Recursive Integral Terminal Sliding Mode Control of MEMS Gyroscopes via Composite Neural Learning and Disturbance Observer

The recursive integral terminal sliding mode control based on disturbance observer and composite neural learning is studied in this paper to control the dynamics of MEMS gyroscopes in the presence of system uncertainties and external disturbances. To achieve more accurate approximation accuracy of system uncertainties, the composite neural learning is employed, where the updating law of the weight vector is designed by the tracking error and the prediction error constructed by serial-parallel estimation model. To further improve the system robustness, the recursive integral terminal sliding mode controller is utilized, where faster convergence can be obtained by forcing the system state to start from the sliding mode manifold at the initial time. To deal with external disturbances with unknown upper bounds, the disturbance observer is designed. Furthermore, simulation-s results are demonstrated to verify that faster convergence and higher tracking accuracy can be achieved under the proposed method.


Introduction
The gyroscopes using micro electro mechanical systems (MEM-S) technology are widely utilized to measure the angular velocity in various fields such as electronics, robots, aeronautics and astronautics. There are two control frames for MEMS gyroscopes: the conventional "one axis drive and one axis detection" control frame using weak signal detection technology to detect the angular velocity, and the "biaxial drive" control frame [1] using an adaptive law to estimate the angular velocity online. Since weak signal detection technology is difficult to be further improved, the detection limit of angular velocity under the first control frame can not be broken. In response to this problem, advanced control methods under the "biaxial drive" control frame have received more attention.
Under the "biaxial drive" control frame, the proportionalintegral (PI) control is widely used in practical applications. However, due to the inevitable existence of system uncertainties and external disturbances in the dynamics of MEMS gyroscopes, the tracking accuracy of PI control is reduced. For the system uncertainties, the adaptive method is introduced in [2] to estimate the uncertainties, which is further fed to the controller for compensation. Owing to the ability of radial basis function (RBF) neural networks (NN) for the approximation of every nonlinear function with any accuracy [3,4], the RBF NN is employed to approximate system uncertainties in [5,6]. To further improve the approximation performance of NN, finite time learning [7] and multiloop recurrent NN [8] are proposed. For the external disturbances, when the upper bounds of the disturbances are known, a robust controller is designed in [9,10] to suppress the influence of disturbances. When the upper bounds of the disturbances exist but are unknown, the disturbance observer (DOB) based on Lyapunov method is designed in [11][12][13]. Furthermore, the active disturbance rejection control is pro-posed in [14,15], where the lumped disturbances including system uncertainties and external disturbances are set as a new state and further introduced to the extended state observers, such that the lumped disturbances are accurately estimated. To improve the tracking accuracy of control system in the applications, more advanced intelligent control methods and disturbance observation control theories need to be studied.
Considering the second order dynamic model of MEMS gyroscopes, the back-stepping control scheme is applied to control MEMS gyroscopes in [16], where the second order dynamics can be decomposed into two first order subsystems and the virtual controllers are proposed step by step. To avoid the parameters expansion, the dynamic surface controllers are further studied in [17,18]. However, the layered design of back-stepping control is too complex. Since sliding mode variable is able to contain the errors of different orders, it is introduced to the control system of MEMS gyroscopes in [19,20]. Following with that, various modified sliding mode control (SMC) are studied. The terminal SMC (TSMC) [21,7] and the super-twisting SMC [22] are addressed to force the sliding mode variable to achieve finite time convergence. In [23,24], the nonsingular TSMC is discussed to avoid the singular problem of conventional TSMC. In addition, various intelligent control methods and disturbance observation control theories are combined with advanced modified SMC to improve the tracking performance of MEMS gyroscopes with system uncertainties and external disturbances. Overall, the improvement of robustness and convergence is the key issue of SMC for MEM-S gyroscopes, which will continue to receive attention in the future research.
Motivated by the above studies on the controller design of MEMS gyroscopes, this paper places emphasis on how to efficiently approximate the compound nonlinearity and how to improve the convergence rate of control system. A high precision and robust control scheme is designed to control the MEMS gyroscopes. The recursive integral terminal sliding mode controller (RTSMC) is utilized to achieve faster convergence and higher tracking accuracy, while the composite neural learning and DOB are designed to handle the system uncertainties and external disturbances. Compared with previous works, the main contributions are listed as follows.
-The RTSMC is designed to control the dynamics of MEM-S gyroscopes in the presence of system uncertainties and external disturbances. When the control system is forced to start from the sliding surface at the initial time, faster convergence and higher tracking accuracy can be obtained. -The composite neural learning works together with the DOB to approximate the lumped uncertainty including system uncertainties and external disturbances. Through updating the NN weight vector by the adaptive law, which is designed by the tracking error and the prediction error constructed by the serial-parallel estimation model (SPEM), higher approximation accuracy is obtained.
The rest of this paper is organized as follows. The dynamics of MEMS gyroscopes and problem formulation are addressed in Section 2. The RTSMC using composite neural learning and DOB is designed in Section 3. The system stability analysis is demonstrated in Section 4. The simulation results are shown in Section 5 to verify the expected faster convergence and higher tracking accuracy. The conclusions are given in Section 6.

