Amplitude control for sensorless self-excited oscillation of cantilever based on a piezoelectric device

Abstract

In this paper, we propose a sensorless amplitude control method for the self-excited oscillation by sensorless nonlinear feedback. Artificial self-excited oscillation by the feedback control is effective for vibrational microsensors because it can eliminate the viscous damping effect in measurement environments and realize the response with the linear eigenmode for the high sensitivity and accuracy of the measurements based on the shift of the linear natural frequency and mode such as AFM (atomic force microscopy). A sensorless feedback method was reported to produce the self-excited oscillation by using the interaction between the mechanical dynamics of the cantilever and the electrical dynamics of the piezoelectric device attached to the cantilever instead of the detection of the oscillation displacement with a sensor. However, there are no propositions of sensorless amplitude control. In the present study, we propose a sensorless nonlinear feedback to achieve the steady-state amplitude of the self-excited oscillation. The nonlinear feedback gain is designed by the theoretical analysis with the aid of the center manifold and the normal form theory. The efficiency is demonstrated through experiments using a macro-scale cantilever with a bimorph-type piezoelectric device. The results show the proposed nonlinear control can settle the oscillation amplitude to be finite in the steady state and change the magnitude depending on the nonlinear feedback gain.

Appendices

A Derivation of nonlinear function N

The nonlinear functions $\left[ N_1, ~N_2, ~N_3, ~N_4 \right] ^\mathrm{T}$ in Sect. 2.2 can be derived as

$$\begin{aligned} \begin{bmatrix} N_1\left( {\varvec{y}}, \epsilon \right) \\ N_2\left( {\varvec{y}}, \epsilon \right) \\ N_3\left( {\varvec{y}}, \epsilon \right) \\ N_4\left( {\varvec{y}}, \epsilon \right) \end{bmatrix} = P^{-1}{\varvec{C}}_\mathrm{nl}\left( P{\varvec{y}} \right) , \end{aligned}$$

(25)

where their elements are:

$$\begin{aligned} N_{1}&= C_{11000} \epsilon y_{1} + C_{10100} \epsilon y_{2} + C_{10010} \epsilon y_{3} + C_{10001} \epsilon y_{4} \nonumber \\&\quad + C_{13000} y_{1}^{3} + C_{12100} y_{1}^{2} y_{2} + \cdots \nonumber \\&\quad + C_{10012} y_{3} y_{4}^{2} + C_{10003} y_{4}^{3}, \end{aligned}$$

(26)

$$\begin{aligned} N_{2}&= C_{21000} \epsilon y_{1} + C_{20100} \epsilon y_{2} + C_{20010} \epsilon y_{3} + C_{20001} \epsilon y_{4} \nonumber \\&\quad + C_{23000} y_{1}^{3} + C_{22100} y_{1}^{2} y_{2} + \cdots \nonumber \\&\quad + C_{20012} y_{3} y_{4}^{2} + C_{20003} y_{4}^{3}, \end{aligned}$$

(27)

$$\begin{aligned} N_{3}&= C_{31000} \epsilon y_{1} + C_{30100} \epsilon y_{2} + C_{30010} \epsilon y_{3} + C_{30001} \epsilon y_{4} \nonumber \\&\quad + C_{33000} y_{1}^{3} + C_{32100} y_{1}^{2} y_{2} + \cdots \nonumber \\&\quad + C_{30012} y_{3} y_{4}^{2} + C_{30003} y_{4}^{3}, \end{aligned}$$

(28)

$$\begin{aligned} N_{4}&= C_{41000} \epsilon y_{1} + C_{40100} \epsilon y_{2} + C_{40010} \epsilon y_{3} + C_{40001} \epsilon y_{4} \nonumber \\&\quad + C_{43000} y_{1}^{3} + C_{42100} y_{1}^{2} y_{2} + \cdots \nonumber \\&\quad + C_{40012} y_{3} y_{4}^{2} + C_{40003} y_{4}^{3}. \end{aligned}$$

(29)

These elements $C_{11000}, C_{10100},\ldots , C_{40003}$ are too massive not to explicitly show in the manuscript, but they consist of the original state vector $\left[ a^{*} ~ \dot{a}^{*} ~ V_\mathrm{LPF}^{*} ~ \dot{V}_\mathrm{LPF}^{*} \right] ^\mathrm{T}$ and the dimensionless parameters defined in Eq. (8).

