Kalman filter
For
This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich.[1][2][3][4] 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.[5]
Kalman filtering
The algorithm works by a two-phase process having a prediction phase and an update phase. For the prediction phase, the Kalman filter produces estimates of the current
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."[13] Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense,[14] although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature[where?]) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian.[15]
Extensions and
History
The filtering method is named for Hungarian
This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).
The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of the very first applications of the Kalman filter) into such a tiny computer is truly remarkable.
— Interview with Jack Crenshaw, by Matthew Reed, TRS-80.org (2009) [1]
Kalman filters have been vital in the implementation of the navigation systems of
Overview of the calculation
Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its
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
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.
Example application
As an example application, consider the problem of determining the precise location of a truck. The truck can be equipped with a
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.
Technical description and context
The Kalman filter is an efficient
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
A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed the "simple" Kalman filter, the
Underlying dynamic system model
Kalman filtering is based on
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:
- Fk, the state-transition model;
- Hk, the observation model;
- Qk, the covariance of the process noise;
- Rk, the covariance of the observation noise;
- and sometimes Bk, the control-input model as described below; if Bk is included, then there is also
- uk, the control vector, representing the controlling input into control-input model.
The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to
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: .
At time k an observation (or measurement) zk of the true state xk is made according to
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: .
The initial state, and the noise vectors at each step {x0, w1, ..., wk, v1, ... ,vk} are all assumed to be mutually
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.[28][29]
Details
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 represents the estimate of at time n given observations up to and including at time m ≤ n.
The state of the filter is represented by two variables:
- , the a posterioristate estimate mean at time k given observations up to and including at time 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 Hk).[30][31]
Predict
Predicted (a priori) state estimate | |
Predicted (a priori) estimate covariance |
Update
Innovation or measurement pre-fit residual | |
Innovation (or pre-fit residual) covariance | |
Optimal Kalman gain | |
Updated (a posteriori) state estimate | |
Updated (a posteriori) estimate covariance | |
Measurement post-fit residual
|
The formula for the updated (a posteriori) estimate covariance above is valid for the optimal Kk 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 Kk is also shown.
A more intuitive way to express the updated state estimate () is:
This expression reminds us of a linear interpolation, for between [0,1]. In our case:
- is the matrix the takes values from (high error in the sensor) to or a projection (low error).
- is the internal state estimated from the model.
- is the internal state estimated from the measurement, assuming is nonsingular.
This expression also resembles the alpha beta filter update step.
Invariants
If the model is accurate, and the values for and accurately reflect the distribution of the initial state values, then the following invariants are preserved:
where is the expected value of . That is, all estimates have a mean error of zero.
Also:
so covariance matrices accurately reflect the covariance of estimates.
Estimation of the noise covariances Qk and Rk
Practical implementation of a Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices Qk and Rk. Extensive research has been done to estimate these covariances from data. One practical method of doing this is the autocovariance least-squares (ALS) technique that uses the time-lagged
Optimality and performance
It follows from theory that the Kalman filter provides an optimal state estimation in cases where a) the model matches the real system perfectly, b) the entering noise is "white" (uncorrelated) and c) the covariances of the noise are known exactly. Correlated noise can also be treated using Kalman filters.[36] Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in the section above. After the covariances are estimated, it is useful to evaluate the performance of the filter; i.e., whether it is possible to improve the state estimation quality. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the innovations measures filter performance. Several different methods can be used for this purpose.[37] If the noise terms are distributed in a non-Gaussian manner, methods for assessing performance of the filter estimate, which use probability inequalities or large-sample theory, are known in the literature.[38][39]
Example application, technical
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 Δt 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 are constant, their time indices are dropped.
The position and velocity of the truck are described by the linear state space
where 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 ak that is normally distributed with mean 0 and standard deviation σa. From Newton's laws of motion we conclude that
(there is no term since there are no known control inputs. Instead, ak is the effect of an unknown input and applies that effect to the state vector) where
so that
where
The matrix is not full rank (it is of rank one if ). Hence, the distribution is not absolutely continuous and has no probability density function. Another way to express this, avoiding explicit degenerate distributions is given by
At each time phase, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also distributed normally, with mean 0 and standard deviation σz.
where
and
We know the initial starting state of the truck with perfect precision, so we initialize
and to tell the filter that we know the exact position and velocity, we give it a zero covariance matrix:
If the initial position and velocity are not known perfectly, the covariance matrix should be initialized with suitable variances on its diagonal:
The filter will then prefer the information from the first measurements over the information already in the model.
