# Parameters Identification for a Composite Piezoelectric Actuator Dynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Prototype of the PZT Bimorph Actuator

Variable | Fiberglass | MFC |
---|---|---|

Length (mm) | L_{f} = 17 | L_{p} = 85 (active length) |

Width (mm) | b_{f} = 75 | b_{p} = 56 |

Height (mm) | h_{f} = 0.5 | h_{p} = 0.3 |

PZT strain constant d_{33} (m/V) | N/A | d_{33} = 4.275 × 10^{−10} |

Electrode spacing e_{s} (mm) | N/A | e_{s} = 0.5 |

## 3. PZT Bimorph-Actuated Smart Fin

**Figure 2.**PZT bimorph-actuated smart fin prototype. (

**a**) Rear view (without the shell); (

**b**) top view; (

**c**) side view (without the shell); (

**d**) assembled fin prototype.

## 4. Modeling

_{i}and ${\varnothing}_{\text{i}}$ are the nodal displacement and slope of node i, respectively. C is the damping matrix that is expressed in the following form [38]:

^{*T}is a unit vector whose (2n−1)-th element is one and the remaining elements are zeros (where n is the number of elements).

_{1}and a

_{2}) and two saturation operators (b

_{1}and b

_{2}), as shown in Figure 3. A detailed explanation on incorporating these parameters in the final model is presented in Appendix A.

#### Identification of the Damping, Hysteresis and Backlash Parameters

_{1}and b

_{2}) are linear functions of the exciting signal’s voltage and frequency. Appendix B lists the values of these saturation parameters. The remaining seven parameters (μ, δ, α, β, γ, a

_{1}and a

_{2}) are determined using the HGANN optimization technique introduced in the following section. The major features of this algorithm are presented in Appendix C.

_{1}and a

_{2}). Each NN neuron has a bounded transfer function (TF) [41]; these TFs are outlined in Appendix C. The overall characteristics of each member of the GA population are coded and compacted within the chromosome in a segmental style using binary representation. The NN uses a subset of four datasets to optimize the parameters of damping, hysteresis and backlash. The remaining experimental datasets are used after the training to validate the effectiveness of the NN. The algorithm is shown in Figure 4. Table 2 shows the GA parameters used in this algorithm. The full range of the optimization parameters can be achieved by scaling the TFs of the output layer using the gain values of Table 3.

Parameter | Value |
---|---|

Encoding technique | Binary |

Number of bits | 3 for discrete, 14 for continuous |

Population | 400 |

Elitism | 50% |

Crossover technique | 2-points |

Mutation rate | 0.01 |

Generation | 1000 |

Parameter | Parameter Range | |||
---|---|---|---|---|

Lower Limit | Upper Limit | Gain | ||

Damping | μ | 0 | 1 | 1 |

δ | 0 | 300 | 300 | |

Hysteresis | α | −0.1 | 0 | −0.1 |

β | −0.1 | 0 | −0.1 | |

γ | −0.1 | 0 | −0.1 | |

Backlash | a_{1} | 0 | 2 | 2 |

a_{2} | 0 | 2 | 2 |

^{−3}. Similar values are used for the bias weights. These values are based on a preliminary evaluation of the system. Table 4 shows the NN parameters and their bounds for this problem.

Parameter | Number of Nodes | Lower Limit | Upper Limit | Binary Coding | Resolution |
---|---|---|---|---|---|

Input nodes | 2 | N/A | N/A | N/A | N/A |

Output nodes | 7 | N/A | N/A | N/A | N/A |

Hidden nodes | Adaptive | 2 | 9 | 3 bits | 1 |

Learning rules | Adaptive | 1 | 8 | 3 bits | 1 |

Connection weights | NA | −10 | 10 | 14 bits | 1.2 × 10^{−3} |

Frequency | 0.25 Hz | 0.5 Hz | 1.0 Hz | |
---|---|---|---|---|

Voltage | ||||

375 V | Case 1 | Case 2 | Case 3 | |

500 V | Case 4 | Case 5 | Case 6 | |

750 V | Case 7 | Case 8 | Case 9 |

_{exp}(i) is the measured fin angle at the i-th sample and β

_{mod}(i) is the fin angle produced by the dynamic model at the same i-th sample.

## 5. Results

Case 1 | Case 5 | Case 6 | Case 7 | ||
---|---|---|---|---|---|

Damping | δ | 0.312 | 0.093 | 0 | 0.128 |

μ | 98.650 | 94.666 | 75.626 | 109.210 | |

Hysteresis | α | 0 | 0 | 0 | 0 |

β | −7.49 × 10^{−2} | −7.88 × 10^{−2} | −8.06 × 10^{−2} | −8.23 × 10^{−2} | |

γ | −6.51 × 10^{−2} | −7.20 × 10^{−2} | −7.84 × 10^{−2} | −5.30 × 10^{−2} | |

Backlash | a_{1} | 0 | 0 | 0 | 0.281 |

a_{2} | 0 | 0 | 0 | 0.294 |

## 6. Model Validation

Case 2 | Case 3 | Case 4 | Case 8 | Case 9 | ||
---|---|---|---|---|---|---|

Damping | δ | 0.124 | 0 | 0.219 | 0.029 | 0 |

μ | 93.534 | 72.812 | 104.47 | 97.144 | 81.792 | |

Hysteresis | α | 0 | 0 | 0 | 0 | 0 |

β | −7.61 × 10^{−2} | −7.92 × 10^{−2} | −7.75 × 10^{−2} | −8.34 × 10^{−2} | −8.32 × 10^{−2} | |

γ | −8.00 × 10^{−2} | −8.44 × 10^{−2} | −6.55 × 10^{−2} | −5.59 × 10^{−2} | −6.57 × 10^{−2} | |

