Consider a _nite-dimensional state space system where xt 2 IRn.
Write a computer program implementing the Kalman recursion equations
(3.13){(3.17).
Given a sample fy1 yng, verify that tjn _ tjt _ t for all t _ n,
where the matrix inequality A _ B means that x0(B A)x _ 0 for all x.
Consider the following state space system:
xt+1 = _xt + "t; yt = _xt + "t;
where "t is white noise with unit variance.
(a) For which values of the parameter _ = (_; _) is this system stable?
(b) For which values of _ is the system observable or controllable?
(c) For which values of _ are the Kalman recursions stable?
(d) Assume that "t is an independent and identically distributed sequence
N(0; 1). Simulate several trajectories of the system for a
sample size n = 1000 and di_erent parameters _.