Kalman filter

 ( Control Theory ) 

In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán.

Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically.

The algorithm works via a two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of the current , including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a Weighted mean, with more weight given to estimates with greater certainty. The algorithm is recursive. It can operate in real time, using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required.

Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán: "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible Linearity estimator in the minimum mean-square-error sense, although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian. See Uhlmann and Julier for roughly a dozen instances of this misconception in the literature.

Extensions and of the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on . The basis is a hidden Markov model such that the state space of the is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in Sensor fusion, and distributed Sensor Networks to develop distributed or consensus Kalman filtering.

This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed by the Soviet mathematician Ruslan Stratonovich.Stratonovich, R. L. (1959). Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise. Radiofizika, 2:6, pp. 892–901.Stratonovich, R. L. (1959). On the theory of optimal non-linear filtering of random functions. Theory of Probability and Its Applications, 4, pp. 223–225.Stratonovich, R. L. (1960) Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp. 1–19.Stratonovich, R. L. (1960). Conditional Markov Processes. Theory of Probability and Its Applications, 5, pp. 156–178. In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before the summer of 1961, when Kalman met with Stratonovich during a conference in Moscow.

This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).

Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarines, and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's Tomahawk missile and the U.S. Air Force's Air Launched Cruise Missile. They are also used in the guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station.

Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a Weighted mean. The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the covariance, a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursive filter and requires only the last "best guess", rather than the entire history, of a system's state to calculate a new state.

The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's gain. The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on the most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness.

When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into matrices because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances.

For this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to the physical laws of motion (the dynamic or "state transition" model). Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds. Next, in the update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull the position estimate back toward the real position but not disturb it to the point of becoming noisy and rapidly jumping.

In most applications, the internal state is much larger (has more degrees of freedom) than the few "observable" parameters which are measured. However, by combining a series of measurements, the Kalman filter can estimate the entire internal state.

For the Dempster–Shafer theory, each state equation or observation is considered a special case of a linear belief function and the Kalman filtering is a special case of combining linear belief functions on a join-tree or Markov chain. Additional methods include which use Bayes or evidential updates to the state equations.

A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed 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 electronics communications equipment.

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 following framework. This means specifying the matrices, for each time-step $k$, following:

• $\backslash mathbf\{F\}\_k$, the state-transition model;
• $\backslash mathbf\{H\}\_k$, the observation model;
• $\backslash mathbf\{Q\}\_k$, the covariance of the process noise;
• $\backslash mathbf\{R\}\_k$, the covariance of the observation noise;
• and sometimes $\backslash mathbf\{B\}\_k$, the control-input model as described below; if $\backslash mathbf\{B\}\_k$ is included, then there is also
• $\backslash mathbf\{u\}\_k$, the control vector, representing the controlling input into control-input model.

As seen below, it is common in many applications that the matrices $\backslash mathbf\{F\}$, $\backslash mathbf\{H\}$, $\backslash mathbf\{Q\}$, $\backslash mathbf\{R\}$, and $\backslash mathbf\{B\}$ are constant across time, in which case their $k$ index may be dropped.

The Kalman filter model assumes the true state at time $k$ is evolved from the state at $k-1$ according to

$\backslash mathbf\{x\}\_k\; =\; \backslash mathbf\{F\}\_k\; \backslash mathbf\{x\}\_\{k-1\}\; +\backslash mathbf\{B\}\_k\; \backslash mathbf\{u\}\_\{k\}\; +\; \backslash mathbf\{w\}\_k$

where

• $\backslash mathbf\{F\}\_k$ is the state transition model which is applied to the previous state x _k−1;
• $\backslash mathbf\{B\}\_k$ is the control-input model which is applied to the control vector $\backslash mathbf\{u\}\_k$;
• $\backslash mathbf\{w\}\_k$ is the process noise, which is assumed to be drawn from a zero mean multivariate normal distribution, $\backslash mathcal\{N\}$, with covariance, $\backslash mathbf\{Q\}\_k$: $\backslash mathbf\{w\}\_k\; \backslash sim\; \backslash mathcal\{N\}\backslash left(0,\; \backslash mathbf\{Q\}\_k\backslash right)$.

If $\backslash mathbf\{Q\}$ is independent of time, one may, following Roweis and Ghahramani, write $\backslash mathbf\{w\}\_\backslash bull$ instead of $\backslash mathbf\{w\}\_k$ to emphasize that the noise has no explicit knowledge of time.

At time $k$ an observation (or measurement) $\backslash mathbf\{z\}\_k$of the true state $\backslash mathbf\{x\}\_k$ is made according to

$\backslash mathbf\{z\}\_k\; =\; \backslash mathbf\{H\}\_k\; \backslash mathbf\{x\}\_k\; +\; \backslash mathbf\{v\}\_k$

where

• $\backslash mathbf\{H\}\_k$ is the observation model, which maps the true state space into the observed space and
• $\backslash mathbf\{v\}\_k$ is the observation noise, which is assumed to be zero mean Gaussian white noise with covariance $\backslash mathbf\{R\}\_k$: $\backslash mathbf\{v\}\_k\; \backslash sim\; \backslash mathcal\{N\}\backslash left(0,\; \backslash mathbf\{R\}\_k\backslash right)$.

