Gated Recurrent Fuzzy Neural Network Sliding Mode Control of a Micro Gyroscope

Abstract

This paper proposes a non-singular fast terminal sliding mode control (NFTSMC) method for micro gyroscopes with unknown uncertainty based on gated recurrent fuzzy neural networks (GRFNNs). First, taking advantage of non-singular fast terminal sliding control, a sliding hyperplane is designed with a nonlinear function to ensure that the tracking error of the system converges to zero within a specified finite time. Then, the unknown model parameters of the micro gyroscope are estimated using a GRFNN. Since the GRFNN can adaptively adjust the base width, center vector, gated recurrent unit parameters, and outer gains, it can achieve accurate approximation to unknown models, enhancing the robustness and accuracy. In addition, due to the introduction of gated recurrent units, The GRFNN can effectively utilize the previous data and avoid the problem of gradient disappearance. The comparison of the simulation results with traditional neural sliding mode control shows that the proposed method can achieve better tracking performance and more accurate estimation of unknown models.

1. Introduction

A micro gyroscope is an important inertial sensor for measuring angular velocity, which is widely used in military, civil, and other fields. The working principle of the micro gyroscope is based on the Coriolis effect, measuring the velocity and Coriolis acceleration of the detected mass in simple harmonic motion, and calculating the input angular velocity. Due to the defects of the manufacturing technology, the actual micro gyroscope cannot achieve the ideal symmetrical structure, preventing the ideal situation where there is only mechanical coupling between the drive axis and the sensing axis. Therefore, considering that the orthogonal coupling between the drive axis and the sensing axis, along with external disturbances and changes in the system parameters, will deteriorate the operation of the micro gyroscope, the investigation of advanced control schemes is necessary in order to improve the performance of micro gyroscopes [1,2,3].

In [4], a harmonic disturbance-observer-based sliding mode controller (SMC) was designed, which exhibited superior robust performance against external disturbances with harmonic characteristics. In [5], a fractional order integral was introduced into the sliding surface to make the controller design and the control law more flexible and maintain a smaller steady-state error. A compound fractional integral terminal sliding mode control and fractional PD control were designed in [6] in order to eliminate the chattering phenomenon. In [7], the super-twisting algorithm was adopted to weaken the chattering and smooth the input of the controller.

Sliding mode control (SMC) has the advantages of being easy to implement, insensitive to parameters, and robust to disturbances. Terminal sliding mode control (TSMC) uses a nonlinear term in the traditional sliding mode surface, which can enable the system state to converge to zero in a limited time [8,9,10]. A composite TSMC method based on uncertainty observer and a composite TSMC method based on disturbance observer were proposed in [11,12] to reduce chattering. Since the controller controlled by the terminal has a negative exponential term, when the base of the term is zero, the output of the controller will be infinite, which is a singular phenomenon. Non-singular terminal sliding mode control (NTSMC) was designed in [13], completely solving the singularity phenomenon. NTSMC was combined with other methods to avoid singular phenomena and achieve high-precision control in [14,15,16]. However, the NTSMC method also has some shortcomings. When the state is far from the equilibrium point, its convergence rate is slow and there is a problem of chattering. In order to solve the above shortcomings of NTSMC, non-singular fast terminal sliding mode control (NFTSMC) was proposed in [17,18,19,20].

For the actual control system, since the model of the control system has unknown items and uncertainties, it is necessary to combine advanced intelligent control strategies [21] to provide an effective approximation of the model. In [22], a model predictive control was viewed as a special multiparameter quadratic programming problem, and a new implementation of a fuzzy control system was proposed. In [23], nonlinear control techniques were introduced to capture the process nonlinearities, but the performance was not significantly improved due to the complex controller structure limiting the invocation of sliding mode control and fuzzy control. In [24], a metaheuristic gray wolf optimization algorithm was employed to train the policy neural network, achieving better overall control performance than the gradient descent algorithm and the metaheuristic particle swarm optimization (PSO) algorithm. In [25], the results of comparing the efficiency of long short-term memory (LSTM) networks and gated recurrent unit (GRU) neural networks as models of dynamic processes in model predictive control are shown, indicating that the GRU network, despite having a lower number of parameters than the LSTM network, can be successfully used in MPC without any significant deterioration of control quality. In [26], a hybrid GRU fully convolutional network (GRU-FCN) was created by replacing the LSTM with a GRU in the LSTM-FCN to offer even better performance on many time-series datasets. A gated recurrent unit neural network was studied to predict the tracking error in [27,28]. Neuro-fuzzy network systems were introduced to handle the unknown nonlinear functions in [29,30,31,32,33].

Therefore, in order to realize finite-time control, avoid singularity, and achieve rapid state convergence, non-singular fast terminal sliding mode control was selected as the controller. In order to track the data with time-varying characteristics and avoid the gradient disappearance problem of recurrent neural networks, we designed a recurrent fuzzy neural network embedded with a GRU.

In this paper, NFTSMC based on the gated recurrent fuzzy neural network (GRFNN) method is proposed for a micro gyroscope to achieve the target tracking and the exact approximation of an unknown model. The main contributions of this paper can be summarized as follows:

(1)

The introduction of NFTSMC completely solves the singular phenomenon of terminal sliding mode control by eliminating the negative exponential term. At the same time, compared with non-singular terminal sliding mode control, the NFTSMC method also can reduce the chattering and speed up the convergence speed of the state away from the equilibrium point.

(2)

The unknown model part of the micro gyroscope is estimated using a GRFNN. For the chosen neural network, it actually embeds a GRU in a fuzzy neural network. A GRFNN has multiple complete gated recurrent units, which can solve the problem of gradient disappearance, so that it can effectively deal with short-term and long-term time-dependent problems and make full use of previous information, so it has better approximation effect for unknown nonlinear models.

(3)

The base width, center vector, outermost weights, and parameters of the GRU of the GRFNN can be set arbitrarily. The neural network can adaptively adjust parameters such as the base width, center vector, etc. Thus, the precise approximation of the unknown uncertainty of the system can be achieved, and the robustness and sensitivity of the design system can be improved.

The rest of this paper is organized as follows: The dynamic model of the micro gyroscope and the dimensionless processing are given in Section 2. In Section 3 and Section 4 discuss the principle of NFTSMC and the structure of GRFNNs for approximating unknown models, respectively, and elaborate the functions and working mechanisms of each layer. Section 5 presents the design of the controller of the control scheme proposed in this paper, and the stability analysis of the proposed control system is presented in Section 6. Section 7 presents simulation studies and performance comparisons. Section 8 presents the conclusions.

2. Dynamics of the Micro Gyroscope

This section briefly introduces the dynamic model of the micro gyroscope. According to the positional relationship between the direction of the input angular velocity and the driving motion plane of the sensitive movable mass, the micro gyroscope can be divided into an xy-axis micro gyroscope and a z-axis micro gyroscope. In this study, the z-axis micro gyroscope was selected as the research object. It can be simplified as shown in Figure 1.

