Simulation-of-Brushless-DC-using-Field-Oriented-Control-and-Extended-Kalman-Filter

Simulation of a brushless dc motor using FOC method and Extended Kalman Filter as estimation method for the velocity of the rotor based on the works of:

• XIA, C. L. Permanent Magnet Brushless DC Motor Drives and Controls. [S. l.: s. n.], 2012. ISBN 9781118188330. DOI: 10.1002/9781118188347.

• XIA, K. et al. Model predictive control method of torque ripple reduction for BLDC Motor. IEEE Transactions on Magnetics, v. 56, n. 1, p. 1–6, 2020. ISSN 19410069. DOI: 10.1109/TMAG.2019.2950953.

• O’ROURKE, C. J. et al. A Geometric Interpretation of Reference Frames and Transformations: Dq0, Clarke, and Park. IEEE Transactions on Energy Conversion, v. 34, n. 4, p. 2070–2083,2019. ISSN 15580059. DOI: 10.1109/TEC.2019.2941175.

• SCHMITZ, C. Projeto E Otimização De Motores Bldc De Imãs Permanentes Superficiais. Dissertação de Mestrado, UFSC, 2017.

• TERZIC, B.; JADRIC, M. Design and implementation of the extended Kalman filter for the speed and rotor position estimation of brushless dc motor. IEEE Transactions on Industrial Electronics, IEEE, v. 48, n. 6, p. 1065–1073, 2001. ISSN 02780046. DOI: 10.1109/41.969385.

• LIU, Y.; ZHU, Z. Q.; HOWE, D. Instantaneous torque estimation in sensorless direct-torque-controlled brushless DC motors. IEEE Transactions on Industry Applications, v. 42, n. 5, p. 1275–1283, 2006. ISSN 00939994. DOI: 10.1109/TIA.2006.880854

Em Portugues meu TCC

• https://repositorio.utfpr.edu.br/jspui/handle/1/29127

The mathematical model for the BRUSHLESS

$$ \frac{d}{dt}x = Fx+Gu\\ y=Hx $$

$$ \frac{d}{dt} \left(\begin{array}{c} I_a\\ I_b \\ I_c \\ \omega_r \\ \theta_e \end{array}\right) = \left(\begin{array}{ccccc} -\frac{R}{L-M} & 0 & 0 & -\frac{P_p\phi_mF_a(\theta_e)}{L-M} & 0\\ 0 & -\frac{R}{L-M} & 0 & -\frac{P_p\phi_mF_b(\theta_e)}{L-M} & 0\\ 0 & 0 & -\frac{R}{L-M} & -\frac{P_p\phi_mF_c(\theta_e)}{L-M} & 0\\ 0 & 0 & -\frac{R}{L-M} & -\frac{P_p\phi_mF_c(\theta_e)}{L-M} & 0\\ \frac{P_p\phi_mF_A(\theta_e)}{J} & \frac{P_p\phi_mF_B(\theta_e)}{J} & \frac{P_p\phi_mF_C(\theta_e)}{J} & -\frac{B}{J} & 0 \\ 0 & 0 & 0 & P_p & 0 \end{array}\right) \left(\begin{array}{c} I_a \\ I_b \\ I_c \\ \omega_r \\ \theta_e \end{array}\right) + \left(\begin{array}{cccc} \frac{1}{L-M} & 0 & 0 & 0\\ 0 & \frac{1}{L-M} & 0 & 0\\ 0 & 0 & \frac{1}{L-M} & 0\\ 0 & 0 & 0 & -\frac{1}{J} \\ 0 & 0 & 0 & 0 \end{array}\right) \left(\begin{array}{c} V_an \\ V_bn \\ V_cn \\ T_c \end{array}\right) $$

$$ \left(\begin{array}{c} I_a \\ I_b \\ I_c \end{array}\right) = \left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 \end{array}\right) \left(\begin{array}{c} I_a \\ I_b \\ I_c \\ \omega_r \\ \theta_e \end{array}\right) $$

