Adaptive Fuzzy Sliding Mode Control of MEMS Gyroscope with Finite Time Convergence

Ren, Jianxin; Zhang, Rui; Xu, Bin

doi:https://doi.org/10.1155/2016/1572303

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Integration of Sensors in Control and Automation Systems

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Research Article | Open Access

Volume 2016 | Article ID 1572303 | https://doi.org/10.1155/2016/1572303

Adaptive Fuzzy Sliding Mode Control of MEMS Gyroscope with Finite Time Convergence

Jianxin Ren,¹Rui Zhang,¹and Bin Xu¹

Academic Editor: José A. Somolinos

Received25 Apr 2016

Accepted18 Jul 2016

Published04 Oct 2016

Abstract

This paper presents adaptive fuzzy finite time sliding mode control of microelectromechanical system gyroscope with uncertainty and external disturbance. Firstly, fuzzy system is employed to approximate the uncertainty nonlinear dynamics. Secondly, nonlinear sliding mode hypersurface and double exponential reaching law are selected to design the finite time convergent sliding mode controller. Thirdly, based on Lyapunov methods, adaptive laws are presented to adjust the fuzzy weights and the system can be guaranteed to be stable. Finally, the effectiveness of the proposed method is verified with simulation.

1. Introduction

MEMS gyroscopes have become the most growing microsensors in recent years due to the characteristics of compact size, low cost, and high sensitivity. Most MEMS gyroscopes sales in the market are vibrating silicon micromechanical gyroscopes, whose basic principle is to generate and detect Coriolis Effect. As depicted in Figure 1, under assumption that the proof mass of gyroscope rotates around -axis at a speed of and makes uniform motion along the -axis at a speed of , a Coriolis force of is produced along -axis.

In the last few years, numerous advanced control approaches with intelligent design have been studied to realize the trajectory tracking in [1–4] and to handle the system parametric uncertainties and disturbances, and the adaptive control can be found in [5–7]. For control of MEMS gyroscope, Park and Horowitz firstly applied adaptive state feedback control method [8]. Both drive shaft and sensitive axis were subjected to feedback control force in this control method, which administered two axial modal vibration track specified reference trajectories, weakening the boundary between drive mode and test mode as well.

Sliding mode control changes its structure to force the system in accordance with a predetermined trajectory. Batur et al. developed a sliding mode control for MEMS gyroscope system in [9]. Since then, adaptive sliding mode control approach with the advantages of variable structure methods and adaptive control strategies are presented to control MEMS gyroscopes in [10, 11].

Due to the necessity of ideal sliding mode, good dynamic quality, and high robustness, several methods are extended to improve the performance. Yu and Man investigated a nonlinear sliding mode hypersurface to ensure that systems from any point of the sliding mode surface were able to reach the balance point in a limited time in [12–16]. Bartoszewicz [17] examined the reaching laws introduced by Gao and Hung in [18] and proposed an enhanced version of those reaching laws, which was more appropriate for systems subject to constraints. Recently, Fallaha et al. studied a novel approach, which allowed chattering reduction on control input while keeping tracking performance in steady-state regime [19]. This approach consisted of designing a nonlinear reaching law by using an exponential function that dynamically adapted to the variations of the controlled system. Mei and Wang in [20] proposed a nonlinear sliding mode surface which converged to the equilibrium point with a higher speed than both linear sliding mode surface and terminal sliding mode surface. In addition, a new two-power reaching law was proposed to make the system move toward the sliding mode faster.

As a matter of fact, the methods mentioned above are highly dependent on the structure of the nonlinearity, while, currently, accurate model is unavailable. Thus, fuzzy model has been widely used to approximate nonlinear objects in [21, 22]. Robust adaptive sliding mode control with on-line identification for the upper bounds of external disturbance and estimator for the nonlinear dynamics of MEMS gyroscope uncertainty parameters was proposed in [23].

In this paper, an adaptive fuzzy sliding mode control strategy with nonlinear sliding mode hypersurface and double exponential reaching law is developed to track MEMS gyroscope. Furthermore, it converges faster compared with strategies using conventional sliding mode surface in [23] and terminal sliding mode surface in [12–16].

The rest of this paper is organized as follows. The dynamics of MEMS gyroscope with parametric uncertainties and disturbances are given in Section 2. Controller design and stability analysis are discussed in Section 3. Numerical simulations are conducted to verify the superiority of the proposed approach in Section 4, compared with conventional adaptive fuzzy sliding mode control. Conclusions are drawn in Section 5.

2. Dynamics of MEMS Gyroscope

The basic principle of -axis vibratory MEMS gyroscope is shown in Figure 2, which can be described as a quality-stiffness-damping system. Owing to mechanical coupling caused by fabrication imperfections, the dynamics can be derived as where is the mass of proof mass; is the input angular velocity; , represent the system generalized coordinates; , represent damping terms; represents asymmetric damping term; , represent spring terms; represents asymmetric spring terms; and , represent the control forces.

On issues related to the study of mechanism, the law described by model is required to be independent of dimensions. So, it is necessary to establish nondimensional vector dynamics. Because of the nondimensional time , both sides of (1) should be divided by reference frequency , reference length , and reference mass . Then the dynamics can be rewritten in vector forms:where , , , , .

