Adaptive control of micro-electro-mechanical system gyroscope using neural network compensator

Abstract

This article proposes an adaptive control scheme with a neural network compensator for controlling a micro-electro-mechanical system gyroscope with disturbance and model errors. The adaptive neural network compensator is used to compensate the nonlinearities in the system based on its universal approximation and improve tracking performance of the gyroscope. The neural compensator, which is trained online, is combined with adaptive control of the Lyapunov framework system to approach the unknown system disturbance and model errors. The system stability is deduced by the Lyapunov stability theory, and the simulation of the micro-electro-mechanical system gyroscope is carried out on Matlab/Simulink, verifying the superior performance of the neural control compensation method.

Introduction

Micro-electro-mechanical system (MEMS) gyroscope is one of the basic measuring elements, which is called an angular velocity meter. However, there are always small manufacturing errors in the manufacturing process of the microgyroscope, which will lead to the asymmetry of the structure, transposition of the left and right drive, and the shift of the mass center. The above influences tend to produce unnecessary interlaced coupling and create inherent system disturbances in the form of mechanical and electrostatic forces, reducing the performance of the microgyroscope. In addition, the microgyroscope is extremely sensitive to environmental changes, including ambient temperature and pressure.

In recent years, some researchers have proposed many advanced adaptive control methods to control MEMS gyroscopes. Some advanced sliding mode control methods were discussed in previous studies^1–7 to improve the control accuracy of the gyroscopes. Hu and Gallacher ⁸ presented a high-precision mode tuning method for the MEMS gyroscope. In Montoya-Cháirez et al., ⁹ an adaptive control scheme was used to control a gyroscope of two degrees of freedom. In Wu et al., ¹⁰ a MEMS gyroscope design was proposed based on a model predictive control method. In Yang et al., ¹¹ a mode-matching control scheme with feedback calibration is proposed for gyroscopes.

In nonlinear systems, there are often unknown nonlinearities. Recently, many researchers are working on neural networks, which do not need the precise models and could approach the nonlinear function with arbitrary precision. In Fei and Yang, ¹² a neural compensation method was presented for a gyroscope. In Xiao and Chen, ¹³ a general projection neural network was applied to solve quadratic programming problem. Hua et al. ¹⁴ and Yin et al. ¹⁵ proposed an adaptive neural network control method for switched systems. Fang et al. ¹⁶ proposed a recurrent cerebellar model articulation neural network controller. In previous studies,^17,18 an adaptive neural network dynamic surface control method was studied for nonlinear systems. In Dong et al., ¹⁹ a fuzzy neural network control method was presented for the unmanned marine vehicle. In previous studies,^20–30 researchers combined neural networks and fuzzy systems with sliding mode control methods for controlling nonlinear systems. In Roh et al., ³¹ a new design method for fuzzy radial basis function (RBF) neural network classifier was investigated and developed.

In the manufacture and use of gyroscopes, there are often model errors and external disturbances. We need to avoid these external factors, which will affect control performance. Motivated by the above studies and research, an adaptive control method combining with an RBF neural network compensator is proposed to control the gyroscope system. The contributions of the proposed controller are listed as follows:

1. This article combines the adaptive control and the nonlinear approximation of neural network compensator. The RBF neural network is a three-layer forward network. The input-to-output mapping is nonlinear and can approximate any continuous function with arbitrary precision. In this article, neural network approximator is used to approximate system disturbance and model errors, improving the robustness of MEMS gyroscope with model uncertainties and external disturbance.

2. The article applies adaptive control to the gyroscope system; weights can be adjusted in real time, increasing the system accuracy, robustness, and adaptability of the gyroscope system; and the estimation of the unknown system is updated in real time.

3. The article compares the compensation effects of RBF neural network and back propagation (BP) neural network for model error and external disturbance. It can be seen that the RBF neural network has better approximation effect. In addition, compared with the common nominal controller, the proposed RBF neural network compensator has smaller tracking error and higher tracking accuracy.

4. Based on the Lyapunov stability theory, the proposed adaptive scheme can ensure the stability of the gyroscope closed-loop system. Two different kinds of external disturbances are considered in the simulation, and the results show good position-tracking performance, proving the superiority of the proposed neural compensation scheme.

