Modelling and control design of the electrostatic linear comb actuator

Received Sep 16, 2021 Revised Mar 16, 2022 Accepted Apr 1, 2022 Control design plays a major part in electrostatic linear comb actuator (ELCA) systems. Most existing control methods of ECLA mainly focus on traditional PID method. However, proportional–integral–derivative (PID) controller is only appropriate for the reference to be known as constant. In this article, the problem of establishing the dynamic model and designing the controller is investigated for ECLA working in the time-varying reference. The tracking effectiveness is constructed for ECLA systems based on the consideration of roots of the equivalent differential equation in tracking error model. Moreover, the convergence velocity is also tuned by changing the parameters in the presented control scheme. Simulation results guarantee the performance of the nonlinear control scheme as well as dynamic model to be established.


INTRODUCTION
Micro electro-mechanical system (MEMS) is a remarkable device and popularly considered in several directions such as robots, vehicles, aerospace, industrial electronics, medicine, etc. Micro actuators can be known as one of the MEMS devices as well as describe an efficient task creating a force to move the micro elements and micro systems. Electrostatic linear comb actuator (ELCA) is a remarkable instrument structure using the working description via electrostatic force for establishing linear motion. The ELCA systems have many advantages including high effectiveness, low power requirement, and simple structure. They are able to explain the fact of easy movement in MEMS elements, such as resonator systems [1], gyroscopes [2] and micro systems of the conveyer or transportation systems [3], micro motors or gripper [4], [5]. Several work of ELCA have been recently developed with many appropriate results including new structures, working effectiveness, manufacturing improvement, smart materials. Particularly, the fabrication effectiveness and the ELCA's movement stability guarantees the operating effectiveness of the elements. The characteristic of suspension beams with a crab-leg, a fold [6], [7] was developed to decrease the stiffness and change the deviation. A different direction on improving deviation was addressed [8], by decreasing the movement of out-of-plane from that generates the in-plane deviation. The work in [9] presented an actuator property after inserting more a module of a sensor as well as a sub-actuator for stability of control systems. The effect of side etching on deviation of ELCA with disability rent increase exemption DRIE method was pointed out in [10]. By computing the ELCA deviation in design objective [3], [4], [10]- [12], it is necessary to utilize the theoretical method to be developed by a static equilibrium equation and decreases a design time. The fact is that these elements manipulate in statement of motion by using AC voltage such as traditional waves. Hence, under the effect of force and damping follows an efficient property in estimating the deviation of ELCA, which was considered in [13], [14] with the EAs to be developed for moving resonators, gyroscopes. Moreover, the motion dynamic model including an efficient mass and factor of conversion of air damping was not considered in these references. To achieve better behavior, the consideration of using control scheme is becoming a promising method [8]. However, the controller of ECLA have been considered with the traditional PID method. It should be noted that conventional PID controller does not satisfy the tracking effectiveness under the time-varying reference. Moreover, nonlinear controller and optimal control scheme have been usually proposed for different systems, such as robotic systems, power electronics [15]- [25]. Motivated by the above-mentioned discussions, this article studies the dynamic model of ECLA and the corresponding controller. To improve the tracking effectiveness under time-varying reference, the controller is carried out based on the consideration of roots of differential equations. Figure 1 describes the structure of ELCA including a moving element  and fixed element ( and ). The moving element with moving comb fingers is pressured by four EB systems driving in y-direction based on the electrostatic force (EF), one terminal of each beam is kept on . The configuration of movable and fixed comb fingers can be seen with the same shape. The work in [4] shows the ECLA geometrical parameters in Figure 1, where b is a beam width, h is the device layer thickness, and ga is the gap. Where 0 = 8.854 × 10 −6 ρF/μm and ε = 1; V is the moving voltage. The stiffness in y-direction with four suspension beams system is described as (2) [4], [10]. In (3) shows the quadratic relation between the deviation of the ELCA and the DC moving voltage. The objective of this article is to establish the dynamic model of ELCA for designing the controller with the reference of output ( ) depending on time.

DYNAMIC MODEL OF ELCA AND CONTROL DESIGN
Under the AC moving voltage, it implies the movement of ELCA's shuttle to be similar with the movement to be described in Figure 2. Where K, C, M are the coefficients of conversion stiffness, conversion damping and conversion effective mass, respectively; Fe(t) is an electrostatic moving force.  (4).
For the requirement of calculating exactly the deviation of the ELCA using dynamic model, the factors K, M, C are necessary to be accurately identified. Moreover, the factor K in y-direction is given in (2) where ρ = 2330 kg/m 3 is the silicon mass density; At is the total top side area of the shuttle. Based on [15], the coefficient C in (4) can be calculated: considering the driving voltage with the square wave ( Figure 3).
considering the driving voltage with the sine wave function (Figure 4).
in this case, the deviation of dynamic model (4) is represented: according to initial condition: 1 (0) = 0anḋ1 (0) = 0, we have: with D to be chosen as: From (17) and (18), it implies that: hence, we only choose the parameter D such that the corresponding polynomial 2 + ( − ) + = 0 have all solutions being on the left side of complex plane.

THE SIMULATION AND EVALUATION OF RESEARCH RESULTS
The following simulations are performed to validate the tracking problem of output with the reference being the sinusoidal signal. The parameters of the proposed controller (18) are given as = 1, = 10, = 9, = 1. Two cases of sinusoidal reference signals are discussed as 1, ef ( ) = 2 ( + 2 ), 2, ef ( ) = 2 ( + 2 ). The simulations will focus on the command tracking effect of the proposed controller (18). It is clearly seen that, although the references are depended on time but the tracking effectiveness is guaranteed (Figures 5 and 6). Moreover, the performance of tracking problem can be adjusted by choosing the parameters in (8) based on the roots of corresponding polynomial 2 + ( − ) + = 0.

CONCLUSION
The article studies the dynamic model and control design of ELCA systems. Through the method of solving differential equation, the control scheme is implemented for ECLA in the presence of time varying reference. The tracking problem is analyzed through considering the solution of differential equation. By carrying out two simulation cases, it is concluded that the tracking problem is absolutely guaranteed with convergence rate to be depended on controller parameters. For future work, the problem of practical ECLA systems can be developed.