Robust Observer Based on Fixed-Time Sliding Mode Control of Position/Velocity for a T-S Fuzzy MEMS Gyroscope

This study focused on a control system of the nonlinear micro-electro-mechanical systems (MEMS) gyroscope. First, sector nonlinearity was used to model a MEMS gyroscope in the Takagi-Sugeno (T-S) fuzzy system. Second, a state observer was designed based on linear matrix inequality (LMI) to identify the optimal eigenvalues of the state tracking error function. Then, full-state fixed-time sliding mode control (FTSMC) was constructed to control the system. Third, a case study of a harmonic disturbance observer was used to address the unknown disturbance of the system. A disturbance observer (DOB) was simply designed based on the error signals of the system outputs and observer outputs. The output signals precisely converged to the predefined trajectories in a very short time, with no overshoots and small of steady-state errors. Moreover, the estimated output states were precisely tracked by the system outputs. These important factors were used to confirm that the control of the T-S fuzzy MEMS was effective and easy to achieve. The study used MATLAB simulation to archive the verification. The maximum of tracking error was <inline-formula> <tex-math notation="LaTeX">$e_{4} \in [-4.657:5.565]\times 10^{-11}$ </tex-math></inline-formula>, and the maximum settling time was <inline-formula> <tex-math notation="LaTeX">$T_{e3} \sim 0.144$ </tex-math></inline-formula> for the error of the <inline-formula> <tex-math notation="LaTeX">$\dot {y}-$ </tex-math></inline-formula> axis and the settling time of the <inline-formula> <tex-math notation="LaTeX">$\dot {x}-$ </tex-math></inline-formula> axis, respectively.


