FPGA Realization of Sensorless PMSM Speed Controller Based on Extended Kalman Filter

Abstract

Based on extended Kalman filter (EKF), the design and FPGA implementation of a sensorless control intellectual property (IP) for permanent magnet synchronous motor (PMSM) drive are presented in this paper. Firstly, the mathematical model for PMSM is derived and the vector control is built up. Secondly, the rotor flux angle (FA) and rotor speed which are estimated by using EKF are described. These estimated values are feedbacked to the current loop for vector control and to the speed loop for speed control. Thirdly, the very-high-speed IC hardware description language (VHDL) is adopted to describe the behavior of the sensorless speed control IP which includes the circuits of space vector pulse width modulation (SVPWM), coordinate transformation, EKF, and PI controller. Finally, to evaluate the effectiveness and correctness of the proposed of system, a cosimulation work performed by Simulink and ModelSim is firstly conducted. Then, an experimental system by FPGA chip and motor driving board is set up to further validate the performance of the proposed EKF-based sensorless speed control IP.

1. Introduction

PMSM has been increasingly used in many automation control fields as actuators, due to its advantages of superior power density, high-performance motion control with fast speed, and better accuracy. However, conventional motor control needs a speed sensor or an optical encoder to measure the rotor speed and feedback it to the controller for ensuring the precision speed control. Such sensor presents some disadvantages such as drive cost, machine size, reliability, and noise immunity. In recent years, a sensorless control without position and speed sensors for PMSM drive become a popular research topic in the literature [1–7]. Those sensorless control strategies have sliding mode observer (SMO), extended Kalman filter (EKF), reduced-order EKF, and so forth. The EKF is basically a full-order stochastic observer for the recursive optimum state estimation of a nonlinear dynamic system in real time by using signals that are in noisy environment [7]. Comparing with SMO, EKF can directly estimate the angular speed and it has high convergence rate which can give a more rapid speed response [5]. However, EKF requires heavy online 4×4 matrix computing; therefore, the complex computation becomes a challenge for a fixed-point processor system. In realization, a fixed-point processor using digital signal processor (DSP) and field programmable gate array (FPGA) both can provide a good solution in this issue. Particularly, FPGA with programmable hard-wired feature, fast computation ability, shorter design cycle, embedding processor, low power consumption, and higher density is better for the implementation of the digital system [8, 9] than DSP.

Recently, a cosimulation work by Electronic Design Automation (EDA) Simulator Link has been gradually applied to verify the effectiveness of the Verilog and VHDL code in the motor drive system [10–12]. The EDA Simulator Link [13] provides a cosimulation interface between MATLAB or Simulink and HDL simulators-ModelSim [14]. Using it you can verify a VHDL, Verilog, or mixed-language implementation against your Simulink model or MATLAB algorithm [13]. Therefore, EDA Simulator Link lets you use MATLAB code and Simulink models as a test bench that generates stimulus for an HDL simulation and analyzes the simulation’s response [13]. In this paper, a cosimulation by EDA Simulator Link is applied to sensorless speed control for PMSM drive. The PMSM, inverter, and speed command are performed in Simulink and the EKF algorithm; current vector controller and speed PI controller described by VHDL code are executed in ModelSim. After successful verification in simulation, an FPGA-based experimental system is established with configuration as in Figure 1 for realizing the proposed IP code again, and experiment results will validate the effectiveness of the sensorless speed control system of PMSM drive.

Figure 1 

Architecture of FPGA-based sensorless PMSM drive system.

2. System Description of PMSM Drive and Sensorless Speed Control

The sensorless speed control block diagram for PMSM drive is shown in Figure 1. The modeling of PMSM, the EKF-based rotor flux angle (FA) and rotor speed estimation, and the PI controller are introduced as follows.

2.1. Mathematical Model of PMSM

The mathematical model of a PMSM is described, in two-axis𝑑-𝑞synchronous rotating reference frame, as follows: 𝑑𝑖_𝑑𝑟𝑑𝑡=−^𝑠𝐿_𝑑𝑖_𝑑+𝜔_𝑒𝐿_𝑞𝐿_𝑑𝑖_𝑞+1𝐿_𝑑𝑣_𝑑,𝑑𝑖_𝑞𝑑𝑡=−𝜔_𝑒𝐿_𝑑𝐿_𝑞𝑖_𝑑−𝑟_𝑠𝐿_𝑞𝑖_𝑞−𝜔_𝑒𝜆_𝑓𝐿_𝑞+1𝐿_𝑞𝑣_𝑞, (1) where 𝑣_𝑑,𝑣_𝑞 are the 𝑑 and 𝑞 axes voltages; 𝑖_𝑑, 𝑖_𝑞, are the𝑑and𝑞axes currents,𝑟_𝑠is the phase winding resistance; 𝐿_𝑑, 𝐿_𝑞 are the 𝑑 and 𝑞 axes inductance;𝜔_𝑒is the rotating speed of magnet flux; and 𝜆_𝑓 is the permanent magnet flux linkage.

