Filtering Methods

Kalman filtering has evolved to a very high state-of-the-art method for recursive state estimation of dynamic systems [14]. It has generated worldwide extensive applications to aerospace system and target tracking problems. It has an intuitively appealing state-space formulation and it gives algorithms that can be easily imple­mented on digital computers, since the filter is essentially a numerical algorithm and is the optimal state observer for the linear systems.

Being a model-based approach, it uses the system’s model in the filter:

x(k + 1) — fx(k) + Bu(k) + Gw(k) z(k) — Hx(k) + v(k)

The process noise w is a white Gaussian sequence with zero mean and covariance matrix Q, the measurement noise v is a white Gaussian noise sequence with zero mean and covariance matrix R, and f is the n x n transition matrix that propagates the states from k to k + 1. Given the mathematical model of the dynamic system, statistics Q and R of the noise processes, the noisy measurement data, and the input, the Kalman filter (KF) obtains the optimal estimate of the state, x, of the system. It is assumed that the values of the elements of f, B, and H are known.

Intuitively, the idea is to incorporate the measurement into the data-filtering process and obtain a refined estimate of the state. The algorithm is given as follows:

State/Covariance Evolution or Propagation

x(k + 1) = fx(k) (9.40)

P(k + 1) = fP(k)fT + GQGT (9.41)

Measurement Data/Covariance Filtering/Update

r(k + 1) = z(k + 1) – H~x(k + 1) (9.42)

K = PHT (HPHT + R)-1 (9.43)

x(k + 1) = x(k + 1) + Kr(k + 1) (9.44)

P = (I – KH)P (9.45)

The matrix S = HpHT + R is the covariance matrix of the residuals/innovations. The actual residuals can be computed from Equation 9.42 and these can be compared with standard deviations obtained by taking the square root of the diagonal elements of S. The performance of the filter can be evaluated by (1) checking the whiteness of the residuals and (2) comparing the computed covariance with the theoretical covariance obtained from the covariance equations of the filter. The test signifies that (1) the residuals being white, no information is left out to be utilized in the filter and (2) the computed covariance from the data matches the filter predictions/ theoretical estimates of the covariance. Thus, a proper tuning has been achieved. The KF could diverge due to many reasons [15]: (1) modeling errors due to the use of a highly approximated model of the nonlinear system; (2) choice of incorrect a priori statistics (P, Q, R); and (3) finite word length computation. For (3) a factorization-based filtering method should be used, or the filter should be imple­mented on a computer with large word length.