7.12. DEKF Technique

AN 773: Drive-On-Chip Design Example for Intel® MAX® 10 Devices

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7.12. DEKF Technique

In the DEKF technique, the design simultaneously executes two cooperating Kalman filters for nonlinear systems: one to estimate the states and the other to estimate the parameters.

The dual Kalman filter approach reduces the computation requirements compared to considering parameter changes as additional states in a single larger Kalman filter.

Equation 1. Parameter EvolutionThis equation describes the parameter evolution that with the measurement equation builds the first EKF.

$p (k + 1) = p (k) + χ (k)$

Equation 2. State of EvolutionThis equation represents the state evolution that combines with the measurement equation to form the second EKF.

$x (k + 1) = F (x (k), i_{L} (k), p (k)) + ξ (k)$

Equation 3. Measurement Equation

$v_{T} (k) = G (x (k), i_{L} (k), p (k)) + ψ (k)$

The measurement equation is the same for both filters. In the above equations:

vT(k) is the vector of measurements at time k
k is the discrete time
iL(k) is the load current at time k
p is parameters vector
x = [SOC; V_RC1] is the battery state vector
χ, ξ and ψ are the parameters, the state and measurement noise, with zero mean and covariance matrix Σχ, Σξ and Σψ, respectively.

Equation 4. Circuit EquationsThe circuit equation describes the actual circuit. OCV(SOC) is the open circuit voltage - the voltage that is measured externally on the battery. It is related to the internal state of charge by the empirical polynomial equation with fixed parameters P1 to P8.

$x_{k} = f ({S O C}_{k}, V_{{R C}_{k}}, i_{L_{k}}) = (\begin{matrix} {S O C}_{k} \\ V_{{R C}_{k}} \end{matrix}) = (\begin{matrix} {S O C}_{k - 1} - \frac{T}{Q_{r}} i_{i_{L_{k}}} \\ v_{{R C}_{k - 1}} e^{- T / τ} + R (1 - e^{- T / τ}) i_{L_{k}} \end{matrix})$ $v_{T_{k}} = g ({S O C}_{k}, V_{{R C}_{k}} {, i}_{L_{k}}) = O C V ({S O C}_{k}) - R_{0} i_{L_{k}} - v_{{R C}_{k}}$ $O C V (S O C) = P_{1} {S O C}^{7} + P_{2} {S O C}^{6} + P_{3} {S O C}^{5} + P_{4} {S O C}^{4} + P_{5} {S O C}^{3} + P_{6} {S O C}^{2} + P_{7} {S O C}^{1} + P_{8}$

Equation 5. DEKF Matrix EquationsThe matrix equations are derived from the circuit equations.

$x_{k} = (\begin{matrix} {S O C}_{k} \\ V_{{R C}_{k}} \end{matrix})$

$q_{k} = (\begin{matrix} {S O C}_{k} \\ 1 / τ \\ R_{k} \end{matrix})$

$A_{k} = (\begin{matrix} 1 & 0 \\ 0 & e^{- T / τ} \end{matrix})$

$C_{x_{k}} = (\frac{d O C V (S O C)}{d S O C} - 1)$

$C_{q_{k}} = (\begin{matrix} i_{L_{k}} & 0 & 0 \end{matrix}) + C_{x_{k}} \frac{d x_{k}^{-}}{d q}$

$\frac{d x_{k}^{-}}{d q} = (\begin{matrix} 1 & 0 \\ 0 & e^{- T / τ} \end{matrix}) \frac{d x_{k - 1}^{+}}{d q} + (\begin{matrix} 0 & 0 & 0 \\ 0 & T e^{- \frac{T}{τ}} (R i_{L_{k}} - V_{{R C}_{k - 1}}) & (1 - e^{- T / τ} i_{L_{k}}) \end{matrix})$

$\frac{d x_{k - 1}^{+}}{d q} = \frac{d x_{k}^{-}}{d q} - L_{x_{k - 1}} C_{q_{k - 1}}$

The following equations are standard Kalman filter equations, which you can calculate after you define the standard Kalman filter variables in terms of the specific variables of the battery model.

Equation 6. Initialization Equation

$x_{0}, P_{0} {, q}_{0} {, P}_{q_{0}}$

Equation 7. Prediction Equations

$q_{k}^{-} = q_{k - 1}^{+}$

$P_{q_{k}}^{-} = P_{q_{k - 1}}^{+} + Q_{q}$

$x_{k}^{-} = f (x_{k - 1}^{+}, u_{k - 1}, q_{k - 1}^{+})$

$P_{k}^{-} = A_{k} P_{k - 1}^{+} A_{k}^{T} + Q$

Equation 8. Correction Equation

$L_{x_{k}} = P_{k}^{-} C_{x_{k}}^{T} {(C_{x_{k}} P_{k}^{-} C_{x_{k}}^{T} + R)}^{- 1}$

$x_{k}^{+} = x_{k}^{-} + {L_{x_{k}} (y}_{k} - g (x_{k}^{-}, u_{k} {, q}_{k - 1}^{+}))$

$P_{k}^{-} = (I - L_{x_{k}} C_{x_{k}}) P_{k}^{-}$

$L_{q_{k}} = P_{q_{k}}^{-} C_{q_{k}}^{T} {(C_{q_{k}} P_{q_{k}}^{-} C_{q_{k}}^{T} + R)}^{- 1}$

$q_{k}^{+} = q_{k}^{+} + {L_{q_{k}} (y}_{k} - g (x_{k}^{-}, u_{k} {, q}_{k - 1}^{+}))$

$P_{q_{k}}^{-} = (I - L_{q_{k}} C_{q_{k}}) P_{q_{k}}^{-}$

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