Improved self-sensing piezoelectric actuator for dither motor rate control

Abstract

A dither motor in self-sensing actuation configuration allows each of the piezoelectric (lead zirconate titanate) elements to be used concurrently for dither rate sensing and dither motion actuation, so that system of improved efficiency and reliability can be achieved. For a self-sensing actuator, bridge circuit either fixed or adaptive is usually required to resolve the mechanical response from the control signal. In this study, a new technique for accurately extracting the mechanical response from the control signal of dither motor is developed. The measured open-loop response of the dither motor in self-sensing actuation configuration compared well with the results from a separate lead zirconate titanate sensor. Closed-loop dither rate control and frequency tracing of the dither motor are implemented and verified upon temperature variations. Method for estimating the equivalent lead zirconate titanate capacitance of the dither motor more accurately online is also provided and confirmed experimentally. The maximum error between the equivalent capacitance measured online and that measured under static conditions is about 1.2%.

Keywords

Piezoelectric materials self-sensing actuator dither motor lock-in amplifier

Introduction

Dither motors have been widely used for reducing the effects of lock-in in ring laser gyro. A dither motor usually have a suspension system, which includes an outer rim, a central hub member, and a plurality of axis-symmetrical spokes connecting the rim to the center hub as shown in Figure 2. A pair of piezoelectric (lead zirconate titanate (PZT)) elements is bonded on each side of the spoke. Conventionally, one pair of PZT elements on a spoke is used for the sensor, while the others are for the actuator. No measurement of dither rate is available when the sensor fails. Using two or more pairs of PZT elements as sensors to improve system reliability will degrade the efficiency of the dither motor. It is therefore advantageous to utilizing all the PZT elements as a self-sensing actuator (SSA).

Yang and Chiu (1994) demonstrated the use of SSA technique to control a dither motor for maximizing the output of PZT actuator under a fixed power supply and improving system reliability. When used as SSA, the control signal is mixed with the sensing signal due to mechanical response. Therefore, the success of SSA relies on the extraction of that mechanical response from the mixed signal. In Yang and Chiu’s (1994) approach, this was performed by a fixed bridge circuit, which was first developed by Ferris and Weitzner (1989) and Dosch et al. (1992). As the control signal tends to be several times greater than the sensing signal and equivalent capacitance of the PZT is temperature sensitive, the fixed design was shown to be insufficient. Tani et al. (1997) demonstrated that the variation in the equivalent PZT capacitance has critical effects on SSA. A fixed bridge circuit would extract a corrupted mechanical response under the variation of the PZT capacitance. This would degrade the system performance or even destabilize the closed-loop system.

In order to take the PZT capacitance variation into account, considerable efforts have been conducted to design robust or adaptive SSAs. Takigami et al. (1997) designed a robust controller against the PZT capacitance variation to suppress the vibration of a cantilever beam. Garnett et al. (2004) also increased the stability of SSA by utilizing capacitors in series or parallel with the PZT patch at the cost of the increase in the power required for the control system. Cole and Clark (1994) and Vipperman and Clark (1996) developed an adaptive filter to estimate the PZT capacitance and compensate parameter changes in self-sensing circuits. Pourboghrat et al. (1999) designed a controller with adaptive compensation technique and verified its vibration suppression capabilities. Law et al. (2003) adopted a low-power random signal as the training signal to the SSA and implemented an adaptive compensation to simultaneously self-tune the bridge circuit and suppress the structural vibration. Qiu and Haraguchi (2006) established an adaptive controller with a finite impulse response filter and the filtered-X least mean squares (LMS) algorithm to balance the bridge circuit of SSA and confirmed its effectiveness. Ji et al. (2011) proposed a method using neural network to identify the strain signal of SSA and verified its effectiveness in beam vibration suppression.

Among the adaptive techniques mentioned, the preferred training signals were low-power random signals, and some kind of adaptive filters or estimators such as LMS, recursive least squares (RLS), and adaptive lattice filters were used. In this article, a technique for utilizing each of the PZT elements concurrently for dither rate sensing and dither motion actuation is developed. A very low-power sinusoidal signal, especially a second harmonic of the fundamental modal of the dither motor, is applied as a training signal. By measuring the phase lag of this sinusoidal signal with a simple embedded digital lock-in amplifier (DLIA), the equivalent capacitance of the PZT elements can be estimated more accurately online. Also with the DLIA, signals due to mechanical dither are extracted; the measured open-loop response functions of the dither motor in SSA configuration compared well with the results from a separate PZT sensor. Additionally, based on the extracted signal, closed-loop dither rate control and frequency tracing of the dither motor are implemented and verified upon temperature variations. All signal processing tasks excluding some necessary analog signal conditioning circuits are fulfilled with a microprocessor.

