An incremental Hammerstein-like modeling approach for the decoupled creep, vibration and hysteresis dynamics of piezoelectric actuator

Abstract

The modeling of the piezoelectric actuator is very important for the fast and accurate nano-positioning control. However, the complex creep, vibration and hysteresis dynamics make the modeling very difficult. Because there are often model uncertainties in the first-principle model, the model identification from the experimental data is very necessary. Most identification approaches only consider partial dynamics of the creep, vibration and hysteresis. Though they can be combined together after they are individually identified from different experiments, it is difficult to design the completely decoupled experiment. In this study, an incremental Hammerstein-like modeling approach is proposed to identify the creep, vibration and hysteresis dynamics from one experiment. The model is called nonlinear–linear–linear-Hammerstein-like model, where one dynamic nonlinear system is used to model the hysteresis and two dynamic linear systems are used to model the creep and vibration. The creep and vibration are assumed to be decoupled, and there is a known upper bound on the order of the creep model. A two-stage incremental modeling approach is proposed to reduce the modeling complexity, where the slow dynamics of the hysteresis and creep are estimated first and then the residual of this model is used to estimate the fast dynamics of the vibration. In each stage, the model structure and order are determined by a locally regularized orthogonal least-squares-based model term selection algorithm and the parameters are estimated using a regularized least-squares method. The effectiveness of the proposed modeling approach is verified by the simulations and experiments on typical piezoelectric actuators.

Appendix

Theorem 1

[56] Let $\hat{{\Theta }}\in R^{n_{bcs} \times n_{fs} }$ have rank $\rho \ge 1$, and let the economy-size SVD of $\hat{{\Theta }}$ be given by

$$\begin{aligned} \hat{{\Theta }}=\mathbf{U}_\rho \varvec{\Sigma }_\rho \mathbf{V}_\rho ^\mathrm{T} =\sum _{i=1}^\rho {\sigma _i \varvec{\mu }_i } \varvec{\upsilon }_i^\mathrm{T} , \end{aligned}$$

(92)

where the singular matrix $\varvec{\Sigma }_\rho =\hbox {diag}\{\sigma _i \}$ such that

$$\begin{aligned} \sigma _1 \ge \cdots \ge \sigma _\rho >0, \end{aligned}$$

and where the matrices $\mathbf{U}_\rho =[\varvec{\mu }_1 ,\ldots ,\varvec{\mu }_\rho ]\in R^{n_{bcs} \times \rho }$ and $\mathbf{V}_\rho =[\varvec{\upsilon }_1 ,\ldots ,\varvec{\upsilon }_\rho ]\in R^{n_{fs} \times \rho }$ contain only the first $\rho$ columns of the unitary matrices $\mathbf{U}\in R^{n_{bcs} \times n_{bcs} }$ and $\mathbf{V}\in R^{n_{fs} \times n_{fs} }$ provided by the full SVD of $\hat{{\Theta }}$,

$$\begin{aligned} \hat{{\Theta }}=\mathbf{U}{\varvec{\Sigma }}\mathbf{V}^\mathrm{T} , \end{aligned}$$

(93)

respectively. Then the vectors $\mathbf{b}_{cs} \in R^{n_{bcs} \times 1}$ and $\varvec{\uptheta }_{fs} \in R^{n_{fs} \times 1}$ that minimize the norm $||\hat{{\Theta }}-\mathbf{b}_{cs} \varvec{\uptheta }_{fs}^\mathrm{T} ||_F^2$, are given by [56]

$$\begin{aligned} (\hat{\mathbf{b}}_{cs} ,\hat{{\varvec{\uptheta }}}_{fs} )=\mathop {\arg \;\min }\limits _{\mathbf{b}_{cs} ,\varvec{\uptheta }_{fs} } \{||\hat{{\Theta }}-\mathbf{b}_{cs} \varvec{\uptheta }_{fs}^\mathrm{T} ||_F^2 \}=(\mathbf{U}_1 ,\mathbf{V}_1 {\varvec{\Sigma }}_1 ),\nonumber \\ \end{aligned}$$

(94)

where $\mathbf{U}_1 \in R^{n_{bcs} \times 1}$, $\mathbf{V}_1 \in R^{n_{fs} \times 1}$ and ${\varvec{\Sigma }}_1 =\hbox {diag}\{\sigma _1 \}$ are given by the following partition of the economy-size SVD in (92)

$$\begin{aligned} \hat{{\Theta }}=\left[ {{\begin{array}{ll} {\mathbf{U}_1 }&{}\quad {\mathbf{U}_2 } \\ \end{array} }} \right] \left[ {{\begin{array}{ll} {\varvec{\Sigma }_1 }&{}\quad 0 \\ 0&{} {\varvec{\Sigma }_2 } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\mathbf{V}_1^\mathrm{T} } \\ {\mathbf{V}_2^\mathrm{T} } \\ \end{array} }} \right] , \end{aligned}$$

(95)

and the approximation error is given by

$$\begin{aligned} ||\hat{{\Theta }}-\hat{\mathbf{b}}_{cs} \hat{{\varvec{\uptheta }}}_{fs}^\mathrm{T} ||_F^2 =\sum _{i=2}^\rho {\sigma _i^2 }. \end{aligned}$$

(96)