# 随机过程IStochastic Processes MATH2012W1-01

with measurement equation
$$z(k)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x^{i}(k)+w(k)$$
The models differ in the control gain parameter $b^{i}$. The process and measurement noises are mutually uncorrelated with zero mean and variances given by
$$E[v(k) v(j)]=0.16 \delta_{k j}$$
and
$$E[w(k) w(j)]=\delta_{k j}$$

The control gain parameters were chosen to be $b^{1}=2$ and $b^{2}=0.5$.
The Markov transition matrix was selected to be
$$\left[\begin{array}{ll} 0.8 & 0.2 \ 0.1 & 0.9 \end{array}\right]$$
For this example $N=7$, and the cost parameters $R(k)$ and $Q(k)$, were selected as
$$R(k)=5.0 \quad k=1,2, \ldots, N-1$$

Proof: Since $f(d)$ is the minimal polynomial of $\boldsymbol{F}, p(d)$ can be factored as
$$p(d)=g(d) . f(d)$$
for some polynomial $g(d)=\sum_{i=0}^{s} b_{i} d^{s-i}$
Let $z_{k}$ and $\bar{z}{k}$ be linear combinations of $y{k}$ defined as in by using polynomials $f(d)$ and $p(d)$, respectively.
$\bar{z}{k} \quad$ can be expressed in terms of $z{k}$ as
$$\bar{z}{k}=\sum{i=0}^{s} b_{i} z_{k-i}$$