SSA

The equivalent electrical model (Ferris and Weitzner, 1989; Yang and Chiu, 1994) for the PZT actuator is adopted here. As shown in the dashed box of Figure 1, C_p is the equivalent capacitance of the PZT element and $i_{m}$ is the current induced from the mechanical deformation and proportional to the strain rate of the dither motor. This model is physically equal to the model (Dosch et al., 1992) of capacitor C_p in series with a voltage generator $V_{p}$ in that $i_{m} = C_{p} {\overset{\cdot}{V}}_{p}$ .

Figure 1.

Self-sensing actuator circuit for rate sensing.

A self-sensing circuit for dither rate sensing employed in this article is shown in Figure 1. It should be noted that the resistor R cannot be connected to the GND hand side as what had been done by Dosch et al. (1992). The reasons are as follows: one side of all the PZT elements is electrically connected to the spokes and thus to the central hub. When the dither motor or the gyro is mounted onto the base, the PZT elements are electrically connected to the base. If R is connected to the GND hand side, it may be grounded by the base. Under this condition, no signal may be detected or spurious interference from other equipments on the base may be detected. $V_{a}$ is the voltage applied to the PZT actuator, and $V_{c}$ is the voltage for control. Between $V_{a}$ and $V_{c}$ , a small resistor R is used for converting current signals to voltage signals.

When $V_{c}$ is a sine wave whose frequency $ω_{1}$ is near the natural frequency $ω_{0}$ of the dither motor, then mechanical dither will occur and

V_{a} = \frac{V_{c} - i_{m} R}{1 + sR C_{p}}

(1)

where $s = j ω_{1}$ . Thus, the voltage proportional to the strain rate of the dither motor can be expressed as

V_{m} = i_{m} R = V_{c} - V_{a} (1 + sR C_{p})

(2)

If $A_{c}$ and $A_{a}$ are the magnitudes of $V_{c}$ and $V_{a}$ , respectively, and $ϕ_{c}$ and $ϕ_{a}$ are the phases of $V_{c}$ and $V_{a}$ , respectively, then the inphase components and quadrature components of $V_{m}$ relative to $V_{a}$ can be written as

\begin{matrix} A_{mI} = A_{c} \cdot \cos (ϕ_{c} - ϕ_{a}) - A_{a} \\ A_{mQ} = A_{c} \cdot \sin (ϕ_{c} - ϕ_{a}) - A_{a} \cdot ω_{1} R C_{p} \\ = A_{c} \cdot \sin (ϕ_{c} - ϕ_{a}) - A_{a} \cdot α_{1} \end{matrix}

(3)

where

α_{1} = ω_{1} R C_{p}

(4)

Hence, the magnitude $A_{m}$ and phase $ϕ_{m}$ of $V_{m}$ (herein $ϕ_{m}$ is also relative to $V_{a}$ ) are as follows

\begin{matrix} A_{m} = \sqrt{A_{mI}^{2} + A_{mQ}^{2}} \\ ϕ_{m} = {\begin{matrix} atan (A_{mQ} / A_{mI}) if A_{mI} > 0 \\ atan (A_{mQ} / A_{mI}) + π otherwise \end{matrix} \end{matrix}

(5)

Observing equations (3) to (5), it is seen that $A_{m}$ and $ϕ_{m}$ are the functions of $A_{c}$ , $A_{a}$ , $ϕ_{c}$ , $ϕ_{a}$ , and C_p. The magnitude and phase of a sine wave can be measured accurately with a DLIA (Dixon and Wu, 1989). What remained unknown is C_p, the equivalent capacitance of the PZT actuator, which is sensitive to ambient temperature. The accurate measurement of PZT capacitance can prove to be difficult. Efforts are made to estimate C_p online and continuously (Cole and Clark, 1994; Law et al., 2003; Vipperman and Clark, 1996). In this work, a new method is adopted to measure C_p.