Assuming that there is a constant input angular velocity, the dynamic equation of the z-axis micro gyroscope can be expressed as follows:

$$\begin{array}{l}m\ddot{x}+d_x\dot{x}+[k_x-m({\mathrm{\Omega }}_y^2+{\mathrm{\Omega }}_z^2)]x+m{\mathrm{\Omega }}_x{\mathrm{\Omega }}_{y}y=u_x+2m{\mathrm{\Omega }}_z\dot{y}\\ m\ddot{y}+d_y\dot{y}+[k_y-m({\mathrm{\Omega }}_x^2+{\mathrm{\Omega }}_z^2)]y+m{\mathrm{\Omega }}_x{\mathrm{\Omega }}_{y}x=u_y-2m{\mathrm{\Omega }}_z\dot{x}\end{array}$$

(1)

where $m$ denotes the mass of a mass block, $x$ and $y$ represent the displacement of the mass on the x-axis and y-axis in the rotating coordinate system, respectively. $k_x$ and $k_y$ represent the spring coefficients, $d_x$ and $d_y$ represent the damping coefficients in the x-axis and y-axis directions, respectively, $u_x$ and $u_y$ are the control inputs for the two axes, and ${\mathrm{\Omega }}_x,{\mathrm{\Omega }}_y,{\mathrm{\Omega }}_z$ denote components of angular velocity for all three axes.

Due to the limitation of manufacturing process, the actual micro gyroscope cannot achieve the ideal symmetrical structure, and there will be certain structural errors, which will affect the measurement accuracy of the micro gyroscope, while its output will also have corresponding errors. At the same time, considering other factors that cause errors, the dynamic equation of the micro gyroscope can be modified as follows:

$$\begin{array}{l}m\ddot{x}+d_{xx}\dot{x}+d_{xy}\dot{y}+k_{xx}x+k_{xy}y=u_x+2m{\mathrm{\Omega }}_z\dot{y}\\ m\ddot{y}+d_{xy}\dot{x}+d_{yy}\dot{y}+k_{xy}x+k_{yy}y=u_y-2m{\mathrm{\Omega }}_z\dot{x}\end{array}$$

(2)

where $k_{xy}$ and $d_{xy}$ represent the coupling and damping coefficients from manufacturing error, respectively, $d_{xx}$ and $d_{yy}$ show the damping coefficients, $k_{xx}$ and $k_{yy}$ represent the spring coefficients, and ${\mathrm{\Omega }}_z$ denotes the angular velocity of the z-axis.

Performing dimensionless processing of Equation (2) by dividing both sides by $m$, $q_0$, and ${\omega }_0^2$, one can derive the following equations:

$$\begin{array}{l}\ddot{x}+d_{XX}\dot{x}+d_{XY}\dot{y}+{\omega }_X^{2}x+{\omega }_{XY}y=u_X+2{\mathrm{\Omega }}_Z\dot{y}\\ \ddot{y}+d_{XY}\dot{x}+d_{YY}\dot{y}+{\omega }_{XY}x+{\omega }_Y^{2}y=u_Y+2{\mathrm{\Omega }}_Z\dot{x}\end{array}$$

(3)

where $q_0$ denotes a reference length, while ${\omega }_0^2$ represents a resonance frequency’s square.

Converting Equation (3) into vector form obtains

$$\ddot{q}+D\dot{q}+kq=u-2\mathrm{\Omega }\dot{q}$$

(4)

In addition, considering the uncertainty error and external disturbance of the system, Equation (4) can be revised as follows:

$$\ddot{q}+(D+2\mathrm{\Omega })\dot{q}+Kq=u+b(t)$$

(5)

where $b(t)$ represents uncertainty errors and external disturbances in the system, bounded by $\left|b\left(t\right)\right|\le L_{\max }$, while $L_{\max }$ is the upper bound of uncertainties and external disturbances, which is a positive number.

$$\begin{array}{l}q=\left[\begin{array}{l}x\\ y\end{array}\right],D=\left[\begin{array}{cc}{d}_{XX}& d_{XY}\\ d_{XY}& d_{YY}\end{array}\right],\mathrm{\Omega }=\left[\begin{array}{cc}0& -{\mathrm{\Omega }}_Z\\ {\mathrm{\Omega }}_Z& 0\end{array}\right],\\ K=\left[\begin{array}{cc}{\omega }_X^2& {\omega }_{XY}\\ {\omega }_{XY}& {\omega }_Y^2\end{array}\right],u=\left[\begin{array}{l}{u}_X\\ u_Y\end{array}\right],b=\left[\begin{array}{l}{b}_X\\ b_Y\end{array}\right]\end{array}$$

(6)

3. Description of Non-Singular Fast Terminal Sliding Mode Control

The second-order uncertain nonlinear dynamic system can be considered as follows:

$$\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=f(x)+g(x)u+b(x)\end{array}$$

(7)

where $x={\left[x_1,x_2\right]}^T$, $g(x)\ne 0$, $b(x)$ represents uncertainty errors and external disturbances of the system.

According to [17,18,19,20], the sliding mode surface of non-singular fast terminal algorithms can be considered as follows:

$$s=x_1+a{x}_1{}^{g/h}+b{x}_2^{p/q}$$

(8)

where $a,b>0$, $p>q$, $g>h$, and the values of $p$, $q$, $g$, and $h$ are all positive odd numbers, satisfying the following conditions:

$$1<\frac{p}{q}<2,\frac{g}{h}>\frac{p}{q}$$

A non-singular fast terminal sliding mode controller was designed as follows:

$$u=-g^{-1}(x)[f(x)+\frac{1}{b}\frac{q}{p}{x}_2{}^{2-p/q}(1+a\frac{g}{h}{x}_1^{g/h-1})+(\epsilon +\eta )\mathrm{sgn}(s)]$$

(9)

where $\eta >0$, $\epsilon >0$.

Next, the stability analysis is discussed. The derivative of $s$ is derived as follows:

$$\begin{array}{l}\dot{s}={\dot{x}}_1+a\frac{g}{h}{x}_1^{g/h-1}{\dot{x}}_1+b\frac{p}{q}{x}_2{}^{p/q-1}{\dot{x}}_2\\ =x_2(1+a\frac{g}{h}{x}_1^{g/h-1})+b\frac{p}{q}{x}_2{}^{p/q-1}\left(f(x)+g(x)u+b(x)\right)\\ =x_2(1+a\frac{g}{h}{x}_1^{g/h-1})+b\frac{p}{q}{x}_2{}^{p/q-1}(f(x)+b(x)-f(x)-\frac{1}{b}\frac{q}{p}{x}_2{}^{2-p/q}(1+a\frac{g}{h}{x}_1^{g/h-1})-(\epsilon +\eta )\mathrm{sgn}(s))\\ =b\frac{p}{q}{x}_2{}^{p/q-1}\left(b(x)-(\epsilon +\eta )\mathrm{sgn}(s)\right)\end{array}$$

(10)

Then

$$s\dot{s}=b\frac{p}{q}{x}_2{}^{p/q-1}\left[sb(x)-(\epsilon +\eta )\left|s\right|\right]$$

(11)

Because $\eta >0$ $b>0$, then $1<\frac{p}{q}<2$, and $p$ and $q$ both all positive odd numbers; $b(x)<\epsilon$, so

$$x_2{}^{p/q-1}>0$$

(12)

$$s\dot{s}\le -b\frac{p}{q}{x}_2^{p/q-1}\eta \left|s\right|=-{\eta }^{\prime }\left|s\right|$$

(13)

where ${\eta }^{\prime }=b\frac{p}{q}{x}_2{}^{\frac{p}{q}-1}\eta$.