The current loop control of PMSM drive in Figure 1 is based on a vector control approach. That is, if the 𝑖_𝑑 is forced to 0, the PMSM will be decoupled, and controlling a PMSM is like controlling a DC motor. After decoupling, the motor’s torque is proportional to 𝑖_𝑞: 𝑇_𝑒=3𝑁_𝑃4𝜆_𝑓𝑖=_𝑞^Δ𝐾_𝑡𝑖_𝑞. (2) Considering the mechanical load, the overall dynamic equation of PMSM drive system is obtained by 𝐽_𝑚𝑑𝜔𝑑𝑡^𝑟+𝐵_𝑚𝜔_𝑟=𝑇_𝑒−𝑇_𝐿, (3) where 𝑇_𝑒 is the motor torque,𝑁_𝑃is pole pairs,𝐾_𝑡is torque constant,𝐽_𝑚 is the inertial value,𝐵_𝑚 is damping ratio,𝑇_𝐿 is the external torque, and 𝜔_𝑟 is rotor speed.

2.2. Extended Kalman Filter (EKF)

For a motor stochastic nonlinear equation, it can be written in the following form: 𝑥̇(𝑡)[𝑥(𝑡(𝑡)(𝑡),𝑦(𝑡)[𝑥(𝑡(𝑡),=𝑓)]+𝐵𝑢+𝜎=ℎ)]+𝜇 (4)(5) where𝑥(𝑡),𝑢(𝑡), and 𝑦(𝑡) are system state, system input, and system output, respectively. The𝜎(𝑡)and𝜇(𝑡)represent system noise and measurement noise which are zero-mean white Gaussian distribution with covariance𝑄(𝑡)and𝑅(𝑡), respectively. Once a nominal solution to a motor nonlinear equation can be found in (4)-(5), the linearized perturbation equation is 𝛿𝑥̇(𝑡)(𝑥(𝑡(𝑡)(𝑡)(𝑡),(𝑡)(𝑥(𝑡(𝑡),=𝐹))𝛿𝑥+𝐵𝛿𝑢+𝜎𝛿𝑦=𝐻))𝛿𝑥+𝜇 (6)(7) where the Jacobian and output matrices are defined as follows: 𝐹(𝑥(𝑡))=𝜕𝑓𝜕𝑥_{𝑥=𝑥(𝑡)},𝐻(𝑥(𝑡))=𝜕ℎ𝜕𝑥_{𝑥=𝑥(𝑡)}. (8) After discretization with sampling time𝑇_𝑐, (6) becomes 𝑥𝑡_𝑛=Φ𝑡_𝑛,𝑡_𝑛−1,𝑥𝑡_𝑛−1𝑥𝑡_𝑛−1+^𝑡𝑛_𝑡𝑛−1Φ𝑡_𝑛,𝑡_𝑛−1,𝑥𝑡_𝑛−1𝐵𝑑𝜏⋅𝑢𝑡_𝑛−1+𝜈𝑡_𝑛−1, (9) whereΦ(𝑡_𝑛,𝑡_𝑛−1,𝑥(𝑡_𝑛−1))is an exponential matrix and its computation can be simplified by Φ𝑡_𝑛,𝑡_𝑛−1,𝑥𝑡_𝑛−1≅I+𝐹𝑇_𝑐. (10) Further, ^𝑡𝑛_𝑡𝑛−1Φ𝑡_𝑛,𝑡_𝑛−1,𝑥𝑡_𝑛−1𝐵𝑑𝜏≅𝐵𝑇_𝑐. (11) Therefore, the discrete model of (6)-(7) becomes 𝑥(𝑛)=𝐼+𝐹𝑇_𝑐𝑥(𝑛−1)+𝐵𝑇_𝑐()(),𝑦(𝑛)(𝑛)(),⋅𝑢𝑛−1+𝜈𝑛−1=𝐻⋅𝑥+𝜉𝑛−1 (12)(13) where𝑡_𝑛=𝑛𝑇_𝑐and𝑡_𝑛−1=(𝑛−1)𝑇_𝑐.

The EKF is an optimal estimator which searches the cost function𝐽=∑^𝑚_𝑛=1𝐸{𝑥²(𝑛)}at the least square sense. The𝑥(𝑛) is defined by𝑥(𝑛)=𝑥(𝑛)−𝑥(𝑛), which is the difference of estimation of state𝑥(𝑛)and system state 𝑥(𝑛). The EKF algorithm is described by the following two-step recursive equations.

