SI&DA - Lecture 21 'State Estimation Problem'

Summary

Introduction to the State Estimation problem, we have seen the linear case and defined the generic problem as:

$$\begin{array}{l}x(t+1)=Ax(t)+Bu(t)+Gw(t)\\ y(t)=Cx(t)+v(t)\end{array}$$

where:

$$\begin{array}{rl}& \left[\begin{array}{c}w(t)\\ v(t)\end{array}\right]\sim WP\left(\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}Q& 0\\ 0& R\end{array}\right)\right)\\ & x(0)\sim (m_0,P_0)\end{array}$$

Where:

• $x(t)$ is the state of the system
• $u(t)$ is the deterministic input
• $y(t)$ is the output of the system
• $w(t)$ is the disturbance process, a stochastic term, which takes into account the discrepancies of the model, whit respect to the real system.
• $v(t)$ is the measurement noise, a stochastic term

We have considered and will consider only the case in which the tre “stochastic sources” $w(t)$, $v(t)$ and $x(0)$ are independent (the initial condition for the state can be chooses at random). Also we have considered $w(t)$ and $v(t)$ White Processes as shown above.

Our objective is to find at every time $t$ an estimate of $x(t)$, based on the knowledge of the input $u(t)$ and the output data:

$$Y^t=\left[\begin{array}{c}y(0)\\ y(1)\\ ⋮\\ y(t)\end{array}\right]$$

This is a Bayesian Estimation Problem, we want to estimate the random variable $x(t)$ based on the observation of another random variable $y(t)$. So from Estimation Theory we know that there are 2 possible solution to minimize the MSE (Mean Square Error):

1. ${\hat{x}}_{MSE}(t)=E\left[x(t)|Y^t\right]$
2. ${\hat{x}}_{MSE}(t)=m_x(t)+R_{x(t)Y^t}\cdot {\left(R_{Y^t}\right)}^{-1}\cdot \left(Y^t-m_{Y^t}\right)$

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$p$ : position $v$ : velocity $u$ : acceleration

#TODO Why $E[\cdot ]=0$ ???