Applying another sine wave $V_{c 2}$ as a training signal to the dither motor and considering

V_{c 2} = A_{c 2} \cdot \sin (ω_{2} t + ϕ_{c 2})

(6)

where $A_{c 2}$ , $ω_{2}$ , and $ϕ_{c 2}$ are the magnitude, frequency, and phase of $V_{c 2}$ , respectively. It is assumed that there is no mechanical dither when $V_{c 2}$ alone is applied. For a high-Q dither motor, it is easy to meet this condition by adjusting $ω_{2}$ . With this restriction, voltage across the PZT actuator due to $V_{c 2}$ can be expressed as

V_{a 2} = \frac{V_{c 2}}{(1 + sR C_{p})}

(7)

where $s = j ω_{2}$ . The phase lags between $V_{c 2}$ and $V_{a 2}$ are

α_{2} = ϕ_{c 2} - ϕ_{a 2} = atan (ω_{2} R C_{p})

(8)

where $ϕ_{a 2}$ denotes the phase of $V_{a 2}$ . If $α_{2}$ can be measured, then C_p and $α_{1}$ can be determined

C_{p} = \frac{tan (α_{2})}{(ω_{2} R)}

(9)

α_{1} = \tan (α_{2}) \cdot \frac{ω_{1}}{ω_{2}}

(10)

Thus, for a self-sensing dither motor, in order to extract the strain rate signal $V_{m}$ (i.e. $A_{m}$ , $ϕ_{m}$ ), the following six quantities must be measured: magnitude and phase of $V_{c}$ at $ω_{1}$ (i.e. $A_{c}$ , $ϕ_{c}$ ), magnitude and phase of $V_{a}$ at $ω_{1}$ (i.e. $A_{a}$ , $ϕ_{a}$ ), and phases of $V_{c 2}$ and $V_{a 2}$ at $ω_{2}$ (i.e. $ϕ_{c 2}$ , $ϕ_{a 2}$ ). All of them can be measured accurately with a DLIA. The reasons for using DLIA are as follows: lock-in amplifiers are used to detect and measure the magnitudes and phases of very small alternating current (AC) signals, and accurate measurements may be made even when the small signal is obscured by noise sources many thousands of times larger. In this article, a simple DLIA is implemented with a microcontroller. With this DLIA, a very low-power sine wave $V_{c 2}$ can be employed as the training signal so as not to overburden PZT driving circuit, and accurate measurements are made.

It should also be noted that the control voltage from $V_{c}$ or $V_{c 2}$ to the PZT actuator is leveled down by a factor $1 / (1 + sR C_{p})$ , which can be seen from equations (1) and (7). To take full advantage of the control voltage under a fixed power supply/amplifier, a small resistor R is chosen to ensure that $| 1 / (1 + sR C_{p}) | > 0.9$ , that is, $ω R C_{p} < 0.48$ $(ω = ω_{1} or ω_{2})$ . Especially when $ω_{1} R C_{p} << 1$ and $ω_{2} R C_{p} << 1$ , the factor $| 1 / (1 + sR C_{p}) | \approx 1$ and

\begin{matrix} α_{2} \approx ω_{2} R C_{p} \\ C_{p} \approx \frac{α_{2}}{(ω_{2} R)} \\ α_{1} \approx α_{2} \cdot \frac{ω_{1}}{ω_{2}} \end{matrix}

(11)

Thus, decreasing resistor R will improve the efficiency of control signal; however, the strain rate signal $V_{m}$ ( $i_{m} R$ ) also decreases. As the strain rate signal $V_{m}$ in R has very high common-mode voltage, traditional skills other than lock-in amplifiers will be inadequate for extracting $V_{m}$ from large control signals. With the DLIA, a resistor R of no less than 30 Ω is enough for sensing.

Experimental results

Experimental setup

Figure 2 is a schematic diagram of self-sensing PZT actuator for dither motor rate control experiment. As shown in Figure 2, the dither motor includes eight spokes and eight pairs of PZT elements. Seven pairs of PZT elements are used as a self-sensing PZT actuator to excite and monitor the dithering motion, while the rest are used as a separate sensor/feedback for monitoring the dithering motion from which one can compare its output with the results of SSA.

Figure 2.

Schematic diagram of self-sensing PZT actuator for dither motor rate control.

Let C_p be the equivalent capacitance of the PZT elements for SSA, $C_{p 1}$ be that of the sensor (about 25 nF at room temperature), $i_{m}$ be the current induced from the mechanical deformation in the PZT elements for SSA, and $i_{m 1}$ be that of the sensor. Ideally, $C_{p} = 7 C_{p 1}, i_{m} = 7 i_{m 1}$ .

