Adaptive double neural network control for micro-gyroscope based on dynamic surface controller

Abstract

An adaptive double neural network with a dynamic surface control scheme is investigated for a micro-gyroscope in the presence of manufacturing errors and disturbances. A dynamic surface controller, where a first-order filter is introduced in each step, is proposed to reduce the parameters and the computational complexity. One radical basis function neural network is used to approximate the lumped dynamics of the micro-gyroscope. Another radical basis function neural network is designed to approximate a sliding-mode controller to compensate the approximation errors in order to weaken the influence of the approximation errors. Simulation studies prove the effectiveness of the proposed strategy, demonstrate the proposed strategy could reduce the chattering, shorten the tracking time, and improve the tracking performance.

Introduction

As an important measuring element and inertial navigation instrument, micro-gyroscope has been widely used in modern aviation, navigation, aerospace, and national defense industry. Recently, there have been serious efforts for controlling micro-gyroscope system, thanks to its significant application in various fields such as control stabilization, navigation, and automobile, which require high precision in position tracking and velocity measurement. However, the inherent nonlinearities and uncertainties caused by fabrication imperfections, ambient conditions, and external disturbances make it difficult to control the gyroscope system. Thus, many researchers have endeavored to study advanced technologies^1–5 applied to micro-gyroscopes like adaptive control, backstepping control, sliding-mode control (SMC), and fuzzy control to enhance performance and robustness of micro-gyroscope.

Dynamic surface method is derived based on the backstepping control technology to provide an effective way for the nonlinear system problems. A first-order filter at each stage is introduced, effectively decreasing the number of the parameters, and lowering the computation complexity. It can reduce high frequency measurement noise by approximating the derivatives via low-pass filters.

Swaroop et al. ⁶ showed a dynamic surface controller to guarantee exponential regulation and bounded tracking error in the presence of Lipschitz mismatched uncertainties in strict-feedback form. Adaptive dynamic surface control (DSC) approaches have been developed for nonlinear systems in the past by many researchers.^7–9 Tong and Li ¹⁰ proposed the adaptive fuzzy control for multiple-input multiple-output (MIMO) nonlinear systems with unknown dead-zone inputs. An adaptive DSC of microelectromechanical systems (MEMS) gyroscope was proposed using fuzzy compensator by Lei et al. ¹¹ An adaptive fuzzy DSC was introduced for uncertain discrete-time nonlinear systems in pure-feedback form by Yoshimura. ¹²

Radial basis function (RBF) neural network (NN) has achieved great practical successes in industrial processes and many other fields. An RBF NN is utilized to approximate the system dynamics in the control system for a class of nonlinear systems. Li and Li ¹³ proposed an adaptive control approach based on the approximation property of NNs for output constraint continuous stirred tank reactor. A single NN was used in an adaptive NN prescribed performance controller for MIMO systems by Theodorakopoulos and Rovithakis. ¹⁴ The concept of DSC has been applied for pure-feedback systems by Wang ¹⁵ and Zhang et al. ¹⁶ with the NN. Zhang et al., ¹⁷ Wang and Huang, ¹⁸ and Gao et al. ¹⁹ proposed the NN-based DSC approaches for adaptive tracking control of strict-feedback systems. Shi et al. ²⁰ developed an adaptive neural DSC for non-strict-feedback systems. Adaptive DSC for a hypersonic aircraft using NNs was described by Shin. ²¹ A modified adaptive neural DSC for morphing aircraft with input and output constraints was investigated by Wu et al. ²² Adaptive neural controllers based on dynamic surface controller for MEMS gyroscope was derived by Lei and Fei. ²³ Adaptive NN with sliding-mode controllers have been developed for dynamic system by Fei and colleagues^24,25 and Chu and Fei. ²⁶

Motivated by the previous research, this article proposed a Lyapunov-based adaptive double neural network with a dynamic surface control (DNNDSC) for a micro-gyroscope to achieve the trajectory tracking because conventional controller cannot realize a desired dynamic behavior. In the presence of parameter uncertainties or even unknown system structure, an RBF NN is proposed to approximate the unknown system dynamics, and another RBF NN is used to eliminate the effect of approximation error of the first NN. By using the DSC technique, the whole computation process and design procedure becomes simple. Successful combination of DSC and NN control strengthens the robustness of the control system in the presence of system uncertainties and external disturbances. Therefore, not only satisfactory tracking performance and convergence of tracking errors to zero can be obtained but also the stability of the whole control system can be guaranteed under the Lyapunov framework. The main motivations compared with existing methods are organized as follows:

1. In the presence of unknown model errors and external disturbances, one RBF NN is employed to compensate for the system nonlinearities, which can eliminate the chattering effectively without losing the robustness property and the precision, and another RBF NN is used to eliminate the effect of approximation error of the first NN . External disturbances can be compensated by using a sliding-mode controller, which is approximated by NN estimator.