Asymptotic form
For simplicity, assume that the control input . Then the Kalman filter may be written:
A similar equation holds if we include a non-zero control input. Gain matrices evolve independently of the measurements . From above, the four equations needed for updating the Kalman gain are as follows:
Since the gain matrices depend only on the model, and not the measurements, they may be computed offline. Convergence of the gain matrices to an asymptotic matrix applies for conditions established in Walrand and Dimakis.[40] Simulations establish the number of steps to convergence. For the moving truck example described above, with . and , simulation shows convergence in iterations.
Using the asymptotic gain, and assuming and are independent of , the Kalman filter becomes a
The asymptotic gain , if it exists, can be computed by first solving the following discrete Riccati equation for the asymptotic state covariance :[40]
The asymptotic gain is then computed as before.
Additionally, a form of the asymptotic Kalman filter more commonly used in control theory is given by
where
This leads to an estimator of the form
Derivations
This section needs additional citations for verification. (December 2010) |
The Kalman filter can be derived as a generalized least squares method operating on previous data.[41]
Deriving the posteriori estimate covariance matrix
Starting with our invariant on the error covariance Pk | k as above
substitute in the definition of
and substitute
and
and by collecting the error vectors we get
Since the measurement error vk is uncorrelated with the other terms, this becomes
by the properties of vector covariance this becomes
which, using our invariant on Pk | k−1 and the definition of Rk becomes
This formula (sometimes known as the Joseph form of the covariance update equation) is valid for any 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
We seek to minimize the expected value of the square of the magnitude of this vector, . This is equivalent to minimizing the
The trace is minimized when its matrix derivative with respect to the gain matrix is zero. Using the gradient matrix rules and the symmetry of the matrices involved we find that
Solving this for Kk yields the Kalman gain:
This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used.
Simplification of the posteriori error covariance formula
The formula used to calculate the a posteriori 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
Referring back to our expanded formula for the a posteriori error covariance,
we find the last two terms cancel out, giving
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.
Sensitivity analysis
This section needs additional citations for verification. (December 2010) |
The Kalman filtering equations provide an estimate of the state and its error covariance recursively. The estimate and its quality depend on the system parameters and the noise statistics fed as inputs to the estimator. This section analyzes the effect of uncertainties in the statistical inputs to the filter.[42] In the absence of reliable statistics or the true values of noise covariance matrices and , the expression
no longer provides the actual error covariance. In other words, . 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.[citation needed] This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices and 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 and respectively, whereas the design values used in the estimator are and respectively. The actual error covariance is denoted by and as computed by the Kalman filter is referred to as the Riccati variable. When and , this means that . While computing the actual error covariance using , substituting for and using the fact that and , results in the following recursive equations for :
and
While computing , by design the filter implicitly assumes that and . The recursive expressions for and are identical except for the presence of and in place of the design values and respectively. Researches have been done to analyze Kalman filter system's robustness.[43]
Square root form
One problem with the Kalman filter is its
Positive definite matrices have the property that they have a
Between the two, the U-D factorization uses the same amount of storage, and somewhat less computation, and is the most commonly used square root form. (Early literature on the relative efficiency is somewhat misleading, as it assumed that square roots were much more time-consuming than divisions,[44]: 69 while on 21st-century computers they are only slightly more expensive.)
Efficient algorithms for the Kalman prediction and update steps in the square root form were developed by G. J. Bierman and C. L. Thornton.[44][45]
The
Parallel form
The Kalman filter is efficient for sequential data processing on
Relationship to recursive Bayesian estimation
The Kalman filter can be presented as one of the simplest dynamic Bayesian networks. The Kalman filter calculates estimates of the true values of states recursively over time using incoming measurements and a mathematical process model. Similarly, recursive Bayesian estimation calculates estimates of an unknown probability density function (PDF) recursively over time using incoming measurements and a mathematical process model.[51]
In recursive Bayesian estimation, the true state is assumed to be an unobserved
Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
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.
Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
However, when a 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 results in the predict and update phases 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 .
The measurement set up to time t is
The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
The denominator
is a normalization term.
The remaining probability density functions are
The PDF at the previous timestep is assumed inductively 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 given the measurements is the Kalman filter estimate.
Marginal likelihood
Related to the recursive Bayesian interpretation described above, the Kalman filter can be viewed as a generative model, i.e., a process for generating a stream of random observations z = (z0, z1, z2, ...). Specifically, the process is
- Sample a hidden state from the Gaussian prior distribution .
- Sample an observation from the observation model .
- For , do
- Sample the next hidden state from the transition model
- Sample an observation from the observation model
This process has identical structure to the hidden Markov model, except that the discrete state and observations are replaced with continuous variables sampled from Gaussian distributions.
In some applications, it is useful to compute the probability that a Kalman filter with a given set of parameters (prior distribution, transition and observation models, and control inputs) would generate a particular observed signal. This probability is known as the
It is straightforward to compute the marginal likelihood as a side effect of the recursive filtering computation. By the chain rule, the likelihood can be factored as the product of the probability of each observation given previous observations,
- ,
and because the Kalman filter describes a Markov process, all relevant information from previous observations is contained in the current state estimate Thus the marginal likelihood is given by
i.e., a product of Gaussian densities, each corresponding to the density of one observation zk under the current filtering distribution . This can easily be computed as a simple recursive update; however, to avoid numeric underflow, in a practical implementation it is usually desirable to compute the log marginal likelihood instead. Adopting the convention , this can be done via the recursive update rule
where is the dimension of the measurement vector.[52]
An important application where such a (log) likelihood of the observations (given the filter parameters) is used is multi-target tracking. For example, consider an object tracking scenario where a stream of observations is the input, however, it is unknown how many objects are in the scene (or, the number of objects is known but is greater than one). For such a scenario, it can be unknown apriori which observations/measurements were generated by which object. A multiple hypothesis tracker (MHT) typically will form different track association hypotheses, where each hypothesis can be considered as a Kalman filter (for the linear Gaussian case) with a specific set of parameters associated with the hypothesized object. Thus, it is important to compute the likelihood of the observations for the different hypotheses under consideration, such that the most-likely one can be found.
Information filter
This section needs additional citations for verification. (April 2016) |
In cases where the dimension of the observation vector y is bigger than the dimension of the state space vector x, the information filter can avoid the inversion of a bigger matrix in the Kalman gain calculation at the price of inverting a smaller matrix in the prediction step, thus saving computing time. In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the
Similarly the predicted covariance and state have equivalent information forms, defined as:
and the measurement covariance and measurement vector, which are defined as:
The information update now becomes a trivial sum.[53]
The main advantage of the information filter is that N measurements can be filtered at each time step simply by summing their information matrices and vectors.
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.[53]
Fixed-lag smoother
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The optimal fixed-lag smoother provides the optimal estimate of for a given fixed-lag using the measurements from to .[54] It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:
where:
- is estimated via a standard Kalman filter;
- is the innovation produced considering the estimate of the standard Kalman filter;
- the various with are new variables; i.e., they do not appear in the standard Kalman filter;
- the gains are computed via the following scheme:
- and
- where and are the prediction error covariance and the gains of the standard Kalman filter (i.e., ).
If the estimation error covariance is defined so that
then we have that the improvement on the estimation of is given by:
Fixed-interval smoothers
The optimal fixed-interval smoother provides the optimal estimate of () using the measurements from a fixed interval to . This is also called "Kalman Smoothing". There are several smoothing algorithms in common use.
Rauch–Tung–Striebel
The Rauch–Tung–Striebel (RTS) smoother is an efficient two-pass algorithm for fixed interval smoothing.[55]
The forward pass is the same as the regular Kalman filter algorithm. These filtered a-priori and a-posteriori state estimates , and covariances , are saved for use in the backward pass (for retrodiction).
In the backward pass, we compute the smoothed state estimates and covariances . We start at the last time step and proceed backward in time using the following recursive equations:
where
is the a-posteriori state estimate of timestep and is the a-priori state estimate of timestep . The same notation applies to the covariance.
