# Actor Critic Neural Network-Based Adaptive Control for MEMS Gyroscopes Suffering from Multiresource Disturbances

Authors: Chao Jing; Gangzhu Qiao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, an actor critic neural network-based adaptive control scheme for micro-electro-mechanical system (MEMS) gyroscopes suffering from multiresource disturbances is proposed. Faced with multiresource interferences consisting of parametric uncertainties, strong couplings between axes, Coriolis forces, and variable external disturbances, an actor critic neural network is introduced, where the actor neural network is employed to estimate the packaged disturbances and the critic neural network is utilized to supervise the system performance. Hence, strong robustness against uncertainties and better tracking properties can be derived for MEMS gyroscopes. Aiming at handling the nonlinearities inherent in gyroscopes without analytically differentiating the virtual control signals, dynamic surface control (DSC) rather than backstepping control method is employed to divide the 2nd order system into two 1st order systems and design the actual control policy. Moreover, theoretical analyses along with simulation experiments are conducted with a view to validate the effectiveness of the proposed control approach.


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## Body

## 1. Introduction

Owing to the microvolume, easy integration, and mass production, micro-electro-mechanical system (MEMS) gyroscopes are indispensable angular rate sensors in a myriad of areas including inertial guidance, aerospace, and national defense industry [1–5]. Typically, the accurate detection of angular rates is highly dependent on the high-precision control of drive axis and sense axis. However, MEMS gyroscopes are inherently accounted for strong uncertainties and nonlinearities, rendering the high-precision control a challenging issue.Ideally, the parameters of MEMS gyroscopes are naturally fixed, including mass, damping parameters, and spring parameters, and mechanical motion of two axes are uncoupled. While in real word applications, manufacturing defects and unpredictable operation environments undesirably result in multiresource interferences, i.e., parametric uncertainties, mechanical coupling between axes, and external disturbances, which inevitably reduce the tracking accuracy of both axes and even lead to system instability. Aiming at reducing the effects of uncertainties and enhancing system behaviours, several advanced schemes have been devised, such as neural networks (NNs) [6–8], extended state observers (ESOs) [9], and fuzzy logic systems (FLSs) [10], where the packaged disturbances are online estimated and compensated in the control policy. In [11], on the basis of universal approximation theorem, the RBFNN is employed to online estimate unknown system dynamics, effectively enhancing system robustness against uncertainties. In [7–10, 12], the tracking errors as well as estimation errors are bounded, while the tracking performance is usually neglected, which plays a central role in the high-accuracy measurement of angular rates. Hence, it is meaningful to devise a control scheme capable of recovering unknown interferences and supervising system tracking properties at the same time.On the contrary, an appropriate stabilizing control policy is crucial to driving the output of MEMS gyroscopes to track the reference signals. In real-world applications, proportional integral derivative (PID) approaches [13, 14] are ubiquitous deployed to control nonlinear systems due to its theorical simplicity and easy implementation. However, in the existence of both system and external uncertainties, the PID-based methods cannot fulfil some high-accuracy tracking requirements, which is a fatal weakness of PID controllers. To simultaneously stabilize high-order nonlinear systems and compensate for the lumped disturbances, backstepping control schemes [15–18] are proposed, where the overall system is divided into several first-order systems and virtual control signals are developed to link up each subsystem. Nevertheless, the repeated analytical differentiations of virtual control signals may result in differentiating explosions, threating the normal operations of nonlinear systems. Hence, it is urgent and necessary to modify existing backstepping control approaches for MEMS gyroscopes without inducing such issues.Oriented by the previous investigations, an actor critic NN-based adaptive control scheme for MEMS gyroscopes suffering from multiresource disturbances is proposed in this paper, the striking features of which can be summarized as follows:Unlike the existing PID controller [13, 14], where the system uncertainties are neglected, the proposed control scheme can reduce the effects of lumped disturbances and stabilize the overall system. And, differing from previous backstepping control approaches [15–18] for MEMS gyroscopes that suffer from differentiating explosions, a low-pass filter is embedded to modify the controller design and avoid the analytical differentiations of virtual control signals.Different from most NN or FLS-based control schemes [7–10,12], which can only ensure the ultimately uniformly bounded (UUB) properties of tracking errors, the proposed ACNN-based control scheme can simultaneously recover unknown dynamics and supervise tracking properties of MEMS gyroscope, resulting better robustness against uncertainties.By resorting Lyapunov analyses, all the existing signals are proven to be UUB and a series of simulation verifications are presented to further validate the effectiveness of our proposed control approach.