Dynamics of MEMS gyroscope

Figure 1 shows a mass-spring-damping system diagram of a MEMS microgyroscope. The motion equation of microgyroscope is evolved from the Lagrange-Maxwell equation ¹²

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\stackrel{.}{x}}_i}\right)-\frac{\partial L}{\partial x_i}+\frac{\partial F}{\partial {\stackrel{.}{x}}_i}=Q_i$$

(1)

where $L=E_K-E_P$ , $E_K$ represents the kinetic energy of sensitive elements and $E_P$ represents the kinetic energy of sensitive elements. $F$ is the general damping force. $Q_i$ is the force for the sensitive element, and $i$ is an integer greater than 0.

Figure 1.

Schematic model of a z-axis MEMS vibratory gyroscope.

We assume that the angular velocity remains constant for a long time, namely, ${\mathrm{\Omega }}_x\approx {\mathrm{\Omega }}_y\approx 0$ , and only ${\mathrm{\Omega }}_z$ leads to a dynamic coupling between x-axis and y-axis. Considering various manufacturing errors causing additional coupling, equation of motion ²² is given as follows

$$\{\begin{array}{c}m\stackrel{··}{x}+d_{\mathrm{xx}}\stackrel{·}{x}+d_{\mathrm{xy}}\stackrel{·}{y}+k_{\mathrm{xx}}x+k_{\mathrm{xy}}y=u_x+d_x+2m{\mathrm{\Omega }}_z\stackrel{·}{y}\hfill \\ \multicolumn{1}{c}{m\stackrel{··}{y}+d_{\mathrm{xy}}\stackrel{·}{x}+d_{\mathrm{yy}}\stackrel{·}{y}+k_{\mathrm{xy}}x+k_{\mathrm{yy}}y=u_y+d_y-2m{\mathrm{\Omega }}_z\stackrel{·}{x}}\end{array}$$

(2)

where $m$ is the proof mass; $d_{\mathrm{xx}},d_{\mathrm{yy}}$ are the damping coefficients; $k_{\mathrm{xx}},k_{\mathrm{yy}}$ are the spring coefficients; $d_{\mathrm{xy}}$ is the coupled damping quadrature error; $k_{\mathrm{xy}}$ is the coupled spring quadrature error; $u_x,u_y$ are the control forces; $d_x,d_y$ represent disturbances as well as the effects of the parameter uncertainties, which are unknown but bounded; and $2m{\mathrm{\Omega }}_z\stackrel{·}{y},2m{\mathrm{\Omega }}_z\stackrel{·}{x}$ are the Coriolis forces for measuring unknown angular velocity ${\mathrm{\Omega }}_z$ . Both sides of equation (2) are divided by proof mass $m$ , reference length $q_0$ , and natural resonance frequency ${\omega }_0^2$ , and the nondimensional equation can be obtained

$$\begin{array}{c}\stackrel{··}{x}+d_{\mathrm{xx}}\stackrel{·}{x}+d_{\mathrm{xy}}\stackrel{·}{y}+{\omega }_x^{2}x+{\omega }_{\mathrm{xy}}y=u_x+2{\mathrm{\Omega }}_z\stackrel{·}{y}+d_x\hfill \\ \stackrel{··}{y}+d_{\mathrm{xy}}\stackrel{·}{x}+d_{\mathrm{yy}}\stackrel{·}{y}+{\omega }_{\mathrm{xy}}x+{\omega }_y^{2}y=u_y-2{\mathrm{\Omega }}_z\stackrel{·}{x}+d_y\hfill \end{array}$$

(3)

where $d_{\mathrm{xx}}/m{\omega }_0\to d_{\mathrm{xx}}$ , $d_{\mathrm{xy}}/m{\omega }_0\to d_{\mathrm{xy}}$ , $d_{\mathrm{yy}}/m{\omega }_0\to d_{\mathrm{yxy}}$ , ${\mathrm{\Omega }}_z/{\omega }_0\to {\mathrm{\Omega }}_z$ , $\sqrt{k_{\mathrm{xx}}/m{{\omega }_0}^2}\to {\omega }_x$ , $\sqrt{k_{\mathrm{yy}}/m{\omega }_0^2}\to {\omega }_y$ , and $k_{\mathrm{xy}}/m{\omega }_0^2\to {\omega }_{\mathrm{xy}}$

Rewriting equation (3) into the following vector form obtains

$$\stackrel{··}{q}+D\stackrel{·}{q}+\mathrm{Kq}=u-2\mathrm{\Omega }\stackrel{·}{q}+d$$

