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

Authors: Bin Xu; Pengchao Zhang
Journal: Complexity
 (2017)

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## Abstract

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-learning-parameter 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.


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## Body

## 1. 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 [1–8], a variety of gyroscopic modal control methods emerged.In [9, 10], one adaptive operation strategy is presented to control the MEMS z-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-model-based 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 [17–19]. 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. Section2 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.


## 2. Problem Formulation

### 2.1. Dynamics of MEMS Gyroscope

As Figure1 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(1)mx¨+dxxx˙+dxy-2mΩz∗y˙+kxx-mΩz∗2x+kxyy=ux∗,my¨+dxxy˙+dxy+2mΩz∗x˙+kyy-mΩz∗2y+kxyx=uy∗,where m represents the mass of proof mass, Ωz∗ represents the input angular velocity, x and y represent the system generalized coordinates, dxx and dyy represent damping terms, dxy represents asymmetric damping term, kxx and kyy represent spring terms, kxy represents asymmetric spring terms, and ux∗ and uy∗ represent the control forces.Figure 1
The basic principle diagram ofz-axis MEMS gyroscope.In (1), the damping terms are affected by atmospheric pressure and spring terms are affected by ambient temperature. That means(2)dxx=dx0+Δdxx,dyy=dy0+Δdyy,kxx=kx0+Δkxx,kyy=ky0+Δkyy,kxy=kxy0+Δkxy,where dx0,dy0, and dxy0 are the damping terms of MEMS gyroscope in normal atmospheric pressure, kx0,ky0, and kxy0 are the spring terms of MEMS gyroscope in room temperature environment, Δdxx,Δdyy, and Δdxy are the deviation in damping coefficients due to the changes of atmospheric pressure, and Δkxx, Δkyy, and Δkxy are the deviation of damping coefficients because of the changes of ambient temperature.Dividing both sides of (1) by reference mass m, reference frequency ωo2, and reference length qo, the dynamics can be derived as(3)q∗qo+D∗mωoq∗qo+2S∗ωoq∗qo-Ωz∗2ωo2q∗qo+K∗mωo2q∗qo=u∗mωo2qo,where q∗=xy, u∗=ux∗uy∗, D∗=dxxdxydxydyy, S∗=0-Ωz∗Ωz∗0, and K∗=kxxkxykxykyy.Define new parameters asq=q∗/qo, u=u∗/mωo2qo, Ωz=Ωz∗/ωo, D=D∗/mωo, K=K∗/mωo2, and S=-S∗/ωo.Then, (3) has the following form:(4)q¨=2S-Dq˙+Ωz2-Kq+u,where D=D0+ΔD,K=K0+ΔK and D0=dx0/mωodxy0/mωodxy0/mωody0/mωo, K0=kx0/mωo2kxy0/mωo2kxy0/mωo2ky0/mωo2, ΔD=Δdxx/mωoΔdxy/mωoΔdxy/mωoΔdyy/mωo, and ΔK=Δkxx/mωo2Δkxy/mωo2Δkxy/mωo2Δkyy/mωo2.With the external disturbances caused by compound maneuvers, (4) is replaced by(5)q¨=2S-D0-ΔDq˙+Ωz2-K0-ΔKq+Cu+dt,where d(t) is the external disturbance and dm=supd(t).Define lumped parametric uncertainties as(6)Pq˙,q=-ΔDq˙-ΔKq.Equation (5) can be written as(7)q¨=Aq˙+Bq+Cu+Pq˙,q+dt,where A=2S-D0 and B=Ωz2-K0.Remark 1.
In order to make the uncertainty of MEMS gyroscope depicted in (6) controllable, there must be unknown matrices of appropriate dimensionsG,H, and p(t) such that ΔA=CG, ΔB=CH, and d(t)=Cp(t). And C is selected as C=1001.


### 2.2. Control Goal

The control goal of this paper is to design a controller to steer the positionq and the speed q˙ to the desired trajectories qm(t)=xmymT and q˙m(t)=x˙my˙mT. Besides, the minimal-learning-parameter technique is further incorporated into the estimation of lumped parametric uncertainties P(q˙,q).