It can be seen that the controller based on the non-singular fast terminal sliding mode design satisfies the Lyapunov stability condition.

4. Structure of the Gated Recurrent Fuzzy Neural Network

The network structure diagram of the GRFNN is shown in Figure 2. There are five layers in the network, of which the third layer is composed of a GRU, and each neuron is a complete gated recurrent unit. There are two gates in the gated recurrent unit: the update gate and the reset gate. The gated recurrent unit can use historical data to avoid the problem of gradient disappearance. Its specific structure is shown in Figure 3.

The signal propagation and main functions of each layer can be expressed as follows:

(1)

Layer 1—Input Layer: The main function of the input layer is to complete the transmission of the input signal.

Input signal:

$$X^1(N)={\left[\begin{array}{ccc}{x}_1^1(N)& \cdots & x_k^1(N)\end{array}\right]}^T$$

(14)

Output signal:

$$Y^1(N)={\left[\begin{array}{ccc}{y}_1^1(N)& \cdots & y_k^1(N)\end{array}\right]}^T$$

(15)

Relationship between the input signal and output signal:

$$y_k^1(N)=x_k^1(N),k=1,2$$

(16)

where $N$ is the number of iterations. The superscript of the input or output node indicates the number of neural network layers, and the subscript indicates the node’s serial number.

(2)

Layer 2—Fuzzification Layer: Gaussian fuzzification calculation is carried out on the input signal to determine the membership degree of the output signal. The connection mode is that each output of the first layer is connected with three neurons of the second layer.

Input signal:

$$x_{{}_i}^2(N)=y_i^1(N),i=1,2$$

(17)

$$X^2(N)={\left[\begin{array}{ccc}{x}_1^2(N)& \cdots & x_i^2(N)\end{array}\right]}^T$$

(18)

Output signal:

$$Y^2(N)=\left[\begin{array}{ccc}{y}_{11}^2(N)& \cdots & y_{1j}^2(N)\\ ⋮& \ddots & ⋮\\ y_{i1}^2(N)& \cdots & y_{ij}^2(N)\end{array}\right]$$

(19)

Relationship between the input signal and output signal:

$$y_{{}_{ij}}^2(N)=\exp \left[-\frac{{\left(x_{{}_i}^2(N)-c_{{}_{ij}}^2\right)}^2}{b_{{}_{ij}}^2}\right],j=1,2,3$$

(20)

where $c_{{}_{ij}}^2$ is the center vector of the j-th node connected to the i-th input of this layer, while $b_{{}_{ij}}^2$ is the base width of the j-th node connected to the i-th input of this layer.

(3)

Layer 3—GRU Layer: There are three nodes in this layer, and each node is a GRU. The GRU has two gates: an update gate and a reset gate. By selectively ignoring and memorizing historical data, the neural network gradient can be solved.

Input signal:

$$x_{{}_n}^3(N)=\left[\begin{array}{c}{y}_{1n}^2(N)\\ ⋮\\ y_{mn}^2(N)\end{array}\right],m=1,2;n=1,2,3$$

(21)

$$X^3(N)={\left[\begin{array}{ccc}{x}_1^3(N)& \cdots & x_n^3(N)\end{array}\right]}^T$$

(22)

Output signal:

$$Y^3(N)={\left[\begin{array}{ccc}{y}_1^3(N)& \cdots & y_n^3(N)\end{array}\right]}^T$$

(23)

Relationships between the input signal and output signal are expressed as follows:

$$z_{{}_n}^3(N)=\sigma (w_{{}_{zn}}^3·x_n^3(N)+u_{{}_{zn}}^3·h_{{}_n}^3(N-1)+b_{zn}^3)$$

(24)

$$r_{{}_n}^3(N)=\sigma (w_{{}_{rn}}^3·x_n^3(N)+u_{{}_{rn}}^3·h_{{}_n}^3(N-1)+b_{rn}^3)$$

(25)

$${\tilde{h}}_{{}_n}^3(N)=\tanh (w_{{}_{hn}}^3·x_{hn}^3(N)+u_{{}_{hn}}^3·(r_{{}_n}^3(N)\odot h_{{}_n}^3(N-1)))$$

(26)

$$h_{{}_n}^3(N)=(1-z_{{}_n}^3(N))\odot h_{{}_n}^3(N-1)+z_{{}_n}^3(N)\odot {\tilde{h}}_{{}_n}^3(N)$$

(27)

$$y_{{}_n}^3(N)=h_{{}_n}^3(N)$$

(28)

where $\sigma (x)$ is the sigmoid function and $\tanh (x)$ is the hyperbolic tangent function.

$$\sigma (x)=\frac{1}{1+e^{-x}},\tanh (x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

(29)

$z_{{}_n}^3(N)$ is the output of the n-th node updating the gate at the N-th iteration, $r_{{}_n}^3(N)$ is the output of the n-th node resetting the gate at the N-th iteration, $h_{{}_n}^3(N)$ is the output of the gated recurrent unit of the n-th node at the N-th iteration. $\left[\begin{array}{ccc}{w}_{zn}^3& u_{zn}^3& b_{zn}^3\end{array}\right]$, $\left[\begin{array}{ccc}{w}_{rn}^3& u_{rn}^3& b_{rn}^3\end{array}\right]$ and $\left[\begin{array}{ccc}{w}_{hn}^3& u_{hn}^3& b_{hn}^3\end{array}\right]$ are the weight vectors and bias values of different computing links in the gated recurrent unit.

(4)

Layer 4—Rule Layer: This layer has three nodes, each of which is a product operation on the output of the previous layer.

Input signal:

$$x_{{}_n}^4(N)=y_n^3(N),n=1,2,3$$

(30)

$$X^4(N)={\left[\begin{array}{ccc}{x}_1^4(N)& \cdots & x_n^4(N)\end{array}\right]}^T$$

(31)

Output signal:

$$Y^4(N)={\left[\begin{array}{ccc}{y}_1^4(N)& \cdots & y_n^4(N)\end{array}\right]}^T$$

(32)

Relationships between the input signal and output signal are written as follows:

$${\mathrm{y}}_1^4(N)=x_1^4(N)·x_2^4(N)$$

(33)

$${\mathrm{y}}_2^4(N)=x_1^4(N)·x_3^4(N)$$

(34)

$${\mathrm{y}}_3^4(N)=x_2^4(N)·x_3^4(N)$$

(35)

(5)

Layer 5—Output Layer: The output layer outputs the final result of the network computation. The output of the rule layer is added according to the weight to obtain the network calculation result.

Input signal:

$$x_{{}_n}^5(N)=y_n^4(N),n=1,2,3$$

(36)

$$X^5(N)={\left[\begin{array}{ccc}{x}_1^5(N)& \cdots & x_n^5(N)\end{array}\right]}^T$$

(37)

Output signal:

$$Y^5(N)=y_{{}^l}^5(N),l=1$$

(38)

Relationships between the input signal and output signal are written as follows:

$$y_l^5(N)={\displaystyle \sum _{n=1}^3{w}_{yn}}{x}_n^5(N)$$

(39)

where $w_{yn}$ is a weight coefficient, and $Y^5(N)$ is the final output of the GRFNN (Algorithm 1).

