Positional Regulation of Electrostatic Micro-electromechanical Actuator via Adaptive Two-stage Sliding Mode Control

In this study, we examine an adaptive two-stage sliding mode control scheme for an electrostatically actuated micro-electromechanical (MEM) actuator. The objective of the control scheme is to regulate the position of the movable plate in the MEM actuator by voltage control. For accurate measurements of system states, direct detection is possible by means of an existing sensing technique. Two main steps are applied to design the closed-loop control system. Firstly, for the dynamical model of the MEM actuator, a normalized time-scale approach and a nonlinear state transformation are introduced. Then, the adaptive two-stage sliding mode control is designed to achieve the control goal. The stability of the closed-loop control is proven and numerical studies to verify the control scheme are also performed.


Introduction
Nowadays, the electrostatic micro-electromechanical (MEM) actuator is one of the most commonly applied devices in practical use, such as in micromirrors, microresonators, microswitches, and accelerometers. (1) Owing to the simplicity of the structure, the rapid response, and the low power consumption, parallel-plate-type electrostatic MEM actuators are mostly used. The one-dimensional parallel-plate electrostatic MEM actuator consists of two fixed plates at the top and bottom of the component and a moving plate in an electrical field. Because the capacitance is directly related to the position of the moving plate, a positional sensor that uses capacitance measurements to measure displacements can be developed. (2) The positional regulation of the electrostatic MEM actuator is important in these applications. To solve the control problem of the parallel-plate-type microactuator, both charge control (3)(4)(5)(6) and voltage control (7,8) approaches are utilized. In practical applications, voltage control is more easily implemented than charge control. (9) Many studies have been published on voltage control approaches, such as static and dynamic output feedback, (10) nonlinear control, (11) passive control, (12) back-stepping control, (13) gain scheduling H∞ control, (14) linear parameter-varying control, (15,16) adaptive fuzzy control, (17) an artificial algorithm method, (18) and extended state observer-based control. (9,19) Among these studies, in Ref. 10, four output feedback control strategies were investigated by the input-output linearization approach. In Refs. 11 and 13, Lyapunov-based back-stepping design methods were used to propose a nonlinear control scheme for an electrostatic microactuator. In Ref. 14, gain-scheduling H∞ control by a linearized model was reported, and linear time-varying techniques were applied to the voltage control problem in Refs. 15 and 16. Intelligent control methods, such as adaptive fuzzy control (17) and the artificial algorithm (18) approach, were introduced for the development of control strategies for electrostatic MEM actuators. In Refs. 9 and 19, active disturbance rejection control schemes, which consist of an extended state observer and PD-type control, were developed in the presence of external disturbance and noise.
Compared with previous studies, (10,11,13) the novelty of this study is the introduction of an adaptive two-stage sliding mode control scheme for the positional regulation of a parallel-plate-type electrostatically actuated MEM actuator. There are three steps to develop the control scheme. Firstly, the dynamical model for the MEM actuator is normalized by means of a normalized time-scale approach. Then, a nonlinear state transformation is defined to transform the normalized dynamical model to a nearly controllable canonical form with a nonlinear cubic-power term. An adaptive two-stage sliding mode control scheme based on the nearly controllable canonical form is introduced. In the designed adaptive sliding mode control scheme, two types of sliding function are involved. The level 1 sliding function is defined by the nonlinear transformed states of the dynamical system, where the asymptotic stability of transformed states is inherent. The level 2 sliding function is formed by the level 1 sliding function, with which finite-time stabilization is guaranteed. The proposed sliding mode control scheme also includes time-varying feedback gains, which are updated according to suitable adaptation rules. This paper is organized as follows. The dynamical model of an electrostatic MEM microactuator is introduced and the normalized time-scale approach and nonlinear state transformation are also discussed in Sect. 2. The design of the control scheme is outlined and the guaranteed stability is proven in Sect. 3. In Sect. 4, numerical studies made to illustrate the effectiveness of the present scheme are reported. Finally, conclusions are given in Sect. 5.

