Page 148 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

EXERCISES 137

7. In exercise 4, can you find the expressions for S(i), K(i), C(iji) and C(iji 1) for
arbitrary i?(*).
8. Autoregressive models and MATLAB’s Control System Toolbox: consider the follow-
ing second order autoregressive (AR) model:

1 1
xði þ 1Þ¼ xðiÞþ xði 1Þþ wðiÞ
2 4
zðiÞ¼ xðiÞþ vðiÞ

Using the functions tf() and ss() from the Control Toolbox, convert this AR model
into an equivalent state space model, that is, x(i þ 1) ¼ Fx(i) þ Gw(i) and
z(i) ¼ Hx(i) þ v(i). (Use help tf and help ss to find out how these functions
should be used.) Assuming that w(i)and v(i) are white noise sequences with variances
2
2
¼ 1and ¼ 1, use the function dlyap() to find the solution of the discrete
v
w
Lyapunov equation, and the function kalman() (or dlqe()) to find the solution for
the steady state Kalman gain and corresponding error covariance matrices. Hint: the
output variable of the command ss() is a ‘struct’ whose fields are printed by typing
struct(ss). (*).
9. Moving average models: repeat exercise 8, but now considering the so-called first
order moving average (MA) model:
1 2
xði þ 1Þ¼ ðwðiÞþ wði 1ÞÞ ¼ 1
w
2 ð Þ
2
zðiÞ¼ xðiÞþ vðiÞ ¼ 1
v
10. Autoregressive, moving average models: repeat exercise 8, but now considering the
so-called ARMA(2, 1) model:
1 1 1 2
xði þ 1Þ¼ xðiÞþ xði 1Þþ ðwðiÞþ wði 1ÞÞ ¼ 1
w
2 2 2 ð Þ
2
zðiÞ¼ xðiÞþ vðiÞ ¼ 1
v
11. Simulate the processes mentioned in exercise 1, 8, 9 and 10, using MATLAB, and
apply the Kalman filters.