Backlash | a_{1} | 0 | 0 | 0.041 | 0.149 | 0 |

a_{2} | 0 | 0 | 0 | 0.344 | 0.255 |

Excitation Frequency (Hz) | |||
---|---|---|---|

Amplitude↓ | 0.25 | 0.50 | 1.00 |

375 V | 1.42 | 2.08 | 2.38 |

500 V | 2.49 | 1.96 | 1.51 |

750 V | 5.64 | 3.23 | 6.78 |

Excitation Frequency (Hz) | |||
---|---|---|---|

Amplitude↓ | 0.25 | 0.50 | 1.00 |

375 V | 0.312 | 0.124 | 0 |

500 V | 0.219 | 0.093 | 0 |

750 V | 0.128 | 0.029 | 0 |

Excitation Frequency (Hz) | |||
---|---|---|---|

Amplitude↓ | 0.25 | 0.50 | 1.00 |

375 V | 98.650 | 93.534 | 72.812 |

500 V | 104.47 | 94.666 | 75.626 |

750 V | 109.210 | 97.144 | 81.792 |

V (volts) | f (Hz) | Y_{intercept} | Coefficient of Determination R^{2} |
---|---|---|---|

0.0020 p = 0.001 | −36.525 p < 0.001 | 102.361 p < 0.001 | 0.988 |

_{1}) is only active at the lower left corner part of the grid (cases with high voltage and low frequency). On the other hand, Table 13 shows the second threshold operator (a

_{2}) is only active when voltage is at the highest amplitude. Consequently, the backlash thresholds of the PZT actuator are activated at cases of higher voltage amplitude, i.e., when the dynamic forces become dominant.

Excitation Frequency (Hz) | |||
---|---|---|---|

Amplitude↓ | 0.25 | 0.50 | 1.00 |

375 V | 0 | 0 | 0 |

500 V | 0.041 | 0 | 0 |

750 V | 0.281 | 0.149 | 0 |

Excitation Frequency (Hz) | |||
---|---|---|---|

Amplitude↓ | 0.25 | 0.50 | 1.00 |

375 V | 0 | 0 | 0 |

500 V | 0 | 0 | 0 |

750 V | 0.294 | 0.344 | 0.255 |

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

## Appendix A: Backlash Operators

_{1}and a

_{2}) and two saturation operators (b

_{1}and b

_{2}). These backlash operators work as follows:

_{1}, which is the control input’s first threshold value. The hysteresis operator at this stage is:

_{1}, continues to be active until a saturation operator, b

_{1}, is reached. The output remains equal to this saturation value. The above equation becomes:

_{1}until the second backlash operator, a

_{2}, is activated as follows:

_{2}, is reached. At this stage, the output is equal to,

## Appendix B: Identification of the Saturation Parameters

Excitation Frequency (Hz) | |||
---|---|---|---|

Amplitude↓ | 0.25 | 0.50 | 1.00 |

375 V | b_{1} = 1.37b _{2} = −1.50 | b_{1} = 1.26b _{2} = −1.26 | b_{1} = 1.16b _{2} = −1.16 |

500 V | b_{1} = 2.34b _{2} = −2.12 | b_{1} = 2.05b _{2} = −1.91 | b_{1} = 1.73b _{2} = −1.66 |

750 V | b_{1} = 4.21b _{2} = −4.03 | b_{1} = 3.74b _{2} = −3.56 | b_{1} = 3.24b _{2} = −3.17 |

Parameter | V (volts) | f (Hz) | Y_{intercept} | Coefficient of Determination R^{2} |
---|---|---|---|---|

b_{1} | 0.00645 p < 0.001 | −0.84771 p = 0.007 | −0.63061 p = 0.070 | 0.985 |

b_{2} | −0.00622 p = 0.007 | 0.68371 p = 0.002 | 0.71011 p = 0.061 | 0.981 |

_{1}= 0.00645 V − 0.84771 f − 0.63061

_{2}= −0.00622 V + 0.68371 f + 0.71011

## Appendix C: Hybrid Genetic Algorithm Neural Network

- Generate a random initial population where the NN characteristics are coded into the genetic material of each individual.
- Expand the compacted information of each individual and decode the genetic material into neural material, including; connection weights, biases, architecture and learning rules.
- Feed the four training cases for each individual at the input layer to identify the parameters of damping, hysteresis and backlash.
- Feed the parameters obtained at Step (3) into the dynamic model of the actuator to generate the simulated output angle.
- Evaluate the fitness of each individual. The individuals are then rearranged according to their fitness.
- Prune the least fitted individuals and select the best fitted ones as members of the new generation, as well as parents who will undergo the reproduction phases of crossover and mutation to produce new offspring for the next generation.
- Combine the new offspring with the best fitted individuals of the current generation to establish the population of the new generation.
- Repeat Steps 2 through 7 to progressively optimize a cost function until either the maximum number of generations is reached or there is no significant improvement recorded within the previous 200 successive generations.

## Appendix D: NN Architecture

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Saadeh, M.; Trabia, M. Parameters Identification for a Composite Piezoelectric Actuator Dynamics. *Actuators* **2015**, *4*, 39-59.
https://doi.org/10.3390/act4010039

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Saadeh M, Trabia M. Parameters Identification for a Composite Piezoelectric Actuator Dynamics. *Actuators*. 2015; 4(1):39-59.
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**Chicago/Turabian Style**

Saadeh, Mohammad, and Mohamed Trabia. 2015. "Parameters Identification for a Composite Piezoelectric Actuator Dynamics" *Actuators* 4, no. 1: 39-59.
https://doi.org/10.3390/act4010039