Sliding Mode Disturbance Observer-Based Fractional Second-Order Nonsingular Terminal Sliding Mode Control for PMSM Position Regulation System

Abstract

This paper investigates the position regulation problem of permanent magnet synchronous motor (PMSM) subject to parameter uncertainties and external disturbances. A novel fractional second-order nonsingular terminal sliding mode control (F2NTSMC) is proposed and the finite time stability of the closed-loop system is ensured. A sliding mode disturbance observer (SMDO) is developed to estimate and make feedforward compensation for the lumped disturbances of the PMSM system. Moreover, the finite-time convergence of estimation errors can be guaranteed. The control scheme combining F2NTSMC and SMDO can not only improve performance of the closed-loop system and attenuate disturbances, but also reduce chattering effectively. Simulation results show that the proposed control method can obtain satisfactory position tracking performance and strong robustness.

Hong-Ru Li 1 and Zhi-Bin Jiang 1 and Nan Kang 2

Academic Editor:Xing-Gang Yan

1, School of Information Science and Engineering, Northeastern University, Shenyang 110819, China
2, Foreign Studies College, Northeastern University, Shenyang 110819, China

Received 20 May 2014; Revised 14 August 2014; Accepted 18 August 2014; 14 January 2015

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.

1. Introduction

Permanent magnet synchronous motor has been widely applied in industrial automation, household appliances, computers, high-speed aerospace drives, and automobiles due to its superior properties such as high power density, high efficiency, low inertia, and reliable operation [1, 2]. However, the PMSM system is a complex nonlinear system with multiple coupled states and unavoidable and unmeasured disturbances, as well as parameter perturbations. To achieve high-performance control, various advanced control methods have been proposed, such as adaptive control [3], robust control [4], sliding mode control [5, 6], optimal control [7], backstepping control [8], predictive control [9], fuzzy control [10], neural network control [11], finite-time control [12], fractional order control [13, 14], and intelligent control [15]. These methods have increased the dynamic and steady state performance of PMSM systems to some degree. Nevertheless, there still exist several obstacles as to complex control laws, conservative or excessive control gains [16], reliance on complete knowledge of the system model, and so forth.

Sliding mode control (SMC) is a well-known and efficient control technique to improve disturbance rejection and robustness of nonlinear systems and parameter estimation [17], and so forth. When system states are in the sliding mode, the closed-loop response becomes totally insensitive to both internal parameter uncertainties and external disturbances [18]. To further improve the transient performance of the closed loop system and ensure the finite-time convergence, terminal sliding mode control (TSMC) that employs nonlinear sliding surface is developed due to some superior properties such as faster, finite-time convergence and higher control precision [19]. The finite-time stabilization can bring dynamical systems to better robustness and disturbance rejection properties [16].

Zhankui and Sun [20] proposed a second-order fast terminal sliding mode control scheme that can not only guarantee tracking errors in approaching and reaching the sliding surface in finite time, but also improve tracking accuracy and eliminate the high frequency chattering of control inputs effectively. However, the second-order fast terminal sliding mode control still has a singularity problem. In Feng et al. [21] and Yu et al. [22], the nonsingular terminal sliding mode controllers (NTSMC) were designed to achieve finite-time tracking control of systems and overcome the singularity problem. Further, Yan et al. [23] combined NTSMC with second-order SMC to design the second-order NTSMC (2NTSMC) for the finite-time convergence of system states. The 2NTSMC possess fast convergence and high control precision; besides, it can eliminate chattering behavior of control signals. In order to improve the control system robustness, the observer-based control method is often adopted, such as [24].

Fractional order calculus extends integer order to nonintegral order and provide an excellent tool for describing complex dynamic features. Recently, some researchers have proposed some fractional-order SMC methodologies. Dadras and Momeni [19] introduced fractional-order TSMC (FTSMC) to integer-order nonlinear systems. However, the chattering problem of control inputs still exists. Aghababa [25] designed a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems. However, to the authors' best knowledge, the order number of FTSMC is usually restricted to one and there is little work in which the order number of FTSMC is second or higher. Therefore, designing a FTSMC whose order number is greater than 1 for nonlinear dynamic systems is still an open problem.

