Robust Predictive Speed Control for SPMSM Drives Based on Extended State Observers

Abstract

The predictive speed control (PSC) strategy can realize the simultaneous control of speed and current by using one cost function. As a model-based control method, the performance of the PSC is vulnerable to model mismatches such as load torque disturbances and parameter uncertainties. To solve this problem, this paper presents a robust predictive speed control (RPSC) strategy for surface-mounted permanent magnet synchronous motor (SPMSM) drives. The proposed RPSC uses extended state observers (ESOs) to estimate the lumped disturbances caused by load torque changes and parameter mismatches. The observer-based prediction model is then compensated by using the estimated disturbances. The introduction of ESOs can achieve robustness against predictive model uncertainties. In addition, a modified cost function is designed to further suppress load torque disturbances. The performance of the proposed RPSC scheme has been corroborated by experimental results under the condition of load torque changes and parameter mismatches.

I. INTRODUCTION

In many motor applications, including washing machines, industrial robots, electric vehicles and airplanes, permanent magnet synchronous motors (PMSMs) have been widely used, due to their high-power density ratio, simple structure and superior control performance [1]. Field-oriented control [2] and direct torque control [3] have become the standard schemes applied for the speed control of PMSM drives. Recently, due to rapid improvements in microprocessor computational power, model predictive control (MPC) has become extensively employed for motor drives and power electronic converters [4]. According to the system model and state variables at the current moment, MPC can predict future state variables. Then it can obtain the control law by optimizing a cost function. When compared with field-oriented control, MPC has the advantages of high dynamic response, easily achieving multi-objective control and direct handling of system constraints.

The three most widely mentioned MPC methods in motor drives are predictive current control [5], predictive torque control [6] and predictive speed control (PSC) [7]-[10]. The predictive current control and predictive torque control schemes use a PI controller as the outer loop speed controller in most cases. Therefore, the overall dynamic performance of the two strategies is affected by the limited bandwidth of cascaded controllers. The PSC is a cascade-free control scheme that is conducive to improving the system bandwidth. Furthermore, some motor applications such as servo drives and spindle drives require fast speed change. Therefore, the PSC strategy used for speed control is quite attractive.

However, the performance of the PSC is easily affected by model accuracy since the prediction model of PSC is generally derived based on a mathematic model of the system. The system model contains unknown load torque and motor parameters such as permanent magnet flux linkage, stator resistance and stator inductance, and these parameters are variable during motor operation, due to temperature change and the magnetic saturation effect [11]. This means that model parameter uncertainties and the unknown load torque disturbances deteriorate the performance of the PSC. In addition, these problems restrict industrial application of the PSC. Thus, it is necessary to enhance the anti-load disturbance capability and parameter robustness of the PSC.

Many improved methods have been reported for model mismatch problems with the MPC. These methods can be divided into three types. The first type is the embedded integral method. In [12], in order to deal with parameter uncertainties and unmodeled dynamics, the integral terms are added to the prediction model. In [13], a predictive current control strategy based on an embedded integral method is proposed to eliminate the mismatches of rotor flux and stator inductance. However, the embedded integral method affects the dynamic performance of the MPC due to the problem of integral saturation. The second type is the online parameter identification method. The parameter robustness of the MPC is improved by using the least squares identification algorithm in [14]. However, the computational burden of the online parameter identification method is heavy and it is difficult to implement. The third type is the disturbance estimation method. In [15], an offset-free MPC based on a disturbance estimator is used to overcome the mismatch of the predictive model caused by model uncertainty and nonlinearity. The authors of [16] employed a disturbance observer to enhance the prediction accuracy of the continuous-time MPC, and this method achieves good offset-free tracking performance. In [17], a discrete-time disturbance observer is intended to simplify the prediction model and to realize robustness against parameter perturbation. Moreover, the disturbance estimation method does not need the exact disturbance model, and it can be flexibly used to estimate multiple disturbances.

The extended state observer (ESO) is a disturbance estimation method that does not depend on system model accuracy. An appropriate ESO only demands that two parameters be tuned [18]. The ESO can precisely estimate the lumped unknown disturbances including load torque, friction torque, parameter mismatch and unmodeled dynamics [19]. In [20], an ESO is used to substitute for the complex disturbance model for handling model uncertainty. In [21], an ESO is utilized to estimate the load disturbances in direct-drive elevator control systems, and experimental results show the good dynamic performance of the ESO.

In this paper, in order to reduce the adverse effects of load torque disturbances and model parameter uncertainties on the performance of the predictive speed control, the proposed robust predictive speed control (RPSC) strategy uses two ESOs to observe the lumped disturbances. The torque ESO is employed to estimate the electromagnetic torque reference, and the current ESO is adopted to estimate the voltage errors caused by parameter mismatch. The observed state variables and disturbances are then utilized to correct the prediction model. In addition, the electromagnetic torque is regarded as a control objective in the modified cost function to further suppress load torque disturbances.