## 2. Problem Statements

### 2.1. Kinetic Model of MEMS Gyroscopes

By resorting to [19–22], the kinetic model of MEMS gyroscopes is typically expressed as(1)mx¨+dxxx˙+dxy−2mΩZy˙+kxx−mΩZ2x+kxyy+ƛx=ux,my¨+dyyy˙+dxy+2mΩZx˙+kyy−mΩZ2y+kxyx+ƛy=uy,where m denotes the mass of internal moving mass, ΩZ represents the rotation velocity, x and y are, respectively, the displacements of drive and sensitive modes, kxx and kyy represent linear spring terms, dxx and dyy refer to damping terms, kxy and dxy are the coupling coefficients, and ux and uy denote the input forces of drive and sensitive modes. Additionally, ƛx and ƛy describe the external disturbances. An overall structure of MEMS gyroscope is depicted in Figure 1.Figure 1
Overall structure of MEMS gyroscope.Dividing both side of (1) by mω02l0, the following nondimensional model can be derived:(2)x¨∗+dxxmω0x˙∗+dxy−2mΩZmω0y˙∗+kxx−mΩZ2mω02x∗+kxymω02y∗+ƛxmω02l0=uxmω02l0,y¨∗+dyymω0y˙∗+dxy+2mΩZmω0y˙∗+kyy−mΩZ2mω02y∗+kxymω02x∗+ƛymω02l0=uymω02l0,with(3)xp=xp1xp2=x∗y∗,xv=xv1xv2=x˙∗y˙∗,u=u1u2=uxmω02l0uymω02l0,Ω=0ΩZω0−ΩZω00,D=dxxmω0dxymω0dxymω0dyymω0,K=kxxmω02kxymω02kxymω02kyymω02,ƛ=ƛxmω02l0ƛymω02l0.In pursuit of description conciseness, (2) is further rewritten into the following strict-feedback vector form:(4)x˙p=xv,x˙v=u+F,with F=F1,F2T=−D−2Ωxv−K−Ω2xp−ƛ defined as the packaged disturbances.


### 2.2. Control Objective

Our objective is to devise a control policyu based on ACNNs and DSC schemes, which is capable of driving the output states of MEMS gyroscopes xp to track the reference signal yr=yr1,yr2T as accurately as possible, in the existence of multisource interferences, i.e., F.


## 2.1. Kinetic Model of MEMS Gyroscopes

By resorting to [19–22], the kinetic model of MEMS gyroscopes is typically expressed as(1)mx¨+dxxx˙+dxy−2mΩZy˙+kxx−mΩZ2x+kxyy+ƛx=ux,my¨+dyyy˙+dxy+2mΩZx˙+kyy−mΩZ2y+kxyx+ƛy=uy,where m denotes the mass of internal moving mass, ΩZ represents the rotation velocity, x and y are, respectively, the displacements of drive and sensitive modes, kxx and kyy represent linear spring terms, dxx and dyy refer to damping terms, kxy and dxy are the coupling coefficients, and ux and uy denote the input forces of drive and sensitive modes. Additionally, ƛx and ƛy describe the external disturbances. An overall structure of MEMS gyroscope is depicted in Figure 1.Figure 1
Overall structure of MEMS gyroscope.Dividing both side of (1) by mω02l0, the following nondimensional model can be derived:(2)x¨∗+dxxmω0x˙∗+dxy−2mΩZmω0y˙∗+kxx−mΩZ2mω02x∗+kxymω02y∗+ƛxmω02l0=uxmω02l0,y¨∗+dyymω0y˙∗+dxy+2mΩZmω0y˙∗+kyy−mΩZ2mω02y∗+kxymω02x∗+ƛymω02l0=uymω02l0,with(3)xp=xp1xp2=x∗y∗,xv=xv1xv2=x˙∗y˙∗,u=u1u2=uxmω02l0uymω02l0,Ω=0ΩZω0−ΩZω00,D=dxxmω0dxymω0dxymω0dyymω0,K=kxxmω02kxymω02kxymω02kyymω02,ƛ=ƛxmω02l0ƛymω02l0.In pursuit of description conciseness, (2) is further rewritten into the following strict-feedback vector form:(4)x˙p=xv,x˙v=u+F,with F=F1,F2T=−D−2Ωxv−K−Ω2xp−ƛ defined as the packaged disturbances.