For the sensor

V_{s} = \frac{i_{m 1} R_{s}}{(1 + s R_{s} C_{p 1})} \approx i_{m 1} R_{s}

(12)

Like the voltage $V_{m}$ , the sensor voltage $V_{s}$ is also proportional to the dither rate of the dither motor. Let $A_{s}$ be the magnitude of $V_{s}$ and $ϕ_{s}$ be the phase of $V_{s}$ relative to $V_{a}$ . If $V_{m}$ has the same characteristic as that of $V_{s}$ around the natural frequency of the dither motor (i.e. $ϕ_{m} = ϕ_{s}$ , and $A_{m}$ is proportional to $A_{s}$ ), then the extracted signal $V_{m}$ can be used for dither motor rate control.

A microcontroller C8051F121 of Silicon Laboratories is employed to generate sine waves at $ω_{1}$ and $ω_{2}$ simultaneously with its internal digital-to-analog (D/A) converter. Two sine wavetables are stored in the flash for sine wave generation. For $ω_{1}$ , the wavetable is $A_{1} \sin (2 π i / N), (i = 0, \dots, N - 1)$ , and for $ω_{2}$ , the wavetable is $A_{2} \sin (2 π j / M), (j = 0, \dots, M - 1, M < N)$ .

As shown in Figure 2, outputs from the two wavetables are first summed and then multiplied by gain $V_{g}$ , which controls the dither rate of the dither motor. Frequencies $ω_{1}$ and $ω_{2}$ can be adjusted by different D/A updating rate. It has a number of advantages to choose $ω_{2} = 2 ω_{1}$ ( $N = 2 M$ ); for instance, for a proper designed high-Q dither motor, the second harmonic will always have no effect on the actuator movement when $ω_{1}$ is near $ω_{0}$ , and $α_{1} = \tan (α_{2}) / 2$ , and it is much easier to calculate the magnitudes and phases of harmonics with a DLIA than that of nonharmonics. $A_{2}$ takes only a small portion of $A_{1}$ (5% in the experiment) so as not to overburden PZT driving circuit.

Due to the microcontroller input limits, signals fed to the analog-to-digital (A/D) converters are pre-scaled down by the level down/shift circuit shown in Figure 2. The level down/shift circuit is made up of several resistors and two matched buffers and scales the inputs to a range of 0–2.5 V. $R_{1}$ and $R_{3}$ and $R_{2}$ and $R_{4}$ are matched resistors with low-temperature coefficients. Here, $R_{1} = R_{3} = 420 k Ω$ and $R_{2} = R_{4} = 50 k Ω$ . Normally, the single-pole double-throw analog switches K connect $R_{3}$ to $V_{a}$ . The difference between the two arms of the level down/shift circuit can be calibrated and compensated on power-up or periodically by turning switch K to $V_{c}$ . The pre-scaled $V_{c}, V_{a}, V_{s}$ are discretized by A/D converters with the same frequency as the D/A converters and fed into the DLIA where $A_{c}, A_{a}, ϕ_{c}, ϕ_{a}$ , $ϕ_{c 2}, ϕ_{a 2}$ , and $α_{2}, α_{1}, A_{s}, ϕ_{s}$ are calculated. Since then, $A_{m}, ϕ_{m}, C_{p}$ can be derived using equations (5) and (8) to (11).

Open-loop response

Fixing $V_{g}$ , the microcontroller steps up $ω_{1}$ with a step of 0.25 Hz near the natural frequency of the dither motor. The results at room temperature (18.5°C) are shown in Figure 3.

Figure 3.

Open-loop response of dither motor sensed by SSA and PZT feedback. (a) amplitude-frequency characteristics of Vc and Va. (b) phase-frequency characteristics of Vc and Va. (c) phase-frequency characteristics of Vc2 and Va2 (left y-axis), and phase lag between Vc2 and Va2 (right y-axis). (d) open-loop amplitude-frequency response curves by SSA and PZT feedback. (e) open-loop phase-frequency response curves by SSA and PZT feedback.