2. Based on the dual NN design, DSC is proposed to overcome the parameter expansion problem in the conventional systems, while ensuring the control system to reach the stable state in a finite period of time from any initial state. The proposed control scheme combined DSC and dual NN technology to reduce the chattering of inputs and improve the system performance against model uncertainties and external disturbances. The presence of the auxiliary first-order filters is the key feature of the algorithm which can remove the need for differentiations in the controller design and reduce the explosion of terms.

Model of micro-gyroscope and preparations

Model of micro-gyroscope

A z-axis micro-gyroscope system is described in Figure 1, in which a proof mass, sensing mechanisms, electrostatic actuation, and velocity of the proof mass are included to force an oscillatory motion and sense the position.

Figure 1.

Schematic diagram of micro-gyroscope.

Thus the motion equation can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{··}{x}+d_{\mathrm{xx}}\stackrel{·}{x}+d_{\mathrm{xy}}\stackrel{·}{y}+{{\omega }_x}^{2}x+{\omega }_{\mathrm{xy}}y=u_x+2{\mathrm{\Omega }}_z\stackrel{·}{y}}\\ \multicolumn{1}{c}{\stackrel{··}{y}+d_{\mathrm{xy}}\stackrel{·}{x}+d_{\mathrm{yy}}\stackrel{·}{y}+{\omega }_{\mathrm{xy}}x+{{\omega }_y}^{2}y=u_y-2{\mathrm{\Omega }}_z\stackrel{·}{x}}\end{array}$$ (1)

where x, and y are the displacements. $d_{\mathrm{xx}}$ , and $d_{\mathrm{yy}}$ are the damping coefficients. $k_{\mathrm{xx}}$ , and $k_{\mathrm{yy}}$ are the spring coefficients. $d_{\mathrm{xy}}$ , and $k_{\mathrm{xy}}$ are the coupling parameters. ${\mathrm{\Omega }}_z$ is the angular velocity. $u_x,\mathrm{and}{u}_y$ are the input control forces.

Dividing both sides of equation (1) by m, $q_0$ and ${\omega }_0^2$ yields the non-dimensional motion equation as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{··}{x}+d_{\mathrm{xx}}\stackrel{·}{x}+d_{\mathrm{xy}}\stackrel{·}{y}+{{\omega }_x}^{2}x+{\omega }_{\mathrm{xy}}y=u_x+2{\mathrm{\Omega }}_z\stackrel{·}{y}}\\ \multicolumn{1}{c}{\stackrel{··}{y}+d_{\mathrm{xy}}\stackrel{·}{x}+d_{\mathrm{yy}}\stackrel{·}{y}+{\omega }_{\mathrm{xy}}x+{{\omega }_y}^{2}y=u_y-2{\mathrm{\Omega }}_z\stackrel{·}{x}}\end{array}$$ (2)

Defining a set of new parameters as follows

$$\frac{d_{xx}}{m{\omega }_0}=D_{xx},\frac{d_{xy}}{m{\omega }_0}=D_{xy},\frac{d_{yy}}{m{\omega }_0}=D_{yy},\frac{{\mathrm{\Omega }}_z}{{\omega }_0}={\mathrm{\Omega }}_Z,\sqrt{\frac{k_{xx}}{m{{\omega }_0}^2}}={\omega }_x,\sqrt{\frac{k_{yy}}{m{{\omega }_0}^2}}={\omega }_y,\frac{k_{xy}}{m{{\omega }_0}^2}={\omega }_{xy},\frac{u_x}{m}={\stackrel{⌢}{u}}_x,\frac{u_y}{m}={\stackrel{⌢}{u}}_y$$

