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Encyclopedia > Kalman filter

The Kalman filter is an efficient recursive filter that estimates the state of a dynamic system from a series of incomplete and noisy measurements. It was developed by Rudolf Kalman. IIR (infinite impulse response) is a property of signal processing systems. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... This article is about noise as in sound. ... Measurement is the estimation of the magnitude of some attribute of an object, such as its length or weight, relative to a unit of measurement. ... Rudolf Emil Kálmán Rudolf Emil Kálmán (born May 19, 1930 in Budapest, Hungary) is an American-Hungarian mathematical system theorist, who is an electrical engineer by training. ...

Contents

Example applications

An example application would be providing accurate continuously-updated information about the position and velocity of an object given only a sequence of observations about its position, each of which includes some error. It is used in a wide range of engineering applications from radar to computer vision. Kalman filtering is an important topic in control theory and control systems engineering. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... This article is about velocity in physics. ... For other uses, see Radar (disambiguation). ... Computer vision is the science and technology of machines that see. ... For control theory in psychology and sociology, see control theory (sociology). ... A control system is a device or set of devices to manage, command, direct or regulate the behaviour of other devices or systems. ...


For example, in a radar application, where one is interested in tracking a target, information about the location, speed, and acceleration of the target is measured with a great deal of corruption by noise at any time instant. The Kalman filter exploits the dynamics of the target, which govern its time evolution, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing). For other uses, see Radar (disambiguation). ... A radar tracker is a component of a radar system that aggregates individual radar observations into tracks. ...


Naming and historical development

The filter is named after Rudolf E. Kalman, though Thorvald Nicolai Thiele[1] and Peter Swerling actually developed a similar algorithm earlier. Stanley F. Schmidt is generally credited with developing the first implementation of a Kalman filter. It was during a visit of Kalman to the NASA Ames Research Center that he saw the applicability of his ideas to the problem of trajectory estimation for the Apollo program, leading to its incorporation in the Apollo navigation computer. The filter was developed in papers by Swerling (1958), Kalman (1960), and Kalman and Bucy (1961). Rudolf Emil Kalman (May 19, 1930 -) is most famous for his invention of the Kalman filter, a mathematical digital signal processing technique widely used in control systems and avionics to extract meaning (a signal) from chaos (noise). ... Thorvald Nicolai Thiele (December 24, 1838 – September 26, 1910) was a Danish astronomer, actuary, and mathematician, most notable for his work in statistics, interpolation, and the three-body problem. ... The Kalman filter is an efficient recursive filter which estimates the state of a dynamic system from a series of incomplete and noisy measurements. ... Dr. Stanley F. Schmidt Stanley F. Schmidt was born in Hollister, Calif. ... Aerial View of Moffett Field and NASA Ames Research Center. ... Project Apollo was a series of human spaceflight missions undertaken by the United States of America (NASA) using the Apollo spacecraft and Saturn launch vehicle, conducted during the years 1961 – 1975. ...


The filter is sometimes called Stratonovich-Kalman-Bucy filter due to the fact that it is a special case of a more general, non-linear filter developed earlier by Ruslan L. Stratonovich[2] [3]. In fact, equations of the special case, linear filter appeared in these papers by Stratonovich that were published before summer 1960, when Rudolf E. Kalman met with Ruslan L. Stratonovich during a conference in Moscow. Ruslan Leontevich Stratonovich was an outstanding physicist, engineer, probabilist. ... Rudolf Emil Kalman (May 19, 1930 -) is most famous for his invention of the Kalman filter, a mathematical digital signal processing technique widely used in control systems and avionics to extract meaning (a signal) from chaos (noise). ... Ruslan Leontevich Stratonovich was an outstanding physicist, engineer, probabilist. ...


In control theory, the Kalman filter is most commonly referred to as linear quadratic estimation (LQE). For control theory in psychology and sociology, see control theory (sociology). ...


A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the simple Kalman filter, to Schmidt's extended filter, the information filter and a variety of square-root filters developed by Bierman, Thornton and many others. Perhaps the most commonly used type of Kalman filter is the phase-locked loop now ubiquitous in radios, computers, and nearly any other type of video or communications equipment. A phase-lock, or phase-locked, loop (PLL) is an electronic control system that generates a signal that is locked to the phase of an input or reference signal. ...


