Saturday, October 17, 2009

Kalman filter

The Kalman filter is an efficient recursive filter that estimates the state of a linear dynamic system from a series of noisy measurements. It is used in a wide range of engineering applications from radar to computer vision, and is an important topic in control theory and control systems engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the linear-quadratic-Gaussian control problem (LQG). The Kalman filter, the linear-quadratic regulator and the linear-quadratic-Gaussian controller are solutions to what probably are the most fundamental problems in control theory.

Example applications

An example application would be providing accurate, continuously updated information about the position and velocity of an object given only a sequence of observations about its position, each of which includes some error. For example, in a radar application where one is interested in tracking a target, information about the location, speed, and acceleration of the target is measured at each time instant with a great deal of corruption by noise. The Kalman filter exploits the trusted model of the dynamics of the target, which describes the kind of movement possible by the target, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing).
Alternatively, consider an old slow car that is known to go from 0 to 60 miles per hour (mph) in no less than 10 seconds. The speedometer on this car however shows very noisy measurements that vary wildly within a 40 mph window around the actual speed of the car. From stop – which is measured with certainty because the wheels are not turning – the driver of the car pushes its gas pedal as far as possible. Five seconds later, the speedometer reads 70 mph. The driver concludes that the slow car cannot be traveling that quickly and uses information about the known speedometer noise to conclude that the car is likely traveling at 30 mph instead. Similarly, a Kalman filter uses information about noise and system dynamics to reduce uncertainty from noisy measurements.

Underlying dynamic system model

Kalman filters are based on linear dynamical systems discretized in the time domain. They are modelled on a Markov chain built on linear operators perturbed by Gaussian noise. The state of the system is represented as a vector of real numbers. At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the visible outputs from the hidden state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space (as opposed to a discrete state space as in the hidden Markov model). Additionally, the hidden Markov model can represent an arbitrary distribution for the next value of the state variables, in contrast to the Gaussian noise model that is used for the Kalman filter. There is a strong duality between the equations of the Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani (1999).[1]
In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the framework of the Kalman filter. This means specifying the following matrices: Fk, the state-transition model; Hk, the observation model; Qk, the covariance of the process noise; Rk, the observation noise; and sometimes Bk, the control-input model for each time-step, k, as described below.
Model underlying the Kalman filter. Squares represent matrices. Ellipses represent multivariate normal distributions (with the mean and covariance matrix enclosed). Unenclosed values are vectors.
The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to
\textbf{x}_{k} = \textbf{F}_{k} \textbf{x}_{k-1} + \textbf{B}_{k} \textbf{u}_{k} + \textbf{w}_{k}
where
  • Fk is the state transition model which is applied to the previous state xk−1;
  • Bk is the control-input model which is applied to the control vector uk;
  • wk is the process noise which is assumed to be drawn from a zero mean multivariate normal distribution with covariance Qk.
\textbf{w}_{k} \sim N(0, \textbf{Q}_k)
At time k an observation (or measurement) zk of the true state xk is made according to
\textbf{z}_{k} = \textbf{H}_{k} \textbf{x}_{k} + \textbf{v}_{k}
where Hk is the observation model which maps the true state space into the observed space and vk is the observation noise which is assumed to be zero mean Gaussian white noise with covariance Rk.
\textbf{v}_{k} \sim N(0, \textbf{R}_k)
The initial state, and the noise vectors at each step {x0, w1, ..., wk, v1 ... vk} are all assumed to be mutually independent.
Many real dynamical systems do not exactly fit this model. In fact, unmodelled dynamics can seriously degrade the filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodelled dynamics depends on the input, and, therefore, can bring the estimation algorithm to unstability (to diverge). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of separating between measurement noise and unmodelled dynamics is a difficult one and is treated in control theory under the framework of robust control.

The Kalman filter

The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows, the notation \hat{\textbf{x}}_{n|m} represents the estimate of \textbf{x} at time n given observations up to, and including time m.
The state of the filter is represented by two variables:
  • \hat{\textbf{x}}_{k|k}, the a posteriori state estimate at time k given observations up to and including at time k;
  • \textbf{P}_{k|k}, the a posteriori error covariance matrix (a measure of the estimated accuracy of the state estimate).
The Kalman filter has two distinct phases: Predict and Update. The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep. In the update phase, the current a priori prediction is combined with current observation information to refine the state estimate. This improved estimate is termed the a posteriori state estimate.

