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Questions tagged [kalman-filter]

For questions about Kalman filter.

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How to aggregate the normal distributions of two Kalman populations?

Suppose I have the following Bayesian Network: It's given by the following probability distributions: $$\begin{aligned}X_1&\sim \mathcal N(\mu, \delta^2)\\ \forall i, 2\leq i\leq n: X_i|X_{i-1}&...
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Kalman Gain from given Variances

$$x[n]=0.6x[n-1]+w[n]$$ $$y[n]=x[n]+v[n]$$ $$\sigma_w^2=0.25, $$$$\sigma_v^2=0.5$$ Find the expression for the Kalman filter equation at convergence and the corresponding mean square error. I know ...
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Extended Kalman Filter for Orientation without Control Input

I'd like to implement an Extended Kalman Filter to estimate the state$$s=[w,x,y,z,av_x,av_y,av_z]$$with $[w,x,y,z]^T$ respresenting the current orientation as a quaternion and $[av_x,av_y,av_z]^T$ ...
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23 views

How is the noise gain function defined for higher order discrete piecewise white noise in a Newtonian system?

Background I have been trying to understand Kalman filters and implement them in a project I have. I have been following Roger Labbe's online book (https://nbviewer.jupyter.org/github/rlabbe/Kalman-...
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69 views

Show that minimizing $Tr(Q)$ equals minimizing $x_0^{T}\:Q\:x_0$

In two different textbooks about Kalman Filter, the so-called Estimator Gain Matrix G is obtained as result of two different minimization problems, i would like to show or at least giustify that the ...
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165 views

Kalman filter, trace of state error covariance matrix, dimensionality problem

One of the derivations of discrete Kalman-filter relies on specifying a $G_k$ matrix gain in the measurement update equation $$\hat{x}_{k}^{+}=\hat{x}_{k}^{-}+G_k\left(y_k-C_k\hat{x}_{k}^{-}\right)$$ ...
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Is $C\left(1+\cos\left(\pi\frac{\sqrt{\delta_x^2 + \delta_y^2}}{R}\right)\right)^2$ a separable filter kernel?

I have a 2D morphology filter with kernel of radius $R$, some scaling constant $C$ and weight function: $$C\left(1+\cos\left(\pi\frac{\sqrt{\delta_x^2 + \delta_y^2}}{R}\right)\right)^2$$ Where $[\...
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38 views

Can I have $Q = R = I$ as covariance matrices for a kalman filter?

Assume that we have no noise in our system. We using a low pass filter to filer away some peaks in the measurements. But our goal is just to estimate the state $X_k$. Can we set the $Q_k$ and $R$ to ...
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Expectation to SDE in filtration problem

I am considering the following system of SDE $\frac{dS_t}{S_t}=Y_tdt+\sigma dB_t$ (observed) $dY_t=\frac{1}{\epsilon}(\theta-Y_t)dt+\frac{\beta}{\sqrt{\epsilon}}dB_t$ (hidden) where $S_t$ denotes a ...
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Kalman filter prediction ahead of next measurement

I want to implement a Kalman Filter for predicting x/y positions. I have a sensor which gives me the current position (noisy). Now I want to smooth and predict the position. Thus Kalman Filter came to ...
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derivation of equation in kalaith's “weiner and kalman filtering” book

Hi: Asked this question ( at the link below ) on the dsp list a few years ago and it didn't really get answered to a point where it clicked for me. Lately, I've been reading the same book again and I'...
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37 views

Value of $x_{k+1}$ and errors in the Kalman filter

I am learning Kalman filter so I have a doubt with respect to the state $x_{k+1}$, I have the following linear system, \begin{align} x_{k+1} & = Ax_{k} + w_{k} \\ y_{k} & = Cx_{k} + v_{k} \...
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Unit used in continuous time process noise matrix in kalman filters, when STD is from discrete time data

I'm trying to make a process noise matrix in continuous time. But i can't seem to find a clear definition of what "unit" the matrix should contain in continuous time. From our control book we have $...
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Kalman filter: the bayesian approach derivation some clarifications

I'm reading the book Methods and algorithms for signal processing from Moon Stirling at page 592 there is a derivation of Kalman filter using the Bayesian approach. I have some issues in understanding ...
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How can I reduce noise from measurement without a Kalman Filter?