## 2.2. Control Objective

Our objective is to devise a control policyu based on ACNNs and DSC schemes, which is capable of driving the output states of MEMS gyroscopes xp to track the reference signal yr=yr1,yr2T as accurately as possible, in the existence of multisource interferences, i.e., F.


## 3. Main Results

### 3.1. AC Neural Networks

In what follows, the actor neural network is employed to reconstruct the packaged disturbanceF, which is formulated as(5)Fix¯=Waiθaix¯+δaix¯.where Wai=Wai,1,Wai,2,…,Wai,lT∈Rl denotes the weight vector and δaix¯ describes the reconstruction errors satisfying δaix¯≤δ¯ai with δ¯ai being a positive finite constant. Define that W^ai=W^ai,1,W^ai,2,…,W^ai,lT∈Rl is estimation of Wai and W˜ai=W^ai−Wai denotes the estimation error. θaix¯ is the active function vector, which is chosen as the commonly used sigmoid function.Herein, in order to supervise the tracking properties in real time, a reinforcement function is devised in the following form similar to [23](6)Si=S¯i1+exp−τiepi−S¯i1+expτiepi,where τi and S¯i are positive design parameters.Subsequently, the critic function is defined as(7)Sci=φiSi+φiSiWciθcix¯,where Wci=Wci,1,Wci,2,…,Wci,lT∈Rl denotes the weight vector. Define that W^ci=W^ci,1,W^ci,2,…,W^ci,lT∈Rl is estimation of Wci and W˜ci=W^ci−Wci denotes the estimation error. θcix¯ is the active function vector, which can be calculated in the following form:Apparently,Si is always bounded within −S¯i,S¯i, and the smaller epi is, the closer Si is to zero. Hence, the control objective of driving xp to track yr boils down to ensuring that Sc is as close to zero as possible. The actual critic signal is formulated as follows:(8)S^ci=φiSi+φiSiW^ciθcix¯,where W^ci is the estimation of Wci.


