# Adaptive Backstepping Fuzzy Neural Network Fractional-Order Control of Microgyroscope Using a Nonsingular Terminal Sliding Mode Controller

Authors: Juntao Fei; Xiao Liang
Journal: Complexity
 (2018)

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## Abstract

An adaptive fractional-order nonsingular terminal sliding mode controller for a microgyroscope is presented with uncertainties and external disturbances using a fuzzy neural network compensator based on a backstepping technique. First, the dynamic of the microgyroscope is transformed into an analogical cascade system to guarantee the application of a backstepping design. Then, a fractional-order nonsingular terminal sliding mode surface is designed which provides an additional degree of freedom, higher precision, and finite convergence without a singularity problem. The proposed control scheme requires no prior knowledge of the unknown dynamics of the microgyroscope system since the fuzzy neural network is utilized to approximate the upper bound of the lumped uncertainties and adaptive algorithms are derived to allow online adjustment of the unknown system parameters. The chattering phenomenon can be reduced simultaneously by the fuzzy neural network compensator. The stability and finite time convergence of the system can be established by the Lyapunov stability theorem. Finally, simulation results verify the effectiveness of the proposed controller and the comparison of root mean square error between different fractional orders and integer order is given to signify the high precision tracking performance of the proposed control scheme.


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## Body

## 1. Introduction

Recently, serious efforts have been paid on the control of the microgyroscope because of its significance in various applications like automobile, navigation, and traffic that require high precision in angular velocity measurement and trajectory tracking. However, the microgyroscope entails inherent uncertainties and nonlinearities produced by the manufacturing process, external disturbances, ambient conditions, and so on, which makes the control very complicated. In order to achieve better control performance, many advanced control methods have been implemented such as backstepping technique, adaptive control, sliding mode control, fuzzy control, and neural network control. Two adaptive controllers were proposed to tune the natural frequency of the drive axis for a vibrational microgyroscope in [1].The sliding mode control (SMC) is considered to be an efficient control scheme for both linear and nonlinear systems and certain and uncertain systems for its insensitivity to parameter uncertainties. In conventional SMC, a linear sliding surface is chosen which only guarantees asymptotic stability of the system in a sliding phase. Namely, no matter how the parameters of the sliding surface are adjusted, it is impossible for the system states to reach the equilibrium point within a finite time. To address this problem, terminal sliding mode control (TSMC) schemes with a nonlinear sliding surface have been proposed in [2] which offer faster and finite time convergence, greater control precision, and stronger robustness regarding uncertainties compared to conventional SMC. A TSMC method with observer-based rotation rate estimation was investigated for a z-axis MEMS gyroscope in [3]. However, there still exist defects in traditional terminal SMC including singularity problem and the requirement for prior knowledge of nonlinear system dynamics. To solve singularity problems, a nonsingular terminal sliding mode control (NTSMC) has been proposed in [4, 5]. In [6], a robust adaptive terminal sliding mode synchronized control was investigated for a class of nonautonomous chaotic systems. In [7], an adaptive NTSMC was presented using fuzzy wavelet networks to approximate unknown dynamics of robots with an adaptive learning algorithm. Adaptive NTSMC strategy techniques were investigated for the microgyroscope in [8, 9].Neural networks and fuzzy systems are capable of learning and approximating any smooth nonlinear function. A disturbance observer-based fuzzy sliding mode controller was designed for a PV grid-connected inverter in [10]. Liu et al. [11, 12] derived an adaptive fuzzy output feedback control and a neural approximation-based adaptive control for nonlinear nonstrict feedback discrete-time systems and nonlinear systems with full state constraint. Wu et al. [13, 14] proposed mixed fuzzy/boundary control schemes consisting of a feedback fuzzy controller and an antidisturbance robust boundary controller for nonlinear coupled ODE systems and nonlinear parabolic PDE systems. Li et al. [15] developed an adaptive fuzzy strategy with prescribed performance for block-triangular-structured nonlinear systems. An adaptive sliding mode control using a double-loop recurrent neural network structure was developed in [16]. Neural networks are employed to