The rest of this paper is arranged as follows. The PSC strategy is introduced in Section II. Then the proposed robust predictive speed controller is designed in Section III. The proposed RPSC is confirmed by experiments in Section IV. Finally, some conclusions are presented in Section V.

II. PREDICTIVE SPEED CONTROL

A. SPMSM Model

The dynamic model of the SPMSM based on the dq-axis rotating synchronously can be derived as follows:

\(\dot{\omega}_{e}=\frac{P_{n} T_{e}}{J}-\frac{P_{n} T_{L}}{J}-\frac{B}{J} \omega_{e}\)       (1)

\(T_{e}=1.5 P_{n} \psi_{f} i_{q}\)       (2)

\(\left\{\begin{array}{l} \dot{i}_{d}=-\frac{R_{s}}{L_{s}} i_{d}+\omega_{e} i_{q}+\frac{1}{L_{s}} u_{d} \\ \dot{i}_{q}=-\frac{R_{s}}{L_{s}} i_{q}-\omega_{e} i_{d}+\frac{1}{L_{s}} u_{q}-\frac{\psi_{f} \omega_{e}}{L_{s}} \end{array}\right.\)       (3)

where ω_e is the electrical rotor angular speed, T_e is the electromagnetic torque, T_L is the load torque that is generally regarded as an unknown external disturbance, B is the friction coefficient, P_n is the number of poles, J is the equivalent rotor inertia constant, u_d is the d-axis voltage, u_q is the q-axis voltage, i_d is the d-axis current, i_q is the q-axis current, R_s is the stator resistance, L_s is the stator inductance, and ψ_f is the rotor flux.

However, it is necessary to use a discrete model of SPMSM to predict the system behavior in each of the sampling periods. The forward Euler discretization method [5] is employed to system models (1), (2) and (3).

\(\omega_{e}(k+1)=\frac{T_{s} P_{n}}{J} T_{e}(k)+\left(1-\frac{T_{s} B}{J}\right) \omega_{e}(k)-\frac{T_{s} P_{n}}{J} T_{L}(k)\)       (4)

\(T_{e}(k+1)=1.5 P_{n} \psi_{f} i_{q}(k+1)\)       (5)

\(i_{d}(k+1)=\left(1-\frac{T_{s} R_{s}}{L_{s}}\right) i_{d}(k)+T_{s} \omega_{e}(k) i_{q}(k)+\frac{T_{s}}{L_{s}} u_{d}(k)\)       (6)

\(\begin{aligned} i_{q}(k+1)=&\left(1-\frac{T_{s} R_{s}}{L_{s}}\right) i_{q}(k)-T_{s} \omega_{e}(k) i_{d}(k)+\frac{T_{s}}{L_{s}} u_{q}(k) \\ &-\frac{T_{s} \psi_{f} \omega_{e}(k)}{L_{s}} \end{aligned}\)       (7)

where T_s is the sample period. In order to control speed without steady-state errors, an internal model of the reference input (i.e., the integral of the speed error [22]) is used to compensate the prediction model. Therefore, the unknown load torque can be eliminated by introducing an internal model to the prediction model (4). Equation (4) is modified as:

\(\omega_{e}(k+1)=\frac{T_{s} P_{n}}{J} T_{e}(k)+\left(1-\frac{T_{s} B}{J}\right) \omega_{e}(k)-\xi_{e} e_{\omega}(k)\)       (8)

\(e_{\omega}(k)=e_{\omega}(k-1)+T_{s}\left(\omega_{e}(k)-\omega_{e}^{*}(k)\right)\)       (9)

where ξ_e is the integral gain, e_ω is the integral of the speed errors, and ω_e^* is the speed reference.

B. Cost Function Design

According to the output optimal control theory, the electrical rotor angular speed ω_e and the d-axis current i_d are taken as output state variables [9]. The cost function is designed as:

\(\begin{aligned} g_{1}=& \lambda_{i 1}\left(i_{d}^{*}(k+1)-i_{d}(k+1)\right)^{2} \\ &+\lambda_{\omega 1}\left(\omega_{e}^{*}(k+2)-\omega_{e}(k+2)\right)^{2} \end{aligned}\)       (10)

where λ_i1 and λ_ω1 are the weight coefficients of the d-axis current error and the speed error, respectively. i_d^*(k+1) and ω_e^*(k+2) are the future references of the d-axis current and speed, respectively. i_d(k+1) and ω_e(k+2) are the future prediction values of the d-axis current and speed, respectively.