## 2.1. Dynamics of MEMS Gyroscope

As Figure1 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(1)mx¨+dxxx˙+dxy-2mΩz∗y˙+kxx-mΩz∗2x+kxyy=ux∗,my¨+dxxy˙+dxy+2mΩz∗x˙+kyy-mΩz∗2y+kxyx=uy∗,where m represents the mass of proof mass, Ωz∗ represents the input angular velocity, x and y represent the system generalized coordinates, dxx and dyy represent damping terms, dxy represents asymmetric damping term, kxx and kyy represent spring terms, kxy represents asymmetric spring terms, and ux∗ and uy∗ represent the control forces.Figure 1
The basic principle diagram ofz-axis MEMS gyroscope.In (1), the damping terms are affected by atmospheric pressure and spring terms are affected by ambient temperature. That means(2)dxx=dx0+Δdxx,dyy=dy0+Δdyy,kxx=kx0+Δkxx,kyy=ky0+Δkyy,kxy=kxy0+Δkxy,where dx0,dy0, and dxy0 are the damping terms of MEMS gyroscope in normal atmospheric pressure, kx0,ky0, and kxy0 are the spring terms of MEMS gyroscope in room temperature environment, Δdxx,Δdyy, and Δdxy are the deviation in damping coefficients due to the changes of atmospheric pressure, and Δkxx, Δkyy, and Δkxy are the deviation of damping coefficients because of the changes of ambient temperature.Dividing both sides of (1) by reference mass m, reference frequency ωo2, and reference length qo, the dynamics can be derived as(3)q∗qo+D∗mωoq∗qo+2S∗ωoq∗qo-Ωz∗2ωo2q∗qo+K∗mωo2q∗qo=u∗mωo2qo,where q∗=xy, u∗=ux∗uy∗, D∗=dxxdxydxydyy, S∗=0-Ωz∗Ωz∗0, and K∗=kxxkxykxykyy.Define new parameters asq=q∗/qo, u=u∗/mωo2qo, Ωz=Ωz∗/ωo, D=D∗/mωo, K=K∗/mωo2, and S=-S∗/ωo.Then, (3) has the following form:(4)q¨=2S-Dq˙+Ωz2-Kq+u,where D=D0+ΔD,K=K0+ΔK and D0=dx0/mωodxy0/mωodxy0/mωody0/mωo, K0=kx0/mωo2kxy0/mωo2kxy0/mωo2ky0/mωo2, ΔD=Δdxx/mωoΔdxy/mωoΔdxy/mωoΔdyy/mωo, and ΔK=Δkxx/mωo2Δkxy/mωo2Δkxy/mωo2Δkyy/mωo2.With the external disturbances caused by compound maneuvers, (4) is replaced by(5)q¨=2S-D0-ΔDq˙+Ωz2-K0-ΔKq+Cu+dt,where d(t) is the external disturbance and dm=supd(t).Define lumped parametric uncertainties as(6)Pq˙,q=-ΔDq˙-ΔKq.Equation (5) can be written as(7)q¨=Aq˙+Bq+Cu+Pq˙,q+dt,where A=2S-D0 and B=Ωz2-K0.Remark 1.
In order to make the uncertainty of MEMS gyroscope depicted in (6) controllable, there must be unknown matrices of appropriate dimensionsG,H, and p(t) such that ΔA=CG, ΔB=CH, and d(t)=Cp(t). And C is selected as C=1001.


## 2.2. Control Goal

The control goal of this paper is to design a controller to steer the positionq and the speed q˙ to the desired trajectories qm(t)=xmymT and q˙m(t)=x˙my˙mT. Besides, the minimal-learning-parameter technique is further incorporated into the estimation of lumped parametric uncertainties P(q˙,q).


## 3. Brief Description of RBF Neural Network

A neural network is established to approximate the lumped parametric uncertaintiesP(q˙,q), which can be expressed as(8)P^q˙,q∣θ=θ^Tμq˙,q,where θ^⊆Rn is the adjustable parameter matrix, μq˙,q is a nonlinear vector function of the inputs, and the RBF has the form(9)μi=exp-q-qmi2σi2,i=1,2,…,n,where qmi is an n-dimensional vector representing the center of the ith basis function and σi is the variance representing the spread of the basis function.Suppose thatθ∗ are the optimal weight parameters; parametric uncertainties could be reexpressed as(10)Pq˙,q=θ∗Tμq˙,q+ε,where ε is the optimal estimation error of RBF neural network and εn=supε.Thus, the estimation error can be written as(11)P-P^=θ∗Tμq˙,q+ε-θ^Tμq˙,q=-θ~Tμq˙,q+ε,where θ~=θ^-θ∗.


