## 1 Introduction
The displacement measurement errors caused by diaphragm vibration in a miniature transducer with a metal membrane may encounter errors due to optical refraction during laser measurements. Hence, there is the need to increase diaphragm reflection accuracy by making a mark on the diaphragm. The measurement of displacement is the basis of measuring position, velocity, acceleration, stress, force, pressure, proximity, thickness, etc. A displacement sensor measures the distance between the sensor and an object to be detected by identifying the displacement over a variety of elements and converting it into a distance. In 2012, [PERSON] et al. [15] proposed a quartz crystal capacitive sensor that consists of a uniplanar scattering-field capacitance and a piezoelectric resonator for high sensitivity response (3.05 Hz/\(\upmu\)m in the range of 20-200 \(\upmu\)m). In the same year, [PERSON] et al. [16] proposed a low-power high-resolution capacitive sensor for fast vibration measurements. As shown in the studies that were reviewed above, the high precision and better resolution of capacitive sensor testing were the main reasons for measuring the diaphragm displacement of a miniature transducer with a composite metal membrane in this study. To identify the nonlinear parameters, among the conventional optimization methods for parameter estimation, such as the Stochastic Gradient Descent (SGM) [17], RMSProp and ADAM [18], Conjugate Gradient method (CGM) [19], and Broyden Fletcher Goldfarb Shanno Method (BFGS) [20], are the incremental gradient descent based iterative method for optimizing a differentiable objective function. The gradient descent algorithm updates the optimized parameters to minimize the objective function by taking small learning steps in the direction of the negative gradient of the loss of the objective function. The CGM searches from the conjugate direction along the negative gradient with the characteristics of double convergence from SGM; moreover, SGM and CGM use a single learning rate for all of the parameters. The RMSProp can improve by using learning steps that are different for different parameters and that can automatically adapt to the objective function being optimized. The BFGS method is one of the famous methods that belongs to quasi-Newton methods [20], which uses an estimation to the inverse Hessian matrix to guide its search through variable space. BFGS generally requires much fewer iterations to reach the local minimum as compared to CGM and SGM. In this paper, the miniature transducer's differential equation was established and the use of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [20] to identify the nonlinear parameters simultaneously during the optimization process was proposed in this study. The Brent method [21] was also utilized to improve the step size for accelerating iteration times. The transducer's equation of impedance curve as a constraint in order to ensure the correctness of optimization parameters and their correspondence with the miniature transducers' physical conditions was used. In order to account for measurement errors in the transducer's displacement, the distribution function with different standard deviations was used to simulate the measurement errors for verifying the results. The experiment system that was employed in this study utilized a high-precision capacitive micro-displacement sensor to measure the diaphragm displacement \(x(t)\). A high-precision electric current instrument was used to measure the transducer's electric current \(i(t)\). In the results and discussion sections, the numerical simulation tests and experiments are both discussed in detail.