Underlying dynamic system model

Kalman filters are based on linear dynamical systems discretised in the time domain. They are modelled on a Markov chain built on linear operators perturbed by Gaussian noise. The state of the system is represented as a vector of real numbers. At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the visible outputs from the hidden state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables are continuous (as opposed to being discrete in the hidden Markov model). Additionally, the hidden Markov model can represent an arbitrary distribution for the next value of the state variables, in contrast to the Gaussian noise model that is used for the Kalman filter. There is a strong duality between the equations of the Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani (1999). In a linear dynamical system, the variation of a state vector (an -dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by . ... In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ... In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Discrete time is non-continuous time. ... State transitions in a hidden Markov model (example) x — hidden states y — observable outputs a — transition probabilities b — output probabilities A hidden Markov model (HMM) is a statistical model in which the system being modeled is assumed to be a Markov process with unknown parameters, and the challenge is to...


In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the framework of the Kalman filter. This means specifying the matrices Fk, Hk, Qk, Rk, and sometimes Bk for each time-step k as described below. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

Model underlying the Kalman filter. Circles are vectors, squares are matrices, and stars represent Gaussian noise with the associated covariance matrix at the lower right.
Model underlying the Kalman filter. Circles are vectors, squares are matrices, and stars represent Gaussian noise with the associated covariance matrix at the lower right.

The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to Diagram of the model underlying the Kalman filter, produced with dia and my own fair hands. ... Diagram of the model underlying the Kalman filter, produced with dia and my own fair hands. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

 textbf{x}_{k} = textbf{F}_{k} textbf{x}_{k-1} + textbf{B}_{k}textbf{u}_{k-1} + textbf{w}_{k-1}

where

  • Fk is the state transition model which is applied to the previous state xk−1;
  • Bk is the control-input model which is applied to the control vector uk;
  • wk is the process noise which is assumed to be drawn from a zero mean multivariate normal distribution with covariance Qk.
textbf{w}_{k} sim N(0, textbf{Q}_k)

At time k an observation (or measurement) zk of the true state xk is made according to In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

textbf{z}_{k} = textbf{H}_{k} textbf{x}_{k} + textbf{v}_{k}

where Hk is the observation model which maps the true state space into the observed space and vk is the observation noise which is assumed to be zero mean Gaussian white noise with covariance Rk.

textbf{v}_{k} sim N(0, textbf{R}_k)

The initial state, and the noise vectors at each step {x0, w1, ..., wk, v1 ... vk} are all assumed to be mutually independent.


Many real dynamical systems do not exactly fit this model; however, because the Kalman filter is designed to operate in the presence of noise, an approximate fit is often good enough for the filter to be very useful. Variations on the Kalman filter described below allow richer and more sophisticated models.


The Kalman filter

The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. It is unusual in being purely a time domain filter; most filters (for example, a low-pass filter) are formulated in the frequency domain and then transformed back to the time domain for implementation. IIR (infinite impulse response) is a property of signal processing systems. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... A low-pass filter is a filter that passes low frequencies but attenuates (or reduces) frequencies higher than the cutoff frequency. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...


The state of the filter is represented by two variables:

  • hat{textbf{x}}_{k|k}, the estimate of the state at time k;
  • textbf{P}_{k|k}, the error covariance matrix (a measure of the estimated accuracy of the state estimate).

The Kalman filter has two distinct phases: Predict and Update. The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. In the update phase, measurement information at the current timestep is used to refine this prediction to arrive at a new, (hopefully) more accurate state estimate, again for the current timestep. “Accuracy” redirects here. ...


The notation hat{textbf{x}}_{n|m} represents the estimate of textbf{x} at time n given observations up to, and including time m.


Predict

Predicted state

hat{textbf{x}}_{k|k-1} = textbf{F}_{k}hat{textbf{x}}_{k-1|k-1} + textbf{B}_{k} textbf{u}_{k-1}

Predicted estimate covariance

textbf{P}_{k|k-1} = textbf{F}_{k} textbf{P}_{k-1|k-1} textbf{F}_{k}^{T} + textbf{Q}_{k-1}

