Continuous Sliding Mode Control for Permanent Magnet Synchronous Motor Speed Regulation Systems Under Time-Varying Disturbances

Abstract

This article explores the speed regulation problem of permanent magnet synchronous motor (PMSM) systems subjected to unknown time-varying disturbances. A continuous sliding mode control (CSMC) technique is introduced for the speed loop to enhance the robustness of PMSM systems and eliminate the chattering phenomenon caused by high-frequency switch function in the conventional control law. However, the high control gain of the CSMC law in the presence of strong disturbances leads to large steady-state speed fluctuations for PMSM systems. In many application fields, PMSM systems are affected by time-varying disturbances instead of constant disturbances. For example, electric bicycles are usually affected by changing environmental disturbances, including wind speeds, road conditions, etc. These disturbances may be in the form of constant, ramp, and parabolic disturbances. Hence, a generalized proportional integral (GPI) observer is employed to estimate these types of disturbances. Then, the disturbance estimation method and the aforementioned CSMC method are combined to establish a composite sliding mode control method called the CSMC+GPI method for the speed loop of PMSM systems. Contrary to the conventional sliding mode control technique, the proposed method completely eliminates the chattering phenomenon caused by the switching function in the conventional control law. Moreover, a small control gain for the CSMC+GPI method is chosen by feed-forwarding estimated values to the speed controller. Hence, the steady-state speed fluctuations are small. The effectiveness of the proposed control scheme is verified by simulation and experimental result.

I. INTRODUCTION

The proportional-integral-derivative control design methodology, which offers simple and easy implementation, is widely used in permanent magnet synchronous motor (PMSM) systems [1], [2]. However, in practical applications, PMSM systems are generally affected by different types of unknown time-varying disturbances and parametric uncertainties, such as parasitic dynamics, unknown plant parameters, and external disturbances. Conventional linear control methods are not always effective in improving the performance of PMSM systems in the presence of these disturbances/uncertainties. Hence, designing control schemes that provide the desired system performance is a challenging task for control engineers. Driven by the significance of rejecting disturbances/uncertainties in practical applications, many authors develop effective nonlinear control algorithms that have been successfully applied to PMSM systems; examples of such algorithms include robust control [3], predictive control [4], [5], disturbance observer-based control [6], [7], finite time control [8], sliding mode control (SMC) [9]-[11], adaptive control [12]-[14], and intelligent control [15], [16].

Among the aforementioned control methods, the SMC method has received increasing attention because it strengthens the robustness of systems against unknown system disturbances and parameter uncertainties and because its tuning and implementation are simple in practice [17]. However, the chattering phenomenon, which is caused by the high-frequency switching control action adopted in the SMC law, is a problem that impedes SMC implementation. Hence, many sophisticated methods have been proposed to reduce or eliminate the effect of chattering. One effective method is to select a suitable switching gain for the SMC law because an unsuitably large switching gain leads to large chattering; examples of methods employing this approach include the learning algorithm-based SMC scheme [15], [16], [18], adaptive SMC scheme [19], [20], and disturbance observer-based SMC control scheme. In [21], an SMC control methodology with an extended state observer is employed to estimate system disturbances and deliver disturbance estimations to the SMC controller for disturbance compensation. This approach is described extensively in the literature [22]-[26]. Through these methods, the switching gain of a sliding mode controller can be set to a small value to reduce chattering. However, these control strategies remain discontinuous.

Another effective method for eliminating the chattering phenomenon is to design a continuous control law. The boundary layer approach employed in [27], [28] is aimed at eliminating the chattering phenomenon completely by replacing the discontinuous control law with a continuous one, but the performance of the disturbance rejection is compromised to some extent. As a control technique for eliminating the chattering phenomenon, a continuous SMC (CSMC) method is proposed in [29], [30] and designed in such a way that the high-frequency switching control is into the higher derivative of the sliding variable.

In this study, we first introduce the CSMC method for PMSM systems under external disturbances and parameter uncertainties. However, the high control gain of the CSMC law in the presence of strong disturbances leads to large steady-state speed fluctuations for PMSM systems. Considering the fact that PMSM systems are generally affected by time-varying disturbances, we employ a generalized proportional integral (GPI) observer to estimate such lumped disturbances. The GPI observer based on differential flatness theory [31] was introduced by Fliess et al. [32], and it is widely applied and reported in the literature [33], [34]. By combining the disturbance estimation method and CSMC method [30], [35], we propose a composite control method for the speed tracking of PMSM systems.