Predict

Predicted (a priori) state \hat{\textbf{x}}_{k|k-1} = \textbf{F}_{k}\hat{\textbf{x}}_{k-1|k-1} + \textbf{B}_{k-1} \textbf{u}_{k-1}
Predicted (a priori) estimate covariance
 \textbf{P}_{k|k-1} = \textbf{F}_{k} \textbf{P}_{k-1|k-1} \textbf{F}_{k}^{\text{T}} + \textbf{Q}_{k-1}

Update

Innovation or measurement residual \tilde{\textbf{y}}_k = \textbf{z}_k - \textbf{H}_k\hat{\textbf{x}}_{k|k-1}
Innovation (or residual) covariance \textbf{S}_k = \textbf{H}_k \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} + \textbf{R}_k
Optimal Kalman gain \textbf{K}_k = \textbf{P}_{k|k-1}\textbf{H}_k^\text{T}\textbf{S}_k^{-1}
Updated (a posteriori) state estimate \hat{\textbf{x}}_{k|k} = \hat{\textbf{x}}_{k|k-1} + \textbf{K}_k\tilde{\textbf{y}}_k
Updated (a posteriori) estimate covariance \textbf{P}_{k|k} = (I - \textbf{K}_k \textbf{H}_k) \textbf{P}_{k|k-1}
The formula for the updated estimate covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the derivations section.

Invariants

If the model is accurate, and the values for \hat{\textbf{x}}_{0|0} and \textbf{P}_{0|0} accurately reflect the distribution of the initial state values, then the following invariants are preserved: all estimates have mean error zero
  • \textrm{E}[\textbf{x}_k - \hat{\textbf{x}}_{k|k}] = \textrm{E}[\textbf{x}_k - \hat{\textbf{x}}_{k|k-1}] = 0
  • \textrm{E}[\tilde{\textbf{y}}_k] = 0
where E[ξ] is the expected value of ξ, and covariance matrices accurately reflect the covariance of estimates
  • \textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k})
  • \textbf{P}_{k|k-1} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1})
  • \textbf{S}_{k} = \textrm{cov}(\tilde{\textbf{y}}_k)

Examples

Consider a truck on perfectly frictionless, infinitely long straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random acceleration. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocity is. We show here how we derive the model from which we create our Kalman filter.
There are no controls on the truck, so we ignore Bk and uk. Since F, H, R and Q are constant, their time indices are dropped.
The position and velocity of the truck is described by the linear state space
\textbf{x}_{k} = \begin{bmatrix} x \\ \dot{x} \end{bmatrix}
where \dot{x} is the velocity, that is, the derivative of position with respect to time.
We assume that between the (k − 1)th and kth timestep the truck undergoes a constant acceleration of ak that is normally distributed, with mean 0 and standard deviation σa. From Newton's laws of motion we conclude that
\textbf{x}_{k} = \textbf{F} \textbf{x}_{k-1} + \textbf{B}a_{k}
where
\textbf{F} = \begin{bmatrix} 1 & \Delta t \\ 0 & 1 \end{bmatrix}
and
\textbf{B} = \begin{bmatrix} \frac{\Delta t^{2}}{2} \\ \Delta t \end{bmatrix}
We find that
\textbf{Q} = \textrm{cov}(\textbf{B}a) = \textrm{E}[(\textbf{B}a)(\textbf{B}a)^{\text{T}}] = \textbf{B} \textrm{E}[a^2] \textbf{B}^{\text{T}} = \textbf{B}[\sigma_a^2]\textbf{B}^{\text{T}} = \sigma_a^2 \textbf{B}\textbf{B}^{\text{T}} (since σa is a scalar).
At each time step, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also normally distributed, with mean 0 and standard deviation σz.
\textbf{z}_{k} = \textbf{H x}_{k} + \textbf{v}_{k}
where
\textbf{H} = \begin{bmatrix} 1 & 0 \end{bmatrix}
and
\textbf{R} = \textrm{E}[\textbf{v}_k \textbf{v}_k^{\text{T}}] = \begin{bmatrix} \sigma_z^2 \end{bmatrix}
We know the initial starting state of the truck with perfect precision, so we initialize
\hat{\textbf{x}}_{0|0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}
and to tell the filter that we know the exact position, we give it a zero covariance matrix:
\textbf{P}_{0|0} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
If the initial position and velocity are not known perfectly the covariance matrix should be initialized with a suitably large number, say L, on its diagonal.
\textbf{P}_{0|0} = \begin{bmatrix} L & 0 \\ 0 & L \end{bmatrix}
The filter will then prefer the information from the first measurements over the information already in the model.

