Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 32
A Tutorial on Kalman Filter-Based Techniques
n
where xk ∈ R nx and yk ∈ R y are the hidden states of the system and
measurements at time k, fk-1(·) and hk(·) are known, possibly nonlinear functions; νk and nk are referred to as process and measurement
noises (assumed mutually independent stochastic processes). The
optimal Bayesian filtering solution [51] is given by the marginal
distribution p(xk|y1:k),1 which gathers all the information about the
system contained in the available observations. This distribution
can be recursively computed in two steps: i) prediction, the predictive distribution p(xk|y1:k−1) is computed using prior information,
p(xk | xk−1), and the previous distribution, and ii) update, the new
measurements yk and the predictive distribution (see Algorithm 1
in [53]) are used to obtain the new filtering distribution p(xk|y1:k).
The standard KF [51], sketched in Algorithm 1,2 provides the
closed-form solution to the optimal Bayesian filtering problem in
linear and Gaussian systems, assumptions that not always hold. A
plethora of alternatives have been proposed in recent decades to
solve the nonlinear estimation problem. Among them are the extended KF (EKF) [51], the family of sigma-point KFs [54] within
the Gaussian framework and the family of sequential Monte Carlo
methods [55] for arbitrary noise distributions. The carrier synchronization problem is just a particular application case of this general
filtering solution.
The probabilistic assumptions made by the KF are vk ∼
(0,Qk) and nk ∼ (0,Rk), with Qk and Rk being the process and
measurement covariance matrices, respectively. For linear dynamic systems, the KF always provides the linear MMSE solution, but
if the process or measurement noises are not Gaussian distributed,
the KF is no longer optimal. The filter uses a Gaussian approximation and only propagates the mean and covariance of the predictive
and posterior distributions, so intuitively the further from Gaussianity, the further from optimality.
The filtering equation in Step 7 of Algorithm 1 reveals how the
KF estimation works
(
)
xˆ k | k = Fk −1xˆ k −1| k −1 + K k y k − yˆ k | k −1 .
state prediction
measurement update
(16)
Algorithm 1 General KF formulation
Require: xˆ 0 , Px ,0|0 , Fk , H k , y k , Q k and R k ∀ k
1: Set k ⇐ 1
Time update (prediction)
2: Estimate the predicted state: xˆ k |k −1 = Fk −1xˆ k −1| k −1.
3: Estimate the predicted error covariance:
Px , k | k −1 = Fk −1Px , k −1| k −1Fk−1 + Q k .
Measurement update (estimation)
4: Estimate the predicted measurement: yˆ k | k −1 = H k xˆ k | k −1.
5: Estimate the innovation covariance matrix:
Py , k | k −1 = H k Px , k | k −1H k + R k .
1
2
32
6: Estimate the Kalman gain: K k = Px , k | k −1H k Py−,1k |k −1.
7: Estimate the updated state: xˆ k | k = xˆ k | k −1 + K k y k − yˆ k | k −1 .
8: Estimate the corresponding error covariance:
Px , k | k = Px , k | k −1 − K k H k Px , k |k −1.
9: Set k ⇐ k + 1 and go to step 2.
(
The first term takes into account the state evolution model to predict the state at the following time step, while the second one corrects this prediction by incorporating the information provided by
the new measurement yk. The term yk − yˆ k | k −1 is called innovation,
which can be seen as an error signal and a key part of the KF theory. If the filter is optimal, the innovations' sequence is a white
Gaussian process, which is a useful theoretical result to build consistency tests [56], [57].
The innovations are weighted by the time-varying Kalman gain
Kk, which is computed by using the uncertainty of the state-space
model (i.e., noise statistics) and the covariance of the estimation
error (i.e., how good the state estimation is)
(
K k = Px , k | k −1H k H k Px , k | k −1H k + R k
)
−1
.
(17)
Observing the terms involved in the gain computation, and
with a slight abuse of language, the following effects can be deduced: i) increasing the measurement noise Rk, or equivalently
the uncertainty on the observation, reduces Kk; thus, the filter is
less confident on the information provided by the observations;
ii) on the other side, increasing the system model uncertainty, Qk,
increases Px,k|k-1 and, in turn, Kk. In this situation, more weight
is given to the observations and less to the state prediction. If
the system is observable (i.e., the system states can be altered
by changing the system input) and controllable (i.e., the value
of the initial state can be determined from the system output),
the filter tends to an asymptotic regime [58], that is, both Kk
and the estimation error covariance matrix tend to steady-state
fixed values,
lim k →∞ K k → K ∞ ; lim k →∞ Px , k | k → P∞ .
The characterization of the posterior distribution allows us to
compute the minimum mean-squared error (MMSE), the maximum a posteriori (MAP), or the median of the posterior (minimax) estimators, addressing optimality in many senses [52].
The standard filtering notation is used, where the subscript k|k−1
stands for prediction at time k using measurements up to time
k−1 and k|k refers to the estimation at time k, including the complete measurements set y1:k.
)
(18)
These values only depend on the transition matrices and both
process and measurement noise statistics and thus can be computed off-line. The steady-state error covariance is obtained by
solving a discrete algebraic ℜiccati equation, and the steadystate gain is straightforwardly derived from this covariance matrix (Fig. 3). The use of such constant gain may be really useful
in applications, where computational complexity is a very critical point, because a steady-state convergence ensures that the
gain in Equation (16) does not need to be recomputed at each
time instant (that is, Kk = K∞). In this case, note that during the
transient time, the filter is no longer optimal, and the ℜicatti
equation may not converge [59].
The KF formulation is only valid for linear systems, but in
many real-life applications, the measurement function, the state
evolution, or both may be nonlinear. A classical solution is to use
the so-called EKF, which uses a linearization of such nonlinear
IEEE A&E SYSTEMS MAGAZINE
JULY 2017, Part II of II
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