Analogously to the situation for $\backslash mathbf\{w\}\_k$, one may write $\backslash mathbf\{v\}\_\backslash bull$ instead of $\backslash mathbf\{v\}\_k$ if $\backslash mathbf\{R\}$ is independent of time.

The initial state, and the noise vectors at each step $\backslash \{\; \backslash mathbf\{x\}\_0,\; \backslash mathbf\{w\}\_1,\; \backslash dots,\; \backslash mathbf\{w\}\_k,\; \backslash mathbf\{v\}\_1,\; \backslash dots,\; \backslash mathbf\{v\}\_k\; \backslash \}$ are all assumed to be mutually independent.

Many real-time dynamic systems do not exactly conform to this model. In fact, unmodeled 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 unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability (it diverges). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of distinguishing between measurement noise and unmodeled dynamics is a difficult one and is treated as a problem of control theory using robust control.

The state of the filter is represented by two variables:

• $\backslash hat\backslash mathbf\{x\}\_\{k\backslash mid\; k\}$, the a posteriori state estimate mean at time k given observations up to and including at time k;
• $\backslash mathbf\{P\}\_\{k\backslash mid\; k\}$, the a posteriori estimate covariance matrix (a measure of the estimated accuracy of the state estimate).

The algorithm structure of the Kalman filter resembles that of Alpha beta filter. The Kalman filter can be written as a single equation; however, it is most often conceptualized as 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 innovation (the pre-fit residual), i.e. the difference between the current a priori prediction and the current observation information, is multiplied by the optimal Kalman gain and combined with the previous state estimate to refine the state estimate. This improved estimate based on the current observation is termed the a posteriori state estimate.

Typically, the two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction procedures performed. Likewise, if multiple independent observations are available at the same time, multiple update procedures may be performed (typically with different observation matrices H _k). 2006 Corrected Version

The formula for the updated ( a posteriori) estimate covariance above is valid for the optimal K_k gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the derivations section, where the formula valid for any K_k is also shown.

A more intuitive way to express the updated state estimate ($\backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash mid\; k\}$) is:

$\backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash mid\; k\}\; =\; (\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k)\; \backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash mid\; k-1\}\; +\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{z\}\_k$

This expression reminds us of a linear interpolation, $x\; =\; (1-t)(a)\; +\; t(b)$ for $t$ between 0,1. In our case:

• $t$ is the matrix $\backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k$ that takes values from $0$ (high error in the sensor) to $I$ or a projection (low error).
• $a$ is the internal state $\backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash mid\; k-1\}$ estimated from the model.
• $b$ is the internal state $\backslash mathbf\{K\}\_k\; \backslash mathbf\{z\}\_k$ estimated from the measurement, assuming $\backslash mathbf\{K\}\_k$ is nonsingular.

$\backslash begin\{align\}$

 \operatorname{E}[\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k}] &= \operatorname{E}[\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k-1}] = 0 \\
                      \operatorname{E}[\tilde{\mathbf{y}}_k] &= 0
     

where $\backslash operatorname\{E\}\backslash xi$ is the expected value of $\backslash xi$. That is, all estimates have a mean error of zero.

Also:

$\backslash begin\{align\}$

   \mathbf{P}_{k\mid k} &= \operatorname{cov}\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k}\right) \\
 \mathbf{P}_{k\mid k-1} &= \operatorname{cov}\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k-1}\right) \\
           \mathbf{S}_k &= \operatorname{cov}\left(\tilde{\mathbf{y}}_k\right)
     

so covariance matrices accurately reflect the covariance of estimates.

]] Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of the truck every Δ t seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and velocity. We show here how we derive the model from which we create our Kalman filter.

Since $\backslash mathbf\{F\},\; \backslash mathbf\{H\},\; \backslash mathbf\{R\},\; \backslash mathbf\{Q\}$ are constant, their time indices are dropped.

The position and velocity of the truck are described by the linear state space

$\backslash mathbf\{x\}\_k\; =\; \backslash begin\{bmatrix\}$

   x \\
   \dot{x}
 \end{bmatrix}
     

where $\backslash dot\{x\}$ is the velocity, that is, the derivative of position with respect to time.

We assume that between the ( k − 1) and k timestep, uncontrolled forces cause a constant acceleration of a _k that is normally distributed with mean 0 and standard deviation σ _a. From Newton's laws of motion we conclude that

$\backslash mathbf\{x\}\_k\; =\; \backslash mathbf\{F\}\; \backslash mathbf\{x\}\_\{k-1\}\; +\; \backslash mathbf\{G\}\; a\_k$

(there is no $\backslash mathbf\{B\}u$ term since there are no known control inputs. Instead, a _k is the effect of an unknown input and $\backslash mathbf\{G\}$ applies that effect to the state vector) where

$\backslash begin\{align\}$

 \mathbf{F} &= \begin{bmatrix}
   1 & \Delta t \\
   0 & 1
 \end{bmatrix} \\[4pt]
 \mathbf{G} &= \begin{bmatrix}
   \frac{1}{2}{\Delta t}^2 \\[6pt]
   \Delta t
 \end{bmatrix}
     

so that

$\backslash mathbf\{x\}\_k\; =\; \backslash mathbf\{F\}\; \backslash mathbf\{x\}\_\{k-1\}\; +\; \backslash mathbf\{w\}\_k$

where

$\backslash begin\{align\}$

 \mathbf{w}_k &\sim N(0, \mathbf{Q}) \\
   \mathbf{Q} &=
 \mathbf{G}\mathbf{G}^\textsf{T}\sigma_a^2 =
 \begin{bmatrix}
   \frac{1}{4}{\Delta t}^4 & \frac{1}{2}{\Delta t}^3 \\[6pt]
   \frac{1}{2}{\Delta t}^3 & {\Delta t}^2
 \end{bmatrix}\sigma_a^2.
     