New parameters are defined as follows: Thus, the final form of the nondimensional vector dynamics is

In presence of parametric uncertainties and external disturbance, based on (4), state equation of dynamics is established aswhere , , are system known matrices; , are parametric uncertainties; and is an external disturbance. Besides, , .

If the system total interference (consisting of parametric uncertainties and external disturbance) is represented by , we know where .

It is vital that (6) must meet the following assumptions.

Assumption 1. The total interference , where is an unknown positive vector.

Assumption 2. The total interference meets sliding mode matching conditions; namely, , , , where , , are unknown matrices with appropriate dimensions.

Assumption 3. , are observability matrices.

Based on the above assumptions, the controller can be designed to compensate the total interference.

3. Adaptive Fuzzy Finite Time Sliding Mode Control

The fuzzy model of could be composed of IF-THEN rules, and the th rule has the form: IF is and is and is and is THEN is , .Based on singleton fuzzifier, product inference, and center-average defuzzifier, its output can be expressed as where , , , , are membership function values of the fuzzy variables , , , with respect to fuzzy sets , , , , respectively.

The fuzzy sets of input variables are defined as , where is negative, is zero, and is positive. Then membership functions of are selected as the following triangular functions:

The corresponding membership functions of these fuzzy sets labels are depicted in Figure 3. In addition, the membership functions of , , and are the same with .

Based on the aforementioned fuzzy sets and membership functions, the fuzzy rules are described in Table 1. Therefore, 81 fuzzy rules are chosen.

The control target for MEMS gyroscope is to maintain the proof mass oscillation at given frequency and amplitude, such as , in the and directions, respectively. So, reference model can be designed as where , .

And the tracking error is defined as Nonlinear sliding mode hypersurface is chosen as where , ; , ; what is more, , , , are odd.

Then the reaching law is designed as the following double exponential function: where , , , .

It should be noted that the converge speed depends on parameters such as , , , , , and , , , .

According to (6), equivalent control law is obtained asAnd the derivative of sliding surface (11) is Then substituting (12) into (14), And substituting (15) into (13),Besides, a robust item is designed to guarantee that the system is asymptotically stable: Thus, the adaptive fuzzy finite time sliding mode controller is obtained asAccording to (10), we have Substituting (18) into (19), Substituting (20) into (14), The optimal parameters are set aswhere is a set of .

And the minimum approximation errors are defined as Substituting (23) into (21), we deriveConsidering (7), (24) can be expressed aswhere .

So adaptive law can be selected asNamely, where .

Lyapunov function is defined asDifferentiate with respect to time yields, and substitute (26) as

Owing to the fuzzy approximation theory, adaptive fuzzy system can approximate nonlinear system closely. Therefore, ; namely, the system is asymptotically stable.

4. Simulation Study

In this section, numerical simulations are investigated to track the position and speed trajectories of MEMS gyroscope, compensate parametric uncertainties and external disturbances, and verify the superiority of the proposed approach compared with conventional adaptive fuzzy sliding mode control strategy using linear sliding mode surface. Those two methods are defined as follows.

Method 1. Define the adaptive fuzzy sliding mode control proposed in this paper as Method 1, whose sliding mode surface is shown in (11), and the reaching law is expressed in (12).

Method 2. Define the conventional adaptive fuzzy sliding mode control as Method 2, whose sliding mode surface is , and the reaching law is .

Parameters of the MEMS gyroscope are as follows:

Since the position of proof mass ranges within the scope of submillimeter and the natural frequency is generally in the range of kilohertz, assume that reference length is m, reference frequency is kHz, and the reference trajectories are , , respectively.

Then set other simulation parameters as

And select the initial state values of the system as .

Then the position and speed trajectories of Method 1 are shown in Figures 4 and 5 and those of Method 2 are depicted in Figures 6 and 7.

The position tracking error and speed tracking error of Methods 1 and 2 are shown in Figures 8–11, respectively.

Through the tracking simulation of MEMS gyroscope, the proposed approach is with satisfying performance; in addition, in comparison to Method 2, the convergence time is shortened to 0.3′′ from 0.6′′.

5. Conclusion and Future Work

An adaptive fuzzy finite time sliding mode control strategy using nonlinear sliding mode hypersurface and double exponential reaching law is proposed to compensate parametric uncertainties and external disturbance of MEMS gyroscope in this paper. Based on Lyapunov methods, the stability of system can be guaranteed. Simulations verify that, compared with conventional adaptive fuzzy linear sliding mode control strategy, the convergence time of finite time convergent control strategy proposed in this paper is shortened to 0.3′′ from 0.6′′; namely, convergence has been significantly improved. For future work, the novel adaptive online constructing fuzzy algorithm [24] can be employed for more efficient learning while disturbance observer based design [25, 26] can be considered to improve system performance.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61304098, 60204005, and 60974109), Aeronautical Science Foundation of China (2015ZA53003), Natural Science Basic Research Plan in Shaanxi Province (2014JQ8326, 2015JM6272, and 2016KJXX-86), Fundamental Research Funds for the Central Universities (3102015AX001, 3102015BJ(II)CG017), Fundamental Research Funds of Shenzhen Science and Technology Project (JCYJ20160229172341417), and International Science and Technology Cooperation Program of China under Grant no. 2014DFA11580.

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Copyright

Copyright © 2016 Jianxin Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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