J. Cent. South Univ. (2016) 23: 1406-1415
DOI: 10.1007/s11771-016-3193-y
A new PMSM speed modulation system with sliding mode based on active-disturbance-rejection control
RONG Zhi-lin(荣智林)1, 2, HUANG Qing(黄庆)2
1. College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China;
2. Zhuzhou CSR Times Electric Co., Ltd., Zhuzhou 412001, China
Central South University Press and Springer-Verlag Berlin Heidelberg 2016
Abstract:
A sliding mode and active disturbance rejection control (SM-ADRC) was employed to regulate the speed of a permanent magnet synchronous motor (PMSM).The major advantages of the proposed control scheme are that it can maintain the original features of ADRC and make the parameters of ADRC transition smoothly. The proposed control scheme also ensures speed control accuracy and improves the robustness and anti-load disturbance ability of the system. Moreover, through the analysis of a d-axis current output equation, a novel current-loop SM-ADRC is presented to improve the system’s dynamic performance and inner ability of anti-load disturbance. Results of a simulation and experiments show that the improved sliding-mode ADRC system has the advantages of fast response, small overshoot, small steady-state error, wide speed range and high control accuracy. It shows that the system has strong anti-interference ability to reduce the influence of variations in rotational inertia, load and internal parameters.
Key words:
disturbance rejection; auto tuning; permanent magnet synchronous motor; speed modulation system;
1 Introduction
Motor control is a key technology in high-performance servo drives. As a typical nonlinear complex control objective, a PMSM has multivariable, strongly coupled nonlinear and variable parameters that make it difficult to meet the requirements of high- performance control for traditional vector control [1-3].
With the development of nonlinear control theory, various effective control strategies based on vector control have made it possible to realize high- performance control of the PMSM governing system. Most of these strategies utilize inner and outer loop controllers, such as adaptive neural network control [4], fuzzy PID control [5-6], sliding-mode variable-structure control (SMVSC), and active-disturbance-rejection control [7]. Among them, SMVSC has stronger adaptability and robustness on variation of parameters for system. However, the discontinuity caused by switch motion, chattering still exists in the SMVSC system [8]. To solve this problem, in Refs. [9-10], an additional integral part is added to the speed loop of the sliding-mode controller. In Refs. [11-12], an adaptive SMVSC is proposed to improve the gain of the controller, which gives the system the ability to self-regulate mismatches and uncertain disturbances. A novel nonlinear speed control that uses sliding-mode control and disturbance compensation techniques is adopted in Ref. [13]. However, the above methods cannot remove the system chattering caused by load torque mutation and parameter changes in low or ultra-low speed control systems. Hence, to obtain more accurate position control results, a more effective control strategy is needed to solve the speed loop control problem in the PMSM.
ADRC is a novel robust control method developed from nonlinear PID that can automatically detect and compensate for external disturbances independent of control objectives. However, although ADRC has strong adaptability and robustness, there are many adjustable parameters that lead to poor operation [14]. In Refs. [15] and [16], a fuzzy logic control is used to optimally estimate and automatically adjust parameters, which can improve the low-speed control performance of the motor. Furthermore, the speed and torque of the system can be effectively controlled without any parameter turning for the no-manual-tuned active-disturbance-rejection control (NMT-ADRC) developed in Ref. [17]. However, because of the influence of load torque mutation, moment of inertia, friction, the two abovementioned methods have poor anti-interference ability and large overshoot in the high-speed control of an AC servo system, which makes it difficult to meet the requirement of high steady-state accuracy.Combining the advantages of SMVSC and ADRC, a sliding mode-ADRC (SM-ADRC) for the PMSM is proposed in this work. In our method, the features of the original ADRC can be well preserved. Meanwhile, the process of parameter adjustment runs more smoothly. Compared with a traditional ADRC, the improved method has the features of fast response speed, non-overshoot, high static precision, wide speed range, and strong robustness with regard to load and system disturbance.