approximate unknown nonlinear functions in [17]. An adaptive neural network output-feedback control was proposed to tackle the unknown nonlinear functions for nonlinear time-delay systems in [18]. The fuzzy neural network (FNN) is a special architecture which integrates the advantages of fuzzy systems and neural networks. An adaptive fuzzy neural network control scheme was proposed to enhance the performance of a shunt active power filter in [19].SMC is applied not only to integer-order systems but also to fractional-order ones. Fractional calculus is an expansion of integer-order differentiation and integration to fractional-order ones which can date back to three hundred years ago [20]. Recently, more and more attention has been focused on its application in control systems rather than a pure theoretical mathematical subject due to its higher control accuracy and additional degree of freedom in comparison with integer-order controllers. A fractional-order controller was proposed for a microgrid in [21], where fractional-order PID controller parameters are tuned with a global optimization algorithm to meet system performance specifications. In recent years, fractional calculus has been merged into SMC in controller design for fractional-order systems and their integer-order counterpart which provides both merits simultaneously. Chen et al. developed an adaptive SMC for a fractional-order nonlinear system with uncertainties in [22]. An adaptive fuzzy SMC with a fractional integration scheme was presented in [23] to tune the parameter which can show better tracking performance and higher degree of robustness to disturbances compared to classical integer-order ones. A fractional-order sliding surface was designed in [24] for both integer- and fractional-order chaotic systems, which has shown an additional degree of freedom in a fractional sliding surface. Nojavanzadeh and Badamchizadeh proposed an adaptive fractional-order NTSMC for robot manipulators with uncertainties solved by adaptive tuning methods which guaranteed finite convergence and better tracking performance in [25]. An adaptive fractional-order sliding controller with a neural estimator was discussed in [26].The backstepping control is well known for its recursive and systematic design for a nonlinear dynamical system by choosing an appropriate function of the state variables as virtual control for subsystems and designing control laws based on Lyapunov functions. Usually, it is combined with other control schemes like SMC and fuzzy control. Adaptive backstepping sliding mode controllers for the dynamic system were proposed in [27]. An adaptive intelligent backstepping SMC was proposed for a finite-time control of fractional-order chaotic systems with uncertainties and external disturbances in [28].Motivated by the above discussion, an adaptive fractional-order nonsingular terminal sliding mode control using a backstepping technique via a fuzzy neural network compensator is proposed for a microgyroscope in this paper. The sliding surface is a fractional-order nonsingular terminal sliding surface, and the dynamics of the microgyroscope is described by integer order not fractional order. The main contributions of this paper are emphasized as follows:(1)
The superior characteristic of the proposed control method is that a fractional-order term is adopted in the sliding manifold which generates an extra degree of freedom, fractional-orderα, so that the performance of the closed-loop system can be improved a lot compared to the integer-order traditional sliding surface.
(2)
The nonsingular terminal sliding mode surface chosen in the controller design ensures the finite convergence without singularity. A fractional-order derivative offers an extra degree of freedom in the terminal sliding surface and makes the corresponding control laws more flexible.
(3)
The backstepping control is a systematic and recursive design method for nonlinear systems. Based on the backstepping fractional-order NTSMC scheme, adaptive algorithms are adopted to estimate the system parameters online automatically including damping and stiffness coefficients and angular velocity.
(4)
A fuzzy neural network compensator is used to approximate the upper bound of the lumped uncertainties of the system which relax the requirement of unknown system dynamics and reduce the chattering phenomenon simultaneously.The rest of this paper is organized as follows. The dynamic of the microgyroscope system is described in Section2. In Section 3, some necessary preliminary knowledge of fractional calculus and fuzzy neural network is introduced. The backstepping fractional-order nonsingular terminal sliding mode controller and the adaptive fractional-order nonsingular terminal sliding mode control using the backstepping technique via the fuzzy neural network compensator are proposed in Section 4 and Section 5, respectively. Simulation results are shown in Section 6, and the final section is the conclusions.