For the SPMSM, the maximum torque per ampere control can be achieved by using the control strategy with i_d=0. Therefore, this paper sets i_d^*(k+1)=0 and uses (6), (7) and (8) as the prediction model. According to the optimization strategies ∂g₁/∂u_d(k)=0 and ∂g₁/∂u_q(k)=0, the control laws can be derived as follows:

\(\begin{aligned} u_{q_{-} \text {ref }}(k)=& \frac{2 \lambda_{\omega 1} J L_{s}}{3 P_{n}^{2} T_{s}^{2} \psi_{f}}\left(\omega_{e}^{*}(k+2)+\xi_{e} e_{\omega}(k)\right.\\ &\left.-\left(1-\frac{T_{s} B}{J}\right) \omega_{e}(k+1)\right)+\lambda_{\omega 1}\left(\psi_{f} \omega_{e}(k)\right.\\ &\left.+\left(R_{s}-\frac{L_{s}}{T_{s}}\right) i_{q}(k)+L_{s} \omega_{e}(k) i_{d}(k)\right) \end{aligned}\)       (11)

\(\begin{aligned} u_{q_{-} \text {ref }}(k)=& \lambda_{\text {i1}}(\frac{L_{s}}{T_{s}} i_{d}^{*}(k+1)+(R_{s}-\frac{L_{s}}{T_{s}}) i_{d}(k)\\ &\left.-L_{s} \omega_{e}(k) i_{q}(k)\right) \end{aligned}\)       (12)

where u_{d_ref}(k) is the d-axis voltage reference and u_{q_ref}(k) is the q-axis voltage reference. In order to minimize the cost function g₁ by using the control laws (11) and (12), the conditions ∂²g₁/∂u_{d_ref}(k)²>0 and ∂²g₁/∂u_{q_ref}(k)²>0 must be satisfied. Thus, the weight coefficients should satisfy λ_i1>0 and λ_ω1>0.

C. Current and Voltage Constraints

In order to limit the current in dynamic processes (such as motor start-up and speed changes), the current is predicted again by using the obtained voltage references u_{d_ref}(k) and u_{q_ref}(k) in (13) and (14), and the predicted state variables are defined as i_dp and i_qp.

\(i_{d p}=\left(1-\frac{T_{s} R_{s}}{L_{s}}\right) i_{d}(k)+T_{s} \omega_{e}(k) i_{q}(k)+\frac{T_{s}}{L_{s}} u_{d_{-} \mathrm{ref}}(k)\)       (13)

\(\begin{aligned} i_{q p}=&\left(1-\frac{T_{s} R_{s}}{L_{s}}\right) i_{q}(k)-T_{s} \omega_{e}(k) i_{d}(k)+\frac{T_{s}}{L_{s}} u_{q_{-} \mathrm{ref}}(k) \\ &-\frac{T_{s} \psi_{f} \omega_{e}(k)}{L_{\mathrm{s}}} \end{aligned}\)       (14)

The maximum current limit is defined as I_max. According to the principle of deadbeat control, let i_q(k+1)=I_max. Then the new control law can be derived as follows by using (7):

\(\begin{aligned} u_{q_{-} \text {sat }}(k)=& \frac{L_{s}}{T_{s}} I_{\max }+\left(R_{s}-\frac{L_{s}}{T_{s}}\right) i_{q}(k) \\ &+L_{s} \omega_{e}(k) i_{d}(k)+\psi_{f} \omega_{e}(k) \end{aligned}\)       (15)

where u_{q_sat}(k) is the saturation value of the q-axis voltage. When i_dp² + i_qp² ≥Imax2 , the control laws are switched from (11) and (12) to (12) and (15). Otherwise the controller continues to use (11) and (12) as control laws. Meanwhile, in order to limit the voltage peak while considering the linear modulation region of SVPWM, the maximum voltage amplitude is defined as |u|_max=U_dc / \(\sqrt{3}\), where U_dc is the DC bus voltage. The current constraint method and the overmodulation strategy are shown as follows:

\(\left|\boldsymbol{u}^{*}\right|=\left\{\begin{array}{ll} \sqrt{u_{d_{-} \mathrm{ref}}^{2}+u_{q_{-} \mathrm{sat}}^{2}}, & i_{d p}^{2}+i_{q p}^{2} \geq I_{\max }^{2} \\ \sqrt{u_{d_{-} \mathrm{ref}}^{2}+u_{q_{-} \mathrm{ref}}^{2}}, & i_{d p}^{2}+i_{q p}^{2}<I_{\max }^{2} \end{array}\right.\)       (16)

\\(\boldsymbol{u}=\left\{\begin{array}{l} \boldsymbol{u}^{*}, & \left|\boldsymbol{u}^{*}\right|<|\boldsymbol{u}|_{\max } \\ |\boldsymbol{u}|_{\max } e^{j \theta}, & \left|\boldsymbol{u}^{*}\right| \geq|\boldsymbol{u}|_{\max } \end{array}\right.\)       (17)

where u is the actual voltage, u^* is the reference voltage, and |u^*| and θ are the magnitude and angle of the reference voltage u^*, respectively. Fig. 1 shows the overall scheme of the PSC strategy, where θ_e is the rotor position.