By equivalent transformation, the non-dimensional representation of the micro-gyroscope system can be depicted in the following expression

$$\stackrel{··}{q}+D\stackrel{·}{q}+\mathrm{Kq}=u-2\mathrm{\Omega }\stackrel{·}{q}$$ (3)

where $q=\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right]$ , $u=\left[\begin{array}{c}{\stackrel{⌢}{u}}_x\hfill \\ {\stackrel{⌢}{u}}_y\hfill \end{array}\right]$ , and $D=\left[\begin{array}{c}{D}_{\mathrm{xx}}{D}_{\mathrm{xy}}\hfill \\ \multicolumn{1}{c}{D_{\mathrm{xy}}{D}_{\mathrm{yy}}}\end{array}\right]$

$$K=\left[\begin{array}{c}{{\omega }_x}^2{\omega }_{\mathrm{xy}}\hfill \\ \multicolumn{1}{c}{{\omega }_{\mathrm{xy}}{{\omega }_y}^2}\end{array}\right],\mathrm{\Omega }=\left[\begin{array}{c}0-{\mathrm{\Omega }}_Z\hfill \\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{Z}0}\end{array}\right]$$

Considering model uncertainties and external disturbances, the model is expressed as

$$\stackrel{··}{q}+(D+\Delta D)\stackrel{·}{q}+(K+\Delta K)q=u-2\mathrm{\Omega }\stackrel{·}{q}+d$$ (4)

where $\Delta K,\Delta D$ represent the parameter variations, and d represents the external disturbances.

After defining $x_1=q,\mathrm{and}{x}_2=\stackrel{·}{q}$ , equation (4) can be written as

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=f+u+d}\end{array}$$ (5)

where f denotes the lumped model uncertainties given as

$$f=-(D+\Delta D+2\mathrm{\Omega })x_2-(K+\Delta K)x_1$$ (6)

and d is external disturbance.

RBF NN

RBF NN has many advantages, such as faster convergence speed, avoiding local minimum problem. The structure of RBF NN is a three-layer feed-forward network consisting of one input layer, one hidden layer, and one output layer as shown in Figure 2.

Figure 2.

Schematic diagram of RBF neural network.

A component of an input vector x represents each input neuron. $\phi$ is the RBF of the NN. The weight between the input layer and the hidden-layer neuron is the center $c_i$ in Gaussian function ${\phi }_i$ , which is given by

$${\phi }_i=\exp (-\frac{{\Vert x(t)-c_i\Vert }^2}{2b_i^2}),i=1,2,\dots ,n$$ (7)

where $c_i$ , and $b_i$ are the center and width of the neuron i, respectively, and $||x(t)-c_i||$ is the Euclidean distance between $x(t)$ and $c_i$ .

The output of RBF is

$$f=W^T\phi$$ (8)

where the weight $W=[w_1,w_2,w_3,\dots ,w_n{]}^T$

Define optimal parameter vector $W^{*}$ as

$$W^{*}=\arg \underset{W\in \mathrm{\Omega }}{min}[sup|\hat{f}-f|]$$ (9)

where $\mathrm{\Omega }$ is a collection of W.

Define estimation error $\tilde{W}$ as

$$\tilde{W}=\hat{W}-W^{*}$$ (10)

$${\tilde{W}}^T={\hat{W}}^T-{W^{*}}^T$$ (11)

From equation (11), equation (8) becomes

$$f=W^{*T}\phi +{\epsilon }_1$$ (12)

$$f-\hat{f}=W^{*T}\phi +{\epsilon }_1-{\hat{W}}^T\phi =-{\tilde{W}}^T\phi +{\epsilon }_1$$ (13)

where ${\epsilon }_1$ is a positive constant, denoting the approximation error in NN. For a given arbitrary small constant ${\epsilon }_1$ , the following condition $|f-W^{*T}\phi |\le {\epsilon }_1$ is satisfied.

Then, the dynamic characteristics of micro-gyroscope and external disturbances can be approximated by the first RBF NN as

$$\hat{f}={\hat{W}}^T{\varphi }_1(x_1,x_2)$$ (14)

Similarly, the sliding-mode switching term $\eta \mathrm{sgn}(s)$ is approximated by the second NN as

$$\hat{h}={\hat{\theta }}^T{\varphi }_2(s,\stackrel{·}{s})$$ (15)

where $W\mathrm{and}\theta$ are the weights of the two NNs, respectively; ${\varphi }_1\mathrm{and}{\varphi }_2$ are the RBF of the two networks, respectively; and s is the sliding surface.