Modified Bryson–Frazier smoother
An alternative to the RTS algorithm is the modified Bryson–Frazier (MBF) fixed interval smoother developed by Bierman.[45] This also uses a backward pass that processes data saved from the Kalman filter forward pass. The equations for the backward pass involve the recursive computation of data which are used at each observation time to compute the smoothed state and covariance.
The recursive equations are
where is the residual covariance and . The smoothed state and covariance can then be found by substitution in the equations
or
An important advantage of the MBF is that it does not require finding the inverse of the covariance matrix.
Minimum-variance smoother
The minimum-variance smoother can attain the best-possible error performance, provided that the models are linear, their parameters and the noise statistics are known precisely.[56] This smoother is a time-varying state-space generalization of the optimal non-causal Wiener filter.
The smoother calculations are done in two passes. The forward calculations involve a one-step-ahead predictor and are given by
The above system is known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward pass may be calculated by operating the forward equations on the time-reversed and time reversing the result. In the case of output estimation, the smoothed estimate is given by
Taking the causal part of this minimum-variance smoother yields
which is identical to the minimum-variance Kalman filter. The above solutions minimize the variance of the output estimation error. Note that the Rauch–Tung–Striebel smoother derivation assumes that the underlying distributions are Gaussian, whereas the minimum-variance solutions do not. Optimal smoothers for state estimation and input estimation can be constructed similarly.
A continuous-time version of the above smoother is described in.[57][58]
In cases where the models are nonlinear, step-wise linearizations may be within the minimum-variance filter and smoother recursions (extended Kalman filtering).
Frequency-weighted Kalman filters
Pioneering research on the perception of sounds at different frequencies was conducted by Fletcher and Munson in the 1930s. Their work led to a standard way of weighting measured sound levels within investigations of industrial noise and hearing loss. Frequency weightings have since been used within filter and controller designs to manage performance within bands of interest.
Typically, a frequency shaping function is used to weight the average power of the error spectral density in a specified frequency band. Let denote the output estimation error exhibited by a conventional Kalman filter. Also, let denote a causal frequency weighting transfer function. The optimum solution which minimizes the variance of arises by simply constructing .
The design of remains an open question. One way of proceeding is to identify a system which generates the estimation error and setting equal to the inverse of that system.[60] This procedure may be iterated to obtain mean-square error improvement at the cost of increased filter order. The same technique can be applied to smoothers.
Nonlinear filters
The basic Kalman filter is limited to a linear assumption. More complex systems, however, can be nonlinear. The nonlinearity can be associated either with the process model or with the observation model or with both.
The most common variants of Kalman filters for non-linear systems are the Extended Kalman Filter and Unscented Kalman filter. The suitability of which filter to use depends on the non-linearity indices of the process and observation model.[61]
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 nonlinear functions. These functions are of differentiable type.
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
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 nonlinear function around the current estimate.
Unscented Kalman filter
When the state transition and observation models—that is, the predict and update functions and —are highly nonlinear, the extended Kalman filter can give particularly poor performance.[62]
Sigma points
For a random vector , sigma points are any set of vectors
attributed with
- first-order weights that fulfill
- for all :
- second-order weights that fulfill
- for all pairs .
A simple choice of sigma points and weights for in the UKF algorithm is
where is the mean estimate of . The vector is the jth column of where . Typically, is obtained via Cholesky decomposition of . With some care the filter equations can be expressed in such a way that is evaluated directly without intermediate calculations of . This is referred to as the square-root unscented Kalman filter.[66]
The weight of the mean value, , can be chosen arbitrarily.
Another popular parameterization (which generalizes the above) is
and control the spread of the sigma points. is related to the distribution of . Note that this is an overparameterization in the sense that any one of , and can be chosen arbitrarily.
Appropriate values depend on the problem at hand, but a typical recommendation is , , and .[67] If the true distribution of is Gaussian, is optimal.[68]
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.
Given estimates of the mean and covariance, and , one obtains sigma points as described in the section above. The sigma points are propagated through the transition function f.
- .
The propagated sigma points are weighed to produce the predicted mean and covariance.
where are the first-order weights of the original sigma points, and are the second-order weights. The matrix is the covariance of the transition noise, .