Fig. 1. Schematic block diagram of the SPMSM drive system with the predictive speed control algorithm.

III. ROBUST PREDICTIVE SPEED CONTROL

The prediction model of PSC is based on the mathematic model of the SPMSM, and the system performance is susceptible to model accuracy. Therefore, model parameter uncertainties and load torque disturbances deteriorate the control system performance. This paper presents a robust predictive speed control (RPSC) strategy to handle the problem of model mismatch. In addition, the proposed RPSC method does not introduce an integral path in the controller to avoid overshoot and windup issues. ESOs are used to estimate the lumped disturbances caused by load torque changes and parameter mismatches. The observer-based prediction model is then derived by using the observed state variables and the lumped disturbances. Furthermore, a modified cost function is designed to further suppress load torque disturbances, and a tuning method of the weight coefficients is also discussed.

A. Extended State Observers Design

The proposed RPSC uses two ESOs to observe lumped unknown disturbances. System external disturbances are estimated by a torque ESO, and internal disturbances caused by parameter uncertainties are estimated by a current ESO.

At the desired equilibrium point of (1), the mechanical dynamics of the SPMSM can be described as:

\(\dot{\omega}_{e}^{*}=\frac{P_{n} T_{e}^{*}}{J}-\frac{P_{n} T_{L}}{J}-\frac{B}{J} \omega_{e}^{*}\)       (18)

where T_e^* is the electromagnetic torque reference. By using (1) to subtract (18), the error model is written as:

\(\dot{\omega}_{\mathrm{err}}=\frac{P_{n} T_{e}}{J}-\frac{P_{n} T_{e}^{*}}{J}-\frac{B}{J} \omega_{\mathrm{err}}\)       (19)

where ω_err= ω_e - ω_e^*.

According to the principle of the discrete-time extended state observer [18], the electromagnetic torque reference T_e^* in (19) is expanded into a new state variable. Therefore, based on the error model (19), the torque ESO is designed as:

\(\left\{\begin{aligned} \hat{\omega}_{\mathrm{err}}(k+1)=& T_{s}\left(\frac{P_{n}}{J} T_{e}(k)-\frac{P_{n}}{J} \hat{T}_{e}^{*}(k)-\frac{B}{J} \omega_{\mathrm{err}}(k)\right.\\ &\left.+c_{1} \omega_{\mathrm{err}}(k)\right)+\left(1-c_{1} T_{s}\right) \hat{\omega}_{\mathrm{err}}(k) \\ \hat{T}_{e}^{*}(k+1)=& \hat{T}_{e}^{*}(k)+\frac{c_{2} T_{s} J}{P_{n}}\left(\hat{\omega}_{\mathrm{err}}(k)-\omega_{\mathrm{err}}(k)\right) \end{aligned}\right.\)       (20)

where \(\hat{\omega}_{\mathrm{err}}(k)\) is the estimated speed error, \(\hat{T}_{e}^{*}(k)\) is the estimated electromagnetic torque reference, and c₁ and c₂ are the torque ESO gains. The estimated \(\hat{T}_{e}^{*}(k)\) is made up of the lumped disturbances caused by load torque, friction torque and mechanical parameter mismatches.

Meanwhile, in the predictive models (6) and (7), the lumped disturbances caused by the dq-axis current cross-coupling and electrical parameter (such as R_s, L_s and ψ_f) mismatches are expanded into new state variables. The new state variables are defined as voltage errors u_{d_comp}(k) and u_{q_comp}(k). Therefore, based on (6) and (7), the current ESO is designed as:

\(\left\{\begin{aligned} \hat{i}_{d}(k+1)=T_{s} &\left(\frac{1}{L_{s}} u_{d_{-} \mathrm{comp}}(k)-\frac{R_{s}}{L_{s}} i_{d}(k)+\omega_{e}(k) i_{q}(k)\right.\\ +&\left.\frac{1}{L_{s}} u_{d}(k)+c_{3} i_{d}(k)\right)+\left(1-c_{3} T_{s}\right) \hat{i}_{d}(k) \\ u_{d_{-} \operatorname{comp}}(k+1)=& u_{d_{-} \operatorname{comp}}(k)+c_{4} T_{s} L_{s}\left(i_{d}(k)-\hat{i}_{d}(k)\right) \end{aligned}\right.\)       (21)

\(\left\{\begin{aligned} \hat{i}_{q}(k+1)=T_{s} &\left(\frac{1}{L_{s}} u_{q_{-} \mathrm{comp}}(k)-\frac{R_{s}}{L_{s}} i_{q}(k)-\omega_{e}(k) i_{d}(k)\right.\\ &\left.+\frac{1}{L_{s}} u_{q}(k)-\frac{\psi_{f} \omega_{e}(k)}{L_{s}}+c_{3} i_{q}(k)\right) \\ &+\left(1-c_{3} T_{s}\right) \hat{i}_{q}(k) \\ u_{q_{-} \operatorname{comp}}(k+1) &=u_{q_{-} \mathrm{comp}}(k)+c_{4} T_{s} L_{s}\left(i_{q}(k)-\hat{i}_{q}(k)\right) \end{aligned}\right.\)       (22)

where c₃ and c₄ are the current ESO gains, \(\hat{i}_d(k)\) and \(\hat{i}_q(k)\) are the estimated dq-axis currents, u_{d_comp}(k) is the compensation value of the d-axis voltage, and u_{q_comp}(k) is the compensation value of the q-axis voltage.

