Parameter Optimization Method for Identifying the Optimal Nonlinear Parameters of a Miniature Transducer with a Metal Membrane
Abstract
:1. Introduction
2. Mathematical Model
2.1. Transmission Equation
2.2. Nonlinear Parameter Fitting Equations
2.3. Calculation of Nonlinear Optimization Parameters
2.4. Broyden–Fletcher–Goldfarb–Shanno (BFGS) Algorithm
2.5. Brent’s Algorithm
- The selected value must be within the [a, c] interval.
- The variation approaching the optimum value cannot exceed the previous variation by more than half, which means that the following inequality must be met: .
2.6. Constraint Equations
2.7. Computational Algorithm
- Step 1
- Let the search index n = 0, and arbitrarily choose a set of initial values of a vector as w(0).
- Step 2
- Substitute w(n) into transmission Equations (1)–(4) using numerical solver as finite difference method to obtain the values of x(t;w) and i(t;w), and use Equation (8) to obtain J(w).
- Step 3
- Solve Equations (13) and (14) for and substitute it into Equation (12) to obtain E(n). Let when k = 0; otherwise, substitute E(n) into Equation (11) to obtain A(n). Then, substitute and A(n) into Equation (10) to get search direction P(n).
- Step 4
- For the step length, the two methods are proposed. The first is to solve Equation (16) to obtain step length . The second is to use the Brent’s method (Equation (17a)), which is described in Section 2.5 to get .
- Step 5
- After obtaining P(n) and , the new w(n+1) can be obtained from Equation (9). If constraint equations in Section 2.6 are applied, substitute , , and w(n+1) into Equations (20)–(23) to make sure w(n+1) are matching the constraints, if not, update the w(n+1) from constraint Equations (20)–(23).
- Step 6
- Set the search iteration to n = n + 1 and determine whether J(w) is smaller than the preset tolerance of convergence criteria or the specified number of iterations. If the constraint of the convergence criteria is satisfied, end the iteration process; otherwise, start from Step 2 again.
3. Simulation Results and Empirical Results Verifications
3.1. Simulation Results
3.2. Experimental Equipment Establishment
3.3. Discussion of Empirical Results
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
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Parameters | Values |
---|---|
Mm | 0.049 ×10−3 (kg) |
Re | 4.560 (Ohm) |
Bl(x) | (N/A) |
Rm(x) | (kg/s) |
Le(x) | (H) |
Km(x) | (N/m) |
Parameters (Unit) | Values | ||
---|---|---|---|
Case1 | Case2 | Case3 | |
Mm (kg) | 0.049 × 10−3 | 0.003 | 0.065× 10−6 |
Re (Ohm) (fixed) | 4.560 | 4.560 | 4.560 |
Bl(x) (N/A) | |||
Rm(x) (kg/s) | |||
Le(x) (H) | |||
Km(x) (N/m) |
Case | Iteration Numbers | Minimal Value of J | Average Error for Each Parameters (%) | ||||
---|---|---|---|---|---|---|---|
Mm | Bl(x) | Rm(x) | Le(x) | Km(x) | |||
1 | 51 | ||||||
2 | 143 | ||||||
3 | 130 |
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Jian, Y.-C.; Tsai, Y.-T.; Pawar, S.J. Parameter Optimization Method for Identifying the Optimal Nonlinear Parameters of a Miniature Transducer with a Metal Membrane. Appl. Sci. 2018, 8, 2647. https://doi.org/10.3390/app8122647
Jian Y-C, Tsai Y-T, Pawar SJ. Parameter Optimization Method for Identifying the Optimal Nonlinear Parameters of a Miniature Transducer with a Metal Membrane. Applied Sciences. 2018; 8(12):2647. https://doi.org/10.3390/app8122647
Chicago/Turabian StyleJian, Yin-Cheng, Yu-Ting Tsai, and S. J. Pawar. 2018. "Parameter Optimization Method for Identifying the Optimal Nonlinear Parameters of a Miniature Transducer with a Metal Membrane" Applied Sciences 8, no. 12: 2647. https://doi.org/10.3390/app8122647
APA StyleJian, Y.-C., Tsai, Y.-T., & Pawar, S. J. (2018). Parameter Optimization Method for Identifying the Optimal Nonlinear Parameters of a Miniature Transducer with a Metal Membrane. Applied Sciences, 8(12), 2647. https://doi.org/10.3390/app8122647