Implementing kalman fillter

Question

Implementing Kalman fillter

Suppose xi is a state-scalar that admits the following recursion (so-called the first-order Gauss-Markov process):

                 x_i+1 = 0.9x_i + n_i
where the process noise n_i is a zero-mean Gaussian random variable with variance Q_i = 1. We assume the initial state x₀ is also a zero-mean Gaussian random variable with variance π₀ = 1. Also, assume the scalar observation y_i and x_i ts the following linear model:
                  y_i = x_i + v_i
where the measurement noise vi is a zero-mean Gaussian random variable with variance Ri. Assume

E(n_in_j^*)= δ_ij , E(v_iv_j^*)=R_iδ_ij , E(n_iv_j^*) = 0, E(n_ix₀^*) = 0, E(v_ix₀^*) = 0.

For (i) R_i = (0:9)ⁱ, (ii)R_i = 1, (iii)Ri = (1.1)ⁱ do the following and explain your results.

1. Plot π_i = E(x_ix_i) for i = 0; 1,....,200.

2. Plot the gains K_p,i and K_f,i for i = 0; 1,...., 200.

3. Plot R_e,i = E(e_ie_i); P_i|i = E(x^~_i|i x^~_i|i), P_i+1|i = E(x^~_i+1|i x^~_i+1|i), ∑_i for i = 0; 1,....,200.