According to the Hurwitz stability criterion [20], ESOs are stable if c₁, c₂, c₃, c₄>0. Taking into account the convergence speed and noise suppression, an ESO parameter selection strategy is presented in [21], where c₁=2ω_c1, c₂ =ω_c1², c3=2ω_c2, c4=ω_c2², ω_c1 and ω_c2 are the desired bandwidths of the torque ESO and the current ESO, respectively.

B. Observer-Based Prediction Model

In this paper, the observer-based prediction model used in the RPSC is corrected in each sampling period by using the torque ESO and the current ESO. Therefore, the prediction errors caused by system disturbances can be compensated in the corrected prediction model.

According to (19), the speed prediction model is modified as (23) by using \(\hat{\omega}_{\mathrm{err}}(k)\) and \(\hat{T}_{e}^{*}(k)\). Based on (6) and (7), i_d(k) and i_q(k) are substituted by the estimated values of \(\hat{i}_d(k)\) and \(\hat{i}_q(k)\), respectively. u_d(k) and u_q(k) are compensated by the estimated values of u_{d_comp}(k) and u_{q_comp}(k), respectively. The current prediction model is modified as (24) and (25).

\(\omega_{\mathrm{err}}(k+1)=\left(1-\frac{T_{s} B}{J}\right) \hat{\omega}_{\mathrm{err}}(k)+\frac{T_{s} P_{n}}{J}\left(T_{e}(k)-\hat{T}_{e}^{*}(k)\right)\)       (23)

\(\begin{aligned} i_{d}(k+1)=&\left(1-\frac{T_{s} R_{s}}{L_{s}}\right) \hat{i}_{d}(k)+\frac{T_{s}}{L_{s}}\left(u_{d}(k)+u_{d_{-} \operatorname{comp}}(k)\right) \\ &+T_{s} \omega_{e}(k) \hat{i}_{q}(k) \end{aligned}\)       (24)

\(\begin{aligned} i_{q}(k+1)=&\left(1-\frac{T_{s} R_{s}}{L_{s}}\right) \hat{i}_{q}(k)+\frac{T_{s}}{L_{s}}\left(u_{q}(k)+u_{q_{-} \operatorname{comp}}(k)\right) \\ &-T_{s} \omega_{e}(k) \hat{i}_{d}(k)-\frac{T_{s} \psi_{f} \omega_{e}(k)}{L_{s}} \end{aligned}\)       (25)

C. Modified Cost Function

When compared with PSC, the electromagnetic torque is also regarded as a control objective to suppress load torque disturbances in the proposed RPSC strategy. It is noteworthy that the PSC method adopts a cascade-free control structure. When compared with the traditional cascade control structures, the q-axis current reference or the electromagnetic torque reference cannot be obtained directly in the PSC since there is no outer loop speed controller. The electrical time constant is generally much smaller than the mechanical time constant of the motion control system, and the slow dynamics of the electromagnetic torque reference can be directly inferred from (18). This means that the electromagnetic torque reference can be observed, and it is reasonable to assume that the observed electromagnetic torque reference of the adjacent sampling period remains constant (i.e., \(​​\hat{T}_{e}^{*}(k+1) \approx \hat{T}_{e}^{*}(k)\) ) when T_s is small enough. Thus, the observed electromagnetic torque reference is used as the future reference of the electromagnetic torque. The cost function (10) is modified as:

\(\begin{aligned} g_{2}=& \lambda_{i}\left(i_{d}^{*}(k+1)-i_{d}(k+1)\right)^{2} \\ &+\lambda_{\omega}\left(\omega_{e}^{*}(k+2)-\omega_{e}(k+2)\right)^{2} \\ &+\lambda_{T}\left(\hat{T}_{e}^{*}(k+1)-T_{e}(k+1)\right)^{2} \end{aligned}\)       (26)

where λ_i, λ_ω and λ_T are the weight coefficients of the d-axis current error term, the speed error term and the electromagnetic torque error term, respectively. T_e(k+1) is the future prediction value of the electromagnetic torque. According to the optimization strategies ∂g₂/∂u_d(k)=0 and ∂g₂/∂u_q(k)=0, the new control laws can be derived by using the new prediction models (23), (24) and (25) as follows:

\(\begin{aligned} u_{d_{-} \mathrm{ref} 2}(k)=& \lambda_{i}\left(\frac{L_{s}}{T_{s}} i_{d}^{*}(k+1)+\left(R_{s}-\frac{L_{s}}{T_{s}}\right) \hat{i}_{d}(k)\right.\\ &\left.-L_{s} \omega_{e}(k) \hat{i}_{q}(k)\right) \end{aligned}\)       (27)

\(\begin{aligned} u_{q_{-} \mathrm{ref} 2}(k)=& \frac{2 \lambda_{\omega} J L_{s}}{3 P_{n}^{2} T_{s}^{2} \psi_{f}}\left(\omega_{e}^{*}(k+2)-\left(1-\frac{T_{s} B}{J}\right) \omega_{e}(k+1)\right) \\ &+\frac{2\left(\lambda_{\omega}+\lambda_{T}\right) L_{s}}{3 P_{n} T_{s} \psi_{f}} \hat{T}_{e}^{*}(k)-\left(\lambda_{\omega}+\lambda_{T}\right)\left(\frac{L_{s}}{T_{s}} \hat{i}_{q}(k)\right.\\ &\left.-R_{s} \hat{i}_{q}(k)-L_{s} \omega_{e}(k) \hat{i}_{d}(k)-\psi_{f} \omega_{e}(k)\right) \end{aligned}\)       (28)

where u_{d_ref2}(k) and u_{q_ref2}(k) are the d-axis voltage reference and the q-axis voltage reference of the proposed RPSC method, respectively. In order to minimize the cost function g₂ using the control laws (27) and (28), the conditions ∂²g₂/∂_{ud_ref2}(k)²>0 and ∂²g₂/∂_{uq_ref2}(k)²>0 must be satisfied. Therefore, the weight coefficients should satisfy λ_i>0, λ_T>0 and λ_ω>0.

The overall scheme of the proposed RPSC strategy based on ESOs is shown in Fig. 2. The specific procedure for the proposed RPSC strategy is described as follows.

Fig. 2. Schematic block diagram of the SPMSM drive system with the proposed RPSC algorithm.

1) Measure the motor phase current, the DC bus voltage and the rotor position, and calculate the electrical angular speed.

2) Observe the electromagnetic torque reference, the current and the voltage errors according to (20), (21) and (22).

3) Predict the electrical angular speed, the d-axis current and the electromagnetic torque according to (5), (23), (24) and(25), and consider the one step delay compensation.

4) Calculate the optimal voltage reference by optimizing the cost function (26).

5) Constrain the motor current and the motor voltage peak according to (15), (16) and (17).

D. Weight Coefficients Tuning

According to the cost function (26), the performance of the proposed RPSC is closely related to the weight coefficients (λ_i, λ_ω and λ_T). In order to ensure both the steady-state performance and the dynamic performance of the RPSC, a tuning method is used to determine the three weight coefficients λ_i, λ_ω and λ_T. First, the weight coefficients (λ_i and λ_T) are calculated by using the normalization method. Thus, the weight coefficients (λ_i and λ_T) can be initialized to λ_i=1/I_N and λ_T=1/T_N, where I_N and T_N are the rated current and rated torque of the motor, respectively. Then in order to achieve an improved dynamic performance without distorting the motor current, only the weight coefficient λ_ω needs to be tuned.

The weight coefficient λ_ω is tuned by conducting a sensitivity analysis as shown in Fig. 3. The simulation operating condition is that the speed reference step is from 0 to 1000 r/min, and the rated load is suddenly added at 0.05s. This shows that the speed responses get faster with a larger λ_ω during the transient process in Fig. 3(a). However, Fig. 3(b) shows that the speed steady-state error and speed ripples increase with a larger λ_ω. In addition, Fig. 3(c) shows that the current ripples increase with a larger λ_ω. Thus, the weight coefficient λ_ω can be determined by considering the steady state performance and dynamic performance of the system.

Fig. 3. Simulation results of speed and current responses under different λ_ω. (a) Speed responses during motor start-up and torque changes. (b) Speed responses in the steady state. (c) Current responses.

IV. EXPERIMENTAL RESULTS

A. Experimental Setup

The main SPMSM parameters are presented in Table I and the laboratory platform is shown in Fig. 4. It is composed of two 2400W SPMSM motors. One is the prototype motor and the other is the load motor. The experiment is conducted using the Expert3 system of MyWay company. The core controller of the Expert3 system consists of a TMS320C6713 DSP chip and a SPARTAN-XC3S1500 FPGA chip. The C6713DSP, which has powerful computing power and can satisfy the need for verifying various algorithms, is mainly used for core algorithm computations. The FPGA is used for the signal sampling, high-speed I/O and PWM signal generation of the control circuit. Considering the operational precision and DSP computing power, the inverter switching frequency is set to 10kHz. The 2500-line incremental encoder integrated into the motor is used to measure the rotor position. The adjustable parameters of the proposed RPSC are presented in Table II. The weight coefficients of the PSC are λ_i1=0.25 and λ_ω1=35.

