Minimal-Learning-Parameter Technique Based Adaptive Neural Sliding Mode Control of MEMS Gyroscope

This paper investigates an adaptive neural sliding mode controller for MEMS gyroscopes with minimal-learning-parameter technique. Considering the system uncertainty in dynamics, neural network is employed for approximation. Minimal-learningparameter technique is constructed to decrease the number of update parameters, and in this way the computation burden is greatly reduced. Sliding mode control is designed to cancel the effect of time-varying disturbance. The closed-loop stability analysis is established via Lyapunov approach. Simulation results are presented to demonstrate the effectiveness of the method.


Introduction
Recently, MEMS gyroscopes have been drawing growing attention because they intend to employ advanced control approaches to realize trajectories tracking and to handle system parametric uncertainties and disturbances.These intelligent control methods improve the performance of gyroscopes, so that the applications of MEMS gyroscopes are expanded.With the vigorous development of nonlinear system control methods [1][2][3][4][5][6][7][8], a variety of gyroscopic modal control methods emerged.
In [9,10], one adaptive operation strategy is presented to control the MEMS -axis gyroscope, which offers a larger operational bandwidth, absence of zero-rate output, self-calibration, and large robustness to parameter variations caused by fabrication defects and ambient conditions.In [11,12], the sliding mode control is proposed to handle the vibrating proof mass, which achieves better estimation of the unknown angular velocity than conventional model reference adaptive feedback controller.Since then, in the presence of significant uncertainties, the regulated model-based and non-modelbased sliding model control approaches are presented to improve tracking control of the drive and sense modes of the vibratory gyroscope in [13].In [14], an adaptive tracking controller with a proportional and integral sliding surface is proposed.
Neural network has an inherent ability to learn and approximate nonlinear functions [15,16], which can be utilized for unstructured uncertainties.Thus, an adaptive control strategy using radial basis function (RBF) network/Fuzzy Logic System compensator is presented for robust tracking of MEMS gyroscope in the presence of model uncertainties and external disturbances to compensate such system nonlinearities and improve the tracking performance in [17][18][19].However, in practical application, large amount of update parameters results in the computation burden of online learning.In [20], the minimal-learning-parameter technique is further incorporated into the high gain observer to greatly reduce the online computation burden.
Inspired by the above-mentioned discussions on designing intelligent controllers and reducing the number of online parameters, this paper will focus on constructing the new control scheme for MEMS gyroscopes to suppress the system parametric uncertainties and disturbances.The main contribution of this paper is that a single parameter is employed to replace weight matrix, which significantly reduces the computation burden.The structure of this paper is organized as follows.Section 2 formulates the dynamics of MEMS gyroscope.Section 3 studies the neural network of lumped parametric uncertainties.In Section 4, an adaptive neural sliding mode control strategy using the minimal-learning-parameter technique is designed and stability analysis is discussed.Numerical simulations are investigated to verify the superiority of the proposed approach in Section 5. Conclusions are given in Section 6.

Dynamics of MEMS Gyroscope.
As Figure 1 shows, the ideal MEMS gyroscope is a quality-stiffness-damping system.Considering the mechanical coupling caused by manufacturing defects, the dynamics of MEMS gyroscopes can be expressed as where  represents the mass of proof mass, Ω *  represents the input angular velocity,  and  represent the system generalized coordinates,   and   represent damping terms,   represents asymmetric damping term,   and   represent spring terms,   represents asymmetric spring terms, and  *  and  *  represent the control forces.In (1), the damping terms are affected by atmospheric pressure and spring terms are affected by ambient temperature.That means where  0 ,  0 , and  0 are the damping terms of MEMS gyroscope in normal atmospheric pressure,  0 ,  0 , and  0 are the spring terms of MEMS gyroscope in room temperature environment, Δ  , Δ  , and Δ  are the deviation in damping coefficients due to the changes of atmospheric pressure, and Δ  , Δ  , and Δ  are the deviation of damping coefficients because of the changes of ambient temperature.
Dividing both sides of (1) by reference mass , reference frequency  2  , and reference length   , the dynamics can be derived as where , and S = −S * /  .Then, (3) has the following form: where and ΔK = [ With the external disturbances caused by compound maneuvers, (4) is replaced by where d(t) is the external disturbance and d m = sup|d(t)|.Define lumped parametric uncertainties as Equation ( 5) can be written as where In order to make the uncertainty of MEMS gyroscope depicted in (6) controllable, there must be unknown matrices of appropriate dimensions G, H, and p(t) such that ΔA = CG, ΔB = CH, and d(t) = Cp(t).And C is selected as

Control Goal.
The control goal of this paper is to design a controller to steer the position  and the speed q to the desired trajectories q m () = [    ]  and q m () = [ ẋ Besides, the minimal-learning-parameter technique is further incorporated into the estimation of lumped parametric uncertainties P( q , q).