Update
Given prediction estimates and , a new set of sigma points with corresponding first-order weights and second-order weights is calculated.[69] These sigma points are transformed through the measurement function .
- .
Then the empirical mean and covariance of the transformed points are calculated.
where is the covariance matrix of the observation noise, . Additionally, the cross covariance matrix is also needed
The Kalman gain is
The updated mean and covariance estimates are
Discriminative Kalman filter
When the observation model is highly non-linear and/or non-Gaussian, it may prove advantageous to apply
where for nonlinear functions . This replaces the generative specification of the standard Kalman filter with a discriminative model for the latent states given observations.
Under a stationary state model
where , if
then given a new observation , it follows that[70]
where
Note that this approximation requires to be positive-definite; in the case that it is not,
is used instead. Such an approach proves particularly useful when the dimensionality of the observations is much greater than that of the latent states[71] and can be used build filters that are particularly robust to nonstationarities in the observation model.[72]
Adaptive Kalman filter
Adaptive Kalman filters allow to adapt for process dynamics which are not modeled in the process model , which happens for example in the context of a maneuvering target when a constant velocity (reduced order) Kalman filter is employed for tracking.[73]
Kalman–Bucy filter
Kalman–Bucy filtering (named for Richard Snowden Bucy) is a continuous time version of Kalman filtering.[74][75]
It is based on the state space model
where and represent the intensities (or, more accurately: the Power Spectral Density - PSD - matrices) of the two white noise terms and , respectively.
The filter consists of two differential equations, one for the state estimate and one for the covariance:
where the Kalman gain is given by
Note that in this expression for the covariance of the observation noise represents at the same time the covariance of the prediction error (or innovation) ; these covariances are equal only in the case of continuous time.[76]
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. Nonlinear generalizations to Kalman–Bucy filters include continuous time extended Kalman filter.
Hybrid Kalman filter
Most physical systems are represented as continuous-time models while discrete-time measurements are made frequently for state estimation via a digital processor. Therefore, the system model and measurement model are given by
where
- .
Initialize
Predict
The prediction equations are derived from those of continuous-time Kalman filter without update from measurements, i.e., . The predicted state and covariance are calculated respectively by solving a set of differential equations with the initial value equal to the estimate at the previous step.
For the case of
Update
The update equations are identical to those of the discrete-time Kalman filter.
Variants for the recovery of sparse signals
The traditional Kalman filter has also been employed for the recovery of sparse, possibly dynamic, signals from noisy observations. Recent works[77][78][79] utilize notions from the theory of compressed sensing/sampling, such as the restricted isometry property and related probabilistic recovery arguments, for sequentially estimating the sparse state in intrinsically low-dimensional systems.
Relation to Gaussian processes
Since linear Gaussian state-space models lead to Gaussian processes, Kalman filters can be viewed as sequential solvers for
Applications
- Attitude and heading reference systems
- Autopilot
- Electric battery state of charge (SoC) estimation[81][82]
- Chaotic signals
- Tracking and vertex fitting of charged particles in particle detectors[83]
- Tracking of objects in computer vision
- Dynamic positioning in shipping
- time series analysis, and econometrics[84]
- Inertial guidance system
- Nuclear medicine – single photon emission computed tomography image restoration[85]
- Orbit determination
- Power system state estimation
- Radar tracker
- Satellite navigation systems
- Seismology[86]
- Sensorless control of AC motor variable-frequency drives
- Simultaneous localization and mapping
- Speech enhancement
- Visual odometry
- Weather forecasting
- Navigation system
- 3D modeling
- Structural health monitoring
- Human sensorimotor processing[87]
See also
- Alpha beta filter
- Inverse-variance weighting
- Covariance intersection
- Data assimilation
- Ensemble Kalman filter
- Extended Kalman filter
- Fast Kalman filter
- Filtering problem (stochastic processes)
- Generalized filtering
- Invariant extended Kalman filter
- Kernel adaptive filter
- Masreliez's theorem
- Moving horizon estimation
- Particle filter estimator
- PID controller
- Predictor–corrector method
- Recursive least squares filter
- Schmidt–Kalman filter
- Separation principle
- Sliding mode control
- State-transition matrix
- Stochastic differential equations
- Switching Kalman filter
References
- ^ 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.
