1096 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010 Function-Based Controller for Linear Motor Control Systems Yi-Sheng Huang, Member, IEEE, and Cheng-Chung Sung, Student Member, IEEE Abstract—This paper presents a new methodology for designing and implementing position control for permanent-magnet linear synchronous motor (PMLSM) systems. We utilize both a new function-based sliding-mode control (SMC) method and direct thrust control (DTC). It has been established that chattering is a problem in conventional SMC and stems from discontinuous control. However, the new function-based controller allows designers to employ fuzzy membership functions directly. The advantage of the proposed method is that the chattering phenomenon can be reduced and/or completely eliminated. The design of the control method for the proposed system can be performed without the need for great expertise as it does not require the use of very complicated techniques. It should be noted that we have managed to control the inherent flux linkage problems associated with PMLSMs. To our knowledge, this is the first work that combines the new SMC with DTC techniques in order to create a position control for a real-world PMLSM. Simulation and experimental results verify that the proposed method can achieve favorable control performance with regard to parameter variations and external disturbances. Index Terms—Direct thrust control (DTC), flux linkage estimation, permanent-magnet linear synchronous motor (PMLSM), sliding-mode control (SMC). I. I NTRODUCTION INEAR MOTORS drive equipment directly and do not need any mechanical couplings such as belts, gears, screws, or crankshafts. Recently, permanent-magnet linear synchronous motors (PMLSMs) have been widely employed in industrial applications involving motor drives. Examples of such applications include linear transmissions or precision servo control [1]–[3]. It is well known that PMLSM control methods can be divided into two categories: scalar and vector controls. In scalar control, only the magnitude and frequency of voltage, the current, and the flux linkage space vector are controlled [4]. In vector control, not only the magnitude and frequency but also the instantaneous position of voltage, the current, and the flux space vector are controlled [4]. Vector control is a general L Manuscript received December, 13, 2008; revised March 27, 2009 and June 22, 2009. First published August 7, 2009; current version published February 10, 2010. This work was supported in part by the National Science Council, Taiwan, under Grant NSC 98-2221-E-606-007. Y.-S. Huang is with the Department of Electrical and Electronic Engineering, Chung Cheng Institute of Technology, National Defense University, Bade City 33448, Taiwan (e-mail: yshuang@ndu.edu.tw; huang.ccit@gmail.com). C.-C. Sung is with the Department of Electrical and Electronic Engineering, Chung Cheng Institute of Technology, National Defense University, Bade City 33448, Taiwan, and also with the Department of Computer Science and Communication Engineering, Army Academy, Jungli 320, Taiwan (e-mail: song4855@gmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2009.2028290 control philosophy that can be implemented in many different ways, the most popular being field-oriented control [5], [6]. However, these control methods are dependent on and are sensitive to changes in the parameters of the motor. Furthermore, such control methods require an inner current loop to serve as a current controller and involve complicated rotating-frame transformations. To solve this problem, direct thrust control (DTC), which is based on the decoupled control of thrust and flux [7]–[9], is often used. The switching-table-based DTC system has the following advantages: 1) It does not require current control loops; therefore, the current need not be regulated directly; 2) complicated rotating-frame transformations are not required; 3) it has a simple and robust control structure; and 4) it provides excellent thrust dynamics [10]–[12]. However, few studies have implemented DTC systems in PMLSMs to this point. This may be because it is difficult to obtain the amplitude and angular position of the actual flux linkage. Therefore, accurate flux estimation is an important issue that must be considered if we are to ensure proper drive operation and stability. Generally, the motor flux is estimated based on the variation of the two types of estimators (i.e., the voltage and current models) [13], [14]. In the current-based method, the air-gap flux