The matrix $\backslash mathbf\{Q\}$ is not full rank (it is of rank one if $\backslash Delta\; t\; \backslash neq\; 0$). Hence, the distribution $N(0,\; \backslash mathbf\{Q\})$ is not absolutely continuous and has no probability density function. Another way to express this, avoiding explicit degenerate distributions is given by

$\backslash mathbf\{w\}\_k\; \backslash sim\; \backslash mathbf\{G\}\; \backslash cdot\; N\backslash left(0,\; \backslash sigma\_a^2\backslash right).$

At each time phase, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise v _k is also distributed normally, with mean 0 and standard deviation σ _z.

$\backslash mathbf\{z\}\_k\; =\; \backslash mathbf\{H\; x\}\_k\; +\; \backslash mathbf\{v\}\_k$

where

and

 \mathbf{R} = \mathrm{E}\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right]
            = \begin{bmatrix} \sigma_z^2 \end{bmatrix}
     

We know the initial starting state of the truck with perfect precision, so we initialize

$\backslash hat\{\backslash mathbf\{x\}\}\_\{0\; \backslash mid\; 0\}\; =\; \backslash begin\{bmatrix\}\; 0\; \backslash \backslash \; 0\; \backslash end\{bmatrix\}$

and to tell the filter that we know the exact position and velocity, we give it a zero covariance matrix:

$\backslash mathbf\{P\}\_\{0\; \backslash mid\; 0\}\; =\; \backslash begin\{bmatrix\}$

   0 & 0 \\
   0 & 0
 \end{bmatrix}
     

If the initial position and velocity are not known perfectly, the covariance matrix should be initialized with suitable variances on its diagonal:

$\backslash mathbf\{P\}\_\{0\; \backslash mid\; 0\}\; =\; \backslash begin\{bmatrix\}$

   \sigma_x^2 & 0 \\
   0          & \sigma_\dot{x}^2
 \end{bmatrix}
     

The filter will then prefer the information from the first measurements over the information already in the model.

$\backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash mid\; k\}\; =\; \backslash mathbf\{F\}\_k\; \backslash hat\{\backslash mathbf\{x\}\}\_\{k-1\backslash mid\; k-1\}\; +\; \backslash mathbf\{K\}\_k\backslash mathbf\{z\}\_k.$

A similar equation holds if we include a non-zero control input. Gain matrices $\backslash mathbf\{K\}\_k$ and covariance matrices $\backslash mathbf\{P\}\_\{k\backslash mid\; k\}$ evolve independently of the measurements $\backslash mathbf\{z\}\_k$. From above, the four equations needed for updating the matrices are as follows:

$\backslash begin\{align\}$

 \mathbf{P}_{k\mid k-1} &= \mathbf{F}_k \mathbf{P}_{k-1\mid k-1} \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k, \\
           \mathbf{S}_k &= \mathbf{H}_k \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k, \\
           \mathbf{K}_k &= \mathbf{P}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}, \\
        \mathbf{P}_{k|k} &= \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \mathbf{P}_{k|k-1}.
     

Since these depend only on the model, and not the measurements, they may be computed offline. Convergence of the gain matrices $\backslash mathbf\{K\}\_k$ to an asymptotic matrix $\backslash mathbf\{K\}\_\backslash infty$ applies for conditions established in Walrand and Dimakis. If the $\backslash mathbf\{P\}\_\{k\backslash mid\; k\}$ series converges, then it converges exponentially to an asymptotic $\backslash mathbf\{P\}\_\backslash infty$, assuming non-zero plant noise.Kalman, Rudolf Emil, T. S. Englar, and Richard S. Bucy. Fundamental study of adaptive control systems. Clearinghouse, US Department of Commerce, 1962. Recent analysis has shown that the rate and nature of this convergence can involve multiple coequal modes, including oscillatory components, depending on the eigenstructure of the Jacobian of the above Riccati map evaluated at $\backslash mathbf\{P\}\_\backslash infty$. For the moving truck example described above, with $\backslash Delta\; t\; =\; 1$ and $\backslash sigma\_a^2=\backslash sigma\_z^2\; =\backslash sigma\_x^2=\; \backslash sigma\_\backslash dot\{x\}^2=1$, simulation shows convergence in $10$ iterations.