- S2CID 53120402.
- S2CID 222355496.
- ISBN 978-1-56347-455-2.
- S2CID 239938971.
- ISSN 2504-2289.
- OCLC 1010658777.
- S2CID 221855079.
- S2CID 736756.
- S2CID 1242324.
- .
- S2CID 251143915.
- ISSN 1026-0226.
- S2CID 204196474.
- JSTOR 1402616.
He derives a recursive procedure for estimating the regression component and predicting the Brownian motion. The procedure is now known as Kalman filtering.
- ISBN 978-0-19-850972-1.
He solves the problem of estimating the regression coefficients and predicting the values of the Brownian motion by the method of least squares and gives an elegant recursive procedure for carrying out the calculations. The procedure is nowadays known as Kalman filtering.
- ISBN 978-1-118-98498-7.
- ^ "Mohinder S. Grewal and Angus P. Andrews" (PDF). Archived from the original (PDF) on 2016-03-07. Retrieved 2015-04-23.
- ^ Jerrold H. Suddath; Robert H. Kidd; Arnold G. Reinhold (August 1967). A Linearized Error Analysis Of Onboard Primary Navigation Systems For The Apollo Lunar Module, NASA TN D-4027 (PDF). National Aeronautics and Space Administration.
{{cite book}}
:|work=
ignored (help) - ISBN 978-1-62410-090-1.
- S2CID 3042206.
- ^ Martin Møller Andreasen (2008). "Non-linear DSGE Models, The Central Difference Kalman Filter, and The Mean Shifted Particle Filter" (PDF).[permanent dead link]
- S2CID 2590898.
- ^ Hamilton, J. (1994), Time Series Analysis, Princeton University Press. Chapter 13, 'The Kalman Filter'
- S2CID 12741796.
- S2CID 8810105.
- ^ Kelly, Alonzo (1994). "A 3D state space formulation of a navigation Kalman filter for autonomous vehicles" (PDF). DTIC Document: 13. Archived (PDF) from the original on December 30, 2014. 2006 Corrected Version Archived 2017-01-10 at the Wayback Machine
- ^ Reid, Ian; Term, Hilary. "Estimation II" (PDF). www.robots.ox.ac.uk. Oxford University. Retrieved 6 August 2014.
- ^ Rajamani, Murali (October 2007). Data-based Techniques to Improve State Estimation in Model Predictive Control (PDF) (PhD Thesis). University of Wisconsin–Madison. Archived from the original (PDF) on 2016-03-04. Retrieved 2011-04-04.
- S2CID 5699674.
- ^ "Autocovariance Least-Squares Toolbox". Jbrwww.che.wisc.edu. Retrieved 2021-08-18.
- ^ Bania, P.; Baranowski, J. (12 December 2016). Field Kalman Filter and its approximation. IEEE 55th Conference on Decision and Control (CDC). Las Vegas, NV, USA: IEEE. pp. 2875–2880.
- ISBN 0-471-41655-X.
- )
- .
- S2CID 21143516.
- ^ a b Walrand, Jean; Dimakis, Antonis (August 2006). Random processes in Systems -- Lecture Notes (PDF). pp. 69–70. Archived from the original (PDF) on 2019-05-07. Retrieved 2019-05-07.
- ^ Sant, Donald T. "Generalized least squares applied to time varying parameter models." Annals of Economic and Social Measurement, Volume 6, number 3. NBER, 1977. 301-314. Online Pdf
- ISBN 978-0-13-638122-8.
- ^ Jingyang Lu. "False information injection attack on dynamic state estimation in multi-sensor systems", Fusion 2014
- ^ a b Thornton, Catherine L. (15 October 1976). Triangular Covariance Factorizations for Kalman Filtering (PhD). NASA. NASA Technical Memorandum 33-798.
- ^ Bibcode:1977fmds.book.....B.
- ^ ISBN 978-0-471-41655-5.
- ISBN 978-0-8018-5414-9.
- ISBN 978-0-89871-521-7.
- S2CID 213695560.
- ^ "Parallel Prefix Sum (Scan) with CUDA". developer.nvidia.com/. Retrieved 2020-02-21.