## 2. Dynamics of the Microgyroscope

The microgyroscope is composed of a proof mass, sensing mechanisms, and electrostatic actuation which are used to force an oscillatory motion and velocity of the proof mass and to sense the position. Referring to [8], the dynamics of the microgyroscope system can be derived under some assumptions: (1) the stiffness of the spring in the z direction is much larger than that in x and y directions, and the motion of the proof mass is constrained to the x- and y-axes as seen in Figure 1; (2) the gyroscope rotates at a constant angular velocity Ωz over a sufficiently long time interval; and (3) the centrifugal forces can be neglected. Then, the dynamics of the microgyroscope can be expressed in the following form:
(1)mx¨+dxẋ+kx−mΩz′2+Ωz′2x+mΩx′Ωy′y=ux′+2mΩz′ẏ,my¨+dyẏ+ky−mΩx′2+Ωz′2y+mΩx′Ωy′x=uy′−2mΩz′ẋ,where m denotes the mass of the proof mass, dx,yandkx,y are vectors representing damping and spring coefficients along the x- and y-axes, respectively. Ωx,y,z′ is the angular velocity along each axis, and ux,y′ is the control force in the x- and y-axes.Figure 1
Schematic diagram of az-axis microgyroscope.Taking fabrication defects into consideration, the dynamics for az-axis microgyroscope can be revised as
(2)mx¨+Dxxẋ+Dxyẏ+Kxxx+Kxyy=ux′+2mΩz′ẏ,my¨+Dxyẋ+Dyyẏ+Kxyx+Kyyy=uy′−2mΩz′ẋ,where Dxx,Dyy and Kxx,Kyy are vectors denoting damping terms and spring coefficient terms, respectively, Dxy and Kxy are coupled damping and spring terms.Dividing both sides of (2) by proof mass m, reference length q0, and natural resonance frequency ω0 simultaneously yields
(3)x¨+dxxẋ+dxyẏ+ωx2x+ωxyy=ux+2Ωzẏ,y¨+dxyẋ+dyyẏ+ωxyx+ωy2y=uy−2Ωzẋ.In (3),
(4)Dxxmω0⟶dxx,Dxymω0⟶dxy,Dyymω0⟶dyy,Kxxmω02⟶ωx2,Kxymω02⟶ωxy,Kyymω02⟶ωy2,ux′mω02⟶ux,uy′mω02⟶uy,Ωz′ω0→Ωz.Through the equivalent transformation, the nondimensional representation of the model can be established as(5)q¨+Dq̇+Kq=u−2Ωq̇,where
(6)q=xy,D=dxxdxydxydyy,K=ωx2ωxyωxyωy2,u=uxuy,Ω=0−ΩzΩz0.


## 3. Preliminaries

Some preliminaries of fractional-order calculus and fuzzy neural network which will be utilized afterward are described in this section.


### 3.1. Fractional-Order Calculus

As the generalization form of the traditional calculus, Caputo (C), Riemann-Liouville (RL), and Grunwald-Letnikov (GL) definitions are three of the most outstanding definitions utilized in engineering, science, and economics fields, especially the Caputo fractional definitions because it takes the same form as integer-order differentiation in the initial condition. So, the following section is based on Caputo derivative definition.According to [26, 28], the Caputo fractional derivative of order α of function fx is denoted as
(7)Datαft=1Γn−α∫atfnτt−τα−n+1dτ,n−1<α<ndndtnft,α=n,where t and a are the upper and lower bounds of the operator, respectively, and Γ is the Gamma function which satisfies
(8)Γγ=∫0∞e−ttγ−1dt.For the sake of convenience in description,Datα is replaced by Dα in the following parts. It is noted that if α=0, then the operation D0fx satisfies D0fx=fx.