of a motor is identified by solving a set of equations. In this paper, the measured motor currents and the speed/position of the motor are required. The main drawback of this method is that the parameters may change when the rotational speed is changed. In the voltage-based method, the motor flux can be obtained directly from its back electromotive force (emf) (i.e., using an integrator). Therefore, the voltage-based method seems to be preferable. The low-pass (LP) estimator is normally used as a pure integrator to avoid integration drift problems [15], [16]. However, phase and magnitude errors are inherent in LP filters (LPFs). The drawback is increasing the estimated error as the motor control system works at a frequency that is close to the cutoff frequency. To overcome this drawback, a previous study [17] proposed an LP estimator with a modified integrator. Uncertainly, the technical skill in the implementation of PMLSMs does not seem to be performed yet. In this paper, we attempt to solve the flux estimation problem in a real PMLSM. Therefore, we propose the use of a modified integrator to obtain both the actual flux and the actual thrust. In this manner, a closed-loop control system can be realized. By using our control method, a high-performance motor drive can be obtained. The conventional sliding-mode control (SMC) approach is based on exactly maintaining system trajectory within a properly chosen constraint by means of high-frequency control switching [18]–[20]. This approach exploits the main features 0278-0046/$26.00 © 2010 IEEE HUANG AND SUNG: FUNCTION-BASED CONTROLLER FOR LINEAR MOTOR CONTROL SYSTEMS of the SMC method: insensitivity to external and internal disturbances, accuracy, and finite-time convergence of the state trajectory to the sliding surface. Unfortunately, the resultant chattering, caused mainly by the discontinuous control, is one of the drawbacks of the conventional SMC approach. It should also be noted that high-frequency control switching may also cause chattering. This is a well-known problem that has been widely investigated in the field. A variety of schemes have been suggested in order to obtain a smoothing of the bang–bangtype discontinuities of the SMC. To overcome this problem, two main research methods have been proposed. One employs the introduction of a boundary layer to reduce the effect [19], [20]. A saturation element is used in the high-gain control to approximate the function of the sign function in a boundary layer around the sliding surface. The other method uses a fuzzy SMC (FSMC) to adjust the control gain [21]–[23]. However, the FSMC method must deal with the two problems by establishing fuzzy rules and determining membership functions. Although the aforementioned studies have dealt with issues related to motor control, they fail to address chattering. Hence, we propose a new function-based SMC method to resolve this problem. The design of control rules for the proposed system can be performed without the need for great expertise as it does not require the use of very complicated techniques. In addition, the new method allows for the location of a continuous fuzzy membership function. As a result, chattering is reduced and/or eliminated by using our control method. The remainder of this paper is organized as follows. A modified flux linkage estimation method is described in Section II, the new function-based sliding mode controller is discussed in Section III, the simulation and experimental results are dealt with in Section IV, and conclusions are drawn in Section V. II. DTC FOR PMLSMs 1097 TABLE I R EQUIRED PARAMETERS OF A PMLSM Fig. 1. PMLSM control block diagram using a DTC system. For convenience, the necessary parameters of the PMLSM are listed in Table I. A. Mathematical Model of the PMLSM The PMLSM used in this paper comprises a long stationary tubular secondary that is supported at both ends and houses a sequence of neodymium–iron–boron (NdFeB) permanent magnets with a guidance rail and linear scale and a full-stroke (0–30 cm) mover called the primary, which contains the core armature winding and Hall sensing elements. The machine model of the PMLSM can be described in the synchronous rotating reference frame as follows [5], [12]: ud = Rs id + pλd − ωe λq (1) uq = Rs iq + pλq + ωe λd (2) where p denotes the differential operator λd = Ld id + λPM (3) λq = Lq iq (4) ωd = np ωr (5) ωr = πν/τ (6) νe = np ν = 2τ fe . (7) B. DTC System Using Space Voltage Vectors Fig. 1 shows a basic DTC scheme for the PMLSM control system. The two