I. INTRODUCTION
In recent years, the development of robotics, artificial intelligence, and automobile devices have resulted in a need for micro-electro-mechanical systems (MEMS) gyroscope need to be used with high-precision requirements. The MEMS gyroscopes can be used to measure the rotation angular velocity due to their size and cost. However, fabrication of a MEMS gyroscope is the main reason for the disturbance sensitivity, parameter variations, and environmental temperature effect inverse problems. To increase the precision of a MEMS gyroscope system, some papers investigated the control methods, such as robust control with active disturbance rejection was proposed for MEMS gyroscope [1]. Robust The associate editor coordinating the review of this manuscript and approving it for publication was Nagarajan Raghavan .
control with a combination of proportional-derivative and fractional-order sliding mode control (SMC) was designed to control MEMS [2]. An adaptive sliding mode control and fuzzy compensator were introduced for the MEMS gyroscope in [3]. The investigations of neural networks for a MEMS gyroscope can be found in [4]- [7]. Wang et al. [8] proposed the control of the z-axis of a MEMS gyroscope by using adaptive fractional-order sliding mode control. Zhang et al. [9] investigated SMC with the updating law of a neural network for the MEMS gyroscope. All of these papers achieved good performances for tracking problem of the MEMS gyroscopes. However, disturbance observers for the MEMS gyroscopes have received a minimal attention. In [10] proposed the output feedback control for MEMS gyroscope by using neural network and DOB. Furthermore, the MEMS can be more precise in tracking if its mathematical model can be corrected as a nonlinear model. In [11], [12] the T-S fuzzy model with the linearization modeling method was introduced. To the best of our knowledge, no investigation of the sector nonlinearity method of T-S fuzzy modelling for MEMS gyroscopes has been conducted. Furthermore, a fixed-time sliding mode control and a harmonic disturbance observer also may not be available. These factors are the main motivations of this work. In this work, a system states observer was obtained by LMI with vertical spaces. These estimated states were used to construct sliding mode surfaces. The output tracking errors were used to design a disturbance observer, and a disturbance observer error was exponentially convergent. The mathematical model of the system was represented by the combination of the outer fuzzy membership functions and inner sublinear systems. The system was then called the T-S fuzzy MEMS gyroscope.
The Takagi-Sugeno model was proposed in 1985 [13]. T-S fuzzy modeling of the nonlinear system was investigated in detail as sector nonlinearity and linearization by Tanaka and Wang [14]. The development of T-S fuzzy modeling was described in previous papers [15]- [22]. By using the T-S fuzzy model, the mathematical model of a MEMS gyroscope can be changed into the combination of four fuzzy membership functions and four sublinear systems. Estimating the system states of MEMS can be easier with the linear observer design. After modeling, the disturbance observers can easily apply the requirements of the linear-based model.
Linear matrix inequality (LMI) is well known as the poles placement control method. With any pole of the control system expressed as λ = −a + bi, a > 0 is required to help the system state remain stable; this can be understood as the control system remaining stable if all its poles are located in the left-half of the complex plane. The eigenvalues are important factors for defining the performance of the control system with the damping, overshoot, settling time, and steady-state [23], [24]. This study used LMI with a vertical boundary to obtain the eigenvalues of the state error function. After estimated states were obtained, a sliding mode for position and velocity control was designed for a MEMS gyroscope system.
Sliding-mode control is the nonlinear control technique that consists of switching and equivalent controls; these control values are used to force the states to converge on the predefined surface and stabilize these states on this surface [25]. Chattering is the main caused of decreased system performance, and it is sourced by the switching control [26]- [30]. This paper applied fixed-time sliding mode control with the aim of obtaining small settling time and small chattering. The basic concept of fixed-time control was introduced in 2012 [31]. Applications of the fixed-time concept can be found in [32]- [36]. To the best of our knowledge, investigations of fixed-time control for a MEMS gyroscope are limited. Furthermore, fixed-time control for the double loops of the position and velocity of the MEMS gyroscope might not be available. This study designed the fixed-time for controlling the position and velocity values of the MEMS gyroscope. Under harsh working condition, the disturbance observer is highly recommended for MEMS gyroscopes.
The disturbance observer is a special case of unknown input estimation, where the disturbance and uncertainty can be suppressed to zero to improve the precision of the control system. In [37], a nonlinear basic disturbance observer (NDOB) was introduced very effectively. The application of a basic disturbance observer was found to achieve synchronization and secure communication in [20] and [22]. The application of NDOB for motor control was described in [38]. The development of NDOB on pendulum system can be found in [39]. Otherwise, some advanced disturbance compensations based on a neural network system were investigated in [6], [40]- [43]. To simplify the procedure of the design of a disturbance observer, this study proposed a new DOB to scope the disturbance and uncertainty of a MEMS gyroscope under the conjunction of an unknown disturbance in exogenous form. The exogenous disturbance observer can be found in previously published papers. An exogenous disturbance observer was proposed for the T-S fuzzy system [44]. The problem of the exogenous inputs of a wind turbine system was investigated [45]. The problem of the exogenous disturbance observer of a robot system was discussed in [46]. In [47], an adaptive DOB was proposed to handle an unknown exogenous disturbance value. In this paper, the disturbance observer was constructed with a high convergence speed. The support of a low-pass filter was presented. These introduced control technique, such as the LMI, the fixed-time sliding mode control, and the disturbance observer, will be introduced to control the MEMS gyroscope system with the motivation of the following published papers. In [48], a new hybrid fractional sliding mode control was introduced to a MEMS gyroscope system. Their paper ignored the disturbance observer. In [9], a neural network was used to compensate for the imprecision of the modeling error of the MEMS system. In [49], the neurodynamic approximation-based quantized control was considered for a MEMS gyroscope subjected to disturbance and uncertainty values. In [50], the disturbance, uncertainty, and chattering could be suppressed by the novel control technique of the state observer-based minimal learning parameter. In [51], the disturbance and uncertainty sigmoid functions were provided to estimate the disturbance. A hysteresis quantizer-based neural estimator to estimate the perturbations of a vehicle was proposed in [52]. The fixed-time of an extended states observer with the function of uncertainty estimation was introduced in [53]. However, the settling time remains high, and there exists overshoot values. Furthermore, among the previously published studies [1]- [9], few investigated disturbance observers. To estimate the disturbance of a MEMS system, a neural network was used to archive the goal [54]. The adaptive neural network with full-states feedback of a MEMS gyroscope was discussed in [55]. In [56], the minimum learning parameter-based neural network was proposed to estimate the perturbations of a MEMS system. The linear extended states observer was used to construct the VOLUME 9, 2021 lumped disturbance of MEMS system [57]. These mentioned papers that investigated a MEMS gyroscope design have few disturbance observers. Furthermore, the discussions of the exogenous disturbance observer for a MEMS gyroscope are limited. Based on these motivations, this paper proposed a new disturbance observer for a MEMS system. The disturbance is finite convergence. The benefits of finite time were described in [58]- [61]. Furthermore, sector nonlinearity modeling for a MEMS gyroscope was not found. To overcome this limitations, this paper addresses the problem of the DOB based on the FTSMC of the T-S fuzzy MEMS gyroscope. Designing the control for a MEMS gyroscope with the nonlinear characteristics of the springs considered is simple work. However, designing a DOB for a nonlinear MEMS gyroscope is a complicated task. To solve this problem, a nonlinear MEMS gyroscope should be considered as sublinear systems. Furthermore, the fixed-time with the states feedback is good control to reject the chattering problem. Obtaining the mass body velocities of MEMS is a complicated task if physical sensors are used. For these values, estimation by using the states observer is easier but costly work.
The contributions of this study are as follows 1. The MEMS gyroscope model was changed into the format of the T-S fuzzy system with the support of the sector nonlinearity method. Nonlinear springs on the MEMS gyroscope were considered with the nonlinear mathematical model of the MEMS system. The new model of the MEMS gyroscope was built to simplify the control design for the MEMS system. The originality of the control of the new model is as follows: a control with full-states and disturbance observer designs is easier to archive. 2. The system states of the MEMS gyroscope as the x-and y-coordinates and the velocity of the mass body can be precisely estimated by the support of the linear matrix inequality method. These estimated states were used to construct the sliding mode surface for the double loops control of position and velocity. This approach is also a new control technique for the MEMS gyroscope. 3. The fixed-time sliding mode control was designed for controlling the MEMS gyroscope with a simple and effective structure. In the sliding mode design, the estimated state and reference input are used to build the sliding mode surface. Furthermore, the generous disturbance observer is based on the errors of the measured and estimated outputs. The disturbance observer was exponentially convergent with the simple structure. 4. To verify the proposed theory, the Lyapunov candidate was used for theoretical proof. Moreover, MATLAB software was used by simulation that to confirm the proposed theories are good for controlling the MEMS gyroscope and robustness with the disturbance and uncertainty effects. The novelties of this paper are as follows. The nonlinear springs of the MEMS gyroscope were mentioned. To design the controller for the nonlinear MEMS system, T-S fuzzy modeling was used to reduce the cost of the control and observer designs. The full states were known by the observer with the support of the linear matrix inequality. The positions and velocity controls are perfectly obtained via the feedback of the estimated states. The fixed-time sliding mode control was perfectly designed to obtain the precision tracking values of position and velocity values. The proposed control algorithms are large suggestions of the full-states feedback control, robustness control, and disturbance rejection control. The outline of this paper is as follows. The introduction of the trends of the research topic, method concepts, and contributions of the paper were given in the first section. In the second section, the mathematical modeling of a MEMS gyroscope into the T-S fuzzy system, the preliminary mathematical operation of the fixed-time sliding mode control, linear matrix inequality and the proposed disturbance observer are shown. In section III, the proposed theories for the T-S fuzzy MEMS gyroscope system is shown. In section IV, an illustrative example is given to show the effectiveness and correctness of the proposed methods for a MEMS gyroscope. Finally, the conclusion and future works are given in the last section.
Notes: A > 0 and A < 0 are the positive and negative matrices, respectively.