The minimum approximation error is defined as

$$w=f(x_1,x_2)-\hat{f}\left(x_1,x_2|W^{*}\right)+\eta \mathrm{sgn}(s)-\hat{h}\left(s,\stackrel{·}{s}|{\theta }^{*}\right)$$ (16)

where $W^{*}$ , and ${\theta }^{*}$ are the optimal approximation constants of the two networks, respectively, and w is a positive constant which is the approximation error of NN. For a given arbitrary small constant $\eta$ , the condition is satisfied as $w\le \eta$ .

Adaptive DNNDSC

As shown in Figure 3, an adaptive DNNDSC strategy for a micro-gyroscope is proposed. Two NN structures are designed, the first is the adaptive RBF NN to approximate the lumped nonlinearities and the second NN is employed to compensate for the error caused by the former NN approximation. Both adaptive RBF NN controller and dynamic surface controller are combined to use in the whole system.

Figure 3.

Block diagram of adaptive double neural network dynamic surface controller.

Define the position error using the following equation

$$z_1=x_1-x_{1d}$$ (17)

where $x_{1d}$ is the desired reference command.

Then the derivative of equation (17) can be expressed as

$${\stackrel{·}{z}}_1={\stackrel{·}{x}}_1-{\stackrel{·}{x}}_{1d}$$ (18)

Define the first Lyapunov function

$$V_1=\frac{1}{2}{z}_1^T{z}_1$$

Then the derivative of Lyapunov function $V_1$ is

$${\stackrel{·}{V}}_1={z_1}^T{\stackrel{·}{z}}_1={z_1}^T({\stackrel{·}{x}}_1-{\stackrel{·}{x}}_{1d})={z_1}^T(x_2-{\stackrel{·}{x}}_{1d})$$ (19)

In order to obtain ${\stackrel{·}{V}}_1={z_1}^T(x_2-{\stackrel{·}{x}}_{1d})\le 0$ , we define ${\overline{x}}_2$ as the virtual term of $x_2$

$${\overline{x}}_2=-c_1{z}_1+{\stackrel{·}{x}}_{1d}$$ (20)

where $c_1$ is a non-zero positive constant.

A first-order low-pass filter was introduced to overcome the “explosion of terms” problem caused by repeated differentiations. The output of the first-order low-pass filters $1/(\tau s+1)$ about ${\overline{x}}_2$ is defined as ${\alpha }_1$ , satisfying the following condition

$$\{\begin{array}{c}\tau {\stackrel{·}{\alpha }}_1+{\alpha }_1={\overline{x}}_2\hfill \\ \multicolumn{1}{c}{{\alpha }_1\left(0\right)={\overline{x}}_2\left(0\right)}\end{array}$$ (21)

where $\tau$ is a non-zero positive constant.

Then we can get

$${\stackrel{·}{\alpha }}_1=\frac{{\overline{x}}_2-{\alpha }_1}{\tau }$$ (22)

The filtering error is defined as

$$y_2={\alpha }_1-{\overline{x}}_2$$ (23)

The error of virtual control is defined as

$$z_2=x_2-{\alpha }_1$$ (24)

Substituting equation (6) into equation (24) yields

$${\stackrel{·}{z}}_2=f+u-{\stackrel{·}{\alpha }}_1$$ (25)

Since the first RBF neural controller may cause the error, the sliding-mode term defined as in equation (26) is introduced to compensate the error

$$s=z_2$$ (26)

That is

$$\stackrel{·}{s}={\stackrel{·}{x}}_2-{\stackrel{·}{\alpha }}_1=f+u-{\stackrel{·}{\alpha }}_1$$ (27)

The second Lyapunov function candidate $V_2$ is designed as

$$V_2=\frac{1}{2}{z_2}^T{z}_2$$ (28)

In order to obtain

$${\stackrel{·}{V}}_2=z_2^T{\stackrel{·}{z}}_2=z_2^T({\stackrel{·}{x}}_2-{\stackrel{·}{\alpha }}_1)=z_2^T(f+u-{\stackrel{·}{\alpha }}_1)\le 0$$ (29)

The controller is designed as

$$u=-f+{\stackrel{·}{\alpha }}_1-c_2{z}_2-\eta \mathrm{sgn}(s)$$ (30)

where $c_2$ is a non-zero positive constant, the sliding-mode term $\eta \mathrm{sgn}(s)$ is employed to compensate for the approximation error of the first RBF network, and $\eta$ is a positive constant.