input signals are the thrust command (Tcmd ) and the flux linkage command (λcmd ). The two input commands (Tcmd and λcmd ) should be respectively compared with the actual values (Tm and |Fm |). Note that the two errors (Terr and λerr ) can be considered input signals of the hysteresis controllers (threshold values ρ1 and ρ2 ). Furthermore, the optimum voltage vector should be generated from the switching table (Table II) by Sθ (the flux linkage position), SF , and Sλ . In DTC systems, the selection of an optimum voltage vector is very important. The voltage vector plane is divided into six regions. In this paper, six effective voltage vectors, i.e., V1 (100), V2 (110), V3 (010), V4 (011), V5 (001), and V6 (101), are defined. We also define the two zero voltage vectors V0 (000) and V7 (111). The nonzero vectors are 60◦ apart from each other. Furthermore, the voltage vectors have different effects on the flux linkage and thrust while the motor is at different positions. Detailed information relating to the voltage vectors is shown in Fig. 2. It should be noted that any two adjacent voltage vectors 1098 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010 TABLE II S WITCHING TABLE Fig. 3. Voltage-based modified integrator block diagram. where em is the back emf. The flux linkage amplitude and angle can then be obtained by the following:  2 + F2 (10) |Fm | = FD Q   FQ Fθ = tan−1 (11) FD where |Fm | and Fθ are the flux linkage amplitude and angular position, respectively. Generally, the developed electromagnetic thrust can be described as Tm = Fig. 2. Movement of the flux linkage and voltage vectors. differ in only 1 b. Therefore, the switching table can provide an increase/decrease in voltage by changing only 1 b. This shows that the voltage vector is changed with the minimum switching frequency [9]. As a result, our motor control system provides fast thrust response. C. Modified Flux Linkage Estimation Method It is well known that the performance of a DTC system strongly depends upon the accuracy of the estimation of the actual flux and thrust. Consequently, the following problems are often associated with the DTC system of a PMLSM [24]: There may be 1) a drift in the estimation of the flux linkage due to offset error in measurement or 2) an error in the estimation of the flux linkage due to variations in the mover resistance. Indeed, flux estimation is important for the implementation of a high-performance motor drive. In the voltage-based method, the mover flux can be directly obtained from its back emf via an integrator. The voltage equation of a PMLSM in the stationary reference frame can be expressed as follows: um = Rm im + dFm dt (8) where Rm is the mover resistance and um = [uD uQ ]T , im = [iD iQ ]T , and Fm = [FD FQ ]T are the mover flux linkage vectors based on the voltage equation. In a DTC scheme, the flux linkage is estimated by integrating the difference between the input voltage and the voltage drop across the mover resistance. As a result, (8) can be rewritten as   (9) Fm = (um − Rm im )dt = em dt 3π kF np (FD iQ − FQ iD ) 2τ (12) where Tm is the output electromagnet thrust, np is the number of pole pairs, and kF is the end-effect correction coefficient for the linear motor thrust. In DTC systems, an accurate prediction of the actual thrust is very important because the control performance of the PMLSM depends upon this estimation. The end effect can be considered a special phenomenon because the linear mover is limited in its stroke. Note that the thrust correction coefficient of the linear motor is defined as a constant (0.9) by [9]. In the estimations of the flux and thrust, the integration of (9) and (12) is the most important factor. If the motor operates in the high-speed region, the voltage drop across the winding resistance can be neglected because um ≫ Rm im . Hence, a value for flux linkage Fm can be obtained from um via an integrator. However, implementing an integrator for motor flux estimation is a difficult task. In the voltage-based method, an LP estimator is normally used in place of a pure integrator to avoid the problem of integration drift. Nevertheless, it is still difficult to implement an integrator for motor flux estimation. Previous studies [15] and [16] have discussed these issues. To overcome these obstacles, a voltage-based flux estimator was proposed in a previous study [17]. However, this does not seem to solve the problem completely. This paper proposes a modified flux linkage estimator based on [17]. However, our estimator has a correction