To improve robustness during the reaching phase of SMC and reduce the conservativeness of selecting switching control gains, a sliding mode disturbance observer (SMDO) is employed to provide feed-forward compensation for parameter uncertainties and external disturbances. Consequently, the closed loop system can achieve global robustness and improve disturbance rejection performance.

In this paper, a new fractional second-order nonsingular terminal sliding mode controller (F2NTSMC) is proposed to ensure fast and finite-time convergence of the PMSM system. Then, a switching control law is determined to drive system states to the designed sliding surface and subsequently constrain system states to the surface hereafter. Meanwhile, the finite-time stability is proved by using fractional Lyapunov theory. Moreover, a SMDO is applied such that uncertainties and disturbance can be estimated and compensated. Eventually, simulation results verify good robustness and fast convergence of the proposed fractional control approach.

The rest of this paper is organized as follows. In Section 2, preliminaries of fractional-order calculus are introduced. In Section 3, the fractional-order PMSM system model and the problem formulation are presented. Section 4 copes with the proposed fractional-order approach and finite-time stability analysis. The effectiveness of the proposed control scheme is illustrated by numerical examples in Section 5. Finally, some concluding remarks are included in Section 6.

2. Preliminaries of Fractional-Order Calculus

Fractional-order integration and differentiation are the generalization of the integer-order ones [19]. Three commonly-used definitions for fractional order calculus are Riemann-Liouville, Caputo, and Grünwald-Letnikov definitions as described below.

Definition 1 (see [26]).

The [figure omitted; refer to PDF] th-order fractional integration of function [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] and the terminal value [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] th-order Riemann-Liouville fractional derivative of function [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is integrable on the closed interval [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is Euler's gamma function.

Property 1 (see [26]).

For the Riemann-Liouville derivative, we have [figure omitted; refer to PDF]

Definition 2 (see [26]).

The Caputo fractional derivative of order [figure omitted; refer to PDF] of a continuous function [figure omitted; refer to PDF] is defined as follows: [figure omitted; refer to PDF]

Definition 3 (see [26]).

The Grünwald-Letnikov fractional derivative of order [figure omitted; refer to PDF] of a continuous function [figure omitted; refer to PDF] is defined as follows: [figure omitted; refer to PDF]

3. Fractional-Order Model of PMSM

Consider a typical PMSM vector control system, as shown in Figure 1. The differential equations of surface-mounted PMSM represented in the rotor reference coordinates based on the assumptions [27] are given as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the stator resistance; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are stator inductances and [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is the rotor flux linkage; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are stator voltages; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are stator currents; [figure omitted; refer to PDF] is the rotor angular velocity; [figure omitted; refer to PDF] is the moment of inertia; [figure omitted; refer to PDF] is the viscous friction coefficient; [figure omitted; refer to PDF] is the number of pole pairs, and [figure omitted; refer to PDF] is the load torque.

Figure 1: A typical PMSM system based on vector control.

[figure omitted; refer to PDF]

In the previous vector control design of PMSM, the [figure omitted; refer to PDF] -axes stator current [figure omitted; refer to PDF] is usually approximately replaced by the [figure omitted; refer to PDF] -axes reference current [figure omitted; refer to PDF] , which degrades the closed-loop system performance. Motivated by the built second-order model of PMSM in [28], we propose a fractional-order model between the [figure omitted; refer to PDF] -axes reference current [figure omitted; refer to PDF] and the position output.

Considering the input and output of the current loop [figure omitted; refer to PDF] in Figure 1, the following Laplace-transform equation can be easily obtained: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the Laplace transformations of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the proportional and integral gains of the [figure omitted; refer to PDF] controller in the current loop [figure omitted; refer to PDF] , respectively; [figure omitted; refer to PDF] is the power of [figure omitted; refer to PDF] in integral actions.

The fractional-order PMSM dynamic equation is derived as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] represents the lumped disturbances including viscous frictions and external load disturbances.

