- #1
Master1022
- 590
- 116
- Homework Statement
- Calculate the Jacobean matrices for the extended Kalman filter
- Relevant Equations
- Partial derivatives
Hi,
I have a question about calculating the Jacobian matrices for the Extended Kalman filter.
Question: If we have a system of the form:
\begin{align*} x_{k+1} =f_k (x_k , u_k) + w_k \\<br /> y_k = h_k (x_k , u_k ) +v_k \end{align*}
where the state x_k comprises of the three variables p_1, p_2, and p_3. The input u_k comprises of the variables q_1 and q_2. Form an extended Kalman filter for this system.
Method:
From what I understand, I need to linearize the system to form the A and C matrices for the state space model. I am, however, confused on how to form these Jacobian matrices. The source I am learning from presents the following formulae:
A_k = \frac{\partial f_k}{\partial x_k} |_{\hat x_{k|k} , u_k}
and
C_k = \frac{\partial h_k}{\partial x_k} |_{\hat x_{k|k} , u_k}
I don't understand how I ought to interpret these formulae. Should my matrices have the following form, or have I misunderstood something?
A_k =<br /> \begin{pmatrix}<br /> \frac{\partial f_1}{\partial p_1} & \frac{\partial f_1}{\partial p_2} & \frac{\partial f_1}{\partial p_3} \\<br /> \frac{\partial f_2}{\partial p_1} & \frac{\partial f_2}{\partial p_2} & \frac{\partial f_2}{\partial p_3} \\<br /> \frac{\partial f_3}{\partial p_1} & \frac{\partial f_3}{\partial p_2} & \frac{\partial f_3}{\partial p_3}<br /> \end{pmatrix}
where f_i refers to the function in the i^{th} row of the vector f (which has three rows) and
C_k =<br /> \begin{pmatrix}<br /> \frac{\partial h_1}{\partial p_1} & \frac{\partial h_1}{\partial p_2} & \frac{\partial h_1}{\partial p_3} \\<br /> \frac{\partial h_2}{\partial p_1} & \frac{\partial h_2}{\partial p_2} & \frac{\partial h_2}{\partial p_3} \\<br /> \end{pmatrix}
where h_i refers to the function in the i^{th} row of the vector h (which has two rows)
Thank you in advance for any help and guidance.
I have a question about calculating the Jacobian matrices for the Extended Kalman filter.
Question: If we have a system of the form:
\begin{align*} x_{k+1} =f_k (x_k , u_k) + w_k \\<br /> y_k = h_k (x_k , u_k ) +v_k \end{align*}
where the state x_k comprises of the three variables p_1, p_2, and p_3. The input u_k comprises of the variables q_1 and q_2. Form an extended Kalman filter for this system.
Method:
From what I understand, I need to linearize the system to form the A and C matrices for the state space model. I am, however, confused on how to form these Jacobian matrices. The source I am learning from presents the following formulae:
A_k = \frac{\partial f_k}{\partial x_k} |_{\hat x_{k|k} , u_k}
and
C_k = \frac{\partial h_k}{\partial x_k} |_{\hat x_{k|k} , u_k}
I don't understand how I ought to interpret these formulae. Should my matrices have the following form, or have I misunderstood something?
A_k =<br /> \begin{pmatrix}<br /> \frac{\partial f_1}{\partial p_1} & \frac{\partial f_1}{\partial p_2} & \frac{\partial f_1}{\partial p_3} \\<br /> \frac{\partial f_2}{\partial p_1} & \frac{\partial f_2}{\partial p_2} & \frac{\partial f_2}{\partial p_3} \\<br /> \frac{\partial f_3}{\partial p_1} & \frac{\partial f_3}{\partial p_2} & \frac{\partial f_3}{\partial p_3}<br /> \end{pmatrix}
where f_i refers to the function in the i^{th} row of the vector f (which has three rows) and
C_k =<br /> \begin{pmatrix}<br /> \frac{\partial h_1}{\partial p_1} & \frac{\partial h_1}{\partial p_2} & \frac{\partial h_1}{\partial p_3} \\<br /> \frac{\partial h_2}{\partial p_1} & \frac{\partial h_2}{\partial p_2} & \frac{\partial h_2}{\partial p_3} \\<br /> \end{pmatrix}
where h_i refers to the function in the i^{th} row of the vector h (which has two rows)
Thank you in advance for any help and guidance.
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