Stability Criterion Routh Hurwitz

Transcription

1 EES404 Fundamental of Control Sytem Stability Criterion Routh Hurwitz DR. Ir. Wahidin Wahab M.Sc. Ir. Arie Subiantoro M.Sc.

2 Stability A ytem i table if for a finite input the output i imilarly finite A ytem which i table mut have ALL it pole in the left half of the -plane

3 Figure 6. Cloed-loop pole and repone: a. table ytem; b. untable ytem Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

4 Figure 6. Common caue of problem in finding cloed-loop pole: a. original ytem; b. equivalent ytem Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

5 Figure 6. Equivalent cloedloop tranfer function Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

6 Routh-Hurwitz Stability 6 Criterion Thi i a mean of detecting untable pole from the denominator polynomial of a t.f. without actually calculating the root. Write the denominator polynomial in the following form and equate to zero - Thi i the characteritic equation. n n n 0 a + a + a a + a = Note that an 0 i.e. remove any zero root n n 0

7 Routh-Hurwitz Stability 7 Criterion If any of the coefficient i zero or negative in the preence of at leat one poitive coefficient there are imaginary root or root in the right half plane i.e. untable root

8 Routh-Hurwitz Stability 8 Criterion if all coefficient are + ve form the Routh Array a a a a a a a a b b b b c c c c.. e e n n n n f g

9 Routh-Hurwitz Stability 9 Criterion b b b = = = aa aa aa a a a aa 0 aa aa 6 0 7

10 Routh-Hurwitz Stability 0 Criterion c c c = = = ba ba ba b b b ab ab 5 ab 7 4

11 Routh-Hurwitz Stability Criterion Thi proce i continued until the nth row i completed The number of root of the characteritic lying in the right half of the - plane (untable root) i equal to the numbe rof ign change in the firt column of the Routh array.

12 Table 6. Initial layout for Routh table Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

13 Table 6. Completed Routh table Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

14 Figure 6.4 a. Feedback ytem for Example 6.; b. equivalent cloed-loop ytem Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

15 Table 6. Completed Routh table for Example 6. Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

16 6 Example Determine if the following polynomial ha root in the right half of the - plane = 0 Firt two row of Routh array formed from coefficient

17 7 Example Form next row = 5 0 =

18 8 Example Form next row = 4 6 5

19 9 Example Form next row = 6 Note two ign change therefore two root in RHP

20 0 Example Apply Routh' criterion to the following polynomial to determine the condition for the exitence of table root a 0 + a + a + a = 0

21 Example Routh Array 0 a a a a a a a a a a a a a a = + + +

22 Example auming all coefficient are poitive the condition for table root i that aa > aa 0

23 Routh Array - Special Cae Cae of a zero in the t column For example = 0 Routh Array 0 Thi preent a problem when we come to obtain the 4th row - divide by zero

24 4 Routh Array - Special Cae Cae of a zero in the t column Define a mall + ve number Routh Array ε and evaluate whole array ε Note no ign change indicating root on imaginary axi

25 5 Routh Array - Special Cae Cae of a zero in the t column 0 Routh array Conider the polynomial 0 ε ε + Two ign change therefore two RHP root

26 Table 6.4 Completed Routh table for Example 6. Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

27 Table 6.5 Determining ign in firt column of a Routh table with zero a firt element in a row Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

28 Table 6.6 Routh table for Example 6. Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

29 9 Routh Array - Special Cae Cae of a row of zero root of equal magnitude but oppoite ign or two conjugate imaginary root = 0 Auxiliary eqn = 0

30 0 Routh Array - Special Cae Cae of a row of zero 4 P () = = 0 P () = replace row with coeficient of P ()

31 Routh Array - Special Cae Cae of a row of zero = One ign change one root with +ve real part

32 Table 6.7 Routh table for Example 6.4 Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

33 Figure 6.5 Root poition to generate even polynomial: A, B, C, or any combination Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

34 Table 6.8 Routh table for Example 6.5 Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

35 Table 6.9 Summary of pole location for Example 6.5 Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

36 Figure 6.8 Feedback control ytem for Example 6.8 Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

37 Table 6. Routh table for Example 6.8 Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

38 Table 6.4 Summary of pole location for Example 6.5 Control Sytem Engineering, Fourth Edition by Norman S. Nie Copyright 004 by John Wiley & Son. All right reerved.

39 9 Ue of Routh Tet Routh tet only tell u whether or not a ytem i table doe not give the DEGREE of tability need to have cloed loop characteritic equation would be more convenient to work from open loop t.f.