Using the asymptotic gain, and assuming $\backslash mathbf\{H\}\_k$ and $\backslash mathbf\{F\}\_k$ are independent of $k$, the Kalman filter becomes a linear time-invariant filter:

$\backslash hat\{\backslash mathbf\{x\}\}\_\{k\}\; =\; \backslash mathbf\{F\}\; \backslash hat\{\backslash mathbf\{x\}\}\_\{k-1\}\; +\; \backslash mathbf\{K\}\_\backslash infty\backslash mathbf\{z\}\_k.$

The asymptotic gain $\backslash mathbf\{K\}\_\backslash infty$, if it exists, can be computed by first solving the following discrete Riccati equation for the asymptotic state covariance $\backslash mathbf\{P\}\_\backslash infty$:

$\backslash mathbf\{P\}\_\backslash infty\; =\; \backslash mathbf\{F\}\backslash left(\backslash mathbf\{P\}\_\backslash infty\; -\; \backslash mathbf\{P\}\_\backslash infty\; \backslash mathbf\{H\}^\backslash textsf\{T\}\; \backslash left(\backslash mathbf\{H\}\backslash mathbf\{P\}\_\backslash infty\backslash mathbf\{H\}^\backslash textsf\{T\}\; +\; \backslash mathbf\{R\}\backslash right)\; ^\{-1\}\; \backslash mathbf\{H\}\backslash mathbf\{P\}\_\backslash infty\backslash right)\; \backslash mathbf\{F\}^\backslash textsf\{T\}\; +\; \backslash mathbf\{Q\}.$

The asymptotic gain is then computed as before.

$\backslash mathbf\{K\}\_\backslash infty\; =\; \backslash mathbf\{P\}\_\backslash infty\; \backslash mathbf\{H\}^\backslash textsf\{T\}\; \backslash left(\; \backslash mathbf\{R\}\; +\; \backslash mathbf\{H\}\; \backslash mathbf\{P\}\_\backslash infty\; \backslash mathbf\{H\}^\backslash textsf\{T\}\; \backslash right)\; ^\{-1\}.$

Additionally, a form of the asymptotic Kalman filter more commonly used in control theory is given by

$\{\backslash displaystyle\; \{\backslash hat\; \{\backslash mathbf\; \{x\}\; \}\}\_\{k+1\}=\backslash mathbf\; \{F\}\{\backslash hat\; \{\backslash mathbf\; \{x\}\; \}\}\_\{k\}+\; \backslash mathbf\; \{B\}\backslash mathbf\{u\}\_k\; +\; \backslash mathbf\; \backslash overline\{K\}\; \_\{\backslash infty\}\backslash mathbf,\}$

where

$\backslash overline\{\backslash mathbf\{K\}\}\_\backslash infty\; =\; \backslash mathbf\{F\}\backslash mathbf\{P\}\_\backslash infty\; \backslash mathbf\{H\}^\backslash textsf\{T\}\; \backslash left(\; \backslash mathbf\{R\}\; +\; \backslash mathbf\{H\}\; \backslash mathbf\{P\}\_\backslash infty\; \backslash mathbf\{H\}^\backslash textsf\{T\}\; \backslash right)\; ^\{-1\}.$

This leads to an estimator of the form

$\{\backslash displaystyle\; \{\backslash hat\; \{\backslash mathbf\; \{x\}\; \}\}\_\{k+1\}=(\backslash mathbf\; \{F\}-\backslash overline\{\backslash mathbf\{K\}\}\; \_\{\backslash infty\}\backslash mathbf\; \{H\})\{\backslash hat\; \{\backslash mathbf\; \{x\}\; \}\}\_\{k\}\; +\; \backslash mathbf\; \{B\}\backslash mathbf\{u\}\_k\; +\backslash mathbf\; \backslash overline\{K\}\; \_\{\backslash infty\}\backslash mathbf\; \{z\}\; \_\{k\},\}$

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash operatorname\{cov\}\backslash left(\backslash mathbf\{x\}\_k\; -\; \backslash hat\{\backslash mathbf\{x\}\}\_\{k\; \backslash mid\; k\}\backslash right)$

substitute in the definition of $\backslash hat\{\backslash mathbf\{x\}\}\_\{k\; \backslash mid\; k\}$

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash operatorname\{cov\}\backslash left\backslash mathbf\{x\}\_k$

and substitute $\backslash tilde\{\backslash mathbf\{y\}\}\_k$

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash operatorname\{cov\}\backslash left(\backslash mathbf\{x\}\_k\; -\; \backslash left\backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash right)$

and $\backslash mathbf\{z\}\_k$

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash operatorname\{cov\}\backslash left(\backslash mathbf\{x\}\_k\; -\; \backslash left\backslash hat\{\backslash mathbf\{x\}\}\_\{k\backslash right)$

and by collecting the error vectors we get

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash operatorname\{cov\}\backslash left\backslash left(\backslash mathbf\{I\}$

Since the measurement error v _k is uncorrelated with the other terms, this becomes

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash operatorname\{cov\}\backslash left\backslash left(\backslash mathbf\{I\}\; +\; \backslash operatorname\{cov\}\backslash left\backslash mathbf\{K\}\_k$

by the properties of vector covariance this becomes

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\backslash right)\backslash operatorname\{cov\}\backslash left(\backslash mathbf\{x\}\_k\; -\; \backslash hat\{\backslash mathbf\{x\}\}\_\{k\; \backslash mid\; k\; -\; 1\}\backslash right)\backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\backslash right)^\backslash textsf\{T\}\; +\; \backslash mathbf\{K\}\_k\backslash operatorname\{cov\}\backslash left(\backslash mathbf\{v\}\_k\backslash right)\backslash mathbf\{K\}\_k^\backslash textsf\{T\}$

which, using our invariant on P _{k |  k−1} and the definition of R _k becomes

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\backslash right)\; \backslash mathbf\{P\}\_\{k\; \backslash mid\; k\; -\; 1\}\; \backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\backslash right)^\backslash textsf\{T\}\; +\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{R\}\_k\; \backslash mathbf\{K\}\_k^\backslash textsf\{T\}$

This formula (sometimes known as the Joseph form of the covariance update equation) is valid for any value of K _k. It turns out that if K _k is the optimal Kalman gain, this can be simplified further as shown below.

