Adaptive Fuzzy Sliding Mode Control of MEMS Gyroscope with Finite Time Convergence

This paper presents adaptive fuzzy finite time sliding mode control of microelectromechanical system gyroscope with uncertainty and external disturbance. Firstly, fuzzy system is employed to approximate the uncertainty nonlinear dynamics. Secondly, nonlinear sliding mode hypersurface and double exponential reaching law are selected to design the finite time convergent sliding mode controller. Thirdly, based on Lyapunov methods, adaptive laws are presented to adjust the fuzzy weights and the system can be guaranteed to be stable. Finally, the effectiveness of the proposed method is verified with simulation.


Introduction
MEMS gyroscopes have become the most growing microsensors in recent years due to the characteristics of compact size, low cost, and high sensitivity.Most MEMS gyroscopes sales in the market are vibrating silicon micromechanical gyroscopes, whose basic principle is to generate and detect Coriolis Effect.As depicted in Figure 1, under assumption that the proof mass  of gyroscope rotates around -axis at a speed of ⃗ Ω and makes uniform motion along the -axis at a speed of ⃗ ^, a Coriolis force of ⃗  = −2 ⃗ Ω × ⃗ ^is produced along -axis.In the last few years, numerous advanced control approaches with intelligent design have been studied to realize the trajectory tracking in [1][2][3][4] and to handle the system parametric uncertainties and disturbances, and the adaptive control can be found in [5][6][7].For control of MEMS gyroscope, Park and Horowitz firstly applied adaptive state feedback control method [8].Both drive shaft and sensitive axis were subjected to feedback control force in this control method, which administered two axial modal vibration track specified reference trajectories, weakening the boundary between drive mode and test mode as well.
Sliding mode control changes its structure to force the system in accordance with a predetermined trajectory.Batur et al. developed a sliding mode control for MEMS gyroscope system in [9].Since then, adaptive sliding mode control approach with the advantages of variable structure methods and adaptive control strategies are presented to control MEMS gyroscopes in [10,11].
Due to the necessity of ideal sliding mode, good dynamic quality, and high robustness, several methods are extended to improve the performance.Yu and Man investigated a nonlinear sliding mode hypersurface to ensure that systems from any point of the sliding mode surface were able to reach the balance point in a limited time in [12][13][14][15][16]. Bartoszewicz [17] examined the reaching laws introduced by Gao and Hung in [18] and proposed an enhanced version of those reaching laws, which was more appropriate for systems subject to constraints.Recently, Fallaha et al. studied a novel approach, which allowed chattering reduction on control input while keeping tracking performance in steady-state regime [19].This approach consisted of designing a nonlinear reaching law by using an exponential function that dynamically adapted to the variations of the controlled system.Mei and Wang in [20] proposed a nonlinear sliding mode surface which converged to the equilibrium point with a higher speed than both linear sliding mode surface and terminal sliding mode surface.In addition, a new two-power reaching law was proposed to make the system move toward the sliding mode faster.
As a matter of fact, the methods mentioned above are highly dependent on the structure of the nonlinearity, while, currently, accurate model is unavailable.Thus, fuzzy model has been widely used to approximate nonlinear objects in [21,22].Robust adaptive sliding mode control with on-line identification for the upper bounds of external disturbance and estimator for the nonlinear dynamics of MEMS gyroscope uncertainty parameters was proposed in [23].
In this paper, an adaptive fuzzy sliding mode control strategy with nonlinear sliding mode hypersurface and double exponential reaching law is developed to track MEMS gyroscope.Furthermore, it converges faster compared with strategies using conventional sliding mode surface in [23] and terminal sliding mode surface in [12][13][14][15][16].
The rest of this paper is organized as follows.The dynamics of MEMS gyroscope with parametric uncertainties and disturbances are given in Section 2. Controller design and stability analysis are discussed in Section 3. Numerical simulations are conducted to verify the superiority of the proposed approach in Section 4, compared with conventional adaptive fuzzy sliding mode control.Conclusions are drawn in Section 5.