Update

Innovation or measurement residual

 tilde{textbf{y}}_{k} = textbf{z}_{k} - textbf{H}_{k}hat{textbf{x}}_{k|k-1}

Innovation (or residual) covariance

textbf{S}_{k} = textbf{H}_{k}textbf{P}_{k|k-1} textbf{H}_{k}^{T}+textbf{R}_{k}

Optimal Kalman gain textbf{K}_{k} = textbf{P}_{k|k-1}textbf{H}_{k}^{T}textbf{S}_{k}^{-1}
Updated state estimate hat{textbf{x}}_{k|k} = hat{textbf{x}}_{k|k-1} + textbf{K}_{k}tilde{textbf{y}}_{k}
Updated estimate covariance textbf{P}_{k|k} = (I - textbf{K}_{k} textbf{H}_{k}) textbf{P}_{k|k-1}

The formula for the updated estimate covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the derivations section. The Kalman filter is an efficient recursive filter that estimates the state of a dynamic system from a series of incomplete and noisy measurements. ...


Invariants

If the model is accurate, and the values for hat{textbf{x}}_{0|0} and textbf{P}_{0|0} accurately reflect the distribution of the initial state values, then the following invariants are preserved: all estimates have mean error zero

  • textrm{E}[textbf{x}_k - hat{textbf{x}}_{k|k}] = textrm{E}[textbf{x}_k - hat{textbf{x}}_{k|k-1}] = 0
  • textrm{E}[tilde{textbf{y}}_k] = 0

where E[ξ] is the expected value of ξ, and covariance matrices accurately reflect the covariance of estimates In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

  • textbf{P}_{k|k} = textrm{cov}(textbf{x}_k - hat{textbf{x}}_{k|k})
  • textbf{P}_{k|k-1} = textrm{cov}(textbf{x}_k - hat{textbf{x}}_{k|k-1})
  • textbf{S}_{k} = textrm{cov}(tilde{textbf{y}}_k)

Example

Consider a truck on perfectly frictionless, infinitely long straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random acceleration. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocity is. We show here how we derive the model from which we create our Kalman filter. This article is about velocity in physics. ...


There are no controls on the truck, so we ignore Bk and uk. Since F, H, R and Q are constant, their time indices are dropped.


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

textbf{x}_{k} = begin{bmatrix} x  dot{x} end{bmatrix}

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


We assume that between the (k − 1)th and kth timestep the truck undergoes a constant acceleration of ak that is normally distributed, with mean 0 and standard deviation σa. From Newton's laws of motion we conclude that The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

textbf{x}_{k} = textbf{F} textbf{x}_{k-1} + textbf{G}a_{k}

where

textbf{F} = begin{bmatrix} 1 & Delta t  0 & 1 end{bmatrix}

and

textbf{G} = begin{bmatrix} frac{Delta t^{2}}{2}  Delta t end{bmatrix}

We find that

 textbf{Q} = textrm{cov}(textbf{G}a) = textrm{E}[(textbf{G}a)(textbf{G}a)^{T}] = textbf{G} textrm{E}[a^2] textbf{G}^{T} = textbf{G}[sigma_a^2]textbf{G}^{T} = sigma_a^2 textbf{G}textbf{G}^{T} (since σa is a scalar).

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

textbf{z}_{k} = textbf{H x}_{k} + textbf{v}_{k}

where

textbf{H} = begin{bmatrix} 1 & 0 end{bmatrix}

and

textbf{R} = textrm{E}[textbf{v}_k textbf{v}_k^{T}] = begin{bmatrix} sigma_z^2 end{bmatrix}

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

hat{textbf{x}}_{0|0} = begin{bmatrix} 0  0 end{bmatrix}

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

textbf{P}_{0|0} = begin{bmatrix} 0 & 0  0 & 0 end{bmatrix}

If the initial position and velocity are not known perfectly the covariance matrix should be initialized with a suitably large number, say B, on its diagonal.

textbf{P}_{0|0} = begin{bmatrix} B & 0  0 & B end{bmatrix}

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


Derivations

Deriving the posterior estimate covariance matrix

Starting with our invariant on the error covariance Pk|k as above

textbf{P}_{k|k} = textrm{cov}(textbf{x}_{k} - hat{textbf{x}}_{k|k})

substitute in the definition of hat{textbf{x}}_{k|k}

textbf{P}_{k|k} = textrm{cov}(textbf{x}_{k} - (hat{textbf{x}}_{k|k-1} + textbf{K}_ktilde{textbf{y}}_{k}))

and substitute tilde{textbf{y}}_k

textbf{P}_{k|k} = textrm{cov}(textbf{x}_{k} - (hat{textbf{x}}_{k|k-1} + textbf{K}_k(textbf{z}_k - textbf{H}_khat{textbf{x}}_{k|k-1})))