II. MATHEMATICAL MODEL OF A MEMS GYROSCOPE AND PRELIMINARY MATHEMATICS
This section is used to show the mathematical model of a MEMS gyroscope and preliminary mathematical operations. In this study, the complexity of the nonlinear springs of the MEMS gyroscope was shown. Designing a control for the nonlinearity of the MEMS gyroscope is still easy. However, designing the disturbance observer or applying the full-states feedback control is complicated work. To meet these difficult requests, the mathematical model of the MEMS gyroscope should be changed into combinations of the sub-linear systems. Because the T-S fuzzy modeling method consists of combinations of the sub-linear systems and outer fuzzy membership functions, the control design for a nonlinear MEMS gyroscope is equivalent to the design of a controller for the sublinear systems. The mathematical modelling of a MEMS gyroscope is first given as follows:

A. MATHEMATICAL MODELING OF THE MEMS GYROSCOPE
This paper reused the mathematic and parameters of [12]. The MEMS gyroscope model [12] is written as where x and y are the coordinates of the system. m is mass of the rigid body of the MEMS gyroscope. d xx and d yy are the damping terms. k xx and k yy are the stiffness coefficients for x-and y-axes, respectively. k x 3 x 3 and k y 3 y 3 are the stiffness coefficients of the x-and y-axes, respectively. u * x and u * y are the control signal for the x-and y-axes, respectively. * z input is the angular velocity. The MEMS gyroscope can be represented as Figure 1 below. Remark 1: In previously published papers, the position controller was introduced to control a MEMS gyroscope. In this work, we proposed the theoretical position and velocity control algorithms.
By dividing both sides of Eq. (1) by the reference mass m, reference length q 0 and resonant frequency 0 , the mathematical model of MEMS gyroscope is written as follows: System (2) can be simplified as follows: where the parameter of system (3) can be calculated as The system parameters are as follows:  (3) with full position and velocity control can be modeled as follows: where u 1 and u 2 are position control for x-and y-axes, respectively. uẋ is velocity control of x-axis and uẏ is velocity control of the y-axis Remark 2: In Eq. (4), the velocity control values uẋ for the x-axis and uẏ for the y-axis are added to help improve the precision control of the full-states feedback.
Designing the controller for system (4) is a simple task. However, the number of controllers that can be applied to system (4) may be limited due to its nonlinear format. Furthermore, the DOB design for this nonlinear system is complicated work. To alleviate these control requests, this paper used the sector nonlinearity to convert system (4) into the form of the T-S fuzzy system, which will be called T-S fuzzy MEMS gyroscope system. To represent the sector nonlinearity method, the system is considered, where z is a state variable vector. g m , h m , and l m are the smooth functions. y is the output vector. The scheduling variable is z j ∈ [z min , z max ], where j = 1, .., p. The weighting functions for z j are These weighting functions are no longer less than zero. The fuzzy membership function is the product of the weighting functions as follows: where ϕ ij (z i ) is either n j 0 (·) or n j 1 (·). Because of these concepts, system (5) can be easily modelled as follows: Hence, the system states in (4) are assumed to be x ∈ [ x max , x min ] and y ∈ [ y max , y min ], and then, system (4) can be converted into the T-S fuzzy system as follows: The MEMS gyroscope mathematical model is now calculated as the combination of four fuzzy membership functions and four sublinear systems. To control the MEMS gyroscope in (9), this paper proposed the fixed-time sliding mode control based on the state feedback. To obtain the system states, the LMI is used to obtain the goal.x andŷ are used to represent the observer system states. The goal of the sliding mode control is to force thesex → x → x r ,ŷ → y → y r , x →ẋ →ẋ r ,ẏ →ẏ →ẏ r , where r is used to mark the reference term. To this end, the preliminary mathematical operation is presented in the next section.

B. PREMILINARY MATHEMATICS
This section is used to describe the mathematical operations of fixed-time sliding mode control, linear matrix inequality, and exponential convergence disturbance observer. First, to consider the fixed-time concept, the systeṁ where x(0) = x 0 .x ∈ R n , and f (x) ∈ R n . Definition 1: Fixed-time stability [32] System (10) is called fixed-time stable if the settling time T is globally bounded and T ≤ T max .T max is a constant value. The system is then called fixed-time stable.
Lemma 1: Consider the equation [32] s = −α 1 sig where δ(0) = 0, a, b, m, n are positively defined. a 1 > b 1 , a 2 < b 2 , α 1 , and α 2 , are positive values, and the settling time is bounded as follows: Proof: The Lyapunov candidate can be chosen with one dimension as follows: Taking derivative of both sides of Eq. (13) yieldṡ BecauseV (s) ≤ 0 system (10) Integrating (17) over the time from zero to T yields or This completes the proof of lemma 1. Remark 3: For help the reader understand the paper, the proof of the lemma 1 needs to be redone. Lemma 1 is used to illustrate the settling times of each control values in Eq. (55).  Definition 2: Linear matrix inequality [23]. Consider the systemẋ where x(t) is the system state vector and A is an approximated matrix of x(t). System (20) is called LMI stable if the eigenvalues of the system are located in the region of the LMI condition, and then, there exists a positive matrix P that satisfies This paper used the vertical area to find the eigenvalues of the system states errors. The vertical area is referred to as The V area can be shown as follows: Lemma 2: For eigenvalues of (20) located in the V area, the eigenvalues of A satisfy −b < eig(A) < −a, which can be represented as follows: Proof: The Lyapunov candidates for system (20) are selected as follows: Taking the derivative of both sides of Eq. (25) yieldṡ or This completes the proof of lemma 2. Remark 4: Lemma 2 is used to define the observer gains of Eq. (40), where the states observer gains are placed in the LMI region.
The advantages of the LMI with the vertical area are as follows: The LMI with the vertical area is a simple control method that can support the system and obtain the stable eigenvalues in the specific area gap, where the eigenvalues can be archived with the best characteristics of overshoot, damping ratio, and settling time performances.
Definition 3: Disturbance observer-based state observer error.
To show the disturbance observer for this state observer error-based system, the state-space equation is as follows: where X ∈ R m×n is the state vector, y ∈ R k×n is the system output vector, A ∈ R m×m , B ∈ R m×p , C ∈ R k×m , and D ∈ R m×q , are the approximated matrices of states, control input, and perturbations vectors, respectively, and u ∈ R p×n is the control input vector. System (29) can work if the disturbance and uncertainty are bounded assumedly as |d| < κ, where κ is positively defined. The estimated system of (29) can be designed as follows: First, consider the case of no disturbance effects on system (29). The state error can be represented aṡ By applying the LMI tool box, the observer gain L is found with where P, a, b are positively defined. Then, the disturbance observer can be design as follows: or By subtracting both sides of (34) by sd(s) yields This paper supposes that the source of disturbance can be modeled as Eq. (35) can then be modified as follows: where a suitable value of the vωv −1 = β * T . The disturbance error goes to zero in finite time. This completes the proof of the proposed disturbance observer stability of an exogenous disturbance observer. The details of the applications of these proposed theories to a MEMS gyroscope are shown in the next section.
Remark 5: Definition 3 is used to estimate the perturbation value of the MEMS gyroscope control system in Eq. (43). The low-pass-filter was used to reduce the high-frequency of disturbance value.