### 3.2. Fuzzy Neural Networks

Fuzzy neural networks (FNN) proposed in this paper have the merits of fuzzy system and neural network simultaneously. Not only can the inference characteristic of the fuzzy system be brought into neural networks but also the learning and computing ability of the neural network can be applied for the adjustment of the if-then fuzzy rules. The block diagram of a four FNNs described in Figure2 which comprises the input, the membership, the rule, and the output layers is used to approximate the unknown upper bound of the lumped uncertainties. The signal propagation and the operation functionalities in each layer of the FNN system are depicted as follows:Figure 2
The configuration of the fuzzy neural network.In the following description, the input and output of a node in theith layer are described as netm and ym, respectively.
(1)
Input layer: All nodes in this layer are input nodes which transmit input values to the next layer directly, and the net input and output are described as
(9)neti1=Xi1,yi1=fi1neti1=neti1,i=1,2,where X11=Q,X21=Q̇.
(2)
Membership function layer: Each node in this layer performs a membership function represented by the Gaussian function in this paper. For thejth node,
(10)netj2=−Xi2−cij2bij2,yj2=fj2netj2=expnetj2,j=1,2,…,n,where cij and bj denote the center mean and variance of the Gaussian function, respectively, and n is the total number of the linguistic variables related to the input nodes.
(3)
Fuzzy rule layer: Each node in this layer is denoted by layer 2. The fuzzy rules are implemented by the product operator. For thekth node,
(11)netk3=∏jwjk3xj3,yk3=fk3netk3=netk3,k=1,2,…,l,where l=n/ii, and wjk3 represents the weights between layer 2 and this layer.
(4)
Output layer: The single node in this layer labeled as∑ computes the overall output as the summation of all input signals
(12)neto4=∑kwko4xk4,yo4=fo4neto4=neto4,where wko4 denotes the connecting weight between the oth and the kth rule. xk4 is the kth input and yo4 is the output of the FNN.For ease of notation, the outputs of FNN are represented as the following vector form,(13)yi=wiTϕi,where wi=wi1,wi2,…,wilT is the weight vector and ϕi=ϕi1,ϕi2,…,ϕilT is the normalized reliability of the FNN.In the following discussion, it is assumed that both the centers and widths of the FNN system have been selected and fixed adequately, and the weight values will be adjusted on line by an adaptive law.


## 3.1. Fractional-Order Calculus

As the generalization form of the traditional calculus, Caputo (C), Riemann-Liouville (RL), and Grunwald-Letnikov (GL) definitions are three of the most outstanding definitions utilized in engineering, science, and economics fields, especially the Caputo fractional definitions because it takes the same form as integer-order differentiation in the initial condition. So, the following section is based on Caputo derivative definition.According to [26, 28], the Caputo fractional derivative of order α of function fx is denoted as
(7)Datαft=1Γn−α∫atfnτt−τα−n+1dτ,n−1<α<ndndtnft,α=n,where t and a are the upper and lower bounds of the operator, respectively, and Γ is the Gamma function which satisfies
(8)Γγ=∫0∞e−ttγ−1dt.For the sake of convenience in description,Datα is replaced by Dα in the following parts. It is noted that if α=0, then the operation D0fx satisfies D0fx=fx.


## 3.2. Fuzzy Neural Networks

Fuzzy neural networks (FNN) proposed in this paper have the merits of fuzzy system and neural network simultaneously. Not only can the inference characteristic of the fuzzy system be brought into neural networks but also the learning and computing ability of the neural network can be applied for the adjustment of the if-then fuzzy rules. The block diagram of a four FNNs described in Figure2 which comprises the input, the membership, the rule, and the output layers is used to approximate the unknown upper bound of the lumped uncertainties. The signal propagation and the operation functionalities in each layer of the FNN system are depicted as follows:Figure 2
The configuration of the fuzzy neural network.In the following description, the input and output of a node in theith layer are described as netm and ym, respectively.
(1)
Input layer: All nodes in this layer are input nodes which transmit input values to the next layer directly, and the net input and output are described as
(9)neti1=Xi1,yi1=fi1neti1=neti1,i=1,2,where X11=Q,X21=Q̇.
(2)
Membership function layer: Each node in this layer performs a membership function represented by the Gaussian function in this paper. For thejth node,
(10)netj2=−Xi2−cij2bij2,yj2=fj2netj2=expnetj2,j=1,2,…,n,where cij and bj denote the center mean and variance of the Gaussian function, respectively, and n is the total number of the linguistic variables related to the input nodes.
(3)
Fuzzy rule layer: Each node in this layer is denoted by layer 2. The fuzzy rules are implemented by the product operator. For thekth node,
(11)netk3=∏jwjk3xj3,yk3=fk3netk3=netk3,k=1,2,…,l,where l=n/ii, and wjk3 represents the weights between layer 2 and this layer.
(4)
Output layer: The single node in this layer labeled as∑ computes the overall output as the summation of all input signals
(12)neto4=∑kwko4xk4,yo4=fo4neto4=neto4,where wko4 denotes the connecting weight between the oth and the kth rule. xk4 is the kth input and yo4 is the output of the FNN.For ease of notation, the outputs of FNN are represented as the following vector form,(13)yi=wiTϕi,where wi=wi1,wi2,…,wilT is the weight vector and ϕi=ϕi1,ϕi2,…,ϕilT is the normalized reliability of the FNN.In the following discussion, it is assumed that both the centers and widths of the FNN system have been selected and fixed adequately, and the weight values will be adjusted on line by an adaptive law.