and textbf{z}_{k}

textbf{P}_{k|k} = textrm{cov}(textbf{x}_{k} - (hat{textbf{x}}_{k|k-1} + textbf{K}_k(textbf{H}_ktextbf{x}_k + textbf{v}_k - textbf{H}_khat{textbf{x}}_{k|k-1})))

and by collecting the error vectors we get

textbf{P}_{k|k} = textrm{cov}((I - textbf{K}_k textbf{H}_{k})(textbf{x}_k - hat{textbf{x}}_{k|k-1}) - textbf{K}_k textbf{v}_k )

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

textbf{P}_{k|k} = textrm{cov}((I - textbf{K}_k textbf{H}_{k})(textbf{x}_k - hat{textbf{x}}_{k|k-1})) + textrm{cov}(textbf{K}_k textbf{v}_k )

by the properties of vector covariance this becomes In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

textbf{P}_{k|k} = (I - textbf{K}_k textbf{H}_{k})textrm{cov}(textbf{x}_k - hat{textbf{x}}_{k|k-1})(I - textbf{K}_k textbf{H}_{k})^{T} + textbf{K}_ktextrm{cov}(textbf{v}_k )textbf{K}_k^{T}

which, using our invariant on Pk|k-1 and the definition of Rk becomes

textbf{P}_{k|k} = (I - textbf{K}_k textbf{H}_{k}) textbf{P}_{k|k-1} (I - textbf{K}_k textbf{H}_{k})^T + textbf{K}_k textbf{R}_k textbf{K}_k^T

This formula (sometimes known as the "Joseph form" of the covariance update equation) is valid no matter what the value of Kk. It turns out that if Kk is the optimal Kalman gain, this can be simplified further as shown below.


Kalman gain derivation

The Kalman filter is a minimum mean-square error estimator. The error in the posterior state estimation is Minimum mean-square error (MMSE) relates to an estimator having estimates with the minimum mean square error possible. ...

textbf{x}_{k} - hat{textbf{x}}_{k|k}

We seek to minimize the expected value of the square of the magnitude of this vector, textrm{E}[|textbf{x}_{k} - hat{textbf{x}}_{k|k}|^2]. This is equivalent to minimizing the trace of the posterior estimate covariance matrix Pk|k. By expanding out the terms in the equation above and collecting, we get: In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

 textbf{P}_{k|k}  = textbf{P}_{k|k-1} - textbf{K}_k textbf{H}_k textbf{P}_{k|k-1} - textbf{P}_{k|k-1} textbf{H}_k^T textbf{K}_k^T + textbf{K}_k (textbf{H}_k textbf{P}_{k|k-1} textbf{H}_k^T + textbf{R}_k) textbf{K}_k^T
 = textbf{P}_{k|k-1} - textbf{K}_k textbf{H}_k textbf{P}_{k|k-1} - textbf{P}_{k|k-1} textbf{H}_k^T textbf{K}_k^T + textbf{K}_k textbf{S}_ktextbf{K}_k^T

The trace is minimized when the matrix derivative is zero: In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. ...

frac{partial ; textrm{tr}(textbf{P}_{k|k})}{partial ;textbf{K}_k} = -2 (textbf{H}_k textbf{P}_{k|k-1})^T + 2 textbf{K}_k textbf{S}_k = 0

Solving this for Kk yields the Kalman gain:

textbf{K}_k textbf{S}_k = (textbf{H}_k textbf{P}_{k|k-1})^T = textbf{P}_{k|k-1} textbf{H}_k^T
 textbf{K}_{k} = textbf{P}_{k|k-1} textbf{H}_k^T textbf{S}_k^{-1}

This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used. Minimum mean-square error (MMSE) relates to an estimator having estimates with the minimum mean square error possible. ...


Simplification of the posterior error covariance formula

The formula used to calculate the posterior error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our Kalman gain formula on the right by SkKkT, it follows that

textbf{K}_k textbf{S}_k textbf{K}_k^T = textbf{P}_{k|k-1} textbf{H}_k^T textbf{K}_k^T

Referring back to our expanded formula for the posterior error covariance,

 textbf{P}_{k|k} = textbf{P}_{k|k-1} - textbf{K}_k textbf{H}_k textbf{P}_{k|k-1} - textbf{P}_{k|k-1} textbf{H}_k^T textbf{K}_k^T + textbf{K}_k textbf{S}_k textbf{K}_k^T

we find the last two terms cancel out, giving

 textbf{P}_{k|k} = textbf{P}_{k|k-1} - textbf{K}_k textbf{H}_k textbf{P}_{k|k-1} = (I - textbf{K}_{k} textbf{H}_{k}) textbf{P}_{k|k-1}.

