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I want to make a kalman filter that will estimate the upwash coefficient $C_{\alpha_{up}}$

my state vector: $ X_k=[u \ v \ w \ C_{\alpha_{up}} ]^T $

My measurement vector: $ Z_k =[\alpha_m \ \beta_m \ V_m]^T $

My control input vector: $ U_k =[\dot u \ \dot v \ \dot w ]^T $ where $ \dot u ,\dot v,\dot w $ are unbiased and noise free.

I have dataset with $ \dot u,\ \dot v, \ \dot w ,\alpha_m ,\ \beta_m, \ V_m $ vectors where each vector contains Nx1 elements

Following are my equations:

$$\begin{align} \alpha_{true} &= \tan^{-1}\left(\dfrac{w}{u}\right) \\ \alpha_m &= \alpha_{true} (1 + C_{\alpha_{up}}) + v_\alpha \\ \\ \beta_{true} &= \tan^{-1}\left(\dfrac{v}{\sqrt{u^2 + w^2}}\right) \\ \beta_m &= \beta_{true} + v_\beta \\ \\ V_{true} &= \sqrt{u^2 + v^2 + w^2} \\ V_m &= V_{true} + v_V \\ \end{align}$$

$v_\alpha, v_\beta \text{ and } v_V$ are white noise sequences


My approach: $$ \begin{bmatrix} \dot u \\ \dot v \\ \dot w \\ \dot C_{\alpha_{up}} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \\ C_{\alpha_{up}} \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix}\dot u \\ \dot v \\ \dot w \end{bmatrix} $$

Assuming I can obtain a linearised non-null Observation matrix $H$ size 3x4 .

  1. Have I defined my state matrix $ \Phi $ properly? If not, what is the mistake?
  2. With a null $ \Phi $ and non-null $H$, I would get the system to be unobservable, violating the condition of Kalman filter

$$O = \begin{bmatrix} H_{k+1,k} \\ H_{k+1,k} \Phi_{k+1,k} \\ H_{k+1,k} \Phi_{k+1,k}^2 \\ \vdots \\ H_{k+1,k} \Phi_{k+1,k}^{n-1} \\ \end{bmatrix}$$

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