Detail Description of Kalman Filter | Real Time System | BSc.CSIT Sixth Semester Kalman Filter – Detail Explanation, Real Time System Notes | Sixth Semester, BSc.CSIT | Tribhuvan University (TU)

Kalman Filter
Kalman filter is a commonly used means to improve the accuracy of measurements and to estimate model parameters in the presence of noise and uncertainty. Consider a simple monitor system that takes a measured value yk every sampling period ‘k’ in order to estimate the value xk of a state variable. Suppose starting from time 0, the value of this state variable is equal to a constant x. Because of noise, the measured value yk is equal to $x + \epsilon_{k}$ where $\epsilon_{k}$ is a random variable whose average value is 0 and standard deviation is $\sigma_{k}$. The Kalman filter starts with the initial estimate $\tilde{x}_{1}=y_1$ and computes a new estimate each sampling period. Specifically, for k>1, the filter computes the estimate $\tilde{x}_{k}$ as follows: $\tilde{x}_{k}=\tilde{x}_{k-1}+K_k(y_{k}-\tilde{x}_{k-1})$ —————(i)
where $K_{k}=\frac{P_k}{\sigma_{k}^2+P_k}$ ———————–(ii)
Here, K= Kalman gain and Pk = variance of the estimation error $\tilde{x}_{k}-x$
and $P_{k}=E[(\tilde{x}_{k}-x)^2]=(1-K_{k-1})P_{k-1}$ ——————(iii)

This value of kalman gain gives the best compromise between the rate at which Pk decreases with k and the steady-state variance i.e; Pk for large k.

In a multivariate system, the state variable xk is an n-dimensional vector where ‘n’ is the no. of variables whose values define the state of the plant. The measured value yk is an $n' -dimensional$vector, if during each sampling period, the readings of $n'$ sensors are taken. Let A denote measurement matrix; which is $n \times n'$ matrix that relates $n'$ measured variables to the n state variables. In other words,

y= Axk + ek
where ek = additive noise in each of the $n'$ measured values
Equation (i) becomes an n-dimensional vector equation $\tilde{x}_{k}=\tilde{x}_{k-1}+K_{k}(y_{k}-A\tilde{x}_{k-1})$
Similarly kalman gain Kk and variance Pk are given by matrix version of equation (ii) and (iii). So the computation involves a few matrix multiplication and addition and one matrix inversion

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