This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with numerical stability, or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the posterior error covariance formula as derived above must be used. In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...


Relationship to the digital filter

The Kalman filter can be regarded as an adaptive low-pass infinite impulse response digital filter, with cut-off frequency depending on the ratio between the process- and measurement (or observation) noise, as well as the estimate covariance. Frequency response is, however, rarely of interest when designing state estimators such as the Kalman Filter, whereas for digital filters such as IIR and FIR filters, frequency response is usually of primary concern. For the Kalman Filter, the important goal is how accurate the filter is, and this is most often decided based on empirical Monte Carlo simulations, where "truth" (the true state) is known. An FIR filter In electronics,nirali a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. ... IIR (infinite impulse response) is a property of signal processing systems. ... A finite impulse response (FIR) filter is a type of a digital filter. ...


Relationship to recursive Bayesian estimation

The true state is assumed to be an unobserved Markov process, and the measurements are the observed states of a hidden Markov model. It has been suggested that this article or section be merged with Sequential_bayesian_filtering. ... It has been suggested that this article or section be merged with Markov property. ...

Hidden Markov model

Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state. Image created by chrislloyd This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

p(textbf{x}_k|textbf{x}_0,dots,textbf{x}_{k-1}) = p(textbf{x}_k|textbf{x}_{k-1})

Similarly the measurement at the k-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.

p(textbf{z}_k|textbf{x}_0,dots,textbf{x}_{k}) = p(textbf{z}_k|textbf{x}_{k} )

Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:

p(textbf{x}_0,dots,textbf{x}_k,textbf{z}_1,dots,textbf{z}_k) = p(textbf{x}_0)prod_{i=1}^k p(textbf{z}_i|textbf{x}_i)p(textbf{x}_i|textbf{x}_{i-1})

However, when the Kalman filter is used to estimate the state x, the probability distribution of interest is that associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalizing out the previous states and dividing by the probability of the measurement set.)


This leads to the predict and update steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is product of the probability distribution associated with the transition from the (k − 1)th timestep to the kth and the probability distribution associated with the previous state, with the true state at (k − 1) integrated out.

 p(textbf{x}_k|textbf{Z}_{k-1}) = int p(textbf{x}_k | textbf{x}_{k-1}) p(textbf{x}_{k-1} | textbf{Z}_{k-1} ) , dtextbf{x}_{k-1}

The measurement set up to time t is

 textbf{Z}_{t} = left { textbf{z}_{1},dots,textbf{z}_{t} right }

The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.

 p(textbf{x}_k|textbf{Z}_{k}) = frac{p(textbf{z}_k|textbf{x}_k) p(textbf{x}_k|textbf{Z}_{k-1})}{p(textbf{z}_k|textbf{Z}_{k-1})}

The denominator

p(textbf{z}_k|textbf{Z}_{k-1}) = int p(textbf{z}_k|textbf{x}_k) p(textbf{x}_k|textbf{Z}_{k-1}) dtextbf{x}_k

is a normalization term.


The remaining probability density functions are

 p(textbf{x}_k | textbf{x}_{k-1}) = N(textbf{F}_ktextbf{x}_{k-1}, textbf{Q}_k)
 p(textbf{z}_k|textbf{x}_k) = N(textbf{H}_{k}textbf{x}_k, textbf{R}_k)
 p(textbf{x}_{k-1}|textbf{Z}_{k-1}) = N(hat{textbf{x}}_{k-1},textbf{P}_{k-1} )

Note that the PDF at the previous timestep is inductively assumed to be the estimated state and covariance. This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for mathbf{x}_k given the measurements mathbf{Z}_k is the Kalman filter estimate.