B Coefficients of the center manifold

In this section, we show the determination process of the coefficients $D_{310}, ~D_{301},\ldots , D_{403}$. Taking derivative with respect to the dimensionless time $t^*$ of Eq. (12) is expressed as

$$\begin{aligned} \dot{y}_{3}&= \frac{\partial y_{3}}{\partial y_{1}} \dot{y}_{1} + \frac{\partial y_{3}}{\partial y_{2}} \dot{y}_{2} \nonumber \\&= (D_{310} \epsilon + 3 D_{330} y_{1}^{2} + 2 D_{321} y_{1} y_{2} + D_{312} y_{2}^{2})\nonumber \\&\quad (\omega y_{2} + N_{1}) \nonumber \\&\quad + (D_{301} \epsilon + D_{321} y_{1}^{2} + 2 D_{312} y_{1} y_{2} + 3 D_{303} y_{2}^{2})\nonumber \\&\quad (- \omega y_{1} + N_{2}) \nonumber \\&= - \omega D_{301} \epsilon y_{1} + \omega D_{310} \epsilon y_{2} - \omega D_{321} y_{1}^{3} \nonumber \\&\quad + \omega (3 D_{330} - 2 D_{312}) y_{1}^{2} y_{2} \nonumber \\&\quad + \omega (2 D_{321} - 3 D_{303}) y_{1} y_{2}^{2} \nonumber \\&\quad + \omega D_{312} y_{2}^{3} + \mathcal {O} \left( \epsilon ^{\frac{5}{2}} \right) , \end{aligned}$$

(30)

which is also expressed in the third row of Eq. (11):

$$\begin{aligned} \dot{y}_{3}&= (\lambda _{r} D_{310} + \lambda _{i} D_{410}) \epsilon y_{1} \nonumber \\&\quad + (\lambda _{r} D_{301} + \lambda _{i} D_{401}) \epsilon y_{2} \nonumber \\&\quad + (\lambda _{r} D_{330} + \lambda _{i} D_{430}) y_{1}^{3} + (\lambda _{r} D_{321} \nonumber \\&\quad + \lambda _{i} D_{421}) y_{1}^{2} y_{2} \nonumber \\&\quad + (\lambda _{r} D_{312} + \lambda _{i} D_{412}) y_{1} y_{2}^{2} \nonumber \\&\quad + (\lambda _{r} D_{303} + \lambda _{i} D_{403}) y_{2}^{3}\nonumber \\&\quad + N_{3} + \mathcal {O} \left( \epsilon ^{\frac{5}{2}} \right) . \end{aligned}$$

(31)

Equating Eqs. (30) and (31) yields following relationships that the coefficients $D_{310}, ~D_{301},\ldots , D_{303}$ satisfy:

$$\begin{aligned}&\epsilon y_{1} : - \omega D_{301} = \lambda _{r} D_{310} + \lambda _{i} D_{410} + C_{31000}, \end{aligned}$$

(32)

$$\begin{aligned}&\epsilon y_{2} : \omega D_{310} = \lambda _{r} D_{301} + \lambda _{i} D_{401} + C_{30100}, \end{aligned}$$

(33)

$$\begin{aligned}&y_{1}^{3} : - \omega D_{321} = \lambda _{r} D_{330} + \lambda _{i} D_{430} + C_{33000}, \end{aligned}$$

(34)

$$\begin{aligned}&y_{1}^{2} y_{2} : \omega (3 D_{330} - 2 D_{312}) = \lambda _{r} D_{321} \nonumber \\&\quad + \lambda _{i} D_{421} + C_{32100}, \end{aligned}$$

(35)

$$\begin{aligned}&y_{1} y_{2}^{2} : \omega (2 D_{321} - 3 D_{303}) = \lambda _{r} D_{312} \nonumber \\&\quad + \lambda _{i} D_{412} + C_{31200}, \end{aligned}$$

(36)

$$\begin{aligned}&y_{2}^{3} : \omega D_{312} = \lambda _{r} D_{303} + \lambda _{i} D_{403} + C_{30300}. \end{aligned}$$

(37)

