Determine initial state using measurement error cost function
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Given dynamic system
$$x(k+1) = Fx(k)$$
$$y(k+1) = Hx(k+1) + v(k+1)$$
where v(k) is zero mean white noise.
I need to derive an estimator for the initial state $hat{x}(0)$ using the cost function
$$J(T) = sum_{k=1}^{T} (y_k - hat{y}_k)^T [R(k)]^{-1} (y_k - hat{y}_k)$$
where R(k) is the covariance matrix, $y_k$ is the actual measurement, and $hat{y}_k$ is the estimator based predicted measurement (also a function of $hat{x}(0)$).
In order to derive the estimator, I know I need to take the partial derivative of J with respect to x and set that equal to zero. My issue is I don't know what $hat{y}_k$ is exactly in terms of $hat{x}(0)$. Is it just $hat{y}_k = HF^{k} hat{x}(0) + v(k)$?
estimation estimation-theory
asked Nov 15 at 3:21
Matt
206
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up vote
0
down vote
favorite
Given dynamic system
$$x(k+1) = Fx(k)$$
$$y(k+1) = Hx(k+1) + v(k+1)$$
where v(k) is zero mean white noise.
I need to derive an estimator for the initial state $hat{x}(0)$ using the cost function
$$J(T) = sum_{k=1}^{T} (y_k - hat{y}_k)^T [R(k)]^{-1} (y_k - hat{y}_k)$$
where R(k) is the covariance matrix, $y_k$ is the actual measurement, and $hat{y}_k$ is the estimator based predicted measurement (also a function of $hat{x}(0)$).
In order to derive the estimator, I know I need to take the partial derivative of J with respect to x and set that equal to zero. My issue is I don't know what $hat{y}_k$ is exactly in terms of $hat{x}(0)$. Is it just $hat{y}_k = HF^{k} hat{x}(0) + v(k)$?
estimation estimation-theory
asked Nov 15 at 3:21
Matt
206
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given dynamic system
$$x(k+1) = Fx(k)$$
$$y(k+1) = Hx(k+1) + v(k+1)$$
where v(k) is zero mean white noise.
I need to derive an estimator for the initial state $hat{x}(0)$ using the cost function
$$J(T) = sum_{k=1}^{T} (y_k - hat{y}_k)^T [R(k)]^{-1} (y_k - hat{y}_k)$$
where R(k) is the covariance matrix, $y_k$ is the actual measurement, and $hat{y}_k$ is the estimator based predicted measurement (also a function of $hat{x}(0)$).
In order to derive the estimator, I know I need to take the partial derivative of J with respect to x and set that equal to zero. My issue is I don't know what $hat{y}_k$ is exactly in terms of $hat{x}(0)$. Is it just $hat{y}_k = HF^{k} hat{x}(0) + v(k)$?
estimation estimation-theory
asked Nov 15 at 3:21
Matt
206
Given dynamic system
$$x(k+1) = Fx(k)$$
$$y(k+1) = Hx(k+1) + v(k+1)$$
where v(k) is zero mean white noise.
I need to derive an estimator for the initial state $hat{x}(0)$ using the cost function
$$J(T) = sum_{k=1}^{T} (y_k - hat{y}_k)^T [R(k)]^{-1} (y_k - hat{y}_k)$$
where R(k) is the covariance matrix, $y_k$ is the actual measurement, and $hat{y}_k$ is the estimator based predicted measurement (also a function of $hat{x}(0)$).
In order to derive the estimator, I know I need to take the partial derivative of J with respect to x and set that equal to zero. My issue is I don't know what $hat{y}_k$ is exactly in terms of $hat{x}(0)$. Is it just $hat{y}_k = HF^{k} hat{x}(0) + v(k)$?
estimation estimation-theory
estimation estimation-theory
asked Nov 15 at 3:21
Matt
206
asked Nov 15 at 3:21
Matt
206
asked Nov 15 at 3:21
Matt
206
asked Nov 15 at 3:21
Matt
206
asked Nov 15 at 3:21
Matt
206
206
add a comment |
add a comment |
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