Dynamical Model of Electrostatic Microactuator
The one-dimensional model of an electrostatic MEM microactuator is shown in Fig. 1. This microactuator has three parallel plates with a middle moving plate (electrode) and top and bottom fixed plates.
In the system, the capacitance of the microactuator is C(t) = εA/y(t), where ε and A are the permittivity and the cross-sectional area of an electrode, respectively. The attractive electrostatic force from the top electrode is f(t) = Q 2 (t)/2εA, where Q(t) is the charge on an electrode. The dynamical equation of the model depicted in Fig. 1 is given as (20) 2 where y(t) is the gap between the moving and bottom electrodes. y 0 is the zero-voltage gap. M, B, and K are the mass of the moving electrode, the damping ratio, and the linear spring constant, respectively. I(t) and V(t) are the current in the device and the control voltage, respectively. The state vector of the system is [ ] . δ 0 denotes the thickness of the insulating layer coated on the bottom electrode; δ 0 is helpful in preventing a singularity in the mathematical model. For the accurate measurement of system states, the direct detection of the velocity ( ) y t  of the moving electrode is possible. (21) The charge Q(t) on the device can be inferred from the measurements of voltage and capacitance. It is possible for the sensing system to detect the voltage between the moving and bottom electrodes and the capacitance across the device. (22) For convenience in performing the normalized time scaling of Eq. (1), we defined a positive constant σ and propose the following normalized terms: Moreover, let u(τ) = V/βR be the normalized control input. The normalized state vector for Eq. (1) is chosen as Then, the normalized state equation of an electrostatic MEM microactuator is described as The goal of the controlled microactuator is to regulate the state vector Z(τ) to the constant desired state vector Z d . For the system given by Eq. (4), the constant vector Z d is defined as [ where z 1d = q d is the normalized desired charge. z 2d and z 3d are the normalized desired gap and velocity, respectively. It can be verified that the normalized desired charge is , when the normalized gap z 2 (τ) reaches the normalized desired gap z 2 = z 2d . To this end, the control objective of the position-regulating problem for the electrostatic MEM microactuator is to determine an appropriate control scheme u(τ), such that the error state vector Z(τ) − Z d converges to zero, that is, To develop the control scheme of the position-regulating problem, we further consider the following nonlinear state transformation: From Eq. (6), it is easy to check that the transformed state vector X(τ) converging to zero implies that the state vector Z(τ) converges to the constant desired state vector Z d . Thus, the positional regulation of the system given by Eq. (1) is equivalent to the state stabilization problem. That is, 3 1 lim ( ) 0 X τ τ × →∞ → for any initial transformed state vector X(0).
In addition, by considering the system uncertainty Δ(τ) and external disturbance d(τ), the transformed state equation of the microactuator system can be presented as follows using the state transformation in Eq. (6): where the appended control input is

Design of Control Scheme
In the following, the rule of thumb for the adaptive two-stage sliding mode control scheme is introduced. There are two basic steps to stabilize the system in Eq. ; then, any initial value of s(τ) converges to and remains at the origin even in the presence of system uncertainty Δ(τ) and external disturbance d(τ). When the status of s(τ) = 0 is maintained, it means that the trajectory in phase space for the transformed states given by Eq. (6) is stabilized under the embedded converging motion. Then, the system in Eq. (7) is stabilized.
The level 1 sliding function s(τ) is defined as where γ is the design parameter and λ i , i = 1, 2, 3 are positive parameters. They are calculated from [ ] where the symmetric matrix P is obtained by solving the following algebraic Riccati equation: where [ ] Because the characteristic equation c 3 + λ 3 c 2 + λ 2 c + λ 1 =0 is Hurwitz, the asymptotic stability of Eq. (12) is guaranteed.
The level 2 sliding function σ(τ) is defined as where k > 0, p > q, p > m, m + q > p, and p, q, m are positive and odd integers. are satisfied such that the transformed states x i , i = 1, 2, 3 tend to zero asymptotically according to Eq. (12).

