Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4720126, 13 pages

http://dx.doi.org/10.1155/2016/4720126

## Adaptive Command-Filtered Backstepping Control for Linear Induction Motor via Projection Algorithm

Institute of Electrical Engineering and Intelligent Equipment, School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China

Received 6 May 2016; Revised 19 August 2016; Accepted 22 August 2016

Academic Editor: Xuejun Xie

Copyright © 2016 Wenxu Yan 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.

#### Abstract

A theoretical framework of the position control for linear induction motors (LIM) has been proposed. First, indirect field-oriented control of LIM is described. Then, the backstepping approach is used to ensure the convergence and robustness of the proposed control scheme against the external time-varying disturbances via Lyapunov stability theory. At the same time, in order to solve the differential expansion and the control saturation problems in the traditional backstepping, command filter is designed in the control and compensating signals are presented to eliminate the influence of the errors caused by command filters. Next, unknown total mass of the mover, viscous friction, and load disturbances are estimated by the projection-based adaptive law which bounds the estimated function and simultaneously guarantees the robustness of the proposed controller against the parameter uncertainties. Finally, simulation results are given to illustrate the validity and potential of the designed control scheme.

#### 1. Introduction

In the past two decades, linear induction motors (LIM) have been broadly used in all walks of life, such as the military, life electric, industrial automation, and transportation, which all have satisfactory performance [1–7]. The most significant advantage of LIM is that it can provide linear motion without any medium machine. Moreover, there are many advantages in LIM, such as simple structure, high-starting thrust force, low noise, high-speed operation, reduction of mechanical losses, and being without any gear between motors and motion devices [8, 9], and as counterpart these advantages introduce the disadvantages of the complicated LIM model presenting the complexity of the control characteristics. Although the driving principles of the LIM and the conventional rotary induction motor (RIM) are actually similar, the control scheme of the LIM is even more intricate than that of the traditional RIM, reason for which is that the parameters of the LIM are time-varying in the process of operation, such as rail configuration, speed of the mover, and temperature [10]. Furthermore, there are more important parameter variations in end effect, slip frequency, saturation of the magnetizing inductance, phase unbalance, and dynamics of the air gap [11, 12], and, because of this, it is hard to get the complete mathematical model of LIM. Despite the fact that the consideration of significant variations for the model of the dynamic performance of the LIM has been researched [11–14], the control performance of LIM is still influenced by uncertainties such as nonlinear dynamics, unknown external load disturbances, and irregular plant parameter variations. The main purpose of this paper is to design a suitable control scheme to deal with the uncertainties existing in the dynamic model of LIM.

In the past decade, indirect field-oriented control (IFOC) technique has been one of the prevalent control techniques widely implemented in industrial LIM drives and it has been proved that it is appropriate for a wide range of technological applications. The main idea of IFOC is to decouple torque and flux, which can be implemented by forcing the secondary flux of the -axis to be a constant and setting the secondary flux of the -axis to zero. Through this method, the system structure can be simplified. However, the performance of the system is sensitive to the variations of motor parameters and the rotor time constant which fluctuates notably with the saturation of the magnetizing inductance and the temperature [14].

Due to the rapid development of nonlinear control theory, backstepping is one of the nonlinear control techniques developed in the 1990s [15] to stabilize the nonlinear dynamic system [16–18], which has been used for the LIM [19, 20]. The backstepping control is a recursive structure, so the design process can be started at the known-stable system and back out new controllers that progressively stabilize each outer subsystem. Although the backstepping alleviates some limitations of other methods and provides an option of design tools to accommodate nonlinearities in the design of the controller, the traditional backstepping requires that exact information of the model is obtained and the parameter uncertainties are not taken into consideration [21]. In order to ensure the stability of most control systems with nonlinearities and parameter uncertainties, adaptive backstepping approach has been investigated, which has been proved to be effective to achieve the satisfactory control performance [10, 14, 22–25]. However, there is a problem called “explosion of terms” in backstepping. That is to say, in the design process of each subsystem, virtual control commands need to be differentiated repeatedly. With the increase of the number of the system orders, it is difficult or even impossible to derive the analytical differential expression of virtual control, so this problem limits the application of backstepping method used in practical engineering. The rest of the problems in backstepping have the following two main points: control saturation problem and the fact that the system must be simplified into the form of parametric strict feedback. In particular the previous drawback may result in some fateful problems in the actual control system, because accumulation of errors may cause system instability if the generated control signal commands are not completely executed by actuators. At present, there are many methods to solve the above-mentioned defects such as dynamic surface control [26] and command-filtered method [27–30], and among them command-filtered backstepping is a more effective way compared to dynamic surface control. Dynamic surface control uses the filter to solve the differential expansion problem, but the introduction of the filter can only ensure that the system is bounded tracking. However, the command-filtered backstepping controller can solve the problem of differential expansion, and it can also realize the asymptotic tracking of the closed-loop signal due to the filter compensation.

In this paper, a command-filtered adaptive backstepping control via projection algorithm designed on indirect field orientation is proposed for the LIM to achieve a position tracking objective under the disturbance of load thrust force and parameter uncertainties. At the same time, Lyapunov stability theory is used to prove that the control system can be maintained closed-loop asymptotically stable. The rest of this paper is organized as follows. In Section 2, the principle of indirect field-oriented control applied to LIM is derived. Section 3 presents the command-filtered adaptive backstepping controller designed for LIM position control via projection algorithm. In Section 4, simulation results are showed to prove the validity of the proposed control scheme. Finally, we come to some conclusions at the end of this paper.

#### 2. Indirect Field-Oriented Control of LIM

The LIM is formed by the primary and secondary, which is shown in Figure 1. The primary of the adopted three-phase LIM is the primary of rotary-motor simply cut open and rolled flat. The secondary usually consists of a sheet conductor using aluminum with an iron back for the return path of magnetic flux. Moreover, a simple linear encoder is adopted for the feedback of the mover position [12, 20]. When the current flowing primary produces a magnetic field though the primary from front to back, the eddy currents are generated on the surface the secondary by this magnetic field. When the primary of the LIM is still, the equivalent circuit of LIM is similar to the rotating motor. However, with the primary moving, the secondary is constantly replaced with new secondary sheet which will resist a sudden increase in flux and only allow the gradual establishment of magnetic flux in air gap. On the contrary, the field in the exit side of the rail will disappear quickly. The phenomenon affecting the air gap flux distribution happens at the entry and exit of the primary called end effect. The parameter is used to simulate the end effect expressed as [11, 20]where is the mover linear velocity, is the primary length, is the secondary inductance per phase, and is the secondary resistance per phase. Then, the inductance can be expressed aswhere , is the magnetizing inductance per phase, is the primary inductance per phase, and is the secondary inductance per phase. And the secondary time constant is expressed by