Step 1 (prediction step). From (4) and using a simple rectangular integration, 𝑥_{𝑛|𝑛−1}=𝑥_𝑛−1+𝑓𝑥_𝑛−1+𝐵⋅𝑢_𝑛−1𝑇_𝑐 (14)
or from (12) 𝑥_{𝑛|𝑛−1}=Φ_𝑛−1𝑥_𝑛−1+𝐵𝑇_𝑐⋅𝑢_𝑛−1. (15)
The covariance is updated by 𝑃_{𝑛|𝑛−1}=Φ_𝑛−1𝑃_𝑛−1Φ^𝑇_𝑛−1+𝑄_𝑑. (16)

Step 2 (innovation step). Consider 𝑥_𝑛=𝑥_{𝑛|𝑛−1}+𝐾_𝑛𝑦_𝑛−𝐻𝑥_{𝑛|𝑛−1}𝑃_𝑛,=𝑃_{𝑛|𝑛−1}−𝐾_𝑛𝐻𝑃_{𝑛|𝑛−1}. (17)(18)
The Kalman gain is calculated by 𝐾_𝑛=𝑃_{𝑛|𝑛−1}𝐻^𝑇𝐻𝑃_{𝑛|𝑛−1}𝐻^𝑇+𝑅⁻¹. (19) Figure 2 shows the procedure of EKF with two step above, after measurement update, it will be changed to time update.

Figure 2 

Demonstration of the EKF operation.

2.3. Design of the Sensorless PMSM Based on EKF

The circuit equation of PMSM on the d-q rotating coordinate in (1) is re-formulated as 𝑣_𝑑𝑣_𝑞=𝑟_𝑠+𝑠𝐿_𝑠−𝜔_𝑒𝐿_𝑠𝜔_𝑒𝐿_𝑠𝑟_𝑠+𝑠𝐿_𝑠𝑖_𝑑𝑖_𝑞+0𝜔_𝑒𝜆_𝑓, (20) where𝐿=_𝑠^Δ𝐿_𝑑=𝐿_𝑞. Transforming (20), the circuit equation of PMSM on the𝛼−𝛽fixed coordinate can be derived by the following equation: 𝑣_𝛼𝑣_𝛽=𝑟_𝑠+𝑠𝐿_𝑠00𝑟_𝑠+𝑠𝐿_𝑠𝑖_𝛼𝑖_𝛽+𝜔_𝑒𝜆_𝑓−sin𝜃_𝑒cos𝜃_𝑒, (21) where𝑣_𝛼𝑣_𝛽^𝑇is voltage on fixed coordinate;𝑖_𝛼𝑖_𝛽^𝑇is current on fixed coordinate;𝜃_𝑒is angular position at magnet flux; and 𝑠 is differential operator. If we choose the 𝑥(𝑡)=[𝑖_𝛼,𝑖_𝛽,𝜔_𝑒,𝜃_𝑒]^𝑇as the state variable of the PMSM mathematical model, then (21) can be expanded to the following formulation: 𝑑𝑖𝑑𝑡^𝛼𝑖_𝛽𝜔_𝑒𝜃_𝑒=−𝑟_𝑠𝑖_𝛼𝐿_𝑠+𝜔_𝑒𝜆_𝑓𝐿_𝑠sin𝜃_𝑒−𝑟_𝑠𝑖_𝛽𝐿_𝑠−𝜔_𝑒𝜆_𝑓𝐿_𝑠cos𝜃_𝑒0𝜔_𝑒+1𝐿_𝑠001𝐿_𝑠𝑣0000^𝛼𝑣_𝛽^Δ𝑣=𝑓+𝐵^𝛼𝑣_𝛽,𝑖_𝛼𝑖_𝛽=𝑖10000100^𝛼𝑖_𝛽𝜔_𝑒𝜃_𝑒^Δ=ℎ, (22)(23) where𝑥(𝑡)=𝑖_𝛼𝑖_𝛽𝜔_𝑒𝜃_𝑒^𝑇,𝑦(𝑡)=𝑖_𝛼𝑖_𝛽, and𝑢(𝑡)=𝑣_𝛼𝑣_𝛽^𝑇. From (8) and (22), the Jacobian and output matrices can be obtained: 𝐹(𝑥(𝑡))=𝜕𝑓𝜕𝑥_{𝑥=𝑥(𝑡)}=−𝑟_𝑠𝐿_𝑠0𝜆_𝑓𝐿_𝑠sin𝜃_𝑒𝜔_𝑒𝜆_𝑓𝐿_𝑠cos𝜃_𝑒𝑟0−^𝑠𝐿_𝑠−𝜆_𝑓𝐿_𝑠cos𝜃_𝑒𝜔_𝑒𝜆_𝑓𝐿_𝑠sin𝜃_𝑒,𝐻(𝑥(𝑡=00000010))𝜕ℎ𝜕𝑥_{𝑥=𝑥(𝑡)}=10000100. (24)(25) Further, substituting (24) into (10), the exponential matrix is shown as follows: Φ𝑡_𝑛,𝑡_𝑛−1,𝑥𝑡_𝑛−1≅𝐼+𝐹𝑇_𝑐=𝑟1−^𝑠𝑇_𝑐𝐿_𝑠0𝜆_𝑓𝑇_𝑐𝐿_𝑠sin𝜃_𝑒𝜔_𝑒𝜆_𝑓𝑇_𝑐𝐿_𝑠cos𝜃_𝑒𝑟01−^𝑠𝑇_𝑐𝐿_𝑠−𝜆_𝑓𝑇_𝑐𝐿_𝑠cos𝜃_𝑒𝜔_𝑒𝜆_𝑓𝑇_𝑐𝐿_𝑠sin𝜃_𝑒001000𝑇_𝑐1_Δ=𝐺0𝜙₁₃𝜙₁₄0𝐺𝜙₂₃𝜙₂₄001000𝑇_𝑐1. (26) As the results, the EKF algorithm in (14)–(19) can be carried out to estimate the state value. The initial values of𝑄_𝑑, 𝑅, and 𝑃₀need to be chosen. Through the recursive calculation, the state value of𝑥(𝑛)=[𝑖_𝛼̂(𝑛),𝑖_𝛽̂(𝑛),𝜔_𝑒(𝑛),𝜃_𝑒(𝑛)]^𝑇 is estimated at each sampling period; then the rotor speed can be derived by 𝜔_𝑟(𝑛)=𝜔_𝑒(𝑛)𝑁_𝑃. (27)