## 4. Backstepping Fractional-Order Nonsingular Terminal Sliding Mode Control

The fractional-order nonsingular terminal sliding mode surface is proposed in this section since it combined the advantages of fractional derivatives and the nonsingular terminal sliding mode control which not only improve the control precision and performance but also guarantee the finite time convergence. The backstepping control is usually applied to a class of special nonlinear dynamical systems which can be built from subsystems by choosing appropriate Lyapunov functions. Thanks to the recursive procedure, good tracking performance and global stability are guaranteed.Consider the dynamic model of microgyroscope with uncertainties and external disturbances as follows:(14)q¨+D+2Ω+ΔDq̇+K+ΔKq=u+d,where q is the position of the microgyroscope system, ΔD and ΔK stand for the parameter uncertainties of the system, and d denotes the disturbance vector.The dynamics of the system can be rewritten as(15)q¨+D+2Ωq̇+Kq=u+f,where
(16)f=f1f2=d−ΔDq̇−ΔKq,denotes the lumped uncertainties of the microgyroscope system.Remark 1.
The lumped uncertainty and disturbancef is input-related, bounded by a positive function of the position and velocity measurements in the form of f<b0+b1q+b2q̇2, where b0, b1, and b2 are positive constants. For the convenience of analysis, it is assumed that f is bounded by a positive constant, that is, fi<ρi, i=1,2, ρ=ρ1ρ2 represent the upper bounds of the lumped uncertainties.By introducing the transformation(17)x1=q,x2=q̇,the dynamics in (2) can be converted as follows:
(18)ẋ1=x2,ẋ2=−D+2Ωx2−Kx1+u+f.The control objective in this paper is to design an appropriate controller for the positionq of the system to track the desired trajectory qr in finite time. The proposed backstepping fractional-order nonsingular terminal sliding mode control (BFONTSMC) strategy is designed to solve the problem of position-tracking and is depicted step by step as follows:Step 1.
For the objective of position-tracking, the tracking error can be defined as(19)e1=x1−qr.
Then the time derivate ofe1 is
(20)ė1=ẋ1−q̇r=x2−q̇r.
Define the virtual control as(21)α1=−c1e1+q̇r,
whereα1 is the estimator of x2; c1 is a positive constant.
The estimator error betweenα1 and x2 is defined as
(22)e2=x2−α1.
Define the first Lyapunov function as(23)V1=12e1Te1.
Differentiating it with respect to time yields(24)V̇1=e1Tė1=e1Tx2−qr=e1Te2−c1e1=e1Te2−c1e1Te1.
Ife2=0, then
(25)V̇1=−c1e1Te1≤0.Step 2.
The time derivative ofe2 is calculated as
(26)ė2=ẋ2−α̇1=−D+2Ωx2−Kx1+u+f−α̇1.
In order to achieve a precise and better control performance for the microgyroscope, a fractional-order nonsingular terminal sliding surface is defined as(27)s=e1+λ1e2p2/p1+λ2Dα−1e1,where λ1andλ2 are positive constants, and p1andp2 are positive odd integers which satisfy the condition 1<p2/p1<2. The third term is the fractional-order term with 0<α<1.
By deriving both sides ofs, we have
(28)ṡ=ė1+λ1p2p1diage2p2/p1−1ė2+λ2Dαe1=ė1+λ1p2p1diage2p2/p1−1−D+2Ωx2−Kx1+u+f−α̇1+λ2Dαe1.
Let(29)λ1p2p1diage2p2/p1−1=R1=r100r2,then
(30)p1λ1p2diage21−p2/p1=R2=1r1001r2,where
(31)R1∗R2=I=1001,r1=λ1p2/p1e21p2/p1−1,r2=λ1p2/p1e22p2/p1−1, and e21 and e22 are the components of e2 along the x- and y-axes.
Define the second Lyapunov function as(32)V2=V1+12sTs.
Taking the derivative ofV2 and using (28), we have
(33)V̇2=V̇1+sTṡ=e1Te2−c1e1Te1+sTė1+R1−D+2Ωx2−Kx1+u+f−α̇1+λ2Dαe1,
In order to guarantee thatV̇2≤0, the proposed backstepping fractional-order nonsingular terminal sliding mode controller is designed as
(34)u1=D+2Ωe2+α1+Ke1+qr−ρ1sgns1ρ2sgns2+α̇1−R2ė1−R2λ2Dαe1−R2ss2e1Te2.
Substituting (34) into (33) yields
(35)V̇2=−c1e1Te1+∑i=12sirifi−ρisgnsi≤−c1e1Te1+∑i=12risifi−ρirisi=−c1e1Te1+∑i=12risifi−ρi≤−c1e1Te1≤0.
The derivative ofV2 keeps negative SEMI definite which can be concluded that the stability of the BFONTSMC strategy is guaranteed and the finite-time reachability is achieved, which means the system trajectory will converge to the fractional-order nonsingular terminal sliding (FONTS) surface within a finite time.


