3 different representations for a dynamic system

1. 3 different representations for a dynamic system 3.2 Equivalent representations

2. DE nth order differential equation TF Transfer function SS State equations 3 different representations 3.2 Equivalent representations

3. DE TF SS ? 3.2 Equivalent representations

4. DE TF SS DE TF Laplace transform of the DE yields: Therefore 3.2 Equivalent representations

5. Remarks Degree of numerator < Degree of denominator STRICTLY PROPER Degree of numerator = Degree of denominator PROPER Degree of numerator > Degree of denominator IMPROPER (Not realizable) 3.2 Equivalent representations

6. DE TF SS SS TF Laplace transform yields: p m 3.2 Equivalent representations

7. Proper SS TF For SISO: Strictly proper Order: n-1 Order: n 3.2 Equivalent representations

8. strictly proper proper Observations 1 If TF is improper, SS representation does not exist! Characteristic polynomial of A = Characteristic polynomial of DE =Denominator of the TF 2 All poles are eigenvalues Reverse may not be true! 3.2 Equivalent representations

9. Equivalent SS representations Let z=Px where |P|≠0 (Coordinate transformation) There are infinitely many equivalent representations! 3.2 Equivalent representations

10. invariants Characteristic equation Characteristic eq’n is the same! Thus eigenvalues are the same! Transfer functions are same! 3.2 Equivalent representations

11. SS DE Special case: Standard choice for state variables =? 3.2 Equivalent representations

12. SS DE A B C 3.2 Equivalent representations

13. Canonical structure Coefficients of the characteristic polynomial in reverse order with negative signs “1”s above the diagonal Zeros elsewhere 3.2 Equivalent representations

14. zeros zeros last entry First entry controllable canonic form A phase-variable form 3.2 Equivalent representations

15. End of Section Next section Restart section Next chapter Restart chapter i The End General index End show 3.2 Equivalent representations