MEMS Gyroscopes Dynamics
MEMS gyroscopes are essentially mechanical gyroscopes, whose dynamics can be described as a mass-stiffness-damping system shown in Fig.1, where x denotes the drive axis, y denotes the sensitive axis, ϑ x and ϑ y denote dimensionless positions, ς xx and ς yy denote dimensionless damping coefficients, ψ xx and ψ yy denote dimensionless stiffness coefficients, ψ x3 and ψ y3 denote dimensionless nonlinear coefficients.  Considering the parameters uncertainties and the external disturbances, the dimensionless dynamics of MEMS gyroscopes with the mechanical coupling caused by machining errors is given as follows [25] ϑ with where ς xy and ς yx denote dimensionless damping coupling coefficients, ψ xy and ψ yx denote dimensionless stiffness coupling coefficients, d x and d y denote dimensionless external disturbances, τ x and τ y denote dimensionless control inputs, Ω denotes the dimensionless angular velocity.
Defining the dimensionless reference signal as ϑ ri , the tracking error is given as Assumption 1 To avoid the mechanical damage, the external disturbance is assumed to be bounded. In other words, there exists constants d mi > 0 andd mi > 0, such that |d i | ≤ d mi , |ḋ i | ≤d mi .

Composite Neural Learning
Since the parameters of dynamics are changing with environmental factors such as temperature, atmospheric pressure, etc., the nonlinear dynamics f i is constantly changing. Considering the RBF NN can approximate every nonlinear function with any accuracy [26], it is employed to approximate f i as is the input vector, ϖ i ∈ ℜ n×1 is the optimal weight vector, θ (·) ∈ ℜ n×1 is basis function vector with η ∈ ℜ 4×1 as the center and κ as the width, ε i is the approximation error with |ε i | ≤ ε m and ε m > 0.
From the dynamics (1), the SPEM is constructed aŝ whereπ i is the estimation of ϖ i , c i > 0 is an user-defined constant, z i is the prediction error implying the approximation error which is constructed as whereθ i is the estimation ofθ i . The updating law ofπ i is designed aṡ where ς 1i > 0, ς 2i > 0 and ι i > 0 are user-defined constants.

Remark 2
The conventional updating law of NN weight is designed asπ i = ς 1i e i θ − ι iπ i . It is observed that only the tracking error e i is utilized to updateπ i , where the approximation error ε i is ignored. Therefore, the approximation performance of NN cannot be evaluated. In response to this problem, the SPEM-based prediction error is constructed, where ε i is implied in z i . Hence, the approximation accuracy can be improved by designing the updating law (7) using e i and z i .

