Reproducing kernel Hilbert space embedding for adaptive estimation of nonlinearities in piezoelectric systems
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ORIGINAL PAPER
Reproducing kernel Hilbert space embedding for adaptive estimation of nonlinearities in piezoelectric systems Sai Tej Paruchuri
· Jia Guo · Andrew Kurdila
Received: 15 February 2020 / Accepted: 7 July 2020 © Springer Nature B.V. 2020
Abstract Nonlinearities in piezoelectric systems can arise from internal factors such as nonlinear constitutive laws or external factors like realizations of boundary conditions. It can be difficult or even impossible to derive detailed models from the first principles of all the sources of nonlinearity in a system. This paper introduces adaptive estimator techniques to approximate the nonlinearities that can arise in certain classes of piezoelectric systems. Here an underlying structural assumption is that the nonlinearities can be modeled as continuous functions in a reproducing kernel Hilbert space (RKHS). This approach can be viewed as a datadriven method to approximate the unknown nonlinear system. This paper introduces the theory behind the adaptive estimator, discusses precise conditions that guarantee convergence of the function estimates, and studies the effectiveness of this approach numerically for a class of nonlinear piezoelectric composite beams. Keywords Reproducing kernels · RKHS · Piezoelectric oscillators · Nonlinear oscillator · Data-driven modeling S. T. Paruchuri (B) · J. Guo · A. Kurdila Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24060, USA e-mail: [email protected] J. Guo e-mail: [email protected] A. Kurdila e-mail: [email protected]
1 Introduction Researchers have studied piezoelectric systems extensively over the past three decades for applications to classical problems like vibration attenuation, which is described in general treatises like [1–4], as well as modern problems like energy harvesting [5,6]. Even though many of these studies model piezoelectric oscillators as linear systems, piezoelectric systems are often inherently nonlinear. At low input amplitudes, the effect of nonlinearity is ordinarily not very pronounced. However, linear models can fail to capture the dynamics of piezoelectric systems that undergo large displacements, velocities, accelerations, or electric field strengths. Researchers have consequently also developed nonlinear models for many examples of piezoelectric oscillators. A general account of nonlinear field theory as it arises in modeling piezoelectric continua can be found in Maugin [7], Yang [8,9], while reference [10] gives a good account of how active nonlinear piezostructural components are incorporated in typical plate or shell models. Some of the models that are perhaps the most relevant to the system considered in this paper are [11–19]. In these studies, researchers investigate case-specific models that include higher-order polynomial terms in the constitutive laws. The models in the above publications by von Wagner and Hagedorn [11,12], von Wagner [13,14], Stanton et al. [15,16], Wolf and Gottlieb [17], Usher and Sim [18], Triplett and Quinn [19]
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