Finite-Time Integral Sliding Mode Control for Motion Control of Permanent-Magnet Linear Motors

Abstract

The finite-time motion control problem of permanent-magnet linear motor (PMLM) is studied in this paper. Firstly, based on finite-time integral sliding mode (FTISM) technique, a finite-time control (FTC) law is proposed such that the PMLM can track the desired trajectory in finite time in the presence of disturbances. Secondly, to alleviate the chattering caused by discontinuous property of the control law, a novel saturation function is introduced to replace the signum function in the proposed FTC law. Finally, the effectiveness of the proposed method is shown by simulation results and comparisons.

1. Introduction

Permanent-magnet linear motor (PMLM) not only has the merits such as high force density achievable, low thermal losses, high precision and accuracy, and simple mechanical structure [1], but also is probably the most natural choice for applications involving high-speed, high-precision motion control among the electric motor drives [2]. Nowadays, PMLM has been widely used in the precision manufacturing industries. Compared with the traditional rotary motors [3–8], linear motors require no indirect coupling mechanisms as in gear boxes, chains, and screws coupling, which greatly reduces the contact type nonlinearities and mechanical disturbances such as friction force and backlash [1, 2, 9]. Therefore, PMLMs meet the increasing demands for high performance servo applications and have been successfully used in machine tools [10], semiconductor manufacturing systems [11], and so forth. Since PMLMs are not equipped with transmission mechanisms, the achievable performance of PMLMs, such as the ability to reduce the effects of uncertainties and external disturbances, is unavoidably lost to some extent. Thus, it is very important to reduce these effects, either through proper mechanical design or via control schemes. The effects may be kept to admissible levels, through the alternative mechanical design approach, but they may be very costly one. It may be an economical feasible method to suppress these effects based on the controller design for PMLM system.

The friction (Coulomb, viscous, and stiction) and ripple forces (detent and reluctance forces) are the two major disturbances that affect the system directly [1]. Friction is a natural phenomenon that arises from the contact between the translator and the track [12]. Moreover, in a closed-loop control system, the friction may result in a steady-state error, a limit cycle, and a low bandwidth. Ripple forces are strong, position-dependent forces arising from the magnetic structure of a PMLM. It comprises of the cogging force and reluctance force, both of which are periodic relationship with respect to the position of the translator relative to the magnets [2]. With the growing interest in improving the precision of position/speed tracking control, more advanced disturbance rejection method is needed in the PMLM control system. To improve the disturbance rejection performance of PMLMs, numerous methods have been presented in the recent literature. Based on sliding mode control and a proportional-integral-based disturbance observer, [13] proposes a control law to guarantee the stability of the tubular linear synchronous motor system regardless of the payload mass. In [11], an Elman neural network controller based on improved particle swarm optimization is developed for a PMLM. In [1], a robust control law which consisted of feedback control law based on integral sliding mode (ISM) method and feedforward compensation part based on adaptive estimation algorithm is proposed. In [2], a new hysteretic relay method is proposed to identify the force ripples and friction in PMLM system, based on which a composite control approach which is consisted of a PID control law with a feedforward compensation term is proposed. Reference [9] proposed a state-periodic adaptive compensation approach, which uses one trajectory period past information along the state axis to update the current adaptation law for PMLM system. In [14], to achieve high control performance, a field-programmable gate array based intelligent complementary sliding mode control is proposed for the PMLM system such that the mover can track periodic reference trajectories. To solve the precision control problem of PMLM motion control system in the presence of significant disturbance, a robust control scheme based on the disturbance observer (DOB) is proposed to reduce the sensitivity of the control performance to disturbance [15]. To provide adequate disturbance rejection performance and overcome the stability problem as well as improve the motion control accuracy, [16] proposes a control approach which combines feedback feedforward controller and a digital DOB for a PMLM drive system. All the above mentioned control methods [1, 2, 9, 13–16] can improve disturbance rejection performance of PMLMs in some extent. However, most of the existing results [1, 2, 9, 14] about PMLMs motion control laws are asymptotically stable ones. This implies that the convergence rate of the system states is at best exponential. In other words, the desired position and velocity cannot be tracked in a finite time. Obviously, the control laws with finite-time convergence are more desirable. Besides faster convergence rates, the closed-loop systems under finite-time control (FTC) laws usually demonstrate higher accuracy and better disturbance rejection properties [17–21]. Due to the above superiorities, FTC technique has been used in robotic manipulator systems [18, 20], inverted pendulum systems [21], spacecraft systems [22], PMSM systems [23], and multiagent systems [24, 25], and so forth.