At this point, the output of the first RBF NN $\hat{f}$ is used to approximate f, which is the sum of dynamic characteristics and external disturbances. At the same time, the output of the second NN $\hat{h}$ is used to approximate $\eta \mathrm{sgn}(s)$ . Then the controller equation (30) can be updated as

$$u=-\hat{f}+{\stackrel{·}{\alpha }}_1-c_2{z}_2-\hat{h}$$ (31)

where $\hat{f}={\hat{W}}^T{\varphi }_1(x_1,x_2)$ , $\hat{h}={\hat{\theta }}^T{\varphi }_2(s,\stackrel{·}{s})$ , W, and $\theta$ are the weight vector of the two NNs, and ${\varphi }_1\mathrm{and}{\varphi }_2$ are the RBFs.

The minimum approximation error is defined as

$$w=f(x_1,x_2)-\hat{f}(x_1,x_2|W^{*})+\eta \mathrm{sgn}(s)-\hat{h}(s,\stackrel{·}{s}|{\theta }^{*})$$ (32)

where $W^{*}$ and ${\theta }^{*}$ are the optimal approximation constants.

Stability analysis

Taking into account the position-tracking error, the virtual control error and the error, as well as parameter errors of the two NNs, then define a Lyapunov function candidate as

$$V=\frac{1}{2}{z}_1^T{z}_1+\frac{1}{2}{z}_2^T{z}_2+\frac{1}{2}{y}_2^T{y}_2+\frac{1}{2r_1}{\tilde{W}}^T\tilde{W}+\frac{1}{2r_2}{\tilde{\theta }}^T\tilde{\theta }$$ (33)

where $z_1$ is the tracking error; $z_2$ is the error of virtual control; $y_2$ is the filtering error; $\tilde{W}=W^{*}-\hat{W}$ and $\tilde{\theta }={\theta }^{*}-\hat{\theta }$ are the parameters’ errors of the two NNs; and $r_1\mathrm{and}{r}_2$ are positive constants.

We consider Lyapunov function as

$$V_a=\frac{1}{2}{z}_1^T{z}_1+\frac{1}{2}{z}_2^T{z}_2+\frac{1}{2}{y}_2^T{y}_2$$ (34)

and V becomes

$$V=V_a++\frac{1}{2r_1}{\tilde{W}}^T\tilde{W}+\frac{1}{2r_2}{\tilde{\theta }}^T\tilde{\theta }$$ (35)

Remark

Assuming $V_a(0)\le p$ , $V(t)\le l$ , where p and l are non-zero positive constants, respectively.

When $V_a=p$ , we can get

$$V_a=\frac{1}{2}{z_1}^T{z}_1+\frac{1}{2}{z_2}^T{z}_2+\frac{1}{2}{y_2}^T{y}_2=p$$

Rewrite the derivative of Lyapunov function as

$$\stackrel{·}{V}={z_1}^T{\stackrel{·}{z}}_1+{z_2}^T{\stackrel{·}{z}}_2+{y_2}^T{\stackrel{·}{y}}_2+\frac{1}{r_1}\mathrm{tr}\left({\tilde{W}}^T\stackrel{·}{\stackrel{~}{W}}\right)+\frac{1}{r_1}\mathrm{tr}\left({\tilde{\theta }}^T\stackrel{·}{\stackrel{~}{\theta }}\right)$$ (36)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{z}}_1={\stackrel{·}{x}}_1-{\stackrel{·}{x}}_{1d}=x_2-{\stackrel{·}{x}}_{1d}}\\ \multicolumn{1}{c}{=z_2+{\alpha }_1-{\stackrel{·}{x}}_{1d}=z_2+y_2+{\overline{x}}_2-{\stackrel{·}{x}}_{1d}}\end{array}$$ (37)

$${\stackrel{·}{z}}_2={\stackrel{·}{x}}_2-{\stackrel{·}{\alpha }}_1=f+u-{\stackrel{·}{\alpha }}_1$$ (38)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{y}}_2={\stackrel{·}{\alpha }}_1-{\stackrel{·}{\overline{x}}}_2=\frac{{\overline{x}}_2-{\alpha }_1}{\tau }-{\stackrel{·}{\overline{x}}}_2=\frac{-y_2}{\tau }-{\stackrel{·}{\overline{x}}}_2}\\ \multicolumn{1}{c}{=\frac{-y_2}{\tau }+c_1{\stackrel{·}{z}}_1-{\stackrel{··}{x}}_{1d}}\end{array}$$ (39)