coefficient (kϕ ) in the flux estimation algorithm. A new modified block diagram is shown in Fig. 3. The estimator structure consists of an LPF and a signal feedback block. Note that the input signals of the estimator are a train of voltage pulses. These pulses are combined with a sine wave that has a fundamental frequency and higher order harmonics. From waveform theory, the input signals can be expressed as a Fourier series   4A0 1 1 (13) sin ω0 t + sin 3ω0 t + sin 5ω0 t + · · · π 3 5 where A0 is the amplitude value and ω0 is the fundamental frequency. The output signals are sine waves having only HUANG AND SUNG: FUNCTION-BASED CONTROLLER FOR LINEAR MOTOR CONTROL SYSTEMS 1099 our modified flux linkage estimator. Based on the experimental results, it can be seen that the new angular position (Fθ ) is more linear and that the new flux linkage (FD ) is less distorted than in the conventional method [17]. Moreover, the position tracking performance of the new method is better than that of the old one. As a result, we can posit that the overall performance of our control method is better than that of the previous study [17]. III. D ESIGN OF A N EW F UNCTION -BASED S LIDING -M ODE C ONTROLLER FOR PMLSMs It is well known that the conventional SMC is robust in the presence of matched uncertainties and that the desired position is perfectly tracked [25]. In this section, the design procedure for matching conditions in a real-world linear motor control system is introduced. Definition 1: Matching Conditions [25]: Consider the system ẋ = Ax + Bu + f (t, x) (14) where x ∈ Rn is the state vector, A ∈ Rn×n , B ∈ Rn×m , and u ∈ Rm×1 are the control inputs, and f (t, x) ∈ Rn×1 is the disturbance term. If there exists g(t, x) ∈ Rn×1 such that f (t, x) = Bg(t, x), n > m and B have the full rank and (14) meets the demands of the matching conditions. The dynamics of the PMLSM can be represented by the following second-order differential equation: Fe = M ẍp + Dẋp + FL Fig. 4. (a) Performance of the flux estimation algorithm [17]. (b) Performance of the new flux estimation algorithm. the fundamental frequency. The proposed estimator plays the role of an LPF and rejects the higher order harmonic components. Hence, the cutoff frequency of the LPF is chosen as ω0 < ωc < 2ω0 (Fig. 3). According to waveform theory, the value of the correction factor ranges between the amplitude of the pulse and sine waves. Hence, the correction coefficient is taken as π/4 in this paper. Fig. 4(a) and (b) show the performances of the control method [17] and our method, respectively; the former does not use the correction coefficient kF , while the latter does. Both performances are shown with the same position input—a triangular signal moving 6 cm at a velocity of 6 cm/s. For convenience, all the experimental results were normalized. For instance, the normalized position output (xp ) is xp /6 cm, angle theta is [(Fθ − 180)/180◦ ], and the flux linkage (fluxD) is (FD /0.325 Wb). Fig. 4(a) clearly shows the disadvantages of the conventional method [17]. From our experimental results, we can infer that the conventional method cannot precisely estimate both the amplitude and the angular position of the flux linkage due to the lack of a correction factor. In other words, the correct voltage vectors are not provided in time for the motor control system by the switching table. Additionally, the incorrect voltage vectors increase the errors and make the control system unstable. Fig. 4(b) shows that the correction factor improves this problem. Indeed, the amplitude and the angular position of the flux linkage can be accurately obtained by using (15) where xp is the position output and FL  ≤ E is the bounded external uncertainty and disturbance term that may comprise dry and viscous friction, as well as any other unknown forces. The uncertainty and disturbance terms must be assumed for the control scheme. Here, the values of the terms are bounded and satisfy the matching conditions. It can be seen that the robustness of the SMC method makes this intensive approach suitable for use in a motor control system. A. Conventional SMC Control Method [18] The aim of the SMC method is to drive the state trajectory onto a sliding surface in the reaching phase as soon as possible. To facilitate the derivation of the SMC law, (15) is written into the following state-space form:        0 x1 0 1 ẋ1 + 1 (u − FL ) (16) = ẋ = D x2 0 −M ẋ2 M where x1 = xp , ẋ1 = x2 , and u = Fe are the control inputs. The position tracking error e = r − x1 and r = xpcmd are defined. Therefore, the error dynamic equation is ė = ṙ − ẋ1 = ṙ − x2 . (17) The error equation that corresponds to (16) can then be described as ë = r̈ − ẋ2 1 1 [Dṙ − Dė + FL ] − u. = r̈ + M M (18) 1100 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010 The aim is to derive the states of the system into the set H, which is defined by H = {e : s(e) = ė + ce = 0} (19) where c is a positive constant that represents the slope and s(e) = 0 is the sliding surface. The derivation of the control law starts with the selection of the Lyapunov function as follows: V (s) = 1 2 s . 