Derivations

Deriving the posterior estimate covariance matrix

Starting with our invariant on the error covariance Pk|k as above
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - \hat{\textbf{x}}_{k|k})
substitute in the definition of \hat{\textbf{x}}_{k|k}
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - (\hat{\textbf{x}}_{k|k-1} + \textbf{K}_k\tilde{\textbf{y}}_{k}))
and substitute \tilde{\textbf{y}}_k
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - (\hat{\textbf{x}}_{k|k-1} + \textbf{K}_k(\textbf{z}_k - \textbf{H}_k\hat{\textbf{x}}_{k|k-1})))
and \textbf{z}_{k}
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - (\hat{\textbf{x}}_{k|k-1} + \textbf{K}_k(\textbf{H}_k\textbf{x}_k + \textbf{v}_k - \textbf{H}_k\hat{\textbf{x}}_{k|k-1})))
and by collecting the error vectors we get
\textbf{P}_{k|k} = \textrm{cov}((I - \textbf{K}_k \textbf{H}_{k})(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1}) - \textbf{K}_k \textbf{v}_k )
Since the measurement error vk is uncorrelated with the other terms, this becomes
\textbf{P}_{k|k} = \textrm{cov}((I - \textbf{K}_k \textbf{H}_{k})(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1})) + \textrm{cov}(\textbf{K}_k \textbf{v}_k )
by the properties of vector covariance this becomes
\textbf{P}_{k|k} = (I - \textbf{K}_k \textbf{H}_{k})\textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1})(I - \textbf{K}_k \textbf{H}_{k})^{\text{T}} + \textbf{K}_k\textrm{cov}(\textbf{v}_k )\textbf{K}_k^{\text{T}}
which, using our invariant on Pk|k-1 and the definition of Rk becomes
\textbf{P}_{k|k} = (I - \textbf{K}_k \textbf{H}_{k}) \textbf{P}_{k|k-1} (I - \textbf{K}_k \textbf{H}_{k})^\text{T} + \textbf{K}_k \textbf{R}_k \textbf{K}_k^\text{T}
This formula (sometimes known as the "Joseph form" of the covariance update equation) is valid no matter what the value of Kk. It turns out that if Kk is the optimal Kalman gain, this can be simplified further as shown below.

Kalman gain derivation

The Kalman filter is a minimum mean-square error estimator. The error in the posterior state estimation is
\textbf{x}_{k} - \hat{\textbf{x}}_{k|k}
We seek to minimize the expected value of the square of the magnitude of this vector, \textrm{E}[|\textbf{x}_{k} - \hat{\textbf{x}}_{k|k}|^2]. This is equivalent to minimizing the trace of the posterior estimate covariance matrix \textbf{P}_{k|k} . By expanding out the terms in the equation above and collecting, we get:
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} - \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} \textbf{K}_k^\text{T} + \textbf{K}_k (\textbf{H}_k \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} + \textbf{R}_k) \textbf{K}_k^\text{T}

= \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} - \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} \textbf{K}_k^\text{T} + \textbf{K}_k \textbf{S}_k\textbf{K}_k^\text{T}
The trace is minimized when the matrix derivative is zero:
\frac{\partial \; \textrm{tr}(\textbf{P}_{k|k})}{\partial \;\textbf{K}_k} = -2 (\textbf{H}_k \textbf{P}_{k|k-1})^\text{T} + 2 \textbf{K}_k \textbf{S}_k = 0
Solving this for Kk yields the Kalman gain:
\textbf{K}_k \textbf{S}_k = (\textbf{H}_k \textbf{P}_{k|k-1})^\text{T} = \textbf{P}_{k|k-1} \textbf{H}_k^\text{T}
\textbf{K}_{k} = \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} \textbf{S}_k^{-1}
This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used.