(4)

where $q=\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right]$ , $u=\left[\begin{array}{c}{u}_x\hfill \\ \multicolumn{1}{c}{u_y}\end{array}\right]$ , $d=\left[\begin{array}{c}{d}_x\hfill \\ \multicolumn{1}{c}{d_y}\end{array}\right]$ , $D=\left[\begin{array}{cc}{d}_{\mathrm{xx}}\hfill & d_{\mathrm{xy}}\hfill \\ \multicolumn{1}{c}{d_{\mathrm{xy}}}& d_{\mathrm{yy}}\hfill \end{array}\right]$ , $K=\left[\begin{array}{cc}{k}_{\mathrm{xx}}\hfill & k_{\mathrm{xy}}\hfill \\ \multicolumn{1}{c}{k_{\mathrm{xy}}}& k_{\mathrm{yy}}\hfill \end{array}\right]$ , and $\mathrm{\Omega }=\left[\begin{array}{cc}0\hfill & -{\mathrm{\Omega }}_z\hfill \\ \multicolumn{1}{c}{{\mathrm{\Omega }}_z}& 0\hfill \end{array}\right]$ . We define that $D=D^{\mathrm{T}},K=K^{\mathrm{T}},\mathrm{\Omega }=-{\mathrm{\Omega }}^T$ . The disturbances $d=[\begin{array}{cc}{d}_1\hfill & d_2\hfill \end{array}{]}^T$ have an upper bound $||d||\le \rho$ , and $\rho$ is a scalar.

Nominal control design and problem formulation

The ideal oscillation trajectory of a MEMS gyroscope is defined as follows

$$x_d=A_1\sin ({\omega }_{1}t),y_d=A_2\sin ({\omega }_{2}t)$$

(5)

Rewriting the ideal trajectory into a vector form, yields

$${\stackrel{··}{q}}_d+K_d{q}_d=0$$

(6)

where $K_d=\mathrm{diag}\{{\omega }_1^2,{\omega }_2^2\}$ , $q_d=[x_d,y_d{]}^T$

We assume that the gyroscope modeling is precise and the external disturbance is nonexistent, namely, $d=0$ . Considering the nominal mathematical model of the MEMS gyroscope, we design the control law $u$ as

$$u={\stackrel{··}{q}}_d-k_v\stackrel{·}{e}-k_{p}e+D\stackrel{·}{q}+\mathrm{Kq}+2\mathrm{\Omega }\stackrel{·}{q}$$

(7)

Substituting the control law (7) into equation (4), we get the closed-loop system equation as

$$\stackrel{··}{e}+k_v\stackrel{·}{e}+k_{p}e=0$$

(8)

where $e=q-q_d$ is the tracking error, and $q_d$ is the reference trajectory. $k_v$ and $k_p$ are the controller parameter matrix, which determine the tracking performance of the microgyroscope. Generally, $k_v$ takes $k_v=\left[\begin{array}{cc}2\alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 2\alpha \hfill \end{array}\right],\alpha >0$ and $k_p$ takes $k_p=\left[\begin{array}{cc}{\alpha }^2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\alpha }^2\hfill \end{array}\right],\alpha >0$ , and the tracking error will converge exponentially to 0.

The equation (8) can be written as a state-space form

$$\stackrel{·}{x}=\mathrm{Ax}$$

(9)

where $x=\left[\begin{array}{c}e\hfill \\ \multicolumn{1}{c}{\stackrel{·}{e}}\end{array}\right]$ , $A=\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-k_p}& -k_v\hfill \end{array}\right]$ , $x\in {\Re }^{4\times 1}$ , and $A\in {\Re }^{4\times 4}$ .

The following is the proof for stability of the nominal controller

Proof

Define the following Lyapunov candidate function as $V=\frac{1}{2}{x}^T\mathrm{Px}$ , where $P$ is a symmetric positive definite matrix and meets the following Lyapunov equation

$$\mathrm{PA}+A^{T}P=-Q$$

(10)

where $Q$ is a symmetric positive definite matrix, too. Because $A=\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-k_p}& -k_v\hfill \end{array}\right]$ , then the eigenvalues of $A$ are all $-\alpha$ , namely, $A$ is steady, which ensure that $P$ is exiting.