Control Goal
The goal is to design a high precision and robust control scheme such that the system state can be forced to follow the dimensionless reference trajectory ϑ ri when system uncertainties caused by environmental factors and external disturbances exist. Under the proposed controller, faster convergence and higher tracking accuracy of system tracking are expected.

Controller Design
For the dimensionless dynamics of MEMS gyroscopes (1), the RTSMC is designed to obtain higher tracking accuracy, while faster convergence is achieved by forcing the control system to start from the sliding surface at the initial time.
Considering the nonlinear dynamics including system uncertainties, RBF NN is adopted in this work. Different from the previous works where the updating law is constructed by the tracking error [5,6], the updating law of composite neural learning is designed by the tracking error and the prediction error constructed by SPEM. To deal with the external disturbances, DOB is further developed. The diagram of the control scheme is presented in Fig. 2.
The recursive integral terminal sliding mode variable [27][28][29] is constructed as where α i > 1, 0 < β i < 1, ρ i > 0, ℓ i > 0 and λ i > 0 are userdefined constants. The derivative of s i is calculated aṡ The controller is designed as Composite neural learning Serial-parallel estimation modelî where µ 1i > 0, µ 2i > 0 and 0 < υ i < 1 are user-defined constants, andD i is the estimation of the compound disturbance From the controller (11),ṡ i is calculated aṡ The prediction error z i is constructed as whereθ i is the estimation ofθ i , andθ i is given by the following SPEM: The derivative of z i is calculated aṡ The updating law is proposed aṡ The DOB is designed aŝ where Γ i > 0 is an user-defined constant, and Φ i is the intermediate variable.
The derivative ofD i is calculated aṡ Remark 3 In the controller (11), τ ai is the adaptive control using composite neural learning to approximate the system nonlinearity, τ di is the DOB designed to deal with the external disturbances, τ ni is the nominal feedback controller, τ si is the RTSMC signal to achieve finite time convergence.
Remark 4 Since the approximation error ε i and its derivativeε i are bounded, the boundedness of the compound disturbance D i and its derivativeḊ i can be guaranteed under Assumption1.
Remark 5 It is observed from the sliding mode manifold (8) that s i (0) = 0 when χ I i (0) = −χ i (0)/ρ i , where s i (0), χ I i (0) and χ i (0) are the initial values of s i , χ I i and χ i , respectively. That implies faster convergence will be achieved when the initial value of χ I i is chosen as χ I i (0) = −[ė i (0) + ℓ i e i (0) + λ i |e i (0)| α i sign (e i (0))]/ρ i , because the control system is forced to start from the sliding surface at the initial time.
Theorem 1 Considering the dimensionless dynamics of MEM-S gyroscopes described by (1), if the controller (11), the composite learning (17) and the DOB (18) are designed, the signals in (21) are bounded, and the tracking error e i will converge to the region ℵ i in the finite time T i . Proof: The Lyapunov function candidate is selected as where The derivative of V s i is calculated aṡ The derivative of V z i is calculated aṡ The derivative of Vπ i is calculated aṡ The derivative of VD i is calculated aṡ Then we obtaiṅ Since the following facts exist where ∥θ ∥ ≤ c θ , and c m > 0 is a constant. V i is rewritten aṡ where Hence, z i ,π i andD i are bounded with If s i ̸ = 0, it is obtained thaṫ From Lemma 1, it is known that |s i | will converge to the region Λ i = min{Λ 1i ,Λ 2i } in finite time T si , with When s i = 0, there exists χ i = −ρ i χ I i , such that χ i is with the same convergence time as When ρ i χ I i ≥ |s i | + C si , where C si > 0 is an unknown constant,V χ I i can be rewritten aṡ Referring to [31,12], it is concluded that χ I i will converge to the region Λ χ I i in finite time T χ i when |s i | is in the region Λ i , where Hence, |χ i | will converge to the region Λ χ i = Λ i + C si in finite time T χ i . Similar to [31,12], it is easy to verify that the above finite time convergence still can be guaranteed when If e i ̸ = 0, it is obtained thaṫ with From Lemma 1, it is known that |e i | will converge to the region ℵ i = min{ℵ 1i , ℵ 2i } in finite time T ei when |χ i | is in the region Λ χ i , with Thus, e i converges from any initial condition to the region ℵ i in the finite time T i = T si + T χ i + T ei .