Algorithm 1: The algorithm of the GRFNN

Input:$e$, $\dot{e}$ // These denote the position tracking error and velocity tracking error, respectively Output: $f$ // Estimation results of the GRFNN for the unknown terms of the micro gyroscopic model. Calculate the affiliation of the two inputs separately (Equation (20)). Each input needs to be substituted into three affiliation functions for calculation. Calculate the GRU output (Equations (24)–(27)). The corresponding affiliation combinations of different inputs are descended into the corresponding GRUs. Calculate the rule layer results (Equations (33)–(35)). Calculate the output layer result (Equation (39)). The calculation result, which is the estimated value of the unknown term of the model, is brought into the controller to participate in the control output (Equation (39)). The tracking error and speed error are calculated and substituted into the adaptive law of GRFNN parameters to achieve the online adjustment of parameters (Equations (63)–(74)).

5. Design of a Non-Singular Fast Terminal Sliding Mode Controller for Micro Gyroscope Based on a Gated Recurrent Fuzzy Neural Network

The configuration of the proposed control scheme is depicted in Figure 4.

Equation (5) can be simplified as follows:

$$\ddot{q}=f+u+b(t)$$

(40)

where $f={\left[\begin{array}{cc}{f}_1& f_2\end{array}\right]}^T=-[(D+2\mathrm{\Omega })\dot{q}+Kq]$ represents the unknown model of the micro gyroscope.

The tracking error is defined as follows:

$$e=q_r-q={[x_r-x,y_r-y]}^T$$

(41)

where $x_r$ and $y_r$ represent the reference trajectory of the x-axis and y-axis, respectively. A non-singular fast terminal sliding surface is designed as follows:

$$s=e+a{e}^{g/h}+b{\dot{e}}^{p/q}$$

(42)

where $a$ and $b$ satisfy $a,b>0$, and the values of $p$, $q$, $g$, and $h$ are all positive odd numbers and satisfy the following conditions:

$$p>q,g>h,1<\frac{p}{g}<2,\frac{g}{h}>\frac{p}{q}$$

Algorithm 2: The calculation process of the control system

Input:$q_r$ // Reference trajectory Output: $q$ // Actual trajectory of model The tracking error and the tracking speed error are calculated according to the reference track and the actual track. The sliding mode value is calculated by substituting two errors into the sliding mode function. The two errors are used as the inputs of the GRFNN to estimate the unknown terms of the model. The calculation process references Algorithm 1. The error and sliding mode value are substituted into the parameter adaptive law to update the parameters. The sliding mode value, estimated value of the GRFNN, error, and parameter are substituted into the controller function to calculate the controller’s output. The controller output is used as the input of the model, and the actual trajectory of the model’s output is obtained.

Using Equations (41) and (42), one can obtain the derivative of $s$:

$$\dot{s}=\dot{e}+a\frac{g}{h}diag(e_1^{g/h-1},e_2^{g/h-1})\dot{e}+b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})({\ddot{q}}_r-\ddot{q})$$

(43)

Substituting Equation (40) into Equation (43) generates

$$\dot{s}=\dot{e}+a\frac{g}{h}diag(e_1^{g/h-1},e_2^{g/h-1})\dot{e}+b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})[{\ddot{q}}_r-f-u-b(t)]$$

(44)

According to the design method of equivalent control, without considering $b(t)$, the equivalent control law of the system can be designed by setting $\dot{s}=0$.

$$u_{eq}={\ddot{q}}_r-f+\frac{1}{b}\frac{q}{p}diag({\dot{e}}_1^{1-p/q},{\dot{e}}_2^{1-p/q})[\dot{e}+a\frac{g}{h}diag(e_1^{g/h-1},e_2^{g/h-1})\dot{e}]$$

(45)

The switching control law is designed as follows:

$$u_{vss}=\epsilon \mathrm{sgn}(s)+ks$$

(46)

where $\epsilon =L_{\max }+\eta$ is the gain coefficient of the switching term, and $\eta$ and $k$ satisfy $\eta ,k>0$.

Therefore, the control law can be designed as follows:

$$\begin{array}{c}u={\ddot{q}}_r-f+\frac{1}{b}\frac{q}{p}diag({\dot{e}}_1^{1-p/q},{\dot{e}}_2^{1-p/q})[\dot{e}+a\frac{g}{h}diag(e_1^{g/h-1},e_2^{g/h-1})\dot{e}]\\ \begin{array}{c}\end{array}+(L_{\max }+\eta )\mathrm{sgn}(s)+ks\end{array}$$

(47)

Since $f$ is an unknown parameter of the micro gyroscope model, the control law described by Equation (47) cannot be used directly. Therefore, the unknown item is estimated by using the GRFNN, and the approximation function of the GRFNN is expressed as follows:

$$\widehat{f}={\widehat{w}}_{\mathrm{y}}{}^T\widehat{\varphi }(x,\widehat{c},\widehat{b},{\widehat{w}}_z,{\widehat{u}}_z,{\widehat{b}}_z,{\widehat{w}}_r,{\widehat{u}}_r,{\widehat{b}}_r,{\widehat{w}}_h,{\widehat{u}}_h,{\widehat{b}}_h)$$

(48)

where the values of $x,\widehat{c},\widehat{b},{\widehat{w}}_z,{\widehat{u}}_z,{\widehat{b}}_z,{\widehat{w}}_r,{\widehat{u}}_r,{\widehat{b}}_r,{\widehat{w}}_h,{\widehat{u}}_h,{\widehat{b}}_h$, and ${\widehat{w}}_{\mathrm{y}}$ can be adaptively adjusted through the adaptive laws.

The actual control law can be expressed as follows:

$$u={\ddot{q}}_r-\widehat{f}+\frac{1}{b}\frac{q}{p}diag({\dot{e}}_1^{1-p/q},{\dot{e}}_2^{1-p/q})[\dot{e}+a\frac{g}{h}diag(e_1^{g/h-1},e_2^{g/h-1})\dot{e}]+(L_{\max }+\eta )\mathrm{sgn}(s)+ks$$

(49)

Assumption 1: There are the optimal center vector $c^{*}$, the optimal basis width $b^{*}$, and the other weights in networks $w_y^{*}$, $w_z^{*}$, $u_z^{*}$, $b_z^{*}$, $w_r^{*}$, $u_r^{*}$, $b_r^{*}$, $w_h^{*}$, $u_h^{*}$, and $b_h^{*}$ to estimate the unknown model $f$; then, one can get $f=w_y^{*}{}^T{\varphi }^{*}+\delta$, where $\delta$ represents the mapping errors and ${\varphi }^{*}={\varphi }^{*}(x,c^{*},b^{*},w_z^{*},u_z^{*},b_z^{*},w_r^{*},u_r^{*},b_r^{*},w_h^{*},u_h^{*},b_h^{*}).$ The approximation errors of the parameters of the GRFNN are defined as follows:

$$\{\begin{array}{l}{\tilde{w}}_z=w_z^{*}-{\widehat{w}}_z\\ {\tilde{u}}_z=u_z^{*}-{\widehat{u}}_z\\ {\tilde{b}}_z=b_z^{*}-{\widehat{b}}_z\end{array}\{\begin{array}{l}{\tilde{w}}_r=w_r^{*}-{\widehat{w}}_r\\ {\tilde{u}}_r=u_r^{*}-{\widehat{u}}_r\\ {\tilde{b}}_r=b_r^{*}-{\widehat{b}}_r\end{array}\{\begin{array}{l}{\tilde{w}}_h=w_h^{*}-{\widehat{w}}_h\\ {\tilde{u}}_h=u_h^{*}-{\widehat{u}}_h\\ {\tilde{b}}_h=b_h^{*}-{\widehat{b}}_h\end{array}\{\begin{array}{c}{\tilde{w}}_{\mathrm{y}}=w_y^{*}-{\widehat{w}}_{\mathrm{y}}\\ \tilde{\varphi }={\varphi }^{\ast }-\widehat{\varphi }\\ \tilde{b}=b^{\ast }-\widehat{b}\\ \tilde{c}=c^{*}-\widehat{c}\end{array}$$