The derivative of $V$ is

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}}& =-{\stackrel{·}{x}}^T\mathrm{Px}+x^{T}P\stackrel{·}{x}\hfill \\ \multicolumn{1}{c}{}& ={\left(\mathrm{Ax}\right)}^T\mathrm{Px}+x^T\mathrm{PAx}\hfill \\ \multicolumn{1}{c}{}& =x^T(A^{T}P+\mathrm{PA})x\hfill \\ \multicolumn{1}{c}{}& =-x^T\mathrm{Qx}\hfill \\ \multicolumn{1}{c}{}& <0\hfill \end{array}$$

(11)

Since $V$ is positive, the derivative of $V$ is negative, so the system is asymptotically stable.

However, in fact, we cannot easily get the accurate gyroscope model, and the external disturbance cannot be ignored. We define the nominal values of $D$ , $K$ , and $\mathrm{\Omega }$ as $D_0$ , $K_0$ , and ${\mathrm{\Omega }}_0$ , respectively. $D_0$ , $K_0$ , and ${\mathrm{\Omega }}_0$ are known values. Then, the design control law is proposed as

$$u={\stackrel{··}{q}}_d-k_v\stackrel{·}{e}-k_{p}e+D_0\stackrel{·}{q}+K_{0}q+2{\mathrm{\Omega }}_0\stackrel{·}{q}$$

(12)

Substituting the control law (12) into equation (4) gets

$$\begin{array}{c}\stackrel{··}{q}+D\stackrel{·}{q}+\mathrm{Kq}\hfill \\ ={\stackrel{··}{q}}_d-k_v\stackrel{·}{e}-k_{p}e+D_0\stackrel{·}{q}+K_{0}q+2{\mathrm{\Omega }}_0\stackrel{·}{q}-2\mathrm{\Omega }\stackrel{·}{q}+d\hfill \end{array}$$

(13)

Defining $\tilde{D}=D_0-D$ , $\tilde{K}=K_0-K$ , and $\tilde{\mathrm{\Omega }}={\mathrm{\Omega }}_0-\mathrm{\Omega }$ , we have the following closed loop equation

$$\stackrel{··}{e}+k_v\stackrel{·}{e}+k_{p}e=\tilde{D}\stackrel{·}{q}+\tilde{K}q+2\tilde{\mathrm{\Omega }}\stackrel{·}{q}+d$$

(14)

From the equations (8) and (11), the tracking performance of the nominal controller based on the nominal model is greatly reduced for the actual MEMS gyroscope, although the system can remain stable. The target is to design a compensation controller to enhance the tracking precision and also ensure the system stable under model errors and external disturbances.

Define the modeling error as follows

$$f=\tilde{D}\stackrel{·}{q}+\tilde{K}q+2\tilde{\mathrm{\Omega }}\stackrel{·}{q}+d$$

(15)

Then, we design the control law as

$$u={\stackrel{··}{q}}_d-k_v\stackrel{·}{e}-k_{p}e+D_0\stackrel{·}{q}+K_{0}q+2{\mathrm{\Omega }}_0\stackrel{·}{q}-f$$

(16)

An ideal closed-loop system in equation (8) is obtained by substituting equation (13) into equation (4). However, the modeling error $f$ is not known actually. As a result, we can use the estimated function $\hat{f}(•)$ to replace the model error $f$ by using the universal approximation of neural networks.

Rewriting the equation (8) as a state space form obtains

$$\stackrel{·}{x}=\mathrm{Ax}+\mathrm{Bf}$$

(17)

where $x=\left[\begin{array}{c}e\hfill \\ \multicolumn{1}{c}{\stackrel{·}{e}}\end{array}\right]$ , $A=\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-k_p}& -k_v\hfill \end{array}\right]$ , $B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{I}\end{array}\right]$ , $x\in {\Re }^{4\times 1}$ , $A\in {\Re }^{4\times 4}$ , $B\in {\Re }^{4\times 2}$ , and $\mathrm{I}$ is an unit matrix.

Because $k_p$ , $k_v$ , and $I$ are all constant, namely, $\mathrm{A}$ and $\mathrm{B}$ are constant, so $f$ is possible to be identified through $x$ . Therefore, the function $f(•)$ could be denoted as $f(x)$ .

• Remark 1. It is true that estimation techniques can only provide estimation with some inaccuracy practically. Actually, we already consider time-varying unknown but bounded parameter uncertainties in the MEMS gyroscope model as equation (4). Note that, the lumped disturbances $d_1$ and $d_2$ could also evolve the effects of the unknown bounded disturbances. Thus, all the related equations in section “Nominal control design and problem formulation” such as equations (10) and (11) have lumped disturbances including the unknown bounded disturbances. We use neural networks to adaptively approximate system uncertainties and external disturbances.