$\backslash mathbf\{x\}\_k\; -\; \backslash hat\{\backslash mathbf\{x\}\}\_\{k\; \backslash mid\; k\}$

We seek to minimize the expected value of the square of the magnitude of this vector, $\backslash operatorname\{E\}\backslash left\backslash left\backslash |\backslash mathbf\{x\}\_\{k\}$. This is equivalent to minimizing the trace of the a posteriori estimate covariance matrix $\backslash mathbf\{P\}\_\{k|k\}$. By expanding out the terms in the equation above and collecting, we get:

$\backslash begin\{align\}$

 \mathbf{P}_{k\mid k} & = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k\right) \mathbf{K}_k^\textsf{T} \\[6pt]
                      &= \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \mathbf{S}_k\mathbf{K}_k^\textsf{T}
     

The trace is minimized when its matrix calculus with respect to the gain matrix is zero. Using the gradient matrix rules and the symmetry of the matrices involved we find that

$\backslash frac\{\backslash partial\; \backslash ;\; \backslash operatorname\{tr\}(\backslash mathbf\{P\}\_\{k\backslash mid\; k\})\}\{\backslash partial\; \backslash ;\backslash mathbf\{K\}\_k\}\; =\; -2\; \backslash left(\backslash mathbf\{H\}\_k\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\backslash right)^\backslash textsf\{T\}\; +\; 2\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{S\}\_k\; =\; 0.$

Solving this for K _k yields the Kalman gain:

$\backslash begin\{align\}$

 \mathbf{K}_k \mathbf{S}_k &= \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1}\right)^\textsf{T} = \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \\
  \Rightarrow \mathbf{K}_k &= \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}
     

This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used.

$\backslash mathbf\{K\}\_k\; \backslash mathbf\{S\}\_k\; \backslash mathbf\{K\}\_k^\backslash textsf\{T\}\; =\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\; \backslash mathbf\{H\}\_k^\backslash textsf\{T\}\; \backslash mathbf\{K\}\_k^\backslash textsf\{T\}$

Referring back to our expanded formula for the a posteriori error covariance,

$\backslash mathbf\{P\}\_\{k\backslash mid\; k\}\; =\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\; -\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\; \backslash mathbf\{H\}\_k^\backslash textsf\{T\}\; \backslash mathbf\{K\}\_k^\backslash textsf\{T\}\; +\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{S\}\_k\; \backslash mathbf\{K\}\_k^\backslash textsf\{T\}$

we find the last two terms cancel out, giving

$\backslash mathbf\{P\}\_\{k\backslash mid\; k\}\; =\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\; =\; (\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k)\; \backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}$

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 a posteriori error covariance formula as derived above (Joseph form) must be used.

$\backslash mathbf\{P\}\_\{k\backslash mid\; k\}\; =\; \backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\backslash mathbf\{H\}\_k\backslash right)\backslash mathbf\{P\}\_\{k\backslash mid\; k-1\}\backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\backslash mathbf\{H\}\_k\backslash right)^\backslash textsf\{T\}\; +\; \backslash mathbf\{K\}\_k\backslash mathbf\{R\}\_k\backslash mathbf\{K\}\_k^\backslash textsf\{T\}$

no longer provides the actual error covariance. In other words, $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; \backslash neq\; E\backslash left\backslash left(\backslash mathbf\{x\}\_k$. In most real-time applications, the covariance matrices that are used in designing the Kalman filter are different from the actual (true) noise covariances matrices. This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices $\backslash mathbf\{F\}\_k$ and $\backslash mathbf\{H\}\_k$ that are fed as inputs to the filter are incorrect. Thus, the sensitivity analysis describes the robustness (or sensitivity) of the estimator to misspecified statistical and parametric inputs to the estimator.

This discussion is limited to the error sensitivity analysis for the case of statistical uncertainties. Here the actual noise covariances are denoted by $\backslash mathbf\{Q\}^a\_k$ and $\backslash mathbf\{R\}^a\_k$ respectively, whereas the design values used in the estimator are $\backslash mathbf\{Q\}\_k$ and $\backslash mathbf\{R\}\_k$ respectively. The actual error covariance is denoted by $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}^a$ and $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}$ as computed by the Kalman filter is referred to as the Riccati variable. When $\backslash mathbf\{Q\}\_k\; \backslash equiv\; \backslash mathbf\{Q\}^a\_k$ and $\backslash mathbf\{R\}\_k\; \backslash equiv\; \backslash mathbf\{R\}^a\_k$, this means that $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}\; =\; \backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}^a$. While computing the actual error covariance using $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}^a\; =\; E\backslash left\backslash left(\backslash mathbf\{x\}\_k$, substituting for $\backslash widehat\{\backslash mathbf\{x\}\}\_\{k\; \backslash mid\; k\}$ and using the fact that $E\backslash left\backslash mathbf\{w\}\_k\backslash mathbf\{w\}\_k^\backslash textsf\{T\}\backslash right\; =\; \backslash mathbf\{Q\}\_k^a$ and $E\backslash left\backslash mathbf\{v\}\_k\; =\; \backslash mathbf\{R\}\_k^a$, results in the following recursive equations for $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}^a$ :