For the convenience of designing the controller, assume that the speed control loop, the current control loop, and the inverter are ideal [29]. Thus, the fractional-order model of PMSM position regulation system is described by [figure omitted; refer to PDF]

The tracking error can be defined as follows: [figure omitted; refer to PDF]

The control objective is to design a F2NTSMC with SMDO to track the reference trajectory [figure omitted; refer to PDF] in finite time. In the next section, the design of F2NTSMC and SMDO will be conducted.

4. Control Design

In this section, a fractional-order nonsingular fast terminal sliding mode controller is proposed to achieve equivalence between fast convergence and nonsingularity. The first step is to develop the F2NTSMC to achieve chattering-free and robust tracking of the position. And then, a sliding mode disturbance observer is designed to estimate and compensate uncertainties and disturbances, which can increase robustness of the control system and improve control performance. Thus, a control scheme with F2NTSMC and SMDO is presented.

4.1. Fractional Second-Order Nonsingular Terminal Sliding Mode Control

To compare the convergence performance between TSMC and NTSMC, the following sliding mode surfaces are considered: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are FTSM surfaces, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are NFTSM surfaces. All their initial states are set as [figure omitted; refer to PDF] . The parameters are given by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . It can be seen from Figure 2 that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have faster convergence rate than [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . For the more detailed comparison between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we consider parameters [figure omitted; refer to PDF] . The parameters of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are tuned with the optimal integrated time absolute error (ITAE) criterion by minimizing the following formula: [figure omitted; refer to PDF]

Convergence conditions of [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The optimal parameters of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] = [figure omitted; refer to PDF] and [figure omitted; refer to PDF] = [figure omitted; refer to PDF] , respectively. The convergence rate of [figure omitted; refer to PDF] is higher than its counterpart of [figure omitted; refer to PDF] .

Therefore, the fractional-order sliding mode surface [figure omitted; refer to PDF] is chosen to be researched in this paper and rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF]

Once the system operates in the sliding manifold, equation [figure omitted; refer to PDF] is satisfied. Since [figure omitted; refer to PDF] holds in (13), it follows that [figure omitted; refer to PDF]

Then, a nonsingular fast terminal sliding surface [29] is selected to ensure [figure omitted; refer to PDF] reach zero in finite time and realize second-order sliding mode control: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are positive odd integers, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Furthermore, the following continuous terminal sliding mode reaching law [29] is introduced to guarantee system states converge to sliding surfaces in finite time and increase system robustness: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are positive odd integers and [figure omitted; refer to PDF] .

Suppose [figure omitted; refer to PDF] is the time when [figure omitted; refer to PDF] reaches zero from [figure omitted; refer to PDF] , that is [figure omitted; refer to PDF] when [figure omitted; refer to PDF] . Solving (17) as [29], the time from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] can be obtained as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is Gauss hypergeometric function, and the Lyapunov function is [figure omitted; refer to PDF] .

Thus, the second-order sliding mode control is achieved. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are driven to reach [figure omitted; refer to PDF] in finite time and then remain on [figure omitted; refer to PDF] to realize the sliding mode motion. Both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] reach zero in finite time [figure omitted; refer to PDF] . After [figure omitted; refer to PDF] reaches zero, the system will stay on the sliding mode motion (13) and the tracking error [figure omitted; refer to PDF] will converge to zero in finite time [figure omitted; refer to PDF] which is calculated in Theorem 5. Next, the second-order sliding mode controller will be derived.

Take the second time derivative of both sides of (13), one obtains [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Differentiating the sliding variable [figure omitted; refer to PDF] , (16) gets [figure omitted; refer to PDF]

Substituting (19) into (20), it follows that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

From (17) and (21), the fractional-order terminal sliding mode controller is designed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes unknown uncertainties and disturbances.

Considering (22), all fractional powers of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive; that is, the control method is nonsingular. After the system state enters the sliding mode [figure omitted; refer to PDF] , there is also no singularity. Therefore, the system is globally nonsingular during both the reaching phase and the sliding phase.

4.2. Sliding Mode Disturbance Observer

In order to increase robustness of the controller and improve control performance, a SMDO is proposed to estimate uncertainties and disturbances.