Similarly, equating

$$\begin{aligned} \dot{y}_{4}&= \frac{\partial y_{4}}{\partial y_{1}} \dot{y}_{1} + \frac{\partial y_{4}}{\partial y_{2}} \dot{y}_{2} \nonumber \\&= (D_{410} \epsilon + 3 D_{430} y_{1}^{2} + 2 D_{421} y_{1} y_{2} + D_{412} y_{2}^{2})\nonumber \\&\quad (\omega y_{2} + N_{1}) \nonumber \\&\quad + (D_{401} \epsilon + D_{421} y_{1}^{2} + 2 D_{412} y_{1} y_{2} + 3 D_{403} y_{2}^{2})\nonumber \\&\quad (- \omega y_{1} + N_{2}) \nonumber \\&= - \omega D_{401} \epsilon y_{1} + \omega D_{410} \epsilon y_{2} - \omega D_{421} y_{1}^{3} \nonumber \\&\quad + \omega (3 D_{430} - 2 D_{412}) y_{1}^{2} y_{2} \nonumber \\&\quad + \omega (2 D_{421} - 3 D_{403}) y_{1} y_{2}^{2} \nonumber \\&\quad + \omega D_{412} y_{2}^{3} + \mathcal {O} \left( \epsilon ^{\frac{5}{2}} \right) , \end{aligned}$$

(38)

and

$$\begin{aligned} \dot{y}_{4}&= (-\lambda _{i} D_{310} + \lambda _{r} D_{410}) \epsilon y_{1} \nonumber \\ {}&\quad + (-\lambda _{i} D_{301} + \lambda _{r} D_{401}) \epsilon y_{2} \nonumber \\&\quad + (-\lambda _{i} D_{330} + \lambda _{r} D_{430}) y_{1}^{3} \nonumber \\ {}&\quad + (-\lambda _{i} D_{321} + \lambda _{r} D_{421}) y_{1}^{2} y_{2} \nonumber \\&\quad + (-\lambda _{i} D_{312} + \lambda _{r} D_{412}) y_{1} y_{2}^{2}\nonumber \\ {}&\quad + (-\lambda _{i} D_{303} + \lambda _{r} D_{403}) y_{2}^{3} \nonumber \\&\quad + N_{4} + \mathcal {O} \left( \epsilon ^{\frac{5}{2}} \right) , \end{aligned}$$

(39)

yields

$$\begin{aligned} \epsilon y_{1}&: - \omega D_{401} = -\lambda _{i} D_{310} + \lambda _{r} D_{410} + C_{41000}, \end{aligned}$$

(40)

$$\begin{aligned} \epsilon y_{2}&: \omega D_{410} = -\lambda _{i} D_{301} + \lambda _{r} D_{401} + C_{40100}, \end{aligned}$$

(41)

$$\begin{aligned} y_{1}^{3}&: - \omega D_{421} = -\lambda _{i} D_{330} + \lambda _{r} D_{430} + C_{43000}, \end{aligned}$$

(42)

$$\begin{aligned} y_{1}^{2} y_{2}&: \omega (3 D_{430} - 2 D_{412}) = -\lambda _{i} D_{321} + \lambda _{r} D_{421} \nonumber \\ {}&+ C_{42100}, \end{aligned}$$

(43)

$$\begin{aligned} y_{1} y_{2}^{2}&: \omega (2 D_{421} - 3 D_{403}) = -\lambda _{i} D_{312} + \lambda _{r} D_{412} \nonumber \\ {}&+ C_{41200}, \end{aligned}$$

(44)

$$\begin{aligned} y_{2}^{3}&: \omega D_{412} = -\lambda _{i} D_{303} + \lambda _{r} D_{403} + C_{40300}. \end{aligned}$$

(45)

C Reduction in the number of nonlinear terms

We select the coefficients $Q_{10}, Q_{01},\ldots , Q_{03}$ in the equation governing $\xi$, Eq. (19). The derivative of z and $\overline{z}$ with respect to $t^*$ is

$$\begin{aligned} \frac{\mathrm{d}z}{\mathrm{d}t^*}&= \frac{\mathrm{d}\xi }{\mathrm{d}t^*} + \frac{\partial q}{\partial \xi }\frac{\mathrm{d}\xi }{\mathrm{d}t^*} + \frac{\partial q}{\partial \overline{\xi }}\frac{\mathrm{d}\overline{\xi }}{\mathrm{d}t^*} \nonumber \\&=j\omega \xi + j\omega q + N_1'', \end{aligned}$$