$$ \dot{x} = f(x,t) + v(t) $$

$$ dF(x,t) = \left(\begin{array}{ccccc} \frac{\partial f_1}{\partial i_a} \frac{\partial f_1}{\partial i_b} \frac{\partial f_1}{\partial i_c} \frac{\partial f_1}{\partial \omega_r} \frac{\partial f_1}{\partial \theta_e} \\ \frac{\partial f_2}{\partial i_a} \frac{\partial f_2}{\partial i_b} \frac{\partial f_2}{\partial i_c} \frac{\partial f_2}{\partial \omega_r} \frac{\partial f_2}{\partial \theta_e} \\ \frac{\partial f_3}{\partial i_a} \frac{\partial f_3}{\partial i_b} \frac{\partial f_3}{\partial i_c} \frac{\partial f_3}{\partial \omega_r} \frac{\partial f_3}{\partial \theta_e} \\ \frac{\partial f_4}{\partial i_a} \frac{\partial f_4}{\partial i_b} \frac{\partial f_4}{\partial i_c} \frac{\partial f_4}{\partial \omega_r} \frac{\partial f_4}{\partial \theta_e} \\ \frac{\partial f_5}{\partial i_a} \frac{\partial f_5}{\partial i_b} \frac{\partial f_5}{\partial i_c} \frac{\partial f_5}{\partial \omega_r} \frac{\partial f_5}{\partial \theta_e} \end{array}\right) $$

$$\begin{array}{ccccc} f_1 &= \frac{1}{L_M}V_{An} -\frac{R}{L_M}i_a -\frac{P_p\phi_mf_A(\theta_e)}{L_M}\omega_m \\ f_2 &= \frac{1}{L_M}V_{An} -\frac{R}{L_M}i_b -\frac{P_p\phi_mf_B(\theta_e)}{L_M}\omega_m \\ f_3 &= \frac{1}{L_M}V_{An} -\frac{R}{L_M}i_c -\frac{P_p\phi_mf_C(\theta_e)}{L_M}\omega_m \\ f_4 &= \frac{P_p\phi_m}{J}(i_af_A(\theta_e)+i_bf_B(\theta_e)+i_cf_C(\theta_e))-\frac{1}{J}(B_v\omega_m+T_c) \\ f_5 &= P_p\omega_m \end{array} $$

Applying the local linearization whe obtain the model we will use for the Extended Kalman Filter.

$$ dF(x,t) = \left(\begin{array}{ccccc} -\frac{R}{L-M} & 0 & 0 & -\frac{P_p\phi_mF_a(\theta_e)}{L-M} & -\frac{P_p\phi_m\omega_r}{L-M}\frac{dF_a(theta_e)}{d\theta_e}\\ 0 & -\frac{R}{L-M} & 0 & -\frac{P_p\phi_mF_b(\theta_e)}{L-M} & -\frac{P_p\phi_m\omega_r}{L-M}\frac{dF_b(theta_e)}{d\theta_e}\\ 0 & 0 & -\frac{R}{L-M} & -\frac{P_p\phi_mF_c(\theta_e)}{L-M} & -\frac{P_p\phi_m\omega_r}{L-M}\frac{dF_c(theta_e)}{d\theta_e}\\ \frac{P_p\phi_mF_a(\theta_e)}{J} & \frac{P_p\phi_mF_b(\theta_e)}{J} & \frac{P_p\phi_mF_c(\theta_e}{J} & -\frac{B}{J} & \frac{P_p\phi_m}{J}(\frac{dF_a(\theta_e)}{d\theta_e}i_a + \frac{dF_b(\theta_e)}{d\theta_e}i_b + \frac{dF_c(\theta_e)}{d\theta_e}i_c) \\ 0 & 0 & 0 & P_p & 0 \end{array}\right) $$

The discrete equation for dF is

$$ dF_d = I + dF.Ts + \frac{dF^2}{2!}Ts $$

The Extended Kalman Equations

$$ \hat{x}{k|k+1} = F\hat{x}{k|k} + Gu_k + \omega_k $$

where:

• $\hat{x}_{k|k+1}$ is the sistem state vector
• F is the state transition matrix
• G is the control matrix
• $u_k$ is control vector
• $\omega_k$ is the process noise

The covariation matrix for the k|k+1 state is calculated as:

$$ P_{k|k+1} =dFP_{k|k}dF^T+Q $$

where:

• $Q$ is the noise process covariance matrix

$$ \hat{y}{k|k} = H\hat{x}{k|k} + v_n $$

where:

• $\hat{y}{k|k}$ is the measurement vector from the model
• $H$ is the observation matrix
• $v_n$ is the measurement noise

$$\tilde{y}k = y_k - \hat{y}{k|k}$$ $$ K_k = P_{k|k-1}H^T(HP_{k|k-1}H^T+R_n)^{-1}$$ $$ \hat{x}{k|k} = \hat{x}{k|k-1}+K_k\tilde{y}k$$ $$ P{k|k} = (I -K_kH)P_{k|k-1}$$

The Results

The results are presented in the Resultados folder.