In this paper, the motion control problem of PMLM is considered in the presence of friction and ripple forces. By introducing a finite-time integral sliding mode (FTISM) manifolds, a FTISM control law is proposed for PMLM system, which can guarantee that the desired trajectories can be tracked in finite time. However, the proposed FTISM control law has a discontinuous term which leads to chattering. Inspired by traditional boundary layer method, a novel saturation function is introduced to replace the signum function in the proposed discontinuous control law. Analysis indicates that under the proposed continuous FTISM control law, the trajectories of the system will converge to a much smaller region than that of under the traditional boundary method; that is, compared with the traditional boundary layer approach, the closed-loop system under the proposed continuous FTISM control law has much smaller steady-state errors. The major merit of the proposed method is that it can not only effectively alleviate chattering, but also maintain relatively good disturbance rejection performance. All the results are supported by simulations.

2. Problem Statement and Preliminaries

The mathematical model of a PMLM can be approximately described as follows [9]: 𝑥̈(𝑡)𝑘=−^𝑓𝑘_𝑒𝑥̇(𝑡)−1𝑅𝑚𝑚𝐹_fric−1𝑚𝐹_ripple+𝑘_𝑓𝑢(𝑡),𝑅𝑚 (1) where𝑢(𝑡)is time-varying motor terminal voltage,𝑥(𝑡)is the motor position,𝑅is the resistance,𝑚is the moving thrust block mass,𝑘_𝑓is the force constant,𝑘_𝑒is the back electromotive force, and𝐹_fricand𝐹_rippledenote the friction and ripple force, respectively.

The friction force affecting the movement of the translator may be modeled as a combination of Coulomb friction, viscous friction, and the component due to Stribeck effect, which can be interpreted as stiction. The friction force is modeled as 𝐹_fric=𝑓_𝑐+𝑓_𝑠−𝑓_𝑐𝑒^{̇̇𝑥−(𝑥/𝑠)2}+𝑓_𝑣𝑥̇sign(𝑥̇), (2) where𝑓_𝑐is the Coulomb friction coefficient,𝑓_𝑠is static friction coefficient,𝑓_𝑣is the viscous friction coefficient, and𝑥̇_𝑠is the lubricant parameter. The ripple force can be described by a sinusoidal function of the load position. In this paper, we assume that the ripple force could be any kind of Fourier expansion such as 𝐹_ripple=^∞_𝑖=1𝐴_𝑖sin𝜔_𝑖𝑥+𝜑_𝑖, (3) where𝐴_𝑖is the amplitude,𝜔_𝑖is the state-dependent ripple force frequency, and𝜑_𝑖is the phase angle [9].

Let𝑎=−𝑘_𝑓𝑘_𝑒/(𝑅𝑚),𝑏=𝑘_𝑓/(𝑅𝑚), and𝑑(𝑡)=𝐹_fric+𝐹_rippledenote lumped disturbance, and let𝑥₁(𝑡)=𝑥(𝑡)and𝑥̇(𝑡)=𝑥₂(𝑡)denote position and velocity, respectively, then we have 𝑥̇₁(𝑡)=𝑥₂(𝑡)𝑥̇₂(𝑡)=𝑎𝑥₂(𝑡)(𝑡)−𝑑(𝑡)+𝑏𝑢𝑚. (4)