### 3.2. Controller Design

The following error dynamics are defined in advance, aiming at promoting the controller design and stability analysis:(9)epi≜xpi−yri,evi≜xvi−ν¯i,φi≜ν¯i−νi,with νi being the virtual control policy and ν¯i is its low-pass filtered signal, which are formulated as(10)χiν¯˙i+ν¯i=νi,νi0=ν¯i0,with χi being a positive time-delay parameter.Taking the time derivative ofφi≜ν¯i−νi and utilizing (10), the following equation can be guaranteed:(11)φ˙i+φiχi=−ν˙i.Note thatν˙i is a continuous and bounded function satisfying ν˙i≤ν0 and ν0 is a positive constant.Step 1.
at this point, devise the Lyapunov candidate of displacement loop asVp=∑i=12epi2+φi2/2. Differentiating it along with time and recalling the kinetic model of MEMS gyroscope (4), one has(12)V˙p=∑i=12e˙piepi+φ˙iφi=∑i=12evi+νi+φi−y˙riepi+φ˙iφi.
In order to stabilizeVp, the virtual control policy is developed as(13)νi=−kpiepi+y˙ri.
According to Young’s inequality, the following inequalities hold:(14)eviepi≤12evi2+12epi2,φiepi≤12φi2+12epi2,φiν˙i≤12φi2+12ν02.
Subsequently, inserting (13) and (14) into (12), it follows that(15)V˙p≤∑i=12−kpiepi2+12evi2+epi2+φi2−1χiφi2+12ν02.Step 2.
construct the Lyapunov candidate of velocity loop asVv=∑i=12S¯iln1+exp−τiepi+ln1+expτiepi/τi+W˜˙aiTW˜ai/2σai+W˜˙ciTW˜ci/2σci. Differentiating it along with time and recollecting the kinetic dynamics of MEMS gyroscope (4), it follows(16)V˙i=∑i=12Sie˙vi+W˜˙aiTW˜aiσai+W˜˙ciTW˜ciσci,=∑i=12Siui+Fi−ν¯˙i+W˜˙aiTW˜aiσai+W˜˙ciTW˜ciσci.
In order to stabilize the velocity loop and supervise the tracking performance in real time for MEMS gyroscopes, the actual control policy as well as adaptive updating policy are presented in (18)–(20):(17)ui=−kviSi−W^aiθai+ν¯˙i,(18)W^˙aiT=σaiS^ciθaiT−κaiW^aiT,(19)W^˙ciT=−σciφiSiθciTW^aiθaiT+κciW^ciT,with umi being the modification control term, which can be derived from (19) and (20). σai, σci, κai, and κci are commonly positive constants.
Via inserting (17)–(19) into (16) and recalling (5) as well as (7), the following inequality can be ensured:(20)V˙i=∑i=12−W˜aiθai+δai−kviSiφiSi−kviSi2+S^ciθaiT−κaiW^aiTW˜ai−φiSiθciTW^aiθaiT+κciW^ciTW˜ci,=∑i=12−W˜aiθai+δai−kviSiφiSi−kviSi2−κaiW^aiTW˜ai+φiSi+φiSiW^ciθciθaiTW˜ai−κciW^ciTW˜ci−φiSiθciTW˜ciW^aiθaiT.


## 3.1. AC Neural Networks

In what follows, the actor neural network is employed to reconstruct the packaged disturbanceF, which is formulated as(5)Fix¯=Waiθaix¯+δaix¯.where Wai=Wai,1,Wai,2,…,Wai,lT∈Rl denotes the weight vector and δaix¯ describes the reconstruction errors satisfying δaix¯≤δ¯ai with δ¯ai being a positive finite constant. Define that W^ai=W^ai,1,W^ai,2,…,W^ai,lT∈Rl is estimation of Wai and W˜ai=W^ai−Wai denotes the estimation error. θaix¯ is the active function vector, which is chosen as the commonly used sigmoid function.Herein, in order to supervise the tracking properties in real time, a reinforcement function is devised in the following form similar to [23](6)Si=S¯i1+exp−τiepi−S¯i1+expτiepi,where τi and S¯i are positive design parameters.Subsequently, the critic function is defined as(7)Sci=φiSi+φiSiWciθcix¯,where Wci=Wci,1,Wci,2,…,Wci,lT∈Rl denotes the weight vector. Define that W^ci=W^ci,1,W^ci,2,…,W^ci,lT∈Rl is estimation of Wci and W˜ci=W^ci−Wci denotes the estimation error. θcix¯ is the active function vector, which can be calculated in the following form:Apparently,Si is always bounded within −S¯i,S¯i, and the smaller epi is, the closer Si is to zero. Hence, the control objective of driving xp to track yr boils down to ensuring that Sc is as close to zero as possible. The actual critic signal is formulated as follows:(8)S^ci=φiSi+φiSiW^ciθcix¯,where W^ci is the estimation of Wci.