Information filter

In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the information matrix and information vector respectively. These are defined as: In statistics, the Fisher information I(θ), thought of as the amount of information that an observable random variable carries about an unobservable parameter θ upon which the probability distribution of X depends, is the variance of the score. ... In statistics and information theory, the Fisher information (denoted ) is the variance of the score. ...

textbf{Y}_{k|k} = textbf{P}_{k|k}^{-1}
hat{textbf{y}}_{k|k} = textbf{P}_{k|k}^{-1}hat{textbf{x}}_{k|k}

Similarly the predicted covariance and state have equivalent information forms, defined as:

textbf{Y}_{k|k-1} = textbf{P}_{k|k-1}^{-1}
hat{textbf{y}}_{k|k-1} = textbf{P}_{k|k-1}^{-1}hat{textbf{x}}_{k|k-1}

as have the measurement covariance and measurement vector, which are defined as:

textbf{I}_{k} = textbf{H}_{k}^{T} textbf{R}_{k}^{-1} textbf{H}_{k}
textbf{i}_{k} = textbf{H}_{k}^{T} textbf{R}_{k}^{-1} textbf{z}_{k}

The information update now becomes a trivial sum.

textbf{Y}_{k|k} = textbf{Y}_{k|k-1} + textbf{I}_{k}
hat{textbf{y}}_{k|k} = hat{textbf{y}}_{k|k-1} + textbf{i}_{k}

The main advantage of the information filter is that N measurements can be filtered at each timestep simply by summing their information matrices and vectors.

textbf{Y}_{k|k} = textbf{Y}_{k|k-1} + sum_{j=1}^N textbf{I}_{k,j}
hat{textbf{y}}_{k|k} = hat{textbf{y}}_{k|k-1} + sum_{j=1}^N textbf{i}_{k,j}

To predict the information filter the information matrix and vector can be converted back to their state space equivalents, or alternatively the information space prediction can be used.

textbf{M}_{k} = [textbf{F}_{k}^{-1}]^{T} textbf{Y}_{k|k} textbf{F}_{k}^{-1}
textbf{C}_{k} = textbf{M}_{k} [textbf{M}_{k}+textbf{Q}_{k}^{-1}]^{-1}
textbf{L}_{k} = I - textbf{C}_{k}
textbf{Y}_{k|k-1} = textbf{L}_{k} textbf{M}_{k} textbf{L}_{k}^{T} + textbf{C}_{k} textbf{Q}_{k}^{-1} textbf{C}_{k}^{T}
hat{textbf{y}}_{k|k-1} = textbf{L}_{k} [textbf{F}_{k}^{-1}]^{T}hat{textbf{y}}_{k|k}

Note that if F and Q are time invariant these values can be cached. Note also that F and Q need to be invertible.


Non-linear filters

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


Extended Kalman filter

In the extended Kalman filter, (EKF) the state transition and observation models need not be linear functions of the state but may instead be (differentiable) functions. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

textbf{x}_{k} = f(textbf{x}_{k-1}, textbf{u}_{k}, textbf{w}_{k})
textbf{z}_{k} = h(textbf{x}_{k}, textbf{v}_{k})

The function f can be used to compute the predicted state from the previous estimate and similarly the function h can be used to compute the predicted measurement from the predicted state. However, f and h cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian) is computed. For the French Revolution faction, see Jacobin. ...


At each timestep the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.


This results in the following extended Kalman filter equations:


Predict

hat{textbf{x}}_{k|k-1} = f(hat{textbf{x}}_{k-1|k-1}, textbf{u}_{k}, 0)
 textbf{P}_{k|k-1} = textbf{F}_{k} textbf{P}_{k-1|k-1} textbf{F}_{k}^{T} + textbf{Q}_{k}

Update

tilde{textbf{y}}_{k} = textbf{z}_{k} - h(hat{textbf{x}}_{k|k-1}, 0)
textbf{S}_{k} = textbf{H}_{k}textbf{P}_{k|k-1}textbf{H}_{k}^{T} + textbf{R}_{k}
textbf{K}_{k} = textbf{P}_{k|k-1}textbf{H}_{k}^{T}textbf{S}_{k}^{-1}
hat{textbf{x}}_{k|k} = hat{textbf{x}}_{k|k-1} + textbf{K}_{k}tilde{textbf{y}}_{k}
 textbf{P}_{k|k} = (I - textbf{K}_{k} textbf{H}_{k}) textbf{P}_{k|k-1}

where the state transition and observation matrices are defined to be the following Jacobians

 textbf{F}_{k} = left . frac{partial f}{partial textbf{x} } right vert _{hat{textbf{x}}_{k-1|k-1},textbf{u}_{k}}
 textbf{H}_{k} = left . frac{partial h}{partial textbf{x} } right vert _{hat{textbf{x}}_{k|k-1}}

Criticism of the EKF

Unlike its linear counterpart, the EKF is not an optimal estimator. In addition, if the initial estimate of the state is wrong, or if the process is modeled incorrectly, the filter may quickly diverge, owing to its linearisation. Another problem with the EKF is that the estimated covariance matrix tends to underestimate the true covariance matrix and therefore risks becoming inconsistent in the statistical sense without the addition of "stabilising noise". In statistics, a consistent estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows. ...


