DERIVATIVE-FREE KALMAN FILTER FOR STATE ESTIMATION

A derivative free Kalman filter (DFKF) [39,40] helps alleviate the problems associated with EKF, especially for nonlinear systems and yields identical performance compared
to EKF when the assumption of local linearity is not violated. It does not require any linearization and uses the deterministic sampling approach to capture the mean and covariance estimates with a minimal set of sample points or sigma points. The emphasis is shifted from linearization of nonlinear systems to the sampling approach of PDF. The fundamental difference is that in EKF, the nonlinear models are linearized to parameterize the PDF in terms of its mean and covariance, whereas in DFKF, the PDF is parameterized through nonlinear transformation of deterministically chosen sample points. The nonlinear transformation is termed as derivative free transformation (DFT) because the transformation does not involve any differentiation expression. Figure 9.16 shows a pictorial representation of DFT. Consider propagation of a random variable x of dimension L(L = 2) through a nonlinear function y = f(x). Assume that mean and covariance of sigma points (black dots on the left side of Figure 9.16) for a random variable are x and Px, respectively. These sigma points and their associated weights are deterministically created using the following equations [39]:

X0 = x

Xi = x + ( (L + A)px){; і = 1 •••, L

Xi = x – (VCL + ffip)^; і = L + 1, …,2L

The associated weights can be positive or negative, but to provide unbiased trans­formation, they must satisfy the condition ^2L1 Wjm or c) = 1. For the square root in Equation 9.104, it is proposed to use a numerically efficient and stable method such as Cholesky decomposition. The scaling parameters used for the creation of sigma points and their associated weights are (1) a to determine the spread of sigma points around x; (2) b to incorporate any prior knowledge about distribution of x, and (3) к to perform the secondary tuning. The sigma points are propagated through the nonlinear function, y, = f (xi), where і = 0,…, 2L, resulting in transformed sigma points. The mean and covariance of transformed points are formulated as

The results are generated for 25 Monte Carlo simulations. Figure 9.17 shows the comparison of true, measured, and estimated observables such as V, a, b, f, U, h. From the plots it is clear that wherever (between 0 and 5 s or around 10 s) the nonlinearity in measurement data is more severe, the performance of UDEKF is degraded compared with DFKF. This can be further proved by comparing the measurement residuals with 1 sigma bounds (i. e., ±VHPH’ + R). It is observed from Figure 9.18 that the theoretical bounds are comparable (due to the same initial conditions) for both filters but in the case of UDEKF its residuals go out of bounds more often than DFKF.