## 3.2. Controller Design

The following error dynamics are defined in advance, aiming at promoting the controller design and stability analysis:(9)epi≜xpi−yri,evi≜xvi−ν¯i,φi≜ν¯i−νi,with νi being the virtual control policy and ν¯i is its low-pass filtered signal, which are formulated as(10)χiν¯˙i+ν¯i=νi,νi0=ν¯i0,with χi being a positive time-delay parameter.Taking the time derivative ofφi≜ν¯i−νi and utilizing (10), the following equation can be guaranteed:(11)φ˙i+φiχi=−ν˙i.Note thatν˙i is a continuous and bounded function satisfying ν˙i≤ν0 and ν0 is a positive constant.Step 1.
at this point, devise the Lyapunov candidate of displacement loop asVp=∑i=12epi2+φi2/2. Differentiating it along with time and recalling the kinetic model of MEMS gyroscope (4), one has(12)V˙p=∑i=12e˙piepi+φ˙iφi=∑i=12evi+νi+φi−y˙riepi+φ˙iφi.
In order to stabilizeVp, the virtual control policy is developed as(13)νi=−kpiepi+y˙ri.
According to Young’s inequality, the following inequalities hold:(14)eviepi≤12evi2+12epi2,φiepi≤12φi2+12epi2,φiν˙i≤12φi2+12ν02.
Subsequently, inserting (13) and (14) into (12), it follows that(15)V˙p≤∑i=12−kpiepi2+12evi2+epi2+φi2−1χiφi2+12ν02.Step 2.
construct the Lyapunov candidate of velocity loop asVv=∑i=12S¯iln1+exp−τiepi+ln1+expτiepi/τi+W˜˙aiTW˜ai/2σai+W˜˙ciTW˜ci/2σci. Differentiating it along with time and recollecting the kinetic dynamics of MEMS gyroscope (4), it follows(16)V˙i=∑i=12Sie˙vi+W˜˙aiTW˜aiσai+W˜˙ciTW˜ciσci,=∑i=12Siui+Fi−ν¯˙i+W˜˙aiTW˜aiσai+W˜˙ciTW˜ciσci.
In order to stabilize the velocity loop and supervise the tracking performance in real time for MEMS gyroscopes, the actual control policy as well as adaptive updating policy are presented in (18)–(20):(17)ui=−kviSi−W^aiθai+ν¯˙i,(18)W^˙aiT=σaiS^ciθaiT−κaiW^aiT,(19)W^˙ciT=−σciφiSiθciTW^aiθaiT+κciW^ciT,with umi being the modification control term, which can be derived from (19) and (20). σai, σci, κai, and κci are commonly positive constants.
Via inserting (17)–(19) into (16) and recalling (5) as well as (7), the following inequality can be ensured:(20)V˙i=∑i=12−W˜aiθai+δai−kviSiφiSi−kviSi2+S^ciθaiT−κaiW^aiTW˜ai−φiSiθciTW^aiθaiT+κciW^ciTW˜ci,=∑i=12−W˜aiθai+δai−kviSiφiSi−kviSi2−κaiW^aiTW˜ai+φiSi+φiSiW^ciθciθaiTW˜ai−κciW^ciTW˜ci−φiSiθciTW˜ciW^aiθaiT.