The scan operation is a simple and powerful parallel primitive with a broad range of applications. In this chapter we have explained an efficient implementation of scan using CUDA, which achieves a significant speedup compared to a sequential implementation on a fast CPU, and compared to a parallel implementation in OpenGL on the same GPU. Due to the increasing power of commodity parallel processors such as GPUs, we expect to see data-parallel algorithms such as scan to increase in importance over the coming years.
- .
- ^ Lütkepohl, Helmut (1991). Introduction to Multiple Time Series Analysis. Heidelberg: Springer-Verlag Berlin. p. 435.
- ^ a b Gabriel T. Terejanu (2012-08-04). "Discrete Kalman Filter Tutorial" (PDF). Archived from the original (PDF) on 2020-08-17. Retrieved 2016-04-13.
- ISBN 978-0-13-638122-8.
- doi:10.2514/3.3166.
- S2CID 15376718.
- S2CID 16218530.
- S2CID 36072082.
- S2CID 16218530.
- S2CID 13569109.
- S2CID 225028760.
- ^ S2CID 9614092.
- S2CID 7937456. Retrieved 2008-05-03.
- S2CID 12606055.
- S2CID 17876531.
- S2CID 7290857.
- )
- S2CID 13992571. Archived from the original(PDF) on 2012-03-03. Retrieved 2010-01-31.
- .
- ^ S2CID 212748230. Retrieved 26 March 2021.
- ^ .
- ^ PMID 30216140. Retrieved 26 March 2021.
- ISBN 0-471-41655-X.
- ISBN 0-8218-3782-6
- ISBN 0-12-381550-9
- .
- S2CID 9282476.
- S2CID 10569233.
- S2CID 18467024.
- arXiv:1504.05994 [stat.ME].
- .
- .
- .
- ISBN 978-0-521-46726-1.
- PMID 18218487.
- Bibcode:2008AGUFM.G43B..01B.
- PMID 12662535.
Further reading
This 'further reading' section may need cleanup. (June 2015) |
- Einicke, G.A. (2019). Smoothing, Filtering and Prediction: Estimating the Past, Present and Future (2nd ed.). Amazon Prime Publishing. ISBN 978-0-6485115-0-2.
- Jinya Su; Baibing Li; Wen-Hua Chen (2015). "On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs". Automatica. 53: 149–154. .
- Gelb, A. (1974). Applied Optimal Estimation. MIT Press.
- Kalman, R.E. (1960). "A new approach to linear filtering and prediction problems" (PDF). Journal of Basic Engineering. 82 (1): 35–45. S2CID 1242324. Archived from the original(PDF) on 2008-05-29. Retrieved 2008-05-03.
- Kalman, R.E.; Bucy, R.S. (1961). "New Results in Linear Filtering and Prediction Theory". Journal of Basic Engineering. 83: 95–108. S2CID 8141345.
- Harvey, A.C. (1990). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press. ISBN 978-0-521-40573-7.
- Roweis, S.; S2CID 2590898.
- Simon, D. (2006). Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley-Interscience. Archived from the original on 2010-12-30. Retrieved 2006-07-05.
- .
- Bierman, G.J. (1977). Factorization Methods for Discrete Sequential Estimation. Vol. 128. Mineola, N.Y.: Dover Publications. )
- Bozic, S.M. (1994). Digital and Kalman filtering. Butterworth–Heinemann.
- Haykin, S. (2002). Adaptive Filter Theory. Prentice Hall.
- Liu, W.; Principe, J.C. and Haykin, S. (2010). Kernel Adaptive Filtering: A Comprehensive Introduction. John Wiley.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - Manolakis, D.G. (1999). Statistical and Adaptive signal processing. Artech House.
- Welch, Greg; Bishop, Gary (1997). "SCAAT: incremental tracking with incomplete information" (PDF). SIGGRAPH '97 Proceedings of the 24th annual conference on Computer graphics and interactive techniques. ACM Press/Addison-Wesley Publishing Co. pp. 333–344. S2CID 1512754.
- Jazwinski, Andrew H. (1970). Stochastic Processes and Filtering. Mathematics in Science and Engineering. New York: ISBN 978-0-12-381550-7.