## 4. Adaptive Sliding Mode Control with Minimal-Learning-Parameter Technique

Define the system tracking error as(12)et=qt-qmt.Select the sliding mode function as (13)st=e˙t+βet,where β is satisfied with Hurwitz condition.The derivative ofs(t) is(14)s˙t=e¨t+βe˙t=q¨t-q¨mt+βe˙t=Aq˙+Bq+Cu+Pq˙,q+dt-q¨mt+βe˙t.Define ϕ=θ∗2 and the estimation error is ϕ~=ϕ^-ϕ, where ϕ^ is the estimation of ϕ.Assume thats˙=0; according to (14), controller could be designed as(15)u=C-1q¨m-Aq˙-Bq-12sϕ^μTμ-η^m∗sgns-βe˙-Ks,where ηm=supη, η=ε+d(t), η^m∗sgns is Hadamard product item, and -C-1Ks is a robust item.Remark 2.
Compared with traditional results of MEMS control [15–17], 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 parameterϕ^ can be designed as(16)ϕ^˙=γ2sTsμTμ-κγϕ^,where γ>0 and κ>0.An adaptive itemη^m is employed to estimate ηm, and the estimated error is η~m=η^m-ηm. The adaptive law of η^m is selected as(17)η^˙m=τs-αη^m,where τ>0 and α>0.Substitute (15) into (14):(18)s˙t=q¨m-Aq˙-Bq-12sϕ^μTμ-η^m∗sgns-βe˙-Ks+Aq˙+Bq+Pq˙,q+dt-q¨mt+βe˙t=-12sϕ^μTμ-η^m∗sgns+θ∗Tμq˙,q+ε+dt-Ks.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(19)L=12sTs+12γϕ~2+12τη~mTη~m.The derivative of Lyapunov function is(20)L˙=sTs˙+1γϕ~ϕ^˙+1τη~mTη^˙m=sT-12sϕ^μTμ-η^m∗sgns+θ∗Tμq˙,q+ε+dt-Ks+1γϕ~ϕ^˙+1τη~mTη^˙m=-12sTsϕ^μTμ+sTθ∗Tμq˙,q+sTε+dt-η^m∗sgns-sTKs+1γϕ~ϕ^˙+1τη~mTη^˙m≤-12sTsϕ^μTμ+12sTsϕμTμ+12+sTη-η^m∗sgns-sTKs+1γϕ~ϕ^˙+1τη~mTη^˙m≤-12sTsϕ~μTμ+12+sTη-ηm∗sgns+ηm∗sgns-η^m∗sgns-sTKs+1γϕ~ϕ^˙+1τη~mTη^˙m≤ϕ~-12sTsμTμ+1γϕ^˙+12+sTηm-η^m∗sgns-sTKs+1τη~mTη^˙m.Substituting (16) and (17) into (20), the following inequality is obtained:(21)L˙≤-κϕ~ϕ^+12-sTKs-sTη~m∗sgns+η~mTs-αη~mTη^m≤-κ2ϕ~2-ϕ2+12-sTKs-αη~mTη^m=-κ2ϕ~2-sTKs+κ2ϕ2+12-αη~mTη^m,where κ=2λ/γ and λ=minλ1λ2,λ1,λ2 are the eigenvalues of matrix K. Furthermore, we have(22)L˙≤-κ2ϕ~2-sTKs+κ2ϕ2+12≤-λγϕ~2-sTKs+κ2ϕ2+12=-2λ12γϕ~2+12sTs+κ2ϕ2+12=-2λL-12τη~mTη~m+κ2ϕ2+12=-2λL+Q,where Q=λ/τη~mTη~m+κ/2ϕ2+1/2.
The solution of (22) is(23)L≤Q2λ+L0-Q2λe-2λt.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 functionsat(s) is used to replace the sign function sgns in (15). The saturation function sat(x) has the form(24)satx=1x>axax≤a-1x<-a,where a is a positive constant.


## 5. 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).Parameters of the MEMS gyroscope are as follows:(25)m=0.57×10-8kg,dxx=0.429×10-6Ns/m,dyy=0.0429×10-6Ns/m,dxy=0.0429×10-6Ns/m,kxx=80.98N/m,kyy=71.62N/m,kxy=5N/m,Ωz=5.0rad/s.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 asqo=10×10-6m and reference frequency is assumed as ωo=1kHz.Suppose that the reference trajectories arexm=sin1.71t, x˙m=1.71cos⁡1.71t, ym=1.2sin⁡1.11t, and y˙m=1.2×1.11cos⁡1.11t, respectively.Then set other simulation parameters as(26)A=-0.0750.0025-0.0175-0.0075,B=-14207-877-877-12564,C=1001,K=8004,β=200015,γ=80,κ=0.01,τ=80,α=0.01.And select the initial state values of the system as0.8010T. The centers of basis function for network are uniformly valued in [-11], and the spreads of the basis function are σi=1. The number of neural network nodes is chosen as 256. The adaptive signals are presented in Figure 2, and the control inputs are shown in Figure 3. As 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.Figure 2
Adaptive signals.Figure 3
Control inputs.Figure 4
Position and speed trajectories.Figure 5
Position and speed tracking errors.


## 6. 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 [21–25] will be tested on the dynamics while the implementation for real systems will be analyzed.



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