2 Mathematical model of PMSM
A salient-pole-rotor PMSM (SPMSM) driven by three-phase sinusoidal current is used as the control objective for this system. Assume that Ld=Lq and that there are no damper windings in the rotor. The positional relationships between the coordinate axes are shown in Fig. 1.
Fig. 1 Schematic diagram of coordinate axes
Given the stator current, when the speed is below the basic speed, the state equations of the PMSM in a synchronous coordinate system can be described as [18-19]
(1)
where id and iq are the stator currents of the d- and q-axes components, respectively; ud and uq, respectively are the stator voltages of the d- and q-axes components; ω is the rotor speed; Rs is the stator resistance; L is the rotor inductance of the d-q axes; ψf is the rotor flux of the permanent magnet; TL is the load torque; Te is the electromagnetic torque; J is the rotor inertia; B is the friction coefficient; np is the pole pair; is the angular speed of the rotor.
From Eq. (1), it can be seen that the d-q axes components of the stator current determine the magnitude of the electromagnetic torque. The torque control of the AC PMSM can be achieved by controlling the stator current in the vector control of the PMSM. Since Te changes with iq in the control system, the vector control can be realized by controlling iq.
3 Mathematical model of SM-ADRC
3.1 Mathematical model of ADRC
The ADRC is composed of a tracking differentiator (TD), extended state observer (ESO), and nonlinear states error feedback (NLSEF). In this work, a second- order ADRC is developed by using an active- disturbance-rejection speed loop that combines the speed and current loops [20-22].
A key component of ADRC is its capability to automatically compensate for disturbance estimations. By selecting an appropriate TD, ESO, nonlinear function, and parameters of NLSEF, the state equations of the second-order control objective can be described as follows:
(2)
where u(t) is the system control value; ω is the output of the control objective; b is the gain of control quantity; the known value f(ωz1, ωz2) is the internal disturbance of the system; the unknown variable is the external disturbance of the system. The sum of the internal and external disturbances constitutes the total disturbance of the system.
Second-order ADRCs of PMSM equations are designed as follows:
1) Given an input signal of motor speed ω*, the linear differential-tracker function is given as follows:
(3)
where ωv1 is the tracking signal of ω*; ωv2 is the differential signal of ωv1; the adjustable parameter R is the speed factor. The greater the value of R, the faster the signal tracking.
2) ESO is used to transform a nonlinear uncertain object with an unknown disturbance into an integral tandem-type object. According to the actual output ω and the d- or q-axis voltage u(t), the nonlinear ESO equations can be described as follows:
(4)
where ωz1 is the track signal of system output ω; ωz2 is the differential signal of ωz1; zω3 is the estimation of the unknown disturbance; εω1 is the error signal. The three adjustable parameters β01, β02, and β03 are output- error-correction gains.
3) NLSEF is the nonlinear combination of errors between derivatives of all orders and estimations of state variables, which come from the TD and ESO, respectively. NLSEF and the total disturbance compensation of the ESO together constitute the control volume. The equations of NLSEF are as follows:
(5)
where eω0, eω1, and eω2 are, respectively, the error, its differential and second differential; β1, β2, and β0 are the error, its differential and second differential gains, respectively. From Eq. (5), if the load disturbance of the motor a(t) can be effectively estimated according to the ESO, the effect of the disturbance can be reduced.
The optimal integrated control function fal(·) can be expressed as
(6)
Abstract: A sliding mode and active disturbance rejection control (SM-ADRC) was employed to regulate the speed of a permanent magnet synchronous motor (PMSM).The major advantages of the proposed control scheme are that it can maintain the original features of ADRC and make the parameters of ADRC transition smoothly. The proposed control scheme also ensures speed control accuracy and improves the robustness and anti-load disturbance ability of the system. Moreover, through the analysis of a d-axis current output equation, a novel current-loop SM-ADRC is presented to improve the system’s dynamic performance and inner ability of anti-load disturbance. Results of a simulation and experiments show that the improved sliding-mode ADRC system has the advantages of fast response, small overshoot, small steady-state error, wide speed range and high control accuracy. It shows that the system has strong anti-interference ability to reduce the influence of variations in rotational inertia, load and internal parameters.