The rest of the paper is organized as follows. The dynamics of PMSM systems are described in Section II. The GPI observer design approach, the composite CSMC law, and corresponding proof are described in Section III. Simulation and experimental results are presented in Section IV to demonstrate the effectiveness of the proposed control algorithm. Finally, the conclusions are given in Section V.

 

II. MODEL OF PMSM SYSTEMS

Suppose that the magnetic circuit is unsaturated, the eddy current and hysteresis loss are negligible, and the three-phase stator windings of a PMSM are sinusoidally distributed in space. The commonly stator (d - q axes) voltage equations of a PMSM are given in terms of its synchronous rotating reference frame as [5], [36].

where id and iq are the stator currents of the d - q axes, ud and uq are the stator voltages of the d - q axes, Ld and Ld are the inductances of the d - q axes, R is the stator resistance, ωe is the electrical angular speed, and ψf is the rotor flux linkage.

Moreover, the motor is assumed to be a surface-mounted PMSM, on which the effects of saliency are ignored. The inductances of the d - q axes are equal to each other and are denoted by L, i.e., Ld = Lq = L ; hence, a reluctance torque component is not present. Using the well-known strategy of the field-oriented control (FOC) (also known as vector control) of a PMSM, we set the d - axis reference current to zero . If two current controllers work effectively, the d - axis current id is controlled to zero to eliminate the couplings between angular velocity and currents and to maximize the output torque. The electromagnetic torque Te is given by

in which Kt = 3npψf / 2 ( np is the number of pole pairs) is the torque constant. The motor torque can be controlled by regulating the q - axis current iq. Therefore, the control of the PMSM in the d - q reference frame is similar to the principle of controlling DC motors. In this way, controller design is simplified. The associated equation of motor dynamics is

where ω=npωe is the mechanical rotor speed, TL is the load torque, B is the viscous friction coefficient, and J is the total inertia (including motor and load).

The general structure of a PMSM system under the FOC strategy is shown in Fig. 1. The overall system consists of a PMSM with load, a three-phase voltage source inverter, a space vector pulse width modulation (SVPWM), an FOC mechanism, and three controllers. The control system features a cascade structure, which includes a speed loop and two current loops. The two current loops adopt two standard proportional integral (PI) controllers to stabilize the current errors of the d - q axes. As shown in Fig. 1, the rotor speed ω (including the rotor position θ) can be obtained from the position and speed sensor. The currents id and iq can be calculated from ia and ib by using the abc / αβ Clarke transform and αβ / dq Park transform. The phase currentsia and ib can be measured with Hall effect current sensors. The reference current is determined by the speed loop. The purpose of this study is to design a composite controller for the speed loop.

Fig. 1.Block diagram of PMSM vector control system.

 

III. CONTROLLER DESIGN

As described in this section, the normalized form of the PMSM system model is obtained for the observer and controller design on the basis of the equation shown in (3). The design procedure is presented as follows.

A. Design of a GPI Observer

PMSM systems are generally affected by time-varying disturbances. For example, an electric bicycle is usually affected by changing environmental disturbances, including wind speeds, road conditions, etc. These disturbances may be constant, ramp, and parabolic disturbances. When an electric bicycle runs uphill or encounters changing winds, the disturbances are in the form of a ramp rather than a constant. Disturbance rejection in practice is an important criterion to evaluate the performance of control systems. If we consider a corresponding feed-forward approach to compensate for disturbances, we can improve the performance of closed-loop systems. Note that a GPI observer can estimate unknown time-varying disturbances [37], [38]. To achieve a good performance in speed command tracking and disturbance rejection, we introduce the GPI observer technique for the PMSM servo system.