As shown in Figure 3(a) to (c), magnitude and phase of $V_{c}$ (i.e. $A_{c}$ , $ϕ_{c}$ ), magnitude and phase of $V_{a}$ (i.e. $A_{a}$ , $ϕ_{a}$ ), phases of $V_{c 2}$ and $V_{a 2}$ (i.e. $ϕ_{c 2}$ , $ϕ_{a 2}$ ), and phase lag $α_{2}$ are calculated with the DLIA. In Figure 3(d) and (e), open-loop response of dither motor sensed by SSA and PZT feedback are shown, and the phases $ϕ_{m}, ϕ_{s}$ herein are referred to $V_{a}$ . From Figure 3(d), an important result is that the envelopes of $A_{m}$ and $A_{s}$ are very similar to each other. It is even clear when the peak of $A_{m}$ is scaled to equal to that of $A_{s}$ , indicating that $A_{m}$ can be used for dither rate control. As the peak value is 516.3 for $A_{m}$ , 652.9 for $A_{s}$ , and

\begin{matrix} A_{m} = | i_{m} R | \\ A_{s} = | i_{m 1} R_{s} | \end{matrix}

(13)

then

\frac{i_{m}}{i_{m 1}} = 6.6

(14)

The deviation between the measured $i_{m} / i_{m 1}$ and the ideal value 7 is partially owing to the fact that only one peak point is used herein. A more accurate method will be introduced later.

As indicated in Figure 3(e), there is some difference between the phases $ϕ_{m}$ and $ϕ_{s}$ when $ω_{1}$ is much different from the natural frequency $ω_{0}$ of the dither motor. However, a dither motor is always operated at $ω_{0}$ , where $ϕ_{m}$ and $ϕ_{s}$ are almost the same. Both $ϕ_{m}$ and $ϕ_{s}$ approach to zero and change quickly at $ω_{0}$ . The slope is about 85°/Hz. Thus, the phase $ϕ_{m}$ can be employed to trace the natural frequency $ω_{0}$ of the dither motor with good sensitivity.

From Figure 3(c), C_p can be calculated with equation (8). An easier and more accurate approach to estimate C_p and $i_{m} / i_{m 1}$ online is to fix $ω_{1}$ and average the results. $α_{2}$ , C_p, and $i_{m} / i_{m 1}$ measured at different temperatures are shown in Table 1. C_p measured by Fluke 15B digital multimeter (DM) under static conditions is also shown for comparison (distinguished by C_p (nF), –DM). The maximum error between the two columns of C_p is less than 1.2%, indicating that using the second harmonic as training signal to stimulate the dither motor and measuring the phase lag $α_{2}$ by DLIA is a good method to estimate the equivalent PZT capacitance online. Although $ω_{1}$ shown in Table 1 is set approximately equal to the natural frequency of the dither motor and changes with temperature, other frequencies such as a fixed frequency of 440 Hz for all temperatures may be used.

Table 1.

C_p and $i_{m} / i_{m 1}$ measured at different temperatures.

T (°C)	$ω_{1}$ (Hz)	$α_{2}$ (°)	C_p (nF)	C_p (nF), −DM	$i_{m} / i_{m 1}$
18.5	447.00	6.85	177.4	179.5	6.79
0	446.50	6.68	173.1	174.4	6.77
60	448.25	7.64	197.3	198.1	6.85

DM: digital multimeter.

Open-loop responses at 0°C and 60°C are shown in Figures 4 and 5. One can see that the curves are similar to functions at room temperature, and the natural frequency of the dither motor changes with temperature.

Figure 4.

Open-loop response at 0°C. (a) open-loop amplitude-frequency response curves by SSA and PZT feedback. (b) open-loop phase-frequency response curves by SSA and PZT feedback.

Figure 5.

Open-loop response at 60°C. (a) open-loop magnitude-frequency response curves by SSA and PZT feedback. (b) open-loop phase-frequency response curves by SSA and PZT feedback.