Simplification of the posterior error covariance formula

The formula used to calculate the posterior error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our Kalman gain formula on the right by SkKkT, it follows that
\textbf{K}_k \textbf{S}_k \textbf{K}_k^T = \textbf{P}_{k|k-1} \textbf{H}_k^T \textbf{K}_k^T
Referring back to our expanded formula for the posterior error covariance,
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} - \textbf{P}_{k|k-1} \textbf{H}_k^T \textbf{K}_k^T + \textbf{K}_k \textbf{S}_k \textbf{K}_k^T
we find the last two terms cancel out, giving
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} = (I - \textbf{K}_{k} \textbf{H}_{k}) \textbf{P}_{k|k-1}.
This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with numerical stability, or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the posterior error covariance formula as derived above must be used.

Relationship to recursive Bayesian estimation

The true state is assumed to be an unobserved Markov process, and the measurements are the observed states of a hidden Markov model.
Hidden Markov model
Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
p(\textbf{x}_k|\textbf{x}_0,\dots,\textbf{x}_{k-1}) = p(\textbf{x}_k|\textbf{x}_{k-1})
Similarly the measurement at the k-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
p(\textbf{z}_k|\textbf{x}_0,\dots,\textbf{x}_{k}) = p(\textbf{z}_k|\textbf{x}_{k} )
Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
p(\textbf{x}_0,\dots,\textbf{x}_k,\textbf{z}_1,\dots,\textbf{z}_k) = p(\textbf{x}_0)\prod_{i=1}^k p(\textbf{z}_i|\textbf{x}_i)p(\textbf{x}_i|\textbf{x}_{i-1})
However, when the Kalman filter is used to estimate the state x, the probability distribution of interest is that associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalizing out the previous states and dividing by the probability of the measurement set.)
This leads to the predict and update steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (k - 1)-th timestep to the k-th and the probability distribution associated with the previous state, over all possible x_{k_-1}.
p(\textbf{x}_k|\textbf{Z}_{k-1}) = \int p(\textbf{x}_k | \textbf{x}_{k-1}) p(\textbf{x}_{k-1} | \textbf{Z}_{k-1} ) \, d\textbf{x}_{k-1}
The measurement set up to time t is
\textbf{Z}_{t} = \left \{ \textbf{z}_{1},\dots,\textbf{z}_{t} \right \}
The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
p(\textbf{x}_k|\textbf{Z}_{k}) = \frac{p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1})}{p(\textbf{z}_k|\textbf{Z}_{k-1})}
The denominator
p(\textbf{z}_k|\textbf{Z}_{k-1}) = \int p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1}) d\textbf{x}_k
is a normalization term.
The remaining probability density functions are
p(\textbf{x}_k | \textbf{x}_{k-1}) = N(\textbf{F}_k\textbf{x}_{k-1}, \textbf{Q}_k)
p(\textbf{z}_k|\textbf{x}_k) = N(\textbf{H}_{k}\textbf{x}_k, \textbf{R}_k)
p(\textbf{x}_{k-1}|\textbf{Z}_{k-1}) = N(\hat{\textbf{x}}_{k-1},\textbf{P}_{k-1} )
Note that the PDF at the previous timestep is inductively assumed to be the estimated state and covariance. This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for \mathbf{x}_k given the measurements \mathbf{Z}_k is the Kalman filter estimate.