(50)

The error between the unknown item of the system and the estimated value of the GRFNN is calculated as follows:

$$\begin{array}{l}f-\widehat{f}=w_y^{*}{}^T{\varphi }^{*}+\delta -{\widehat{w}}_y^T\widehat{\varphi }\\ =w_y^{*}{}^T(\widehat{\varphi }+\tilde{\varphi })+\delta -{\widehat{w}}_y^T\widehat{\varphi }\\ =(w_y^{*}{}^T-{\widehat{w}}_y^T)\widehat{\varphi }+({\widehat{w}}_y^T+{\tilde{w}}_y^T)\tilde{\varphi }+\delta \\ ={\tilde{w}}_y^T\widehat{\varphi }+{\widehat{w}}_y^T\tilde{\varphi }+{\tilde{w}}_y^T\tilde{\varphi }+\delta \\ ={\tilde{w}}_y^T\widehat{\varphi }+{\widehat{w}}_y^T\tilde{\varphi }+{\delta }_0\end{array}$$

(51)

where ${\delta }_0={\tilde{w}}_y^T\tilde{\varphi }+\delta$ denotes the approximation error.

Then, expanding $\tilde{\varphi }$ by Taylor expansion obtains

$$\begin{array}{l}\tilde{\varphi }={\frac{d\varphi }{d{w}_z}|}_{w_z={\widehat{w}}_z}(w_z^{*}-{\widehat{w}}_z)+{\frac{d\varphi }{d{u}_z}|}_{u_z={\widehat{u}}_z}(u_z^{*}-{\widehat{u}}_z)+{\frac{d\varphi }{d{b}_z}|}_{b_z={\widehat{b}}_z}(b_z^{*}-{\widehat{b}}_z)\\ +{\frac{d\varphi }{d{w}_r}|}_{w_r={\widehat{w}}_r}(w_r^{*}-{\widehat{w}}_r)+{\frac{d\varphi }{d{u}_r}|}_{u_r={\widehat{u}}_r}(u_r^{*}-{\widehat{u}}_r){+\frac{d\varphi }{d{b}_r}|}_{b_r={\widehat{b}}_r}(b_r^{*}-{\widehat{b}}_r)\\ +{\frac{d\varphi }{d{w}_h}|}_{w_h={\widehat{w}}_h}(w_h^{*}-{\widehat{w}}_h)+{\frac{d\varphi }{d{u}_h}|}_{u_h={\widehat{u}}_h}(u_h^{*}-{\widehat{u}}_h){+\frac{d\varphi }{d{b}_h}|}_{b_h={\widehat{b}}_h}(b_h^{*}-{\widehat{b}}_h)\\ +{\frac{d\varphi }{db}|}_{b=\widehat{b}}(b^{*}-\widehat{b})+{\frac{d\varphi }{dc}|}_{c=\widehat{c}}(c^{*}-\widehat{c})+O\\ ={\varphi }_{w_z}\cdot {\tilde{w}}_z+{\varphi }_{u_z}\cdot {\tilde{u}}_z+{\varphi }_{b_z}\cdot {\tilde{b}}_z+{\varphi }_{w_r}\cdot {\tilde{w}}_r\\ +{\varphi }_{u_r}\cdot {\tilde{u}}_r+{\varphi }_{b_r}\cdot {\tilde{b}}_r+{\varphi }_{w_h}\cdot {\tilde{w}}_h+{\varphi }_{u_h}\cdot {\tilde{u}}_h\\ +{\varphi }_{b_h}\cdot {\tilde{b}}_h+{\varphi }_b\cdot \tilde{b}+{\varphi }_c\cdot \tilde{c}+O\end{array}$$

(52)

where $O$ is a high-order term, and ${\varphi }_{w_z}$, ${\varphi }_{u_z}$, ${\varphi }_{b_z}$, ${\varphi }_{w_r}$, ${\varphi }_{u_r}$, ${\varphi }_{b_r}$, ${\varphi }_{w_h}$, ${\varphi }_{u_h}$, ${\varphi }_{b_h}$, ${\varphi }_b$, and ${\varphi }_c$ are matrices, expressed as follows:

$$\begin{gathered} \frac{d\varphi }{d{w}_z}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{w}_{z11}^3}& \cdots & \frac{d{\varphi }_1}{d{w}_{z23}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{w}_{z11}^3}& \cdots & \frac{d{\varphi }_3}{d{w}_{z23}^3}\end{array}\right]|}_{w_z={\widehat{w}}_z}\in R^{3\times 6},\frac{d\varphi }{d{u}_z}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{u}_{z1}^3}& \cdots & \frac{d{\varphi }_1}{d{u}_{z3}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{u}_{z1}^3}& \cdots & \frac{d{\varphi }_3}{d{u}_{z3}^3}\end{array}\right]|}_{u_{\mathrm{z}}={\widehat{u}}_z}\in R^{3\times 3} \\ \frac{d\varphi }{d{b}_z}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{b}_{z1}^3}& \cdots & \frac{d{\varphi }_1}{d{b}_{z3}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{b}_{z1}^3}& \cdots & \frac{d{\varphi }_3}{d{b}_{z3}^3}\end{array}\right]|}_{b_z={\widehat{b}}_z}\in R^{3\times 3},\frac{d\varphi }{d{w}_r}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{w}_{r11}^3}& \cdots & \frac{d{\varphi }_1}{d{w}_{r23}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{w}_{r11}^3}& \cdots & \frac{d{\varphi }_3}{d{w}_{r23}^3}\end{array}\right]|}_{w_r={\widehat{w}}_r}\in R^{3\times 6} \\ \frac{d\varphi }{d{u}_r}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{u}_{r1}^3}& \cdots & \frac{d{\varphi }_1}{d{u}_{r3}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{u}_{r1}^3}& \cdots & \frac{d{\varphi }_3}{d{u}_{r3}^3}\end{array}\right]|}_{u_r={\widehat{u}}_r}\in R^{3\times 3},\frac{d\varphi }{d{b}_r}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{b}_{r1}^3}& \cdots & \frac{d{\varphi }_1}{d{b}_{r3}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{b}_{r1}^3}& \cdots & \frac{d{\varphi }_3}{d{b}_{r3}^3}\end{array}\right]|}_{b_r={\widehat{b}}_r}\in R^{3\times 3} \\ \frac{d\varphi }{d{w}_h}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{w}_{h11}^3}& \cdots & \frac{d{\varphi }_1}{d{w}_{h23}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{w}_{h11}^3}& \cdots & \frac{d{\varphi }_3}{d{w}_{h23}^3}\end{array}\right]|}_{w_h={\widehat{w}}_h}\in R^{3\times 6},\frac{d\varphi }{d{u}_h}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{u}_{h1}^3}& \cdots & \frac{d{\varphi }_1}{d{u}_{h3}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{u}_{h1}^3}& \cdots & \frac{d{\varphi }_3}{d{u}_{h3}^3}\end{array}\right]|}_{u_h={\widehat{u}}_h}\in R^{3\times 3} \\ \frac{d\varphi }{d{b}_h}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{b}_{h1}^3}& \cdots & \frac{d{\varphi }_1}{d{b}_{h3}^3}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{b}_{h1}^3}& \cdots & \frac{d{\varphi }_3}{d{b}_{h3}^3}\end{array}\right]|}_{b_h={\widehat{b}}_h}\in R^{3\times 3},\frac{d\varphi }{db}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{b}_{11}^2}& \cdots & \frac{d{\varphi }_1}{d{b}_{23}^2}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{b}_{11}^2}& \cdots & \frac{d{\varphi }_3}{d{b}_{23}^2}\end{array}\right]|}_{b=\widehat{b}}\in R^{3\times 6} \\ \frac{d\varphi }{dc}={\left[\begin{array}{ccc}\frac{d{\varphi }_1}{d{c}_{11}^2}& \cdots & \frac{d{\varphi }_1}{d{c}_{23}^2}\\ ⋮& \ddots & ⋮\\ \frac{d{\varphi }_3}{d{c}_{11}^2}& \cdots & \frac{d{\varphi }_3}{d{c}_{23}^2}\end{array}\right]|}_{c=\widehat{c}}\in R^{3\times 6} \end{gathered}$$