Define the tracking errors 𝑒₁(𝑡)=𝑥₁(𝑡)−𝑥_𝑑(𝑡),𝑒₂(𝑡)=𝑥₂(𝑡)−𝑥̇_𝑑(𝑡), (5) where𝑥_𝑑(𝑡)and𝑥̇_𝑑(𝑡)are the desired position and velocity, respectively. The control objective is to track the given desired position𝑥_𝑑(𝑡)and the corresponding desired velocity𝑥̇_𝑑(𝑡)in finite time, that is, to design an FTC law for the following error system: 𝑒₁̇(𝑡)=𝑒₂(𝑡)𝑒₂̇(𝑡)=𝑎𝑒₂(𝑡)(𝑡)−𝑑(𝑡)+𝑏𝑢𝑚+𝑎𝑥̇_𝑑(𝑡)−𝑥̈_𝑑(𝑡) (6) such that𝑒₁(𝑡)→0and𝑒₂(𝑡)→0in finite time.

Next, we review the concept of finite-time stability and several lemmas, which will serve as basis for this study.

Definition 1 (see [19]). Consider system 𝑥̇=𝑓(𝑥),𝑓(0)=0,𝑥∈𝑅^𝑛, (7) where𝑓(⋅)∶𝐷→𝑅^𝑛is nonLipschitz continuous on a neighborhood𝐷of the origin𝑥=0in𝑅^𝑛. The equilibrium𝑥=0of (7) is (locally) finite-time stable if there are an open neighborhood𝑈of the origin and a function𝑇∶𝑈{0}→[0,∞), such that every solution trajectory𝑥(𝑡,𝑥₀)of system (7) starting from the initial point𝑥₀∈𝑈{0}is well defined and unique in forward time for𝑡∈[0,𝑇(𝑥₀)) andlim_{𝑡→𝑇(𝑥0)}𝑥(𝑡,𝑥₀)=0. Here𝑇(𝑥₀)is called the convergence time (of the initial state𝑥₀). The equilibrium of system (7) is finite-time stable if it is Lyapunov stable and finite-time convergent. If𝑈=𝐷=𝑅^𝑛, the origin is a globally finite-time stable equilibrium.

Lemma 2 (see [17]). Assume that there exists a continuous function𝑉(𝑥)∶𝑈→𝑅such that the following conditions hold.(i)𝑉(𝑥)is positive definite. (ii)There exist real numbers𝑐>0and𝛼∈(0,1) and an open neighborhood𝑈₀⊂𝑈of the origin such that𝑉̇(𝑥)+𝑐𝑉^𝛼(𝑥)≤0, 𝑥∈𝑈₀⧵{0}.
Then the origin is a finite-time stable equilibrium of system (7), and the finite convergence time𝑇satisfies𝑇≤𝑉(𝑥(0))^1−𝛼/𝑐(1−𝛼). If𝑈=𝑈₀=𝑅^𝑛, the origin is a globally finite-time stable equilibrium of (7).

Lemma 3 (see [26]). The origin of system 𝑥̇=𝑦,𝑦̇=𝑢 (8) is a globally finite-time-stable equilibrium under the feedback control law 𝑢=−𝑘₁sig^𝛼1(𝑥)−𝑘₂sig^𝛼2𝑦, (9) where𝑘₁,𝑘₂>0,0<𝛼₁<1 and 𝛼₂=2𝛼₁/(1+𝛼₁).

3. Main Results

Assumption 4. Assume that both the friction force𝐹_fricand the ripple force𝐹_rippleare bounded, and there exists a positive constant scalar𝑙such that|𝐹_fric+𝐹_ripple|≤𝑙, that is, |𝑑(𝑡)|≤𝑙.
For system (6), a finite-time integral sliding mode is designed as [21] 𝑠=𝑒₂+^𝑡₀𝑘₁sign𝑒₁𝑒₁^𝛼1+𝑘₂sign𝑒₂𝑒₂^𝛼2𝑑𝜏, (10) where𝑘₁,𝑘₂>0,𝛼₁>0, and𝛼₂=2𝛼₁/(1+𝛼₁).

Theorem 5. If system (6) satisfies Assumption 4, then under the control law 𝑢(𝑡)1=−𝑏𝑘₁sign𝑒₁𝑒₁^𝛼1+𝑘₂sign𝑒₂𝑒₂^𝛼2+𝑎𝑒₂+𝑎𝑥̇_𝑑−𝑥̈_𝑑+𝜂sign(𝑠), (11) with 𝑠 defined in (10), and𝜂>𝑙/𝑚, and the states of system (6) will be stabilized to the origin in finite time; that is, 𝑒₁→0, 𝑒₂→0 in finite time.