Finally, a summary for the estimation of rotor FA and rotor speed based on EKF is presented as follows.

Step 1. Set the initial values of 𝑄_𝑑, 𝑅, 𝑃₀, and 𝑛=1.

Step 2. Measure the values of𝑖_𝛼(𝑛),𝑖_𝛽(𝑛),𝑣_𝛼(𝑛), and 𝑣_𝛽(𝑛) from PMSM system.

Step 3. Estimate the temporary state variables from (14). In addition, refer to (22), and the scalar form of the prediction equation can be expressed as follows: 𝑖_𝛼̂(𝑛∣𝑛−1)=1−𝑇_𝑐𝑟_𝑠𝐿_𝑠𝑖_𝛼̂()+𝜔𝑛−1_𝑒(𝑛−1)𝑇_𝑐𝜆_𝑓sin𝜃_𝑒(𝑛−1)𝐿_𝑠+𝑇_𝑐𝑣_𝛼(𝑛−1)𝐿_𝑠,𝑖_𝛽̂(𝑛∣𝑛−1)=1−𝑇_𝑠𝑟_𝑠𝐿_𝑠𝑖_𝛽̂()−𝜔𝑛−1_𝑒(𝑛−1)𝑇_𝑠𝜆_𝑓cos𝜃_𝑒(𝑛−1)𝐿_𝑠+𝑇_𝑐𝑣_𝛽(𝑛−1)𝐿_𝑠,𝜔_𝑒(𝑛𝑛−1)=𝜔_𝑒(),𝜃𝑛−1_𝑒(𝑛𝑛−1)=𝜃_𝑒(𝑛−1)+𝜔_𝑒(𝑛−1)𝑇_𝑐. (28)

Step 4. Calculate theΦ_𝑛−1from (26).

Step 5. Obtain the temporary covariance matrix𝑃_{𝑛∣𝑛−1} from (16). Because matrix𝑃_{𝑛∣𝑛−1}is symmetric matrix by 𝑝_𝑖𝑗=𝑝_𝑗𝑖, it can be chosen by the following form: 𝑃_{𝑛∣𝑛−1}=𝑝₁₁𝑝₁₂𝑝₁₃𝑝₁₄𝑝₂₁𝑝₂₂𝑝₂₃𝑝₂₄𝑝₃₁𝑝₃₂𝑝₃₃𝑝₃₄𝑝₄₁𝑝₄₂𝑝₄₃𝑝₄₄. (29)

