1. Similarity Transformation

2. Basic Sets Use a new basis set for state space. Obtain the state-space matrices for the new basis set. Similarity transformation. 2

3. Transformation 3

4. Realization 4

5. Transformation to Diagonal Form 5

6. Example 7.18 6

7. Solution 7

8. Companion Form 8

9. Diagonal Form 9

10. Invariance of TF & Characteristics equation Theorem 7.1: Similar systems have identical transfer functions and characteristic polynomial. 10

11. Proof 11

12. Equivalent Systems Systems with the same transfer function Example 7.20 Show that the following system is equivalent to the system of 7.17(2). x(k + 1) =0 8187 x(k) + 9 0635 × 10−2 u(k) y(k) = x(k) Solution: The transfer function of the system is G(z) = 9.0635 × 10 − 2/(z − 0.8187) Identical to the reduced transfer function of Example 7.17(2). 12