III. PROPOSED APPROACH
This section used to show the application of these proposed method on the T-S fuzzy MEMS gyroscope. The structure of this part is as follows: First, states and disturbance observers is designed to the MEMS gyroscope. Second, the fixed-time sliding mode control is designed with full states feedback for controlling the system.

A. STATES AND DISTURBANCE OBSERVER FOR T-S FUZZY MEMS GYROCSOPE
The MEMS gyroscope with full disturbance and uncertainty can be written as follows: . System (38) can work if the variation of approximated matrices of the disturbance and uncertainty is bounded as assumption 1 below.
Assumption 1: System (38) can work as its original char- System (38) can be modified as follows: Remark 6: Matrix E i should be identity defined and all these E i should be identical. The variations of these parameters are unknown.
Remark 7: The estimated disturbance observer was used to compensate the disturbance of the MEMS gyroscope via the control input channel.
By combining of systems (39) and (40), the tracking error equation can be modeled as follows: where L ij is the observer gain andl is the disturbance error.  (43) By applying the LMI to solve the stability of Eq. (43) without disturbance effects, the tracking errors of the measure states and observer converge to each other if where b ij < eig(A i − L ij C) < −a ij , and P is positively defined. After archiving the precision of the state observer, the disturbance need to be converged to zero, such asl = 0, in finite time. By applying the disturbance observer in Eq. (40) to the system (43) Taking the Laplace transform of both sides of Eq. (46) leads tol With the exogenous disturbance as Eq. (36), where l(s) = vξ (s) and sξ (s) = −ωξ (s), system (47) can be changed to Subtracting both sides of (48) by the sl(s) yields Eq. (49) can be simplified as follows: With the condition of Eq. (44) and vωv −1 ∼ ς T → ρ, the disturbance error goes to zero in finite time. This completes the proof of the disturbance observer stability. This study used the estimated states to design the sliding mode control, which is shown in the next section.

B. FIXED-TIME SLIDING MODE CONTROL FOR THE MEMS GYROSCOPE
To archive the fixed time of the reaching phase, the sliding mode surface is proposed as follows: where i = 1 ÷ 4 is used to represent the element of the states and estimated states. Note that X r = x r y rẋrẏ r T , X = x yẋẏ T andX = xŷẋẏ T .
Differentiating for both sides of Eq. (52) leads tȯ ω ij (x i ,ŷ j ) = 1, the equivalent control value is To archive the fixed time for the reaching phase, the switching control value is proposed as follows: where i = 1 ÷ 4 is used to represent the element of sliding mode surfaces. The control values for velocity controls are for the x-axis, and for the y-axis. The position controls are and for the x-and y-axes, respectively. A i1 , A i2 , A i3 , A i4 are rows 1 to 4 of matrix A i .L ij1 , L ij2 , L ij3 , L ij4 , are rows 1 to 4 of matrix L ij .