Hence, from (16), individual elements of𝑃_{𝑛|𝑛−1} could be updated as follows: 𝑝₁₁⟸𝐺²𝑝₁₁+2𝐺𝜙₁₄𝑝₁₄+2𝐺𝜙₁₃𝑝₁₃+2𝜙₁₄𝜙₁₃𝑝₃₄+𝜙²₁₄𝑝₄₄+𝜙²₁₃𝑝₃₃+𝑞₁₁,𝑝₁₂⟸𝐺²𝑝₁₂+𝐺𝜙₁₄𝑝₂₄+𝐺𝜙₁₃𝑝₂₃+𝜙₂₃𝐺𝑝₁₃+𝜙₁₃𝑝₃₃+𝜙₁₄𝑝₃₄+𝜙₂₄𝐺𝑝₁₄+𝜙₁₃𝑝₃₄+𝜙₁₄𝑝₄₄,𝑝₁₃⟸𝐺𝑝₁₃+𝜙₁₃𝑝₃₃+𝜙₁₄𝑝₃₄,𝑝₁₄⟸𝐺𝑝₁₃+𝜙₁₃𝑝₃₃+𝜙₁₄𝑝₃₄𝑇_𝑐+𝐺𝑝₁₄+𝜙₁₃𝑝₃₄+𝜙₁₄𝑝₄₄,𝑝₂₂⟸2𝐺𝜙₂₃𝑝₂₃+𝜙₂₄𝑝₂₄+𝐺²𝑝₂₂+𝜙²₂₃𝑝₃₃+𝜙²₂₄𝑝₄₄+2𝜙₂₃𝜙₂₄𝑝₃₄+𝑞₂₂,𝑝₂₃⟸𝐺𝑝₂₃+𝜙₂₃𝑝₃₃+𝜙₂₄𝑝₃₄,𝑝₂₄⟸𝐺𝑝₂₃+𝜙₂₃𝑝₃₃+𝜙₂₄𝑝₃₄𝑇_𝑐+𝐺𝑝₂₄+𝜙₂₄𝑝₄₄+𝜙₂₃𝑝₃₄,𝑝₃₃⟸𝑝₃₃+𝑞₃₃,𝑝₃₄⟸𝑇_𝑐𝑝₃₃+𝑝₃₄,𝑝₄₄⟸𝑝₄₄+𝑇²_𝑐𝑝₃₃+2𝑇_𝑐𝑝₃₄+𝑞₄₄. (30)

Step 6. Calculate the Kalman gain from (19). Using (25) and (29), the formulation is further simplified as follows. Firstly, 𝐻𝑃_{𝑛∣𝑛−1}𝐻^𝑇+𝑅⁻¹=𝑝10000100¹¹𝑝₁₂𝑝₁₃𝑝₁₄𝑝₂₁𝑝₂₂𝑝₂₃𝑝₂₄𝑝₃₁𝑝₃₂𝑝₃₃𝑝₃₄𝑝₄₁𝑝₄₂𝑝₄₃𝑝₄₄+𝑅10010000₁₁00𝑅₂₂⁻¹=𝑝₁₁𝑝₁₂𝑝₂₁𝑝₂₂+𝑅₁₁00𝑅₂₂⁻¹, (31) where𝑅represents the uncertainties of measurement variable which are very difficult to know. Hence, 𝑅is often chosen as unity matrix for simplifying the task but does not affect the outcome (31). Since𝑃₁₂=𝑃₂₁, (31) can be rewritten in the following form: 𝐻𝑃_{𝑛∣𝑛−1}𝐻^𝑇+𝑅⁻¹=1𝑝₁₁𝑝₂₂+𝑝₁₁+𝑝₂₂−𝑝²₁₂𝑝+1²²+1−𝑝₁₂−𝑝₁₂𝑝₁₁.+1 (32) To normalize the input value and to prevent the numerical overflow condition which occurred during computation, (32) is normalized and those values are all divided by 4. Therefore, (32) becomes 𝐻𝑃_{𝑛∣𝑛−1}𝐻^𝑇+𝑅⁻¹=1𝑀𝑝₂₂+14−𝑝₁₂4−𝑝₁₂4𝑝₁₁+14^Δ=𝑡₁₁𝑡₁₂𝑡₂₁𝑡₂₂, (33) where𝑀=(𝑝₁₁𝑝₂₂+𝑝₁₁+𝑝₂₂−𝑝²₁₂+1)/4. Then, from (19), the Kalman gain can be calculated by 𝐾=_𝑛^Δ𝑘₁₁𝑘₂₁𝑘₃₁𝑘₄₁𝑘₁₂𝑘₂₂𝑘₃₂𝑘₄₂=𝑝₁₁𝑝₂₁𝑝₃₁𝑝₄₁𝑝₁₂𝑝₂₂𝑝₃₂𝑝₄₂𝑡₁₁𝑡₁₂𝑡₂₁𝑡₂₂=𝑝₁₁𝑡₁₁+𝑝₁₂𝑡₂₁𝑝₂₁𝑡₁₁+𝑝₂₂𝑡₂₁𝑝₃₁𝑡₁₁+𝑝₃₂𝑡₂₁𝑝₄₁𝑡₁₁+𝑝₄₂𝑡₂₁𝑝₁₁𝑡₁₂+𝑝₁₂𝑡₂₂𝑝₂₁𝑡₁₂+𝑝₂₂𝑡₂₂𝑝₃₁𝑡₁₂+𝑝₃₂𝑡₂₂𝑝₄₁𝑡₁₂+𝑝₄₂𝑡₂₂. (34)