According to the design fundamentals of a GPI observer, we can rewrite the motor dynamic equation (3) into the following canonical form:

where b = Kt / J,

is the lumped time-varying disturbance, including friction, external load disturbances, and the tracking error of the iq current loop. Unknown time-varying disturbances dω(t) are assumed to satisfy the following conditions. (1) The time-varying disturbance dω(t) and the finite number of its time derivatives are uniformly and absolutely bounded. (2) The time-varying disturbance dω(t) can be expressed as a family of Taylor polynomials of (r - 1) degree, dω(t) = where all ai > 0 are unknown constant coefficients, r = 1, 2, ⋯, m, and m is a positive integer [33]. Then, we can design a GPI observer for system (4) as follows

where (λ0, λ1,⋯, λm) are the observer coefficients, which are chosen so that the roots of the characteristic polynomial

in the complex plane are located in the left-half side of the complex plane and are thus sufficiently far from the imaginary axis. According to the analysis in [33], [39], the estimation errors e = ω - and disturbance estimation errors , i = 0, 1, ⋯, m converge to the equilibrium point asymptotically. Hence, ,z1,z2, ⋯,zm are the estimations of .

B. Design of Speed Controller

The speed tracking error e is given as follows:

where ωr represents the reference speed signal. By taking the derivative of e and substituting (4) into it, we obtain the following error equation:

The continuous sliding mode speed control law with disturbance estimation is designed as [30]

with the corresponding sliding mode surface

where c ˃ 0, k ˃ 0, and sgn(·) denotes the standard sign function; b = Kt / J is the control gain, and z1 given by the GPI observer is the estimate of the lumped time-varying system disturbance dω.

Assumption 1: Assume that the derivative of ed is bounded and that ked ˃ 0 exists such that

where ed(t) = dω - z1 is the disturbance estimation error and dω is the lumped varying-time disturbance of system (4).

Theorem 1: Assume that system (4) satisfies Assumption 1. Under the proposed continuous control law (9), the speed error of the system converges to the desired equilibrium point asymptotically if the gain satisfies k ˃ ked.

Proof: Substituting equation (8) into sliding mode surface (10) yields

Substituting control law (9) into (11) yields

Consider a candidate Lyapunov function as V = (1/2)σ2. Taking the derivative of sliding mode surface (12) yields

According to Assumption 1, the above equation can be rewritten as follows:

According to the above analysis process, the speed error arrives at the sliding mode surface σ = 0 in finite time if k ˃ ked. When the speed error states reach the sliding surface ce+ė=0 with c ˃ 0, the speed error converges in an asymptotically exponential manner to zero along the sliding surface σ = 0. This completes the proof.

The continuous sliding mode speed controller with the GPI observer is illustrated in Fig. 2.

Fig. 2.Control scheme of the CSMC+GPI method.

Remark 1: Basically, a simple PI-like controller could be employed for the speed regulation problem of PMSM systems. Such types of servo control systems have been widely used in many low-level industries that do not aim for high servo accuracy. Although the dynamic model (4) of the speed loop of a PMSM appears relatively simple, PMSM systems are essentially nonlinear systems subject to multiple disturbances, including external disturbances, unmodeled dynamics, and parametric disturbances. These disturbances/uncertainties generally exhibit different dynamics and features. In general, the high-precision speed regulation of PMSM systems should adequately consider the complicated disturbance/uncertainty dynamics, such as that achieved with the control law (9) proposed in this study. This complicated disturbance/uncertainty dynamics is not as straightforward as that described by the simple dynamics in equation (4). The PI and many other nonlinear control algorithms, such as the back-stepping control and predictive control, could not guarantee the advantageous servo control performance of PMSMs because they do not directly consider disturbance rejection in controller design. In many advanced industrial systems, such as robotics, machine tools, and aircraft, achieving a high servo control performance is difficult because simple PI-like control algorithms are not satisfactory. Consequently, new generations of servo control design for PMSMs have been put forward, and they include those in [5], [12], [25], [28], and [39]. In addition to academic research, advanced control algorithms have been extensively applied to the practical industrial production of of AC motors and drivers, such as the Panasonic MINAS A5-series AC servo motors and Estun EDC-08 APE [40] and the EDB-10 AMA series AC servo drivers [41].

Remark 2: The sliding variable σ in control law (9) is not available because the acceleration signal in σ could not be measured directly. To calculate the sign of the sliding mode variable σ in (10), we define a function g(t) as [30]

Then, sgn(σ) can be obtained as

where τ is the fundamental sampling time and

Remark 3: The GPI observer designed in (5) is employed to estimate time-varying system disturbances. The GPI observer parameters are (λ0, λ1, ⋯, λm) . When (λ0, λ1, ⋯, λm) are designed appropriately, z1 becomes a good estimate of dω. By feed-forward compensating disturbances, the CSMC+GPI method need not reject such disturbances, and the gain k can be set to a small value. This approach can reduce steady-state speed fluctuations for the system, as shown in the following simulation and experimental results.