Information filter

In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the information matrix and information vector respectively. These are defined as:
\textbf{Y}_{k|k} = \textbf{P}_{k|k}^{-1}
\hat{\textbf{y}}_{k|k} = \textbf{P}_{k|k}^{-1}\hat{\textbf{x}}_{k|k}
Similarly the predicted covariance and state have equivalent information forms, defined as:
\textbf{Y}_{k|k-1} = \textbf{P}_{k|k-1}^{-1}
\hat{\textbf{y}}_{k|k-1} = \textbf{P}_{k|k-1}^{-1}\hat{\textbf{x}}_{k|k-1}
as have the measurement covariance and measurement vector, which are defined as:
\textbf{I}_{k} = \textbf{H}_{k}^{\text{T}} \textbf{R}_{k}^{-1} \textbf{H}_{k}
\textbf{i}_{k} = \textbf{H}_{k}^{\text{T}} \textbf{R}_{k}^{-1} \textbf{z}_{k}
The information update now becomes a trivial sum.
\textbf{Y}_{k|k} = \textbf{Y}_{k|k-1} + \textbf{I}_{k}
\hat{\textbf{y}}_{k|k} = \hat{\textbf{y}}_{k|k-1} + \textbf{i}_{k}
The main advantage of the information filter is that N measurements can be filtered at each timestep simply by summing their information matrices and vectors.
\textbf{Y}_{k|k} = \textbf{Y}_{k|k-1} + \sum_{j=1}^N \textbf{I}_{k,j}
\hat{\textbf{y}}_{k|k} = \hat{\textbf{y}}_{k|k-1} + \sum_{j=1}^N \textbf{i}_{k,j}
To predict the information filter the information matrix and vector can be converted back to their state space equivalents, or alternatively the information space prediction can be used.
\textbf{M}_{k} = [\textbf{F}_{k}^{-1}]^{\text{T}} \textbf{Y}_{k-1|k-1} \textbf{F}_{k}^{-1}
\textbf{C}_{k} = \textbf{M}_{k} [\textbf{M}_{k}+\textbf{Q}_{k}^{-1}]^{-1}
\textbf{L}_{k} = I - \textbf{C}_{k}
\textbf{Y}_{k|k-1} = \textbf{L}_{k} \textbf{M}_{k} \textbf{L}_{k}^{\text{T}} + \textbf{C}_{k} \textbf{Q}_{k}^{-1} \textbf{C}_{k}^{\text{T}}
\hat{\textbf{y}}_{k|k-1} = \textbf{L}_{k} [\textbf{F}_{k}^{-1}]^{\text{T}}\hat{\textbf{y}}_{k-1|k-1}
Note that if F and Q are time invariant these values can be cached. Note also that F and Q need to be invertible.

Fixed-lag smoother

The optimal fixed-lag smoother provides the optimal estimate of \hat{\textbf{x}}_{k - N | k} for a given fixed-lag N using the measurements from \textbf{z}_{1} to \textbf{z}_{k}. It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:
\begin{bmatrix} \hat{\textbf{x}}_{t|t} \\ \hat{\textbf{x}}_{t-1|t} \\ \vdots \\ \hat{\textbf{x}}_{t-N+1|t} \\ \end{bmatrix} = \begin{bmatrix} I \\ 0 \\ \vdots \\ 0 \\ \end{bmatrix} \hat{\textbf{x}}_{t|t-1} + \begin{bmatrix} 0 & \ldots & 0 \\ I & 0 & \vdots \\ \vdots & \ddots & \vdots \\ 0 & \ldots & I \\ \end{bmatrix} \begin{bmatrix} \hat{\textbf{x}}_{t-1|t-1} \\ \hat{\textbf{x}}_{t-2|t-1} \\ \vdots \\ \hat{\textbf{x}}_{t-N|t-1} \\ \end{bmatrix} + \begin{bmatrix} K^{(1)} \\ K^{(2)} \\ \vdots \\ K^{(N)} \\ \end{bmatrix} y_{t|t-1}
where:
1) \hat{\textbf{x}}_{t|t-1} is estimated via a standard Kalman filter;
2) y_{t|t-1} = z(t) - \hat{\textbf{x}}_{t|t-1} is the innovation produced considering the estimate of the standard Kalman filter;
3) the various \hat{\textbf{x}}_{t-i|t} with i = 0,\ldots,N are new variables, i.e. they do not appear in the standard Kalman filter;
4) the gains are computed via the following scheme:
K^{(i)} = P^{(i)} H^{T} \left[ H P H^{T} + R \right]^{-1}
and
P^{(i)} = P \left[ \left[ F - K H \right]^{T} \right]^{i}
where P and K are the prediction error covariance and the gains of the standard Kalman filter.
Note that if we define the estimation error covariance
P_{i} := E \left[ \left( \textbf{x}_{t-i} - \hat{\textbf{x}}_{t-i|t} \right)^{*} \left( \textbf{x}_{t-i} - \hat{\textbf{x}}_{t-i|t} \right) | z_{1} \ldots z_{t} \right]
then we have that the improvement on the estimation of \textbf{x}_{t-i} is given by:
P - P_{i} = \sum_{j = 0}^{i} \left[ P^{(j)} H^{T} \left[ H P H^{T} + R \right]^{-1} H \left( P^{(i)} \right)^{T} \right]