Step 7. Tune the present state variables from (17), and its scalar form is shown as follows: 𝑖_𝛼̂(𝑛)=𝑖_𝛼̂(𝑛𝑛−1)+𝑘₁₁𝑖_𝛼̃(𝑛)+𝑘₁₂𝑖_𝛽̃(𝑛),𝑖_𝛽̂(𝑛)=𝑖_𝛽̂(𝑛𝑛−1)+𝑘₂₁𝑖_𝛼̃(𝑛)+𝑘₂₂𝑖_𝛽̃(𝑛),𝜔_𝑒(𝑛)=𝜔_𝑒(𝑛𝑛−1)+𝑘₃₁𝑖_𝛼̃(𝑛)+𝑘₃₂𝑖_𝛽̃(𝑛),𝜃_𝑒(𝑛)=𝜃_𝑒(𝑛∣𝑛−1)+𝑘₄₁𝑖_𝛼̃(𝑛)+𝑘₄₂𝑖_𝛽̃(𝑛), (35) where𝑘_𝑖𝑗is the element in Kalman gain𝐾_𝑛⋅𝑖_𝛼̃(𝑛)=𝑖_𝛼(𝑛)−𝑖_𝛼̂(𝑛∣𝑛−1)and 𝑖_𝛽̃(𝑛)=𝑖_𝛽(𝑛)−𝑖_𝛽̂(𝑛∣𝑛−1).

Step 8. Update the present covariance matrix𝑃_𝑛from (18), where𝐾_𝑛𝐻𝑃_{𝑛∣𝑛−1}can be directly obtained as follows:𝐾_𝑛𝐻𝑃_{𝑛∣𝑛−1}=𝑘₁₁𝑝₁₁+𝑘₁₂𝑝₂₁𝑘₁₁𝑝₁₂+𝑘₁₂𝑝₂₂𝑘₁₁𝑝₁₃+𝑘₁₂𝑝₂₃𝑘₁₁𝑝₁₄+𝑘₁₂𝑝₂₄𝑘₂₁𝑝₁₁+𝑘₂₂𝑝₂₁𝑘₂₁𝑝₁₂+𝑘₂₂𝑝₂₂𝑘₂₁𝑝₁₃+𝑘₂₂𝑝₂₃𝑘₂₁𝑝₁₄+𝑘₂₂𝑝₂₄𝑘₃₁𝑝₁₁+𝑘₃₂𝑝₂₁𝑘₃₁𝑝₁₂+𝑘₃₂𝑝₂₂𝑘₃₁𝑝₁₃+𝑘₃₂𝑝₂₃𝑘₃₁𝑝₁₄+𝑘₃₂𝑝₂₄𝑘₄₁𝑝₁₁+𝑘₄₂𝑝₂₁𝑘₄₁𝑝₁₂+𝑘₄₂𝑝₂₂𝑘₄₁𝑝₁₃+𝑘₄₂𝑝₂₃𝑘₄₁𝑝₁₄+𝑘₄₂𝑝₂₄. (36)

Step 9. Calculate the rotor speed from (27); then set𝑛=𝑛+1and go back to Step 2.