2 (20) It has been shown that [18], if there exists a positive constant ε such that V̇ (s) = sṡ ≤ −ε|s| (21) then the state trajectories will hit the sliding surface in finite time. Equation (21) shows a global hitting condition and guarantees that all initial states lying off H will converge to the sliding surface. Hence, if the control law is chosen as (22), asymptotic stability is guaranteed   D x2 + cė + Ksgn(s) = ueq + usw , u = M r̈ + M   D x2 + cė ; usw = Ksgn(s). ueq = M r̈ + (22) M In general, K is chosen as E/M + ε. It can be seen that the resultant chattering is one of the drawbacks of the SMC method. Indeed, the sign function is the main factor for the occurrence of chattering. B. FSMC Control Method [21] To design the FSMC system, both the membership functions and the fuzzy inference rules of a control system can be located by the SMC method. The feedback control gain can then be determined from the fuzzy inference rules. Next, the control gain should be smoothed by fuzzifying; this will attenuate the chattering. Thus, FSMC has the advantages of both SMC and FLC policies. Additionally, the associated fuzzy sets are defined as follows: Rule 1 : If s is Z, then u is ueq . (23) Rule 2 : If s is N Z, then u is ueq + usw . (24) By defuzzifying the aforementioned control rules, we can obtain the output of FSMC as follows: u= αZ ueq + αNZ (ueq + usw ) = ueq + αNZ usw αZ + αNZ αZ + αNZ = 1 (25) (26) where αZ and αNZ are the degrees of the fuzzy membership functions. Obviously, the FSMC method is the same as the conventional SMC law when αNZ = 1. However, the chattering can be reduced by the fuzzy membership function when Fig. 5. Structure of the proposed SMC method. αNZ = 1. One can realize that considerable expertise are required in FSMC. It is well known that the knowledge base required for FSMC is the minimum. The main reason for the aforementioned assertion is that one does not need to know the parameters and the structure of the controlled systems. It should be noted that two problems need to be overcome by the FSMC theory: establishing the fuzzy rules and deciding what the membership functions are. However, the system performance cannot be accurately obtained from the conventional control method [21]. In the following section, we propose a solution for the aforementioned problems. C. Our Proposed Function-Based SMC Control Method In this section, we propose a function-based SMC technique to deal with the chattering problem. Our method controls both the switching gain (usw ) and the sat function at the same time. Fig. 5 shows the new structure. Here, the control law can be rewritten as u = ueq + αNZ K. (27) An adaptive factor for the switching gain and a modified boundary layer width are included in the parameter αNZ . Therefore, the fuzzy membership functions can be used directly. It should be noted that parameter αNZ is dependent upon the difference between the real value (s) and the sliding surface (s = 0). Furthermore, this method allows for the avoidance of problems related to high gain. Considering the requirement, function αNZ is designed as αNZ = F (s) = 2 −1 1 + e−f ·s (28) where f is a positive constant. Fig. 6 shows that the system’s dynamic responses are varied due to the variety of the parameter f . It can be seen that the convergence speed for approaching the sliding surface and the sliding width is completely dependent on parameter f . Furthermore, the control system should output a positive/negative gain when the state trajectory is over (s > 0)/under (s < 0) the sliding surface. In addition, the switching gain (αNZ ) is variant, and its values change according to its distance from the sliding surfaces (s = 0). Hence, the design requirement can be met. However, a larger steady-state error will be generated due to a small f value. Additionally, high-frequency switching cannot be avoided in the event of a large f value. Therefore, the designer needs to select the value of f carefully. HUANG AND SUNG: FUNCTION-BASED CONTROLLER FOR LINEAR MOTOR CONTROL SYSTEMS Fig. 6. Fig. 7. 