(53)

6. Stability Analysis

In order to prove that the state will approach zero, one can choose a Lyapunov function candidate as follows:

$$\begin{array}{c}V=\frac{1}{2}{s}^{T}s+\frac{1}{2{\eta }_1}tr({\tilde{w}}_y^T{\tilde{w}}_y)+\frac{1}{2{\eta }_2}tr({\tilde{w}}_z^T{\tilde{w}}_z)+\frac{1}{2{\eta }_3}tr({\tilde{u}}_z^T{\tilde{u}}_z)\\ +\frac{1}{2{\eta }_4}tr({\tilde{b}}_z^T{\tilde{b}}_z)+\frac{1}{2{\eta }_5}tr({\tilde{w}}_r^T{\tilde{w}}_r)+\frac{1}{2{\eta }_6}tr({\tilde{u}}_r^T{\tilde{u}}_r)\\ +\frac{1}{2{\eta }_7}tr({\tilde{b}}_r^T{\tilde{b}}_r)+\frac{1}{2{\eta }_8}tr({\tilde{w}}_h^T{\tilde{w}}_h)+\frac{1}{2{\eta }_9}tr({\tilde{u}}_h^T{\tilde{u}}_h)\\ +\frac{1}{2{\eta }_{10}}tr({\tilde{b}}_h^T{\tilde{b}}_h)+\frac{1}{2{\eta }_{11}}tr({\tilde{b}}^T\tilde{b})+\frac{1}{2{\eta }_{12}}tr({\tilde{c}}^T\tilde{c})\end{array}$$

(54)

The above expression can be simplified by defining the following expression:

$$\begin{array}{c}G=\frac{1}{2{\eta }_1}tr({\tilde{w}}_y^T{\tilde{w}}_y)+\frac{1}{2{\eta }_2}tr({\tilde{w}}_z^T{\tilde{w}}_z)+\frac{1}{2{\eta }_3}tr({\tilde{u}}_z^T{\tilde{u}}_z)\\ +\frac{1}{2{\eta }_4}tr({\tilde{b}}_z^T{\tilde{b}}_z)+\frac{1}{2{\eta }_5}tr({\tilde{w}}_r^T{\tilde{w}}_r)+\frac{1}{2{\eta }_6}tr({\tilde{u}}_r^T{\tilde{u}}_r)\\ +\frac{1}{2{\eta }_7}tr({\tilde{b}}_r^T{\tilde{b}}_r)+\frac{1}{2{\eta }_8}tr({\tilde{w}}_h^T{\tilde{w}}_h)+\frac{1}{2{\eta }_9}tr({\tilde{u}}_h^T{\tilde{u}}_h)\\ +\frac{1}{2{\eta }_{10}}tr({\tilde{b}}_h^T{\tilde{b}}_h)+\frac{1}{2{\eta }_{11}}tr({\tilde{b}}^T\tilde{b})+\frac{1}{2{\eta }_{12}}tr({\tilde{c}}^T\tilde{c})\end{array}$$

(55)

Therefore, the simplified form of Equation (54) can be written as follows:

$$V=\frac{1}{2}{s}^{T}s+G$$

(56)

Then, solving the derivative of the Lyapunov function:

$$\dot{V}=s^T\dot{s}+\dot{G}$$

(57)

where

$$\begin{array}{c}\dot{G}=\frac{1}{{\eta }_1}tr({\tilde{w}}_y^T{\dot{\tilde{w}}}_y)+\frac{1}{{\eta }_2}tr({\dot{\tilde{w}}}_z^T{\tilde{w}}_z)+\frac{1}{{\eta }_3}tr({\dot{\tilde{u}}}_z^T{\tilde{u}}_z)\\ +\frac{1}{{\eta }_4}tr({\dot{\tilde{b}}}_z^T{\tilde{b}}_z)+\frac{1}{{\eta }_5}tr({\dot{\tilde{w}}}_r^T{\tilde{w}}_r)+\frac{1}{{\eta }_6}tr({\dot{\tilde{u}}}_r^T{\tilde{u}}_r)\\ +\frac{1}{{\eta }_7}tr({\dot{\tilde{b}}}_r^T{\tilde{b}}_r)+\frac{1}{{\eta }_8}tr({\dot{\tilde{w}}}_h^T{\tilde{w}}_h)+\frac{1}{{\eta }_9}tr({\dot{\tilde{u}}}_h^T{\tilde{u}}_h)\\ +\frac{1}{{\eta }_{10}}tr({\dot{\tilde{b}}}_h^T{\tilde{b}}_h)+\frac{1}{{\eta }_{11}}tr({\dot{\tilde{b}}}^T\tilde{b})+\frac{1}{{\eta }_{12}}tr({\dot{\tilde{c}}}^T\tilde{c})\end{array}$$

(58)

From Equations (44) and (49), the time derivative of $s$ can be expressed as follows:

$$\dot{s}=-b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})[f-\widehat{f}+(L_{\max }+\eta )\mathrm{sgn}(s)+ks+b(t)]$$

(59)

Then, substituting Equation (59) into Equation (57) generates

$$\dot{V}=-s^{T}F[f-\widehat{f}+(L_{\max }+\eta )\mathrm{sgn}(s)+ks+b(t)]+\dot{G}$$

(60)

where

$$F=b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})$$

(61)