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k-1\}^a\; =\; \backslash mathbf\{F\}\_k\backslash mathbf\{P\}\_\{k-1\; \backslash mid\; k-1\}^a\; \backslash mathbf\{F\}\_k^\backslash textsf\{T\}\; +\; \backslash mathbf\{Q\}\_k^a$

and

$\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}^a\; =\; \backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\backslash right)\backslash mathbf\{P\}\_\{k\; \backslash mid\; k-1\}^a\; \backslash left(\backslash mathbf\{I\}\; -\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{H\}\_k\backslash right)^\backslash textsf\{T\}\; +\; \backslash mathbf\{K\}\_k\; \backslash mathbf\{R\}\_k^a\; \backslash mathbf\{K\}\_k^\backslash textsf\{T\}$

While computing $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}$, by design the filter implicitly assumes that $E\backslash left\backslash mathbf\{w\}\_k\; =\; \backslash mathbf\{Q\}\_k$ and $E\backslash left\backslash mathbf\{v\}\_k\; =\; \backslash mathbf\{R\}\_k$. The recursive expressions for $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}^a$ and $\backslash mathbf\{P\}\_\{k\; \backslash mid\; k\}$ are identical except for the presence of $\backslash mathbf\{Q\}\_k^a$ and $\backslash mathbf\{R\}\_k^a$ in place of the design values $\backslash mathbf\{Q\}\_k$ and $\backslash mathbf\{R\}\_k$ respectively. Researches have been done to analyze Kalman filter system's robustness.Jingyang Lu. target="_blank" rel="nofollow"> "False information injection attack on dynamic state estimation in multi-sensor systems", Fusion 2014

Positive definite matrices have the property that they have a factorization into the product of a non-singular, lower-triangular matrix S and its Matrix transpose : P =  S· S^T . The factor S can be computed efficiently using the Cholesky factorization algorithm. This product form of the covariance matrix P is guaranteed to be symmetric, and for all 1 <= k <= n, the k-th diagonal element P_kk is equal to the euclidean norm of the k-th row of S, which is necessarily positive. An equivalent form, which avoids many of the square root operations involved in the Cholesky factorization algorithm, yet preserves the desirable numerical properties, is the U-D decomposition form, P =  U· D· U^T, where U is a unit triangular matrix (with unit diagonal), and D is a diagonal matrix.

Between the two, the U-D factorization uses the same amount of storage, and somewhat less computation, and is the most commonly used triangular factorization. (Early literature on the relative efficiency is somewhat misleading, as it assumed that square roots were much more time-consuming than divisions, while on 21st-century computers they are only slightly more expensive.)

Efficient algorithms for the Kalman prediction and update steps in the factored form were developed by G. J. Bierman and C. L. Thornton.

The L· D· L^T decomposition of the innovation covariance matrix S_k is the basis for another type of numerically efficient and robust square root filter.

     p\left(\mathbf{x}_k \mid \mathbf{x}_{k-1}\right) &= \mathcal{N}\left(\mathbf{F}_k\mathbf{x}_{k-1}, \mathbf{Q}_k\right) \\
         p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) &= \mathcal{N}\left(\mathbf{H}_k\mathbf{x}_k, \mathbf{R}_k\right) \\
 p\left(\mathbf{x}_{k-1} \mid \mathbf{Z}_{k-1}\right) &= \mathcal{N}\left(\hat{\mathbf{x}}_{k-1}, \mathbf{P}_{k-1}\right)
     

 p(\mathbf{z}) &= \prod_{k=0}^T \int p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) p\left(\mathbf{x}_k \mid \mathbf{z}_{k-1}, \ldots,\mathbf{z}_0\right) d\mathbf{x}_k\\
               &= \prod_{k=0}^T \int \mathcal{N}\left(\mathbf{z}_k; \mathbf{H}_k\mathbf{x}_k, \mathbf{R}_k\right) \mathcal{N}\left(\mathbf{x}_k; \hat{\mathbf{x}}_{k \mid k-1}, \mathbf{P}_{k \mid k-1}\right) d\mathbf{x}_k\\
               &= \prod_{k=0}^T \mathcal{N}\left(\mathbf{z}_k;  \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}, \mathbf{R}_k + \mathbf{H}_k \mathbf{P}_{k \mid k-1} \mathbf{H}_k^\textsf{T}\right)\\
               &= \prod_{k=0}^T \mathcal{N}\left(\mathbf{z}_k;  \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}, \mathbf{S}_k\right),
     

       \mathbf{Y}_{k \mid k} &= \mathbf{P}_{k \mid k}^{-1} \\
 \hat{\mathbf{y}}_{k \mid k} &= \mathbf{P}_{k \mid k}^{-1}\hat{\mathbf{x}}_{k \mid k}
     

       \mathbf{Y}_{k \mid k-1} &= \mathbf{P}_{k \mid k-1}^{-1} \\
 \hat{\mathbf{y}}_{k \mid k-1} &= \mathbf{P}_{k \mid k-1}^{-1}\hat{\mathbf{x}}_{k \mid k-1}
     

 \mathbf{I}_k &= \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-1} \mathbf{H}_k \\
 \mathbf{i}_k &= \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-1} \mathbf{z}_k
     