Closed-loop dither rate control upon temperature variation

In this experiment, the dither motor is put into a temperature chamber, and temperature is changed from 60°C to 0°C while the circuits are kept outside. Upon temperature variation, the natural frequency $ω_{0}$ of the dither motor and PZT capacitance C_p will change. As it has been mentioned, $ϕ_{m}$ can be employed to trace the natural frequency $ω_{0}$ of the dither motor and $A_{m}$ can be used for dither rate control. A simple method is adopted to trace $ω_{0}$ . First, $ω_{0}$ is found by scanning $ω_{1}$ linearly near the natural frequency of the dither motor. Second, the following cycles are repeated: if $ϕ_{m} > 3 \circ$ (other threshold than 3 may be used), $ω_{1}$ is increased by a small step; if $ϕ_{m} < - 3 \circ$ , $ω_{1}$ is decreased by a small step. A proportional–integration (PI) control law is also used to control $A_{m}$ to 600 mV by adjusting gain $V_{g}$ . The results are shown in Figure 6. $ω_{0}$ and C_p at three temperature points (0°C, 18.5°C, and 60°C) are consistent with the values shown in Table 1. As the control voltage $V_{c}$ is proportional to gain $V_{g}$ , the closed-loop response of $V_{g}$ upon temperature variation indicates that the dither motor is less effective at high temperatures. When $A_{m}$ is stabilized to 600 mV, the PZT feedback voltage $A_{s}$ shown in Figure 6(a) is also stabilized. The fluctuation of $A_{s}$ originates from the slight difference between $A_{s}$ and $A_{m}$ around $ω_{0}$ , which is neglectable for most applications.

Figure 6.

Closed-loop response upon temperature variation: (a) closed-loop control of $A_{m}$ with $V_{g}$ and (b) $ω_{0}$ and $C_{p}$ upon temperature variation.

Conclusion

A technique for accurately extracting the mechanical response from the control signal of dither motor in SSA configuration is developed and validated upon temperature variations in this study. The measured open-loop frequency response of the dither motor in SSA configuration compared well with the results from a separate PZT sensor. Closed-loop dither rate control and frequency tracing of a dither motor may be done with the extracted signal. In addition, a new method for estimating the equivalent PZT capacitance of the dither motor more accurately online is provided. The effectiveness of the new method is confirmed experimentally, and a maximum error of 1.2% between the equivalent capacitance measured online and that measured under static conditions has resulted. The developed techniques and methods can be employed to improve the efficiency and reliability of the dither motor control.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Cole

Clark

(1994) Adaptive compensation of piezoelectric sensoriactuators. Journal of Intelligent Material Systems and Structures 5: 665–672.

Dixon

(1989) Broadband digital lock-in amplifier techniques. Review of Scientific Instruments 60: 3329–3336.

Dosch

Inman

Garcia

(1992) A self-sensing piezoelectric actuator for collocated control. Journal of Intelligent Material Systems and Structures 3: 166–185.

Ferris

Weitzner

(1989) Velocity control system using piezoelectric transducers. US patent no. 4856901.

Garnett

Jeffrey

David

. (2004) Improved piezoelectric self-sensing actuation. Journal of Intelligent Material Systems and Structures 15: 941–953.

Qiu

. (2011) Novel approach of self-sensing actuation for active vibration control. Journal of Intelligent Material Systems and Structures 22: 449–459.

Law

Liao

W-H

Huang

(2003) Vibration control of structures with self-sensing piezoelectric actuators incorporating adaptive mechanisms. Smart Materials and Structures 12: 720–730.

Pourboghrat

Pongpairoj

Youn

. (1999) Vibration control of flexible structures using self-sensing actuators. In: Proceeding SPIE 3668, Smart Structures and Materials 1999- Smart Structures and Integrated Systems, 9 June, pp. 950-959, International Society for Optics and Photonics. DOI:10.1117/12.350770

Qiu

Haraguchi

(2006) Vibration control of a plate using a self-sensing piezoelectric actuator and an adaptive control approach. Journal of Intelligent Material Systems and Structures 17: 661–669.

10.

Takigami

Oshima

Hayakawa

(1997) Application of self-sensing actuator to control of a cantilever beam. In: Proceedings of the 1997 American control conference (AACC), Albuquerque, NM, 4–6 Jun 1997, pp. 1867–1872. DOI: 10.1109/ACC.1997.610909.

11.

Tani

Cheng

Qiu

(1997) Effectiveness and limits of self-sensing piezoelectric actuators. In: Proceedings of 1st structural health monitoring workshop, September 18–20, 1997, Stanford, CA, pp. 502–514. Lancaster - Basel, Technomic Publishing Company.

12.

Vipperman

Clark

(1996) Implementation of an adaptive piezoelectric sensoriactuator. AIAA Journal 34: 2102–2109.

13.

Yang

Chiu

(1994) Dither-motor design with concurrent sensing and actuating piezoelectric materials. Smart Materials and Structures 3: 248–253.