Kalman filtering is a classic state estimation technique usedin application areas such as signal processing and autonomouscontrol of vehicles. It is now being used to solve problemsin computer systems such as controlling the voltage andfrequency of processors.Although there are many presentations of Kalman filteringin the literature, they usually deal with particular systemslike autonomous robots or linear systems with Gaussiannoise, which makes it difficult to understand the generalprinciples behind Kalman filtering. In this paper, we firstpresent the abstract ideas behind Kalman filtering at a levelaccessible to anyone with a basic knowledge of probabilitytheory and calculus, and then show how these concepts canbe applied to the particular problem of state estimation inlinear systems. This separation of concepts from applicationsshould make it easier to understand Kalman filtering and toapply it to other problems in computer systems.
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 explore the situations where Kalman filters are commonly used. When the state of a system can only be measured indirectly, you can use a Kalman filter to optimally estimate the states of that system. And when measurements from different sensors are available but subject to noise, you can use a Kalman filter to combine sensory data from various sources (known as sensor fusion) to find the best estimate of the parameter of interest.
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.
Finally, an example demonstrates how the states of a linear system can be estimated using Kalman filters,
KALMAN FILTERS FOR LINEAR SYSTEMS
we apply the algorithms developed in to the particular problem of state estimation in linear systems, which is the classical application of Kalman filtering shows how the evolution of the state of such a system over time can be computed if the initial state x0 and the model of the system dynamics are known precisely.Time advances in discrete steps. The state of the system at any time step is a function of the state of the system atthe previous time step and the control inputs applied to thesystem during that interval. This is usually expressed byan equation of the form xt=ft(xt−1,ut) where ut is the control input. The function ft is non linear in the general case, and can be different for different steps. If the systemis linear, the relation for state evolution over time can be written as xt=Ftxt−1+Bt ut,
where Ft and Bt are time-dependent matrices that can be determined from the physicsof the system. Therefore, if the initial statex0is knownexactly and the system dynamics are modeled perfectly by the Ft and Bt matrices, the evolution of the state over time can be computed precisely.In general however, we may not know the initial state ex-actly, and the system dynamics and control inputs may notbe known precisely. These inaccuracies may cause the statecomputed by the model to diverge unacceptably from theactual state over time.To avoid this, we can make measure-ments of the state after each time step. If these measurementswere exact, there would of course be no need to model thesystem dynamics. However, in general, the measurementsthemselves are imprecise.
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