C. STABILITY OF THE PROPOSED METHODS
To theoretically verify that the proposed methods are correct, the Lyapunov candidate is assumed as follows: Differentiating of both sides of Eq. (56) yieldṡ Combining Eqs. (54), (59), and (61) leads tȯ By using lemma 1 for Eq. (58), the settling time for the reaching phase is then the fixed-time stability. The general settling time can be represented as follows: The illustration of the proposed methods by MATLAB simulation is given in the next section.
Remark 8: The control parameters selection is the main factor effecting the control output performances. This paper has procedures for parameter selection, as LMI gains should be used to obtain small overshoots, damped, small settling times. The source of chattering and overshoot of sliding-mode control is switching control gain. Therefore, the fixed-time sliding mode control gain should be suitably chosen to obtain small settling times, small chattering. The high disturbance observer gains will occur the high oscillation but a fast response, while small gains lead to a slower the disturbance response.
The details of the proposed control algorithms for the MEMS gyroscope are represented in the below diagram.

IV. AN ILLUSTRATIVE EXAMPLE
Through the simulation sign ∧ is replace by the word ''hat''. The simulation performances of the proposed method, named the disturbance observer based on fixed-time sliding mode control for the T-S fuzzy MEMS gyroscope is shown in this section. With the system parameters as Eq.   [3], an adaptive SMS and fuzzy compensator were introduced to a MEMS gyroscope; the settling was given as approximately 0.2 second, and the magnitude scale of the tracking error was 10 −3 . With the high-magnitudes tracking error values, the energy consumption values of the position controls in [3] are smaller than those of our paper. However, our paper can obtain better control factors as small as of those of steady states and settling times. In [5], the results are intuitively larger than our obtained results. In [5], the adaptive neural backstepping PID global SMC was introduced to a MEMS with high-oscillation control input, while the control input for the position control values of this paper is smooth, as shown Figure 8a. Their paper showed that the simple controller consumed less energy. Thus, the energy consumptions values are mostly smaller and smoother than that of [5]. Otherwise, the comparison of this study with the earlier studies [5] is shown in the table below.
The performance of the disturbance is shown in Figure 7 below.
The functions of d x = sin(2πt) and d y = 1.5 sin(2.75πt) were tested on the x-and y-axes of the MEMS system, respectively. To completely reject the disturbances on the xaxis and y-axes is difficult for a nonlinear MEMS gyroscope.
In this paper, the tested disturbances were mostly deleted by the estimated disturbances compensation. The control inputs are shown in Figure 8.
The position control was precisely archived by the good position tracking error values on x-, and y-axes. Beyond that point, the inner velocity control input signals are very small compared with the control inputs of the position control. This result shows that the proposed disturbance observer has a strong effect on the unknown exogenous disturbances value, and the proposed fixed-time control method is effective with the T-S fuzzy MEMS gyroscope system. In Figure 8 (b), the control system of MEMS gyroscope exhibits chattering. However, using double loops of position and velocity controls obtains smooth position tracking responses on the x-and yaxes, respectively. The chattering in this paper is very small by using suitable fixed-time control gains.
In the comparison of the two cases, the sine function was estimated more easily because the original point of the sine functions and the disturbance observer are identical. For the cosine function, the oscillation will occur for the estimated disturbance in response to the tested disturbance. The oscillation of the estimated disturbance is realistic. To show the  independence of the tracking error on the initial states, this case tested the initial on x-and y-axes are ten times in compare with these initial in the first case. The tracking errors are shown as follows: The settling times on the position of the x-axis and y-axis are T e1 < 0.17 seconds and T e2 < 0.1 seconds, respectively. The settling times on the velocity of the x-axis and y-axis are T e3 < 0.151 seconds and T e4 ∼ 0 second, respectively. These values are used to confirm that the tracking error values are weaken dependence on the initial states.

V. CONCLUSION
This paper modeled a nonlinear MEMS in the T-S fuzzy model, and this is an important step for the design of the robust control techniques for MEMS gyroscope systems. Furthermore, a new exogenous disturbance observer based on the FTSMC for a MEMS gyroscope system was investigated with exponentially convergent speed. The novelties of this study are the design of the sliding mode control by the reference and estimated states and the disturbance was simple and effective disturbance observer via the tracking error values of the measured and estimated states. The tracking error values are very small, and without overshoots, and the settling times are very short. A MATLAB simulation was used to verify that the proposed control theories are correct and powerful for T-S fuzzy MEMS gyroscopes. In the near future, the fixed-disturbance based on robust control of a T-S MEMS gyroscope will be considered to improve the performance of MEMS control systems. The direction of a new state observer and new state feedback control will also be considered.