Fixed-interval filters

The optimal fixed-interval smoother provides the optimal estimate of \hat{\textbf{x}}_{k | n} (k \leq n) using the measurements from a fixed interval \textbf{z}_{1} to \textbf{z}_{n}. This is also called Kalman Smoothing.
There exists an efficient two-pass algorithm, Rauch-Tung-Striebel Algorithm, for achieving this. The main equations of the smoother is the following (assuming \textbf{B}_{k} = \textbf{0}):
  • forward pass: regular Kalman filter algorithm
  • backward pass: \hat{\textbf{x}}_{k|n} = \tilde{F}_k \hat{\textbf{x}}_{k+1|n} + \tilde{K}_k \hat{\textbf{x}}_{k+1|k} , where
    • \tilde{\textbf{F}}_k = \textbf{F}_k^{-1} (\textbf{I} - \textbf{Q}_k \textbf{P}_{k+1|k}^{-1})
    • \tilde{\textbf{K}}_k = \textbf{F}_k^{-1} \textbf{Q}_k \textbf{P}_{k+1|k}^{-1}

Non-linear filters

The basic Kalman filter is limited to a linear assumption. However, most non-trivial systems are non-linear. The non-linearity can be associated either with the process model or with the observation model or with both.

Extended Kalman filter

Main article: Extended Kalman filter
In the extended Kalman filter, (EKF) the state transition and observation models need not be linear functions of the state but may instead be (differentiable) functions.
\textbf{x}_{k} = f(\textbf{x}_{k-1}, \textbf{u}_{k}) + \textbf{w}_{k}
\textbf{z}_{k} = h(\textbf{x}_{k}) + \textbf{v}_{k}
The function f can be used to compute the predicted state from the previous estimate and similarly the function h can be used to compute the predicted measurement from the predicted state. However, f and h cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian) is computed.
At each timestep the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.

