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A Kalman filter is a mathematical algorithm that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone. The filter is a type of recursive Bayesian filter that estimates the state of a system using a set of equations.
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A **Kalman filter** is a mathematical algorithm that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone. The filter is a type of recursive **Bayesian filter** that estimates the state of a system using a set of equations.
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A Kalman filter can be represented as a two-step process:
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A **Kalman filter** can be represented as a two-step process:
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1. Prediction step: The system state at the current time step is estimated using the state at the previous time step and any control inputs that may have been applied to the system. This step is represented mathematically using the following equation: $x̂_{k} = A_{k} x̂_{k-1} + B_{k} u_{k}$ where $x̂_{k}$ is the estimate of the system state at time $k$, $x̂_{k-1}$ is the estimate of the state at the previous time step, $u_{k}$ is the control input at time $k$, $A_{k}$ is the state transition matrix, and $B_{k}$ is the control input matrix.
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**1. Prediction step:** The system state at the current time step is estimated using the state at the previous time step and any control inputs that may have been applied to the system.
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This step is represented mathematically using the following equation:  where  is the estimate of the system state at time ,  is the estimate of the state at the previous time step,  is the control input at time ,  is the state transition matrix, and  is the control input matrix.
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2. Update step: A new measurement is taken, and the estimate of the state is updated using a weighting factor that takes into account the uncertainty of the prediction and the measurement. This step is represented mathematically using the following equations: $$K_{k} = P_{k-1} H_{k}^{T} (H_{k} P_{k-1} H_{k}^{T} + R_{k})^{-1} x̂_{k} = x̂_{k} + K_{k} (z_{k} - H_{k} x̂_{k}) P_{k} = (I - K_{k} H_{k}) P_{k-1}$$ where $K_{k}$ is the Kalman gain matrix, $P_{k}$ is the estimate of the error covariance, $H_{k}$ is the measurement matrix, $z_{k}$ is the measurement, and $R_{k}$ is the measurement noise covariance.
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**2. Update step:** A new measurement is taken, and the estimate of the state is updated using a weighting factor that takes into account the uncertainty of the prediction and the measurement. This step is represented mathematically using the following equations:
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 where  is the Kalman gain matrix,  is the estimate of the error covariance,  is the measurement matrix,  is the measurement, and  is the measurement noise covariance.
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In these equations the terms $x̂_{k}$ , $P_{k}$, and $K_{k}$ are called the estimates of the state, the error covariance, and the Kalman gain respectively. This filter is called Linear Kalman Filter as it's based on Linear System Model and Gaussian Noise.
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In these equations the terms  , , and  are called the estimates of the state, the error covariance, and the Kalman gain respectively. This filter is called Linear Kalman Filter as it's based on Linear System Model and Gaussian Noise.
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In summary, the Kalman filter algorithm uses a prediction-update approach to estimate the state of a system using a set of mathematical equations. It uses a recursive Bayesian algorithm to produce estimates that are more precise than those based on a single measurement alone. |
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[](https://fr.wikipedia.org/wiki/Filtre_de_Kalman)
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In summary, the **Kalman filter** algorithm uses a prediction-update approach to estimate the state of a system using a set of mathematical equations. It uses a recursive **Bayesian algorithm** to produce estimates that are more precise than those based on a single measurement alone. |
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