Equation (21) is rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Select the auxiliary sliding variable as [figure omitted; refer to PDF]

Construct a disturbance observer as follows: [figure omitted; refer to PDF]

By differentiating [figure omitted; refer to PDF] with respect to time, one obtains [figure omitted; refer to PDF]

Then, the auxiliary super-twisting sliding mode control is designed to drive the sliding variable [figure omitted; refer to PDF] to zero: [figure omitted; refer to PDF]

Equation (26) closed by control (27) results in [figure omitted; refer to PDF]

Theorem 4.

Suppose that the uncertainties of the system (28) are globally bounded by [figure omitted; refer to PDF] Then, sliding variable [figure omitted; refer to PDF] of (28) converges in finite time to the origin [figure omitted; refer to PDF] if the gains satisfy the following relations: [figure omitted; refer to PDF] The convergence time [figure omitted; refer to PDF] is upperbounded by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the initial state and [figure omitted; refer to PDF] [30].

Proof.

We propose the following Lyapunov function and its quadratic form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Its time derivative along the solution of (31) is [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Applying (29), it yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

[figure omitted; refer to PDF] is negative definite if [figure omitted; refer to PDF] . It is easy to see that this is the case if the gains are as in (30). The sliding variable [figure omitted; refer to PDF] is driven to zero in finite time [figure omitted; refer to PDF] . Namely, the lumped disturbances [figure omitted; refer to PDF] are exactly estimated by [figure omitted; refer to PDF] in finite time [figure omitted; refer to PDF] . This completes the proof.

Eventually, the fractional-order terminal sliding mode controller with SMDO is designed as [figure omitted; refer to PDF]

In the next section, the stability of the proposed controller will be proved.

4.3. Stability Analysis

The stability analysis consists of two parts. The first part is to prove that position tracking error (10) of the system (9) converges to [figure omitted; refer to PDF] in finite time. The second part is to verify whether or not the proposed approach (37) can ensure the system trajectories (21) converge to the sliding surface [figure omitted; refer to PDF] in finite time and have no singularity.

Theorem 5.

Consider the sliding mode surface (13). The tracking errors converge to the origin [figure omitted; refer to PDF] in finite time [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the reaching time from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive constants.

Proof.

Motivated by [25, 31], assume that the following inequality is valid: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are optional positive constants.

The Lyapunov function is defined as [figure omitted; refer to PDF]

Taking fractional-order derivative of both sides of (39) with respect to time, it yields [31] [figure omitted; refer to PDF]

The value of [figure omitted; refer to PDF] is set as [figure omitted; refer to PDF] . Substituting (15) and inequality (38) into (40), one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is chosen appropriately such that [figure omitted; refer to PDF] is satisfied. Consequently, by Theorem 5, the system error [figure omitted; refer to PDF] will converge to zero asymptotically. Next, the convergence of [figure omitted; refer to PDF] to zero in finite time will be proved.

Taking fractional-order integral of (38) from reaching time [figure omitted; refer to PDF] to stopping time [figure omitted; refer to PDF] , one obtains [figure omitted; refer to PDF]

According to [25], there exists a positive constant [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Noting that [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF]

Solving the inequality (43), it yields [figure omitted; refer to PDF]

Therefore, the tracking error [figure omitted; refer to PDF] (10) will converge to zero in finite time. This completes the proof.

Theorem 6.

Consider the fractional-order PMSM system (9). If the system is controlled by the control input (37), then system states will converge to the sliding surface [figure omitted; refer to PDF] in finite time [figure omitted; refer to PDF] (18).

Proof.

Consider the following Lyapunov function [figure omitted; refer to PDF]

Differentiating [figure omitted; refer to PDF] with respect to time, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Since the sliding variable [figure omitted; refer to PDF] converges to zero only after the lumped disturbances [figure omitted; refer to PDF] is estimated in finite time [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] when [figure omitted; refer to PDF] , therefore, for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Owing to [figure omitted; refer to PDF] , it gets that [figure omitted; refer to PDF] .

