The contributions of this research are as follows:

(1) Propose a new fractional sliding mode surface for use in the FOSMC to suppress the external perturbations. (2) A new compound FOSMC and STC is proposed, in which the STC controller will calculate an error value and apply a correction value to the system. Therefore, the proposed compound control method, reduces the oscillation, increases tracking performance, and reduces the tracking error.

Section 2 presents the dynamic modeling of a MEMS gyroscope. In Sec. 3, the FOSMC is described. Section 4, compound FOSMC + STC has been delineated. Section 5 presents simulation results. Section 6 concludes the paper.

A z-axis MEMS gyroscope is shown in Fig. 1. The conventional MEMS vibratory gyroscope consists of a proof mass (m) suspended by springs, where x and y are the coordinates of the proof mass with respect to the gyro frame in a cartesian coordinate system, sensing mechanisms, and an electrostatic actuation for forcing an oscillatory motion and velocity of the proof mass and sensing the position. Ω x,y,z are the angular rate components along each axis of the gyro frame. The frame where the proof mass is mounted moves with a constant velocity, and the gyroscope rotates at a slowly changing angular velocity Ω z . The centrifugal forces m Ω 2 z x and m Ω 2 z y are assumed to be negligible because of small displacements. The Coriolis force is generated perpendicular to the drive and rotational axes [25].

The dynamics equations of the gyroscope are given by

The origin for x and y coordinates is at the center of the proof mass without force employed. The stiffness and damping terms may vary slightly from nominal values [1,25]. However, the magnitude of the proof mass m can be obtained precisely.

x and u * y are the control forces in the x-and y-directions. Dividing gyroscope dynamics (Eqs. ( 1) and ( 2)) by the reference mass results in the following vector forms:

where

The final form of the non-dimensional equation of motion as follows:

we determine a set of new parameters as follows:

where

Equation ( 8) can be rearranged as follows:

where E is an external disturbance. Since the disturbance is considered unknown, the model from Eq. ( 9) used to generate the control signal must be modified by setting

where M = (D + 2Ω) and N = K b .

Selecting a fractional sliding mode surface is the central part of FOSMC design. The fractional derivative and integral order in sliding mode surface provide the flexibility of having fractional type of error in controller design. The best performance will be obtained if a fractional sliding mode surface is chosen correctly. The fractional sliding mode surface can be selected as follows:

r mdτ (11) where r, m, α, β, and γ are positive constants, and D is a fractional-order operator (D = d/dt, and µ > 2). The fractional-order

The derivative of the fractional sliding mode surface is

Equivalent control u eq can be obtained by setting ṡ(t) = 0.

u eq (t) = M q + Nq + qd -αD μ e(t) -βD μ-1 e(t) -γe(t)

The FOSMC can be shown as

The equivalent control cannot compensate for external perturbation and unmodeled dynamic uncertainties. A reaching control law can be designed to remove those problems as u s (t), which can be defined as:

where K s is a positive constant. Considering the following Lyapunov function candidate (V ), continuous and non-negative [26][27][28].

The time derivative of V yields

By substituting Eqs. ( 15) into ( 18) generates

Simplifying Eq. ( 19) results in

Therefore, Eq. ( 20) can be expressed as

Equation (21) shows that V < 0, which expressed that the proposed control law is stable.

Fractional sliding mode control is one of the techniques that can enhance the robustness of the control system and improve tracking performance. As discussed before, its main drawback is creating a chattering phenomenon. However, STC can be used in conjunction with FOSMC to minimize chattering of the system, improve trajectory tracking, and remove singularity problems. By combining both FOSMC and STC, a better control method will be obtained, combining both controllers' benefits. The Block diagram of the proposed controller is illustrated in Fig. 2. The compound control law can be defined as

where u STC (t)is

where k 1 , k 2 , r,a ndm are positive constants. The stability proving of the proposed control law can be arranged by substituting Eqs. ( 22) into (18) as follows:

The stability of FOSMC was proved in Sec. 3. Therefore, the main controller is stable. Also, the error was reduced by using the compound controller. This shows that the proposed controller will improve the system's stability. Therefore, Eq. ( 26) can be written as

where K s is positive, which leads to V < 0.

The most important part of the controller design procedure is the selection of proposed controller parameters (α, β, K s , μ, r, m, k 1 , and k 2 ). If parameters are chosen inappropriately, the proposed control (20,20). The desired motion trajectory is determined by q d1 = sin (4.17t) and q d2 = 1.2sin (5.11t). The initial values of the system are selected as follows:

q 1 (0) = 0.5, q 2 (0) = 0.5, q1 (0) = 0 and q2 (0) = 0

The parameters of the MEMS gyroscope are selected as Typically, the natural frequency of each axis of a MEMS gyroscope is in the kHz range. Thus, ω 0 is selected as 1 kHz. It is suitable to choose 1μm as the reference length q 0 when the displacement range of the MEMS gyroscope in each axis is sub-micrometer level. The unknown angular velocity is assumed Ω z = 100 rad/s. Figure 3 shows position tracking of the x-axis and y-axis under FOSMC and FOSMC + STC. It can be seen clearly that tracking performance under the proposed controllers is consistent with the desired tracking of the MEMS gyroscope. Figure 4 illustrates the tracking error of the x-axis and y-axis under FOSMC and proposed control. FOSMC creates a chattering phenomenon, which by using STC is reduced. In addition, FOSMC + STC has a lower maximum overshoot and undershoot than FOSMC. Figure 5 shows the velocity of the x-axis and y-axis under FOSMC and the proposed control law. The robustness of the proposed control method was verified by applying the random noise as 0.5*randn (1,1). Figure 6 shows that the proposed control method is robust against external disturbances.

This study proposed a novel FOSMC + STC law to control a MEMS gyroscope. First, a new FOSMC is applied to control the x-axis and y-axis of a MEMS gyroscope. It has high tracking performance, but its main drawback was creating a chattering phenomenon. To solve this problem, an STC is proposed in parallel with