 

IV. SIMULATION AND EXPERIMENTAL RESULTS

To illustrate the superiority of the proposed CSMC+GPI control design methodology, we perform simulations and experiments under the MATLAB real-time workshop and DSP-based test environment. The CSMC and CSMC+GPI schemes are applied to the PMSM servo system. The CSMC method can be described as

where the sliding variable σ is the same as that in equation (10).

The nominal parameters of the PMSM used in the simulation and experiment are shown in Table I.

TABLE IMOTOR PARAMETERS

A. Simulation Results

The speed regulation system under the CSMC and CSMC+GPI approaches is simulated using MATLAB R2012a. The control gains of the CSMC and CSMC+GPI schemes are selected as k =600000 and k =200000, respectively. The parameters of the GPI observer are given as m = 2, λ0 = , λ1 = , λ2 = 3ωo, ωo = 8000. The two current loops adopt conventional PI control algorithms. The saturation limit of is set to 9.42A.

The dynamic speed response curves of the PMSM speed regulation system are shown in Fig. 3. The CSMC and CSMC+GPI control methods exhibit small overshoots and short settling times. That is, the proposed CSMC+GPI method retains its nominal performance because the GPI observer functions like a patch for the baseline CSMC controller and does not cause any adverse effects on the PMSM system in the absence of load disturbance, as shown in the following experimental results.

Fig. 3.Response curves of the PMSM under the CSMC and CSMC+GPI schemes. (a) Speed. (b) .

The response curves of disturbance rejection performance are shown in Figs. 4–6. Fig. 4 shows that when a load torque TL = 4.7Nm is applied at t = 0.1s , the speed response under the proposed method recovers rapidly after the intrusion of the load torque disturbance. As shown in Fig. 5, the CSMC+GPI method performs better than the CSMC method in terms of its disturbance rejection performance when a ramp torque varying from 0 to 4.7Nm is added at t = 0.1s . We consider a generic time-varying disturbance torque TL that is given by cos2(40t) [39]. When this time-varying parabolic disturbance is applied to the motor during t ϵ (0,0.2s] , the CSMC method with a large gain of k = 600000 cannot suppress this type of disturbance effectively, whereas the CSMC+GPI control law achieves effective disturbance rejection even with a small gain of k = 200000 (Fig. 6).

Fig. 4.Response curves of the PMSM under the CSMC and CSMC+GPI schemes with constant disturbance. (a) Speed. (b) .

Fig. 5.Response curves of the PMSM under the CSMC and CSMC+GPI schemes with ramp disturbance. (a) Speed. (b) .

Fig. 6.Response curves of the PMSM under the CSMC and CSMC+GPI schemes with generic time-varying disturbance. (a) Speed. (b) .

B. Experimental Results

To further verify the validity and feasibility of the proposed control scheme, we build an experimental system for the speed control of a PMSM. Figs. 7 and 8 illustrate the configuration of the experimental system and the DSP-based test setup, respectively. The algorithm includes one speed controller, two current controllers, abc / αβ Clarke transform, αβ / dq Park transform, dq / αβ inverse Park transform, and SVPWM. These components are implemented with the program of the DSP TMS320F2808 with a clock frequency of 100 MHz. The proposed GPI observer-based CSMC control method is employed for the speed loop, and the execution period is set to 250 µs. Other programs consisting of the inner q-axis and d-axis current loop programs, all coordinate transform programs, and the SVPWM program are adopted at 60 µs. The saturation limit of the q - axis reference current is ± 9.42A . The phase currents are measured by the Hall effect devices with 12-bit A/D conversion accuracy. An incremental position encoder of 2,500 lines used in this case measures the rotor speed and absolute rotor position.

Fig. 7.Configuration of the experimental system.

Fig. 8.Experimental test setup.

For the speed loop control, the control gains are chosen as k = 35000 for the CSMC scheme and k = 1500 for the CSMC+GPI scheme.