       \mathbf{Y}_{k \mid k} &= \mathbf{Y}_{k \mid k-1} + \mathbf{I}_k \\
 \hat{\mathbf{y}}_{k \mid k} &= \hat{\mathbf{y}}_{k \mid k-1} + \mathbf{i}_k
     

       \mathbf{Y}_{k \mid k} &= \mathbf{Y}_{k \mid k-1} + \sum_{j=1}^N \mathbf{I}_{k,j} \\
 \hat{\mathbf{y}}_{k \mid k} &= \hat{\mathbf{y}}_{k \mid k-1} + \sum_{j=1}^N \mathbf{i}_{k,j}
     

                \mathbf{M}_k &=
   \left[\mathbf{F}_k^{-1}\right]^\textsf{T} \mathbf{Y}_{k-1 \mid k-1} \mathbf{F}_k^{-1} \\
                \mathbf{C}_k &=
   \mathbf{M}_k \left[\mathbf{M}_k + \mathbf{Q}_k^{-1}\right]^{-1} \\
                  \mathbf{L}_k &=
   \mathbf{I} - \mathbf{C}_k \\
       \mathbf{Y}_{k \mid k-1} &=
   \mathbf{L}_k \mathbf{M}_k +
   \mathbf{C}_k \mathbf{Q}_k^{-1} \mathbf{C}_k^\textsf{T} \\
 \hat{\mathbf{y}}_{k \mid k-1} &=
   \mathbf{L}_k \left[\mathbf{F}_k^{-1}\right]^\textsf{T} \hat{\mathbf{y}}_{k-1 \mid k-1}
     

\begin{bmatrix}
 \hat{\mathbf{x}}_{t \mid t} \\
 \hat{\mathbf{x}}_{t-1 \mid t} \\
 \vdots \\
 \hat{\mathbf{x}}_{t-N+1 \mid t} \\
\end{bmatrix}
=
\begin{bmatrix}
 \mathbf{I} \\
 0 \\
 \vdots \\
 0 \\
\end{bmatrix}
\hat{\mathbf{x}}_{t \mid t-1}
+
\begin{bmatrix}
 0          & \ldots & 0 \\
 \mathbf{I} & 0      & \vdots \\
 \vdots     & \ddots & \vdots \\
 0          & \ldots & \mathbf{I} \\
\end{bmatrix}
\begin{bmatrix}
 \hat{\mathbf{x}}_{t-1 \mid t-1} \\
 \hat{\mathbf{x}}_{t-2 \mid t-1} \\
 \vdots \\
 \hat{\mathbf{x}}_{t-N+1 \mid t-1} \\
\end{bmatrix}
+
\begin{bmatrix}
 \mathbf{K}^{(0)} \\
 \mathbf{K}^{(1)} \\
 \vdots \\
 \mathbf{K}^{(N-1)} \\
\end{bmatrix}
\mathbf{y}_{t \mid t-1}
     

 \mathbf{K}^{(i+1)} =
 \mathbf{P}^{(i)} \mathbf{H}^\textsf{T}
 \left[
   \mathbf{H} \mathbf{P} \mathbf{H}^\textsf{T} + \mathbf{R}
 \right]^{-1}
     

   \mathbf{P}^{(i)} =
   \mathbf{P}
   \left[
     \left(
       \mathbf{F} - \mathbf{K} \mathbf{H}
     \right)^\textsf{T}
   \right]^i
 
     

   E \left[
     \left(
       \mathbf{x}_{t-i} - \hat{\mathbf{x}}_{t-i \mid t}
     \right)^{*}
     \left(
       \mathbf{x}_{t-i} - \hat{\mathbf{x}}_{t-i \mid t}
      \right)
      \mid
      z_1 \ldots z_t
   \right],
     

 \mathbf{P} - \mathbf{P}_i =
 \sum_{j = 0}^i
 \left[
   \mathbf{P}^{(j)} \mathbf{H}^\textsf{T}
   \left(
     \mathbf{H} \mathbf{P} \mathbf{H}^\textsf{T} + \mathbf{R}
   \right)^{-1}
   \mathbf{H} \left( \mathbf{P}^{(i)} \right)^\textsf{T}
 \right]
     

 \hat{\mathbf{x}}_{k \mid n} &= \hat{\mathbf{x}}_{k \mid k} + \mathbf{C}_k \left(\hat{\mathbf{x}}_{k+1 \mid n} - \hat{\mathbf{x}}_{k+1 \mid k}\right) \\
       \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k} + \mathbf{C}_k \left(\mathbf{P}_{k+1 \mid n} - \mathbf{P}_{k+1 \mid k}\right) \mathbf{C}_k^\textsf{T}
     

   \tilde{\Lambda}_k &= \mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \mathbf{H}_k + \hat{\mathbf{C}}_k^\textsf{T} \hat{\Lambda}_k \hat{\mathbf{C}}_k \\
 \hat{\Lambda}_{k-1} &= \mathbf{F}_k^\textsf{T}\tilde{\Lambda}_k\mathbf{F}_k \\
     \hat{\Lambda}_n &= 0 \\
   \tilde{\lambda}_k &=  -\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \mathbf{y}_k + \hat{\mathbf{C}}_k^\textsf{T} \hat{\lambda}_k \\
 \hat{\lambda}_{k-1} &= \mathbf{F}_k^\textsf{T}\tilde{\lambda}_k \\
     \hat{\lambda}_n &= 0
     

 \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k} - \mathbf{P}_{k \mid k}\hat{\Lambda}_k\mathbf{P}_{k \mid k} \\
 \mathbf{x}_{k \mid n} &= \mathbf{x}_{k \mid k} - \mathbf{P}_{k \mid k}\hat{\lambda}_k
     

 \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k-1} - \mathbf{P}_{k \mid k-1}\tilde{\Lambda}_k\mathbf{P}_{k \mid k-1} \\
 \mathbf{x}_{k \mid n} &= \mathbf{x}_{k \mid k-1} - \mathbf{P}_{k \mid k-1}\tilde{\lambda}_k.
     