TABLE I MAIN PARAMETERS OF THE SPMSM

Fig. 4. Photo of the laboratory platform.

TABLE II MAIN PARAMETERS OF THE PROPOSED RPSC ALGORITHM

B. Performance Comparison of PSC and RPSC

First, the motor parameters used by the controllers are initialized to the rated values in Table I. The experimental results shown in Fig. 5 are the motor start-up and speed change responses of the PSC method and the proposed RPSC method. In order to evaluate the control performances of the two methods, some speed reference steps have been applied to the systems. The first speed reference step is from 0 to 500 r/min, the second step is from 500 to 1000 r/min and the third step is from 1000 to 500 r/min. Fig. 5 shows that the dynamic responses of the electromagnetic torque, speed and current of the proposed RPSC method are similar to those of the PSC method. However, the steady state response of the proposed RPSC method is better than that of the PSC method. Fig. 5(a) shows that the d-axis current of the PSC method has steady state errors under different speeds, and that this is caused by the model uncertainty that is not considered in the PSC method. It can be observed that the steady-state error of the d-axis current of the proposed RPSC method is almost zero in Fig. 5(c). This confirms that the proposed RPSC method has better steady-state performance than the PSC method.

Fig. 5. Experimental results of motor start-up and speed changes comparisons between: (a) PSC, (b) PSC, (c) RPSC, (d) RPSC.

Fig. 6 shows experimental responses of the PSC method and the proposed RPSC method to the rated load. The experimental operating condition is that the rated load is suddenly added at 0.8 s, and then suddenly removed at 3.15 s. When compared with Fig. 6(a), Fig. 6 (b) shows that the speed of the proposed RPSC method has a shorter recovery time than the PSC method when suddenly adding or removing the rated load. The speed recovery times of the PSC method and the proposed RPSC method are 0.08 s and 0.03 s, respectively. The phase current THD of the PSC method is 4.73% and the phase current THD of the proposed RPSC method is 3.31%, when motor speed is 2000 r/min under full load. The current THD is calculated using experimental data. These results illustrate that the proposed RPSC method has better load disturbance rejection capability than the PSC method. This is due to the use of the modified cost function and the observer-based prediction model.

Fig. 6. Experimental results of load torque changes comparison between: (a) PSC, (b) RPSC.

Experimental results of the speed reversal responses of the proposed RPSC method are shown in Fig. 7, the speed reference step applied to the system is from -1000 to 1000 r/min. Fig. 7 shows the good dynamic and steady responses of the speed, current and electromagnetic torque of the proposed RPSC method. Moreover, the d-axis current of the proposed RPSC method can remain zero under different speeds, and the phase current and the q-axis current both show the desired behavior without exceeding the maximum current limit of 10A.

Fig. 7. Experimental results of the speed reversal of the proposed RPSC method.

C. Parameter Robustness Comparison Between PSC and the Proposed RPSC

In Fig. 8, 9 and 10, the speed reference step is from 0 to 1000 r/min. |ψ_s| is the stator flux amplitude.

Fig. 8. Experimental results of the rotor flux robustness comparison between PSC and RPSC when the rotor flux is 2.5ψ_f: (a) PSC. (b) PSC. (c) RPSC. (d) RPSC.

Fig. 9. Experimental results of the stator inductance robustness comparison between PSC and RPSC when the stator inductance is 2.5L_s. (a) PSC. (b) PSC. (c) RPSC. (d) RPSC.

Fig. 10. Experimental results of the stator resistance robustness comparison between PSC and RPSC when the stator resistance is 10R_s. (a) PSC. (b) PSC. (c) RPSC. (d) RPSC.

In Fig. 8, the robustness of the rotor flux mismatch is compared between the PSC and RPSC. The rotor flux used by the controllers is 2.5ψ_f, and other motor parameters used by controllers are the rated values. Fig. 8(a) shows that the q-axis current of the PSC method is larger than the maximum current limit of 10 A during the transient process. This means that the mismatch of rotor flux deteriorates the dynamic performance of the PSC method. When compared with the PSC method, it can be observed in Fig. 8(c) that the q-axis current of the proposed RPSC method is constrained within the maximum current limit when the rotor flux is mismatched. A quantitative comparison of the steady state performance of the two methods under rotor flux mismatch is shown in Table III, where the steady-state errors and the ripples are both calculated using the acquired experimental data. It can be seen that the RPSC method has better dynamic performance and steady state performance than the PSC method under the condition of rotor flux mismatch. These results show that the proposed RPSC method can effectively weaken the adverse effects of rotor flux mismatch on the system performance.