1101 Corresponding curve of function αNZ . Structure of a closed-loop PMLSM control system. For better motor control, the function-based SMC method and DTC are both used in our control method. The structure of our control system is shown in Fig. 7. Here, an optimum voltage vector of the switching table can be determined according to the flux linkage error, thrust error, and flux linkage position (Fθ ). The corresponding voltage vector can be chosen from the new switching table for the improvement of the system’s performance. Additionally, a new function-based SMC method is used to produce the thrust command in this paper. The advantage of the proposed method is that chattering can be reduced or even eliminated. Moreover, our method allows designers to employ fuzzy membership functions directly. It should also be noted that the computation cost can be alleviated because a defuzzifying procedure is not required. When we compare our proposed method with the FSMC [21], it can be seen that the rise and settle times are considered under a unit step input with no overshoot condition. The comparison results are shown in Fig. 8(a). Obviously, our system’s performance is better than that of the FSMC. Indeed, the new method is more flexible than conventional SMC and FSMC methods. It is our opinion that determining a set of parameters for the new SMC control method is an important issue. The set of parameters can meet the required performance. It should be noted that the conventional method uses trial and error to obtain the parameters required by the control system. However, an efficient control methodology has not been proposed as of yet. In this paper, we propose a control method that is able to select the parameters for the controller automatically in order to Fig. 8. (a) Two unit step responses of the FSMC method and the proposed method. (b) State trajectory of our function-based SMC method. meet the requirements of the control system. The performance indexes (i.e., the rise time, settle time, and steady-state error) are satisfied using our control system. IV. S IMULATION AND E XPERIMENTAL R ESULTS In this paper, the nonlinear control design (NCD) toolbox is used to obtain the vector [c, ε, f, K] for the new controller. The NCD procedure is divided into four steps: 1) specifying the desired response; 2) selecting the tuned parameters; 3) setting up the ranges of the corresponding parameters; and 4) executing the program until all specifications are met [26]. To do so, the specifications of the control system are set as follows: Rise time (tr ) is less than 0.1 s; settle time (ts ) is less than 0.2 s; and overshoot is less than 2%. The parameter vector [c, ε, f, K] can then be ranged between its lower and upper bounds (i.e., [1, 1, 1, 1] and [100, 100, 100, 200]). Next, the suitable parameter vector [25.7895, 3.33, 25.1286, 120.84] can be located. A. Simulation Results The performances of the proposed integration algorithms were simulated in Matlab/Simulink software. To do so, the parameters of the motor control system had to be identified. The related parameters of the PMLSM are listed in Table III. Herein, we attempt to trace the performance of the control system under a triangular wave input. The slope of the rising ramp of the triangular wave is 6, and that of the descending ramp is −6. The simulation results [i.e., the position response 1102 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010 TABLE III PARAMETERS OF THE PMLSM (xp ) and the error state phase trajectory (e and ė)] are shown in Fig. 9(a) and (b), respectively. Fig. 9(a) shows that the trace of the output response closely follows the input command. Fig. 9(b) shows that the reaching phase can be reached from its starting position. Additionally, the system state trajectory can be held on the sliding phase. Moreover, Fig. 9(c) shows that the position tracking error of the full stroke is within 50 µm, except when in the transient state. As a result, we can infer that the trace of the output response signal closely follows the input command in our control method. To demonstrate the effect of the variation parameters and external disturbance, two cases are examined. One is a nominal case that is without loading or disturbance, while the other is a case where the parameters vary (an 8-kg load on the mover and an external disturbance of 10 Nt after 1 s). Additionally, we again used a square wave as a reference input. The simulation results of the