(46)

$$\begin{aligned} \frac{\mathrm{d}\overline{z}}{\mathrm{d}t^*}&= \frac{\mathrm{d}\overline{\xi }}{\mathrm{d}t^*} + \frac{\partial \overline{q}}{\partial \xi }\frac{\mathrm{d}\overline{\xi }}{\mathrm{d}t^*} + \frac{\partial q}{\partial \overline{\xi }}\frac{\mathrm{d}\overline{\xi }}{\mathrm{d}t^*} \nonumber \\&=j\omega \overline{\xi } + j\omega \overline{q} + N_2''. \end{aligned}$$

(47)

Substituting Eq. (47) into Eq. (46) yields

$$\begin{aligned} \frac{\mathrm{d}\xi }{\mathrm{d}t^*}&= \left( 1 + \frac{\partial q}{\partial \xi } \right) ^{-1} \left[ j\omega \xi + j\omega q + N_1'' - \frac{\partial q}{\partial \overline{\xi }}\frac{\mathrm{d}\overline{\xi }}{\mathrm{d}t^*} \right] \nonumber \\&= i\omega \xi + j\omega q + i\omega \frac{\partial q}{\partial \overline{\xi }}\overline{\xi } - j\omega \frac{\partial q}{\partial \xi }\xi + N_1'' \nonumber \\ {}&\quad + \mathcal {O}\left( \epsilon ^{\frac{5}{2}} \right) \end{aligned}$$

(48)

$$\begin{aligned}&= i\omega \xi + R_{110}\epsilon \xi + \left( R_{101} + 2j\omega Q_{01} \right) \epsilon \overline{\xi } \nonumber \\&\quad + \left( R_{130} - 2j\omega Q_{30} \right) \xi ^3 + R_{121} \xi ^2\overline{\xi } \nonumber \\&\quad + \left( R_{112} + 2j\omega Q_{12} \right) \xi \overline{\xi }^2\nonumber \\&\quad + \left( R_{103} + 4j\omega Q_{03} \right) \overline{\xi }^3. \end{aligned}$$

(49)

To reduce the number of nonlinear terms in the equation, we select the coefficients as

$$\begin{aligned}&Q_{01} = -\frac{R_{101}}{2j\omega }, ~ Q_{30} = \frac{R_{130}}{2j\omega }, \nonumber \\&Q_{12} = -\frac{R_{112}}{2j\omega }, ~ Q_{03} = -\frac{R_{103}}{4j\omega }. \end{aligned}$$

(50)

Then, Eq. (16) is simplified as Eq. (21).

Fig. 7

Experimentally obtained characteristics for nonlinear feedback gain $\beta _\mathrm{NL}$ versus oscillation amplitude of the cantilever when the linear feedback gain $\beta _\mathrm{NL}$ is fixed near the critical value for the self-excited oscillation; $\beta _\mathrm{lin} = 109 \times 10^{-10}$. Blue and red markers show the experimentally obtained and theoretically calculated response amplitudes, respectively. The theoretical amplitudes are calculated by using Eqs. (10), (14), (15), (19) and $v(x,t) = a_1(t) \varPhi _1(x)$

D Quantitative comparison between experimental and theoretical results

We compare the experimentally and theoretically obtained amplitudes quantitatively. Figure 7 shows the oscillation amplitude when the linear proportional and derivative feedback gains are set at $\alpha = -2.68 \times 10^{-5}$ and $\beta _\mathrm{lin} = 109 \times 10^{-10}$, where the blue and red markers denote the experimentally and theoretically obtained values, respectively. Namely, the blue markers correspond to the amplitude at $\beta _\mathrm{lin} = 109 \times 10^{-10}$ in Fig. 6 . The experimental and theoretical oscillation amplitudes have a quantitative discrepancy. Such a difference is caused by the difficulty in quantitative identification of the parameter $\psi$. Overcoming this difficulty is an open problem. Moreover, the experimentally obtained trend of the response amplitude depending on the nonlinear feedback is in a qualitative good agreement with the theoretical one. It is not necessary to know the nonlinear feedback gain precisely because we can flexibly tune the nonlinear feedback gain to realize the desired magnitude of the steady-state amplitude in practical systems.