Reproducing kernel Hilbert space embedding for adaptive estimation of nonlinearities in piezoelectric systems

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 data-driven 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.

Appendices

Piezoelectric oscillator—governing equations

In this section, we go over the detailed steps involved in the derivation of the infinite-dimensional governing equation of the piezoelectric oscillator as shown in Fig. 1. The kinetic energy and the electric potential are given by Eqs. 3 and 5, respectively. Using Hamilton’s principle, we get the variational identity

$$\begin{aligned}&\delta \int _{t_0}^{t_1} (T - {\mathcal {V}}_{\mathcal {H}}) \mathrm{d}t \nonumber \\&\quad = \delta \int _{t_0}^{t_1} \left\{ \left[ \frac{1}{2} \mathrm {m} \int _0^l (\dot{w} + \dot{\mathrm {z}})^2 \mathrm{d}x \right] \right. \nonumber \\&\qquad - \left[ \frac{1}{2} C_b I_b \int _0^l (w'')^2 \mathrm{d}x + 2 a_{(0,2)} \int _{a}^{b} (w'')^2 \mathrm{d}x \right. \nonumber \\&\qquad + 2 a_{(2,4)} \int _{a}^{b} (w'')^4 \mathrm{d}x + 2 b_{(1,1)} \left[ \int _{a}^{b} w'' \mathrm{d}x \right] E_z \nonumber \\&\qquad \left. \left. + 2 b_{(3,1)} \left[ \int _{a}^{b} (w'')^3 \mathrm{d}x \right] E_z - 2 b_{(0,2)} E_z^2 \right] \right\} \mathrm{d}t = 0. \end{aligned}$$

(21)

The above variational statement can be rewritten as

$$\begin{aligned}&\delta \int _{t_0}^{t_1} (T - \mathcal {V}_\mathcal {H}) \mathrm{d}t = \int _{t_0}^{t_1} \left\{ \int _0^l \mathrm {m} \dot{w} \delta \dot{w} + \mathrm {m} \dot{\mathrm {z}} \delta \dot{w} ) \mathrm{d}x \right. \nonumber \\&\quad - \,\int _0^l C_b I_b w'' \delta w'' \mathrm{d}x - 4a_{(0,2)} \int _{a}^{b} w'' \delta w'' \mathrm{d}x \nonumber \\&\quad - \,8 a_{(2,4)} \int _{a}^{b} (w'')^3 \delta w'' \mathrm{d}x - 2b_{(1,1)} \left( \int _{a}^{b} (\delta w'') \mathrm{d}x \right) E_z \nonumber \\&\quad - \,2b_{(1,1)} \left( \int _{a}^{b} w'' \mathrm{d}x \right) \delta E_z \nonumber \\&\quad - \,6b_{(3,1)} \left( \int _{a}^{b} (w'')^2 \delta w'' \mathrm{d}x \right) E_z \nonumber \\&\quad \left. -\, 2b_{(3,1)} \left( \int _{a}^{b} (w'')^3 \mathrm{d}x \right) \delta E_z \right. \nonumber \\&\quad \left. + \,4b_{(0,2)} E_z \delta E_z \right\} \mathrm{d}t = 0 \end{aligned}$$

(22)

After integrating the above statement by parts, we get

$$\begin{aligned}&\int _{t_0}^{t_1} \Bigg \{ - \int _0^l \bigg [ \mathrm {m} \ddot{w} + \mathrm {m} \ddot{\mathrm {z}} + C_b I_b w'''' \\&\quad + 4 a_{(0,2)} \left( \chi _{[a,b]} w'' \right) '' + 8 a_{(2,4)} (\chi _{[a,b]} (w'')^3)'' \\&\quad + 2 b_{(1,1)} \chi _{[a,b]}'' E_z + 6 b_{(3,1)} (\chi _{[a,b]} (w'')^2 )'' E_z \bigg ] \delta w \mathrm{d}x \\&\quad - \left[ 2b_{(1,1)} \left( \int _0^l \chi _{[a,b]} w'' \mathrm{d}x \right) \right. \\&\quad \left. + 2b_{(3,1)} \left( \int _{0}^{l} \chi _{[a,b]} (w'')^3 \mathrm{d}x \right) - 4b_{(0,2)} E_z \right] \delta E_z \\&\quad - \left\{ C_b I_b w'' + 4a_{(0,2)} \chi _{[a,b]} w'' + 8a_{(2,4)} \chi _{[a,b]} (w'')^3 \right. \\&\quad \left. + 2 b_{(1,1)} \chi _{[a,b]} E_z + 6 b_{(3,1)} \chi _{[a,b]} (w'')^2 \right\} \delta w' \big |_0^l \\&\quad + \left\{ C_b I_b w''' + 4a_{(0,2)} \left( \chi _{[a,b]} w''\right) ' \right. \\&\left. \quad + 8a_{(2,4)} \left( \chi _{[a,b]} (w'')^3 \right) ' \right. \\&\quad \left. + 2 b_{(1,1)} \chi _{[a,b]}' E_z + 6 b_{(3,1)} \right. \\&\left. \qquad \left( \chi _{[a,b]} (w'')^2 \right) ' \right\} \delta w \bigg |_0^l \Bigg \} \mathrm{d}t = 0. \end{aligned}$$