Adaptive neural compensation scheme

Introduction of neural network

Figure 2 is a block diagram of a three-layer RBF neural network structure, which mainly includes an input layer, hidden layer, and output layer. The hidden layer maps the signal from the input space to a higher dimensional space, and the output layer performs a weighted summation operation to generate an RBF network output value.

Figure 2.

Structure of RBF neural network.

The RBF neural network model is described by mathematical symbols as follows

$$y_i=\sum _{j=1}^{n^{*}}{w}_{\mathrm{ij}}{\varphi }_j,i=1,2,\dots ,L$$

(18)

$${\varphi }_j\left(x\right)=g\left(\frac{\Vert x-c_j\Vert }{{\sigma }_j}\right),j=1,2,\dots ,n^{*}$$

(19)

where $n^{*}$ and $L$ represent the number of hidden layer and output layer nodes, respectively. $w_{\mathrm{ij}}$ is the weight between the jth hidden node and the ith output node. $y_i$ is the output of the neural network. ${\varphi }_j(x)$ is the output value of the jth hidden layer node. Differing from other neurons, the net input to the jth radial basis activation function is the vector distance between its center vector $c_j$ and the input vector $x$ , divided by the width ${\sigma }_j$ . We chose $g(\alpha )=\exp (-{\alpha }^2)$ as the activation function $g$ . This Gaussian activation function $g$ has a maximum of 1 when its input is 0, namely, the maximum output of the neural network is achieved when the input coincides with the center vector. As the distance between $x$ and $c$ increases, the output decreases.

• Remark 2. In order to achieve a predetermined mapping relationship, there should be more hidden layer nodes for a single hidden layer network to increase the tunable parameters of the network; however, too much nodes will increase the complexity of the network structure. According to the experienced formula $m_1=\sqrt{n_1+l_1}+{\alpha }_1$ , where $m_1$ is the number of hidden layer nodes, $n_1$ is the number of input layer nodes, $l_1$ is the number of output layer nodes, and ${\alpha }_1$ is a constant between 1 and 10. Since the adaptive law of neural network is designed based on the Lyapunov stability theory, the initial value of network weight is relatively loose.

Design of neural network compensator

In addition to the global approximation properties, the RBF neural network has the optimal approximation performance, namely, the RBF network can approach any nonlinear function over a compact set with arbitrary precision. The block diagram of adaptive neural compensation system is designed in Figure 3. We make the following assumptions for our study:

Figure 3.

Block diagram of adaptive neural compensation system.

Defining ${\epsilon }_0$ is a smaller positive constant, and there exists a set of optimal network weights $\theta ={\theta }^{*}$ to make the output $\hat{f}(x,{\theta }^{*})$ of the neural network to satisfy the following inequality

$$max\Vert \hat{f}(x,{\theta }^{*})-f(x)\Vert \le {\epsilon }_0$$

(20)

where $x$ is a compact set and $n^{*}$ is the number of hidden layer nodes.

Based on the above assumptions, rewrite the equation (14) as

$$\stackrel{·}{x}=\mathrm{Ax}+B\{\hat{f}(x,{\theta }^{*})+[f\left(x\right)-\hat{f}(x,{\theta }^{*})]\}$$

(21)

where ${\theta }^{*}$ represents the optimal weight for the approximation.

We assume $\eta$ as the modeling approximation error with the optimal RBF neural network

$$\eta =f\left(x\right)-f(x,{\theta }^{*})$$

(22)

And ${\eta }_0$ is the upper bound of the $\eta$

$${\eta }_0=\underset{t\ge 0}{sup}\Vert f\left(x\right)-\hat{f}(x,{\theta }^{*})\Vert$$

(23)

Then, the equation (18) can be expressed as

$$\stackrel{·}{x}=\mathrm{Ax}+B[f(x,{\theta }^{*})+\eta ]$$

(24)

According to the characters of the RBF network, the function $\hat{\mathrm{f}}(\mathrm{x},{\theta }^{*})$ can be expressed in the form

$$\hat{f}(x,{\theta }^{*})={\theta }^{*T}\varphi \left(x\right)$$

(25)

where ${\theta }^{*}$ is the optional weight value for approximation and $\varphi (x)\in {\Re }^{n^{*}}$ is the hidden layer output vector.