nominal and parameter variation cases are shown in Fig. 10(a) and (b). Fig. 10(a) shows that the system response is without any overshoots or oscillations. Interestingly, it can be seen in Fig. 10(b) that the system response is still without any overshoots or oscillations even if there are disturbances. Here, our controller responds quickly and has a well trace performance, easily meeting designer requirements. Moreover, three different input signals (triangular, sine, and trapezoidal waves) are used to examine the performances starting at different positions. To do so, one started at 0◦ and 180◦ with a triangular wave input signal (i.e., the slope of the rising ramp is 6, while that of the descending ramp is −6). Another was then started at 120◦ and 240◦ with a sine wave input signal (i.e., a 1-cm amplitude and a frequency of 1 Hz). Additionally, the other was started at 60◦ and 300◦ with a trapezoidal wave input signal (i.e., the slope of the rising ramp was 12 and that of the descending ramp was −12). The three simulation results are shown in Fig. 11. Based on the simulation results, it can be asserted that the traces of the output responses closely follow the input commands. As a result, our proposed control method seems as though it were started at arbitrary positions. In the following section, a practical PMLSM control system will be introduced, and the aforementioned parameters will be applied. B. Experimental Results In this real motor control system, a number of pieces of equipment were used to support the practical PMLSM control system. The system hardware consisted of a digital signal Fig. 9. (a) Simulation results for triangular wave input tracking (xp ). (b) Simulation results for the error state trajectory. (c) Simulation results for the position tracking error (e). processor (TMS320LF2400), a pulsewidth modulation driver with a three-phase inverter (UTP10), a PMLSM (LM210-2AWD3), a personal computer (Pentium II 300 MHz with 512-MB RAM), and several interface cards. The primary function of the experiment was to verify the performance of the control system in a real-world scenario. Here, we attempt to compare the simulation results with the experimental results. For this reason, the same reference input signal (i.e., a triangular wave) was used. The experimental results (i.e., the position response (xp ) and the flux linkage amplitude [(i.e., FD and FQ )] are shown in Fig. 12(a) and (b), respectively. Fig. 12(a) shows that the trace of the output response closely follows the input command. In particular, the system still responds quickly at the apex of the triangular signal. This means that the heading HUANG AND SUNG: FUNCTION-BASED CONTROLLER FOR LINEAR MOTOR CONTROL SYSTEMS 1103 Fig. 10. (a) Simulation results for the square wave input (xp ) without FL . (b) Simulation results for the square wave input (xp ) with 10-Nt FL . Fig. 12. (a) Experimental results for a triangular wave input (xp ). (b) Experimental results for the flux phase (FD , FQ ) trajectory. (c) Experimental results for the requirement of the sliding surface. Fig. 11. Simulation results when starting at the three different positions. of the mover can be changed at any time. Fig. 12(b) shows a circle. It can be seen that the D- and Q-axes are orthogonal. Moreover, the flux remains within the hysteresis band. Hence, the values of the flux magnitude can be accurately obtained using our proposed estimator. Fig. 12(c) shows the error in the state trajectory around with the specified sliding surface. Based on the comparison results, we can infer that the whole system response almost matched the simulation results. In addition, the three different speeds and load conditions (i.e., 6 cm/s with 8 kg, 3 cm/s with 4 kg, and 2 cm/s with 2 kg) were successfully examined for the real motor control system. The experimental results relating to the position output and the position tracking error are shown in Fig. 13(a) and (b). The position tracking errors of the full stroke are within 50 µm, except when in a transient state. Obviously, our method is robust. To further explain the transient performance, various requirements were considered (i.e., 6 cm/s with 8 kg, 3 cm/s with 4 kg, and 2 cm/s with 2 kg) for the real motor control system. The transient responses in each of the three cases are shown in Fig. 13(c) (i.e., 0.32, 0.17, and 0.09 cm). In short, the transient error is with a small value for each command. Indeed, it seems as though the transient error can be neglected. As a result, one can infer that our control system