3. Design of Sensorless Speed Controller IP

Figure 3 illustrates the internal architecture of the proposed FPGA implementation of a sensorless speed control IP for PMSM drive system. The internal circuit comprises a Nios II embedded processor IP and an application IP. The Nios II processor is depicted to generate the speed command, collect the response data, and communicate with external device. All programs in Nios II processor are coded in the C language. The application IP includes mainly a circuit of an EKF-based rotor FA and rotor speed estimation, a speed PI controller, a circuit for current controllers and coordinate transformation (CCCT), a SVPWM circuit, and an ADC interface circuit. The sampling frequencies of the speed control loop and current control loop are designed with 2 kHz and 16 kHz, respectively. The frequency divider generates 50 Mhz (Clk) and 12.5 Mhz (Clk-step) clock to supply all circuits in Figure 3. The internal circuit of CCCT performs the function of two PI controllers, table look-up for sin/cos function, and the coordinate transformation for Clark, Park, inverse Park, and modified inverse Clarke. The detailed circuit design of CCCT, speed PI controller, ADC interface, and SVPWM generation in Figure 3 refers to [8]. Herein, only the digital hardware circuit design of the EKF-based rotor FA and rotor speed estimation is described. To reduce the FPGA resource usage, a finite-state machine (FSM) is employed to model the EKF-based algorithm whose design procedure and mathematical formulation are presented in the previous section. The design result is shown in Figure 4 which uses one adder, one multiplier, one divider, two sin/cos look-up table, shifters, registers, and so forth and manipulates 131 steps machine to carry out the overall computation. The data type adopts 16-bit length with Q15 format and 2’s complement operation. The multiplier, adder, and divider apply Altera LPM (Library Parameterized Modules) standard. In Figure 4, the steps 𝑠₀∼𝑠₁₂ execute the calculation of Jacobian matrix and predict temporary state variables; 𝑠₁₃∼𝑠₆₈ perform the computation of temporary covariance matrix; steps 𝑠₆₉∼𝑠₇₀ are for state error calculation; steps 𝑠₇₁∼𝑠₁₂₀ update of the present covariance matrix 𝑃_𝑛 and calculate the Kalman gain 𝐾_𝑛; 𝑠₁₂₁∼𝑠₁₃₀ describe the present state tuning as well as executing the computation of rotor FA and rotor speed. According to (30), the computation of temporary covariance matrix𝑃_{𝑛∣𝑛−1}in Figure 4 is described in Figure 5. Further, according to (18), (33)-(34), and (36), the update of the present covariance matrix 𝑃_𝑛 and calculation the Kalman gain 𝐾_𝑛 in Figure 4 are illustrated in Figure 6. The operation of each step in Figure 4 is 80 ns (12.5 MHz) in FPGA; therefore a total of 131 steps need only 10.48 μs operation times. The FPGA (Altera) resource usages of EKF-based rotor FA and rotor speed estimation in Figure 4 is 4,158 LEs (logic elements) and 2,890 RAM bits. Additionally, the FPGA resource usages of the circuits of CCCT, speed PI controller, ADC interface, and SVPWM in Figure 3 are 864 LEs and 24,576 RAM, 2,043 LEs and 0RAM bits, 136 LEs and 0RAM bits, and 1,221 LEs and 0RAM bits, respectively.

Figure 3 

Internal architecture of a sensorless speed control IP in FPGA.

Figure 4 

State diagram of FSM for describing the EKF-based algorithm.

Figure 5 

State diagram of FSM for prediction of covariance matrix𝑃𝑛∣𝑛−1.

(a)

(b)

Figure 6 

State diagram of FSM for (a) the calculation of matrix 𝐾𝑛 and (b) the update of covariance matrix𝑃𝑛.

4. Simulink/ModelSim Cosimulation and Its Simulation Results

The Simulink/ModelSim cosimulation architecture for sensorless PMSM speed control system is shown in Figure 7. The SimPowerSystem-Blockset in the Simulink executes the PMSM and the IGBT-based inverter. The EDA simulator link for ModelSim executes the cosimulation using VHDL code running in ModelSim program. It executes the function of the sensorless speed controller by three works. The work-1 to work-3 of ModelSim in Figure 7, respectively, perform the function of speed loop PI controller, the function of current controller and coordinate transformation (CCCT) and SVPWM, and the function of EKF-based rotor FA and rotor speed estimation. The sampling frequency in current control loop and EKF-based estimation algorithm is designed with 16 kHz, but in speed control loop is 2 kHz. The clocks of 50 MHz and 12.5 MHz will supply all works of ModelSim. The FPGA (Altera) resource usages of work-1 to work-3 are 2,043 LEs and 0 RAM bits, 2,085 LEs and 24,576 RAM bits, and 4,158 LEs and 2,890 RAM bits, respectively.

Figure 7 

The Simulink/ModelSim cosimulation architecture for sensorless speed control of PMSM drive.

In simulation, the designed PMSM parameters used in Figure 7 are as follows: pole pairs are 4, stator phase resistance is 1.3 Ω, stator inductance is 6.3 mH, inertia is 𝐽=0.000108 kg*m², and friction factor is 𝐹=0.0013 N*m*s. PI gains in speed control loop are designed with 𝐾𝑝=0.2442 and 𝐾𝑖=0.00458. To evaluate the proposed controller performance, PMSM running at low speed (200 rpm) and inverse speed (from 200 rpm to −200 rpm or vice versa) condition is firstly considered, and its simulation results regarding as the speed step response, the current response as well as the actual rotor flux angle (FA) and the estimated rotor flux angle are shown in Figure 8. It shows that the motor speed gives a good dynamic response performance with a little overshoot and 25 ms rising time. It also shows that the estimated rotor speed can track actual rotor speed well. Figure 8(b) shows a successful vector control because𝑖_𝑑 is controlled to zero. Figure 8(c) shows that the estimated rotor FA can fast track the actual rotor FA even in the inverse speed command condition. Further, another simulation case while the PMSM running at wide-range high speed (from 0 rpm → 1000 rpm → 2000 rpm → 1000 rpm → −1000 pm → −2000 rpm) and inverse speed (from 1000 rpm to −1000 rpm) condition is tested, and its simulation results are shown in Figure 9. It presents that not only at speed tracking but also at rotor FA tracking, the estimated value adopted by EKF method can give a good follow to the real rotor value. However, except in the initial condition, the actual rotor speed gives a fast step response by 18 ms rising time, near 0 mm steady-state value, and maximum 17.5% overshoot.