Having stated this, the EKF can give reasonable performance, and is arguably the de facto standard in navigation systems and GPS. De facto is a Latin expression that means in fact or in practice. It is commonly used as opposed to de jure (meaning by law) when referring to matters of law or governance or technique (such as standards), that are found in the common experience as created or developed without...


Unscented Kalman filter

When the state transition and observation models – that is, the predict and update functions f and h (see above) – are highly non-linear, the extended Kalman filter can give particularly poor performance [JU97]. This is because only the mean is propagated through the non-linearity. The unscented Kalman filter (UKF) [JU97] uses a deterministic sampling technique known as the unscented transform to pick a minimal set of sample points (called sigma points) around the mean. These sigma points are then propagated through the non-linear functions and the covariance of the estimate is then recovered. The result is a filter which more accurately captures the true mean and covariance. (This can be verified using Monte Carlo sampling or through a Taylor series expansion of the posterior statistics.) In addition, this technique removes the requirement to analytically calculate Jacobians, which for complex functions can be a difficult task in itself. Monte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. ...


Predict


As with the EKF, the UKF prediction can be used independently from the UKF update, in combination with a linear (or indeed EKF) update, or vice versa.


The estimated state and covariance are augmented with the mean and covariance of the process noise.

 textbf{x}_{k-1|k-1}^{a} = [ hat{textbf{x}}_{k-1|k-1}^{T} quad E[textbf{w}_{k}^{T}]  ]^{T}
 textbf{P}_{k-1|k-1}^{a} = begin{bmatrix} & textbf{P}_{k-1|k-1} & & 0 &  & 0 & &textbf{Q}_{k} & end{bmatrix}

A set of 2L+1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.

chi_{k-1|k-1}^{0} = textbf{x}_{k-1|k-1}^{a}

The sigma points are propagated through the transition function f.

The weighted sigma points are recombined to produce the predicted state and covariance.

where the weights for the state and covariance are given by:

Typical values for α, β, and κ are 10 − 3, 2 and 0 respectively. (These values should suffice for most purposes.)


Update


The predicted state and covariance are augmented as before, except now with the mean and covariance of the measurement noise.

As before, a set of 2L + 1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.

Alternatively if the UKF prediction has been used the sigma points themselves can be augmented along the following lines

where

The sigma points are projected through the observation function h.

The weighted sigma points are recombined to produce the predicted measurement and predicted measurement covariance.

The state-measurement cross-covariance matrix,

is used to compute the UKF Kalman gain.

As with the Kalman filter, the updated state is the predicted state plus the innovation weighted by the Kalman gain,

And the updated covariance is the predicted covariance, minus the predicted measurement covariance, weighted by the Kalman gain.

Kalman-Bucy filter

The Kalman-Bucy filter is a continuous time version of the Kalman filter.[4] [5]


It is based on the state space model

where the covariances of the noise terms and are given by 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.[6]


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. In mathematics, a Riccati equation is any ordinary differential equation that has the form It is named after Count Jacopo Francesco Riccati (1676-1754). ...


Applications

An autopilot is a mechanical, electrical, or hydraulic system used to guide a vehicle without assistance from a human being. ... Offshore Support Vessel Toisa Perseus with, in the background, the fifth-generation deepwater drillship Discoverer Enterprise, at the Thunder Horse location. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Circulation in macroeconomics Macroeconomics is a branch of economics that deals with the performance, structure, and behavior of a national economy as a whole. ... In statistics, signal processing, and econometrics, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. ... Econometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. ... An inertial guidance system consists of an Inertial Measurement Unit (IMU) combined with a set of guidance algorithms and control mechanisms, allowing the path of a vehicle to be controlled according to the position determined by the inertial navigation system. ... A radar tracker is a component of a radar system that aggregates individual radar observations into tracks. ... It has been suggested that this article or section be merged into Global Navigation Satellite System. ... Simultaneous localization and mapping (SLAM) is a technique used by robots and autonomous vehicles to build up a map within an unknown environment while at the same time keeping track of their current position. ...

