Implicit Euler Implementation of Twisting Controller and Super-Twisting Observer without Numerical Chattering: Precise Quasi-Static MEMS Mirrors Control

The quasi-static operations of MEMS mirror are very sensitive to undesired oscillations due to its very low damping. It has been shown that closed-loop control can be superior to reduce those oscillations than open-loop control in the literature. For the closed-loop control, the conventional way of implementing sliding mode control (SMC) algorithm is forward Euler method, which results in numerical chattering in the control input and output. This paper proposes an implicit Euler implementation scheme of super twisting observer and twisting control for a commercial MEMS mirror actuated by an electrostatic staggered vertical comb (SVC) drive structure. The famous super-twisting algorithm is used as an observer and twisting SMC is used as a controller. Both are discretized by an implicit Euler integration method, and their implementation algorithms are provided. Simulations verify that, as compared to traditional sliding mode control implementation, the proposed scheme reduces the chattering both in trajectory tracking output and control input in presence of model uncertainties and external disturbances. The comparison demonstrates the potential applications of the proposed scheme in industrial applications in terms of feasibility and


Introduction
MEMS (Micro-electro-mechanical systems) mirrors are micro mechanical systems that can perform high dynamic and precise beam positioning for many applications such as 1D/2D light detection and ranging (LIDAR), tunable laser spectrometer and micro laser projection displays. In comparison with traditional laser beam positioning devices such as galvanometer scanners and deflection mirror driven by voice coil motor, MEMS scanners have the advantages of small size, high repeatability, rapid scanning frequency, light-weight and low power consumption. The low mass and dimensional make MEMS mirrors suitable for compact designs.
In recent years, the close-loop control shows its advantages over open-loop control in some aspects, such as larger bandwidth and robustness of the external dynamic effects. Increasing bandwidth of MEMS mirrors means it can perform higher speed scanning and deflection of laser beam. Robustness to external disturbance indicates the trajectory and magnitude will be less affected by factors such as temperature changes and model uncertainties. Those advantages are desired for many applications. For example, MEMS mirror based LIDAR and micro laser projection display can be used as automotive sensor and head-up display, respectively, both of which need to keep its performances in very dynamical environments. Constructing a close-loop control needs the feedback signal of mirror angular position, which can be achieved onboard fabricated sensors such as piezo-resistivity or capacitances, or external sensors such as quad photodiode-based backside position sensor or PSD-based position detection module. The quality of feedback signal for various sensor fabrications can be very different and a robust control algorithm should be resistant to the feedback signal variance. This paper focus on sliding mode control to suppress the model uncertainties of MEMS mirror and external disturbance.
Sliding mode control (SMC) has been recognized as one of potentially useful control schemes due to its finite-time convergence, tracking accuracy and robustness against uncertainty [1], [2], [3]. In practice, the main drawback of SMC is numerical chattering which could cause damages to the actuators of systems and deteriorate the control performance. Several solutions have been proposed to alleviate the numerical chattering, such as higher order sliding mode (HOSM) [3], [4], adaptive sliding mode designs [5], [6], and implicit Euler methods [7], [8].
The HOSM is known to enable reduction of chattering by integrating the signum function just like the super-twisting algorithm. It, however, requires the derivatives of the sliding variable. It comes at the price of tolerating only a smaller class of disturbance than the first order SMC. Furthermore, by implementing with the conventional forward Euler integration method, it can still suffer from severe chattering effects. The adaptive sliding mode is to render gains adaptive in the conventional SMC. Since the magnitude of chattering is proportional to the gains, the chattering effect can be reduced if the gains automatically fit themselves to perturbations the SMC needs to counteract. The adaptive SMC can reduce the numerical chattering but it cannot totally remove it.
This paper applies implicit Euler integration method to the conventional sliding mode control for the MEMS mirror close-loop control. The super twisting algorithm is employed to obtain the velocity estimation while the twisting SMC is used for the close-loop control of the MEMS mirror. Both are discretized by using implicit Euler integration method. It is the first time to apply the implicit Euler implementation of SMC to MEMS mirror control for reducing chattering and improving tracking performance. The present work is the first time to show implicit Euler integration for the twisting SMC based on graph of the multi-valued signum function sgn(•). Some researchers give the implementation of implicit Euler integration based on ZOH (Zero-Order Hold) discretization and AVI (Affine Variational Inequality) [9].

Problem statement
The aim is to track a desired deflection trajectory by controlling the applied driving voltage 1 and 2 at the comb electrodes. The relationship between the deflection angle and the voltages 1 and 2 is shown as Fig. 1. The physical model of the MEMS (Micro-electro-mechanical systems) mirrors is as follows [10]: where ∶= ( 1 , 2 ) = ( ,) with the output ∶= 1 = , ∅( ) represents external disturbances and model uncertain-ties, J is the mirror inertia, b linear viscous damping, and ( 1 ) is the nonlinear spring torque: with positive linear spring coefficient 0 ， 2 > 0. Due to the dual comb structure of the MEMS mirror shown in Fig. 1(a), the input u is split into two parts , = 1,2: Where ′ ( ), = 1,2 are the capacitance derivatives and can be approximated by using  (1) as The objective is to design a close-loop control with flatness-based feedforward to calculate the command voltages ( 1 , 2 ) so that the 1 can follow the desired trajectory with small mean tracking error and high repeatability. The designed control * is the sum of feedforward and feedback control u: * = + (5) It should be noted that due to the staggered comb design, only an unidirectional torque can be produced. This means, the mirror combs can only pull but not push the mirror. This leads to a switch structure for the control design. A comb switch toggles between the comb electrode according to the sign of the desired generalized input * as follows:

Simulation validation
To validate the proposed algorithm with simulations, all parameters are set as the real experimental system of MEMS mirror described in [10]. A jerk-limited trajectory design is employed to reduce undesired oscillations: Let us design the control as follows: where the first three terms are used as the equivalent control while u is for sliding mode control. Substituting (28) into (27) leads to the formulation as (18) and the analysis follows that in Section 3.2. The control input u is updated by using Algorithm 1.  The comparison is shown in Fig. 3. In Fig. 3(a)-(f), they results shown both explicit and implicit Euler integrations can track the desired trajectory. The acceleration rate is changed at crossing points. Chattering means the actual angular is oscillated and unstable, which is a disaster to some precise instruments such as lidar and optical communicating system. Fig.  3(a)-(f) show that in the control input and control output, the chattering is drastically reduced by using the proposed implicit Euler method.

Conclusion
This paper proposed an implicit Euler implementation algorithm for MEMS mirrors driven by the electrostatic staggered vertical comb (SVC) drive structure. The system model is a typical second order system and only the position or angular can be obtained. The velocity is observed by using the conventional super-twisting algorithm while its implementation is done by using the proposed implicit Euler method. The implicit Euler implementation for the twisting control is relative more complicate than that of super-twisting and therefore a special algorithm is introduced. The simulation results and comparison show that the proposed implicit Euler method is superior than the conventional explicit Euler method in term of chattering magnitude in both sliding mode control input and output. Further study may focus on the proof of the convergence of the proposed implicit Euler integration and estimate the accuracy with disturbance existence. The experiments can be demonstrated with commercial MEMS mirrors integrated with some suitable angular position feedback sensor. Another interesting topic is to apply the proposed algorithm in other kind of higher order sliding mode control such as terminal sliding mode control with nested signum functions.