Proof. Consider the Lyapunov function 1𝑉=2𝑠². (12) By differentiating𝑉with respect to time𝑡, we have 𝑉̇=𝑠𝑠̇=𝑠𝑎𝑒₂𝑑(𝑡)+𝑏𝑢−𝑚+𝑎𝑥̇_𝑑−𝑥̈_𝑑+𝑘₁sign𝑒₁𝑒₁^𝛼1+𝑘₂sign𝑒₂𝑒₂^𝛼2. (13) Substituting (11) into (13) yields 𝑉̇=𝑠𝑠̇=−𝑑(𝑡)𝑚(𝑠)𝑙−𝜂sign𝑠≤−𝜂−𝑚|𝑠|=−√2𝜂−𝑙𝑉^1/2. (14) By Lemma 2, (14) implies that the FTISM𝑠=0can be reached in finite time.
As𝑠=0, we have 𝑒₂=−^𝑡₀𝑘₁sign𝑒₁𝑒₁^𝛼1+𝑘₂sign𝑒₂𝑒₂^𝛼2𝑑𝜏, (15) that is, 𝑒₁̈=−𝑘₁sign𝑒₁𝑒₁^𝛼1−𝑘₂sign𝑒₂𝑒₂^𝛼2. (16)
According to Lemma 3, the states of system (6) will converge to zero in finite time. This completes the proof.

Remark 6. The proposed control law (11) can not only suppress the lumped disturbance (friction and ripple forces) effectively, but also guarantee that the states of closed-loop error system is stabilized to origin in finite time. However, the signum function in control law (11) leads to undesired chattering of the closed-loop system. In order to eliminate chattering, the signum function (𝑠)=𝑠sign|𝑠|,|𝑠||𝑠|≠0,0,=0 (17) can be replaced by the saturation function 𝑠sat𝜖=𝑠|𝑠|,|𝑠|𝑠≥𝜖,𝜖,|𝑠|<𝜖, (18) where𝜖>0represents the width of the boundary layer [20], according to the conventional method. In this manner, we obtain a continuous control law that can be expressed as 𝑢(𝑡)1=−𝑏𝑘₁sign𝑒₁𝑒₁^𝛼1+𝑘₂sign𝑒₂𝑒₂^𝛼2+𝑎𝑒₂+𝑎𝑥̇_𝑑−𝑥̈_𝑑𝑠+𝜂sat𝜖, (19) which can guarantee the finite time reachability to the region|𝑠|≤Δ₁≜𝑙𝜖/(𝑚𝜂)<𝜖(as𝜂>𝑙/𝑚).

Next, we introduce a class of novel nonlinear saturation functions as follows: sat_𝛼(𝑥)=(𝑥),|𝑥|(𝑥𝑥|sign>1,sign)|^𝛼,|𝑥|≤1, (20) where0≤𝛼≤1. It is easy to show that sat_𝛼(𝑥)=(𝑥)(𝑥)sign,𝛼=0sat,𝛼=1. (21) In the case of𝑥∈{𝑥∣0<|𝑥|<1}and0<𝛼<1, we have|sat_𝛼(𝑥)|>|sat(𝑥)|, andsat_𝛼(𝑥)=sat(𝑥)as|𝑥|=1or𝑥=0. In order to facilitate comprehension, the standard saturation functionsat(𝑥), the classical signum functionsign(𝑥), and the novel saturation functionsat_𝛼(𝑥)with two different𝛼(1/3 and 2/3) are shown in Figure 1. It can be observed from Figure 1 that|sat_𝛼(𝑥)|>|sat(𝑥)|as𝑥∈{𝑥∣0<|𝑥|<1} andsat_𝛼(𝑥)=sat(𝑥)as|𝑥|≥1or𝑥=0, which verifies the validity of the above analysis.

Figure 1 

Classical signum functionssign(𝑥), standard saturation functionssat(𝑥)andsat𝛼(𝑥)with different𝛼.