Unscented Kalman filter

When the state transition and observation models – that is, the predict and update functions f and h (see above) – are highly non-linear, the extended Kalman filter can give particularly poor performance.[2] This is because the mean and covariance are propagated through linearization of the underlying non-linear model. The unscented Kalman filter (UKF) [2] uses a deterministic sampling technique known as the unscented transform to pick a minimal set of sample points (called sigma points) around the mean. These sigma points are then propagated through the non-linear functions, from which the mean and covariance of the estimate are then recovered. The result is a filter which more accurately captures the true mean and covariance. (This can be verified using Monte Carlo sampling or through a Taylor series expansion of the posterior statistics.) In addition, this technique removes the requirement to explicitly calculate Jacobians, which for complex functions can be a difficult task in itself (i.e., requiring complicated derivatives if done analytically or being computationally costly if done numerically).
Predict
As with the EKF, the UKF prediction can be used independently from the UKF update, in combination with a linear (or indeed EKF) update, or vice versa.
The estimated state and covariance are augmented with the mean and covariance of the process noise.
\textbf{x}_{k-1|k-1}^{a} = [ \hat{\textbf{x}}_{k-1|k-1}^{T} \quad E[\textbf{w}_{k}^{T}] \ ]^{T}
\textbf{P}_{k-1|k-1}^{a} = \begin{bmatrix} & \textbf{P}_{k-1|k-1} & & 0 & \\ & 0 & &\textbf{Q}_{k} & \end{bmatrix}
A set of 2L+1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
\chi_{k-1|k-1}^{0} = \textbf{x}_{k-1|k-1}^{a}
\chi_{k-1|k-1}^{i} =\textbf{x}_{k-1|k-1}^{a} + \left ( \sqrt{ (L + \lambda) \textbf{P}_{k-1|k-1}^{a} } \right )_{i} i = 1..L \,\!
\chi_{k-1|k-1}^{i} = \textbf{x}_{k-1|k-1}^{a} - \left ( \sqrt{ (L + \lambda) \textbf{P}_{k-1|k-1}^{a} } \right )_{i-L} i = L+1,\dots{}2L \,\!
where
\left ( \sqrt{ (L + \lambda) \textbf{P}_{k-1|k-1}^{a} } \right )_{i}
is the ith column of the matrix square root of
(L + \lambda) \textbf{P}_{k-1|k-1}^{a}
using the definition: square root A of matrix B satisfies
B \equiv A A^T.
The matrix square root should be calculated using numerically efficient and stable methods such as the Cholesky decomposition.
The sigma points are propagated through the transition function f.
\chi_{k|k-1}^{i} = f(\chi_{k-1|k-1}^{i}) \quad i = 0..2L
The weighted sigma points are recombined to produce the predicted state and covariance.
\hat{\textbf{x}}_{k|k-1} = \sum_{i=0}^{2L} W_{s}^{i} \chi_{k|k-1}^{i}
\textbf{P}_{k|k-1} = \sum_{i=0}^{2L} W_{c}^{i}\ [\chi_{k|k-1}^{i} - \hat{\textbf{x}}_{k|k-1}] [\chi_{k|k-1}^{i} - \hat{\textbf{x}}_{k|k-1}]^{T}
where the weights for the state and covariance are given by:
W_{s}^{0} = \frac{\lambda}{L+\lambda}
W_{c}^{0} = \frac{\lambda}{L+\lambda} + (1 - \alpha^2 + \beta)
W_{s}^{i} = W_{c}^{i} = \frac{1}{2(L+\lambda)}
\lambda = \alpha^2 (L+\kappa) - L \,\!
Typical values for α, β, and κ are 10 − 3, 2 and 0 respectively. (These values should suffice for most purposes.)[citation needed]
Update
The predicted state and covariance are augmented as before, except now with the mean and covariance of the measurement noise.
\textbf{x}_{k|k-1}^{a} = [ \hat{\textbf{x}}_{k|k-1}^{T} \quad E[\textbf{v}_{k}^{T}] \ ]^{T}
\textbf{P}_{k|k-1}^{a} = \begin{bmatrix} & \textbf{P}_{k|k-1} & & 0 & \\ & 0 & &\textbf{R}_{k} & \end{bmatrix}
As before, a set of 2L + 1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
\chi_{k|k-1}^{0} = \textbf{x}_{k|k-1}^{a}
\chi_{k|k-1}^{i} =\textbf{x}_{k|k-1}^{a} + \left ( \sqrt{ (L + \lambda) \textbf{P}_{k|k-1}^{a} } \right )_{i} i = 1..L \,\!
\chi_{k|k-1}^{i} = \textbf{x}_{k|k-1}^{a} - \left ( \sqrt{ (L + \lambda) \textbf{P}_{k|k-1}^{a} } \right )_{i-L} i = L+1,\dots{}2L \,\!
Alternatively if the UKF prediction has been used the sigma points themselves can be augmented along the following lines
\chi_{k|k-1} := [ \chi_{k|k-1}^T \quad E[\textbf{v}_{k}^{T}] \ ]^{T} \pm \sqrt{ (L + \lambda) \textbf{R}_{k}^{a} }
where
\textbf{R}_{k}^{a} = \begin{bmatrix} & 0 & & 0 & \\ & 0 & &\textbf{R}_{k} & \end{bmatrix}
The sigma points are projected through the observation function h.
\gamma_{k}^{i} = h(\chi_{k|k-1}^{i}) \quad i = 0..2L
The weighted sigma points are recombined to produce the predicted measurement and predicted measurement covariance.
\hat{\textbf{z}}_{k} = \sum_{i=0}^{2L} W_{s}^{i} \gamma_{k}^{i}
\textbf{P}_{z_{k}z_{k}} = \sum_{i=0}^{2L} W_{c}^{i}\ [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}] [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}]^{T}
The state-measurement cross-covariance matrix,
\textbf{P}_{x_{k}z_{k}} = \sum_{i=0}^{2L} W_{c}^{i}\ [\chi_{k|k-1}^{i} - \hat{\textbf{x}}_{k|k-1}] [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}]^{T}
is used to compute the UKF Kalman gain.
K_{k} = \textbf{P}_{x_{k}z_{k}} \textbf{P}_{z_{k}z_{k}}^{-1}
As with the Kalman filter, the updated state is the predicted state plus the innovation weighted by the Kalman gain,
\hat{\textbf{x}}_{k|k} = \hat{\textbf{x}}_{k|k-1} + K_{k}( \textbf{z}_{k} - \hat{\textbf{z}}_{k} )
And the updated covariance is the predicted covariance, minus the predicted measurement covariance, weighted by the Kalman gain.
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - K_{k} \textbf{P}_{z_{k}z_{k}} K_{k}^{T}

