**7.4. Exercises for Chapter 7**

*Exercise 7.1.*

Given the state equation *X*_{K+1} = *A* *X*_{K} + *N*_{K}

where the state matrix *A* is the “identity” matrix of dimension 2 and *N*_{K} the system noise whose covariance matrix is written *Q* = *σ*^{2}*I*_{d} (*I*_{d} : identity matrix).

The system is observed by the scalar equation:

where and are the components of the vector *X*_{K} and where *W*_{K} is the measurement noise of the variance .

and are the initial conditions.

1) Give the expression of the Kalman gain *K*(1) at instant “1” according to *σ*^{2} and .

2) Give the estimate of of *X*_{1} at instant “1” according to the first measurement of *K*(1) and the first measurement *Y*_{1}.

*Solution 7.1*

1)

2)

*Exercise 7.2.*

We are considering the movement of a ...