Then, the equation (18) becomes

$$\stackrel{·}{x}=\mathrm{Ax}+B[{\theta }^{*T}\varphi \left(x\right)+\eta ]$$

(26)

Then, we design a new neural compensator based on the proposed neural network for the MEMS gyroscope to eliminate the influence of uncertainty and external disturbance. Design the new control law as

$$u=u_1+u_2$$

(27)

where

$$u_1={\stackrel{··}{q}}_d-k_v\stackrel{·}{e}-k_{p}e+D_0\stackrel{·}{q}+K_{0}q+2{\mathrm{\Omega }}_0\stackrel{·}{q}$$

(28)

$$u_2=-{\hat{\theta }}^T\varphi \left(x\right)$$

(29)

where $\hat{\theta }$ is the estimated value of the optimal weight ${\theta }^{*}$ .

Then, the closed-loop system equation becomes

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{x}}& =\mathrm{Ax}+B[{\theta }^{*T}\varphi \left(x\right)+\eta -{\hat{\theta }}^T\varphi \left(x\right)]\hfill \\ \multicolumn{1}{c}{}& =\mathrm{Ax}+B[\eta -{\tilde{\theta }}^T\varphi \left(x\right)]\hfill \end{array}$$

(30)

where $\tilde{\theta }=\hat{\theta }-{\theta }^{*}$ is the estimation error.

Analysis of system stability

Select the following Lyapunov function

$$V=\frac{1}{2}{x}^T\mathrm{Px}+\frac{1}{2\gamma }\mathrm{tr}\left({\tilde{\theta }}^T\tilde{\theta }\right)$$

(31)

where $\gamma >0$ and $P$ is a symmetric positive definite matrix and meets the following Lyapunov equation

$$\mathrm{PA}+A^{T}P=-Q$$

(32)

with $Q$ is a symmetric positive definite matrix, too.

Because $A=\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-k_p}& -k_v\hfill \end{array}\right]$ , we design $k_p=\left[\begin{array}{cc}{\alpha }^2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\alpha }^2\hfill \end{array}\right]$ , $k_v=\left[\begin{array}{cc}2\alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 2\alpha \hfill \end{array}\right]$ , with $\alpha >0$ , then the eigenvalues of $A$ are all $-\alpha$ , namely, $A$ is steady, which ensure that $P$ is exiting.

Differentiating $V$ , and we can get

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}}& =\frac{1}{2}(x^{T}P\stackrel{·}{x}+{\stackrel{·}{x}}^T\mathrm{Px})+\frac{1}{\gamma }\mathrm{tr}\left({\stackrel{·}{\stackrel{~}{\theta }}}^T\tilde{\theta }\right)\hfill \\ \multicolumn{1}{c}{}& =\frac{1}{2}{x}^T(\mathrm{PA}+A^{T}P)x+{[-{\tilde{\theta }}^T\varphi \left(x\right)+\eta ]}^T{B}^T\mathrm{Px}+\frac{1}{\gamma }\mathrm{tr}\left({\stackrel{·}{\stackrel{~}{\theta }}}^T\tilde{\theta }\right)\hfill \\ \multicolumn{1}{c}{}& =-\frac{1}{2}{x}^T\mathrm{Qx}+{\eta }^T{B}^T\mathrm{Px}-\varphi {\left(x\right)}^T\tilde{\theta }{B}^T\mathrm{Px}+\frac{1}{\gamma }\mathrm{tr}\left({\stackrel{·}{\stackrel{~}{\theta }}}^T\tilde{\theta }\right)\hfill \end{array}$$

(33)

Considering that

$$\varphi {\left(x\right)}^T\tilde{\theta }{B}^T\mathrm{Px}=\mathrm{tr}\left(B^T\mathrm{Px}\varphi {\left(x\right)}^T\tilde{\theta }\right)$$

(34)

then, we can get

$$\stackrel{·}{V}=-\frac{1}{2}{x}^T\mathrm{Qx}+{\eta }^T{B}^T\mathrm{Px}+\frac{1}{\gamma }\mathrm{tr}({\stackrel{·}{\stackrel{~}{\theta }}}^T\tilde{\theta }-\gamma B^T\mathrm{Px}\varphi {\left(x\right)}^T\tilde{\theta })$$

(35)

Set the adaptive law as

$${\stackrel{·}{\stackrel{~}{\theta }}}^T={\stackrel{·}{\hat{\theta }}}^T=\gamma B^T\mathrm{Px}{\varphi }^T\left(x\right)$$

(36)