Substituting equations (37)–(39) into equation (36) yields equation (41), define $B_2=c_1{\stackrel{·}{z}}_1-{\stackrel{··}{x}}_{1d}$ . Specifically

$$\begin{array}{c}\multicolumn{1}{c}{B_2=c_1({\overline{x}}_2-{\stackrel{·}{x}}_{1d})-{\stackrel{··}{x}}_{1d}}\\ \multicolumn{1}{c}{=c_1(z_2+{\alpha }_1-{\stackrel{·}{x}}_{1d})-{\stackrel{··}{x}}_{1d}}\\ \multicolumn{1}{c}{=c_1(z_2+y_2+{\overline{x}}_2-{\stackrel{·}{x}}_{1d})-{\stackrel{··}{x}}_{1d}}\\ \multicolumn{1}{c}{=c_1(z_2+y_2-c_1{z}_1)-{\stackrel{··}{x}}_{1d}}\end{array}$$ (40)

Equation (40) illustrates $B_2$ is the function of $z_1$ , $z_2$ , $y_2$ , and ${\stackrel{··}{x}}_{1d}$ . The limit of $B_2$ is $M_2$ , we can get $({B_2}^2/{M_2}^2)-1\le 0$ .

Choose $c_1⩾1+r$ , $r>0$ , $c_2⩾(1/2)+r$ , $(1/\tau )⩾(1/2)M_2+(1/2)+r$ . Equation (36) can be rewritten as equation (41)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}={z_1}^T{\stackrel{·}{z}}_1+{y_2}^T{\stackrel{·}{y}}_2}\\ \multicolumn{1}{c}{+s^T[{\tilde{W}}^T{\varphi }_1(x_1,x_2)+{\tilde{\theta }}^T{\varphi }_2(s,\stackrel{·}{s})+w-\eta \mathrm{sgn}(s)-c_2{z}_2]}\\ \multicolumn{1}{c}{+\frac{1}{r_1}\mathrm{tr}({\tilde{W}}^T\stackrel{·}{\stackrel{~}{W}})+\frac{1}{r_2}\mathrm{tr}({\tilde{\theta }}^T\stackrel{·}{\stackrel{~}{\theta }})}\\ \multicolumn{1}{c}{\le (1-c_1){z_1}^T{z}_1+(\frac{1}{2}-c_2){z_2}^T{z}_2}\\ \multicolumn{1}{c}{+(\frac{1}{2}{B_2}^2+\frac{1}{2}-\frac{1}{\tau }){y_2}^T{y}_2}\\ \multicolumn{1}{c}{+\frac{1}{2}+\frac{1}{r_1}\mathrm{tr}[{\tilde{W}}^T(r_1{\varphi }_1(x_1,x_2)s^T+\stackrel{·}{\stackrel{~}{W}})]}\\ \multicolumn{1}{c}{+\frac{1}{r_2}\mathrm{tr}[{\tilde{\theta }}^T(r_2{\varphi }_2(s,\stackrel{·}{s})s^T+\stackrel{·}{\stackrel{~}{\theta }})]+s^T[w-\eta \mathrm{sgn}(s)]}\end{array}$$ (41)

Then, we continuously obtain the following condition

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -{\mathrm{rz}}_1^T{z}_1-{\mathrm{rz}}_2^T{z}_2+(\frac{1}{2}{B_2}^2-\frac{1}{2}{M_2}^2-r)y_2^T{y}_2+\frac{1}{2}}\\ \multicolumn{1}{c}{+\frac{1}{r_1}\mathrm{tr}[{\tilde{W}}^T(r_1{\varphi }_1(x_1,x_2)s^T+\stackrel{·}{\stackrel{~}{W}})]}\\ \multicolumn{1}{c}{+\frac{1}{r_2}\mathrm{tr}[{\tilde{\theta }}^T(r_2{\varphi }_2(s,\stackrel{·}{s})s^T+\stackrel{·}{\stackrel{~}{\theta }})]+s^T(w-\eta \mathrm{sgn}(s))}\\ \multicolumn{1}{c}{\le -2r{V}_a+\frac{1}{2}+\frac{1}{r_1}\mathrm{tr}[{\tilde{W}}^T(r_1{\varphi }_1(x_1,x_2)s^T+\stackrel{·}{\stackrel{~}{W}})]}\\ \multicolumn{1}{c}{+\frac{1}{r_2}\mathrm{tr}[{\tilde{\theta }}^T(r_2{\varphi }_2(s,\stackrel{·}{s})s^T+\stackrel{·}{\stackrel{~}{\theta }})]+s^{T}w-\Vert s^T\Vert \eta }\end{array}$$ (42)