Brief Description of RBF Neural Network
A neural network is established to approximate the lumped parametric uncertainties P( q , q), which can be expressed as where θ ⊆ R n is the adjustable parameter matrix, ( q , q) is a nonlinear vector function of the inputs, and the RBF has the form where q mi is an -dimensional vector representing the center of the th basis function and   is the variance representing the spread of the basis function.
Suppose that  * are the optimal weight parameters; parametric uncertainties could be reexpressed as P ( q , q) =  * T  ( q , q) + , (10) where  is the optimal estimation error of RBF neural network and   = sup||.Thus, the estimation error can be written as where θ = θ −  * .

Adaptive Sliding Mode Control with Minimal-Learning-Parameter Technique
Define the system tracking error as e (t) = q (t) − q m (t) .
Select the sliding mode function as where  is satisfied with Hurwitz condition.The derivative of s(t) is = A q + Bq + Cu + P ( q , q) + d (t) − q m (t) Define  = ‖ * ‖ 2 and the estimation error is φ = φ − , where φ is the estimation of .
Remark 2. Compared with traditional results of MEMS control [15][16][17], in this paper, minimal-learning-parameter technique is employed for controller design to reduce computation burden.
Remark 3. When strong time-varying disturbances exist, the boundary layer is bigger than before and the estimation errors are increased.
The adaptive law of single parameter φ can be designed as where  > 0 and  > 0.
An adaptive item η is employed to estimate   , and the estimated error is η = η −   .The adaptive law of η is selected as where  > 0 and  > 0. Substitute ( 15) into ( 14): −  ė − Ks] + A q + Bq + P ( q , q) + d (t) − q m (t) Theorem 4. Considering that the nonlinear system ( 7) is with parametric uncertainties and disturbances, if controller (15) and updating laws ( 16) and ( 17) are designed, then the boundedness of all the closed-loop system signals included in (19) can be guaranteed.
Proof.Lyapunov function is selected as The derivative of Lyapunov function is Substituting ( 16) and ( 17) into (20), the following inequality is obtained: where  = 2/ and  = min { 1  2 } ,  1 ,  2 are the eigenvalues of matrix K. Furthermore, we have where  = (/)η T m ηm + (/2) 2 + 1/2.The solution of ( 22) is Then all the signals included in the Lyapunov function are bounded.This concludes the proof.
Remark 5.In practical application, the high-frequency switching control signals of MEMS gyroscopes result in serious chattering.Therefore, the saturation function sat() is used to replace the sign function sgn(s) in (15).The saturation function sat() has the form where  is a positive constant.

Numerical Simulation
In this section, the aforementioned control scheme of MEMS gyroscope is simulated, the controller of which is designed as (15), and the adaptive laws are proposed as ( 16) and (17).
Since the position of proof mass ranges within the scope of submillimeter and the natural frequency is generally in the range of kilohertz, the reference length is assumed as   = 10 × 10 −6 m and reference frequency is assumed as   = 1 kHz.
Suppose that the reference trajectories are   = sin(1.71),ẋ  = 1.71 cos(1.71),  = 1.2 sin(1.11),and ẏ  = 1.2 × 1.11 cos(1.11),respectively.depicted in Figure 4, the system using adaptive neural network sliding mode control with minimal-learning-parameter technique could track the reference signals very well.The position tracking errors and speed tracking errors are shown in Figure 5. Through the tracking simulations of MEMS gyroscope, the proposed approach has satisfying performance.

Conclusions
In this paper, an adaptive neural network sliding mode control strategy is proposed to compensate parametric uncertainties and external disturbances of MEMS gyroscopes.Based on Lyapunov criterion, system stability is guaranteed.With minimal-learning-parameter technique, the online computation burden is significantly reduced.Numerical simulations verify that the novel control scheme could track reference trajectories very well, which is similar to conventional adaptive neural network sliding mode control scheme.Therefore, the control scheme proposed in this paper could force the mass moves along reference trajectories, so that the performances of MEMS gyroscopes are improved.
For future work, more efficient learning methods [21][22][23][24][25] will be tested on the dynamics while the implementation for real systems will be analyzed.

Figure 1 :
Figure 1: The basic principle diagram of -axis MEMS gyroscope.

Figure 5 :
Figure 5: Position and speed tracking errors.