Methods  Pros  Cons 

RadioSurgery  
Chemos  
Radiation Therapy (RT) 
Results  Solution drawbacks 




Problem Statement: Given the continuousvalued state of the measurements, statistically model how the observations from each sensor change over time.
where $\textbf{F}(k) \in \mathbb{R}^{2\times 2}$ is the state transition matrix given by \begin{equation} \textbf{F} = \begin{bmatrix} 1 & \Delta T \\ 0 & 1 \end{bmatrix} \end{equation}
$\textbf{u}(k) \in \mathbb{R}^2$ is the control input, $\textbf{B}(k)$ is the control input matrix that maps inputs to system states, $\textbf{G}(k) \in \mathbb{R}^{ 2 \times 2}$ is the process noise matrix, and $\textbf{w}(k) \in \mathbb{R}^2$ is a random variable that models the state uncertainty
\begin{align}\label{eq.accelmodel} \textbf{x}_k = \textbf{F}_k \textbf{x}_{k1}+ \textbf{G}_k \textbf{w}_k \end{align}
$ \textbf{w}_k$ is the effect of an unknown input causing an acceleration $a_k$ in the head position and $ \textbf{G}_k$ applies that effect to the state vector, $ \textbf{x}_k$
setting $\textbf{G}_k$ to identity and set $\textbf{w}(k) \sim \mathcal{N}(0, \textbf{Q}(k))$,
the covariance matrix $\textbf{Q}(k)$ to a random walk sequence defined as $\textbf{W}_k={[\frac{{\Delta T}^2}{2}, \Delta T ]}^T$
\begin{equation}\label{eq:sensors_obs} {z}_s= \textbf{H}_s(k)\textbf{x}(k)+{v}_s(k) \qquad \qquad s = 1,2 \end{equation}
set $\textbf{H}_s(k) ={\begin{bmatrix} 1 & 0 \end{bmatrix} }^T$ maps the system's state space into the observation space
${v}_s(k) \in \mathbb{R}$ is a normally distributed random variable that models the sensors uncertainty with zero mean and variance $\sigma_{rs}^2$
\begin{equation} \label{eqn:Riccati} \begin{split} A^T P A \mbox{}(A^T P B \mbox{+} N)(R \mbox{+} B^T P B)^{1}(B^T P A \mbox{+} N) \mbox{+} & Q, \end{split} \end{equation}
allows us to solve for the optimal control law through a minimization of the resulting LQ problem\begin{align} Q = \begin{bmatrix} 1.0566 & 0 \\ 0 & 1.0566 \end{bmatrix} \hspace{0.4em} R = \begin{bmatrix} 0.058006 \end{bmatrix} \end{align}
full online state estimator with noise processes assumed to be independednt, white Gaussian
Slides available at the author's website.