Nonlinear control design of piezoelectric actuators with micro positioning capability

Abstract

Inherently, piezoelectric actuator is one of the devices equipped with the micrometer positioning capability and characterized by small size, fast response, high stiffness, and large blocking force. These advantages give piezoelectric actuator the possibility of being high-accuracy industrial machineries. However, factors of nonlinear hysteresis, modeling uncertainties, and environmental disturbances result in unacceptable positioning errors and greatly increase the control difficulties. In this paper, a hybrid nonlinear robust control design that integrates a feedback linearization control method and a robust compensator is proposed. It aims to eliminate above-mentioned impacts and tackle the micrometer (μm) positioning design of piezoelectric actuators. The feedback linearization controller is developed for converging of positioning errors exponentially. The robust compensator is used to mitigate the total impact of hysteresis, modeling uncertainties and environmental disturbances and carry out the fine tuning of positioning errors to zero. Simulation results and practical tests reveal that the controlled piezoelectric actuator reaches 1 μm positioning accuracy, and the proposed robust control law delivers promising positioning performance under impacts of hysteresis, modeling uncertainties and environmental disturbances.

Appendix

Proof of Theorem 1

The mini-max positioning performance index in Eq. (23) can be rewritten as:

$$\mathop {\hbox{min} }\limits_{{u_{r} (t) \in L_{2} [0,t_{f} ]}} \mathop {\hbox{max} }\limits_{{\tilde{w}(t) \in L_{2} [0,t_{f} ]}} \int_{0}^{{t_{f} }} {\left[ {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)Q}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \right]} dt \le 0$$

(28)

Considering the nonzero initial condition of ${\mathbf{\tilde{e}(0)}} \ne 0$, the mini-max positioning performance index (28) is modified as:

$$\mathop {\hbox{min} }\limits_{{u_{r} (t) \in L_{2} [0,t_{f} ]}} \mathop {\hbox{max} }\limits_{{\tilde{w}(t) \in L_{2} [0,t_{f} ]}} \int_{0}^{{t_{f} }} {\left[ {{\tilde{\mathbf{e}}}^{T} ({\mathbf{t)Q}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \right]} dt \le {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}}$$

(29)

For deriving the robust compensator u_r(t) and the Riccati-like equation, a cost function is defined as:

$$J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w}) = \int_{0}^{{t_{f} }} {\left[ {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)Q}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \right]} dt$$

(30)

Equation (30) is calculated as:

$$\begin{aligned} J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w}) & = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \\ & \quad + \int_{0}^{{t_{f} }} {\left[ {\begin{array}{*{20}l} {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)Q}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \hfill \\ { + \frac{d}{dt}\left( {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \right)} \hfill \\ \end{array} } \right]} dt \\ & = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \\ & \quad + \int_{0}^{{t_{f} }} {\left[ {\begin{array}{*{20}l} {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)Q}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \hfill \\ { + {\mathbf{\dot{\tilde{e}}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + {\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{\dot{\tilde{e}}(t)}}} \hfill \\ \end{array} } \right]} dt \\ \end{aligned}$$

(31)

Substitute the positioning error dynamics ${\mathbf{\dot{\tilde{e}}(t)}} = {\mathbf{A}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + {\mathbf{B}}_{{\mathbf{r}}} u_{r} (t) + {\mathbf{B}}_{{\mathbf{r}}} \tilde{w}(t)$ into Eq. (31), then we have

$$\begin{aligned} J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w}) & = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \\ & \quad + \int_{0}^{{t_{f} }} {\left[ {\begin{array}{*{20}l} {{\tilde{\mathbf{e}}}^{T} (t)\left[ {{\mathbf{A}}_{r}^{T} {\mathbf{P}}_{{\mathbf{r}}} + {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{A}}_{{\mathbf{r}}} + {\mathbf{Q}}_{{\mathbf{r}}} } \right]{\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \hfill \\ { + u_{r}^{T} (t){\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + {\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} u_{r} (t) + {\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} \tilde{w}(t) + \tilde{w}^{T} (t){\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \hfill \\ \end{array} } \right]} dt \\ \end{aligned}$$

(32)