## 5. Adaptive Fractional-Order Nonsingular Terminal Sliding Mode Control Using the Backstepping Technique via a Fuzzy Neural Network Compensator

In the previous procedure, the control law (34) is derived under the condition that parameter variations D,K,Ω and uncertain upper boundary ρ are known in advance. However, the fact is that it is difficult to measure these variations in practice. So, an adaptive algorithm and a fuzzy neural network compensator are integrated to estimate the system parameters with D^,K^,Ω^ and approximate the unknown upper bound of lumped uncertainties ρi, respectively, which attenuates the chattering phenomenon caused by a sign function in reaching the phase of the nonsingular terminal sliding surface simultaneously, and then the adaptive backstepping fractional-order nonsingular terminal sliding mode controller via a fuzzy neural network (ABFONTSMC-FNN) compensator is proposed. Suppose that position q and its derivate q̇ are the inputs of the fuzzy neural network. The block diagram of the proposed ABFONTSMC-FNN is depicted in Figure 3.Figure 3
Block diagram of the ABFONTSMC-FNN controller.In the presence of unknown system parameters and the lumped uncertainties, the control law in (34) can be modified as
(36)u'=D^+2Ω^e2+α1+K^e1+qr+α̇1−h^1sgns1h^2sgns2−R2ė1−R2λ2Dαe1−R2ss2e1Te2,where D^,K^,andΩ^ are the estimators of system parameters D,K,andΩ, and h^i is the fuzzy neural network compensator which satisfies
(37)h^i=wiTϕi.Assuming that the optimal output of FNN is in the form(38)h^i∗=wi∗Tϕi=ρi,where wi∗ denotes the optimal weight vector.Define the parameter estimation error as(39)D~=D^−D,K~=K^−K,Ω~=Ω^−Ω.Consider the third Lyapunov function candidate as(40)V3=12e1Te1+12sTs+12trD~M−1D~T+12trK~N−1K~T+12trΩ~P−1Ω~T+12η∑i=12w~iTwi,where M=MT,N=NT,andP=PT are positive definite matrices, w~i=wi−wi∗s, η is a positive constant, and tr⋅ denotes the matrix trace operator.Using (36) and taking the time derivation on both sides of V3 yield