I'm going to create an adaptive Model Predictive Controller (MPC). The model is a state space model. Due to noise, it's very difficult to determine the model order. I'm using subspace identification ...
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Noisy Brownian Noise

Suppose that $W_t$ and $B_t$ are independent Brownian motions, and define the process $X_t\triangleq W_t + B_t$. What is the conditional expectation of $W_t$ given the $\sigma$-algebra generated by $...
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Trajectory interpolation between two known states

I have a trajectory made of several state vectors $\mathbf{x}_n$ (position and speed). One step forward in time is done with : $$\mathbf{x}_{n+1} = M_n\mathbf{x}_n + q_n$$ where $M_n$ is a matrix and $...
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How to properly reset 1 ambiguity state of a LS filter when a new sensor is being read?

I am using a least squares filter with multiple states coming in per epoch of time. I have up to 100 sensors available. But only 10 are usually active at any given epoch. So I am resolving roughly ...
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Kalman Filter and Covariance Matrix

In the Kalman filter, these equations represent the error on the state $x(k)$ a priori and a posteriori (discrete time). \begin{align} e_k^- &= x_k - \hat{x}_k^- \\ e_k &= x_k - \hat{x}_k \...
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Modified Bryson-Frazier (MBF) smoother explain

I'm reading about MBF smoother on Wikipedia. I'm confused of the quantity $\hat{\lambda}_k$ and $\tilde{\lambda}_k$. What does they really mean intuitively ? Why the update formula has the form $\...
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Measurement Noise Estimation using a Genetic Algorithm and Statistical Consistency Test

I hope someone can help me understand this paper: here It is explaining how to estimate both process and measurement noise covariance Q and R of a Kalman Filter. I am just starting to dig into this ...
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How to estimate process noise for a Kalman Filter

I hope this question is posted in the correct forum: Consider a small robot car equipped with an ultrasonic distance sensor to measure its distance from the car in front, and an encoder used to ...
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Kalman Filter with same A, B and C, D matrices

I am trying to work on a toy problem of EKF which consists of a Kinematic Model of a bicycle. The model is as follows:Kinematic Model I have linearized the model with inputs as ...
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Help with transforming a value so that it better fits within a recursive filter

I am working on an application where I need to evaluate a metric that determines how 'well distributed' a set of points ($X) are in the 2D plane that is a camera image. By evaluating this metric for ...
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1answer
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Kalman Filter applied to linear discrete time process and interpretation of the estimated covariance matrix

I want to have a deeper understanding of the discrete time Kalman Filter. As a part of this I have modeled a forced, damped, mass spring system numerically in the Jupyter Notebook available here: ...
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How to Modify Measurement-Noise in Kalman Filter from 2D Const-Velocity to 2D Const-Acceleration

After extending a Kalman Filter from 2D Linear Velocity (code) to 2D Constant Acceleration, I realized the State-Predictions have the Y-Position pinned to roughly zero. As you can see, the yellow-...
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Derivation of Kalman Gain for the Unscented Kalman Filter (UKF)

I recently went through the mathematical derivations of the Kalman filter (KF), the extended Kalman filter (EKF) and the Unscented Kalman filter (UKF). My question is concerned with some detail ...
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Kalman Filter Modeling to Get Lat/Lon Position

I know this probably asked many times but i am struggling integrating navaid based positioning outputs (gives lat & lon) with very accurate ins outputs ( roll pitch yaw, accelerations x y z and ...
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Confusion regarding usage of Mahalanobis distance for update rejection in Kalman filtering

I recently came across some material that discussed a method for performing update rejection in Kalman filters when bad measurements are received. [Paper 1] [Paper 2: see Section III(E)] This method ...
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How to quickly resolve a value from noise around a signal?

Kalman filter seems to be slow in resolving a steady signal. Is there a way to apply Ito calculus (stochastic calculus) to resolve the signal quicker? The data takes a minute or longer to ...
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The name of a mathematical property

When two Gaussian equations are multipled together, the outcome is another Gaussian distribution. Roger R. Labbe Jr., author of "Kalman and Bayesian Filters in Python" calls this property "rare" and ...
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38 views

Kalman smoother equations

I'm trying to find the Kalman filter equations, i.e $\mathbb{E}[X_k \mid y_0,...,y_n]$ with $k \lt n$ by figuring out the law of the the density $p(x_k \mid y_0,...,y_n)$ under the Hidden Markov Chain ...
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Analytical value of multivariate normal posterior

Suppose I have the following Bayesian Network: It's given by the following relations: $$\begin{aligned}X_1&\sim \mathcal N(\mu, 1/\sigma^2)\\ \forall k, 2\leq k\leq n: X_k|X_{k-1}&\sim \...
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31 views