Reorganize Eq. (32), then

$$\begin{aligned} J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w}) & = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \\ & \quad + \int_{0}^{{t_{f} }} {\left[ {\begin{array}{*{20}l} {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)}}\left[ {{\mathbf{A}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} + {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{A}}_{{\mathbf{r}}} + {\mathbf{Q}}_{{\mathbf{r}}} - {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} \left[ {R_{r}^{ - 1} - \frac{1}{{\rho^{2} }}I} \right]{\mathbf{B}}_{{\mathbf{r}}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{r}}} } \right]\tilde{e}(t)} \hfill \\ { + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t) + u_{r}^{T} (t){\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + \tilde{w}^{T} (t){\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \hfill \\ \end{array} } \right]} dt \\ & = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \\ & \quad + \int_{0}^{{t_{f} }} {\left[ {\begin{array}{*{20}l} {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)}}\left[ {{\mathbf{A}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} + {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{A}}_{{\mathbf{r}}} + {\mathbf{Q}}_{{\mathbf{r}}} - {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} \left[ {R_{r}^{ - 1} - \frac{1}{{\rho^{2} }}I} \right]{\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} } \right]{\mathbf{\tilde{e}(t)}}} \hfill \\ { + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t) + u_{r}^{T} (t){\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + \tilde{w}^{T} (t){\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \hfill \\ { + {\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} u_{r} (t) + {\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} \tilde{w}(t) + {\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} R_{r}^{ - 1} {\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} - \frac{1}{{\rho^{2} }}{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \hfill \\ \end{array} } \right]} dt \\ \end{aligned}$$

(33)

Completing the squares, the following result can be obtained

$$\begin{aligned} J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w}) & = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \\ & \quad + \int_{0}^{{t_{f} }} {\left[ {\begin{array}{*{20}l} {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)}}\left[ {{\mathbf{A}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} + {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{A}}_{{\mathbf{r}}} + {\mathbf{Q}}_{{\mathbf{r}}} - {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} \left[ {R_{r}^{ - 1} - \frac{1}{{\rho^{2} }}I} \right]{\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} } \right]{\mathbf{\tilde{e}(t)}}} \hfill \\ {{ + }\left( {R_{r} u_{r} (t)} \right. + {\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} \left. {{\mathbf{\tilde{e}(t)}}} \right)^{T} R_{r}^{ - 1} \left( R \right._{r} u_{r} (t) + {\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} \left. {{\mathbf{\tilde{e}(t)}}} \right)} \hfill \\ { - \left( {\rho \tilde{w}(t) - \frac{1}{\rho }{\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \right)^{T} \cdot \left( {\rho \tilde{w}(t) - \frac{1}{\rho }{\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}} \right)} \hfill \\ \end{array} } \right]} dt \\ \end{aligned}$$

(34)

Choosing the robust compensator $u_{r} (t) = - R_{r}^{ - 1} {\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}$ for Eq. (34), the Riccati like equation ${\mathbf{A}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} + {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{A}}_{{\mathbf{r}}} + {\mathbf{Q}}_{{\mathbf{r}}} - {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{B}}_{{\mathbf{r}}} \left[ {R_{r}^{ - 1} - \frac{1}{{\rho^{2} }}I} \right]{\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} = 0$, and the worst disturbance $\tilde{w}(t) = \frac{1}{{\rho^{2} }}{\mathbf{B}}_{{\mathbf{r}}}^{T} {\mathbf{P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}}$, Eq. (34) can be further expressed as

$$\mathop {\hbox{min} }\limits_{{u_{r} (t)}} \mathop {\hbox{max} }\limits_{{\tilde{w}(t)}} J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w}) = {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}} - {\tilde{\mathbf{e}}}^{T} {\mathbf{(t}}_{{\mathbf{f}}} {\mathbf{)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t}}_{{\mathbf{f}}} {\mathbf{)}} \le {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}}$$

(35)

According the definition of the cost function $J({\tilde{\mathbf{e}}},u_{r} ,\tilde{w})$, we have

$$\mathop {\hbox{min} }\limits_{{u_{r} (t) \in L_{2} [0,t_{f} ]}} \mathop {\hbox{max} }\limits_{{\tilde{w}(t) \in L_{2} [0,t_{f} ]}} \int_{0}^{{t_{f} }} {\left[ {{\tilde{\mathbf{e}}}^{T} {\mathbf{(t)Q}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(t)}} + u_{r}^{T} (t)R_{r} u_{r} (t) - \rho^{2} \tilde{w}^{T} (t)\tilde{w}(t)} \right]} dt \le {\tilde{\mathbf{e}}}^{T} {\mathbf{(0)P}}_{{\mathbf{r}}} {\mathbf{\tilde{e}(0)}}$$

(36)

From Eq. (36), if the initial condition is ${\tilde{\mathbf{e}}}\left( 0 \right) = 0$, the mini-max positioning performance in Eq. (23) can be proven.□