Extended Kalman Filter Jacobians

[Master1022]

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:
$$\begin{align*} x_{k+1} =f_k (x_k , u_k) + w_k \\ y_k = h_k (x_k , u_k ) +v_k \end{align*}$$
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?
$$A_k = \begin{pmatrix} \frac{\partial f_1}{\partial p_1} & \frac{\partial f_1}{\partial p_2} & \frac{\partial f_1}{\partial p_3} \\ \frac{\partial f_2}{\partial p_1} & \frac{\partial f_2}{\partial p_2} & \frac{\partial f_2}{\partial p_3} \\ \frac{\partial f_3}{\partial p_1} & \frac{\partial f_3}{\partial p_2} & \frac{\partial f_3}{\partial p_3} \end{pmatrix}$$
where $f_i$ refers to the function in the $i^{th}$ row of the vector $f$ (which has three rows) and
$$C_k = \begin{pmatrix} \frac{\partial h_1}{\partial p_1} & \frac{\partial h_1}{\partial p_2} & \frac{\partial h_1}{\partial p_3} \\ \frac{\partial h_2}{\partial p_1} & \frac{\partial h_2}{\partial p_2} & \frac{\partial h_2}{\partial p_3} \\ \end{pmatrix}$$
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.

 

[Master1022]

Previously, we used a Kalman Filter which could only model linear ... It can be very difficult to implement an EKF correctly: you will most likely get the Jacobian.

Thanks for your reply. Do my expressions above seem correct though?