When $r⩾(1/4p)$ , equation (42) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -2\frac{1}{4p}p+\frac{1}{2}+\frac{1}{r_1}\mathrm{tr}[{\tilde{W}}^T(r_1{\varphi }_1(x_1,x_2)s^T+\stackrel{·}{\stackrel{~}{W}})]}\\ \multicolumn{1}{c}{+\frac{1}{r_2}\mathrm{tr}[{\tilde{\theta }}^T(r_2{\varphi }_2(s,\stackrel{·}{s})s^T+\stackrel{·}{\stackrel{~}{\theta }})]+s^{T}w-\Vert s^T\Vert \eta }\\ \multicolumn{1}{c}{=\frac{1}{r_1}\mathrm{tr}[{\tilde{W}}^T(r_1{\varphi }_1(x_1,x_2)s^T+\stackrel{·}{\stackrel{~}{W}})]}\\ \multicolumn{1}{c}{+\frac{1}{r_2}\mathrm{tr}[{\tilde{\theta }}^T(r_2{\varphi }_2(s,\stackrel{·}{s})s^T+\stackrel{·}{\stackrel{~}{\theta }})]+s^{T}w-\Vert s^T\Vert \eta }\end{array}$$ (43)

Design adaptive law as the following equation

$$\stackrel{·}{\hat{W}}=-\stackrel{·}{\stackrel{~}{W}}=r_1{\varphi }_1(x_1,x_2)s^T$$ (44)

$$\stackrel{·}{\hat{\theta }}=-\stackrel{·}{\stackrel{~}{\theta }}=r_2{\varphi }_2(s)s^T$$ (45)

Substituting equations (44) and (45) into equation (43) yields $\stackrel{·}{V}\le s^{T}w-||s^T||\eta$ . According to the RBF NN approximation theory, the RBF NN system can realize the approximation error $w\le \eta$ .We can get $\stackrel{·}{V}\le 0$ . Then $z_1,z_2,y_2$ and $\tilde{\theta }\mathrm{and}\tilde{W}$ are all bounded.

From equation (42) we can get

$$\begin{array}{c}\multicolumn{1}{c}{V_a=\frac{1}{2}{z_1}^T{z}_1+\frac{1}{2}{z_2}^T{z}_2+\frac{1}{2}{y_2}^T{y}_2}\\ \multicolumn{1}{c}{\le \frac{1}{2r}(-\stackrel{·}{V}+{\left|w\right|}_{max}-\eta +\frac{1}{2})\le -\frac{1}{2r}\stackrel{·}{V}}\end{array}$$ (46)

Converted to

$$\underset{0}{\overset{t}{\int }}{V}_a(\tau )d\tau \le \frac{1}{2r}(V(0)-V(t))$$ (47)

Since both $V(0)$ and $V(t)$ are all non-increasing and bounded, then $\underset{t\to \infty }{lim}{\int }_0^t{V}_a(\tau )d\tau <\infty$ . Simultaneously, $V_a(t)$ is also uniformly continuous and bounded. We can obtain $\underset{t\to \infty }{lim}s=0,\underset{t\to \infty }{lim}{V}_a(t)=0$ according to Barbalat’s lemma, which can imply that $z_1,z_2,y_2$ and $\tilde{\theta },\tilde{W}$ will approach zero as $t\to \infty$ .

Simulation study

In this section, the proposed adaptive double NN dynamic surface controller is applied for a z-axis micro-gyroscope using MATLAB/Simulink. The parameters of the micro-gyroscope are set as follows

$$\begin{array}{c}\multicolumn{1}{c}{m=1.8\times {10}^{-7}\mathrm{kg},k_{\mathrm{xx}}=63.955\mathrm{N}/\mathrm{m}}\\ \multicolumn{1}{c}{k_{\mathrm{yy}}=95.92\mathrm{N}/\mathrm{m},k_{\mathrm{xy}}=12.779\mathrm{N}/\mathrm{m}}\\ \multicolumn{1}{c}{d_{\mathrm{xx}}=1.8\times {10}^{-6}\mathrm{Ns}/\mathrm{m},d_{\mathrm{yy}}=1.8\times {10}^{-6}\mathrm{Ns}/\mathrm{m}}\\ \multicolumn{1}{c}{d_{\mathrm{xy}}=3.6\times {10}^{-7}\mathrm{Ns}/\mathrm{m}}\end{array}$$

