be an issue.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…, the convergence is guaranteed. The signals appear to be well approximated as soon as M ≥ 3 (slight errors on the linear approximations are visible in the zoomed figures on periods [20,22] and [30,32]). As expected, signals built with N=16 modes (right column) are sharper (with a richer spectral content) than with N=2 modes (left column).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Consider the trajectory x of the nonlinear beam model (30) with initial condition x ini and input u respectively in X and U. Then, we define the pseudomodal truncation of x on X as the solution x of problem…”

Section: Projection Of the Nonlinear Termsmentioning

confidence: 99%

“…One important issue raised by this approach in practical applications is to find a bound on the system's input magnitude for which the series representation is convergent, and on the series truncation error. This is still an active domain of research as evidenced in the recent paper [30] and references therein (see also [2,25,20] for frequency domain criteria). This paper relies on computable convergence results valid for the class of bounded input signals, first established for linear-analytic ODE systems in [14], and improved in [15] where tighter convergence bounds and criterion are provided for both ODEs and PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

Hélie

Laroche

2021

Nonlinear Dyn

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This paper addresses the model reduction and the simulation of a damped Euler -Bernoulli -von Kármán pinned beam excited by a distributed force. This nonlinear problem is formulated as a PDE and reformulated as a well-posed state-space system. The model order reduction and simulation are derived by combining two approaches: a Volterra series expansion and truncation; a pseudo-modal truncation defined from the eigenbasis of the linearized problem. The interest of this approach lies in the large class of input waveshapes that can be considered and in the simplicity of the simulation structure. This structure only involves cascades of finite dimensional decoupled linear systems and multilinear functions. Closed-form bounds depending on the model coefficients and the truncation orders are provided for the Volterra convergence domain and the approximation error. These theoretical results are generalized to a large class of nonlinear models and refinement of bounds are also proposed for a large subclass. Numerical experiments confirm that the beam model is well approximated by the very first Volterra terms inside the convergence domain.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Projection Of the Nonlinear Termsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

Hélie

Laroche

2021

Nonlinear Dyn

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“…This question is still difficult to answer [28] . The latest related research is due to Zhu and Lang [29] , who have deduced a new criterion to judge the convergence of the Volterra series of nonlinear systems. In this paper, the problem of convergence is not studied.…”

Section: Psd Analysis Of Nonlinear Random Vibration Based On Volterra Series 221 Basic Theory Of Psd Analysismentioning

confidence: 99%

Power spectral density analysis for nonlinear systems based on Volterra series

Zhao

2021

Appl. Math. Mech.-Engl. Ed.

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A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.

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Beneficial effects of antisymmetric nonlinear damping with application to energy harvesting and vibration isolation under general inputs

Zhu

Lang

2022

Nonlinear Dyn

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Beneficial effects of nonlinear damping on energy harvesting and vibration isolation under harmonic inputs have been investigated showing that the introduction of nonlinear damping can increase the harvested energy and reduce the vibration over both the resonant and higher frequency ranges. However, the scenario becomes more complicated when the loading inputs are of more general form such as multi-tone and random inputs, which can produce system responses that are induced by an interaction of system input components of different frequencies. In the present study, by introducing the concept of power transmissibility, the study of the beneficial effects of nonlinear damping is extended to the systems subject to general inputs including both multi-tone and random inputs. A rigorous analysis is conducted based on single degree of freedom systems subject to general inputs. The analysis reveals the conditions under which the antisymmetric nonlinear damping is beneficial for improving energy harvester performance and reducing of the power of system output in vibration isolation. Moreover, the beneficial effects are demonstrated by two case studies.

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A new convergence analysis for the Volterra series representation of nonlinear systems

Cited by 16 publications

References 31 publications

Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

Power spectral density analysis for nonlinear systems based on Volterra series

Beneficial effects of antisymmetric nonlinear damping with application to energy harvesting and vibration isolation under general inputs

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