Dynamics of MEMS Gyroscope
The basic principle of -axis vibratory MEMS gyroscope is shown in Figure 2, which can be described as a qualitystiffness-damping system.Owing to mechanical coupling caused by fabrication imperfections, the dynamics can be derived as where  is the mass of proof mass; Ω *  is the input angular velocity; ,  represent the system generalized coordinates;   ,   represent damping terms;   represents asymmetric damping term;   ,   represent spring terms;   represents asymmetric spring terms; and  *  ,  *  represent the control forces.
On issues related to the study of mechanism, the law described by model is required to be independent of dimensions.So, it is necessary to establish nondimensional vector dynamics.Because of the nondimensional time  * =   , both sides of (1) should be divided by reference frequency  2  , reference length   , and reference mass .Then the dynamics can be rewritten in vector forms: where New parameters are defined as follows: Thus, the final form of the nondimensional vector dynamics is In presence of parametric uncertainties and external disturbance, based on (4), state equation of dynamics is established as q = ( + Δ) q + ( + Δ)  +  +  () , where  ∈  2×2 ,  ∈  2×2 ,  ∈  2×2 are system known matrices; Δ, Δ are parametric uncertainties; and () is an external disturbance.Besides,  = 2 − ,  = Ω 2  −  1 .If the system total interference (consisting of parametric uncertainties and external disturbance) is represented by (), we know q =  q +  +  +  ( q , , ) , where ( q , , ) = Δ q + Δ + ().
It is vital that (6) must meet the following assumptions.
Based on the above assumptions, the controller can be designed to compensate the total interference.

Adaptive Fuzzy Finite Time Sliding Mode Control
The fuzzy model of () could be composed of  IF-THEN rules, and the th rule has the form Based on singleton fuzzifier, product inference, and centeraverage defuzzifier, its output can be expressed as where ( q , ) = ( The fuzzy sets of input variables are defined as {, , }, where  is negative,  is zero, and  is positive.Then  membership functions of ẋ are selected as the following triangular functions: The corresponding membership functions of these fuzzy sets labels are depicted in Figure 3.In addition, the membership functions of ẏ  ,   , and   are the same with ẋ  .
Based on the aforementioned fuzzy sets and membership functions, the fuzzy rules are described in Table 1.Therefore, 81 fuzzy rules are chosen.
The control target for MEMS gyroscope is to maintain the proof mass oscillation at given frequency and amplitude, such as   =   sin(  ),   =   sin(  ) in the  and  directions, respectively.So, reference model can be designed as where And the tracking error is defined as Nonlinear sliding mode hypersurface is chosen as where  > 0,  > 0; Then the reaching law is designed as the following double exponential function: where  1 > 0,  2 > 0, 0 <  < 1,  > 1.
According to (6), equivalent control law is obtained as And the derivative of sliding surface ( 11) is Then substituting ( 12) into ( 14), And substituting ( 15) into (13), Besides, a robust item is designed to guarantee that the system is asymptotically stable: Thus, the adaptive fuzzy finite time sliding mode controller is obtained as According to (10), we have ë = q − q  =  q +  +  +  ( q , , ) − q  .
Substituting ( 23) into (21), we derive Considering ( 7), ( 24) can be expressed as where   =  *  −   .So adaptive law can be selected as Namely, where φ  = − θ  .Lyapunov function is defined as Differentiate  with respect to time yields, and substitute ( 26) as Owing to the fuzzy approximation theory, adaptive fuzzy system can approximate nonlinear system closely.Therefore, V ≤ 0; namely, the system is asymptotically stable.

Simulation Study
In this section, numerical simulations are investigated to track the position and speed trajectories of MEMS gyroscope, compensate parametric uncertainties and external disturbances, and verify the superiority of the proposed approach compared with conventional adaptive fuzzy sliding mode control strategy using linear sliding mode surface.Those two methods are defined as follows.
Method 1. Define the adaptive fuzzy sliding mode control proposed in this paper as Method 1, whose sliding mode surface is shown in (11), and the reaching law is expressed in (12).
And select the initial state values of the system as [0.8 0 1 0]  .
Then the position and speed trajectories of Method 1 are shown in Figures 4 and 5 and those of Method 2 are depicted in Figures 6 and 7.
The position tracking error and speed tracking error of Methods 1 and 2 are shown in Figures 8-11, respectively.
Through the tracking simulation of MEMS gyroscope, the proposed approach is with satisfying performance; in addition, in comparison to Method 2, the convergence time is shortened to 0.3  from 0.6  .

Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategy using nonlinear sliding mode hypersurface and double exponential reaching law is proposed to compensate parametric uncertainties and external disturbance of MEMS gyroscope in this paper.Based on Lyapunov methods, the stability of system can be guaranteed.Simulations verify that, compared with conventional adaptive fuzzy linear sliding mode control strategy, the convergence time of finite time convergent control strategy proposed in this paper is shortened to 0.3  from 0.6  ; namely, convergence has been significantly improved.
For future work, the novel adaptive online constructing fuzzy algorithm [24] can be employed for more efficient learning while disturbance observer based design [25,26] can be considered to improve system performance.

Figure 2 :
Figure 2: The basic principle diagram of -axis vibratory MEMS gyroscope.