Page 147 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

136 STATE ESTIMATION

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4.6 EXERCISES

1. Consider the following model for a random constant:

xði þ 1Þ¼ xðiÞ
zðiÞ¼ xðiÞþ vðiÞ vðiÞ is white noise with variance 2 v

The prior knowledge is E[x(0)] ¼ 0 and 2 ¼1. Give the expression for the solu-
x(0)
tion of the discrete Lyapunov equation. (0).
2. For the random constant model given in exercise 1, give expressions for the innovation
matrix S(i), the Kalman gain matrix K(i), the error covariance matrix C(iji) and the
prediction matrix C(iji 1) for the first few time steps. That is, for i ¼ 0, 1, 2 and 3.
Explain the results. (0).
3. In exercise 2, can you find the expressions for arbitrary i. Can you also prove that these
expressions are correct? Hint: use induction. Explain the results. (*).
4. Consider the following time-invariant scalar linear-Gaussian system

xði þ 1Þ¼ xðiÞþ wðiÞ wðiÞ is white noise with variance 2
w
zðiÞ¼ xðiÞþ vðiÞ vðiÞ is white noise with variance 2 v
The prior knowledge is Ex0 ¼ 0 and 2 ¼1. What is the condition for the
x0
existence of the solution of the discrete Lyapunov equation? If this condition is
met, give an expression for that solution. (0).
5. For the system described in exercise 4, give the steady state solution. That is, give
expressions for S(i), K(i), C(iji) and C(iji 1) if i !1. (0).
6. For the system given in exercise 4, give expressions for S(i), K(i), C(iji) and C(iji 1)
for the first few time steps, that is, i ¼ 0, 1, 2 and 3. Explain the results. (0).