Note, in the above statement, the term $\chi _{[a,b]}$ is called the characteristic function of [a, b] and is defined as

$$\begin{aligned} \chi _{[a,b]}(x) := \left\{ \begin{array}{ll} 1 &{} \text {if } x \in [a,b], \\ 0 &{} \text {if } x \notin [a,b]. \end{array} \right. \end{aligned}$$

(23)

Since the variation of w and $E_z$ are arbitrary, we can conclude that the equations of motion of the nonlinear piezoelectric cantilevered bimorph have the form shown in Eqs. 6 and 7.

Single mode approximation of the piezoelectric oscillator equations

As mentioned earlier, the effects of nonlinearity in piezoelectric oscillators are most noticeable near the natural frequency of the system. Hence, single-mode models are sufficient to model the dynamics as long as the range of input excitation is restricted to a band around the first natural frequency. Let us introduce the single-mode approximation $w(x,t) = \psi (x) u(t)$. To make calculations easier, let us introduce this approximation into the variational statement shown in Eq. 22. Further, note that

$$\begin{aligned} \int _{t_0}^{t_1} \int _0^l \mathrm {m} \dot{w} \delta {\dot{w}} \mathrm{d}x \mathrm{d}t&= - \int _{t_0}^{t_1} \int _0^l \mathrm {m} \ddot{w} \delta w \mathrm{d}x \mathrm{d}t, \\ \int _{t_0}^{t_1} \int _0^l \mathrm {m} \dot{\mathrm {z}} \delta \dot{w} \mathrm{d}x \mathrm{d}t&= - \int _{t_0}^{t_1} \int _0^l \mathrm {m} \ddot{\mathrm {z}} \delta w \mathrm{d}x \mathrm{d}t. \end{aligned}$$

After introducing the approximation for w(x, t) into the variational statement in Eq. 22 and using the equations shown above, we get the variational statement

$$\begin{aligned} 0&= \int _{t_0}^{t_1} \Bigg \{ - \bigg [ \mathrm {m} \left( \int _0^l \psi ^2(x) \mathrm{d}x \right) \ddot{u} + \mathrm {m} \left( \int _0^l \psi (x) \mathrm{d}x \right) \ddot{\mathrm {z}} \\&\quad + C_b I_b \left( \int _0^l \left( \psi ''(x) \right) ^2 \mathrm{d}x \right) u \\&\quad + 4a_{(0,2)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^2 \right) u \\&\quad + 8a_{(2,4)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^4 \mathrm{d}x \right) u^3 \\&\quad + 2b_{(1,1)} E_z \left( \int _0^l \chi _{[a,b]} \psi ''(x) \mathrm{d}x \right) \\&\quad + 6b_{(3,1)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^3 \right) u^2 E_z \bigg ] \delta u \\&\quad - \bigg [ 2b_{(1,1)} \left( \int _0^l \chi _{[a,b]} \psi ''(x) \mathrm{d}x \right) u \\&\quad + 2b_{(3,1)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^3 \right) u^3 \\&\quad - 4b_{(0,2)} E_z \bigg ] \delta E_z \Bigg \} \mathrm{d}t \end{aligned}$$

Thus, the approximated equation of motion are

$$\begin{aligned}&\underbrace{\mathrm {m} \left( \int _0^l \psi ^2(x) \mathrm{d}x \right) }_{M} \ddot{u} + \underbrace{\mathrm {m} \left( \int _0^l \psi (x) \mathrm{d}x \right) }_{P} \ddot{\mathrm {z}} \\&\quad + \underbrace{C_b I_b \left( \int _0^l \left( \psi ''(x) \right) ^2 \mathrm{d}x \right) }_{K_b} u \\&\quad + \underbrace{4a_{(0,2)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^2 \right) }_{K_p} u \\&\quad + \underbrace{8a_{(2,4)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^4 \mathrm{d}x \right) }_{K_N} u^3 \\&\quad + \underbrace{2b_{(1,1)} \left( \int _0^l \chi _{[a,b]} \psi ''(x) \mathrm{d}x \right) }_{\mathcal {B}} E_z \\&\quad + \underbrace{6b_{(3,1)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^3 \right) }_{Q_{N}} u^2 E_z = 0, \\&\underbrace{2b_{(1,1)} \left( \int _0^l \chi _{[a,b]} \psi ''(x) \mathrm{d}x \right) }_{\mathcal {B}} u(t) \\&\quad + \underbrace{2b_{(3,1)} \left( \int _0^l \chi _{[a,b]} \left( \psi ''(x) \right) ^3 \right) }_{{\mathcal {B}}_N} u^3(t) = \underbrace{4b_{(0,2)}}_{{\mathcal {C}}} E_z. \end{aligned}$$

These calculations generate the approximated Eqs. 8 and 9.