Substituting (36) into (35) yields

$$\stackrel{·}{V}=-\frac{1}{2}{x}^T\mathrm{Qx}+{\eta }^T{B}^T\mathrm{Px}$$

(37)

Due to $||{\eta }^T||\le {\eta }_0$ , $||{\mathrm{B}}^{\mathrm{T}}\mathrm{P}||\le ||\mathrm{P}||$ , we have

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -\frac{1}{2}{\lambda }_{min}\left(Q\right)\Vert x{\Vert }^2+{\eta }_0{\lambda }_{max}\left(P\right)\Vert x\Vert }\\ \multicolumn{1}{c}{=-\frac{1}{2}\Vert x\Vert [{\lambda }_{min}\left(Q\right)\Vert x\Vert -2{\eta }_0{\lambda }_{max}\left(P\right)]}\end{array}$$

(38)

where ${\lambda }_{min}(Q)$ is the minimum eigenvalue of $Q$ and ${\lambda }_{max}(P)$ is the maximum eigenvalue of $P$ . When the approximation error ${\eta }_0$ approaches 0, and ${\lambda }_{min}(Q)\ge (2{\lambda }_{max}(P)/||x||){\eta }_0$ , the derivative of $V$ satisfies $\stackrel{·}{V}\le 0$ . Therefore, the adaptive law defined by equation (36) can guarantee the stability of the system, making the tracking error eventually bounded by $R_0=2{\lambda }_{max}(P){\eta }_0/{\lambda }_{min}(Q)$ . It can be guaranteed that the tracking error will eventually fall into the residue $R_0$ , which depends on the approximation error $\eta$ and the constant $\beta$ . Only if the bound is equal to zero, the zero tracking can be obtained.

Simulation study

We verify the proposed neural network adaptive control method on the MEMS gyroscope model. The parameters of the microgyroscope are set as follows

$$\begin{array}{c}{k}_{\mathrm{xx}}=63.955\mathrm{N}/\mathrm{m},k_{\mathrm{yy}}=95.92\mathrm{N}/\mathrm{m},\hfill \\ k_{\mathrm{xy}}=12.779\mathrm{N}/\mathrm{m},m=1.8\times {10}^{-7}\mathrm{kg}\hfill \\ d_{\mathrm{xx}}=1.8\times {10}^{-6}\mathrm{N}\mathrm{s}/\mathrm{m},d_{\mathrm{yy}}=1.8\times {10}^{-6}\mathrm{N}\mathrm{s}/\mathrm{m},\hfill \\ d_{\mathrm{xy}}=3.6\times {10}^{-7}\mathrm{N}\mathrm{s}/\mathrm{m}\hfill \end{array}$$

We set $1\mu \mathrm{m}$ and $1\mathrm{kHz}$ as the value of $q_0$ and ${\omega }_0$ . We suppose that the unknown angular velocity ${\mathrm{\Omega }}_z$ is $100\mathrm{rad}/\mathrm{s}$ . The following numerical values are the dimensionless parameters of the gyroscope

$$\begin{array}{c}{d}_{\mathrm{xx}}=0.01,d_{\mathrm{yy}}=0.01,d_{\mathrm{xy}}=0.002\hfill \\ {\mathrm{\Omega }}_z=0.1,{{\omega }_x}^2=355.3,{{\omega }_y}^2=532.9,{\omega }_{\mathrm{xy}}=70.99\hfill \end{array}$$

We define $q_{dx}=\cos (6.17t)$ and $q_{dy}=\cos (5.11t)$ as the ideal trajectory and set $D_0=0.9*D$ , $K_0=0.9*K$ , and ${\mathrm{\Omega }}_0=0$ as the nominal values of $D$ , $K$ , and $\mathrm{\Omega }$ . The initial value of $q(0)$ is 0. Two kinds of external interference forms are considered in the simulation experiment: resonant frequency noise such as $d_1(t)=[10*((\sin (6.17*t){)}^2+2*\cos (6.17*t));10*((\sin (5.11*t){)}^2+2*\cos (5.11*t))]$ and white noise such as $d_2(t)=[3.0*\mathrm{randn}(1,1);3.0*\mathrm{randn}(1,1)]$ .

Figures 4 and 5 are tracking error plots for two noises under the ordinary controller with $k_v=2\alpha I$ , $k_p={\alpha }^{2}I$ , where $\alpha =2$ and $I$ is a unite matrix.

Figure 4.

Tracking errors under $d_1$ .