The parameters of the GPI observer are m = 2 , λ0 = , λ1 = , λ2 = 3ωo, and ωo = 200. The experiment results are shown in Figs. 9–13.

Fig. 9.Response curves of the PMSM. (a) Speed under the CSMC and CSMC+GPI control schemes. (b) q - axis current under the CSMC control scheme. (c) d - axis current under the CSMC control scheme . (d) q - axis current under the CSMC+GPI control scheme. (e) d - axis current under the CSMC+GPI control scheme .

Fig. 10.Response curves of the PMSM under the CSMC control scheme with load disturbance (k = 35,000). (a) Speed. (b) q - axis current. (c) d - axis current .

Fig. 11.Response curves of the PMSM under the CSMC control scheme with load disturbance (k = 50,000). (a) Speed. (b) q - axis current. (c) d - axis current .

Fig. 12.Response curves of the PMSM under the CSMC control scheme with load disturbance (k = 70,000). (a) Speed. (b) q - axis current. (c) d - axis current .

Fig. 13.Response curves of the PMSM under the CSMC+GPI control scheme with load disturbance (k = 1,500). (a) Speed. (b) q - axis current. (c) d - axis current .

The dynamic speed response curves of the closed loop system under the CSMC and CSMC+GPI schemes are shown in Fig. 9. The CSMC and CSMC+GPI control schemes both show small overshoots and short settling times that are the same as the corresponding simulation results.

The disturbance rejection ability curves of the CSMC and CSMC+GPI control schemes are presented in Figs. 10–13. We provide detailed descriptions about how to improve the disturbance rejection performance of the CSMC control method. According to the theoretical analysis in Theorem 1, the disturbance rejection performance of the system mainly depends on the control gain k . That is, a large gain k equates to a satisfactory disturbance rejection performance. Thus, we tune k of the CSMC control law from a small value to a large value. After the motor runs at a steady state, the rated load TL = 2.4N·m is added. When the control gain is set to k = 35000 (Fig. 10), the motor speed can recover to its reference value as fast as possible. However, the speed decrease amplitude is large at approximately 25.75 rpm. To further enhance the disturbance rejection performance of the closed-loop system, k is tuned to a large value, i.e., k = 50000 (Fig. 11). The closed-loop system exhibits a small speed decrease of about 17 rpm, as well as large steady state fluctuations. When k is tuned to a large value, i.e., k = 70000 (Fig. 12), the motor speed decreases further to about 13.25 rpm but shows even larger speed fluctuations. Thus, a good balance between disturbance rejection and steady-state fluctuations under the CSMC scheme is difficult to achieve by simply tuning the control gain k . The experiment results of the disturbance rejection performance of the CSMC+GPI control scheme are presented. The control gain is adjusted to k = 1500 , which is much smaller than that of the CSMC method. The speed response is shown in Fig. 13. When load disturbance is added, the motor speed can quickly return to the steady state, and the maximum speed drop is only about 9.75 rpm, which is much smaller than that under the CSMC method. The system also shows few steady-state fluctuations because the gain k can be set to a smaller value than that under the CSMC method.

To further demonstrate the superior performance of the proposed CSMC+GPI control scheme, we introduce a number of performance indices, such as the maximum decrease in speed caused by a load torque, speed fluctuation, fluctuation ratio, and standard deviation (STD). A detailed comparison of the performance indices of the two methods under different conditions is shown in Table II.

TABLE IISPEED RESPONSE COMPARISONS

 

V. CONCLUSIONS

A composite speed controller is proposed for PMSM system subjected to time-varying disturbances. To eliminate the chattering phenomenon in conventional sliding mode control, we introduce a continuous sliding mode control method based on a motor dynamic equation. However, the high control gain of the CSMC law in the presence of strong disturbances can lead to large steady-state speed fluctuations for PMSM systems. Thus, a GPI observer-based CSMC speed controller is formulated. The GPI observer is employed to estimate disturbances, and the estimation is delivered to the speed controller for feed-forward compensating the effect of the system disturbances on the PMSM system. Compared with the traditional SMC method, the proposed method offers the following advantages. (a) The proposed method, in theory, can reduce and ultimately eliminate the chattering caused by the switching function in the control law. (b) The proposed method can significantly reduce the fluctuations of the steady-state speed of systems. The simulation and experimental results show the feasibility and applicability of the proposed method.