# Adaptive and Robust Operation with Active Fuzzy Harvester under Nonstationary and Random Disturbance Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Harvester Model

_{s}. The second configuration is the charge inversion circuit composed of a switch device S and an inductor L. R

_{L}is the parasite resistance in the inductor L. The harvester circuit has three circuit equations based on the conditions of the two semiconductor elements, a switch element, and a rectifier, thereby dominating the circuit connection. The switch element changes its status in accordance with the control signals determined by the switching strategy. The rectifier used in this paper is passive and conducts both input and output circuits only when the amplitude of the input AC voltage is larger than that of the output DC voltage.

_{p}, and Q

_{s}denote the mass displacement, the electric charge stored in the piezoelectric transducer, and that stored in the smoothing capacitor, respectively. The governing equations of electric charges depend on the circuit connections (Figure 2). The electromechanical coupling equation of the harvester in the state space expression is expressed as

_{dist}denotes the disturbance. The elements in each block matrix are described in Appendix A. Each component in the electromechanical model with a piezoelectric transducer used in this paper employs the following assumptions:

- The mechanical components are composed of two masses connected in series by two springs. This system is modeled as a linear 2-DOF structure and has two vibration modes.
- The piezoelectric transducer mechanically deforms only in one direction. Both direct and inverse piezoelectric effects are discussed for a range of small deformations and do not consider hysteresis properties. The transducer is installed between the first vibrating mass and the fixed base. The deformation of the transducer corresponds to the displacement of the first mass.
- The electrical components are modeled using only passive components. Semiconductor elements in the electric circuit are modeled linearly as either open or closed electrical conditions. Because the forward voltage in diode elements is sufficiently small compared to the piezoelectric voltage, the forward voltage of the diodes is neglected.

#### 2.2. Mechanism of Charge Inversion Circuit and Switching Action

_{s}denotes a constant value of the displacement when the switch element changes from open to closed. Although the displacement is a variable of time, this variable is handled as the constant while the switch element is closed owing to the assumption regarding the magnitude relation of both the mechanical and electrical vibration frequencies. Here, we assume ${\zeta}_{\mathrm{e}}\ll 1$. The resulting electric charge Q

_{after}is expressed as

_{before}and x

_{s}satisfies the following inequality:

- A switching action should be performed when the signs between the displacement and charge are opposite.
- A switching action should be performed when the magnitude of displacement is sufficiently larger than the right term depending on the piezoelectric charge in inequality (6).

#### 2.3. Threshold-Based Switching Strategy for Complex Vibration

_{1}, T

_{s}, and k denote the higher resonance period of the two vibrating masses, sampling period, and discrete time, respectively. The RMS values are used with the previous state values to indirectly detect the trend of changes in the present condition of the harvester. When a short resonant period is used for RMS calculation, a small change in the harvester conditions is strongly reflected in the calculation result. Small changes in the harvester are undesirable to tune the time-varying threshold because it induces control chatter; therefore, the longer resonance period is adopted. In this research, the piezoelectric voltage-based threshold is adopted as a criterion for switching execution. Hence, the threshold is

- The STDH control type, which occurs by the exceedingly large threshold coefficient magnitude, does not perform switching actions in the harvesting process. This type relinquishes the opportunities of the increment of the piezoelectric charge by switching actions and should be avoided by decreasing the threshold magnitude.
- The SSHI control type, which occurs because of the exceedingly small threshold coefficient magnitude, performs switching actions, including both desirable and undesirable actions. This type leads to the attenuation of the piezoelectric charge because of the undesirable switching actions and should be avoided by increasing the threshold magnitude.
- The SCVS control type, which occurs because of the adequate threshold coefficient magnitude, accomplishes appropriate intermittent switching actions for effective harvesting. The present threshold design is aimed at this type.

## 3. Experiment

_{p}, was calculated from the least-squares estimation. The voltage was measured through the voltage follower to prevent the influence of the outflowing piezoelectric charge. The capacitance in the transducer, which is C

_{p}, was measured by the impedance analyzer. Because the deformation of the piezoelectric transducer during impedance analysis was minute, the measured capacitance was dealt with as the constant-strain capacitance. The inductor in the charge inversion circuit was selected to satisfy the frequency requirement under switching actions. The values of the piezoelectric and electric parameters denoted by b

_{p}, C

_{p}, C

_{s}, L, R

_{L}, and R

_{load}, were 4.5 × 10

^{5}V/m, 4.3 × 10

^{−7}F, 4.7 × 10

^{−5}F, 2.0 × 10

^{−2}H, 2.0 × 10

^{1}Ω, and 2.2 × 10

^{6}Ω, respectively.