Substituting Equations (51), (52), and (58) into Equation (60) generates

$$\begin{array}{c}\dot{V}=-s^{T}F{\widehat{w}}_y^T({\varphi }_{w_z}\cdot {\tilde{w}}_z+{\varphi }_{u_z}\cdot {\tilde{u}}_z+{\varphi }_{b_z}\cdot {\tilde{b}}_z+{\varphi }_{w_r}\cdot {\tilde{w}}_r+{\varphi }_{u_r}\cdot {\tilde{u}}_r\\ +{\varphi }_{b_r}\cdot {\tilde{b}}_r+{\varphi }_{w_h}\cdot {\tilde{w}}_h+{\varphi }_{u_h}\cdot {\tilde{u}}_h+{\varphi }_{b_h}\cdot {\tilde{b}}_h+{\varphi }_b\cdot \tilde{b}+{\varphi }_c\cdot \tilde{c}+O)\\ -s^{T}F{\tilde{w}}_y^T\widehat{\varphi }-s^{T}F[(L_{\max }+\eta )\mathrm{sgn}(s)+ks+b(t)+{\delta }_0]\\ +\frac{1}{{\eta }_1}tr({\tilde{w}}_y^T{\dot{\tilde{w}}}_y)+\frac{1}{{\eta }_2}tr({\dot{\tilde{w}}}_z^T{\tilde{w}}_z)+\frac{1}{{\eta }_3}tr({\dot{\tilde{u}}}_z^T{\tilde{u}}_z)\\ +\frac{1}{{\eta }_4}tr({\dot{\tilde{b}}}_z^T{\tilde{b}}_z)+\frac{1}{{\eta }_5}tr({\dot{\tilde{w}}}_r^T{\tilde{w}}_r)+\frac{1}{{\eta }_6}tr({\dot{\tilde{u}}}_r^T{\tilde{u}}_r)\\ +\frac{1}{{\eta }_7}tr({\dot{\tilde{b}}}_r^T{\tilde{b}}_r)+\frac{1}{{\eta }_8}tr({\dot{\tilde{w}}}_h^T{\tilde{w}}_h)+\frac{1}{{\eta }_9}tr({\dot{\tilde{u}}}_h^T{\tilde{u}}_h)\\ +\frac{1}{{\eta }_{10}}tr({\dot{\tilde{b}}}_h^T{\tilde{b}}_h)+\frac{1}{{\eta }_{11}}tr({\dot{\tilde{b}}}^T\tilde{b})+\frac{1}{{\eta }_{12}}tr({\dot{\tilde{c}}}^T\tilde{c})\end{array}$$

(62)

Setting $-s^{T}F{\tilde{w}}_y^T\widehat{\varphi }+\frac{1}{{\eta }_1}tr({\tilde{w}}_y^T{\dot{\tilde{w}}}_y)=0$ generates

$$-{\dot{\widehat{w}}}_y={\dot{\tilde{w}}}_y={\eta }_1{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})\widehat{\varphi }$$

(63)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{w_z}{\tilde{w}}_z+\frac{1}{{\eta }_2}tr({\dot{\tilde{w}}}_z^T{\tilde{w}}_z)=0$ generates

$$-{\dot{\widehat{w}}}_z^T={\dot{\tilde{w}}}_z^T={\eta }_2{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{w_z}$$

(64)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{u_z}{\tilde{u}}_z+\frac{1}{{\eta }_3}tr({\dot{\tilde{u}}}_z^T{\tilde{u}}_z)=0$ generates

$$-{\dot{\widehat{u}}}_z^T={\dot{\tilde{u}}}_z^T={\eta }_3{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{u_z}$$

(65)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{b_z}{\tilde{b}}_z+\frac{1}{{\eta }_4}tr({\dot{\tilde{b}}}_z^T{\tilde{b}}_z)=0$ generates

$$-{\dot{\widehat{b}}}_z^T={\dot{\tilde{b}}}_z^T={\eta }_4{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{b_z}$$

(66)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{w_r}{\tilde{w}}_r+\frac{1}{{\eta }_5}tr({\dot{\tilde{w}}}_r^T{\tilde{w}}_r)=0$ generates

$$-{\dot{\widehat{w}}}_r^T={\dot{\tilde{w}}}_r^T={\eta }_5{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{w_r}$$

(67)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{u_r}{\tilde{u}}_r+\frac{1}{{\eta }_6}tr({\dot{\tilde{u}}}_r^T{\tilde{u}}_r)=0$ generates

$$-{\dot{\widehat{u}}}_r^T={\dot{\tilde{u}}}_r^T={\eta }_6{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{u_r}$$

(68)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{b_r}{\tilde{b}}_r+\frac{1}{{\eta }_7}tr({\dot{\tilde{b}}}_r^T{\tilde{b}}_r)=0$ generates

$$-{\dot{\widehat{b}}}_r^T={\dot{\tilde{b}}}_r^T={\eta }_7{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{b_r}$$

(69)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{w_h}{\tilde{w}}_h+\frac{1}{{\eta }_8}tr({\dot{\tilde{w}}}_h^T{\tilde{w}}_h)=0$ generates

$$-{\dot{\widehat{w}}}_h^T={\dot{\tilde{w}}}_h^T={\eta }_8{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{w_h}$$

(70)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{u_h}{\tilde{u}}_h+\frac{1}{{\eta }_9}tr({\dot{\tilde{u}}}_h^T{\tilde{u}}_h)=0$ generates

$$-{\dot{\widehat{u}}}_h^T={\dot{\tilde{u}}}_h^T={\eta }_9{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{u_h}$$

(71)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_{b_h}{\tilde{b}}_h+\frac{1}{{\eta }_{10}}tr({\dot{\tilde{b}}}_h^T{\tilde{b}}_h)=0$ generates

$$-{\dot{\widehat{b}}}_h^T={\dot{\tilde{b}}}_h^T={\eta }_{10}{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_{b_h}$$

(72)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_b\tilde{b}+\frac{1}{{\eta }_{11}}tr({\dot{\tilde{b}}}^T\tilde{b})=0$ generates

$$-{\dot{\widehat{b}}}^T={\dot{\tilde{b}}}^T={\eta }_{11}{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_b$$

(73)

Setting $-s^{T}F{\tilde{w}}_y^T{\varphi }_c\tilde{c}+\frac{1}{{\eta }_{12}}tr({\dot{\tilde{c}}}^T\tilde{c})=0$ generates

$$-{\dot{\widehat{c}}}^T={\dot{\tilde{c}}}^T={\eta }_{12}{s}^{T}b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1}){\widehat{w}}_y^T{\varphi }_c$$

(74)

Substituting the adaptive laws represented by Equations (63)–(74) into Equation (62) yields

$$\begin{array}{l}\dot{V}=-s^{T}F[(L_{\max }+\eta )\mathrm{sgn}(s)+ks+b(t)+{\delta }_0]-s^{T}F{\widehat{w}}_y^{T}O\\ =-s^{T}F[(L_{\max }+\eta )\mathrm{sgn}(s)+ks+b(t)+{\delta }_0+O_0]\\ =-s^{T}F(ks+{\delta }_0+O_0)-F\eta \left|s\right|-F(L_{\max }\left|s\right|+b(t)s)\\ \le -s^{T}F(ks+{\delta }_0+O_0)-F\eta \left|s\right|\\ \le -b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})[|s|(\eta -\left|{\delta }_0\right|-\left|O_0\right|)+k{s}^2]\end{array}$$