Figure 5.

Tracking errors under $d_2$ .

As is shown in the Figures 4 and 5 that the ordinary computed force control method can maintain the system stable, however, the tracking error is large owing to the modeling error and external disturbance. Note that the tracking errors displayed in the Figure 5 are nondimensional, namely, no units present. Their value characterizes their relative measurement respect to the reference length $q_0$ .

The following are the parameters of the RBF neural network: $n^{*}=11$ , $\alpha =2$

$$\begin{array}{c}P=\left[\begin{array}{cccc}-225/2\hfill & 0\hfill & 25/2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -225/2\hfill & 0\hfill & 25/2\hfill \\ \multicolumn{1}{c}{25/2}& 0\hfill & -125/8\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 25/2\hfill & 0\hfill & -125/8\hfill \end{array}\right],\hfill \\ \mathrm{Q}=100*\mathrm{I}.\hfill \end{array}$$

The center in each dimension is selected from −1 to 1 with the step $\mathrm{\Delta }=0.2$ . The initial conditions of gyroscope and the initial weights are set to one. We use a neural network compensator to control the gyroscope under two kinds of noise such as $d_1(t)$ and $d_2(t)$ . Note that the RBF compensation begins to work after fifth second, and during the first 5 s, only the nominal controller works.

The simulation results of the neural network compensator under $d_1(t)$ are shown from Figures 6–10. Figures 6–8 show the vibrational trajectories of the $x$ -axis and $y$ -axis and the tracking errors in the case of resonant frequency disturbance, respectively. Figures 9 and 10 are the approximation curves of the $x$ -axis and $y$ -axis for modeling error $f(x)$ using RBF network.

Figure 6.

Position tracking of $x$ - axis under $d_1$ using RBF neural network compensator.

Figure 7.

Position tracking of $y$ -axis under $d_1$ using RBF neural network compensator.

Figure 8.

Tracking errors under $d_1$ using RBF neural compensator.

Figure 9.

RBF network approximation of $x$ -axis under $d_1$ .

Figure 10.

RBF network approximation of $y$ -axis under $d_1$ .

The simulation results of the neural network compensator under white noise disturbance $d_2$ are shown from Figures 11–15. Figures 11–13 show the vibrational trajectories of the $x$ -axis and $y$ -axis and the tracking errors, respectively. Figures 14 and 15 are the approach curves of the $x$ -axis and $y$ -axis for modeling error $f(x)$ using the RBF neural network, respectively. The Figures 6, 7, 11, and 12 show that the position-tracking performance is quite good under both disturbances with the adaptive RBF compensation. After the fifth second, the tracking errors begin to diminish obviously, and the RBF NN outputs begin to approximate the unknown modeling error. After about 3-s regulation, the RBF neural network could estimate the modeling errors well.

Figure 11.

Position tracking of $x$ - axis under $d_2$ using RBF neural network compensator.

Figure 12.

Position tracking of $y$ -axis under $d_2$ using RBF neural network compensator.

Figure 13.

Tracking errors under $d_2$ using RBF neural network compensator.

Figure 14.

RBF network approximation of $x$ -axis under $d_2$ .

Figure 15.

RBF network approximation of $y$ -axis under $d_2$ .

As is shown in the above figures, the modeling error $f(x)$ can be estimated effectively by the RBF neural network compensator, and the tracking error of the gyroscope is significantly improved under two different external disturbances, demonstrating the effectiveness of the proposed scheme.

To demonstrate the effectiveness of the proposed RBF neural network compensator, we conduct a comparative study between RBF neural network compensator and BP neural network compensator. ³² The simulation results of the BP neural network compensator under $d_1(t)$ are shown in Figure 16. Figure 16 is the tracking error of the x-axis and y-axis. It can be found that the proposed RBF neural network control method has higher approximation accuracy and better robustness than the BP neural network compensator.

Figure 16.

Tracking errors under $d_1$ using BP neural network compensator.

Conclusion

In this article, an adaptive control method based on the RBF neural network is proposed for MEMS gyroscope. An adaptive RBF neural network compensator, which is trained online, is used to approximate the model uncertainties and external disturbances. The control law and adaptive law are derived based on the Lyapunov theory, ensuring the stability of the MEMS gyroscope system. The ordinary controller method and the proposed neural network controller are applied to the microgyroscope system under two different interferences, respectively. The simulation results show that the proposed control scheme has a smaller control error, the better tracking performance, and strong robustness.