State space

State vector

Assuming the target’s motion on the two-dimensional own-ship’s tracking plane, then two different state-space descriptions can be used depending on the expression of the velocity vector:

• Cartesian velocity model $$\mathbf{x} = \left[x,y,\dot{x}, \dot{y},\omega\right]^{T} $$

• Polar velocity model $$\mathbf{x} = \left[x,y,\upsilon, \psi,\omega\right]^{T} $$

where

$$ \begin{align*} \upsilon &= \sqrt{\dot{x}^2 + \dot{y}^2} \\ \psi &= \arctan\left(\frac{\dot{y}}{\dot{x}}\right) \\ \omega &= \dot{\psi} \end{align*} $$

• $(x,y)$ is the displacement relative to the local self-body fixed coordinate system

• $\upsilon$ is the target’s linear velocity

• $\psi$ is the heading angle w.r.t the tracking system’s x-axis.

• $\omega$ is the turn-rate of the target

all expressed w.r.t the own-ship’s tracking coordinate system.