## 4. Stability Analyses

Herein, the main theorem of our paper is presented as, on the basis of the virtual control policy, the actual control policy and the adaptive updating policy; all the error dynamics involved in the closed-loop MEMS gyroscope system are UUB.Proof.
In the first place, constructing the general Lyapunov function asV=Vp+Vv, differentiating it along with time and employing (15), (20), it can be derived that(21)V˙=V˙p+V˙v≤∑i=12−kpiepi2+12evi2+epi2+φi2−1χiφi2+12ν02+∑i=12−W˜aiθai+δai−umiφiSi−kviSi2−κaiW^aiTW˜ai+φiSi+φiSiW^ciθciθaiTW˜ai−κciW^ciTW˜ci−φiSiθciTW˜ciW^aiθaiT.
Due to Young’s inequality and the fact thatθci≤θ¯ci, the following inequalities hold:(22)−W˜aiθai+δaiφiSi+φiSi+φiSiW^ciθciθaiTW˜ai−φiSiθciTW˜ciW^aiθaiT≤φiSiW^aiθ¯aiθ¯ciWci+WaiTθ¯aiθ¯ciW^ci+δ¯ai,−W^ciTW˜ci≤12Wci2−W˜ci2,−W^aiTW˜ai≤12Wai2−W˜ai2.
Hence, (21) can be further rewritten as(23)V˙≤∑i=12−kpi−1epi2−1χi−1φi2−kviSi2−12κciW˜ci2−12κaiW˜ai2+δaiφiSi+φiSiGi+12ν02+12Wci2+12Wai2,with Gi=W^aiθ¯aiθ¯ciWci+WaiTθ¯aiθ¯ciW^ci+δ¯ai.
Moreover, employing Young’s inequality, it can be achieved that(24)δaiφiSi≤12φiSi2+12φiδ¯ai2,φiSiGi≤12φiSi2+12φiGi2.
Substituting (24) into (23) further produces(25)V˙≤∑i=12−kpi−1epi2−1χi−1φi2−kvi−1Si2−12κciW˜ci2−12κaiW˜ai2+12ν02+12Wci2+12Wai2+12φiδ¯ai2+12φiGi2.
From (25), it can be easily discovered that, as long as kpi−1>p,1/χi−1>p,kvi−1>p,κci/2>p,κai/2>p and Γ/V0<p hold with p being a positive constant and Γ=ν02/2+Wci2/2+1/2Wai2/2+φiδ¯ai2/2+φiGi2/2, V˙≤0 always holds for ∀t≥0. Hence, all the error signals considered in V are proven to be UUB, and the proof is completed.To verify the practicability of the proposed control scheme, a series of simulations are presented for MEMS gyroscopes in the proceeding section. By resorting to [20], the physical parameters of MEMS gyroscopes are typically chosen as(26)m=1.8×10−7kg,kxx=63.955N/m,kyy=95.92N/m,kxy=12.779N/m,dxx=1.8×10−6Ns/m,dyy=1.8×10−6Ns/m,dxy=3.6×10−7Ns/m,kx2=355.3,ky2=355.3,kxy=70.99,Dx=0.01,Dy=0.01,Dxy=0.002,Ωz=0.1.And, the following time-varying functions are utilized to simulate the unknown external disturbances:(27)ƛt=6cos3t+6sin2texp−t+33cos3texp−0.5t+4sin2t−2,t≥0.Define the reference signal asyr=sin4.17t,0,T. All the design parameters existing in the proposed control method are given in Table 1.Table 1
Design parameters of this simulation.
SectionParametersACNNσa1=1200,σa2=1100,κa1=15,κa2=12σc1=670,σc2=840,κc1=8,κc2=7,τ1=τ2=0.5Controllerkp1=5,kp2=5,χ1=χ2=0.05,kv1=10,kv2=10The tracking performance of displacement loop and velocity loop is presented in Figure2, where satisfactory tracking accuracy can be discovered, which can be attributed to the effectiveness of DSC approach. Even in the presence of unknown dynamics, the tracking performance is rarely degraded, mainly due to the implementation of ACNN, which is able to recover system uncertainties and supervise system tracking performance at the same time.Figure 2
Tracking performance of both displacement and velocity loops.Herein, it is notable that the issue of differentiation explosion is effectively avoided, mainly owing to the filtered signal is involved in the velocity loop controller design rather than the analytically differentiated signals. Moreover, although, in the existence of unknown multiresource interferences, the tracking performance is not degraded severely by resorting to the employed ACNN scheme and unlike most existing control schemes based on NN or FLS [7–10,12], which can only ensure the UUB behaviours of tracking errors, the proposed ACNN-based control scheme can recover unknown dynamics and supervise tracking properties of MEMS gyroscope at the same time, validating the robustness of our algorithm against uncertainties.


## 5. Conclusions

In this paper, an actor critic neural network-based adaptive control scheme for MEMS gyroscopes subject to multiresource disturbances is proposed. By combining an actor NN with a critic NN, the packaged disturbances can be effectively suppressed and system performance can be supervised at the same time, such that better tracking properties and stronger robustness against unknown dynamics can be guaranteed. Through inserting a first-order filter between the displacement and velocity loops of MEMS gyroscope, a DSC controller is constructed to ensure the stabilization for both displacement and velocity loops and avoid unexpected differentiation explosion. Moreover, all system dynamics are proven to be UUB via Lyapunov synthesis and a series of simulations through MATLAB/SIMULINK are presented in the end of this paper, validating the effectiveness of our algorithm. In the near future, we may modify the ACNN by combining the fuzzy logics with neural networks [24, 25], aiming at further enhancing its generalization ability. At the same time, prescribed performance of tracking error will be considered for MEMS gyroscope to improve transient and steady-state tracking accuracy.



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