TABLE III PERFORMANCE COMPARISON BETWEEN PSC AND RPSC UNDER ROTOR FLUX MISMATCH

In Fig. 9, the robustness of the stator inductance mismatch is compared between PSC and RPSC. The stator inductance used by the controllers is 2.5L_s, and the other motor parameters used by controllers are the rated values. In Fig. 9(a) and 9(b), the d-axis current, the q-axis current, the electromagnetic torque and the stator flux of PSC have severe ripples and the phase current of PSC also has serious distortion, when the stator inductance used by the controller is mismatched. These results show that the PSC method is very sensitive to the stator inductance, and that stator inductance mismatch leads to serious deterioration of the system performance. When compared with the PSC method, Fig. 9(c) and 9(d) show that the ripples of the d-axis current, the q-axis current, the electromagnetic torque andthe stator flux of RPSC are noticeably smaller and that the phase current of RPSC has less distortion. A quantitative comparison of the steady state performancesof PSC and RPSC under the stator inductance mismatch is given in Table IV. These results indicate that the proposed RPSC method based on ESOs has stronger stator inductance robustness than the PSC method.

TABLE IV PERFORMANCE COMPARISON BETWEEN PSC AND RPSC UNDER STATOR INDUCTANCE MISMATCH

In Fig. 10, the robustness of the stator resistance mismatch is compared between PSC and RPSC. It can be seen that the stator resistance used by controllers is 10R_s, and that the other motor parameters used by controllers are the rated values. Fig. 10(a) and 10(b) show that the q-axis current and the phase current of PSC have exceeded the maximum current limit of 10 A during the dynamic process, when the stator resistance is mismatched. However, Fig. 10(c) and 10(d) show that the q-axis current and the phase current of RPSC can be constrained within the maximum current limit of 10 A under the condition of stator resistance mismatch. A quantitative comparison of the steady state performance of PSC and RPSC under the stator resistance mismatch is shown in Table V. When compared with the PSC method, these results verify that the RPSC method has better dynamic performance and steady state performance than the PSC method under stator resistance mismatch. This means that the introduction of ESOs can improve robustness against stator resistance mismatch.

TABLE V PERFORMANCE COMPARISON BETWEEN PSC AND RPSC UNDER STATOR RESISTANCE MISMATCH

Robustness of inertia constant mismatch experimental results are shown in Fig. 11, and the other motor parameters used by the controllers are the rated values. As Fig. 11(a) and Fig. 11(b) show, the motor runs stably at 500 r/min when the inertia constant used by controller changes suddenly. It can be observed in Fig. 11 that the inertia constant mismatch has little impact on the performance of the proposed RPSC.

Fig. 11. Experimental results of the inertia constant robustness of the proposed RPSC method.

D. Performance Comparison of Field Oriented Control and the Proposed RPSC

The following experimental test is carried out to compare the performance of the proposed RPSC with field oriented control (FOC). FOC is a cascade control scheme that contains a speed-loop and a current-loop. The speed-loop uses a PI controller with K_p=12 and K_I=0.03, which is tuned by using the Ziegler-Nichols method. The current-loop consists of two PI controllers, whose parameters K_pi and K_ii are designed based on the internal model principle [23], where K_pi=L_sω_c3 and K_ii=R_sω_c3, and ω_c3 is the bandwidth of the current-loop.

In Fig. 12, the experimental operating condition is to start the motor to 1000 r/min under a load torque of 5N·m. Experimental results show that the speed adjustment times of FOC and RPSC are approximately 0.096 s and 0.052 s. When the motor speed is 1000 r/min, with a load torque of 5N·m, the phase current THDs of FOC and RPSC are 5.07% and 4.42%, respectively. When compared with Fig. 12(a), Fig. 12(b) shows that the speed and current of the proposed RPSC have faster responses than the speed and current of FOC. This means that the proposed RPSC has better steady state performance and dynamic performance than FOC.

Fig. 12. Experimental results of the motor starting under load. (a) FOC. (b) RPSC.

From the obtained experimental results, when compared with the PSC method, the proposed RPSC method has better dynamic performance when the load torque suddenly changes. The proposed RPSC method has better dynamic performance and steady state performance than the PSC method under the condition of parameter mismatch. Since the proposed RPSC strategy uses state variables estimated by ESOs to correct the prediction model in every sampling period, it reduces the sensitivity of the controller to parameters and decreases the adverse effects of the model mismatch on the control system performance. Furthermore, the proposed RPSC strategy considers the electromagnetic torque as the control target to further suppress load torque disturbance.

V. CONCLUSIONS

This paper proposes a robust predictive speed control (RPSC) strategy for dealing with the model mismatches caused by load torque changes and parameter uncertainties. The prediction model used by the RPSC strategy is compensated by two extended state observers to enhance algorithm robustness against parameter mismatch without introducing an integral path in the controller. A modified cost function is designed to further suppress the load torque disturbance. When compared with the PSC strategy, experimental results show that the proposed RPSC strategy has better load disturbance rejection capability and stronger parameter robustness. Future works will discuss and reduce the adverse effects of cogging torque, dead-time effects and encoder quantization under the proposed control strategy.