(75)

where $O_0={\widehat{w}}_y^{T}O$. Because $\left|{\delta }_0\right|\le {\delta }_M$, $\left|O_0\right|\le O_M$, Equation (75) can then be modified as follows:

$$\dot{V}\le -b\frac{p}{q}diag({\dot{e}}_1^{p/q-1},{\dot{e}}_2^{p/q-1})[|s|(\eta -{\delta }_M-O_M)+k{s}^2]$$

(76)

Because $1<(p/q)<2$, $b>0$, and $p$ and $q$ are positive odd numbers, ${\dot{e}}_1^{p/q-1}\ge 0$,${\dot{e}}_2^{p/q-1}\ge 0$. Therefore, as long as the following condition is satisfied:

$$\eta \ge {\delta }_M+O_M$$

(77)

then $\dot{V}\le 0,\dot{V}\in R^{2\times 1}$, and it can be deduced that $\dot{V}$ is semi-negative in the x-axis and y-axis directions of the micro gyroscope. According to the second method of Lyapunov, $V(x)$ is chosen to be positive definite. If $\dot{V}(x)$ is negative definite, or although $\dot{V}(x)$ is semi-negative definite for any initial state $x(t_0)\ne 0$ except $x=0$, $\dot{V}(x)$ is not constantly zero for $x\ne 0$, then the origin equilibrium state is asymptotically stable. According to the third line of Equation (75), it satisfies the condition in the second method of Lyapunov that for any initial state $x=0$, except $x=0$, $\dot{V}(x)$ is not constant zero for $x\ne 0$, which implies that the system is asymptotically stable. It also means that $s$ will converge to zero, as along with $e$ and $\dot{e}$. On the other hand, since $V\ge 0$ and $\dot{V}\le 0$, $V$ and $s$ are both bounded. This indicates that the output trajectory can track the reference trajectory.

The purpose of control is to make $s(t)\dot{s}(t)\to 0$. Based on this purpose, by designing the Lyapunov function described in Equation (54), the adaptive law of the neural network parameters, which is also the gradient of parameters derived from $s(t)\dot{s}(t)$, can be derived and the stability of the system can be proved.

7. Experimental Results

In order to demonstrate the applicability and superiority of the proposed control scheme, two control schemes were used on the MATLAB platform—namely, the GRFNN-based NFTSMC method and the FNN-based SMC method—to conduct simulation experiments. The GRFNN-based NFTSMC method was simplified as NFTSMC-GRFNN, and the FNN-based SMC method was simplified as SMC-FNN. The parameters of the micro gyroscope model can be considered as follows:

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

(78)

After dimensionless processing, the parameters of the micro gyroscope were as follows:

$$\{\begin{array}{l}{\omega }_x{}^2=355.3\\ {\omega }_y{}^2=532.9\\ {\omega }_{xy}=70.99\end{array}\{\begin{array}{l}{d}_{xx}=0.01\\ d_{yy}=0.01\\ d_{xy}=0.002\end{array}$$

(79)

The initial values of the state variables of the system were set as $q_1(0)=0$, ${\dot{q}}_1(0)=0$, $q_2(0)=0$, and ${\dot{q}}_2(0)=0$; the reference trajectories were chosen as $x_m=-\sin (0.5\pi t)$, $y_m=-\sin (0.5\pi t)$, the angular velocity was assumed as ${\mathrm{\Omega }}_z=0.1$, and the external disturbance signal was selected as ${\mathrm{b}\left(\mathrm{t}\right)}_x{=\mathrm{b}\left(\mathrm{t}\right)}_y=0.5\mathrm{randn}(1,1)$. For NFTSMC-GRFNN, the sliding surface parameters were taken as $a=1$, $b=1$, $p=7$, $q=5$, $g=5$, $h=3$, $\epsilon =40$, and $k=20$. The initial values of the parameters of the GRFNN were chosen arbitrarily. For SMC-FNN, the sliding surface was selected as $s=\dot{e}+\alpha e$, where the value of the parameter $\alpha$ is 5, and switching control law $u_{sw}=-D\mathrm{sgn}(s)$, where the value of the coefficient $D$ is 10. Additionally, the base width and center vector of the FNN were fixed values: $b=2$,$c=[-0.100.1;-0.100.1]$.

Under the control of NFTSMC-GRFNN, the trajectory tracking, speed tracking, and tracking errors are shown in Figure 5, Figure 6, Figure 7 and Figure 8. The results show that the x-axis and y-axis output trajectories of the system can track the reference trajectory within a limited time, which means that the mass can move according to the specified amplitude and frequency. Figure 9 shows that the sliding manifold converges to zero in finite time, which implies that the system is converged and stable. Figure 10 and Figure 11 demonstrate the output curve of the control and the approximate curve of the output of the GRFNN, respectively, where the controller output is bounded with less chattering, and the GRFNN can quickly learn and identify unknown parameters of the system model.

It can be seen from Figure 12 that, under both methods, the system can track the reference trajectory in a short time, but it can be clearly observed that NFTSMC-GRFNN can guarantee higher tracking accuracy. From the sliding surface comparison in Figure 13, under the control of NFTSMC-GRFNN, the system state reaches the sliding mode manifold at a faster speed, and after reaching the sliding mode manifold the variation range of the system state is smaller. Figure 14 shows the comparison of the identification performance of the two methods. It can be seen that the GRFNN maintains a smaller error in the approximation of the unknown parameters of the system model compared to the FNN, and the learning speed is faster. In order to more intuitively reflect the differences between the tracking errors of the two methods, the RMSE of the x-axis and y-axis was calculated under the control of the two methods. The RMSE was used to measure the deviation between the observed value and the true value. The smaller the RMSE, the smaller the deviation.

It can be seen from

Table 1. Comparison of the RMSE between NFTSMC-GRFNN and SMC-FNN.

	Scheme	X	Y
RMSE
SMC-FNN		0.003	0.004
NFTSMC-GRFNN		9.0108 × 10−5	1.0398 × 10−4

that, under the control of NFTSMC-GRFNN, the steady-state error of the system is significantly smaller than that under the control of SMC-FNN, which indicates that the control accuracy of NFTSMC-GRFNN is remarkably higher.

8. Conclusions

In this study, a GRFNN-based NFTSMC was designed to control the vibration of a micro gyroscope. The non-singular fast terminal sliding mode was adopted, so that the system not only completely avoids the singular phenomenon of TSMC but also makes the tracking error converge to zero in a limited time. At the same time, it has a positive effect on accelerating the convergence speed and reducing the chattering problem to a certain extent. Aiming at the unknowns and uncertainties of the model, a fuzzy neural network embedded with gated recurrent units was studied to approximate the true value. Due to the introduction of the GRUs, the GRFNN can effectively utilize the previous data and solve the problem of gradient disappearance, so the estimated value is closer to the real value. Simulation results showed that the proposed control system has superior tracking performance, fast response performance, and good convergence performance. Furthermore, compared with SMC-FNN, the proposed control scheme can achieve higher micro gyroscope control accuracy. The theoretical verification was completed through numerical simulation with physical parameters, connecting theory and application. The next research step will be to verify it on the hardware platform.