Theorem 7. If system (6) satisfies Assumption 4, then under the control law 𝑢(𝑡)1=−𝑏𝑘₁sign𝑒₁𝑒₁^𝛼1+𝑘₂sign𝑒₂𝑒₂^𝛼2+𝑎𝑒₂+𝑎𝑥̇_𝑑−𝑥̈_𝑑+𝜂sat_𝛼𝑠𝜖, (22) with𝑠defined in (10),𝜂>𝑙/𝑚, and sat_𝛼𝑠𝜖=𝑠sign𝜖,|𝑠|𝑠≥𝜖,sign𝜖𝑠𝜖^𝛼,|𝑠|<𝜖 (23) with0<𝛼<1and𝜖>0, and the trajectory of system (6) will converge to the neighborhood of FTISM𝑠=0as |𝑠|≤Δ₂,Δ₂=𝑙𝑚𝜂^1/𝛼𝜖 (24) in finite time.

Proof. Substituting control law (22) into (13) yields 𝑉̇=𝑠𝑠̇=𝑑(𝑡)−𝜂sat_𝛼𝑠𝜖𝑠=𝑑(𝑡)𝑠−𝜂sign𝜖𝑠𝑑(𝑡)𝑠−𝜂sign𝜖𝑠𝜖^𝛼≤−𝑙𝑠,𝜂−𝑚|𝑠|−𝜂|𝑠|^𝛼𝜖^𝛼−𝑙𝑚|𝑠|=−√2𝑙𝜂−𝑚𝑉^1/2,|𝑠|−√≥𝜖2𝜂|𝑠|^𝛼𝜖^𝛼−𝑙𝑚𝑉^1/2,|𝑠|<𝜖. (25) By Lemma 2, we can have that the regionΔ₂=(𝑙/𝑚𝜂)^1/𝛼𝜖can be reached in finite time. This completes the proof.

Remark 8. One may argue that through adjusting control gains𝜂to be large enough, both convergence regions, that is, Δ₁ and Δ₂can be rendered to be as small as desired, which means that the proposed novel saturation functionsat_𝛼(⋅)does not have any prominent advantage over the standard saturation functionsat(⋅)disturbance rejection performance. However, high-gain feedback control system often exhibits instability in the actual operation. From the considerations as well as control saturation constraints,𝜂cannot be selected to be sufficiently large. In this case for the control law (22) has an additional parameter𝛼that can be adjusted to enhance the disturbance rejection performance without obvious increase in the control input. In fact,0<𝑙/(𝑚𝜂)<1, we can select𝛼to approximate to0such that1/𝛼in (24) is sufficiently larger than 1, which means(𝑙/(𝑚𝜂))^1/𝛼𝜖≪(𝑙/(𝑚𝜂))𝜖. That is,Δ₂can be designed to be much smaller thanΔ₁, which implies that we can regulate the parameters𝛼and𝜂such that under the control law (22), the closed-loop system can have smaller steady-state errors compared to that under the control law (19).

Remark 9. Obviously, if𝛼₁=1in the proposed FTISM (10), then it degenerates to the traditional linear integral sliding mode (LISM), and the FTISM control law (11) becomes 𝑢(𝑡)1=−𝑏𝑘₁𝑒₁+𝑘₂𝑒₂+𝑎𝑒₂+𝑎𝑥̇_𝑑−𝑥̈_𝑑+𝜂sign(𝑠). (26) It is easy to prove that under the control law (26), the states of system (6) will be stabilized to the origin asymptotically. Similarly, in order to alleviate chattering of the LISM control law (26), we take saturation functions (18) and (23) to replace the signum function in the LISM control law (26), respectively. We obtain two continuous LISM control law: 𝑢(𝑡)1=−𝑏𝑘₁𝑒₁+𝑘₂𝑒₂+𝑎𝑒₂+𝑎𝑥̇_𝑑−𝑥̈_𝑑𝑠+𝜂sat𝜖,𝑢(𝑡)1=−𝑏𝑘₁𝑒₁+𝑘₂𝑒₂+𝑎𝑒₂+𝑎𝑥̇_𝑑−𝑥̈_𝑑+𝜂sat_𝛼𝑠𝜖 (27)(28) for system (6). And under continuous control laws (27) and (28), the states of the system (6) will converge to the small regionsΔ₁andΔ₂, respectively, in finite time.