Equation (47) belongs to the following type of inequality: [figure omitted; refer to PDF]

Based on [32, Lemma 2], the sliding variables [figure omitted; refer to PDF] will converge to zero in finite time driven by the control input (37). Then, [figure omitted; refer to PDF] reaches zero in finite time [figure omitted; refer to PDF] . Hence, the proof is achieved.

5. Simulation Results

This section presents simulation results to investigate control performance of the proposed method. The parameters of both current loops are set as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The parameter values of the PMSM system are as follows: rated power is [figure omitted; refer to PDF] ; rated voltage is [figure omitted; refer to PDF] ; rated torque is [figure omitted; refer to PDF] ; number of poles is [figure omitted; refer to PDF] ; moment of inertia is [figure omitted; refer to PDF] ; stator resistance is [figure omitted; refer to PDF] ; rotor flux linkage is [figure omitted; refer to PDF] ; viscous damping coefficient is [figure omitted; refer to PDF] ; rated speed is [figure omitted; refer to PDF] , and both stator inductances are [figure omitted; refer to PDF] . The saturation limit of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] .

5.1. Simulation of TSMC and 2NTSMC

In this simulation, the performance of the conventional TSMC method and the 2NTSMC method is simulated for comparison. We only consider the tracking control problem for the nominal model, that is, no parameter uncertainties and external disturbances. The integral-order sliding surface of conventional TSMC is designed as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are set as positive constants and [figure omitted; refer to PDF] .

The constant rate reaching law is chosen as [figure omitted; refer to PDF]

Taking the time derivative on both sides of (49) and substituting (9), (10), and (49) into (50), the conventional sliding mode control input can be obtained as [figure omitted; refer to PDF]

The integral-order sliding surface of 2NTSMC is designed as follows: [figure omitted; refer to PDF]

Taking the second derivatives with respect to time on both sides of (52) and substituting (9), (10), and (52) into (17) and (20), the control input of 2NTSMC can be gotten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The following control parameters are all tuned with the optimal integrated time absolute error (ITAE) criterion.

For the nominal system, the optimal control parameters in (51) are selected as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The optimal integral-order 2NTSMC (I2NTSMC) parameters in (53) are selected as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Simulation results are shown in Figure 3, respectively.

Control performance of ITSMC and I2NTSMC for the nominal system. (a) Position tracking performance, (b) tracking errors, (c) sliding surfaces, (d) control inputs, and (e) current [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

Figure 3 displays time diagrams of position tracking performance, tracking errors, sliding surfaces, control inputs, and current [figure omitted; refer to PDF] , respectively. The PMSM system controlled by both I2NTSM and ITSM controllers is stable. The control input [figure omitted; refer to PDF] is continuous and chattering-free from Figure 3(d). In comparison with the ITSMC, the I2NTSMC can improve the transient performance substantially and offer higher tracking precision as seen in Figures 3(a), 3(b) and 3(c). It is because that I2NTSMC combines advantages of ITSMC and those of second-order SMC, which increases the convergence rate of the PMSM system and tracking precision.

5.2. Simulation of ITSMC and FTSMC

For comparison between ITSMC and FTSMC, we design the following fractional order sliding surface as [figure omitted; refer to PDF]

Taking the time derivative on both sides of (53) and substituting (9), (10), and (55) into (50), the fractional-order sliding mode control input is obtained as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

The simulation results are presented in Figure 4 and demonstrate that the FTSMC has better performance of faster and higher tracking precision. However, the chattering phenomenon remains during the FTSMC process as shown in Figure 4(d). The position tracking performance of FTSMC and ITSMC is given in Figure 4(a). It can be seen from Figures 4(b) and 4(c) that tracking errors of FTSMC reach zero within finite time, but the convergence rate of ITSMC is far too slow. Furthermore, less chattering occurs during the FTSMC process than ITSMC according to Figures 4(d) and 4(e).