Simulation
Note that the proposed controller (11) is denoted as "CL-RTSMC". The highlights of "CL-RTSMC" are the RTSMC (8) and the composite neural learning (17). To verify faster convergence and higher tracking accuracy of "CL-RTSMC", it is compared with "CL-TSMC" (the controller not using the RTSMC) and "NN-TSMC" (the controller not using the RTSMC and the composite neural learning) in simulations, while "CL-TSMC" and "NN-TSMC" are presented as follows: (a) "CL-TSMC": The sliding mode manifold is designed as The controller is designed as where the DOB is designed as (18) and the updating law of NN weight is designed aṡ with z i =θ i −θ i constructed by the dynamics (1) and the SPEM (15).
Through comparing the tracking performance index of "CL-TSMC" and "NN-TSMC', it is observed that, under  "CL-TSMC', the average of steady-state errors in drive axis and sensitive axis are shortened by 66.76% and 36.11% respectively, while the RMSE of steady-state errors are shortened by 66.36% and 69.46% respectively. Higher tracking accuracy is obtained under "CL-TSMC" because the prediction error z i constructed from the SPEM is introduced to the updating law (17), such that the approximation error ε i of NN is indirectly considered. The estimation of the lumped uncertainty F i = f i + d i shown in Fig.6∼Fig.7 further confirms the above conclusion, where smaller estimation error F i =π T i θ +D i is obtained under "CL-TSMC"'. Through comparing the tracking performance index of "CL-RTSMC" and "CL-TSMC', it is observed that, under "CL-RTSMC', the transient time is shortened by 16%, the average of steady-state errors in drive axis and sensitive axis are shortened by 99.30% and 97.27% respectively, and the RMSE of steady-state errors are shortened by 99.63% and 99.60% respectively. Higher tracking accuracy and faster convergence are achieved under RTSMC.
The norm of the RBF NN weight vectorπ i is shown in Fig.8, which is bounded in the steady state. From the s- liding mode manifold s i presented in Fig.9, it is observed that the system states under "CL-RTSMC" are closer to the sliding mode. The control input τ i under two methods are depicted in Fig.10, where τ i has reached 100000 in some cases. It is noted that τ i is the dimensionless control amount rather than the control force directly implemented on the MEMS gyroscopes. The dimensionless control amount τ i is transformed from the control force τ * i to avoid the effect of the dimension on gyroscopic motion characteristics by using the reference frequency ω 0 = 1kHz, reference length ϑ 0 = 10 −5 m and the mass m = 5.7 × 10 −9 kg, where the deducing process is presented in [25]. Thus, it is referred that τ * i = mω 2 0 ϑ 0 τ i = 5.7 × 10 −8 τ i . When τ i reaches 100000, the control forces are calculated as τ * i = 5.7 × 10 −3 N. For MEMS gyroscopes, the actual control force τ * i meets the requirement. Hence, τ i reaching 100000 in some cases is reasonable. Thus the simulation results verified that the expected higher tracking accuracy of the proposed RTSMC using the composite neural learning and the DOB can be obtained.

Conclusions
In this paper, the RTSMC has been designed to control the dynamics of MEMS gyroscopes with faster convergence and higher tracking accuracy in the presence of system uncertainties and external disturbances. In the proposed method, system states have been forced to start from the sliding mode surface at the initial time. The composite neural learning based on SPEM has been proposed to improve the approximation accuracy of system nonlinearity, such that higher tracking accuracy can be obtained. The DOB has been employed to handle the external disturbances. Simulation results have been demonstrated to verify that the proposed control can nicely achieve faster convergence and higher accuracy, compared with the schemes of "NN-TSMC" and "CL-TSMC".