responds very quickly. Our motor control system can also run at low speeds under loaded conditions (i.e., 0.1 cm/3 s with 0 kg, 0.05 cm/3 s with 2 kg, 0.01 cm/3 s with 1104 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 3, MARCH 2010 Fig. 14. Experimental results for the position response (xp ) at different speeds and loads. proposed approach is that it allows one to directly employ the fuzzy membership function of the controlled system. The design of control rules for the proposed system can be performed without the need for great expertise as it does not require the use of very complicated techniques. Moreover, chattering can be reduced and/or eliminated in the reaching and sliding phases. In addition, a new flux linkage estimator, which is able to obtain the actual flux magnitude and the value of its angular position, is also proposed. The proposed method achieves favorable results with regard to parameter variations and external disturbances. Finally, the simulation and the experimental results have shown that our proposed control method can be used in a real PMLSM system. R EFERENCES Fig. 13. (a) Experimental results for the position response (xp ) at different speeds and loads. (b) Experimental results for the position tracking error (e) at different speeds and loads. (c) Transient experimental results. 4 kg, and 0.05 cm/s with 8 kg). Fig. 14 shows the position tracking performance. Based on the experimental results, we can conclude that the new method can be implemented in the real motor system. V. C ONCLUSION In this paper, we have proposed a new function-based SMC controller and a DTC method for PMLSM systems. In particular, the parameters of a real motor can be automatically generated by an optimization algorithm. The advantage of the [1] L. N. Tutelea, M. C. Kim, M. Topor, J. Lee, and I. Boldea, “Linear permanent magnet oscillatory machine: Comprehensive modeling for transients with validation by experiments,” IEEE Trans. Ind. Electron., vol. 55, no. 2, pp. 492–500, Feb. 2008. [2] C. C. Sung and Y. S. Huang, “Based on direct thrust control for linear synchronous motor systems,” IEEE Trans. Ind. Electron., vol. 56, no. 5, pp. 1629–1640, May 2009. [3] Y. 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Available: http://www.mathworks.com 1105 Yi-Sheng Huang (M’05) received the B.S. degree in automatic control engineering from Feng Chia University, Taichung, Taiwan, in 1989, the M.S. degree in electronic engineering from Chung Yuan Christian University, Jhongli City, Taiwan, in 1991, and the Ph.D. degree in electrical engineering from National Taiwan University of Science and Technology, Taipei, Taiwan, in 2001. He was an Associate Professor in the Department of Aeronautical Engineering, Chung Cheng Institute of Technology, National Defense University, Bade City, Taiwan, where he is currently a Full Professor in the Department of Electrical and Electronic Engineering. He has been serving as a Reviewer for Automatica, IET Control Theory and Applications, IET Intelligent Transport Systems, the International Journal of Production Research, The Computer Journal, the International Journal of Production Research, the International Journal of Applied Management and Technology, the Asian Journal of Control, the Journal of the Chinese Institute of Engineers, and the Journal of Information Science and Engineering. His research interests include DESs, Petri nets, computer-integrated manufacturing, automation, reactive systems, air traffic control, intelligent transport systems, and motor control systems. Prof. Huang has been serving as a Reviewer for the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART A: S YSTEMS AND H U MANS , IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS — PART C: A PPLICATIONS AND R EVIEWS, IEEE T RANSACTIONS ON AUTOMATION AND E NGINEERING, and IEEE T RANSACTIONS ON I NDUS TRIAL E LECTRONICS . Cheng-Chung Sung (S’08) was born in Taoyuan, Taiwan, in 1959. He received the B.S. degree in electronic engineering from National Taiwan Ocean Institute of Technology, Keelung, Taiwan, in 1981, and the M.S. degree in electronic engineering from National Central University, Jhongli City, Taiwan, in 1990. Since 2005, he has been working toward the Ph.D. degree in electrical engineering at National Defense University, Bade City, Taiwan. He is also currently an Instructor in the Department of Computer Science and Communication Engineering, Army Academy, Jhongli City, Taiwan. His research interests include motor drives, robust control, and intelligent control.