This paper presents an application of the Kalman filter in signal processing in instrumentation systems when the conditions of the environment generate a large amount of interference for the acquisition of signals from measurement systems. The unwanted interferences make important use of the instrumentation system resources and do not represent useful information under any aspect. A simulation is presented using the Matlab tool, which remarkably facilitates the information processing so that the corresponding actions are taken according to the information obtained, taking advantage of the current resources offered by the embedded systems and the required measurements are obtained with enough accuracy.
Discover real-world situations in which you can use Kalman filters. Kalman filters are often used to optimally estimate the internal states of a system in the presence of uncertain and indirect measurements. Learn the working principles behind Kalman filters by watching the following introductory examples.
You will also learn about state observers by walking through a few examples that include simple math. This will help you understand what a Kalman filter is and how it works. At a high level, Kalman filters are a type of optimal state estimator. The videos also include a discussion of nonlinear state estimators, such as extended and unscented Kalman filters.
Part 1: Why Use Kalman Filters?Discover common uses of Kalman filters by walking through some examples. A Kalman filter is an optimal estimation algorithm used to estimate states of a system from indirect and uncertain measurements.
An Unscented Kalman Filer is one of the versions of nonlinear Kalman filter ( together with Extended KF). They solve problems that are non-linear in nature. While the Extended KF uses gradual expansion of linear algorithm, Unscented KF take a unique approach to eliminate linearisation proces [Kim 2011].
These two filters make an assumption (among others) that the uncertainty of the state is gaussian and can be fully (adequately?) described with the mean and a 1-sigma point (along each estimated dimension with cross-covariance - how one dimension varies relative to another dimension changing). The operation of forward projecting the mean (read: state estimate) and covariance (read: uncertainty) is linear and thus only scales and translates these values.
where is the state vector, is the measurements, is the process noise, and is the measurement noise. Kalman filter assumes that and are zero-mean, independent random variables with known variances , , and . Here, the A, G, and C matrices are:
The C matrix represents that only position measurements are available. A position sensor, such as GPS, provides these measurements at the sample rate of 1Hz. The variance of the measurement noise , the R matrix, is specified as . Since R is specified as a scalar, the Kalman filter block assumes that the matrix R is diagonal, its diagonals are 50 and is of compatible dimensions with y. If the measurement noise is Gaussian, R=50 corresponds to 68% of the position measurements being within or the actual position in the east and north directions. However, this assumption is not necessary for the Kalman filter.
The Kalman filter velocity estimates track the actual velocity trends correctly. The noise levels decrease when the vehicle is traveling at high velocities. This is in line with the design of the Q matrix. The large two spikes are at t=50s and t=200s. These are the times when the car goes through sudden deceleration and a sharp turn, respectively. The velocity changes at those instants are much larger than the predictions from the Kalman filter, which is based on its Q matrix input. After a few time-steps, the filter estimates catch up with the actual velocity.
Configuring the Kalman filter can be very challenging. Besides basic understanding of the Kalman filter, it often requires experimentation in order to come up with a set of suitable configuration parameters. The trackSingleObject function, defined above, helps you to explore the various configuration options offered by the configureKalmanFilter function.
Typically objects do not move with constant acceleration or constant velocity. You use the MotionNoise to specify the amount of deviation from the ideal motion model. When you increase the motion noise, the Kalman filter relies more heavily on the incoming measurements than on its internal state. Try experimenting with MotionNoise parameter to learn more about its effects.
This text is a practical guide to building Kalman filters and shows how the filtering equations can be applied to real-life problems. Numerous examples are presented in detail, showing the many ways in which Kalman filters can be designed. Computer code written in FORTRAN, MATLAB®, and True BASIC accompanies all of the examples so that the interested reader can verify concepts and explore issues beyond the scope of the text. Sometimes mistakes are introduced intentionally to the initial filter designs to show the reader what happens when the filter is not working properly. The text spends a great deal of time setting up a problem before the Kalman filter is actually formulated to give the reader an intuitive feel for the problem being addressed. Real problems are seldom presented in the form of differential equations and they usually do not have unique solutions. Therefore, the authors illustrate several different filtering approaches for tackling a problem. Readers will gain experience in software and performance tradeoffs for determining the best filtering approach for the application at hand.The second edition has two new chapters and an additional appendix. In the first new chapter, a recursive digital filter known as the fading memory filter is introduced and it is shown that for some radar tracking applications the fading memory filter can yield similar performance to a Kalman filter at far less computational cost. A second new chapter presents techniques for improving Kalman filter performance. Included is a practical method for preprocessing measurement data when there are too many measurements for the filter to utilize in a given amount of time. The chapter also contains practical methods for making the Kalman filter adaptive. A new appendix has been added which serves as a central location and summary for the text's most important concepts and formulas.MATLAB is a registered trademark of The MathWorks, Inc.
obj = unscentedKalmanFilter(StateTransitionFcn,MeasurementFcn) creates an unscented Kalman filter object using the specified state transition and measurement functions. Before using the predict and correct commands, specify the initial state values using dot notation. For example, for a two-state system with initial state values [1;0], specify obj.State = [1;0].
obj = unscentedKalmanFilter(Name,Value) creates an unscented Kalman filter object with properties specified using one or more Name,Value pair arguments. Before using the predict and correct commands, specify the state transition function, measurement function, and initial state values using Name,Value pair arguments or dot notation.
If you want a filter with single-precision floating-point variables, specify InitialState as a single-precision vector variable. For example, for a two-state system with state transition and measurement functions vdpStateFcn.m and vdpMeasurementFcn.m, create the unscented Kalman filter object with initial states [1;2] as follows:
The unscented Kalman filter algorithm treats the state of the system as a random variable with mean value State and variance StateCovariance. To compute the state and its statistical properties at the next time step, the algorithm first generates a set of state values distributed around the mean State value by using the unscented transformation. These generated state values are called sigma points. The algorithm uses each of the sigma points as an input to the state transition and measurement functions to get a new set of transformed state points and measurements. The transformed points are used to compute the state and state estimation error covariance value at the next time step.
The initial value of State is the value youspecify in the InitialState input argument duringobject creation. If you specify InitialState asa column vector, then State is also a column vector,and the predict and correct commandsreturn state estimates as a column vector. Otherwise, a row vectoris returned. If you want a filter with single-precision floating-pointvariables, you must specify State as a single-precisionvariable during object construction using the InitialState inputargument.
Enabling the HasMeasurementWrapping property wraps the measurement residuals in a defined bound, which helps to prevent the filter from divergence due to incorrect measurement residual values. For an example, see State Estimation with Wrapped Measurements Using Extended Kalman Filter. 2b1af7f3a8