Complexity

Volume 2018, Article ID 5246074, 12 pages

https://doi.org/10.1155/2018/5246074

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

College of IoT Engineering, Hohai University, Changzhou 213022, China

Correspondence should be addressed to Juntao Fei; moc.oohay@ieftj

Received 14 January 2018; Revised 16 June 2018; Accepted 21 June 2018; Published 10 September 2018

Academic Editor: Matilde Santos

Copyright © 2018 Juntao Fei and Xiao Liang. 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.

#### 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.

#### 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 -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-order , 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 Section 2. 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 direction is much larger than that in and directions, and the motion of the proof mass is constrained to the - and -axes as seen in Figure 1; (2) the gyroscope rotates at a constant angular velocity 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: where denotes the mass of the proof mass, are vectors representing damping and spring coefficients along the - and -axes, respectively. is the angular velocity along each axis, and is the control force in the - and -axes.