(41)V̇3=e1Tė1+sTṡ+trD~M−1D~̇T+trΩ~P−1Ω~̇T+trK~N−1K~̇T+1η∑i=12w~iTw~̇i=−c1e1Te1+e1Te2+sTė1+λ2Dαe1+sTR1D~+2Ω~e2+α1+K~e1+qr−h^1sgns1h^2sgns2+f−R2ė1−R2λ2Dαe1−R2ss2e1Te2+trD~M−1D~̇T+trΩ~P−1Ω~̇T+trK~N−1K~̇T+1η∑i=12w~iTw~̇i=−c1e1Te1+sTR1D~+2Ω~e2+α1+K~e1+qr+∑i=12sirifi−h^isgnsi+trD~M−1D~̇T+trΩ~P−1Ω~̇T+trK~N−1K~̇T+1η∑i=12w~iTw~̇i=−c1e1Te1+sTR1D~e2+α1+trD~M−1D~̇T+sTR1K~e1+qr+trK~N−1K~̇T+2sTR1Ω~e2+α1+trΩ~P−1Ω~̇T+∑i=12sirifi−wiTϕisgnsi+1η∑i=12wi−wi∗Tw~̇iSinceD=DT,K=KT,Ω=−ΩT and sTR1D~e2+α1=e2+α1TD~R1s are scalar, one can obtain
(42)sTR1D~e2+α1=12sTR1D~e2+α1+e2+α1TD~R1s=tr12D~e2+α1sTR1+D~R1se2+α1T.Simultaneously, we have(43)sTR1K~e1+qr=12sTR1K~e1+qr+e1+qrTK~R1s=tr12K~e1+qrsTR1+K~R1se1+qrT2sTR1Ω~e2+α1=sTR1Ω~e2+α1−e2+α1TΩ~R1s=trΩ~e2+α1sTR1−Ω~R1se2+α1T.Substituting (42) and (43) into (41) results
(44)V̇3=−c1e1Te1+trD~M−1D~̇T+12e2+α1sTR1+R1se2+α1T+trK~N−1K~̇T+12e1+qrsTR1+R1se1+qrT+trΩ~P−1Ω~̇T+e2+α1sTR1−R1se2+α1T+∑i=12sirifi−wiTϕisgnsi+1η∑i=12wi−wi∗Tw~̇i.In order to ensureV̇3≤0, the online adapting laws for parameters and the weight adaption law are derived, respectively, as follows:
(45)D^̇T=D~̇T=−12Me2+α1sTR1+R1se2+α1T,K^̇T=K~̇T=−12Ne1+qrsTR1+R1se1+qrT,Ω^̇T=Ω~̇T=−Pe2+α1sTR1−R1se2+α1T,ẇi=w~̇i=ηrisiϕi.Substituting (45) into (44) yields
(46)V̇3=−c1e1Te1+∑i=12sirifi−wiTϕisgnsi+∑i=12wi−wi∗Tϕirisi=−c1e1Te1+∑i=12sirifi−wi∗Tϕirisi≤−c1e1Te1+∑i=12sirifi−ρi≤−c1e1Te1≤0.V̇3 is proved to be negative semidefinite which implies that V3,s,D~,K~,andΩ~ are all bounded and converge to zero. Integrating V̇3 with respect to time, we have ∫0tc1e1Te1+∑i=12siriρi−fi≤V0−Vtdt. Since V0 is bounded and Vt is bounded and nonincreasing, then ∫0tc1e1Te1dt and ∫0t∑i=12siriρi−fidt are all bounded. According to Barbalat’s lemma, limt→∞e1t=0 and limt→∞st=0, which means the FONTSM surface converges to zero in finite time and D,K,andΩ converge to their true values if the persistent excitation condition is satisfied.