Kalman–Bucy filter

The Kalman–Bucy filter is a continuous time version of the Kalman filter.[3][4]
It is based on the state space model
\frac{d}{dt}\mathbf{x}(t) = \mathbf{F}(t)\mathbf{x}(t) + \mathbf{w}(t)
\mathbf{z}(t) = \mathbf{H}(t) \mathbf{x}(t) + \mathbf{v}(t)
where the covariances of the noise terms \mathbf{w}(t) and \mathbf{v}(t) are given by \mathbf{Q}(t) and \mathbf{R}(t), respectively.
The filter consists of two differential equations, one for the state estimate and one for the covariance:
\frac{d}{dt}\hat{\mathbf{x}}(t) = \mathbf{F}(t)\hat{\mathbf{x}}(t) + \mathbf{K}(t) (\mathbf{z}(t)-\mathbf{H}(t)\hat{\mathbf{x}}(t))
\frac{d}{dt}\mathbf{P}(t) = \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}^{T}(t) + \mathbf{Q}(t) - \mathbf{K}(t)\mathbf{R}(t)\mathbf{K}^{T}(t)
where the Kalman gain is given by
\mathbf{K}(t)=\mathbf{P}(t)\mathbf{H}^{T}(t)\mathbf{R}^{-1}(t)
Note that in this expression for \mathbf{K}(t) the covariance of the observation noise \mathbf{R}(t) represents at the same time the covariance of the prediction error (or innovation) \tilde{\mathbf{y}}(t)=\mathbf{z}(t)-\mathbf{H}(t)\hat{\mathbf{x}}(t); these covariances are equal only in the case of continuous time.[5]
The distinction between the prediction and update steps of discrete-time Kalman filtering does not exist in continuous time.
The second differential equation, for the covariance, is an example of a Riccati equation.

Naming and historical development

The filter is named after Rudolf E. Kalman, though Thorvald Nicolai Thiele[6][7] and Peter Swerling developed a similar algorithm earlier. Stanley F. Schmidt is generally credited with developing the first implementation of a Kalman filter. It was during a visit of Kalman to the NASA Ames Research Center that he saw the applicability of his ideas to the problem of trajectory estimation for the Apollo program, leading to its incorporation in the Apollo navigation computer. This Kalman filter was first described and partially developed in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).
Kalman filters have been vital in the implemention of the navigation systems of U.S. Navy nuclear ballistic missile submarines; and in the guidance and navigation sustems of cruise missiles such as the U.S. Navy's Tomahawk missile; the U.S. Air Force's Air Launched Cruise Missile; It is also used in the guidance and navigation systems of the NASA Space Shuttle and the attitude control and navigation systems of the International Space Station.
This digital filter is sometimes called the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, non-linear filter developed somewhat earlier by the Soviet mathematician Ruslan L. Stratonovich.[8][9] In fact, some of the equations of the special case linear filter appeared in these papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow.
In control theory, the Kalman filter is most commonly referred to as linear quadratic estimation (LQE).
A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the simple Kalman filter, the Kalman-Bucy filter, Schmidt's extended filter, the information filter, and a variety of square-root filters that were developed by Bierman, Thornton and many others. Perhaps the most commonly used type of very simple Kalman filter is the phase-locked loop, which is now ubiquitous in radios, especially frequency modulation (FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment.

Applications

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