Errata For "Max-Algebraic System Theory For Discrete Event Systems" (PhD Thesis).pdf

## Errata for "Max-Algebraic System Theory for Discrete Event Systems" by Bart De Schutter

This document summarizes the errata and corrections for Bart De Schutter's PhD thesis, "Max-Algebraic System Theory for Discrete Event Systems" (K.U.Leuven, 1996).

I. List of Errata:

1. Page 6: Equation number " (1.5)" at the end of line -12 should be removed.
2. Page 24: "to only" should be replaced by "the only".
3. Page 37: "Ax = b" should be replaced by "A ⊗x = b".
4. Page 42: "(Rε, ⊕) is not a group" - (Rε, ⊕) is correctly identified as not a group in the original thesis. No change needed.
5. Page 47:
"det⊕A ∇ε" should be replaced by "det⊕A ∇/ ε".
"det⊕A ∇ε" should be replaced by "det⊕A ∇/ ε".
6. Page 57, Figure 3.2:
Upper part of dotted line representing Gα and Gδ: Change to a full line.
Upper part of dotted line representing Gδ and Gγ: Change to a full line.
7. Page 58, Line 15: Remove the words "are contained".
8. Page 62, Line -8: "{1, 2, . . . , p}" should be replaced by "{1, 2, . . . , n}".
9. Page 66, Line -2: "j = 1, 2, . . . , n" should be replaced by "i = 1, 2, . . . , n".
10. Page 75, Line -1: "e2" should be replaced by "e2".
11. Page 76, Line 2: "e2" should be replaced by "e2".
12. Page 93, Line -10:
"cu" should be replaced by "xc"
"eu" should be replaced by "xe" (twice).
13. Page 157, Line -10: "[54, 54]" should be replaced by "[54, 56]".
14. Page 158, Line -6: "αjA.,ij" should be replaced by "αj ⊗A.,ij".
15. Page 161, Lines 1 and 15: "triple" should be replaced by "4-tuple".
16. Page 169, Line 12: "State Realization" should be replaced by "State Space Realization".
17. Page 171, Line 23: "numbers" should be changed to "number".
18. Page 193, Line -4: "1,2" should be replaced by "1, 2", i.e. the font style should be changed.
19. Page 195, Line -14: "cancelation" should be replaced by "cancellation".
20. Page 196, Line 11: "If" should be replaced by "If µ3µ6 −µ2µ5 ̸= 0 and if".
21. Page 197, Line -14: "algebra" should be capitalized to "Algebra".
22. Page 198: The sentence "for all i" might be added before "lim i→∞ai = ε" in the formula immediately after (7.12).
23. Page 199: In formula (7.13), the entry on the last row and last column of the matrix G(n, i, k, c, s) should be changed from 0 to 1.
24. Page 208, Line 5: "exponentials" should be replaced by "exponents".
25. Page 209, Line 8: "then" should be replaced by "that".
26. Page 235: The word "Chapter" in the heading of this page (and subsequent even-numbered pages) needs to be changed to "Appendix". This also applies to:
Pages 236–242
Pages 288–304
Pages 308–314
27. Page 236, Line -7: The words "that contains" may be added before "zmax{dom⊕Aϕϕ| ϕ∈Ck n}".
28. Page 238, Line 2: "1 ∈I" should be replaced by "1 ∈J".
29. Page 238, Line 10: Add "⊗" between "ck" and "A⊗n−k" (both on the left-hand side and right-hand side of the equation).
30. Page 238, Line 10: Add "⊗" between "ck" and "λ⊗n−k" (both on the left-hand side and right-hand side of the equation).
31. Page 275:
Lemma C.1.4: There is an error in the formulation and proof. See Section 2.1 for a corrected version.
Proof of Lemma 6.3.7: Since there was an error in Lemma C.1.4, the proof of Lemma 6.3.7 is no longer entirely correct. However, by considering the corrected Lemma 2.4 (from Section 2.1) and using similar reasoning, it can be shown that Lemma 6.3.7 still holds.
32. Page 287, Lines -4 and -2:
"with a negative dominant exponent" should be added after the word "series".
"ai ∈R" should be replaced by "ai < 0".
Note: This correction does not affect the validity of Lemmas 7.3.2 and Proposition 7.3.3, as these lemmas only relied on series with a negative dominant exponent in their proofs.
33. Page 288, Lines -4 to 10: The text should be removed. In lines 5–8, "ci" should be replaced by "ai", "γi" by "αi" and "i = 1" by "i = 0".
Lines 9–10: ". , which . . . in [K, ∞)" should be replaced by just ".".
34. Page 288, Line 6: Replace "<" with "⩽".
35. Page 327, Line 3: "the the" should be corrected to "the".

II. Corrections (Lemmas)

2.1 Corrected Lemma C.1.4: (See page 275 for the original formulation with errors.)

New Lemma 2.1: See text on pages 275-276.
* New Lemma 2.2: See text on page 276.