## 6. Simulation Study

In this section, the proposed ABFONTSMC-FNN is applied for trajectory tracking of az-axis microgyroscope by Matlab/Simulink. Referring to [8], the parameters of the microgyroscope are chosen as follows:
(47)m=1.8×10−7kg,dxx=1.8×10−6Ns/m,dyy=1.8×10−6Ns/m,dxy=3.6×10−7Ns/m,kxx=63.955N/m,kyy=95.92N/m,kxy=12.779N/m.Assume that the unknown angular velocity isΩz=100rad/s. Then, the nondimensional gyroscope parameter matrices can be derived as follows:
(48)D=0.010.0020.0020.01,K=355.370.9970.99532.9,Ω=0−0.10.10.We will track the reference trajectoryqr1=sin4.17t and qr2=1.2sin5.11t with the initial states of the system q10=0.5,q̇10=0,q20=0.5,q̇20=0. Choose the initial conditions of D,K,Ω as D0=0.95∗D,K0=0.95∗K,Ω0=0.95∗Ω. Select the sliding surface parameters p1=3,p2=5,λ1=λ2=1, the control parameter c1=10, and the adaptive gain M=N=P=diag150,150.The architecture of FNN is specified as 2, 6, 9, and 1 neurons at the input, membership, rule, and output layer, respectively. It is assumed thatqiq̇iT is the input of the fuzzy neural network. The initial values of the Gaussian width vector and the center vector are set as
(49)C=cij=−5−2.502.55−5−2.502.55and B=bij=33333T. The initial values of the weights are selected as random numbers between −1 and 1. The learning rate in the adaptive law of wi in (45) is designed as η=0.01. When the fractional order is set as α=0.9 and the lumped uncertainty is applied as random signal f=0.5∗randn1,1;0.5∗randn1,1, the corresponding simulation results are shown in Figures 4–10.Figure 4
Tracking trajectory using ABFONTSMC-FNN.Figure 5
Tracking error using ABFONTSMC-FNN.Figure 6
Tracking error using the adaptive sliding mode control.Figure 7
Fractional-order nonsingular terminal sliding surface.Figure 8
Control input signals.Figure 9
Adaption of damping coefficients of the microgyroscope.Figure 10
Adaption of spring constants of the microgyroscope.The reference trajectories are well tracked as seen from Figure4 which demonstrates that the proposed ABFONTSMC-FNN strategy is satisfactory as expected. Figure 5 describes the tracking error of the microgyroscope system along the x- and y-axes. It is obvious that the tracking errors under the ABFONTSMC-FNN scheme are bounded and converge to zero in finite time compared to the tracking errors under the conventional adaptive sliding mode control scheme as shown in Figure 6. Tracking errors under the conventional adaptive sliding controller converges with small fluctuations near zero.Figure7 plots the convergence of the fractional-order nonsingular terminal sliding surface. It is intuitive that the designed sliding surface converges to zero within finite time which insures that the trajectory of the system attains to the sliding surface. Figure 8 illustrates the smooth control input signals of the MEMS gyroscope along the x- and y-directions without any chattering because of the approximation for the gain of switching function using the fuzzy neural network compensator. Figures 9 and 10 draw the estimation of the system parameter matrices D and K, respectively, which verified that the estimations converge to their true values with persistent sinusoidal signals. The adaption of angular velocity is described in Figure 11 whose estimate value also converges to its true value.Figure 11
Adaption of angular velocity.In order to observe the tracking performance under different fractional orders and integer orders visually, a universal standard is adopted here to quantify the tracking error by calculating the root mean square error (RMSE) which reflects how much the measured value deviates from its true value. The smaller the RMSE is, the higher the measurement accuracy is. Thus, RMSE can be a criterion to evaluate the tracking performance of the proposed control strategy under different fractional orders. The RMSE along thex- and y-axes under different orders is shown in Table 1.Table 1
RMS errors of thex- and y-axes under different fractional orders.
RMSE
X
Y
α
0.1
0.2082
0.1253
0.3
0.1952
0.1188
0.5
0.2470
0.1389
0.7
0.1058
0.0644
0.9
0.1050
0.0644For fairness, the fractional-orderα is added in the form of a ladder-type increase and the integer order is α=1. It is easy to see that the value of the fractional-order α will make a difference on system tracking errors. So, a better tracking performance can be achieved by adjusting the value α. When the fractional-order α is selected as 0.9, the RMSE seems to be minimal; this is the reason we choose α=0.9 in the previous control design. In the case of the integer order, the RMSE values along the x- and y-axes are 0.2243 and 0.1590, respectively, which are larger than the case of α=0.9. This effectively verified that the adaptive backstepping fractional-order nonsingular terminal sliding mode controller via a fuzzy neural network control scheme is superior to the conventional integer-order ones.


## 7. Conclusions

In this paper, a novel fractional-order intelligent control scheme combining adaptive control, nonsingular terminal sliding mode control, fuzzy neural network, and backstepping control is proposed for the microgyroscope system in the presence of uncertainties and external disturbances. The finite time convergence, nonsingularity, and extra freedom of degree are achieved by the fractional-order nonsingular terminal sliding mode controller. The fuzzy neural network is employed to approximate the upper bound of the lumped uncertainties which also reduces the chattering phenomenon. In addition, adaptive estimators are utilized to identify the angular velocity and other unknown system parameters. Finally, the effectiveness of the proposed controller and the best value of the fractional-orderα can be confirmed by simulation results. In this paper, the dynamics of the system is represented by integer-order calculus; we hope to study the fractional-order system and apply an intelligent method to it in the future.



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