Since the general displacement range of the micro-gyroscope in each axis is sub-micrometer level, it is reasonable to choose $q_0=1\mathrm{\mu }\mathrm{m}$ as the reference length. Given that the usual natural frequency of each axis of a micro-gyroscope is in the KHz range, so, choose the ${\omega }_0$ as 1 KHz. The unknown angular velocity is set to be ${\mathrm{\Omega }}_z=100\mathrm{rad}/\mathrm{s}$ . Then, the non-dimensional parameters are obtained as

$$\begin{array}{c}\multicolumn{1}{c}{{{\omega }_x}^2=355.3,{{\omega }_y}^2=532.9,{\omega }_{\mathrm{xy}}=70.99,D_{\mathrm{xx}}=0.01}\\ \multicolumn{1}{c}{D_{\mathrm{yy}}=0.01,D_{\mathrm{xy}}=0.002,{\mathrm{\Omega }}_Z=0.1}\end{array}$$

The desired motion trajectories are given as

$$r_1=\sin (4.17t),r_2=1.2\sin (5.11t)$$

The initial values of conditions of the micro-gyroscope are set as

$$x_{11}(0)=0.01,x_{12}(0)=0,x_{12}(0)=0.01,x_{22}(0)=0$$

We choose the external disturbances as

$$d_1=\sin (5t);d_2=\sin (2t)$$

The related parameters are selected as

$$\begin{array}{c}\multicolumn{1}{c}{c_{11}=1500,c_{12}=1500,c_{21}=30,c_{22}=60;}\\ \multicolumn{1}{c}{r_1=1,r_2=1}\\ \multicolumn{1}{c}{{\gamma }_1=500,{\gamma }_2=500;\mathrm{to}{l}_1=0.01,\mathrm{to}{l}_2=0.01}\end{array}$$

The center of the RBF NN in each dimension is chosen from −1 to +1, with step $\Delta =0.2$ and therefore $n^{*}=11$ and the Gaussian width vector $b_j=0.015$ . The control law is as equations (44) and (45). The simulation results are shown as below.

In Figure 4, we compare the tracking curve between adaptive double NN DSC system and SMC system, where DNNDSC represents the tracking curve of adaptive DNNDSC, and SMC represents the tracking curve of general SMC. It can be seen that after about 0.02 s, the actual trajectory of micro-gyroscope using DNNDSC strategy coincides with the reference trajectory perfectly, and the tracking performance is satisfactory. While ordinary sliding-mode controller takes about 4 s to be able to track the reference trajectory, it is concluded that the micro-gyroscope with the DNNDSC strategy can track the expected trajectory quickly, and the adaptive DNNDSC method can improve the tracking speed of the micro-gyroscope obviously.

Figure 4.

Position-tracking error comparison between DNNDSC and SMC.

We can see from the position-tracking error curve as Figure 5, the trajectory can also be a good track on the reference trajectory and the orders of magnitude of error are less than ${10}^{-3}$ for both axes in the proposed DNNDSC controller which indicates that the method achieves a good tracking performance.

Figure 5.

Position-tracking errors of both axes using DNNDSC.

Figures 6 and 7 are the control input curves of adaptive DNNDSC method compared with SMC, respectively. Compared with the SMC, the control force based on the adaptive DNNDSC method is smoother, and the chattering phenomenon is not obvious.

Figure 6.

Control input of x-axis comparison between DNNDSC and SMC.

Figure 7.

Control input of y-axis comparison between DNNDSC and SMC.

Conclusion

In this article, an adaptive DNNDSC strategy for the micro-gyroscope is presented. DSC is used to reduce the introduction of parameters and eliminate the problems of parameter expansion, decreasing the computational complexity significantly. Two RBF NN controllers are utilized to approximate unknown nonlinear dynamics and compensating approximation errors. Simulation study is implemented to verify that the proposed adaptive double NN dynamic surface controller could reduce the chattering, compensate the manufacturing errors and environmental disturbances, and improve the sensitivity and robustness of the closed-loop system effectively.