4. Simulations

In this section, we take a PMLM as target to illustrate the effectiveness of the proposed methods. The model parameters of PMLM are given as𝑚=5.4 kg,𝑅=16.8 ohms,𝑘_𝑓=130 N/A, and𝑘_𝑒=123 V/m/s, and the same values are used in [9]. Among the friction forces, the viscous force has the greatest impact to PMLM control system, which is proportional to the speed in the motion process [27], and the parameters for the friction are taken as𝑓_𝑐=10 N, 𝑓_𝑠=20, and𝑓_𝑣=10 N. To represent the high order Fourier expansion, the ripple force is chosen as 𝐹_ripple=𝐴₁sin(𝜔𝑥)+𝐴₂sin(3𝜔𝑥)+𝐴₃sin(5𝜔𝑥), (29) where𝐴₁=8.5, 𝜔=314 rad/m, 𝐴₂=4.25, and𝐴₃=2.0. The following reference trajectory is used [9]: 𝑥_𝑑(𝑡)𝑡=0.25sin2𝜋𝑇_𝑑−𝜋2+0.25 (30) with𝑇_𝑑=4 s.

For convenience, in the simulation abbreviations FTISM1, FTISM2, LISM1 and LISM2 denote control laws (19), (22), (27), and (28), respectively.

Consider the initial states of system (4) as(𝑥₁(0),𝑥₂(0))=(0,0). The control parameters of all the four control laws (19), (22), (27), and (28) are listed in Table 1. The desired position and velocity are shown in Figure 2. The position and velocity tracking error curves under control laws (19), (22), (27), and (28) are shown in Figures 3 and 4, respectively. In this simulation, the switching gain𝜂and width of the boundary layer 𝜖 for all the four control laws FTISM1, FTISM2, LISM1, and LISM2 are the same. However, as clearly shown in the simulation results, the proposed continuous FTISM control laws FTISM1 and FTISM2 can achieve better control performance than that of the continuous LISM control laws LISM1 and LISM2, and the control laws with novel saturation function FTISM2 and LISM2 can obtain much smaller tracking errors than that with standard saturation function FTISM1 and LISM1. Under the control law FTISM2, the system not only has the best disturbance rejection performance, but also has the fastest convergence rate, which is an evidence as stated in Remark 8. The input signals of control laws FTISM1, FTISM2, LISM1, and LISM2 are shown in Figure 5. It can be observed from Figure 5 that all of the four control laws FTISM1, FTISM2, LISM1, and LISM2 can alleviate chattering effectively, and the amplitudes of the control laws FTISM1, FTISM2 are smaller than that of the control laws LISM1 and LISM2.

Table 1 

Control parameters for the simulation.

(a)

(b)

Figure 2 

The desired position (a) and velocity (b).

Figure 3 

Position tracking errors under control laws (19), (22), (27), and (28).

Figure 4 

Velocity tracking errors under control laws (19), (22), (27), and (28).

Figure 5 

Input signals under control laws (19), (22), (27), and (28).

5. Conclusions

In this paper, a finite-time integral sliding mode (FTISM) control method has been proposed to study the motion control problem of PMLMs. Under the proposed FTISM control law, the desired trajectory can be tracked in finite time in the presence of friction and ripple forces. Note that the signum function could lead to chattering. To alleviate chattering, a novel saturation function has been introduced to replace the signum function in the proposed FTISM control law. Both theoretical analysis and simulation results show that under the proposed FTISM control law with the novel saturation function, the trajectories of the system have better control performance (convergence rate and disturbance rejection ability) than that of FTISM control law with the standard saturation function. In summary, the major merit of the proposed method is that it can not only effectively alleviate chattering, but also maintain relatively good disturbance rejection performance.

Acknowledgment

This work is supported by the Natural Science Foundation of Jiangsu Province, China (Grant no. BK2011089).