Theorem
If the input control u(τ) in the system in Eq. (7) in the nonlinear feedback control scheme is designed to be the appended control input U(τ) has the form ( ) where s(τ) and σ(τ) are defined in Eqs. (8) and (13), respectively. The positive and adaptive feedback gains K i (τ), i = 0, 1, 2, 3 are updated according to the following adaptation algorithms: where K i (0) = 0 and ρ i , i = 0, 1, 2, 3 are the positive constants determined in the adaptation process. Then, the level 1 sliding function σ(τ) will be asymptotically stabilized and remain at σ(τ) = 0 and ( ) 0 t σ =  . It follows that the level 2 sliding function s(τ) converges to zero in the finite time T s defined in Eq. (16). When the status of s(τ) = 0 is maintained, the trajectory in phase space for the transformed states x(τ) i , i = 1, 2, 3 of the system in Eq. (7) is stabilized in the inherent converging motion given by Eq. (12), where the positive parameters λ i , i = 1, 2, 3 are obtained from Eqs. (9)- (11). Then, the position regulation control problem of the system in Eq. (4) can be solved.

Proof
The Lyapunov candidate function is selected to be where , 0,1, 2,3 i K i = , i = 0, 1, 2, 3 are sufficiently large positive constants and satisfy the following inequalities: Taking the derivative of Eq. (21) with respect to τ along with the solutions of the system in Eq. (7), the two sliding functions Eqs. (8) and (13), and the control law in Eq. (17), we obtain Therefore, the condition of 0 V <  is satisfied. The level 2 sliding function σ(τ) can reach σ(τ) = 0 and ( ) 0 σ τ =  asymptotically. Then, the level 1 sliding function s(τ) approaches zero in the finite time T s given by Eq. (16). The conditions s(τ) = 0 and ( ) 0 are satisfied such that the transformed states x i (τ), i = 1,2,3 defined in Eq. (12) tend to zero asymptotically. It is thus proven that the positional regulation is accomplished.

Numerical Studies and Discussion
In the following section, the effectiveness of the proposed adaptive two-stage sliding mode control scheme for the positional regulation of the system in Eq. (4) Figure 3 shows the time responses of the adaptive feedback gains K i (τ), i = 0, 1, 2, 3 and the control input u(τ). When compared with the past work (23) for the robust controller design of an electrostatic MEM actuator, the proposed control input u(τ) is clearly continuous and chatter-free in the appearance of system uncertainties and external disturbances. Figure 4 demonstrates the time responses of transformed states x i (τ), i = 1, 2, 3. It is clear that the system in Eq. (7) is stabilized. Figure 5 depicts the time responses of the normalized charge, z 1 (τ), the normalized gap z 2 (τ), and the normalized velocity z 3 (τ). When z 2 (∞) → z 2d = 0.6 is achieved, the problem of the positional regulation of the system in Eq. (4) is solved.
In the other case of the normalized desired gap, z 2d = 0.2, the normalized desired charge is z 1d = 1 1.6 1.2649 d z = = = 1.2649. Figure 6 depicts the time responses of s(τ), σ(τ), and the state trajectory of the controlled system in the s(τ) vs σ(τ) plane. In Fig. 7, it is shown that the positional regulation of the system in Eq. (4) is completed for z 2 (∞) → z 2d = 0.2.   The proposed control scheme can be implemented similarly to those in previous studies. (21,22) The direct detection of the velocity ( ) y t  for the moving electrode is possible by the accurate measurement of system states. (21) The charge Q(t) on the MEM actuator can be inferred from the measurements of voltage and capacitance. The sensing system can be set to sense the voltage between the moving and bottom electrodes, and the capacitance across the device. (22)

Conclusion
In this study, an adaptive two-stage sliding mode control scheme for achieving the positional regulation of an electrostatic MEM actuator has been proposed. To develop the control scheme, a normalized time-scale approach and a nonlinear state transformation are applied to the dynamical model. It is proved that the proposed control scheme can guarantee the regulation of the states of the system in Eq. (4) to the desired values. Numerical studies are performed for two cases to demonstrate the effectiveness of the presented control scheme.