I have this state equations for Discrete Kalman Filter:
1) $x_{k+1}=Ax_{k}+Bu_{k}+Gw_{k}$
2) $y_{k}=Cx_{k}+v_{k}$
I don't understand/know what is $G$ in 1).
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2 Answers
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A lot of textbooks define $w_{k}$ as white noise, where $w_{k}$ may or may not be a vector. The $G$ then is the matrix that modifies the variance of the noise $w_{k}$, and is assumed to be known. Note that in a Kalman filter, the noise going into the system (is this case , $w_{k}$) is assumed unknown, only its characteristics are known, one of which is $G$.
Maybe useful information: If $w_{k}$ is a zero mean Gaussian white noise signal with unit variance, then $\sqrt{G}w_{k}$ is a white noise signal with variance $G$.
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Sometimes $G=B$, when there is actuation noise. Eg. $u$ is motor torque command from the controller. But the actual motor torque may be $u+\Delta u$, where the uncertainty of $\Delta u$ can be described by $w$.