Discretization of continuous model with white noise to use Kalman filter later

I have this system which describes dynamics of a car in 2D space. The dynamics are governed by Newton's law g(t) = ma(t). The final task is to use Kalman filter on discretized system to estimate it's ...
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Kalman filter implementation for a driving simulation in a final project

Currently I am designing a Kalman filter-based steering for my final paper in a driving simulator. I'm actually new to the Kalman filtering method but I've studied a couple of journals I can find, ...
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LQG with bias rejection for quadcopter attitude control

We're trying to design an attitude controller for a quadcopter. The system dynamics are given: $$ \boldsymbol{\dot{q}} = \frac{1}{2} \boldsymbol{q} \otimes \begin{pmatrix} 0 \\ \vec{\omega} \end{...
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Does there exist a closed-form solution to the Kalman filter problem with auto-regressive measurement errors?

Suppose the state variable evolves as $x_t = \rho \cdot x_{t-1} + u_t$ where $u_t$ is mean-zero, normally distributed iid noise. Suppose the observation is $y_t = x_t + \epsilon_t$, where, ...
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1answer
82 views

Optimisation vs. Bayes' Theorem not coinciding

Suppose I have the following Bayesian Network: It's given by the following relations: $$\begin{aligned}X_1&\sim \mathcal N(\mu, 1/\sigma^2)\\ \forall k, 2\leq k\leq n: X_k|X_{k-1}&\sim \...
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33 views

The Riccati equation and its asymptotic behavior

Consider matrices $A\in\mathbb{R}^{n\times n},B\in\mathbb{R}^{n\times m}$, a positive semidefinite symmetric matrix $Q\in\mathbb{R}^{n\times n}$ and a positive definite symmetric matrix $R\in\mathbb{R}...
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1answer
22 views

Extended Kalman Filter measurement residual computation

I am trying to follow the computation of EKF presented in this paper http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=C9CB210A45F0D7ED5CA7DE174F1A5490?doi=10.1.1.681.8390&rep=rep1&type=...
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32 views

Extended Kalman Filter practical application

I have navigation data to which I need to apply the Extended Kalman Filter in order to predict the next state of a vessel. I have the longitude, latitude (converted to ECEF format) and velocity ...
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UD decomposition in continuous-discrete kalman filter

My problem is basically this: you have a matrix A, positive semi-definite matrix P. Now, find matrix M such that: $AP+PA^T=MPM^T$ why I need this? in the continuous-discrete Kalman formulation, ...
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Likelihood Estimation from Transfer Function

Given a sequence of observations presumed to be produced by an autonomous linear dynamical system, is it possible (and if so how) to determine the likelihood of the sequence without inferring the ...
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Why are outputs of my UKF filter divided by 2pi and 2pi**2

I am feeding a synthetic sine into a UKF filter and plotting the position,velocity and acceleration of the filter state. To make velocity sensible, I need to multiply by 2*pi and for acceleration, by ...
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Kalman decomposition of given system.

The following question is from a System Theory test without answers or solutions: Consider the continuous-time state-space representation $\frac{d}{dt}x(t)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t\in \...
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The invariant property of Kalman filter

I came a cross a property for Kalman filter known as invariant property. I could only find some information about it on a wikipedia article but I still struggle to understand it. The property is ...
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41 views

What is the class of this optimization problem?

I have the following optimization problem: Find $\mathbf{w}$ such that the following error measure is minimised: $E_u = \dfrac{1}{N_u}\sum_{i=0}^{N_u-1}\lVert \mathbf{w}^Tx(t_{i+1})-\mathbf{F}(\{\...
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31 views

Writing the Extended Kalman Filter

Suppose that you have a system: \begin{cases}\dot{x}(t)=A(u(t))x(t)+d_1(t)\\y(t)=f(x(t))+d_2(t),\end{cases} where $A:\mathbb{R}\longrightarrow\mathbb{R}^{n\times n}$ is a matrix function, $u:\mathbb{R}...
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Bounding the solution to a Riccati equation

I have the following continuous-time matrix Riccati equation $$A X + X A' - X b b' X + Q = 0$$ where $Q>0$, $A$ is a diagonal matrix with strictly negative eigenvalues and $b$ is a (column) ...
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Bayesian Filtering Smoothing over multiple classes

I am classifying images over time in categories such as office, bathroom, living room and so on. The idea is to use all these classification to categorize the room where a robot is. I want to use a ...