 \hat{\mathbf{x}}_{k+1 \mid k} &= (\mathbf{F}_k - \mathbf{K}_k\mathbf{H}_k)\hat{\mathbf{x}}_{k \mid k-1} + \mathbf{K}_k\mathbf{z}_k \\
                      \alpha_k &= -\mathbf{S}_k^{-\frac{1}{2}}\mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1} + \mathbf{S}_k^{-\frac{1}{2}}\mathbf{z}_k
     

 \mathbf{x}_k &= f(\mathbf{x}_{k-1}, \mathbf{u}_k) + \mathbf{w}_k \\
 \mathbf{z}_k &= h(\mathbf{x}_k) + \mathbf{v}_k
     

  \hat{\mathbf{x}}_{k \mid k-1} &= \sum_{j=0}^{2L} W_j^a \mathbf{x}_j \\
        \mathbf{P}_{k \mid k-1} &= \sum_{j=0}^{2L} W_j^c \left(\mathbf{x}_j - \hat{\mathbf{x}}_{k \mid k-1}\right)\left(\mathbf{x}_j - \hat{\mathbf{x}}_{k \mid k-1}\right)^\textsf{T}+\mathbf{Q}_k
     

 \hat{\mathbf{z}} &= \sum_{j=0}^{2L} W_j^a \mathbf{z}_j \\[6pt]
 \hat{\mathbf{S}}_k &= \sum_{j=0}^{2L} W_j^c (\mathbf{z}_j-\hat{\mathbf{z}})(\mathbf{z}_j-\hat{\mathbf{z}})^\textsf{T} + \mathbf{R_k}
     

 \mathbf{C_{xz}} &= \sum_{j=0}^{2L} W_j^c (\mathbf{x}_j-\hat\mathbf{x}_{k|k-1})(\mathbf{z}_j-\hat\mathbf{z})^\textsf{T}.
     

 \mathbf{K}_k=\mathbf{C_{xz}}\hat{\mathbf{S}}_k^{-1}.
     

 \frac{d}{dt}\mathbf{x}(t) &= \mathbf{F}(t)\mathbf{x}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{w}(t) \\
             \mathbf{z}(t) &= \mathbf{H}(t) \mathbf{x}(t) + \mathbf{v}(t)
     

 \frac{d}{dt}\hat{\mathbf{x}}(t) &= \mathbf{F}(t)\hat{\mathbf{x}}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{K}(t) \left(\mathbf{z}(t) - \mathbf{H}(t)\hat{\mathbf{x}}(t)\right) \\
       \frac{d}{dt}\mathbf{P}(t) &= \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}^\textsf{T}(t) + \mathbf{Q}(t) - \mathbf{K}(t)\mathbf{R}(t)\mathbf{K}^\textsf{T}(t)
     

 \dot{\mathbf{x}}(t) &= \mathbf{F}(t)\mathbf{x}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{w}(t),
          &\mathbf{w}(t) &\sim N\left(\mathbf{0}, \mathbf{Q}(t)\right) \\
        \mathbf{z}_k &= \mathbf{H}_k\mathbf{x}_k + \mathbf{v}_k,
          &\mathbf{v}_k  &\sim N(\mathbf{0},\mathbf{R}_k)
     

                 \dot{\hat{\mathbf{x}}}(t) &= \mathbf{F}(t) \hat{\mathbf{x}}(t) + \mathbf{B}(t) \mathbf{u}(t)
   \text{, with } \hat{\mathbf{x}}\left(t_{k-1}\right) = \hat{\mathbf{x}}_{k-1 \mid k-1} \\
 \Rightarrow \hat{\mathbf{x}}_{k \mid k-1} &= \hat{\mathbf{x}}\left(t_k\right) \\
         \dot{\mathbf{P}}(t) &= \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}(t)^\textsf{T} + \mathbf{Q}(t)
   \text{, with } \mathbf{P}\left(t_{k-1}\right) = \mathbf{P}_{k-1 \mid k-1} \\
       \Rightarrow \mathbf{P}_{k \mid k-1} &= \mathbf{P}\left(t_k\right)
     

                \mathbf{K}_k &= \mathbf{P}_{k \mid k-1}\mathbf{H}_k^\textsf{T} \left(\mathbf{H}_k\mathbf{P}_{k \mid k-1}\mathbf{H}_k^\textsf{T} + \mathbf{R}_k\right)^{-1} \\
 \hat{\mathbf{x}}_{k \mid k} &= \hat{\mathbf{x}}_{k \mid k-1} + \mathbf{K}_k\left(\mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}\right) \\
       \mathbf{P}_{k \mid k} &= \left(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k\right)\mathbf{P}_{k \mid k-1}