(a)

(b)

(c)

Figure 8 

PMSM running at low speed (200 rpm) and inverse speed (±200 rpm) condition. (a) Speed step response, (b) current response, and (c) actual rotor FA and estimated rotor FA.

(a)

(b)

(c)

Figure 9 

PMSM running at wide-range high speed (2000 rpm) and inverse speed (±1000 rpm) condition. (a) Speed step response, (b) current response, AND (c) actual rotor FA and estimated rotor FA.

5. Experimental System and Results

After confirming the correctness of the proposed EKF-based sensorless control IP by simulation, the VHDL codes are directly applied to the experimental FPGA-based sensorless PMSM drive system. The block diagram and real experimental system are depicted in Figure 10. The main devices include a PMSM, a DE2 board with Altera CycloneII FPGA, a motor driver, and a power supplier. The parameters of the PMSM are 𝑟_𝑠=0.63 Ω, 𝐿=2.77 mH, and 4 pole pairs. The input voltage, continuous current, rating torque, rating speed, and continuous power of the PMSM are 220 V, 12 A, 2.3 N-m, 3000 rpm, and 750 W, respectively. The Altera CycloneII EP2C35 chip adopted in the design possesses 33,216 LEs, maximum 475 available I/O pins, 483,840 RAM, and 35 embedded multipliers. The chip can be embedded with a Nios II multicore processor that is equipped with several 32-bit CPU, flexibility of core size, 1 to 16 Mbytes of flash memory in the available memory chip, 1 Mbyte SRAM, 16 M byte SDRAM, and 4 Gbytes memory outside of the chip. In implementation, except the VHDL codes of CCCT, EKF, and PI controller, the VHDL code of ADC interface circuit should be added and integrated and then downloaded into FPGA. The PI gains of speed control loop in experiment are designed with 𝐾𝑝=0.512 and 𝐾𝑖=0.021.

(a)

(b)

Figure 10 

Experimental system (a) with block diagram and (b) in real system.

In the experimental case, PMSM running at low speed (200 rpm) and inverse speed (from 200 rpm to −200 rpm or vice versa) condition are firstly considered, and its experiment results regarding the speed step response as well as the actual rotor FA and the estimated rotor FA are shown in Figure 11. Figure 11(a) shows that the motor speed gives a good dynamic response performance with a little oscillation and overshoot and it has 15 ms rising time. Figure 11(b) shows that the estimated FA can fast track the actual rotor FA even in the inverse speed command condition. Further, another experimental case while the PMSM running at medium speed (from 0 rpm → 300 rpm → 600 rpm → 900 rpm → 600 rpm → 900 rpm) condition is evaluated, and its experiment results are shown in Figure 12. The Figure 12(a) presents a fast speed tracking with 15 ms rising time and no occurrence of overshoot condition. Figure 12(b) shows a successful vector control due to the𝑖_𝑑 being controlled to zero. Therefore, from the simulation results in Figures 8 and 9 and the experimental results in Figures 11 and 12, it is demonstrated that the EKF-based estimation algorithm used in sensorless PMSM drive and the behavior description by using VHDL are effective and correct.

(a)

(b)

Figure 11 

(a) Step speed response of actual rotor speed and estimated rotor speed and (b) actual rotor FA and estimated rotor FA.

(a)

(b)

Figure 12 

(a) Step speed response of actual rotor speed and estimated rotor speed and (b) current response.

6. Conclusions

This study has presented a sensorless PMSM drive based on EKF and successfully demonstrated its performance through cosimulation by using Simulink/ModelSim and implementation by using FPGA. In realization aspect, the VHDL is used to describe the behavior of EKF algorithm, and FSM is used to reduce the FPGA resource usage; therefore, it only needs 4,158 LEs and 2,890 RAM bits. In computational power aspect, the operation time to complete the computation of EKF algorithm is only 10.48 μs, which is less than the 62.5 μs (16 KHz) sampling time in current control loop. In simulated and experimental results, it shows that the use of EKF in sensorless PMSM drive can accurately estimate the rotor FA and rotor speed, and it can give a good step response performance in case of low speed control, inverse speed control, and high speed control as well.

Acknowledgment

This work was supported by the National Science Council of China under Grant no. NSC 101-2221-E-218-039.