Control performance of ITSMC and FTSMC for the nominal system. (a) Position tracking performance, (b) tracking errors, (c) sliding surfaces, (d) control inputs, and (e) current [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

5.3. Simulation of I2NTSMC and F2NTSMC

In order to further accelerate the convergence rate and eliminate chattering effect, we adopt F2NTSMC combining 2NTSMC with FSMC. The control scheme takes advantages of less chattering of 2NTSMC and faster convergence rate, more precise tracking of FSMC.

The optimum parameters of F2NTSMC without [figure omitted; refer to PDF] in (22) are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively.

It is noticed from Figures 5(d) and 5(e) that there is neither singularity nor chattering during the F2NTSMC process. The actual position tracks the desired reference value more quickly than in the cases of I2NTSMC as seen in Figures 5(a), 5(b), and 5(c). To sum up, the F2NTSMC is better than the 2NTSMC.

Control performance of I2NTSMC and F2NTSMC for the nominal system. (a) Position tracking performance, (b) tracking errors, (c) sliding surfaces, (d) control inputs, and (e) current [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

5.4. Simulation of SMDO-Based F2NTSMC

5.4.1. SMDO-Based F2NTSMC for the Model with [figure omitted; refer to PDF]

The block diagram of the proposed SMDO-based F2NTSMC method is shown in Figure 6.

Figure 6: Block diagram of the SMDO-based F2NTSMC system.

[figure omitted; refer to PDF]

The position tracking control of the PMSM system with parameter uncertainties and external disturbances is considered. The optimal parameters of the F2NTSMC in (37) are calculated as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and other parameters are as same as ones of F2NTSMC without [figure omitted; refer to PDF] . The optimal parameters of SMDO are given by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Simulation results are illustrated in Figure 7. The SMDO-based F2NTSMC shows excellent tracking performance depicted in Figure 7(a). The position errors and sliding surface [figure omitted; refer to PDF] converge to the equilibrium points in finite time in spite of lumped disturbances including parameter uncertainties and external disturbances as seen from Figures 7(b) and 7(c), respectively.

SMDO-based F2NTSMC for the system with [figure omitted; refer to PDF] . (a) Position tracking performance, (b) tracking errors, (c) sliding surfaces, (d) control inputs, (e) current [figure omitted; refer to PDF] , and (f) lumped disturbances and their estimation values.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

(e) [figure omitted; refer to PDF]

(f) [figure omitted; refer to PDF]

The tracking performance of the F2NTSMC without SMDO is deteriorated due to too big lumped disturbances. Therefore, SMDO is applied to realize estimation and compensation in order to improve control performance and increase robustness of the system. From Figure 7(f), we can see that lumped disturbances can be effectively estimated. Moreover, Figures 7(d) and 7(e) demonstrate that the chattering phenomenon becomes effectively weakened.

5.4.2. Robustness of SMDO-Based F2NTSMC

In order to verify the robustness of SMDO, the disturbance simulation is carried out. Figure 8 shows the system response as the load disturbances vary from 0 N· m to 2.5 N· m during the time range [figure omitted; refer to PDF] sec and from 0 N· m to -2.5 N· m during the time range [figure omitted; refer to PDF] sec.

Robustness of SMDO-based F2NTSMC for the PMSM system with [figure omitted; refer to PDF] . (a) Position tracking performance, (b) current [figure omitted; refer to PDF] , and (c) the lumped disturbances and their estimation.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

The parameters of SMDO-based F2NTSMC are set as same as ones in Section 5.4.1. Figures 8(a) and 8(b) show the stability and robustness of SMDO-based F2NTSMC method. Figure 8(c) demonstrates that the SMDO can primely track the lumped disturbances.

6. Conclusion

In this paper, a SMDO-based F2NTSMC with strong robustness is developed to solve the position tracking control problem for the PMSM system in spite of parameter uncertainties and external disturbances. Simulation results show that the closed-loop system under the proposed F2NTSMC method has achieved fast convergence and high tracking precision. To further improve the disturbance rejection ability, SMDO is introduced to estimate and make compensation for the lumped disturbances. The combination of F2NTSMC and SMDO can obtain strong robustness and good dynamic performance. Simulation results have demonstrated the effectiveness and superiority of the proposed method.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant no. 61374147).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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