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**A**and input matrix

**B**are fixed during the harvesting process. Those matrices are described as

_{p}denotes the piezoelectric coefficient; subscripts 1 and 2 in each variable indicate both masses, and subscript p in each variable indicates the piezoelectric transducer.

- The first circuit connection indicates that the switch element and rectifier are open and nonconductive. The piezoelectric charge does not change during this process because the piezoelectric transducer is an open circuit. In contrast, the storage charge Q
_{s}in the smoothing capacitor is consumed during this process by the resistor assumed as an electric device. - The second circuit equation indicates that the switch element and the rectifier are open and conductive. The piezoelectric charge stored in the transducer flows out to the smoothing capacitor and the resistor. Because of the rectifies functions, the sign of the circuit equations is changed depending on the polarity of the piezoelectric voltage.
- The third circuit equation, which is a unique connection on an active harvester, is that the switch element and rectifier are closed and nonconductive. The piezoelectric charge vibrates because of the LC series resonance.

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**Figure 1.**Schematic of the piezoelectric vibration energy harvester composed of a piezoelectric transducer, 2-DOF vibration structure, and harvesting circuit with a charge inversion circuit.

**Figure 2.**Circuit connections under each circuit equation. The black and gray lines show the conductive and nonconductive lines. (

**a**) The piezoelectric charge is constant because of the nonconductive conditions of the switch and rectifier. (

**b**) The piezoelectric charge outflows from clamped storage in the piezoelectric transducer to the smoothing capacitor owing to the conductive rectifier. This diagram indicates the condition in which the piezoelectric voltage sign is positive. (

**c**) The piezoelectric charge vibrates through the inductor and capacitor. This connection is maintained for half of the LC resonance frequency time.

**Figure 3.**Time histories of the piezoelectric charge, and the piezoelectric voltage and the threshold under each control type. (

**a**) The STDH control type, which occurs because of the exceedingly large threshold coefficient magnitude, does not perform switching actions. Although the charge moves slightly in accordance with the harvesting circuit state, the change in the charge under STDH conditions is hardly visible because the scale of the vertical axis is matched with the other control types. (

**b**) The SSHI control type, which occurs because of the exceedingly small threshold coefficient magnitude, performs switching actions at all peaks. (

**c**) The SCVS control type, which occurs because of the adequate threshold coefficient magnitude, accomplishes appropriate intermittent switching actions.

**Figure 5.**The frequency response between the input force and the first mass displacement of the 2-DOF vibrating structure under the open-circuit condition.

**Figure 6.**Three nonstationary and random disturbance conditions in the experiments. (

**a**) Frequency sweep vibration experiment: the upper figure shows the time history of displacement 1; the lower figures show each PSD in response to vibration changes. (

**b**) Change in the number of dominant frequency vibration experiment: each figure shows the same information as (

**a**). (

**c**) Random vibration experiment: sample PSD is shown.

**Figure 7.**Time histories of output with each switching strategy. (

**a**) Frequency sweep experiment. (

**b**) Experiment on change in the number of dominant frequencies. (

**c**) Random experiment.

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**MDPI and ACS Style**

Hara, Y.; Otsuka, K.; Makihara, K. Adaptive and Robust Operation with Active Fuzzy Harvester under Nonstationary and Random Disturbance Conditions. *Sensors* **2021**, *21*, 3913.
https://doi.org/10.3390/s21113913

**AMA Style**

Hara Y, Otsuka K, Makihara K. Adaptive and Robust Operation with Active Fuzzy Harvester under Nonstationary and Random Disturbance Conditions. *Sensors*. 2021; 21(11):3913.
https://doi.org/10.3390/s21113913

**Chicago/Turabian Style**

Hara, Yushin, Keisuke Otsuka, and Kanjuro Makihara. 2021. "Adaptive and Robust Operation with Active Fuzzy Harvester under Nonstationary and Random Disturbance Conditions" *Sensors* 21, no. 11: 3913.
https://doi.org/10.3390/s21113913