See also

// The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. ... This article contains information that has not been verified and thus might not be reliable. ... The Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published [1]. // Description Unlike the typical filtering theory of designing a filter for a desired frequency response the Wiener filter approaches filtering from a different angle. ... This article is about the statistical method. ... A non-linear filter is a signal-processing device whose output is not a linear function of its input. ... There are very few or no other articles that link to this one. ... The Zakai equation is a linear recursive filtering equation for the un-normalized density of a hidden state. ... // Description Recursive least squares algorithm is used in adaptive filters to find the filter coefficients that relate to producing the recursively least squares of the error signal (difference between the desired and the actual signal) Performance This algorithm converges faster than the LMS algorithm. ... Data assimilation (DA) is a method used in the weather forecasting process in which observations of the current (and possibly, past) weather are combined with a previous forecast for that time to produce the meteorological `analysis; the best estimate of the current state of the atmosphere. ... A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. ... In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. ...

References

General

  • Gelb A., editor. Applied optimal estimation. MIT Press, 1974.
  • Kalman, R. E. "A New Approach to Linear Filtering and Prediction Problems," Transactions of the ASME - Journal of Basic Engineering Vol. 82: pp. 35-45 (1960).
  • Kalman, R. E., Bucy R. S., "New Results in Linear Filtering and Prediction Theory", Transactions of the ASME - Journal of Basic Engineering Vol. 83: pp. 95-107 (1961).
  • [JU97] Julier, Simon J. and Jeffery K. Uhlmann. "A New Extension of the Kalman Filter to nonlinear Systems." In The Proceedings of AeroSense: The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Controls, Multi Sensor Fusion, Tracking and Resource Management II, SPIE, 1997.
  • Harvey, A.C. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press, Cambridge, 1989.
  • Simon, D. Optimal state estimation: Kalman, H-infinity, and nonlinear approaches. John Wiley & Sons, 2006. (Book web site at http://academic.csuohio.edu/simond/estimation/)
  • Bierman, Gerald J., Factorization Methods for Discrete Sequential Estimation. Dover Publications, 1977
  • Bozic, S. M., Digital and Kalman Filtering. Edward Arnald Publications, 1979 (1st edition), 1994 (2nd edition)
  • Amir Vasebi, Maral Partovibakhsha and S. Mohammad Taghi Bathaee, A novel combined battery model for state-of-charge estimation in lead-acid batteries based on extended Kalman filter for hybrid electric vehicle applications doi:10.1016/j.jpowsour.2007.04.011.
  • A. Vasebi, S.M.T. Bathaeea and M. Partovibakhsh, Predicting state of charge of lead-acid batteries for hybrid electric vehicles by extended Kalman filter doi:10.1016/j.enconman.2007.05.017.

Notes

  1. ^ Steffen L. Lauritzen, Thiele: Pioneer in Statistics, Oxford University Press, 2002. ISBN 0-19-850972-3.
  2. ^ 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.
  3. ^ Stratonovich, R.L. (1960) Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp.1-19.
  4. ^ Bucy, R.S. and Joseph, P.D., Filtering for Stochastic Processes with Applications to Guidance, John Wiley & Sons, 1968; 2nd Edition, AMS Chelsea Publ., 2005. ISBN 0821837826
  5. ^ Jazwinski, Andrew H., Stochastic processes and filtering theory, Academic Press, New York, 1970. ISBN 0123815509
  6. ^ Kailath, Thomas, "An innovation approach to least-squares estimation Part I: Linear filtering in additive white noise", IEEE Transactions on Automatic Control, 13(6), 646-655, 1968

Oxford University Press (OUP) is a highly-respected publishing house and a department of the University of Oxford in England. ...

External links


  Results from FactBites:
 
Kalman Filtering (1335 words)
Kalman filtering is a relatively recent (1960) development in filtering, although it has its roots as far back as Gauss (1795).
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Kalman filtering is a huge field whose depths we cannot hope to begin to plumb in such a brief paper as this.
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We design the filter under the assumption that we are trying to estimate a constant bias state.
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The filter is designed to be robust to changes in the variance of the process noise and measurement noise.
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