- Maybeck, Peter S. (1979). "Chapter 1" (PDF). Stochastic Models, Estimation, and Control. Mathematics in Science and Engineering. Vol. 141–1. New York: ISBN 978-0-12-480701-3.
- Moriya, N. (2011). Primer to Kalman Filtering: A Physicist Perspective. New York: ISBN 978-1-61668-311-5.
- Dunik, J.; Simandl M.; Straka O. (2009). "Methods for Estimating State and Measurement Noise Covariance Matrices: Aspects and Comparison". 15th IFAC Symposium on System Identification, 2009. France. pp. 372–377. ISBN 978-3-902661-47-0.)
{{cite book}}
: CS1 maint: location missing publisher (link - Chui, Charles K.; Chen, Guanrong (2009). Kalman Filtering with Real-Time Applications. Springer Series in Information Sciences. Vol. 17 (4th ed.). New York: ISBN 978-3-540-87848-3.
- Spivey, Ben; Hedengren, J. D. and Edgar, T. F. (2010). "Constrained Nonlinear Estimation for Industrial Process Fouling". Industrial & Engineering Chemistry Research. 49 (17): 7824–7831. doi:10.1021/ie9018116.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - ISBN 978-0-13-022464-4.
- ISBN 978-0-470-25388-5.
External links
This article's use of external links may not follow Wikipedia's policies or guidelines. (June 2015) |
- A New Approach to Linear Filtering and Prediction Problems, by R. E. Kalman, 1960
- Kalman and Bayesian Filters in Python. Open source Kalman filtering textbook.
- How a Kalman filter works, in pictures. Illuminates the Kalman filter with pictures and colors
- Kalman–Bucy Filter, a derivation of the Kalman–Bucy Filter
- MIT Video Lecture on the Kalman filter on YouTube
- Kalman filter in Javascript. Open source Kalman filter library for node.js and the web browser.
- An Introduction to the Kalman Filter Archived 2021-02-24 at the Wayback Machine, SIGGRAPH 2001 Course, Greg Welch and Gary Bishop
- Kalman Filter webpage, with many links
- Kalman Filter Explained Simply, Step-by-Step Tutorial of the Kalman Filter with Equations
- "Kalman filters used in Weather models" (PDF). SIAM News. 36 (8). October 2003. Archived from the original (PDF) on 2011-05-17. Retrieved 2007-01-27.
- Haseltine, Eric L.; Rawlings, James B. (2005). "Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation". Industrial & Engineering Chemistry Research. 44 (8): 2451. .
- Gerald J. Bierman's Estimation Subroutine Library: Corresponds to the code in the research monograph "Factorization Methods for Discrete Sequential Estimation" originally published by Academic Press in 1977. Republished by Dover.
- Matlab Toolbox implementing parts of Gerald J. Bierman's Estimation Subroutine Library: UD / UDU' and LD / LDL' factorization with associated time and measurement updates making up the Kalman filter.
- Matlab Toolbox of Kalman Filtering applied to Simultaneous Localization and Mapping: Vehicle moving in 1D, 2D and 3D
- The Kalman Filter in Reproducing Kernel Hilbert Spaces A comprehensive introduction.
- Matlab code to estimate Cox–Ingersoll–Ross interest rate model with Kalman Filter Archived 2014-02-09 at the Wayback Machine: Corresponds to the paper "estimating and testing exponential-affine term structure models by kalman filter" published by Review of Quantitative Finance and Accounting in 1999.
- Online demo of the Kalman Filter. Demonstration of Kalman Filter (and other data assimilation methods) using twin experiments.
- Botella, Guillermo; Martín h., José Antonio; Santos, Matilde; Meyer-Baese, Uwe (2011). "FPGA-Based Multimodal Embedded Sensor System Integrating Low- and Mid-Level Vision". Sensors. 11 (12): 1251–1259. PMID 22164069.
- Examples and how-to on using Kalman Filters with MATLAB A Tutorial on Filtering and Estimation
- Explaining Filtering (Estimation) in One Hour, Ten Minutes, One Minute, and One Sentence by Yu-Chi Ho
- Simo Särkkä (2013). "Bayesian Filtering and Smoothing". Cambridge University Press. Full text available on author's webpage https://users.aalto.fi/~ssarkka/.